Strong and Weak Convexity of Closed Sets in a Hilbert Space

Goncharov, Vladimir V.; Ivanov, Grigorii Evgen'evich

doi:10.1007/978-3-319-51500-7_12

Cited by 18 publications

(6 citation statements)

References 42 publications

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“…Moreoever, we note that a gauge • K is 2-convex iff the set K is strong convex with respect to itself. Despite the extensive literature on strongly convex sets (see the survey [19]), we could not find a reference for this result. We present its simple proof for completeness.…”

Section: Equivalence Of Strongly Convex and Gauge Bodiesmentioning

confidence: 90%

Curvature of Feasible Sets in Offline and Online Optimization

Molinaro¹

2020

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It is known that the curvature of the feasible set in convex optimization allows for algorithms with better convergence rates, and there has been renewed interest in this topic both for offline as well as online problems.In this paper, leveraging results on geometry and convex analysis, we further our understanding of the role of curvature in optimization:• We first show the equivalence of two notions of curvature, namely strong convexity and gauge bodies, proving a conjecture of Abernethy et al. As a consequence, this show that the Frank-Wolfe-type method of Wang and Abernethy has accelerated convergence rate O( 1 t 2 ) over strongly convex feasible sets without additional assumptions on the (convex) objective function.• In Online Linear Optimization, we show that the logarithmic regret guarantee of the algorithm Follow the Leader (FTL) over strongly convex sets recently proved by Huang et al. follows directly from a partial lipschitzness of the support function of such sets. We believe that this provides a simpler explanation for the good performance of FTL in this context.• We provide an efficient procedure for approximating convex bodies by curved ones, smoothly trading off approximation error and curvature, allowing one to extend the applicability of algorithms for curved set to non-curved ones.

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Section: Equivalence Of Strongly Convex and Gauge Bodiesmentioning

confidence: 90%

Curvature of Feasible Sets in Offline and Online Optimization

Molinaro¹

2020

Preprint

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“…As we will see, these local properties are important as they explain empirical globally accelerated convergence rates in optimization problems where the functions or constraints do not satisfy global regularity assumptions such as, e.g., strong convexity [Dun79,KdP20]. The local modulus of smoothness [GI17,(15…”

Section: Spaces Sets Functions Uniform Smoothness and Convexitymentioning

confidence: 99%

“…This is a direct consequence of Theorem 4.1 (b). In the particular case where the set is strongly convex, this is a variation of [GI17,(i) Hence, if the norms of the d i for i = 1, 2 are lower bounded by c > 0, and the set is (α, p)-uniformly convex with p ∈ [2, 3], we obtain that the condition described in [Lan13] is valid and of the form v 1 − v 2 ≤ 1/(2αc) 1/(p−1) d 1 − d 2 1/(p−1) .…”

Section: Applicationsmentioning

confidence: 99%

“…We refer to [Zǎ83,AP95,Zǎ02] for the study of functional uniform convexity and smoothness, to [LT13,Bea11,DGZ93,BGHV09] for the study of the geometry of Banach spaces in terms of uniform convexity and smoothness, and to [Pis11] for results on type/cotype properties of a Banach space. We also invoke [GI17] for practical local characterizations of the strong convexity of sets. Finally, we rely on [Roc70] for convex analysis references and on [Sch14] for convex geometry in finite dimensions.…”

Section: Introductionmentioning

confidence: 99%

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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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“…In his paper, Vial shows ([37, Theorem 5.1]) in the finite dimensional setting that a weakly convex set of constant r (a concept equivalent to the r-prox-regularity in Hilbert spaces) and an R-strongly convex set can always be separated by a ball whenever a certain condition holds between the radii r, R. Vial also provides an estimate for the radius of the involved ball, depending on r and R. Such a result has been successfuly extended to the Hilbert framework by G.E. Ivanov [23] (see also the survey [22]) and by M.V. Balashov and G.E.…”

Section: Prox-regularity and Functional Separationmentioning

confidence: 99%