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Reinforcement learning 2: action selection

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1 Reinforcement learning 2: action selection
Peter Dayan (thanks to Nathaniel Daw)

2 Global plan Reinforcement learning I: (Wednesday)
prediction classical conditioning dopamine Reinforcement learning II: dynamic programming; action selection sequential sensory decisions vigor Pavlovian misbehaviour Chapter 9 of Theoretical Neuroscience

3 Learning and Inference
Learning: predict; control ∆ weight  (learning rate) x (error) x (stimulus) dopamine phasic prediction error for future reward serotonin phasic prediction error for future punishment acetylcholine expected uncertainty boosts learning norepinephrine unexpected uncertainty boosts learning

4 Action Selection Evolutionary specification Immediate reinforcement:
leg flexion Thorndike puzzle box pigeon; rat; human matching Delayed reinforcement: these tasks mazes chess Bandler; Blanchard

5 Immediate Reinforcement
stochastic policy: based on action values:

6 Indirect Actor use RW rule: switch every 100 trials

7 Direct Actor

8 Direct Actor

9 Could we Tell? correlate past rewards, actions with present choice
indirect actor (separate clocks): direct actor (single clock):

10 Matching: Concurrent VI-VI
Lau, Glimcher, Corrado, Sugrue, Newsome

11 Matching income not return approximately exponential in r
alternation choice kernel

12 Action at a (Temporal) Distance
learning an appropriate action at u=1: depends on the actions at u=2 and u=3 gains no immediate feedback idea: use prediction as surrogate feedback

13 Action Selection start with policy: evaluate it: improve it:
0.025 -0.175 -0.125 0.125 thus choose L more frequently than R

14 Policy value is too pessimistic action is better than average

15 actor/critic m1 m2 m3 mn dopamine signals to both motivational & motor striatum appear, surprisingly the same suggestion: training both values & policies

16 Variants: SARSA Morris et al, 2006

17 Variants: Q learning Roesch et al, 2007

18 Summary prediction learning actor-critic indirect method (Q learning)
Bellman evaluation actor-critic asynchronous policy iteration indirect method (Q learning) asynchronous value iteration

19 Sensory Decisions as Optimal Stopping
consider listening to: decision: choose, or sample

20 Optimal Stopping equivalent of state u=1 is and states u=2,3 is

21 Transition Probabilities

22 Evidence Accumulation
Gold & Shadlen, 2007

23 Current Topics Vigour & tonic dopamine
Priors over decision problems (LH) Pavlovian-instrumental interactions impulsivity behavioural inhibition framing Model-based, model-free and episodic control Exploration vs exploitation Game theoretic interactions (inequity aversion)

24 Vigour Two components to choice: real-valued DP what:
lever pressing direction to run meal to choose when/how fast/how vigorous free operant tasks real-valued DP

25 The model ? 1 time 2 time S0 S1 S2 vigour cost unit cost (reward) UR
PR how fast ? LP NP Other S0 S1 S2 1 time 2 time choose (action,) = (LP,1) Costs Rewards choose (action,) = (LP,2) Costs Rewards goal

26 The model Goal: Choose actions and latencies to maximize the average rate of return (rewards minus costs per time) S0 S1 S2 1 time 2 time choose (action,) = (LP,1) Costs Rewards choose (action,) = (LP,2) Costs Rewards ARL

27 Average Reward RL Compute differential values of actions
Differential value of taking action L with latency  when in state x ρ = average rewards minus costs, per unit time Future Returns QL,(x) = Rewards – Costs + Mention that the model has few parameters (basically the cost constants and the reward utility) but we will not try to fit any of these, but just look at principles steady state behavior (not learning dynamics) (Extension of Schwartz 1993)

28 Average Reward Cost/benefit Tradeoffs
1. Which action to take? Choose action with largest expected reward minus cost How fast to perform it? slow  less costly (vigour cost) slow  delays (all) rewards net rate of rewards = cost of delay (opportunity cost of time) Choose rate that balances vigour and opportunity costs explains faster (irrelevant) actions under hunger, etc masochism

29 Optimal response rates
1st Nose poke Niv, Dayan, Joel, unpublished Experimental data seconds seconds since reinforcement 1st Nose poke seconds since reinforcement seconds Model simulation

30 % Reinforcements on lever A
Optimal response rates 20 40 60 80 100 Pigeon A Pigeon B Perfect matching % Reinforcements on key A % Responses on key A Herrnstein 1961 Experimental data Model simulation 50 % Reinforcements on lever A % Responses on lever A Model Perfect matching More: # responses interval length amount of reward ratio vs. interval breaking point temporal structure etc.

