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2:1: Input: K ∈ N+ copies of independently initialized Q*-functions {Qk}k=1· Let B be a replay buffer storing transitions for training
for each episode do do
3: 4:Obtain initial state from environment so
5:for step t = 1,... until end of episode do
6: Pick an action aording toat = MajorityVote({argmaxq Qk(st,α)}-1)
7:Execute at. Receive state St+1 and reward rt from the environment
8:Add (st, at, rt, St+1) to replay buffer B
9:At learning interval, sample random minibatch and update {Qk }
10: 11: end forend for
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2:
Let B be a replay buffer storing experience for training.
3: for each episode do
Obtain initial state from environment so
for step t = 1,... until end of episode do
Pick an action according to at ∈ argmaxa {μ(St,a) + λ · δ(st, a)}
6: 7: Receive state St+1 and reward rt from environment, having taken action ät
8: Add (St,at, rt, St+1) to replay buffer B
9: At learning interval, sample random minibatch and update {Qk}
10: end for
11: end for
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value
hyperparameter total training frames40 milliondescriptionsLength of training for each game.
minibatch size32
replay buffer size1000000parameter update.The number of most recent frames
agent history length4
length.concatenated as input to the Q net- work. Total number of iterations = total training frames /agent history
target network update10000 frequency
discount factor0.99
action repeat4
update frequency 4
optimizerAdamOptimizer for parameter updates.
β1 0.9Adam optimizer parameter.
β0.99Adam optimizer parameter.
E 10-4Adam optimizer parameter.
learning rate schedule 210-4 Interp(10-4,5 * 10-5) 5*10-5t≤106 otherwise t>5*106Learning rate for Adam optimizer, as a function of iteration t.
exploration scheduleInterp(1,0.1) Interp(0.1,0.01) 0.01t<106 otherwise Probability of random action in e- t>5*106
greedy exploration, as a function of the iteration t . Number of uniform random ac- tions taken before learning starts.
replay start size50000
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UCB-Exploration
Bootstrapped DQNDoubleDQNEnsemble Voting 2282.8
Alien1445.12059.72817.6
Amidar430.58667.5683.72663.8
Assault2519.062820.613213.583702.76
Asterix3829.07639.58740.08732.0
Asteroids1009.51002.31149.31007.8
Atlantis1314058.01982677.01786305.02016145.0
Bank Heist795.1789.9869.4906.9
Battle Zone26230.024880.027430.026770.0
Beam Rider8006.587743.747991.99188.26
Bowling28.6230.9232.9238.06
Boxing85.9194.0794.4798.08
Breakout400.22467.45426.78411.31
Centipede5328.775177.516153.286237.18
Chopper Command2153.03260.03544.03677.0
Crazy Climber110926.0124456.0126677.0127754.0
Demon Attack9811.4523562.5530004.459861.9
Double Dunk-10.82-14.58-11.94-4.08
Enduro1314.311439.591999.882752.55
Fishing Derby21.8923.6930.0229.71
Freeway33.5732.9333.9233.96
Frostbite1284.8529.21196.01903.0
Gopher7652.212030.010993.212910.8
Gravitar227.5279.5371.5318.0
Ice Hockey-4.62-4.63-1.73-4.71
Jamesbond594.5594.0602.0710.0
Kangaroo8186.07787.08174.014196.0
Krull8537.528517.918669.179171.61
Kung Fu Master24153.032896.030988.031291.0
Montezuma Revenge2.04.01.04.0
Ms Pacman2508.72498.13039.73425.4
Name This Game8212.49806.99255.19570.5
Pitfall-5.99-7.57-3.37-1.47
Pong21.020.6721.020.95
Private Eye1815.19788.631845.281252.01
Qbert10557.256529.512036.514198.25
Riverraid11528.011834.712785.815622.2
Road Runner52489.049039.054768.053596.0
Robotank21.0329.831.8341.04
Seaquest9320.718056.420458.624001.6
Space Invaders1549.91917.51890.82626.55
Star Gunner20115.052283.041684.047367.0
Tennis-15.11-14.04-11.63-7.8
Time Pilot5088.05548.06153.06490.0
Tutankham167.47223.43208.61200.76
Up N Down9049.111815.319528.319827.3
Venture115.096.078.067.0
Video Pinball364600.85374686.89343380.29372564.11
Wizard Of Wor2860.03877.05451.05873.0
Zaxxon592.08903.03901.03695.0
Times best17930
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Ensemble VotingUCB-ExplorationA3C+
Alien2282.82817.61848.33
Amidar683.72663.8964.77
Assault3213.583702.762607.28
Asterix8740.08732.07262.77
Asteroids1149.31007.82257.92
Atlantis1786305.02016145.01733528.71
Bank Heist869.4906.9991.96
Battle Zone27430.026770.07428.99
Beam Rider7991.99188.265992.08
Bowling32.9238.0668.72
Boxing94.4798.0813.82
Breakout426.78411.31323.21
Centipede6153.286237.185338.24
Chopper Command3544.03677.05388.22
Crazy Climber126677.0127754.0104083.51
Demon Attack30004.459861.919589.95
Double Dunk-11.94-4.08-8.88
Enduro1999.882752.55749.11
Fishing Derby30.0229.7129.46
Freeway33.9233.9627.33
Frostbite1196.01903.0506.61
Gopher10993.212910.85948.40
Gravitar371.5318.0246.02
Ice Hockey-1.73-4.71-7.05
Jamesbond602.0710.01024.16
Kangaroo8174.014196.05475.73
Krull8669.179171.617587.58
Kung Fu Master30988.031291.026593.67
Montezuma Revenge1.04.0142.50
Ms Pacman3039.73425.42380.58
Name This Game9255.19570.56427.51
Pitfall-3.37-1.47-155.97
Pong21.020.9517.33
Private Eye1845.281252.01100.0
Qbert12036.514198.2515804.72
Riverraid12785.815622.210331.56
Road Runner54768.053596.049029.74
Robotank31.8341.046.68
Seaquest20458.624001.62274.06
Space Invaders1890.82626.551466.01
Star Gunner41684.047367.052466.84
Tennis-11.63-7.8-20.49
Time Pilot6153.06490.03816.38
Tutankham208.61200.76132.67
Up N Down19528.319827.38705.64
Venture78.067.00.00
Video Pinball343380.29372564.1135515.92
Wizard Of Wor5451.05873.03657.65
Zaxxon3901.03695.07956.05
Times Best102810
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CategoryTotalBootstrapped DQNDouble DQNEnsemble VotingUCB-Exploration
Human Optimal2303515
Score Explicit80215
Dense Reward80116
Sparse Reward51022
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1: : Input: Value function networks Q with K outputs {Qk}K=1. Hyperparameters T,X,and p.
2: Let B be a replay buffer storing experience for training. for each episode do
3:
4:Obtain initial state from environment so
5:for step t = 1,... until end of episode do
6:Pick an action according to at ∈ argmaxa{μ(st,a) + λ ·δ(st,a)}
7:Receive state St+1 and reward rt from environment, having taken action at
8:Calculate exploration bonus br(st) according to (12)
9:Add (St,at,rt + ρ : br(St), St+1) to replay buffer B
10:At learning interval, sample random minibatch and update {Qk}
11: end for 12: end for
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