Clipped Action Policy Gradient
Yasuhiro Fujita and Shin-ichi Maeda
Preferred Networks, Inc.
A variance reduction technique for policy gradient
with bounded actions
Mismatch between action space and policy
l Policy gradient with continuous actions
– the action space is bounded (ex. ,)
– but, a policy to optimize has unbounded support (ex. Gaussian)
action space
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Action clipping
l Actions sampled from a policy need to be clipped
l The environment uses , not , to decide next state
and reward
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u = max(min(u, ), ↵)<latexit sha1_base64="hfspXzBFYV0X/poTaCqn7jh1Fsc=">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</latexit><latexit sha1_base64="hfspXzBFYV0X/poTaCqn7jh1Fsc=">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</latexit><latexit sha1_base64="hfspXzBFYV0X/poTaCqn7jh1Fsc=">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</latexit><latexit sha1_base64="hfspXzBFYV0X/poTaCqn7jh1Fsc=">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</latexit>
s0
⇠ P(·|s, u)<latexit sha1_base64="0UXWTuU0v5YNdUdSQQUChh6vku0=">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</latexit><latexit sha1_base64="0UXWTuU0v5YNdUdSQQUChh6vku0=">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</latexit><latexit sha1_base64="0UXWTuU0v5YNdUdSQQUChh6vku0=">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</latexit><latexit sha1_base64="0UXWTuU0v5YNdUdSQQUChh6vku0=">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</latexit>
r = R(s, u, s0
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<latexit sha1_base64="5W55hMsCWqzvIEcoiS8G3ARQPMA=">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
Policy gradient with action clipping
l Policy gradient (PG) uses actions before clipping, not
l It is unbiased
l It works well
l But, why not use the knowledge of actions being clipped?
E[Q⇡✓
(s, u) (s, u)] = r✓ E
h X
t
t
rt
i
| {z }
expected return<latexit sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">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</latexit><latexit sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">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</latexit><latexit sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">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</latexit><latexit sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">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</latexit>
u<latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">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</latexit><latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">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</latexit><latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">AAAHgHicjZVZa9wwEMed9Nh0e7ePfTENgdCyh0NLQyCQJgTal5AeOSBegizPrs3KkiPJORD+EH1tP1m/TUdet83aCq3BMJ75/6TRWBpFOUuVHg5/Lizeun3nbmfpXvf+g4ePHj95+uxQiUJSOKCCCXkcEQUs5XCgU83gOJdAsojBUTTdsfGjc5AqFfyrvsphlJEJT8cpJRpdR2FEpCnK0yfLw/6wevy2EdTGslc/+6dP79IwFrTIgGvKiFInwTDXI0OkTimDshsWCnJCp2QCJ2hykoEamSrf0l9BT+yPhcSXa7/yXicMyZS6yiJUZkQnqhmzTlfspNDj9ZFJeV5o4HQ20bhgvha+XbwfpxKoZldoECpTzNWnCZGEaizR3CyxsqnhCD7mjsVUfmini6JsTkYTPsnRmFuwsUotBFPz7kiIqSYRervhH4kCbVQiLgRnVxLGalPLAqyCwwUVWUZ4/MqETFyUJgSuCgkWNSFheULKsqFL0knSEEag27rdhmi2NrNbtpSHRLq1hw7tfyt3xLlDKzNjAy31F000KAdACTNfUO/jf1JW5CssNjTw99Tu9Zv4g5onlco5wGf3uj7XJJ431iD23MReTXCiC9mCiJxk5NJdlzrWKg1lae4mqkhLn6c6sRtiDrHu03AWaBHbQgK7oXbbbTmc4b7kWmQNpPYaU26a0pEXK9S/wNcu8KMjtViZoK50ymPb62yrKXj1h5sFJPkkV2nz0GBbDK3bWfAc4txN3AzsO4F9BxDDGM4a4nNqGy32Sb6J+u6Kn5Ep+OE5UFP6kWBxN2Sg7fdsq427M8ds39VFsRyOMPHt3GgnWudqYzDQcNnH00OncIndkE+gj8kMzgpQ1bkZrK0Fw+EAt0cPuR4m2RN4pdgbp1dnjb1Twt9FVIsrT4IR/o4pSI43yNusCH9TtbdXeZeD61/ldcBeS0HzEmobh2v9YNgPPr1Z3lqvL6gl74X30lv1Au+dt+V98Pa9A496U++b99370VnsrHYGnWAmXVyomefe3NPZ+AXd/b4L</latexit><latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">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</latexit>
PG: Q⇡✓
(s, u)
| {z }
action value
r✓ log ⇡✓(u|s)
| {z }
(s, u)<latexit sha1_base64="WyXT3TDFerJaT275SAvIWc9D4WA=">AAAIHHicjZXdjtw0FMczBTpl+NrCDRI3pqtKy8d8ZAWiQlqptKoEN9W2ZbeVNsPIcc5MrHHsrO3M7mLyBjwDT8Md4haJ1+AJOE7SdibJCiKN5Dn+/46Pj+1z4lxwY2ezvwc33njzrZvDW2+P3nn3vfc/2Lv94alRhWZwwpRQ+kVMDQgu4cRyK+BFroFmsYDn8fqhn3++AW24kj/aqxzmGV1JvuSMWjQt9n6NCpmAjjVl4J785KKcLyKbgqXlgfmy+KxcuCijNhVUr0C7yMKldZR5mGyoKKAsS7LtI5I0FrTxQSKhVuS1z4PiF9NxmRteL1Uu9vZnk1n1ke4gbAb7QfMdL27fvIwSxYoMpGWCGnMWznI7d1RbzgSUo6gwkFO2pis4w6GkGZi5q9JWkrtoSchSafxJSyrrNuFoZsxVFqPSx2vac97YN3dW2OW9ueMyLyxIVi+0LASxivgzIAnXwKy4wgFlmmOshKUU02fxpHZWSYwPDT0Q8HkFQ6rUxXG2I2OpXOU42Nmw80qrlDC75liptaVxy6pBGP4zehhFrzgD1plUXSgprjQszZHVRaWQcMFUllGZfO7wiC9KF4E0hQaPuoiKPKVl2dKlfJW2hLG/Z23do5ao3rB7VHaUp1T3a097tP9b+VBterQ6c36io35mqQXTAzAq3DP/OvDwjBcRg7mGFv5d9ZSu408avnlwfQ6e9u/raUNiLRAt4nE/8bghJLWF7kD4VjN62Z+XZq6TGiZ43k9UMx19zuvCs4NsV6QO8UDhtb0mdw+6cjjHeymtylpIY3WuPHJlT1yiMP8FftEH/tATWmJc2GSay8TXYV9/ClmdcDuBNF9hdWw/GloXzf6E55Dk/cT1wHEvcNwDJLCE85Z4w3z1xeIpj1A/uksyugYSbYC5ksRKJKNIgPX/66u2HNWG+t41SfEcesBmgWvjOLU2N99Op9hvJvh62BousUTKFUwwmOl5AaZ6N9PDw3A2m+L1GCM3xiDHCtud74bjJmosqBpeb6LaXHkWzvE41qAltpWvsyJ6STXWcWXdD7f/lduA71VhuzN1B6eHk3A2CZ98tX//XtO1bgWfBHeCgyAMvgnuB98Hx8FJwIJ/Bh8PPh3cGf42/H34x/DPWnpj0DAfBTvf8K9/AfQW/b8=</latexit><latexit sha1_base64="WyXT3TDFerJaT275SAvIWc9D4WA=">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</latexit><latexit sha1_base64="WyXT3TDFerJaT275SAvIWc9D4WA=">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</latexit><latexit sha1_base64="WyXT3TDFerJaT275SAvIWc9D4WA=">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</latexit>
Clipped action policy gradient (CAPG)
l For scalar actions, use
instead of PG
l Similarly defined for vector actions that are conditionally independent
Q⇡✓
(s, u) r✓ log ⇡✓(u|s)
| {z }
(s, u)<latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit><latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit><latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit><latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit>
