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⇡⇤
2 argmax
⇡
E⇡
" 1X
t=0
t
rt
#
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R : S ⇥ A ! R<latexit sha1_base64="sOKQo7jhJQ7IhqYjp0gmf22f2Ig=">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</latexit><latexit sha1_base64="sOKQo7jhJQ7IhqYjp0gmf22f2Ig=">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</latexit><latexit sha1_base64="sOKQo7jhJQ7IhqYjp0gmf22f2Ig=">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</latexit><latexit sha1_base64="sOKQo7jhJQ7IhqYjp0gmf22f2Ig=">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</latexit>
(S, A, P, R, ⇢0, )<latexit sha1_base64="UJRWbmNDTVAAncv103Huefh9uvo=">AAACsXichVFNSxtRFD2OWm1qa9RNoZuhQbFQwh0VFFd+bLpwERPjByrpm/GZDJkvZl4COuQP+AdcuFIoUtz5F9z4B4S61J10aaGbLnozGZBU2t5h3jvnvHvuu+89M3DsSBHd9mi9ff0vBgZfZl4NvX4znB0ZXY/8RmjJsuU7frhpikg6tifLylaO3AxCKVzTkRtmfbm9vtGUYWT73po6COSuK6qevW9bQrFUya5M6juuUDVLOHGppX98YotdrNDFigkLa34lphajqnBdoX+oZHOUpyT058BIQQ5pFPzsOXawBx8WGnAh4UExdiAQ8bcNA4SAtV3ErIWM7GRdooUMexucJTlDsFrnscpsO1U95u2aUeK2eBeH/5CdOsbphr7SI13TBT3Qr7/WipMa7V4OeDY7XhlUho/eln7+1+XyrFB7cv2zZ4V9zCW92tx7kCjtU1gdf/Pw+LE0XxyPJ+iMvnP/p3RLV3wCr/nD+rIqiyfI8AMYf173c7A+lTcob6zO5BaW0qcYxDu8xyTf9ywW8AkFlHnfS3zDHe61aW1L+6yZnVStJ/WMoSu0+m9f1KWm</latexit><latexit sha1_base64="UJRWbmNDTVAAncv103Huefh9uvo=">AAACsXichVFNSxtRFD2OWm1qa9RNoZuhQbFQwh0VFFd+bLpwERPjByrpm/GZDJkvZl4COuQP+AdcuFIoUtz5F9z4B4S61J10aaGbLnozGZBU2t5h3jvnvHvuu+89M3DsSBHd9mi9ff0vBgZfZl4NvX4znB0ZXY/8RmjJsuU7frhpikg6tifLylaO3AxCKVzTkRtmfbm9vtGUYWT73po6COSuK6qevW9bQrFUya5M6juuUDVLOHGppX98YotdrNDFigkLa34lphajqnBdoX+oZHOUpyT058BIQQ5pFPzsOXawBx8WGnAh4UExdiAQ8bcNA4SAtV3ErIWM7GRdooUMexucJTlDsFrnscpsO1U95u2aUeK2eBeH/5CdOsbphr7SI13TBT3Qr7/WipMa7V4OeDY7XhlUho/eln7+1+XyrFB7cv2zZ4V9zCW92tx7kCjtU1gdf/Pw+LE0XxyPJ+iMvnP/p3RLV3wCr/nD+rIqiyfI8AMYf173c7A+lTcob6zO5BaW0qcYxDu8xyTf9ywW8AkFlHnfS3zDHe61aW1L+6yZnVStJ/WMoSu0+m9f1KWm</latexit><latexit sha1_base64="UJRWbmNDTVAAncv103Huefh9uvo=">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</latexit><latexit sha1_base64="UJRWbmNDTVAAncv103Huefh9uvo=">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</latexit>
S, A<latexit sha1_base64="QA65atGbL276ApMeusSM2vOeAGQ=">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</latexit><latexit sha1_base64="QA65atGbL276ApMeusSM2vOeAGQ=">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</latexit><latexit sha1_base64="QA65atGbL276ApMeusSM2vOeAGQ=">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</latexit><latexit sha1_base64="QA65atGbL276ApMeusSM2vOeAGQ=">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</latexit>
⇢0 : S ! R<latexit sha1_base64="fPJoGXIseznf6aFKaAxHtRZIcNc=">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</latexit><latexit sha1_base64="fPJoGXIseznf6aFKaAxHtRZIcNc=">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</latexit><latexit sha1_base64="fPJoGXIseznf6aFKaAxHtRZIcNc=">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</latexit><latexit sha1_base64="fPJoGXIseznf6aFKaAxHtRZIcNc=">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</latexit>
