…p…^†[…fiflF”¯ Pattern Recognition
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1 Pattern Recognition Shin ichi Satoh National Institute of Informatics June 11, 2019
2 (Support Vector Machines) (Support Vector Machines: SVM) SVM Vladimir N. Vapnik and Alexey Ya. Chervonenkis 1963 SVM (Structural Risk) : ( : empirical risk) (VC ) (overfitting) SVM Bernhard E. Boser, Isabelle M. Guyon and Vladimir N. Vapnik 1992 Corinna Cortes and Vapnik 1993
3 SVM SVM (Linear Support Vector Machines) margin
4 SVM X = {x i R d } (i = 1,, n) y i { 1, 1} ( ) w x b = 0 w x b = 1 and w x b = 1 2 w
5 SVM w w x i b 1 for x i with y i = 1 w x i b 1 for x i with y i = 1 y i (w x i b) 1 for all i = 1,, n Minimize w subject to y i (w x i b) 1 for all i = 1,, n.
6 (Primal Form) (quadratic programming optimization problem) 1 arg min (w,b) 2 w 2 subject to (for any i = 1,, n) y i (w x i b) 1
7 (Dual Form) (Lagrange multipliers) α i { } 1 n L = 2 w 2 α i [y i (w x i b) 1] i=1 arg min max w,b α Karush-Kuhn-Tucker (KKT Karush-Kuhn-Tucker conditions) L L = 0, w b = 0 α i 0, α i [y i (w x i b) 1] = 0 L
8 Maximize f (x) subject to g i (x) 0, h j (x) = 0 Karush-Kuhn-Tucker Karush-Kuhn-Tucker f (x) α i g i (x) λ j h j (x) = 0 g i (x) 0, h j (x) = 0 α i 0 α i g i (x) = 0
9 Maximize f (x) subject to h j (x) = 0 : (Lagrange Multipliers Method) (Lagrangian): L = f (x) λ j h j (x) : L = f (x) λ j h j (x) = 0 h j (x) = 0
10 L w = 0 w = α i y i x i L b = 0 α i y i = 0 α i [y i (w x i b) 1] = 0 if y i (w x i b) 1 > 0 then α i = 0, otherwise (y i (w x i b) 1 = 0) α i 0 α i > 0 x i b = 1 {α i > 0} (w x i y i ) α i >0
11 Maximize: n L(α) = α i 1 α i α j y i y j xi T x j 2 subject to: i=1 α i 0 and i,j n α i y i = 0 i=1
12 Minimize subject to (Quadratic Programming Optimization) 1 2 x T Qx + p T x Cx b C eq x = b eq LB x UB x = α, Q i,j = y i y j x T i x j, p = [1 1 1], LB = 0, C eq = y, b eq = 0
13 SVM (Python) h=x*l qpp = cvxopt.matrix(h.t.dot(h)) qpq = cvxopt.matrix(-np.ones(n), (n, 1)) qpg = cvxopt.matrix(-np.eye(n)) qph = cvxopt.matrix(np.zeros(n), (n, 1)) qpa = cvxopt.matrix(l.astype(float), (1, n)) qpb = cvxopt.matrix(0.) cvxopt.solvers.options[ abstol ] = 1e-5 cvxopt.solvers.options[ reltol ] = 1e-10 cvxopt.solvers.options[ show_progress ] = False res=cvxopt.solvers.qp(qpp, qpq, qpg, qph, qpa, qpb) alpha = np.reshape(np.array(res[ x ]),-1) w=np.sum(x*(np.ones(n)*(l*alpha)),axis=1) sv=alpha>1e-5 isv=np.where(sv)[-1] b=np.sum(w.t.dot(x[:,isv])-l[isv])/np.sum(sv)
14 SVM (Matlab) h=x; h(:,l<0)=-h(:,l<0); options=optimset( Algorithm, interior-point-convex ); alpha=quadprog(h *h,-ones(1,size(x,2)),[],[],l,0,... zeros(1,size(x,2)),[],[],options) ; w=sum(x.*(ones(size(x,1),1)*(l.*alpha)),2); sv=alpha>1e-5; isv=find(sv); b=sum(w *x(:,isv)-l(isv))/sum(sv);
15 SVM (Scilab) h=x; h(:,l<0)=-h(:,l<0); alpha=quapro(h *h,-ones(size(x,2),1),l,0,... zeros(size(x,2),1),[],1) ; w=sum(x.*(ones(size(x,1),1)*(l.*alpha)),2); sv=alpha>1e-5; isv=find(sv); b=sum(w *x(:,isv)-l(isv))/sum(sv);
16 (linear)
17 (slinear)
18 (slack variables) ξ i y i (w x i b) 1 ξ i { 1 arg min w,ξ,b 2 w 2 + C subject to } n ξ i i y i (w x i b) 1 ξ i, ξ 0
19 arg min w,b { 1 2 w 2 + C } n max(1 y i (w x i + b), 0) max(1 y i (w x i + b), 0) (hinge loss) i hinge loss 0-1 loss
20 KKT Maximize subject to L(α) = n α i 1 2 i=1 0 α i C, i,j α i α j y i y j xi T x j n α i y i = 0 : x i with 0 < α i < C (x i with α i = C ). i=1
21 SVM
22 (slinear)
23 (qlinear)
24 x ϕ(x) : x = [x 1 x 2 ] T, ϕ(x) = [x 1 x 2 x 2 1 x 1x 2 x 2 2 ]T. ϕ(x) x
25 SVM k(x, y) = ϕ(x) ϕ(y) : (Polynomial Kernel) k(x, y) = (x y + 1) p, k(x, y) = (x y) p (Gaussian Kernel, Radial Basis Function (RBF) Kernel) x y 2 k(x, y) = exp( ) 2σ 2
26 SVM Maximize n L(α) = α i 1 α i α j y i y j xi T x j 2 subject to i=1 0 α i C, i,j n α i y i = 0 i=1 x i x j
27 Maximize L(α) = = n α i 1 α i α j y i y j ϕ(x i ) T ϕ(x j ) 2 i=1 i,j n α i 1 α i α j y i y j k(x i, x j ) 2 i=1 i,j subject to 0 α i C, n α i y i = 0 i=1
28 α i w = α i y i ϕ(x i ) ϕ(x i ) w b = 1 #sv = 1 #sv = 1 #sv (w ϕ(x i ) y i ) i sv i sv j i sv j α j y j ϕ(x j ) T ϕ(x i ) y i α j y j k(x j, x i ) y i
29 x f (x) = w ϕ(x) b = α i y i ϕ(x i ) T ϕ(x) b = α i y i k(x i, x) b f (x) x
30 (qlinear, Polynomial kernel)
31 (nonlinear, C=1, RBF kernel)
32 (nonlinear, C=1000, RBF kernel)
33 1 SVM : SVM SVM Python: cvxopt ( pip install cvxopt ) Matlab: quadprog (optimization toolbox ) Scilab: quapro (quapro toolbox ) linear, slinear, qlinear, nonlinear
34 w x b = 1 w x b = 1 2 w 2
35 SVM : arg min w,ξ,b { 1 2 w 2 + C } n ξ i i 2 subject to : Maximize subject to y i (w x i b) 1 ξ i, ξ 0 L(α) = n α i 1 α i α j y i y j xi T x j 2 i=1 0 α i C, i,j n α i y i = 0 i=1
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
() 2009 TD(0) Q AC 2009 1/42 2009 2/42 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
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