31 Effects of motivation (in the model)
RR25 low utility high utility mean latency LP Other energizing effect energizing effect

32 seconds from reinforcement seconds from reinforcement
Effects of motivation (in the model) RR25 response rate / minute seconds from reinforcement directing effect 1 response rate / minute seconds from reinforcement UR 50% low utility high utility mean latency LP Other energizing effect 2

33 Relation to Dopamine Phasic dopamine firing = reward prediction error
What about tonic dopamine? more less

34 Tonic dopamine = Average reward rate
explains pharmacological manipulations dopamine control of vigour through BG pathways Aberman and Salamone 1999 # LPs in 30 minutes 1 4 16 64 500 1000 1500 2000 2500 Control DA depleted Model simulation # LPs in 30 minutes Control DA depleted eating time confound context/state dependence (motivation & drugs?) less switching=perseveration NB. phasic signal RPE for choice/value learning

35 Tonic dopamine hypothesis
♫ $ Satoh and Kimura 2003 Ljungberg, Apicella and Schultz 1992 reaction time firing rate …also explains effects of phasic dopamine on response times

36 Pavlovian & Instrumental Conditioning
learning values and predictions using TD error Instrumental learning actions: by reinforcement (leg flexion) by (TD) critic (actually different forms: goal directed & habitual)

37 Pavlovian-Instrumental Interactions
synergistic conditioned reinforcement Pavlovian-instrumental transfer Pavlovian cue predicts the instrumental outcome behavioural inhibition to avoid aversive outcomes neutral Pavlovian cue predicts outcome with same motivational valence opponent Pavlovian cue predicts opposite motivational valence negative automaintenance

38 -ve Automaintenance in Autoshaping
simple choice task N: nogo gives reward r=1 G: go gives reward r=0 learn three quantities average value Q value for N Q value for G instrumental propensity is

39 -ve Automaintenance in Autoshaping
Pavlovian action assert: Pavlovian impetus towards G is v(t) weight Pavlovian and instrumental advantages by ω – competitive reliability of Pavlov new propensities new action choice

40 -ve Automaintenance in Autoshaping
basic –ve automaintenance effect (μ=5) lines are theoretical asymptotes equilibrium probabilities of action

41 Impulsivity & Hyperbolic Discounting
humans (and animals) show impulsivity in: diets addiction spending, … intertemporal conflict between short and long term choices often explained via hyperbolic discount functions alternative is Pavlovian imperative to an immediate reinforcer framing, trolley dilemmas, etc

42 Kalman Filter Markov random walk (or OU process) no punctate changes
additive model of combination forward inference

43 Kalman Posterior ^ ε

44 Assumed Density KF Rescorla-Wagner error correction
competitive allocation of learning P&H, M

45 Blocking forward blocking: error correction
backward blocking: -ve off-diag

46 Mackintosh vs P&H under diagonal approximation: for slow learning, E
effect like Mackintosh E

47 Summary Kalman filter models many standard conditioning paradigms
elements of RW, Mackintosh, P&H but: downwards unblocking negative patterning L→r; T→r; L+T→· recency vs primacy (Kruschke) predictor competition stimulus/correlation rerepresentation (Daw)

48 Uncertainty (Yu) expected uncertainty - ignorance
amygdala, cholinergic basal forebrain for conditioning ?basal forebrain for top-down attentional allocation unexpected uncertainty – `set’ change noradrenergic locus coeruleus part opponent; part synergistic interaction

49 Experimental Data ACh & NE have similar physiological effects
suppress recurrent & feedback processing enhance thalamocortical transmission boost experience-dependent plasticity (e.g. Kimura et al, 1995; Kobayashi et al, 2000) (e.g. Gil et al, 1997) (e.g. Bear & Singer, 1986; Kilgard & Merzenich, 1998) ACh & NE have distinct behavioral effects: ACh boosts learning to stimuli with uncertain consequences NE boosts learning upon encountering global changes in the environment (e.g. Bucci, Holland, & Gallagher, 1998) (e.g. Devauges & Sara, 1990)

50 Model Schematics context cortical processing sensory inputs top-down
unexpected uncertainty expected uncertainty top-down processing NE ACh cortical processing prediction, learning, ... bottom-up processing sensory inputs

51 Attention Example 1: Posner’s Task 0.2-0.5s
attentional selection for (statistically) optimal processing, above and beyond the traditional view of resource constraint sensory input Example 1: Posner’s Task stimulus location cue high validity low (Phillips, McAlonan, Robb, & Brown, 2000) cue target response 0.1s 0.1s s 0.15s generalize to the case that cue identity changes with no notice

52 Formal Framework ACh NE avoid representing full uncertainty
variability in identity of relevant cue variability in quality of relevant cue cues: vestibular, visual, ... target: stimulus location, exit direction... avoid representing full uncertainty Sensory Information