2 [0, 1)<latexit sha1_base64="N+lsnfCp40f/+in4nSeXK9ok3Wk=">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</latexit><latexit sha1_base64="N+lsnfCp40f/+in4nSeXK9ok3Wk=">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</latexit><latexit sha1_base64="N+lsnfCp40f/+in4nSeXK9ok3Wk=">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</latexit><latexit sha1_base64="N+lsnfCp40f/+in4nSeXK9ok3Wk=">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</latexit>
⇡ : S ⇥ A ! R<latexit sha1_base64="uHGwoARb8hndxeCH/tMmlEre6lc=">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</latexit><latexit sha1_base64="uHGwoARb8hndxeCH/tMmlEre6lc=">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</latexit><latexit sha1_base64="uHGwoARb8hndxeCH/tMmlEre6lc=">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</latexit><latexit sha1_base64="uHGwoARb8hndxeCH/tMmlEre6lc=">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</latexit>
⇢⇡
(s) =
1X
t=0
t
Pr(st = s|⇡, ⇢0)
<latexit sha1_base64="kUS6hK9oeVifqwOPqSmM1kq5/eQ=">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</latexit><latexit sha1_base64="kUS6hK9oeVifqwOPqSmM1kq5/eQ=">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</latexit><latexit sha1_base64="kUS6hK9oeVifqwOPqSmM1kq5/eQ=">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</latexit><latexit sha1_base64="kUS6hK9oeVifqwOPqSmM1kq5/eQ=">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</latexit>
P : S ⇥ A ⇥ S ! R<latexit sha1_base64="7T1RClzAdnC0cw1ysfXv3VjN/5A=">AAACs3ichVG7ThtBFD1eHiFOAg5pItGsYhFRWXcRiIjKQJMqMnaMLdnI2l0PMGJf2h1bclb+AX6AgiqRoiii5Q9o8gMpTAtNlJJIaVLker2KRayQu9qZc8/cc+fMjBU4MlJEg4w2NT0z+2DuYfbR4yfzC7mni3uR3wltUbV9xw/rlhkJR3qiqqRyRD0IhelajqhZxzvD9VpXhJH0vbeqF4h91zz05IG0TcVUK/em6ZrqyDaduNTXN/U/WaWvN5V0RTSmtiappMof5ZYVl/utXJ4KlIQ+CYwU5JFGyc99QhNt+LDRgQsBD4qxAxMRfw0YIATM7SNmLmQkk3WBPrKs7XCV4AqT2WMeDzlrpKzH+bBnlKht3sXhP2SljmX6Sp/plr7QOX2jX//sFSc9hl56PFsjrQhaCyfPKz//q3J5Vjgaq+71rHCAV4lXyd6DhBmewh7pu+9Obyub5eX4JX2g7+z/PQ3okk/gdX/YH3dF+QxZfgDj7+ueBHurBYMKxu5avridPsUclvACK3zfGyjiNUqo8r4XuMI1brR1raFZWntUqmVSzTPcCc39DR2QqSw=</latexit><latexit sha1_base64="7T1RClzAdnC0cw1ysfXv3VjN/5A=">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</latexit><latexit sha1_base64="7T1RClzAdnC0cw1ysfXv3VjN/5A=">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</latexit><latexit sha1_base64="7T1RClzAdnC0cw1ysfXv3VjN/5A=">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</latexit>
⌧ = (s0, a0, s1, a1, · · · , s1, a1)<latexit sha1_base64="Ta91/z+3/Mhcg0VLicc2H3EDnbw=">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</latexit><latexit sha1_base64="Ta91/z+3/Mhcg0VLicc2H3EDnbw=">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</latexit><latexit sha1_base64="Ta91/z+3/Mhcg0VLicc2H3EDnbw=">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</latexit><latexit sha1_base64="Ta91/z+3/Mhcg0VLicc2H3EDnbw=">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</latexit>
