1. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1 Organizing Principles for Learning in the Brain Associative Learning: Hebb rule and variations, self-organizing maps Adaptive Hedonism: Brain seeks pleasure and avoids pain: conditioning and reinforcement learning Imitation: Brain specially set up to learn from other brains? Imitation learning approaches Supervised Learning: Of course brain has no explicit teacher, but timing of development may lead to some circuits being trained by others

2. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 2 Classical Conditioning and Reinforcement Learning Outline: 1.classical conditioning and its variations 2.Rescorla Wagner rule 3.instrumental conditioning 4.Markov decision processes 5.reinforcement learning Note: this presentation follows a chapter of “Theoretical Neuroscience” by Dayan&Abbott

3. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3 Example of embodied models of reward based learning: Skinnerbots in Touretzky’s lab at CMU: http://www-2.cs.cmu.edu/~dst/Skinnerbots/index.html

4. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4 Project Goals We are developing computational theories of operant conditioning. While classical (Pavlovian) conditioning has a well-developed theory, implemented in the Rescorla-Wagner model and its descendants (work by Sutton & Barto, Grossberg, Klopf, Gallistel, and others), there is at present no comprehensive theory of operant conditioning. Our work has four components: 1. Develop computationally explicit models of operant conditioning that reproduce classical animal learning experiments with rats, dogs, pigeons, etc. 2. Demonstrate the workability of these models by implementing them on mobile robots, which then become trainable robots (Skinnerbots). We originally used Amelia, a B21 robot manufactured by Real World Interface, as our implementation platform. We are moving to the Sony AIBO.AmeliaReal World Interface 3. Map our computational theories onto neuroanatomical structures known to be involved in animal learning, such as the hippocampus, amygdala, and striatum. 4. Explore issues in human-robot interaction that arise when non-scientists try to train robots as if they were animals. also at: http://www-2.cs.cmu.edu/~dst/Skinnerbots/index.html

5. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 5 Classical Conditioning Pavlov’s classic finding: (classical conditioning) Initially, sight of food leads to dog salivating: food salivating unconditioned stimulus, US unconditioned response, UR (reward) Sound of bell consistently precedes food. Afterwards, bell leads to salivating: bell salivating conditioned stimulus, CS conditioned response, CR (expectation of reward)

6. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6 Variations of Conditioning 1 Extinction: Stimulus (bell) repeatedly shown without reward (food): conditioned response (salivating) reduced. Partial reinforcement: Stimulus only sometimes preceding reward: conditioned response weaker than in classical case. Blocking (2 stimuli): First: stimulus S1 associated with reward: classical conditioning. Then: stimulus S1 and S2 shown together followed by reward: Association between S2 and reward not learned.

7. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7 Variations of Conditioning 2 Inhibitory Conditioning (2 stimuli): Alternate 2 types of trials: 1. S1 followed by reward. 2. S1+S2 followed by absence of reward. Result: S2 becomes predictor of absence of reward. To show this use for example the following 2 methods: A. train animal to predict reward based on S2. Result: learning slowed B. train animal to predict reward based on S3, then show S2+S3. Result: conditioned response weaker than for S3 alone.

8. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 8 Variations of Conditioning 3 Overshadowing (2 stimuli): Repeatedly present S1+S2 followed by reward. Result: often, reward prediction shared unequally between stimuli. Example (made up): red light + high pitch beep precede pigeon food. Result: red light more effective in predicting the food than high pitch beep. Secondary Conditioning: S1 preceding reward (classical case). Then, S2 preceding S1. Result: S2 leads to prediction of reward. But: if S1 following S2 showed too often: extinction will occur

9. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9 Summary of Conditioning Findings (incomplete, has been studied extensively for decades, many books on topic) figure taken from Dayan&Abbott

10. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 10 Modeling Conditioning The Rescorla Wagner rule (1972): Consider stimulus variable u representing presence (u=1) or absence (u=0) of stimulus. Correspondingly, reward variable r represents presence or absence of reward. The expected reward v is modeled as “stimulus x weight”: v = wu Learning is done by adjusting the weight to minimize error between predicted reward and actual reward.

11. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 11 Rescorla Wagner Rule Denote the prediction error by δ (delta): δ = r-v Learning rule: w := w + ε δ u, where ε is a learning rate. Q: Why is this useful? A: This rule does stochastic gradient descent to minimize the expected squared error (r-v) 2, w converges to. R.W. rule is variant of the “delta rule” in neural networks. Note: in psychological terms the learning rate is measure of associability of stimulus with reward.

12. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 12 Rescorla Wagner Rule Example prediction error δ = r-v; learning rule: w := w + ε δ u figure taken from Dayan&Abbott

13. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 13 Multiple Stimuli Essentially the same idea/learning rule: In case of multiple stimuli: v = w·u (predicted reward = dot product of stimulus vector and weight vector) Prediction error: δ = r-v Learning rule: w := w + ε δ u

14. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14 In how far does Rescorla Wagner rule account for variants of classical conditioning? (prediction: v = w·u; error: δ = r-v; learning: w := w + ε δ u) figure taken from Dayan&Abbott

15. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15 (prediction: v = w·u; error: δ = r-v; learning: w := w + ε δ u) Extinction, Partial Reinforcement: o.k., since w converges to Blocking: during pre-training, w 1 converges to r. During training v=w 1 u 1 +w 2 u 2 =r, hence δ=0 and w 2 does not grow. Inhibitory Conditioning: on S1 only trials, w 1 gets positive value. on S1+S2 trials, v=w 1 +w 2 must converge to zero, hence w 2 becoming negative. Overshadow: v=w 1 +w 2 goes to r, but w 1 and w 2 may become different if there are different learning rates ε i for them. Secondary Conditioning: R.-W.-rule predicts negative S2 weight! Rescorla Wagner rule qualitatively accounts for wide range of conditioning phenomena but not secondary conditioning.

16. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 16 Temporal Difference Learning Motivation: need to keep track of time within a trial Idea: (Sutton&Barto, 1990) Try to predict the total future reward expected from time t onward to the time T of end of trial. Assume time is in discrete steps. Predicted total future reward from time t (one stimulus case): Problem: how to adjust the weight? Would like to adjust w(τ) to make v(t) approximate the true total future reward R(t) (reward that is yet to come) but this is unknown since lying in future.

17. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 17 TD Learning cont’d. Solution: (Temporal Difference Learning Rule), with temporal difference To see why this makes sense: We want v(t) to approximate left hand side but also: v(t+1) should approximate 2 nd term of right hand side. Hence: or

18. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 18 TD Learning Rule Example figure taken from Dayan&Abbott ; Note: temporal difference learning rule can also account for secondary conditioning (sorry, no example) reward and time course of reward correctly predicted!

19. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 19 Dopamine and Reward Prediction figure taken from Dayan&Abbott (VTA= ventral tegmental area (midbrain)) VTA neurons fire for unex- pected reward: seem to re- present the prediction error δ

20. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 20 Instrumental Conditioning So far: only concerned with prediction of reward. Didn’t consider agent’s actions. Reward usually depends on what you do! Skinner boxes, etc. Distinguish two scenarios: A.Rewards follow actions immediately (Static Action Choice) Example: n-armed bandit (slot machine) B. Rewards may be delayed (Sequential Action Choice) Example: playing chess Goal: choose actions to maximize rewards

21. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 21 Static Action Choice Consider bee foraging: Bee can choose to fly to blue or yellow flowers, wants to maximize nectar volume. Bees learn to fly to “better” flower in single session (~40 flowers)

22. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 22 Simple model of bee foraging When bee chooses blue: reward ~p(r b ) or yellow: reward ~p(r y ) Assume model bee has stochastic policy: chooses to fly to blue or yellow flower with p(b) or p(y) respectively. A “convenient” assumption: p(b), p(y) follow softmax decision rule: Notes: p(b)+p(y)=1; m b, m y are action values to be adjusted; β: inverse temperature: big β  deterministic behavior, where

23. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 23 Exploration-Exploitation dilemma Why use softmax action selection? Idea: bee could also choose “better” action all the time. But: bee can’t be sure that better action is really better action. Bee needs to test and continuously verify which action leads to higher rewards. This is the famous exploration-exploitation dilemma of reinforcement learning: Need to explore to know what’s good. Need to exploit what you know is good to maximize reward. Generalization of softmax to many possible actions:

24. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 24 The Indirect Actor Question: how to adjust the action values m a ? Idea: have action values adapt to average reward for that action: m b = and m y = This can be achieved with simple delta rule: m b  m b + εδ, where δ = r b -m b This is indirect actor because action choice is mediated indirectly by expected amounts of rewards.

25. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 25 Indirect Actor Example figure taken from Dayan&Abbott

26. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 26 The Direct Actor figure taken from Dayan&Abbott Idea: choose action values directly to maximize expected reward Maximize this expected reward by stochastic gradient ascent: This leads to the following learning rule: where δ ab is the “Kronecker delta” and r is a parameter often chosen to be an estimate of the average reward per time. ¯

27. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 27 Direct Actor Example figure taken from Dayan&Abbott again: nectar volumes reversed after first 100 visits

28. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 28 Sequential Action Choice So far: immediate reward after each action (n-armed bandit problem) Now: delayed rewards, can be in different states Example: Maze Task figure taken from Dayan&Abbott Amount of reward after decision at second intersection depends on action taken at first intersection.

29. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 29 Policy Iteration Big body of research on how to solve this and more complicated tasks, easily filling an entire course by itself. Here we just consider one example method: policy iteration. Assumption: state is fully observable (in contrast to only partially observable), i.e. the rat knows exactly where it is at any time. Idea: maintain and improve a stochastic policy, determining actions at each decision point (A,B,C) using action values and softmax decision. Two elements: critic: use temporal difference learning to predict future rewards from A,B,C if current policy is followed actor: maintain and improve the policy figure taken from Dayan&Abbott

30. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 30 Policy Iteration cont’d. How to formalize this idea? Introduce state variable u to describe whether rat is at A,B,C. Also introduce action value vector m(u) describing the policy. (softmax rule assigns probability of action a based on action values) Immediate reward for taking action a in state u: r a (u) Expected future reward for starting in state u and following current policy: v(u) (state value). The rat’s estimate for this is denoted by w(u). Policy Evaluation (critic): estimate w(u) using temporal difference learning. Policy Improvement (actor): improve action values m(u) based on estimated state values. figure taken from Dayan&Abbott

31. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 31 Policy Evaluation Initially, assume all action values are 0, i.e. left/right equally likely everywhere. True value of each state can be found by inspection: v(B) = ½(5+0)=2.5; v(C) = ½(2+0)=1; v(A) = ½(v(B)+v(C))=1.75. These values can be learned with temporal difference learning rule: with where u’ is the state that results from taking action a in state u. figure taken from Dayan&Abbott

32. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 32 Policy Evaluation Example with figures taken from Dayan&Abbott

33. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 33 Policy Improvement figures taken from Dayan&Abbott How to adjust action values? where and p(a’;u) is the softmax probability of chosing action a’ in state u as determined by m a’ (u). Example: consider starting out from random policy and assume state value estimates w(u) are accurate. Consider u=A, leads to for left turn for right turn rat will increase probability of going left in A

34. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 34 Policy Improvement Example figures taken from Dayan&Abbott

35. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 35 Some Extensions -Introduction of a state vector u -discounting of future rewards: put more emphasis on rewards in the near future than rewards that are far away. Note: reinforcement learning is big subfield of machine learning. There is a good introductory textbook by Sutton and Barto.

36. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 36 Questions to discuss/think about 1.Even at one level of abstraction there are many different “Hebbian”, or Reinforcement learning rules; is it important which one you use? What is the right one? 2.The applications we discussed in Hebbian and Reinforcement learning considered networks passively receiving simple sensory input and learning to code it or behave “well”; how can we model learning through interaction with complex environments? Why might it be important to do so? 3.The problems we considered so far are very “low-level”, no hint of “complex behaviors” yet. How can we bridge this huge divide? How can we “scale up”? Why is it difficult?