Summary of policy gradient and value gradient with Q-Prop

6. St+1 ~ P( ′s | St ,At )
rt+1 = r(St ,At ,St+1)
At ~ π( ′a | St )

7. St+1 ~ P( ′s | St ,At )
rt+1 = r(St ,At ,St+1)
At ~ π( ′a | St )
π∗
= argmax
π
Eπ [ γ τ
rτ ]
τ =0
∞
∑

8. π∗
= argmax
π
Eπ [ γ τ
rτ ]
τ =0
∞
∑
= J
∇θ J

9. ∇θ J = Eπθ
[∇θ log(πθ (at | st ))Qt ]
∇θ J = Es∼ρ ∇aQµ
s,a( )a=µθ s( )
∇θ µθ s( )⎡
⎣⎢
⎤
⎦⎥

10. ∇θ J = ∇θ Eπθ
[ γ τ
rτ ]
τ =0
∞
∑
= ∇θ Es0 ~ρ,s'~p πθ at ,st( ) γ τ
rτ
τ =0
∞
∑t=0
∏
⎡
⎣
⎢
⎤
⎦
⎥
= Es0 ~ρ,s'~p ∇θ πθ at ,st( ) γ τ
rτ
τ =0
∞
∑t=0
∏
⎡
⎣
⎢
⎤
⎦
⎥
= Es~ρ πθ at ,st( )
∇θ πθ at ,st( )
t=0
∏
πθ at ,st( )
t=0
∏
γ τ
rτ
τ =0
∞
∑
t=0
∏
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= Es~ρ πθ (at | st ) ∇θ log(πθ (at | st ))
t=0
∑t=0
∏ γ τ
rτ
τ =0
∞
∑
⎡
⎣
⎢
⎤
⎦
⎥
= Eπθ
[ ∇θ log(πθ (at | st ))
t=0
∑ γ τ
rτ
τ =t
∞
∑ ]

11. ∇log p x( )( ) f x( )

12. ∇log p x( )( ) f x( )

14. J = Es∼ρ [Qµθ
s,µθ s( )( )]
∇θ J = Es∼ρ ∇θQµ
s,µθ s( )( )⎡⎣ ⎤⎦
= Es∼ρ ∇aQµ
s,a( )a=µθ s( )
∇θ µθ s( )⎡
⎣⎢
⎤
⎦⎥

21. f st ,at( )= f st ,at( )+ ∇a f st ,a( )a=at
at − at( )
∇θ J = Eρ,π ∇θ logπθ at st( ) Q st ,at( )− f st ,at( )( )⎡
⎣
⎤
⎦ + Eρ,π ∇θ logπθ at st( ) f st ,at( )⎡
⎣
⎤
⎦
= Eρ,π ∇θ logπθ at st( ) Q st ,at( )− f st ,at( )( )⎡
⎣
⎤
⎦ + Eρ,π ∇a f st ,a( )a=at
∇θ µθ st( )⎡
⎣
⎤
⎦

23. ∇θ J = Eρ,π ∇θ logπθ at st( ) Q st ,at( )−Qw st ,at( )( )⎡
⎣
⎤
⎦ + Eρ,π ∇aQw st ,a( )a=at
∇θ µθ st( )⎡
⎣
⎤
⎦
∇θ J = Eρ,π ∇θ logπθ at st( ) A st ,at( )− Aw st ,at( )( )⎡
⎣
⎤
⎦ + Eρ,π ∇aQw st ,a( )a=at
∇θ µθ st( )⎡
⎣
⎤
⎦
a

24. ∇θ J = Eρ,π ∇θ logπθ at st( ) A st ,at( )− Aw st ,at( )( )⎡
⎣
⎤
⎦ + Eρ,π ∇aQw st ,a( )a=at
∇θ µθ st( )⎡
⎣
⎤
⎦
Aw = Qw st ,at( )− Eπ Qw st ,at( )⎡⎣ ⎤⎦
= Qw st ,µθ st( )( )+ ∇aQw st ,a( )a=µθ st( )
at − µθ st( )( )− Eπ Qw st ,µθ st( )( )+ ∇aQw st ,a( )a=µθ st( )
at − µθ st( )( )⎡
⎣⎢
⎤
⎦⎥
= ∇aQw st ,a( )a=µθ st( )
at − µθ st( )( )
rt+1 +γV st+1( )−V st( )
Eπ at[ ]= µθ st( )

26. m*
= m −η(t −τ )
E m*
⎡⎣ ⎤⎦ = E m[ ]
Var m*
⎡⎣ ⎤⎦ = Var m[ ]− 2ηCov m,t[ ]+η2
Var t[ ]
η*
=
Cov m,t[ ]
Var t[ ]

27. ∇θ J = Eρ,π ∇θ logπθ at st( ) A st ,at( )−η st( )Aw st ,at( )( )⎡
⎣
⎤
⎦ +
Eρ,π η st( )∇aQw st ,a( )a=at
∇θ µθ st( )⎡
⎣
⎤
⎦
Var A −ηAw⎡⎣ ⎤⎦ = Var A[ ]− 2ηCov A,Aw( )+η2
Var Aw( )
η*
=
Cov A,Aw( )
Var Aw( )