Machine Learning I Continuous Reinforcement Learning

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1 Machine Learning I Continuous Reinforcement Learning Thomas Rückstieß Technische Universität München January 7/8, 2010

2 RL Problem Statement (reminder) state s t+1 ENVIRONMENT reward r t+1 new step r t s t AGENT action a t Definition (agent, environment, state, action, reward) An agent interacts with an environment at discrete time steps t = 0, 1, 2,... At each time step t, the agent receives state s t S from the environment. It then chooses to execute action a t A(s t ) where A(s t ) is the set of available actions in s t. At the next time step, it receives the immediate reward r t+1 R and finds itself in state s t+1.

3 Different Types of RL (reminder) state state action state action π Q P a ss R a ss action value next state reward Direct RL Value-based RL Model-based RL data is cheap computation is limited e.g. embedded systems data is expensive computation doesn't matter e.g. medical trials

4 General Assumptions (reminder) For now, we assume the following: Both states and actions are discrete and finite. Our problem fulfills the Markov property (MDP) the current state information summarizes all relevant information from the past (e.g. chess, cannonball) the next state is only determined by the last state and the last action, not the entire history the environment has a stationary P a ss and Ra ss.

5 Continuous Reinforcement Learning Why continuous reinforcement learning? Problems with too many states/actions Generalization for similar states/actions Continuous domains, like robotics, computer vision,... Let s loosen the restrictive assumptions from last week: Both states and actions are discrete and finite. What changes when we allow s, a R n? No transition graphs anymore No Q-table anymore Q-function? Q(s, a) R is possible, but max a Q(s, a) difficult

6 Continuous RL Overview Value-based Reinforcement Learning Continuous states, discrete actions NFQ Continuous states and actions NFQCA Direct Reinforcement Learning (Policy Gradients) Finite Difference methods Likelihood Ratio methods REINFORCE 1D controller example Application to Neural Networks

7 NFQ Neural Fitted Q-iteration We want to apply Q-Learning to continuous states (but discrete actions for now). Instead of a Q-table, we have a Q-function (or function approximator, e.g. neural network), that maps Q(s t, a t ) R. We sample from the environment and collect (s t, a t, r t )-tuples Q-Learning Update Rule ( Q π (s t, a t ) Q π (s t, a t ) + α r t+1 + γ max a ) Q π (s t+1, a) Q π (s t, a t ) How do we get the maximum over all actions in a certain state s?

8 NFQ Neural fitted Q-iteration Maximum over discrete actions: 1. Use several neural networks, one for each action Q Q Q S S S S S S action 1 action 2 action 3 2. or encode the action as additional input to the network Q S S A A A one-of-n coding 0, 0, 1 0, 1, 0 1, 0, 0

9 NFQ Neural fitted Q-iteration a forward pass in the network returns Q π (s t, a t ) to train the net, convert the (s t, a t, r t )-tuples to a dataset with input (s t, a t ) target Q π (s t, a t ) + α (r t+1 + γ max a Q π (s t+1, a) Q π (s t, a t )) train network with dataset (until convergence) collect new samples by experience and start over Unfortunately, there is no guarantee of convergence, because the Q-values change during training. But in many cases, it works anyway.

10 NFQCA NFQ with continuous actions With continuous actions, getting the maximum value of a state over all actions is infeasable. Instead, we can use an actor / critic architecture: One network (the actor) predicts actions from states The second network (the critic), predicts values from states and actions state state hidden_actor value action hidden_critic action value

11 NFQCA Training 1 Backprop TD error through critic network 2 Backprop resulting error further through actor network state θi π θi π + α Q t(s t,a t ) π(s t ) π(s t ) θi π value action θ Q i θ Q i + α(r t + γ max Q t (s t+1,a) Q t (s t,a t )) Q t(s t,a t ) a θ Q i

12 More value-based continuous RL There are other methods of using function approximation with value-based RL ( Sutton&Barto, Chapter 8).

13 Continuous RL Overview Value-based Reinforcement Learning Continuous states, discrete actions NFQ Continuous states and actions NFQCA Direct Reinforcement Learning (Policy Gradients) Finite Difference methods Likelihood Ratio methods REINFORCE 1D controller example Application to Neural Networks

14 Direct Reinforcement Learning Key aspects of direct reinforcement learning: skip value functions (change policy directly) sample from experience (like MC methods) calculate gradient of parameterized policy follow gradient to local optimum Methods that follow the above description are called Policy Gradient Methods or short Policy Gradients.

15 Policy Gradients Notation For now, we will even loosen our second assumption: Our problem fulfills the Markov property. The next state can now depend on the whole history h, not just the last state-action pair (s, a). Policy π(a h, θ) probability of taking action a when encountering history h. The policy is parameterized with θ. History h π history of all states, actions, rewards following policy π. h π 0 = {s 0} h π t = {s 0, a 0, r 0, s 1,..., a t 1, r t 1, s t } Return R(h π ) = T t=0 γt r t

16 Performance Measure J(π) We need a way to measure the performance for the whole policy π. We define the overall performance of a policy as: J(π) = E π {R(h π )} = p(h π )R(h π ) dh π (1) h π Optimize the parameters θ of the policy to improve J: θ J(π) = θ p(h π )R(h π ) dh π h π = θ p(h π )R(h π ) dh π. (2) h π Knowing the gradient, we can update θ as θ t+1 = θ t + α θ J(π) (3)

