Reinforcement Learning. Value Function Updates

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1 Reinforcement Learning Value Function Updates Manfred Huber

2 Value Function Updates Different methods for updating the value function Dynamic programming Simple Monte Carlo Temporal differencing (TD, Q-learning, SARSA Character of value function updates varies across methods Different branching over updates Different depth of update Manfred Huber

3 Value Function Backups Temporaldifference learning width of backup Dynamic programming height (depth of backup Monte Carlo Exhaustive search... Manfred Huber

4 Value Function Backups Wider (branching backups allow to back up multiple outcomes at the same time Requires to generate multiple (probabilistic outcomes according to the underlying distribution Can only be performed in simulation (off-line Deeper backups allow to include information from further in the future More precise estimate of the value of the outcome Requires to track rewards over multiple time steps Manfred Huber

5 N-Step TD Backups Using multiple steps for backup TD (1-step 2-step 3-step n-step Monte Carlo to: Reinforcement Learning: An Introduction Manfred Huber

6 N-Step TD Backups Different outcome utility estimates 1-step outcome utility prediction: Uˆ (1 t = r t +!V+1 Simple Monte Carlo outcome utility: Uˆ (! t = # T! " "t r " " =t n-step outcome utility prediction: Uˆ (n t = " t+n!1! "!t r " +" n V t +n! =t ( Corresponding value backup!v (k t =! U ˆ (k t "V t ( Manfred Huber

7 N-Step TD Backups TD and MC converge With limited data to different results N-step on-policy updates converge to the correct utility for on-policy exploration Precision of expected value of n-step update depends on precision of estimate after the n th step max s E! " U ˆ (k (s! # $ %V! (s &! n max s V t (s%v! (s Expected error for n-step backup is at least as small as for 1-step (TD backup and thus converges Manfred Huber

8 N-Step TD Backups RMS for 10 episodes of 19 state! problem ! On-line ON-LINE n-step TD Manfred Huber ! Off-line

9 Multi-Horizon Backups Optimal n depends on data, learning rate, and the problem Larger n usually require smaller learning rates Off-line generated data usually benefits more from larger n No a priori optimal value Combining multiple horizon lengths Average the outcome estimate for multiple n Uˆ w = " k!k w ˆ k U (k Manfred Huber

10 Exponentially Weighted Complex Backups Assign exponentially decreasing weights w k = (1!!! k!1 w T =! T!1 Longer horizon estimates have lower weights All weights add up to 1 Terminal length gets complete remaining weight TD and MC are special cases =0 : TD =1 : Simple Monte Carlo Manfred Huber

11 Exponential Averaging Exponentially weighted complex backups ( ( "V (s! =1 t t!v t =! ( 1" "# T"t"1! " "1 Uˆ (! t +! T"t"1 U ˆ (T"t (s t t Update is based on all horizons regulates the amount of update due to future rewards Also changes how much emphasis is put on representing value of training data sequence versus on following local Markov assumption Manfred Huber

12 Exponential Averaging.55 "=1 "=.99 "= "=0.45 "=.2 "=.4 s. 4 "=.975 "=.6.35 "=.95 "=.9 "= ! Manfred Huber

13 Incremental Complex Backups Forward-looking n-step backups can only be performed after the future rewards are known No fully incremental learning Not feasible for non-episodic tasks Apply complex backups in parts At each step apply a backup to all previous states Make backups add up to the complete complex backup over time Manfred Huber

14 Incremental Complex Backups Complex backup can be broken into components due to particular states / steps ( ( "V! =1 t T"t"1!V t =! ( 1" "! " "1 Uˆ (! # t +! T"t"1 U ˆ (T"t (s t t (!!!!!!!!!!!!!=!!!!(1" "( r t +!V t +1 ( (!!!!!!!!!!!!!!!!!!!!!!+(1"!! r t +!r t+1 +! 2 V t +2!!!!!!!!!!!!!!!!!!!!!!+(1"!! 2 r t +!r t+1 +! 2 r t+2 +! 3 V t +3!!!!!!!!!!!!!!!!!!!!!!+!!!!!!!!!!!!!!!!!!!!!!!!+! T"t"1 r t +!r t+1 + +! T"t V t (s T"t!!!!!!!!!!!!!!!!!!!!!!"V t ( Manfred Huber