•
•
•
A(s, a) = Q(s, a) V (s)<latexit sha1_base64="raNSrebX08dBbiMuQZLYSyttRrE=">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</latexit><latexit sha1_base64="raNSrebX08dBbiMuQZLYSyttRrE=">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</latexit><latexit sha1_base64="raNSrebX08dBbiMuQZLYSyttRrE=">AAACe3ichVHLSsNAFD2N7/qqCiK4CRallVpuRFEEwcfGpVVbBS0liVMNpklI0oJWf8AfcOFKQUT0L9z4Ay78BHGp4EbB2zQgKuodZubMmXvunJnRHNPwfKKHiNTQ2NTc0toWbe/o7OqO9fTmPLvs6iKr26btbmiqJ0zDElnf8E2x4bhCLWmmWNf2Fmv76xXheoZtrfn7jsiX1B3LKBq66jNViPXPJ7yUmpRn5UwdjMm5hJcsxOKUpiDkn0AJQRxhLNuxS2xhGzZ0lFGCgAWfsQkVHrdNKCA4zOVRZc5lZAT7AkeIsrbMWYIzVGb3eNzh1WbIWryu1fQCtc6nmNxdVsoYpnu6ome6o2t6pLdfa1WDGjUv+zxrda1wCt3HA6uv/6pKPPvY/VT96dlHEdOBV4O9OwFTu4Ve11cOTp5XZ1aGqyN0Tk/s/4we6JZvYFVe9IuMWDlFlD9A+f7cP0FuPK1QWslMxOcWwq9oxSCGkOD3nsIclrCMLJ97iHNc4ybyLsWlUSlVT5UioaYPX0Ka/ABIvI+7</latexit><latexit sha1_base64="raNSrebX08dBbiMuQZLYSyttRrE=">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</latexit>
0 1 2 10 11 12
1 2 3 9 10 11
2 3 4 7 8 9 10
5 6 7
2 3 4 5 4 3 2
1 2 3 3 2 1
0 1 2 2 1 0
⇡⇤
(a|s) 2 argmax
⇡
⌘(⇡), where ⌘(⇡) =
X
s2S
⇢⇡
(s)V ⇡
(s)
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V ⇡
(s) = E⇡
 1X
t=0
t
R(st, at)|s0 = s
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Q⇡
(s, a) = E⇡
 1X
t=0
t
R(st, at)|s0 = s, a0 = a
= R(s, a) +
X
s2S
P(s0
|s, a)V ⇡
(s0
)
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•
•
Policy or
trajectory
Reward
function
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Inverse Reinforcement Learning
Reinforcement Learning
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複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習
複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習

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複数の報酬関数を推定可能なタスク条件付き敵対的模倣学習

  • 1.
  • 5. • s<latexit sha1_base64="aOnLHDE1okdAPO1KhHwMbMtaXa0=">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</latexit><latexit 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  • 6. • • • • • • • • • R : S ⇥ A ! 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  • 7. • • • A(s, a) = Q(s, a) V (s)<latexit sha1_base64="raNSrebX08dBbiMuQZLYSyttRrE=">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</latexit><latexit 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V ⇡ (s) = E⇡  1X t=0 t R(st, at)|s0 = s <latexit sha1_base64="GLUrXhyF7wMzTk+zT/akKoMbOlI=">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</latexit><latexit 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sha1_base64="AkBzV91eLlL+QlOElmz7KH/Prfk=">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</latexit>
  • 8.
  • 9. • Policy or trajectory Reward function ⇡(a|s)<latexit 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Reinforcement Learning
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