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• W2964164027 abstract "Motivated by recent work of Renegar (A framework for applying subgradient methods to conic optimization problems, arXiv:1503.02611v2 , 2015) we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization problem $$f^* = min _{x in Q} f(x)$$ , where we presume knowledge of a strict lower bound $$f_{mathrm{slb}}< f^*$$ . [Indeed, $$f_{mathrm{slb}}$$ is naturally known when optimizing many loss functions in statistics and machine learning (least-squares, logistic loss, exponential loss, total variation loss, etc.) as well as in Renegar’s transformed version of the standard conic optimization problem arXiv:1503.02611v2 , 2015; in all these cases one has $$f_{mathrm{slb}}= 0 < f^*$$ .] We introduce a new functional measure called the growth constant G for $$f(cdot )$$ , that measures how quickly the level sets of $$f(cdot )$$ grow relative to the function value, and that plays a fundamental role in the complexity analysis. When $$f(cdot )$$ is non-smooth, we present new computational guarantees for the Subgradient Descent Method and for smoothing methods, that can improve existing computational guarantees in several ways, most notably when the initial iterate $$x^0$$ is far from the optimal solution set. When $$f(cdot )$$ is smooth, we present a scheme for periodically restarting the Accelerated Gradient Method that can also improve existing computational guarantees when $$x^0$$ is far from the optimal solution set, and in the presence of added structure we present a scheme using parametrically increased smoothing that further improves the associated computational guarantees." @default.
• W2964164027 created "2019-07-30" @default.
• W2964164027 creator A5004737240 @default.
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• W2964164027 date "2017-06-22" @default.
• W2964164027 modified "2023-10-01" @default.
• W2964164027 title "New computational guarantees for solving convex optimization problems with first order methods, via a function growth condition measure" @default.
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• W2964164027 doi "https://doi.org/10.1007/s10107-017-1164-1" @default.
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