TD(0) Q AC (Reward): () Pr(r t+1 s t+1 = s,s t = s, a t = a) t R a ss = E(r t+1 s t+1 = s,s t = s, a t = a) R t = r t+1 + γr t γ T r t+t +1 = T

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1 () 2009 TD(0) Q AC / /42 TD(0) Q AC (Renforcement Learnng) : (polcy) Acton: a t Agent (= Controller) Envronment (= Controlled object) State: s t Reward: r t TD(0) Q AC (Envronment) (Markov decson process:mdp) t s t a t t +1 s t+1 P a ss = Pr(s t+1 = s s t = s, a t = a) s t a t Pss a (Transton probabltes) / /42

2 TD(0) Q AC (Reward): () Pr(r t+1 s t+1 = s,s t = s, a t = a) t R a ss = E(r t+1 s t+1 = s,s t = s, a t = a) R t = r t+1 + γr t γ T r t+t +1 = T γ k r t+k+1 k=0 TD(0) Q AC (Polcy) s t = s a t = a π(s, a) R t a ɛ ɛ- π s t = s R t (Value functon) V π (s) =E{R t s t = s} 0 <γ<1 T P a ss, Ra ss / /42 =Bellman TD(0) Q AC Pss a, Ra ss T = V π (s) =E π {R t s t = s} = E π {r t+1 + γr t+1 s t = s} = π(s, a) Pss a [Ra ss + γe π{r t+1 s t+1 = s }] a s = a Bellman : V π (s) = a π(s, a) s P a ss [Ra ss + γv π (s )] π(s, a) s P a ss [Ra ss + γv π (s )] TD(0) Q AC : T = Q π (s, a) =E π {R t s t = s, a t = a} Q π (s, a) =E π {r t+1 + γv π (s t+1 ) s t = s, a t = a} Bellman : Q (s, a) = ) Pss (R a a ss + γ max Q (s,a ) a s Bellman : V (s) = max Pss a a [Ra ss + γv (s )] s : argmax Q (s t,a) a : V π (s) = π(s, a)q π (s, a), V (s) =maxq (s, a) a a / /42

3 TD(0) Q AC 2 : V (s) : Q(s, a) TD(0) Q,, Actor-Crtc TD(λ) Q(λ), (λ), Actor-Crtc(λ) TD(0) Q AC π V (s) V (s) V () (s) γ<1 t R t V π (s) =E π {R t s t = s} V (s) R t (Monte-Carlo method; MC ): V (+1) (s t )=V () (s t )+α[r t V () (s t )] 0 <α<1 t s t (R t ) / /42 TD(0) (1) TD(0) (2) TD(0) Q AC (= ) s t = s s t+1 = s r t+1 = r R t = r t+1 + γr t+2 + γ 2 r t+3 + = r t+1 + γr t+1 E π {R t s t = s, s t+1 = s,r t+1 = r} = r + γe π {R t+1 s t+1 = s } = r + γv π (s ) R t r t+1 + γv () (s t+1 ) TD(0) Q AC TD(0) : V (s) s π a a r s V (s) V (s)+α[r + γv (s ) V (s)] s s TD(0) (Temporal-Dfference Learnng): V (+1) (s t )=V () (s t )+α[r t+1 + γv () (s t+1 ) V () (s t )] r t+1 + γv () (s t+1 ) V () (s t ) TD / /42

4 (1) (2) TD(0) Q AC V π (s) Q π (s, a) Q π (s t,a t ) Q (+1) (s t,a t )=Q () (s t,a t )+α[q π (s t,a t ) Q () (s t,a t )] Q π (s t,a t ) Q π (s, a) =E{r t + γv π (s t+1 ) s t = s, a t = a} s t+1 = s r t+1 = r E{r t+1 + γv π (s t+1 ) s t = s, a t = a, s t+1 = s,r t+1 = r} = r + γv π (s )=r + E{Q π (s t+1,a ) s t+1 = s } TD(0) Q AC : Q (+1) (s t,a t )=Q () (s t,a t ) + α t [r t+1 + γq () (s t+1,a t+1 ) Q () (s t,a t )] a t+1 1 π Q π (s, a) Q s a ɛ- ɛ Q Q on TD / /42 (3) Q (1) TD(0) Q AC : Q(s, a) s a a r s s a Q(s, a) Q(s, a)+α t [r + γq(s,a ) Q(s, a)] s s, a a TD(0) Q AC Q Q (s t,a t ) Q (+1) (s t,a t )=Q () (s t,a t )+α[q (s t,a t ) Q () (s t,a t )] Q (s t,a t ) Q (s, a) =E{r t + γv (s t+1 ) s t = s, a t = a} s t+1 = s r t+1 = r E{r t+1 + γv (s t+1 ) s t = s, a t = a, s t+1 = s,r t+1 = r} = r + γv (s )=r +maxq (s,a ) a s r / /42

5 Q (2) Q (3) TD(0) Q AC Q : Q (+1) (s t,a t )=Q () (s t,a t )+α t [r t+1 +γ max a Q () (s t+1,a ) Q () (s t,a t )] a t Q (s t,a t ) off TD s a α t, t=0 αt 2 < 1 Q (s t,a t ) a t ɛ- t=0 TD(0) Q AC Q : Q(s, a) s a a r s a Q(s,a ) max a Q(s, a) Q(s, a)+α t [r + γ max a s s Q(s,a ) Q(s, a)] Q(s,a ) / /42 (1) (2) TD(0) Q AC 1 : : TD(0) Q AC : TD TD : δ t = r t+1 + γv (s t+1 ) V (s t ) TD : r t+1 = a t TD : r t+1 = a t : (Gbbs ) : p(s, a) π t (s, a) =Pr{a t s t = s} = p(s t,a t ) p(s t,a t )+βδ t ep(s,a) b ep(s,b) / /42

