Revisiting the Approximate Carathéodory Problem via the Frank-Wolfe Algorithm

Combettes, Cyrille W.; Pokutta, Sebastian

doi:10.48550/arxiv.1911.04415

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…A similar assumption is made in Asi & Duchi (2019), to build optimization algorithms that have a robust performance with respect to different step sizes and algorithm parameters. Another important example is the approximate Carathéodory problem (Mirrokni et al, 2017;Combettes & Pokutta, 2019) where we want to find the decomposition of a point x 0 in a polytope P as a convex combination of the vertices of P and where the objective…”

Section: Quadratic Subproblem Stopping Criterionmentioning

confidence: 99%

Second-order Conditional Gradient Sliding

Carderera¹,

Pokutta²

2020

Preprint

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Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to the quadratic convergence rates these methods enjoy when close to the optimum. These algorithms require the solution of a constrained quadratic subproblem at every iteration. In the case where the feasible region can only be accessed efficiently through a linear optimization oracle, and computing first-order information about the function, although possible, is costly, the coupling of constrained second-order and conditional gradient algorithms leads to competitive algorithms with solid theoretical guarantees and good numerical performance.

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Section: Quadratic Subproblem Stopping Criterionmentioning

confidence: 99%

Second-order Conditional Gradient Sliding

Carderera¹,

Pokutta²

2020

Preprint

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“…Other projection-free algorithms exist with improved guarantees on strongly convex sets, e.g., for nonconvex optimization [RBWM19], min-max problems [GJLJ17,WA18] or approximate Carathéodory results [CP19]. The various equivalent definitions of strongly convex sets have also stimulated an interest in designing and analysing affine-invariant first-order methods.…”

Section: Introductionmentioning

confidence: 99%

Local and Global Uniform Convexity Conditions

Kerdreux,

d'Aspremont,

Pokutta

2021

Preprint

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We review various characterizations of uniform convexity and smoothness on norm balls in finitedimensional spaces and connect results stemming from the geometry of Banach spaces with scaling inequalities used in analyzing the convergence of optimization methods. In particular, we establish local versions of these conditions to provide sharper insights on a recent body of complexity results in learning theory, online learning, or offline optimization, which rely on the strong convexity of the feasible set. While they have a significant impact on complexity, these strong convexity or uniform convexity properties of feasible sets are not exploited as thoroughly as their functional counterparts, and this work is an effort to correct this imbalance. We conclude with some practical examples in optimization and machine learning where leveraging these conditions and localized assumptions lead to new complexity results.

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“…However, there is general interest of solving matrix games with general norms. For instance, matrix games are closely related to the Carathéodory problem for finding a sparse linear combination in the convex hull of given data points, where all the p -metrics with p ≥ 2 have been well-studied (Barman 2015; Mirrokni et al 2017;Combettes and Pokutta 2019). In addition, matrix games are common in machine learning especially support vector machines (SVMs), and general p -margin SVMs have also been considered by previous literature, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Sublinear Classical and Quantum Algorithms for General Matrix Games

Wang

Chakrabarti

et al. 2021

AAAI

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We investigate sublinear classical and quantum algorithms for matrix games, a fundamental problem in optimization and machine learning, with provable guarantees. Given a matrix, sublinear algorithms for the matrix game were previously known only for two special cases: (1) the maximizing vectors live in the L1-norm unit ball, and (2) the minimizing vectors live in either the L1- or the L2-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed q between 1 and 2, we solve, within some additive error, matrix games where the minimizing vectors are in an Lq-norm unit ball. We also provide a corresponding sublinear quantum algorithm that solves the same task with a quadratic improvement in dimensions of the maximizing and minimizing vectors. Both our classical and quantum algorithms are optimal in the dimension parameters up to poly-logarithmic factors. Finally, we propose sublinear classical and quantum algorithms for the approximate Carathéodory problem and the Lq-margin support vector machines as applications.

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Revisiting the Approximate Carathéodory Problem via the Frank-Wolfe Algorithm

Cited by 4 publications

References 10 publications

Second-order Conditional Gradient Sliding

Second-order Conditional Gradient Sliding

Local and Global Uniform Convexity Conditions

Sublinear Classical and Quantum Algorithms for General Matrix Games

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