(s, u) :=
8
><
>:
r✓ log ⇧✓(↵|s) if u  ↵
r✓ log ⇡✓(u|s) if ↵ < u <
r✓ log(1 ⇧✓( |s)) if  u<latexit 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Q⇡✓
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Clipped action policy gradient (CAPG)
l For scalar actions, use
instead of PG
l Similarly defined for vector actions that are conditionally independent
Q⇡✓
(s, u) r✓ log ⇡✓(u|s)
| {z }
(s, u)<latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit><latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit><latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">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</latexit><latexit sha1_base64="RhoUS88//m+CAO024kLkLG432oY=">AAAH7HicjVXdjtw0FE4LdMrw0y1ccmN1VWkLzE9WraiQViqtKtGbatuy20qbYeQ4ZybWOHbWdvYHk7foXdVbHoS34A24LU/AcSbQncQrsBTp5Jzvs48/2+ekpeDGTqd/XLn6wYcfXRtc/3j4yaeffX5j6+YXh0ZVmsEBU0LpVyk1ILiEA8utgFelBlqkAl6mq0c+/vIEtOFK/mTPS5gVdCn5gjNq0TXfSp797JKSzxObg6X1jvm2ukOSSmagU00ZuETSVNA2ThKhluQ9fqf61dyp5y4pqM0F1UvQOJvhzTR1Pd/ano6nzSB9I26N7agd+/Ob186STLGqAGmZoMYcxdPSzhzVljMB9TCpDJSUregSjtCUtAAzc40MNbmNnowslMZPWtJ4LzIcLYw5L1JE+nxNN+adodhRZRf3Z47LsrIg2XqhRSWIVcRrSjKugVlxjgZlmmOuhOUU5bOo/MYqmfGp4QwEvK5gSCNdmhYbMJbLZYnGxoadR1qlhNl0p0qtLE07Xg3C8F9whmHyL8+AdSZXp0qKcw0Ls2d11SAknDJVFFRmXzs84tPaJSBNpcFTXUJFmdO67uByvsw7wNTfoS7ucQe03rB7XPeQh1SHsYcB7P9GPlInAawunA/00C8stWACBEaFe4F4godnPIgY1Bo69B+Yf1eX8Q9aPm1QwQmeh/f1vGXi2xYdxtMw42nLkNRWukfCt1rQs7AubawnDRO8DDOaSA9f8nVR2aBcrDY9xkOF1/YS7R724XCM91JaVXQorde5es/VgbxEZf6L+E2I+CSQWmZc3CrNZebrqq8/lWxOuCsgLZdYHbuPhq6LZljwErIyzLicsB8k7AcIGSzguAM+Yb76YvGUe4gf3iYFXQFJToC5mqRKZMNEgPX/66u2GK4d63vXiuJ5OAM2C1wb7dza0nw/mVg4G+PrYSs4wxIplzDGZCbHFZjm3Ux2d+PpdILXY4S8ESY5Uti+fHcbtVljQdXwfhPN5uqjeIbHsQItsa3cK6rkH1brHTXe7fjiX32R4HtV3O1MfeNwdxxPx/Gzu9sP7rdd63r0VXQr2oni6LvoQfRjtB8dRCz6Pfozehf9NZCD14M3g7dr6NUrLefLaGMMfvsbYl3qxw==</latexit>
(s, u) :=
8
><
>:
r✓ log ⇧✓(↵|s) if u  ↵
r✓ log ⇡✓(u|s) if ↵ < u <
r✓ log(1 ⇧✓( |s)) if  u<latexit 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sha1_base64="Jjg6jTyJZ/B9WjPlF0rsqoWVTZM=">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</latexit><latexit sha1_base64="Jjg6jTyJZ/B9WjPlF0rsqoWVTZM=">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</latexit>
Q⇡✓
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equal
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Clipped action policy gradient (CAPG)
l For scalar actions, use
instead of PG
l Similarly defined for vector actions that are conditionally independent
⇧✓(↵|s) :=
Z ↵
1
⇡✓(u|s)du
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Q⇡✓
(s, u) r✓ log ⇡✓(u|s)
| {z }
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(s, u) :=
8
><
>:
r✓ log ⇧✓(↵|s) if u  ↵
r✓ log ⇡✓(u|s) if ↵ < u <
r✓ log(1 ⇧✓( |s)) if  u<latexit 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Q⇡✓
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What is good about CAPG?