6 (1) (2) TD(0) Q AC 1 TD : n TD : R t = r t+1 + γv t (s t+1 ) R [n] t = r t+1 + γr t γ n 1 r t+n + γ n V t (s t+n ) s t E[R [n] t ]=E[R t ] λ (λ return): Rt λ =(1 λ) λ n 1 R [n] t n=1 TD(0) Q AC λ : V (+1) (s t )=V () (s t )+α[r λ t V () (s t )] λ =0TD(0) λ =1() s t E[R t ]=E[R λ t ] / /42 (3) (4) TD(0) Q AC : (elgblty trace): s { γλe t 1 (s) (s s t ) e t (s) = γλe t 1 (s)+1 (s = s t ) s t 1 TD(0) Q AC TD(λ) ( ): V (s) s e(s) s π a a r s δ t = r + γv (s ) V (s) e(s) e(s)+1 σ : V (σ) V (σ)+αδe(σ) e(σ) γλe(σ) s s / /42

7 (5) TD(0) Q AC s t V λ ( ) TD(0) Q AC Q(s, a) e(s, a) s a (λ) : Q(s, a) s, a e(s, a) =0 s a a r s s a δ r + γq(s,a ) Q(s, a) e(s, a) e(s, a)+1 s, ā : Q( s, ā) Q( s, ā)+αδe( s, ā) e( s, ā) γλe( s, ā) s s, a a Q(λ) Watkns Q(λ) / /42 2 HM- SM- SM- NN 2 HM- SM- SM- NN : x 2 () 2 (x 1,y 1 ), (x 2,y 2 ), (x 3,y 3 ),... x y (±1) : y = sgn[w T x + b] = sgn[w 1 x w n x n + b] { 1 (u 0) sgn[u] = 1 (u<0) y = 1 y = 1 w T x + b = / /42

8 2 2 HM- SM- SM- NN d d w T x + b d = mn w w T x + b = 0 d w b (Support Vector Machne; ) 2 HM- SM- SM- NN y =1 w T x + b =1, y = 1 w T x + b = 1 d = 1 w ( ): L(w) = 1 2 w 2 mn subject to y (w T x + b) ( ) / /42 (1) (2) 2 HM- SM- SM- NN α ( 0) (): L(w, b, α) = 1 N 2 w 2 α {y (w T x + b) 1} mn, α 0 w, b ( ) T L(w, b, α) N = w α y x =0 w ( ) T L(w, b, α) N = α y =0 b 2 HM- SM- SM- NN (): N L D (α) = α 1 N α α j y y j x T x j max 2,j=1 N subject to α y =0,α 0 α N w = α y x b = y w T x, such that α / /42

9 (3) (1) 2 HM- SM- SM- NN Karush-Kuhn-Tucker : N α y =0 α 0, =1,...,N y (w T x + b) 1, =1,...,N N w = α y x α {y (w T x + b) 1} =0, =1,...,N 2 HM- SM- SM- NN Plus[s] = 1 (s + s ) { 2 s (s 0) = 0 (s<0) C Plus[1 y (w T x + b)] w T x + b = 1 Surface for green ponts w T x + b = 0 w T x + b = 1 Surface for red ponts. α α α x / /42 (2) 2 HM- SM- SM- NN : 1 2 w 2 + C w, b N Plus[1 y (w T x + b)] mn w, b, ξ 2 HM- SM- SM- NN Lagrange : L = 1 2 w 2 + C w = α y x, ξ α (ξ 1+y (w T x + b)) β ξ y α =0, α + β = C α C 1 N 2 w 2 + C ξ mn subject to ξ 1 y (w T x + b) ξ 0 : max α 1 α α j y y j x T x j α 2,j subject to 0 α C y α = / /42

10 2 HM- SM- SM- NN y = 1 y = 1 x z = φ(x) 2 HM- SM- SM- NN : max α 1 α α j y y j φ(x ) T φ(x j ) α 2,j subject to 0 α C, α y =0 1 k(x,x j )=φ(x ) T φ(x j ) [ ] x =(x 1,x 2 ) T z =(x 2 1,x 1 x 2,x 2 2,x 1,x 2, 1) T 2 () 1 b y α = / /42 2 HM- SM- SM- NN : ( ) k(x,x j )=k(x j,x ) ( ) x 1, x 2,... k(x 1,x 1 ) k(x 1,x p ) [k(x,x j )] (,j) =.. k(x p,x 1 ) k(x p,x p ) k(x, y) =φ(x) T φ(y) 2 HM- SM- SM- NN : k(x, y) =(1+x T y) p Gaussan : ( ) x y 2 k(x, y) = exp 2σ 2 : ( ) k(x, y) = tanh(ax T y b) / /42

11 2 HM- SM- SM- NN 1. () max α 1 α α j y y j k(x,x j ) α 2,j subject to 0 α C, α y = HM- SM- SM- NN 3 ( sgn ) I H H O Gaussan RBF () sgn S = { 0 <α <C}, O = { α = C}, I = { α =0} 3. [ ] y = sgn α k(x,x)+b, b = y j S O α j k(x j,x ) ( S) / /42

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....

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