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Proximal Gradient Descent - CMU Statistics

Proximal Gradient Descent (and Acceleration)Ryan TibshiraniConvex Optimization 10-725 Last time: subgradient methodConsider the problemminxf(x)withfconvex, anddom(f) =Rn. Subgradient method: choosean initialx(0) Rn, and repeat:x(k)=x(k 1) tk g(k 1), k= 1,2,3,..whereg(k 1) f(x(k 1)). We use pre-set rules for the step sizes( , diminshing step sizes rule)Iffis Lipschitz, then subgradient method has a convergence rateO(1/ 2)Upside: very generic. Downside: can be slow addressed today2 OutlineToday: Proximal Gradient Descent Convergence analysis ISTA, matrix completion Special cases Acceleration3 Composite functionsSupposef(x) =g(x) +h(x) gis convex, differentiable,dom(g) =Rn his convex, not necessarily differentiableIffwere differentiable, then Gradient Descent update would be:x+=x t f(x)Recall motivation: minimize quadratic approximation tofaroundx, replace 2f(x)by1tI,x+= argminzf(x) + f(x)T(z x) +12t z x 22 ft(z)4In our casefis not differentiable, butf=g+h, don t we make quadratic approximation tog, leavehalone?

Proximal Gradient Descent (and Acceleration) Ryan Tibshirani Convex Optimization 10-725

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