15 Incremental Complex Backups Complex backup broken by time steps!v t =! (!!!r t +!V t +1 "!"V t +1 (!!!!!!!!!!!!!!!!!!!!!!+!" r t+1 +!V t +2 "!"V t +2!!!!!!!!!!!!!!!!!!!!!!+!! (!!!!!!!!!!!!!!!!!!!!!!+! T"t"1! T"t"1 r T"t +!V t (s T"t!!!!!!!!!!!!!!!!!!!!!!"V t!!!!!!!!!!!!!=! (!!!r t +"V t +1!V t!!!!!!!!!!!!!!!!!!!!!!+#" r t+1 +"V t +2!V t +1!!!!!!!!!!!!!!!!!!!!!!+!! ( (!!!!!!!!!!!!!!!!!!!!!!+# T!t!1 " T!t!1 r T!t +"V t (s T!t!V t (s T!t!1 Manfred Huber !!!!!!!!!!!!!!!!!!!

16 Eligibility Traces Apply partial backups retroactively as transitions occur!v t = # T"t"1!V TD t,!! =0!V t,i TD =! "# ( i r t+i +!V t +i+1 "V t +i (!!!!!!!!!!!!!!=!E t+i " t,i Eligibility trace E keeps track how much a TD error should affect an earlier state Manfred Huber

17 Eligibility Traces To precisely reflect complex updates the retroactive updates would have to depend on the non-updated value function More practical (approximately equal for small T "!V t =!V! TD! =t!v TD t+i+1 =! ("# i ( r t+i +!V t+i +i+1 "V t+i +i!!!!!!!!!!!!!!=!e t+i " t+i Manfred Huber

18 Eligibility Traces Alternative view of eligibility traces and complex backups Updates to previous states reflect anticipated changes that would happen if the current state would have already been updated Individual propagation assumes that the states would happen in the same sequence the next time, too reflects the incremental, exponentially weighted averaging constant for subsequent states Addresses the fact that successor states do not follow deterministically Manfred Huber

19 TD( Temporal difference learning can be generalized with complex updates Single step error:! t = r t +"V t +1!V t Eligibility trace: E t (s = " $ # %$!"E t!1 (s+1!"e t!1 (s Traditional TD is TD(0 s = s t otherwise Manfred Huber

20 TD( s "=1 "=.975 "=.95 "=.99 "=.975 "= ! Batch update "=0 "=.8 "=.2 "=.4 "= Manfred Huber O " 1 "=1 "=.99 "=.975 "=.95 "=.9 "=.95 "=.8 Online TD(" on Random Walk ! "=.6 "=.4 "=.2 On-line update "=0

21 Eligibility Traces To use eligibility traces transition samples have to be generated consecutively Equations assume update sequences that are On-policy On-line (i.e. which form a consecutive state action sequence Extending this to state/action value function learning requires additional considerations What if policy changes? Manfred Huber

22 SARSA( SARSA requires on-policy data learning and its update depends only on the action taken Q t+1 = Q t +! r t +!Q t +1 +1!Q t SARSA( Single step error:! t = r t +"Q t +1 +1!Q t Eligibility trace: E t (s, a = " $ # %$ ( Q t+i+1 = Q t+i +!E t+i " t+i!"e t!1 (s, a+1 s = s t, a = a t!"e t!1 (s, a otherwise Manfred Huber

23 Q( Q-learning includes a policy improvement operation that changes the policy Q t+1 = Q t +! r t +! max b Q t +1, b!q t Single step Q-learning error: Handling non-policy actions Resetting eligibility traces & ( ( E t (s, a = ' ( (!"E t!1 (s, a+1 s = s t, a = a t " argmax a Q t, a 1 s = s t, a = a t # argmax a Q t, a 0 s $ s t %a $ a t # argmax a Q t, a!"e t!1 (s, a (! t = r t +" max b Q t +1, b!q t otherwise Manfred Huber

24 Eligibility Traces Eligibility traces can produce instabilities when their values get too high Replacing traces " $ 1 s = s t E t (s = # %$!"E t!1 (s otherwise Replacing and resetting can be combined % 1 s = s t, a = a t ' E t (s, a = & 0 s! s t "a! a t # argmax a Q t, a ' ('!"E t$1 (s, a otherwise Manfred Huber

25 Replacing Traces 0.5 RMS error at best " accumulating traces 0.2 replacing traces ! Manfred Huber

26 Value Function Updates Methods for updating the value function differ in the way they propagate values Eligibility traces provide a means to combine different depths updates Different depths provide different properties Low depths provide good bootstrapping using Markov property assumption High depths provide more precision but do not use local Markov assumption as efficiently High depths can be better if system is not truly Markov Manfred Huber

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