Proximal Online Gradient Is Optimum for Dynamic Regret: A General Lower Bound

Zhao, Yawei; Qiu, Shuang; Li, Kuan; Luo, Lailong; Yin, Jianping; Liu, Ji

doi:10.1109/tnnls.2021.3087579

Cited by 11 publications

(13 citation statements)

References 17 publications

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“…where T is the maximal number of rounds, and D is the given budget of dynamics. But, the dynamic regret for an OCO method is O √ T D + √ T , which is same with the case of no switching cost [20,21,50,51]. Furthermore, we provide a lower bound of dynamic regret, namely Ω √ T D + √ T for the OCO se ing.…”

Section: Introductionmentioning

confidence: 58%

Understand Dynamic Regret with Switching Cost for Online Decision Making

Zhao¹,

Zhao²,

Zhang³

et al. 2019

Preprint

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As a metric to measure the performance of an online method, dynamic regret with switching cost has drawn much a ention for online decision making problems. Although the sublinear regret has been provided in many previous researches, we still have li le knowledge about the relation between the dynamic regret and the switching cost. In the paper, we investigate the relation for two classic online se ings: Online Algorithms (OA) and Online Convex Optimization (OCO). We provide a new theoretical analysis framework, which shows an interesting observation, that is, the relation between the switching cost and the dynamic regret is di erent for se ings of OA and OCO. Speci cally, the switching cost has signi cant impact on the dynamic regret in the se ing of OA. But, it does not have an impact on the dynamic regret in the se ing of OCO. Furthermore, we provide a lower bound of regret for the se ing of OCO, which is same with the lower bound in the case of no switching cost. It shows that the switching cost does not change the di culty of online decision making problems in the se ing of OCO.

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Section: Introductionmentioning

confidence: 58%

Understand Dynamic Regret with Switching Cost for Online Decision Making

Zhao¹,

Zhao²,

Zhang³

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We now separately bound the terms on the right-hand side of (13). In order to bound the first term of the above inequality, we add and subtract several terms as follows:…”

Section: B Proof Of Theoremmentioning

confidence: 99%

“…where the last line follows from the fact that D r (x, y) is nonnegative when r(x) is convex, and the Lipschitz condition stated in (7). We now proceed to bound the other term on the right-hand side of (13). We add and subtract f t+1 (x t+1 ) to obtain…”

Section: B Proof Of Theoremmentioning

confidence: 99%

“…which measures the variation in the comparator sequence, where • could be any norm. Several online learning algorithms provide an O( √ T (1 + C T )) upper bound on the dynamic regret of convex cost functions [3], [11], [12], which can be improved to O( √ T C T ) when prior knowledge of C T and T is available [13]. The path length has also been recently used in the study of online convex optimization with constraint violation [14], where upper bounds of O( T (1 + C T )) and O( √ T ) are derived on the dynamic regret and cumulative constraint violation, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Regret of Online Mirror Descent for Relatively Smooth Convex Cost Functions

Eshraghi¹,

Liang²

2022

Preprint

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The performance of online convex optimization algorithms in a dynamic environment is often expressed in terms of the dynamic regret, which measures the decision maker's performance against a sequence of time-varying comparators. In the analysis of the dynamic regret, prior works often assume Lipschitz continuity or uniform smoothness of the cost functions. However, there are many important cost functions in practice that do not satisfy these conditions. In such cases, prior analyses are not applicable and fail to guarantee the optimization performance. In this letter, we show that it is possible to bound the dynamic regret, even when neither Lipschitz continuity nor uniform smoothness is present. We adopt the notion of relative smoothness with respect to some user-defined regularization function, which is a much milder requirement on the cost functions. We first show that under relative smoothness, the dynamic regret has an upper bound based on the path length and functional variation. We then show that with an additional condition of relatively strong convexity, the dynamic regret can be bounded by the path length and gradient variation. These regret bounds provide performance guarantees to a wide variety of online optimization problems that arise in different application domains. Finally, we present numerical experiments that demonstrate the advantage of adopting a regularization function under which the cost functions are relatively smooth.

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“…It is well known that assuming convexity of the c k and boundedness of the gradients, online gradient descent achieves O( √ K) regret bound and this bound is improved to O log(K) assuming strong convexity of the c k [9]. More general classes of OCO algorithms have been studied [11,21], notably (accelerated) proximal gradient descent algorithms concerned about composite convex functions of the form φ k = f k + g where only the f k are smooth. Improved regret bounds again hold assuming strong convexity of the φ k .…”

Section: Connection With Online Convex Optimizationmentioning

confidence: 99%

Learning Large DAGs by Combining Continuous Optimization and Feedback Arc Set Heuristics

Gillot¹,

Parviainen²

2021

Preprint

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Bayesian networks represent relations between variables using a directed acyclic graph (DAG). Learning the DAG is an NP-hard problem and exact learning algorithms are feasible only for small sets of variables. We propose two scalable heuristics for learning DAGs in the linear structural equation case. Our methods learn the DAG by alternating between unconstrained gradient descent-based step to optimize an objective function and solving a maximum acyclic subgraph problem to enforce acyclicity. Thanks to this decoupling, our methods scale up beyond thousands of variables.

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Proximal Online Gradient Is Optimum for Dynamic Regret: A General Lower Bound

Cited by 11 publications

References 17 publications

Understand Dynamic Regret with Switching Cost for Online Decision Making

Understand Dynamic Regret with Switching Cost for Online Decision Making

Dynamic Regret of Online Mirror Descent for Relatively Smooth Convex Cost Functions

Learning Large DAGs by Combining Continuous Optimization and Feedback Arc Set Heuristics

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