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m�CJ∗ (·)

W@qMD;q9FOFcfP3TD (λ)

A9:ZR9JMLETaFGS0tKCË_�J94YFEJMCGA@4È8�D�Fh¤;COTZ4P?9: �EAgf�¶8�R96\kKW¦]^8;: ÇgHKJ9R96

x1, x2, . . . , xN

8;R96+�E?@FGm�W¦?9:=:BA9_`J9FGA9TZ4PA@WjAjCOJMLEJ9DcA@FGAgf b 8�:ZqM: �OkKW�A@Dxde8;NgTZW�m�C

x + 1d�8;NKT=W�?9:=:BA9_`J9FEiPA94�<@?9LOHPA@FGQ¨�OtgFGA9:ZSVA9W>W9qMDoFGT!WYkK:=qM4YiKCEm�RM8�F

dt

d�<M8�:zÇXA9D;A9Dw:Z8�R96M8�4

dt := J̃ (xt+1, w) − J̃ (xt, w) .

� 8;:aFG8;Hg8�FÕ�OkgW`ÇPHgJMR96J̃ (xN , w) = r (xN )

f2�eR96

dN−1 = J̃ (xN , w) − J̃ (xN−1, w) = r (xN ) − J̃ (xN−1, w) .

3¨D+T=R7A9D;T�8;:=FEm;LOm;C�AjW9m�Fh?9:=:BA9_`J9FEiPA94�<@?9LOHPA@FGQ¨�OtgFGA9:ZJ9S±W@qMD;q9FOF�FG8;LOSVm�CED�8�FG8�CE8�40f

Ext+1|xt[J∗ (xt+1) − J∗ (xt)] = 0.

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�eR96ÈAJ̃ (·, w)

A@W\W@JML-�OQVW@qMD;8�:Zn=FEm;CO8J∗ (·)

de4PA9WXÇXHPAExt+1|xt

dt

W@qMD;8�:Zn=FhA0d�HKJMD9fX3�E?@FEm;W¦<Mm�RMm;4�A9D

A9:=RMJMLETaFGS0tgC2A/W9q�<M8oFGW@8;D;[¦]^J9LES0tK:ZA0CED�8;LET=47F�]^LOTZCOCEn=FET�A/CElg:=6MJ9W@AUF�FGA9LOFGA9:ZSVA9D�Q)<M8�WYFEJMLOFQB

w := w + αN−1∑

t=1

∂J̃ (xt, w)

[

N−1∑

j=t

λj−tdt

]

,

A9HKJ9:∂J̃ (xt, w)

AJ̃ w

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λ ∈ [0, 1]_KA9LGA9S>m�FE8;LjA9DoF�i`8�]^J9:=67?@CEJM: �EAgÇ�HKJ9R96«SV8�:=6

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td�8;NKT=WTZNg[M_KTZ:=:BA94KA@FGiPA@4 :=m�<M[

J̃m�LOFGm�W9moF�A

(t + 1)d�8;NKT=W m�LOFEm;WYHK8;D W9q9D;8;:=n=F^�OkKWXÇ

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tanh (x) = e2x−1

e2x+1

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