Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems

Veliov, Vladimir M.; Vuong, Phan Tu

doi:10.1007/s00245-018-9528-3

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…Another sufficient assumption for nonempty X ⋆ of VIs is that X is sufficiently strongly con-vex. This condition has recently been used to show fast convergence of mirror descent and conditional gradient descent (Garber and Hazan, 2015;Veliov and Vuong, 2017). We leave this discussion to Appendix B.…”

Section: Ep and VI Perspectivesmentioning

confidence: 98%

Online Learning with Continuous Variations: Dynamic Regret and Reductions

Cheng

Goldberg

Boots

2019

Preprint

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Online learning is a powerful tool for analyzing iterative algorithms. However, the classic adversarial setup sometimes fails to capture certain regularity in online problems in practice. Motivated by this, we establish a new setup, called Continuous Online Learning (COL), where the gradient of online loss function changes continuously across rounds with respect to the learner's decisions. We show that COL covers and more appropriately describes many interesting applications, from general equilibrium problems (EPs) to optimization in episodic MDPs. In particular, we show monotone EPs admits a reduction to achieving sublinear static regret in COL. Using this new setup, we revisit the difficulty of sublinear dynamic regret. We prove a fundamental equivalence between achieving sublinear dynamic regret in COL and solving certain EPs. With this insight, we offer conditions for efficient algorithms that achieve sublinear dynamic regret, even when the losses are chosen adaptively without any a priori variation budget. Furthermore, we show for COL a reduction from dynamic regret to both static regret and convergence in the associated EP, allowing us to analyze the dynamic regret of many existing algorithms.

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Section: Ep and VI Perspectivesmentioning

confidence: 98%

Online Learning with Continuous Variations: Dynamic Regret and Reductions

Cheng

Goldberg

Boots

2019

Preprint

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“…Their analysis rely on assumptions that combine scaling inequalities for strongly convex feasible sets and an affine-invariant characterization of smoothness [Jag13]. Finally, strong convexity for sets was also used outside of projection-free optimization techniques in, e.g., [VV20,Bac20].…”

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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Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems

Cited by 2 publications

References 23 publications

Online Learning with Continuous Variations: Dynamic Regret and Reductions

Online Learning with Continuous Variations: Dynamic Regret and Reductions

Local and Global Uniform Convexity Conditions

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