17 Finite Differences One method to approximate the gradient of the performance is Finite Differences: J(θ) θ i J(θ + δθ) J(θ) δθ i J(π) θ J J(θ + δθ) J(θ) δθ θ

18 Finite Differences Or even better: take many samples with different δθ s and run a linear regression ( pseudo inverse) J(π) θ J θ matrix Θ i = [ δθ i 1 ], vector J i = [ J(θ + δθ i ) ] β = (Θ T Θ) 1 Θ T J

19 Finite Differences Problems with Finite Differences For Finite Differences, the chosen action can be written as a = f (h, θ + ɛ), where ɛ N (0, σ 2 ) is some exploratory noise. We change the policy parameters θ directly the resulting controller is not predictable. Example robot control: changing the parameters randomly can damage the robot or cause a risk for nearby humans In some recent publications, finite differences perform badly in probabilistic settings most real problems are probabilistic.

20 Likelihood Ratio The safer (and currently more popular) method is to estimate the gradient with likelihood ratio methods. Policy Gradients explore by perturbing the resulting action instead of the parameters a = f (h, θ) + ɛ, again with some exploratory noise ɛ N (0, σ 2 ). The policy, that causes this behavior is unknown (and might not even exist). J(θ + δθ) cannot be measured. Another method of estimating θ J is needed.

21 Likelihood Ratio We start from the performance gradient equation: θ J(π) = θ p(h π )R(h π ) dh π h π where the probability of encountering history h under policy π is: p(h π ) = p(s 0 )π(a 0 h0 π )p(s 1 h0 π, a 0 )π(a 1 h1 π )p(s 2 h1 π, a 1 )... T = p(s 0 ) π(a t 1 ht 1) π p(s t ht 1, π a t 1 ) t=1 Multiplying with 1 = p(hπ ) p(h π ) gives p(h π ) θ J(π) = h p(h π ) θp(h π )R(h π ) dh π π

22 Likelihood Ratio p(h π ) θ J(π) = h p(h π ) θp(h π )R(h π ) dh π π can be simplified by applying 1 x x = log(x): θ J(π) = p(h π ) θ log p(h π ) R(h π ) dh π h π where after a few more steps we get θ log p(h π ) = T θ log π(a t 1 ht 1) π t=1 which we will insert into above equation.

23 Likelihood Ratio REINFORCE This leads to the likelihood ratio gradient estimate θ J(π) = p(h π ) T θ log π(a t 1 ht 1) π R(h π ) dh π t=1 { T } = E π θ log π(a t 1 ht 1) π R(h π ) t=1 Just like in the classical case, the expectation cannot be calculated directly. We use Monte-Carlo Sampling of episodes to approximate and get Williams REINFORCE algorithm (1992): θ J(π) 1 N h π T θ log π(a t 1 ht 1) π R(h π ) t=1

24 Example: Linear Controller (1D) After this general derivation, we now go back to an MDP Here with a linear controller: The actions are distributed like and the policy is thus π(a t h t, θ) = π(a t s t, θ) a = f (s, θ) + ɛ = θs + ɛ, ɛ N (0, σ 2 ) a N (θs, σ 2 ) π(a s) = p(a s, θ, σ) = 1 2πσ exp ( ) (a θs)2 2σ 2

25 Example: Linear Controller (1D) Policy from last slide: π(a s) = p(a s, θ, σ) = 1 2πσ exp ( ) (a θs)2 2σ 2 Deriving the policy with respect to the free parameters θ and σ results in θ log π(a s) = (a θs)s σ 2 σ log π(a s) = (a θs)2 σ 2 σ 3

26 Example: Linear Controller (1D) REINFORCE Algorithm 1 initialize θ randomly 2 run N episodes, draw actions a π(a s, θ), remember all st n, at n, rt n 3 approximate gradient with REINFORCE θ J(π) 1 N 1 T 1 θ log π(at n st n ) Rt n N n=0 t=0 4 update the parameter θ θ + α θ J(π) 5 goto 2

27 Application to Neural Network Controllers How does the policy for a NN controller look like? Probabilistic Output Layer Deterministic Output Layer { {... action... σ k u k µ k z k a k u k N (µ k,σ 2 k) z k = f act (a k ) a k = θ kj z j j θ kj Neuron k Hidden Layer {... θ ji z Input Layer { i... z j a j Neuron j state gaussian unit squashing unit summing unit neuron

28 Application to Neural Network Controllers Again we need to derive the log of the policy with respect to the parameters, which here are the weights θ ij of the network The factor µ k θ ji log π(a s) θ ji = k O log π k (a k s) µ k µ k θ ji describes the back-propagation through the network. use existing NN implementation, but back-propagate the log likelihood derivatives log π k (a k s) µ k instead of the error from supervised learning. use REINFORCE to approximate θ J(π) which results in the weight update θ θ + α θ J(π).

29 Where did the exploration go? no explicit exploration probabilistic policy π(s, a) = p(a s) covers two random concepts: non-deterministic policies and exploration this is actually not very efficient State-Dependent Exploration a = f (s, θ + ɛ) a = f (s, θ) + ɛ a = f (s, θ) + ɛ(s)

30 Conclusion Does it work? Yes, for few parameters and many episodes Policy Gradients converge to a local optimum There are ways to improve REINFORCE: baselines, pegasus, state-dependent exploration,... New algorithms use data more efficiently: ENAC

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