l Under mild assumptions, for any , CAPG is
– unbiased
– elementwise lower-variance than PG
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Eu[Q⇡✓
(s, u) (s, u)
| {z }
CAPG
|s] = Eu[Q⇡✓
(s, u) (s, u)
| {z }
PG
|s]
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Vu[Q⇡✓
(s, u) (s, u)
| {z }
CAPG
|s]  Vu[Q⇡✓
(s, u) (s, u)
| {z }
PG
|s]
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Numerical analysis
l Estimated gradients with a policy
– (left) Mean gradients by CAPG and PG are almost the same
– (center and right) Standard dev. of gradients are smaller for CAPG
u ⇠ N(✓mean, (exp(✓log std))2
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0 1 2
mean of policy
0.10
0.12
0.14
0.16
0.18
stdevofgradients
10 1
100
101
variance of policy
0.0
0.1
0.2
0.3
0.4
stdevofgradients
0 1 2
mean of policy
0.1
0.0
0.1
0.2
meanofgradients
(PG overlaps CAPG)
CAPG ✓mean CAPG ✓logstd PG ✓mean PG ✓logstd
Experiments on MuJoCo environments
l We replaced PG in existing algorithms with CAPG
– Proximal policy optimization (PPO)
– Trust region policy optimization (TRPO)
l Ten MuJoCo environments in OpenAI Gym
Experimental results
l CAPG either significantly outperformed PG, or made no significant
difference in average returns throughout training (50 random seeds)
0.2 0.4 0.6 0.8 1.0
number of timesteps ⇥107
1
2
3
4
5
return
⇥103 Humanoid-v1
TRPO CAPG
TRPO PG
0.2 0.4 0.6 0.8 1.0
number of timesteps ⇥107
0.6
0.8
1.0
1.2
1.4
return
⇥105 HumanoidStandup-v1
TRPO CAPG
TRPO PG
Large improvement Small but significant improvement
Experimental results
l CAPG either significantly outperformed PG, or made no significant
difference in average returns throughout training (50 random seeds)
No significant difference
0.2 0.4 0.6 0.8 1.0
number of timesteps ⇥107
0
1
2
3
return
⇥103 Ant-v1
TRPO CAPG
TRPO PG
0.2 0.4 0.6 0.8 1.0
number of timesteps ⇥106
0.0
0.5
1.0
return
⇥103 HalfCheetah-v1
TRPO CAPG
TRPO PG
Summary
l For a bounded action space and a policy with unbounded support,
use CAPG
instead of PG for lower variance.
action spaceclipped to clipped to
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(s, u)r✓ log ⇧✓(↵|s) if u  ↵
Q⇡✓
(s, u)r✓ log ⇡✓(u|s) if ↵ < u <
Q⇡✓
(s, u)r✓ log(1 ⇧✓( |s)) if  u<latexit 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Q⇡✓
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Poster: #175

Clipped Action Policy Gradient

  • 1.
    Clipped Action PolicyGradient Yasuhiro Fujita and Shin-ichi Maeda Preferred Networks, Inc. A variance reduction technique for policy gradient with bounded actions
  • 2.
    Mismatch between actionspace and policy l Policy gradient with continuous actions – the action space is bounded (ex. ,) – but, a policy to optimize has unbounded support (ex. 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  • 3.
    Action clipping l Actionssampled from a policy need to be clipped l The environment uses , not , to decide next state and reward u ⇠ ⇡✓(·|s)<latexit 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u = max(min(u, ), ↵)<latexit 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s0 ⇠ P(·|s, u)<latexit 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r = R(s, u, s0 )<latexit 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sha1_base64="5W55hMsCWqzvIEcoiS8G3ARQPMA=">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</latexit> ↵<latexit 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sha1_base64="i8ketnqi3IZJQzl4OPc8UzaE5yM=">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</latexit><latexit 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<latexit sha1_base64="5W55hMsCWqzvIEcoiS8G3ARQPMA=">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
