7bregman

7bregman - EE236C(Spring 2008-09 7 Gradient methods with...

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Unformatted text preview: EE236C (Spring 2008-09) 7. Gradient methods with generalized distances • Bregman distances • variant of Nesterov’s method • example 7–1 Gradient method and extension basic gradient method for minimizing f (lecture 1) x + = argmin z parenleftbigg f ( x ) + ∇ f ( x ) T ( z − x ) + 1 2 t bardbl z − x bardbl 2 2 parenrightbigg extension for minimizing f + g over C (lectures 4-5) x + = argmin z ∈ C parenleftbigg f ( x ) + ∇ f ( x ) T ( z − x ) + 1 2 t bardbl z − x bardbl 2 2 + g ( z ) parenrightbigg Δ = S t ( x − t ∇ f ( X )) • g a simple nondifferentiable function; C a simple convex set • interesting if projection/thresholding operation S t is inexpensive Gradient methods with generalized distances 7–2 Generalization replace (1 / 2) bardbl z − x bardbl 2 2 with ‘generalized distance function’ d ( z, x ) • basic gradient update argmin z parenleftbigg f ( x ) + ∇ f ( x ) T ( z − x ) + 1 t d ( z, x ) parenrightbigg • extension with projection/thresholding argmin z ∈ C parenleftbigg f ( x ) + ∇ f ( x ) T ( z − x ) + 1 t d ( z, x ) + g ( z ) parenrightbigg potential benefits • select d ( z, x ) to fit the curvature of f , or geometry of C • simplify the thresholding/projection Gradient methods with generalized distances 7–3 Bregman distance functions Bregman distance associated with strictly convex, differentiable h : d ( x, y ) = h ( x ) − h ( y ) − ∇ h ( y ) T ( x − y ) h is called the kernel function of d properties • convex in x for fixed y • d ( x, y ) ≥ for all x, y ; d ( x, y ) = 0 if and only if x = y • not a real distance (not symmetric) • d ( x, y ) ≥ ( μ/ 2) bardbl x − y bardbl 2 2 if h is strongly convex with constant μ first two properties follow from (strict) convexity of h Gradient methods with generalized distances 7–4 Examples quadratic function: h ( x ) = bardbl x bardbl 2 2 / 2 d ( x, y ) = 1 2 bardbl x − y bardbl 2 2 negative entropy: h ( x ) = ∑ n i =1 x i log x i with dom h = R n ++ d ( x, y ) = n summationdisplay i =1 ( x i log( x i /y i ) − x i + y i ) the relative entropy or Kullback-Leibler divergence Gradient methods with generalized distances 7–5 logarithm barrier: h ( x ) = − ∑ n i =1 log x i with dom h = R n ++ d ( x, y ) = n summationdisplay i =1 ( x i /y i − log( x i /y i )) − n inverse barrier: h ( x ) = ∑ n i =1 1 /x i with...
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7bregman - EE236C(Spring 2008-09 7 Gradient methods with...

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