Adapting to Smoothness: A More Universal Algorithm for Online Convex Optimization

Wang, Guanghui; Lu, Shiyin; Hu, Yao; Zhang, Lijun

doi:10.1609/aaai.v34i04.6081

Cited by 5 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One milestone is MetaGrad of van Erven and Koolen (2016), which can handle general convex functions as well as expconcave functions. Later, Wang et al (2019) propose Maler, which further supports strongly convex functions explicitly. In a subsequent work, Wang et al (2020b) develop UFO, which exploits smoothness to deliver small-loss regret bounds, i.e., regret bounds that depend on the minimal loss.…”

Section: Introductionmentioning

confidence: 80%

“…Later, Wang et al (2019) propose Maler, which further supports strongly convex functions explicitly. In a subsequent work, Wang et al (2020b) develop UFO, which exploits smoothness to deliver small-loss regret bounds, i.e., regret bounds that depend on the minimal loss. However, the three aforementioned methods need to design one surrogate loss for each possible type of functions, which is both tedious and challenging.…”

Section: Introductionmentioning

confidence: 80%

“…is the cumulative loss of the best point in X (Srebro et al, 2010;Orabona et al, 2012;Wang et al, 2020b). These small-loss bounds could be much tighter when L * T is small, and still ensure the minimax optimality otherwise.…”

Section: Traditional Algorithmsmentioning

confidence: 99%

“…Similar to MetaGrad, it gets rid of the priori knowledge of strong convexity. To improve the regret bound for general convex functions, Wang et al (2019) further propose the following surrogate loss:…”

Section: Universal Algorithmsmentioning

confidence: 99%

“…When the functions are convex and smooth, it yields a small-loss bound. Compared to previous universal methods (van Erven and Koolen, 2016;Wang et al, 2019Wang et al, , 2020b, the only limitation of our algorithm is that it needs to query the gradient multiple times in each round. However, it does not change the order of complexity, which remains O(log T ) per iteration, and more importantly, has the following advantages.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Simple yet Universal Strategy for Online Convex Optimization

Zhang,

Wang,

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Recently, several universal methods have been proposed for online convex optimization, and attain minimax rates for multiple types of convex functions simultaneously. However, they need to design and optimize one surrogate loss for each type of functions, which makes it difficult to exploit the structure of the problem and utilize the vast amount of existing algorithms. In this paper, we propose a simple strategy for universal online convex optimization, which avoids these limitations. The key idea is to construct a set of experts to process the original online functions, and deploy a meta-algorithm over the linearized losses to aggregate predictions from experts. Specifically, we choose Adapt-ML-Prod to track the best expert, because it has a second-order bound and can be used to leverage strong convexity and exponential concavity. In this way, we can plug in off-the-shelf online solvers as black-box experts to deliver problem-dependent regret bounds. Furthermore, our strategy inherits the theoretical guarantee of any expert designed for strongly convex functions and exponentially concave functions, up to a double logarithmic factor. For general convex functions, it maintains the minimax optimality and also achieves a smallloss bound.

show abstract

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 80%

Section: Traditional Algorithmsmentioning

confidence: 99%

Section: Universal Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Simple yet Universal Strategy for Online Convex Optimization

Zhang,

Wang,

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Projection-free Online Learning in Dynamic Environments

Wan

Xue

Zhang

2021

AAAI

Self Cite

View full text Add to dashboard Cite

To efficiently solve high-dimensional problems with complicated constraints, projection-free online learning has received ever-increasing research interest. However, previous studies either focused on static regret that is not suitable for dynamic environments, or only established the dynamic regret bound under the smoothness of losses. In this paper, without the condition of the smoothness, we propose a novel projection-free online algorithm, and achieve an O(max{T^{2/3}V_T^{1/3},T^{1/2}}) dynamic regret bound for convex functions and an O(max{(TV_Tlog T)^{1/2},log T}) dynamic regret bound for strongly convex functions, where T is the time horizon and V_T denotes the variation of loss functions. Specifically, we first improve an existing projection-free algorithm called online conditional gradient (OCG) to enjoy small dynamic regret bounds with the prior knowledge of V_T. To work with unknowable V_T, we maintain multiple instances of the improved OCG that can handle different functional variations, and combine them with a meta-algorithm that can track the best one. Experimental results validate the efficiency and effectiveness of our algorithm.

show abstract

Dual Adaptivity: A Universal Algorithm for Minimizing the Adaptive Regret of Convex Functions

Zhang¹,

Wang²,

Tu³

et al. 2019

Preprint

View full text Add to dashboard Cite

To deal with changing environments, a new performance measure-adaptive regret, defined as the maximum static regret over any interval, is proposed in online learning. Under the setting of online convex optimization, several algorithms have been successfully developed to minimize the adaptive regret. However, existing algorithms lack universality in the sense that they can only handle one type of convex functions and need apriori knowledge of parameters. By contrast, there exist universal algorithms, such as MetaGrad, that attain optimal static regret for multiple types of convex functions simultaneously. Along this line of research, this paper presents the first universal algorithm for minimizing the adaptive regret of convex functions. Specifically, we borrow the idea of maintaining multiple learning rates in MetaGrad to handle the uncertainty of functions, and utilize the technique of sleeping experts to capture changing environments. In this way, our algorithm automatically adapts to the property of functions (convex, exponentially concave, or strongly convex), as well as the nature of environments (stationary or changing). As a by product, it also allows the type of functions to switch between rounds.

show abstract

Adapting to Smoothness: A More Universal Algorithm for Online Convex Optimization

Cited by 5 publications

References 12 publications

A Simple yet Universal Strategy for Online Convex Optimization

A Simple yet Universal Strategy for Online Convex Optimization

Projection-free Online Learning in Dynamic Environments

Dual Adaptivity: A Universal Algorithm for Minimizing the Adaptive Regret of Convex Functions

Contact Info

Product

Resources

About