  • 4.
    Policy gradient withaction clipping l Policy gradient (PG) uses actions before clipping, not l It is unbiased l It works well l But, why not use the knowledge of actions being clipped? E[Q⇡✓ (s, u) (s, u)] = r✓ E h X t t rt i | {z } expected return<latexit 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sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">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</latexit><latexit sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">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</latexit><latexit sha1_base64="yaabuIKwpJjR5SUJfgS5PLKhDoM=">AAAH9HicjZXfj9w0EMezhXbLAu0VHnmxOFUqP/ZHTiAqpJPKVSfBS3VtuWul9XblOLNJtI6ds529W1n5T3hDvPL/8MS/wjgb4DbJCSJFmoy/n3g8tmeiQmTGzmZ/DO689/7de8P7H4w+/OjjBw8PHn1yYVSpOZxzJZR+GzEDIpNwbjMr4G2hgeWRgDfR+rkff7MBbTIlf7bbAhY5S2S2yjiz6FoebOnpnLx852iRLalNwbLqifm6/ILQwmQ7a0GOCZUsEqxREFrKGHSkGQdHT+lJlsypKfOlJTRhec7eWaKX1vsX1dJRC9fWwXUB3EJMNNhSy6paHhzOJrP6IV0jbIzDoHnOlo/ucRorXuYgLRfMmHk4K+zCMW0zLqAa0dJAwfiaJTBHU7IczMLVOarIY/TEZKU0vtKS2nuTcCw3ZptHqMyZTU17zDv7xualXT1duEwWpQXJdxOtSkGsIj7hJM40Llxs0WBcZxgr4SnD3Fnclr1ZYuNDwz8Q8OkGQ6ifLoryPRlPZVKgsbdg55VWKWH23ZFSa8si9I7oPxID1plUXSkpthpW5tjqErxCwhVXuIEy/tJRoa4qR0GaUoNHHWWiSFlVtXRplqQtYeTPUVt32hLt1uZOq47ygul+7UWP9n8rn6tNj1bnzg901K8ts2B6AM6Ee416gvtkvIgYTDa08B+4v1+38ecNz2pV7w9e9a/rVUPiHRct4kU/8aIhJMN714GYTnJ23Z+XZqyTGi6yop+oRzr6ItsVlj3kZsXpECdKg7gldyddOVziuZRW5S2k8TpXHbuqJy5Rmv8Cv+oDf+oJLTYubDKdydjXV19qSlnvcDuBrEiwvLYvDdPOV93+hBcQF/3E7cBZL3DWA8SwgsuWeMN9ocU6KY9RP3pMcrYGQjfAXUUiJeIRFWD99+6orUY7x+7cNUnxHP4hIX5utFNrC/P9dIo9YYK3h6/hGquhTGCCwUwvSzD1vZkeHYWz2RSPxxi5MQY5VtjGfJcbN1Fj7dTw7yLqxVXzcIHbsQYtsYN8m5f0b6rxjmvvYXjzq7oJ+LYUtptQ17g4moSzSfjym8NnT5sGdT/4LPg8eBKEwXfBs+DH4Cw4D3jw5+Du4MHg4XAz/GX46/C3nfTOoGE+Dfae4e9/AXue6bU=</latexit> u<latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">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</latexit><latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">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</latexit><latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">AAAHgHicjZVZa9wwEMed9Nh0e7ePfTENgdCyh0NLQyCQJgTal5AeOSBegizPrs3KkiPJORD+EH1tP1m/TUdet83aCq3BMJ75/6TRWBpFOUuVHg5/Lizeun3nbmfpXvf+g4ePHj95+uxQiUJSOKCCCXkcEQUs5XCgU83gOJdAsojBUTTdsfGjc5AqFfyrvsphlJEJT8cpJRpdR2FEpCnK0yfLw/6wevy2EdTGslc/+6dP79IwFrTIgGvKiFInwTDXI0OkTimDshsWCnJCp2QCJ2hykoEamSrf0l9BT+yPhcSXa7/yXicMyZS6yiJUZkQnqhmzTlfspNDj9ZFJeV5o4HQ20bhgvha+XbwfpxKoZldoECpTzNWnCZGEaizR3CyxsqnhCD7mjsVUfmini6JsTkYTPsnRmFuwsUotBFPz7kiIqSYRervhH4kCbVQiLgRnVxLGalPLAqyCwwUVWUZ4/MqETFyUJgSuCgkWNSFheULKsqFL0knSEEag27rdhmi2NrNbtpSHRLq1hw7tfyt3xLlDKzNjAy31F000KAdACTNfUO/jf1JW5CssNjTw99Tu9Zv4g5onlco5wGf3uj7XJJ431iD23MReTXCiC9mCiJxk5NJdlzrWKg1lae4mqkhLn6c6sRtiDrHu03AWaBHbQgK7oXbbbTmc4b7kWmQNpPYaU26a0pEXK9S/wNcu8KMjtViZoK50ymPb62yrKXj1h5sFJPkkV2nz0GBbDK3bWfAc4txN3AzsO4F9BxDDGM4a4nNqGy32Sb6J+u6Kn5Ep+OE5UFP6kWBxN2Sg7fdsq427M8ds39VFsRyOMPHt3GgnWudqYzDQcNnH00OncIndkE+gj8kMzgpQ1bkZrK0Fw+EAt0cPuR4m2RN4pdgbp1dnjb1Twt9FVIsrT4IR/o4pSI43yNusCH9TtbdXeZeD61/ldcBeS0HzEmobh2v9YNgPPr1Z3lqvL6gl74X30lv1Au+dt+V98Pa9A496U++b99370VnsrHYGnWAmXVyomefe3NPZ+AXd/b4L</latexit><latexit sha1_base64="dwQi/GSg/5Ve+tLQrQww/MqnfJE=">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</latexit> PG: Q⇡✓ (s, u) | {z } action value r✓ log ⇡✓(u|s) | {z } (s, u)<latexit 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  • 5.