53 Simulation Results: Posner’s Task
nicotine validity effect concentration scopolamine (Phillips, McAlonan, Robb, & Brown, 2000) vary cue validity  vary ACh fix relevant cue  low NE increase ACh validity effect % normal level 100 120 140 decrease ACh 80 60

54 example 2: attentional shift
Maze Task example 2: attentional shift (Devauges & Sara, 1990) reward cue 1 cue 2 relevant irrelevant no issue of validity

55 Simulation Results: Maze Navigation
fix cue validity  no explicit manipulation of ACh change relevant cue  NE % Rats reaching criterion No. days after shift from spatial to visual task experimental data model data (Devauges & Sara, 1990)

56 Simulation Results: Full Model
true & estimated relevant stimuli neuromodulation in action trials validity effect (VE)

57 Simulated Psychopharmacology
50% NE ACh compensation 50% ACh/NE NE can nearly catch up

58 Inter-trial Uncertainty
single framework for understanding ACh, NE and some aspects of attention ACh/NE as expected/unexpected uncertainty signals experimental psychopharmacological data replicated by model simulations implications from complex interactions between ACh & NE predictions at the cellular, systems, and behavioral levels activity vs weight vs neuromodulatory vs population representations of uncertainty

59 Aston-Jones: Target Detection
detect and react to a rare target amongst common distractors elevated tonic activity for reversal activated by rare target (and reverses) not reward/stimulus related? more response related? no reason to persist as unexpected uncertainty argue for intra-trial uncertainty explanation

60 Vigilance Model variable time in start η controls confusability
one single run cumulative is clearer exact inference effect of 80% prior

61 Phasic NE NE reports uncertainty about current state
state in the model, not state of the model divisively related to prior probability of that state NE measured relative to default state sequence start → distractor temporal aspect - start → distractor structural aspect target versus distractor

62 Phasic NE onset response from timing uncertainty (SET)
growth as P(target)/0.2 rises act when P(target)=0.95 stop if P(target)=0.01 arbitrarily set NE=0 after 5 timesteps (small prob of reflexive action)

63 Four Types of Trial 19% 1.5% 1% 77% fall is rather arbitrary

64 Response Locking slightly flatters the model – since no further
response variability

65 Task Difficulty set η=0.65 rather than 0.675
information accumulates over a longer period hits more affected than cr’s timing not quite right

66 Interrupts PFC/ACC LC

67 Intra-trial Uncertainty
phasic NE as unexpected state change within a model relative to prior probability; against default interrupts ongoing processing tie to ADHD? close to alerting (AJ) – but not necessarily tied to behavioral output (onset rise) close to behavioural switching (PR) – but not DA farther from optimal inference (EB) phasic ACh: aspects of known variability within a state?

68 Learning and Inference
Learning: predict; control ∆ weight  (learning rate) x (error) x (stimulus) dopamine phasic prediction error for future reward serotonin phasic prediction error for future punishment acetylcholine expected uncertainty boosts learning norepinephrine unexpected uncertainty boosts learning

69 Conditioning Ethology Psychology Computation Algorithm Neurobiology
prediction: of important events control: in the light of those predictions Ethology optimality appropriateness Psychology classical/operant conditioning Computation dynamic progr. Kalman filtering Algorithm TD/delta rules simple weights Neurobiology neuromodulators; amygdala; OFC nucleus accumbens; dorsal striatum

70 Computational Neuromodulation
dopamine phasic: prediction error for reward tonic: average reward (vigour) serotonin phasic: prediction error for punishment? acetylcholine: expected uncertainty? norepinephrine unexpected uncertainty; neural interrupt?

71 Markov Decision Process
class of stylized tasks with states, actions & rewards at each timestep t the world takes on state st and delivers reward rt, and the agent chooses an action at

72 Markov Decision Process
World: You are in state 34. Your immediate reward is 3. You have 3 actions. Robot: I’ll take action 2. World: You are in state 77. Your immediate reward is -7. You have 2 actions. Robot: I’ll take action 1. World: You’re in state 34 (again).

73 Markov Decision Process
Stochastic process defined by: reward function: rt ~ P(rt | st) transition function: st ~ P(st+1 | st, at)

74 Markov Decision Process
Stochastic process defined by: reward function: rt ~ P(rt | st) transition function: st ~ P(st+1 | st, at) Markov property future conditionally independent of past, given st

75 The optimal policy Definition: a policy such that at every state, its expected value is better than (or equal to) that of all other policies Theorem: For every MDP there exists (at least) one deterministic optimal policy.  by the way, why is the optimal policy just a mapping from states to actions? couldn’t you earn more reward by choosing a different action depending on last 2 states?

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