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Gradient Descent - CMU Statistics

Gradient DescentRyan TibshiraniConvex Optimization 10-725 Last time: canonical convex programs Linear program (LP): takes the formminxcTxsubject toDx dAx=b Quadratic program (QP): like LP, but with quadratic criterion Semidefinite program (SDP): like LP, but with matrices Conic program: the most general form of all2 Gradient descentConsider unconstrained, smooth convex optimizationminxf(x)That is,fis convex and differentiable withdom(f) =Rn. Denoteoptimal criterion value byf?= minxf(x), and a solution byx? Gradient Descent : choose initial pointx(0) Rn, repeat:x(k)=x(k 1) tk f(x(k 1)), k= 1,2,3.

Gradient descent has O(1= ) convergence rate over problem class of convex, di erentiable functions with Lipschitz gradients First-order method: iterative method, which updates x(k) in x(0) + spanfrf(x(0));rf(x(1));:::rf(x(k 1))g Theorem (Nesterov): For any k (n 1)=2 and any starting point x(0), there is a function fin the problem class such that

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