L21_sgdmet - Lecture 21 Subgradient Methods April 9 2007...

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Lecture 21: Subgradient Methods April 9, 2007
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Lecture 21 Outline Subgradients and Level Sets Subgradient Method Convergence and Convergence Rate Convex Optimization 1
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Lecture 21 Subgradients and Level Sets A vector s is a subgradient of a convex function f : R n R at ˆ x dom f when subgradient inequality holds f ( z ) f x ) + s T ( z - ˆ x ) for all z dom f We have interpreted the subgradient inequality in terms of a hyperplane in R n +1 supporting the epigraph epi f at x, f x )) Convex Optimization 2
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Lecture 21 Projecting the hyperplane and the epigraph on the set { ( x, w ) | x R n , w = f x ) } The subgradient inequality can be interpreted in terms of a hyperplane in R n supporting the level set L γ ( f ) at ˆ x for γ = f x ) For z in the level set L f x ) ( f ) = { z | f ( z ) f x ) } , the subgradient inequality implies s T ( z - ˆ x ) 0 for all z L f x ) ( f ) and the inequality is tight at z = ˆ x . Convex Optimization 3
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Lecture 21 Another Interpretation of Subgradients A subgradient of f at any ˆ x defines a hyperplane that cuts R n in two regions and one of them contains the level sets L γ ( f ) for all γ f x ) The optimal set X * lies entirely on one side of each such hyperplane Cutting-plane method can be interpreted as successively “reducing” the size of a region where the “optimal solutions” are located, by increasing the number of hyperplanes that “cut” the space R n Convex Optimization 4
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Lecture 21 Subgradient Method Constrained Convex Minimization minimize f ( x ) subject to x X General Assumption The function f ( x ) is convex and dom f = R n The set X is nonempty closed and convex Subgradient Method Having the current iterate x k , a subgradient s k of f at x k , and a stepsize value α k > 0 , the next iterate is given by x k +1 = P X [ x k - α k s k ] Polyak 1969, Ermoliev 1969, Shor 1985 Convex Optimization 5
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Lecture 21 Property “ensuring” that subgradient methods work Along the negative subgradient direction - s k , there is a range of stepsize (0 , ˜ a ) , such that
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