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

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EE236C (Spring 2008-09) 4. Gradient methods for nonsmooth problems motivation example: 1-norm regularization gradient mapping gradient method Nesterov’s gradient method examples 4–1 Motivation complexity results from previous lectures bounds on number of iterations to reach accuracy f ( x ) f ǫ : subgradient method: O (1 2 ) gradient method: O (1 ) Nesterov’s optimal gradient method: O (1 / ǫ ) can the faster gradient methods be extended to nonsmooth problems? no, if we consider the problem class and the (oracle) algorithm model for which the subgradient method is known to be optimal yes, if we can take advantage of additional structure in the problem Gradient methods for nonsmooth problems 4–2
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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
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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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