    Clipped action policygradient (CAPG) l For scalar actions, use instead of PG l Similarly defined for vector actions that are conditionally independent Q⇡✓ (s, u) r✓ log ⇡✓(u|s) | {z } (s, u)<latexit 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(s, u) := 8 >< >: r✓ log ⇧✓(↵|s) if u  ↵ r✓ log ⇡✓(u|s) if ↵ < u < r✓ log(1 ⇧✓( |s)) if  u<latexit 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Q⇡✓ (s, u) (s, u)<latexit 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  • 6.
    Clipped action policygradient (CAPG) l For scalar actions, use instead of PG l Similarly defined for vector actions that are conditionally independent Q⇡✓ (s, u) r✓ log ⇡✓(u|s) | {z } (s, u)<latexit 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(s, u) := 8 >< >: r✓ log ⇧✓(↵|s) if u  ↵ r✓ log ⇡✓(u|s) if ↵ < u < r✓ log(1 ⇧✓( |s)) if  u<latexit 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Q⇡✓ (s, u) (s, u)<latexit 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  • 7.
    Clipped action policygradient (CAPG) l For scalar actions, use instead of PG l Similarly defined for vector actions that are conditionally independent ⇧✓(↵|s) := Z ↵ 1 ⇡✓(u|s)du <latexit 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Q⇡✓ (s, u) r✓ log ⇡✓(u|s) | {z } (s, u)<latexit 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(s, u) := 8 >< >: r✓ log ⇧✓(↵|s) if u  ↵ r✓ log ⇡✓(u|s) if ↵ < u < r✓ log(1 ⇧✓( |s)) if  u<latexit 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Q⇡✓ (s, u) (s, u)<latexit 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  • 8.
    What is goodabout CAPG? l Under mild assumptions, for any , CAPG is – unbiased – elementwise lower-variance than PG s<latexit 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Vu[Q⇡✓ (s, u) (s, u) | {z } CAPG |s]  Vu[Q⇡✓ (s, u) (s, u) | {z } PG |s] <latexit 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  • 9.
    Numerical analysis l Estimatedgradients with a policy – (left) Mean gradients by CAPG and PG are almost the same – (center and right) Standard dev. of gradients are smaller for CAPG u ⇠ N(✓mean, (exp(✓log std))2 )<latexit 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0 1 2 mean of policy 0.10 0.12 0.14 0.16 0.18 stdevofgradients 10 1 100 101 variance of policy 0.0 0.1 0.2 0.3 0.4 stdevofgradients 0 1 2 mean of policy 0.1 0.0 0.1 0.2 meanofgradients (PG overlaps CAPG) CAPG ✓mean CAPG ✓logstd PG ✓mean PG ✓logstd
  • 10.
    Experiments on MuJoCoenvironments l We replaced PG in existing algorithms with CAPG – Proximal policy optimization (PPO) – Trust region policy optimization (TRPO) l Ten MuJoCo environments in OpenAI Gym
  • 11.
    Experimental results l CAPGeither significantly outperformed PG, or made no significant difference in average returns throughout training (50 random seeds) 0.2 0.4 0.6 0.8 1.0 number of timesteps ⇥107 1 2 3 4 5 return ⇥103 Humanoid-v1 TRPO CAPG TRPO PG 0.2 0.4 0.6 0.8 1.0 number of timesteps ⇥107 0.6 0.8 1.0 1.2 1.4 return ⇥105 HumanoidStandup-v1 TRPO CAPG TRPO PG Large improvement Small but significant improvement
  • 12.
    Experimental results l CAPGeither significantly outperformed PG, or made no significant difference in average returns throughout training (50 random seeds) No significant difference 0.2 0.4 0.6 0.8 1.0 number of timesteps ⇥107 0 1 2 3 return ⇥103 Ant-v1 TRPO CAPG TRPO PG 0.2 0.4 0.6 0.8 1.0 number of timesteps ⇥106 0.0 0.5 1.0 return ⇥103 HalfCheetah-v1 TRPO CAPG TRPO PG
  • 13.
    Summary l For abounded action space and a policy with unbounded support, use CAPG instead of PG for lower variance. action spaceclipped to clipped to ⇡✓(u|s)<latexit 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PDF action u<latexit 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Q⇡✓ (s, u)r✓ log ⇡✓(u|s)<latexit 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Poster: #175