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4nsmooth

# 4nsmooth - EE236C(Spring 2008-09 4 Gradient methods for...

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Interpretation of gradient update recall the gradient update for convex differentiable f : x + = x t f ( x ) interpretation x + = argmin z parenleftbigg f ( x ) + f ( x ) T ( z x ) + 1 2 t bardbl z x bardbl 2 2 parenrightbigg x + minimizes a quadratic approximation of f , consisting of the first-order linearization f ( x ) + f ( x ) T ( z x ) of f ( z ) at x a proximity term bardbl z x bardbl 2 2 with weight 1 / (2 t ) Gradient methods for nonsmooth problems 4–3 Extension to nondifferentiable problems split f in a smooth and a nonsmooth component: minimize f ( x ) = g ( x ) + h ( x ) g convex, differentiable; h convex, nondifferentiable generalized gradient update x + = argmin z parenleftbigg g ( x ) + g ( x ) T ( z x ) + 1 2 t bardbl z x bardbl 2 2 + h ( z ) parenrightbigg we make a quadratic approximation to g only complexity of computing x + depends on structure of h repeating the update provides a ‘gradient method’ for minimizing f Gradient methods for nonsmooth problems 4–4
Example: 1-norm regularization minimize f ( x ) = g ( x ) + bardbl x bardbl 1 generalized gradient update x + = argmin z parenleftbigg g ( x ) + g ( x ) T ( z x ) + 1 2 t bardbl z x bardbl 2 2 + bardbl z bardbl 1 parenrightbigg = argmin z parenleftbigg 1 2 t bardbl z x + t g ( x ) bardbl 2 2 + bardbl z bardbl 1 parenrightbigg = S t ( x t g ( x )) where S t ( y ) Δ = argmin z parenleftbigg 1 2 t bardbl z y bardbl 2 2 + bardbl z bardbl 1 parenrightbigg Gradient methods for nonsmooth problems 4–5 computing S t : solve a simple separable problem in z R n minimize n summationdisplay k =1 parenleftbigg 1 2 t ( z k y k ) 2 + | z k | parenrightbigg solution: S t ( y ) k = y k t y k t 0 t y k t y k + t y k ≤ − t S t is often called the shrinkage or

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