Lipschitz gradient f R n R has a L Lipschitz gradient if f x f y 2 L x y 2 If f

# Lipschitz gradient f r n r has a l lipschitz gradient

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Lipschitz gradient f : R n R has a L Lipschitz gradient if ||∇ f ( x ) - ∇ f ( y ) || 2 L || x - y || 2 If f ( x ) = Ax , L = || A || 2 . If f is twice differentiable L = sup x || H f ( x ) || 2 , i.e. , the highest eigenvalue of H f ( x ) among all possible x . Convexity f : R n R is convex if for all x , y and λ , f ( λx + (1 - λ ) y ) λf ( x ) + (1 - λ ) f ( y ) f is strictly convex if the inequality is strict. f convex and twice differentiable iif H f ( x ) is Hermitian non-negative definite. f is strictly convex and twice differentiable iif H f ( x ) is Hermitian positive definite. If f is convex, f has only global minima if any. We write the set of minima as argmin x f ( x ) = { x \ for all z R n f ( x ) f ( z ) } Gradient descent Let f : R n R be differentiable with L Lipschitz gradient then, for 0 < γ 1 /L , the sequence x k +1 = x k - γ f ( x k ) converges towards a stationary point x in ) f ( x ) = 0 If f is moreover convex then x argmin x f ( x ) . f : n k k f ( k ) 1 f ( k ) f 2 ) . f ( { p ( ) f ( ) } . p ( ) f . f 0 ( ) . f ( f ( ) } . f g g 0 1 , k ( k γ g ( k f ( ) 1 ( ) z 1 2 || z || 2 ( z ) f . f : n f * ( z x f ( ) if f f f . f ( g ( ( ) f ( ) x z g ( * ( z ) z * ( z ) * z ( ) Cookbook for data scientists Multi-variate differential calculus Partial and directional derivatives Let f : R n R m . The ( i, j ) -th partial derivative of f , if it exists, is ∂f i ∂x j ( x ) = lim ε 0 f i ( x + εe j ) - f i ( x ) ε where e i R n , ( e j ) j = 1 and ( e j ) k = 0 for k j . The directional derivative in the dir. d R n D d f ( x ) = lim ε 0 f ( x + εd ) - f ( x ) ε Jacobian and total derivative J f = ∂f ∂x = ∂f i ∂x j i,j ( m × n Jacobian matrix) d f ( x ) = tr ∂f ∂x ( x ) d x (total derivative) Gradient, Hessian, divergence, Laplacian f = ∂f ∂x i i (Gradient) H f = ∇∇ f = 2 f ∂x i ∂x j i,j div f = t f = n i =1 ∂f i ∂x i = tr J f (Divergence) Δ f = div f = n i =1 2 f ∂x 2 i = tr H f (Laplacian) Properties and generalizations f = J t f (Jacobian div = -∇ * (Integration by part) d f ( x ) = tr [ J f d x ] (Jacob. character. I) D d f ( x ) = J f ( x ) × d f ( x + h )= f ( x ) + D h f ( x ) + o ( || h || ) ( 1 st order exp.) f ( x + h )= f ( x ) + D h f ( x ) + 1 2 h * H f ( x ) h o ( || || 2 ) ( f g ) ∂x = ∂f ∂x g ∂g ∂x (Chain rule) a b ) ) * ) * 1 1 ) 1 ] ] * * ( * ] 1 1 1 ] n n 1 ] e e ] | | | | 1 ] | * | | * | 1 ] | n | | n | 1 ] | | 1 ] | * | ] f : n n f ( a n b ( ) g g ( a b n a f ( ( i ( ( ( 1 i ( ( f ( ( ) y , y g ( ) ( ) 2 y 2 . Cookbook for data scientists Probability and Statistics Kolmogorov’s probability axioms Let Ω be a sample set and A an event P [Ω] = 1 , P [ A ] 0 P i =1 A i = i =1 P [ A i ] with A i j Basic properties P [ ] = 0 , P [ A ] [0 , 1] , P [ A c ] = 1 [ ] P [ A ] P [ B ] if A B P [ A B ] = P [ A ] + P [ B ] - P [ A ] Conditional probability P [ A | B ] = P [ A B ] P [ B ] subject to P [ ] 0 Bayes’ rule P [ A | B ] = P [ B | A ] P [ A ] P [ B ] Independence Let A and B be two events, X and Y be two rv A B if P [ A B ] = P [ A ] P [ B ] X Y if ( X x ) ( Y y ) If X and Y admit a density, then X Y if f X,Y ( x, y ) = f X ( x ) f Y ( y ) Properties of Independence and uncorrelation P [ A | B ] = P [ A ] A B X Y ( E [ XY * ] = E [ X ] E [ Y * ] Cov [ Independence uncorrelation correlation dependence uncorrelation Independence dependence correlation n [ ] i k k [ i k ] f : k [ k ] .

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