Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization

Zhao, Peng; Zhang, Yujie; Zhang, Lijun; Zhou, Zhi‐Hua

doi:10.48550/arxiv.2112.14368

Cited by 2 publications

(4 citation statements)

References 15 publications

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“…Bound on dR worker * T ({u t } t∈[T ] ): The regret bound for the worker is very similar to wellknown results (c.f. or Zhao et al (2021)). However, typically it contains a variational term V T .…”

Section: Analysis Of Dynmetagradmentioning

confidence: 98%

Accelerated Rates between Stochastic and Adversarial Online Convex Optimization

Sarah¹,

Hadiji²,

Erven³

et al. 2023

Preprint

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Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the related expert and bandit settings. In the fully i.i.d. case, our regret bounds match the rates one would expect from results in stochastic acceleration, and we also recover the optimal stochastically accelerated rates via online-to-batch conversion. In the fully adversarial case our bounds gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes in terms of the stochastic variance and the adversarial variation of the loss gradients.

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Section: Analysis Of Dynmetagradmentioning

confidence: 98%

Accelerated Rates between Stochastic and Adversarial Online Convex Optimization

Sarah¹,

Hadiji²,

Erven³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…, u T ∈ X are allowed to change over time. Therefore, this measure is more attractive in non-stationary online learning (Zhao et al, 2021). Notably, the static regret can be treated as its special case with a fixed comparator, i.e., u 1 = .…”

Section: Extension To Dynamic Regret Minimizationmentioning

confidence: 99%

“…Remark 8. Our algorithm design and regret analysis follow the recent work of Zhao et al (2021), where the correction term λ x t,i − x t−1,i 2 2 in the meta-algorithm (both feedback loss and optimism) plays an important role. Technically, in this two-layer structure, to cancel the additional positive term T t=2 x t − x t−1 2 2 appearing in the derivation of σ 2 1:T and Σ 2 1:T , one needs to simultaneously exploit negative terms of the regret upper bounds in both base level and meta level as well as additional negative terms introduced by the above correction term.…”

Section: Extension To Dynamic Regret Minimizationmentioning

confidence: 99%

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Optimistic Online Mirror Descent for Bridging Stochastic and Adversarial Online Convex Optimization

Chen¹,

Tu²,

Zhao³

et al. 2023

Preprint

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Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. (2022) as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic followthe-regularized-leader (FTRL) depends on the cumulative stochastic variance σ 2 1:T and the cumulative adversarial variation Σ 2 1:T for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance σ 2 max and the maximal adversarial variation Σ 2 max for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same O( σ 2 1:T + Σ 2 1:T ) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log(σ 2 1:T +Σ 2 1:T ), (σ 2 max +Σ 2 max ) log T }) bound, better than their O((σ 2 max + Σ 2 max ) log T ) bound. For exp-concave and smooth functions, we achieve a new O(d log(σ 2 1:T + Σ 2 1:T )) bound. Owing to the OMD framework, we can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

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Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization

Cited by 2 publications

References 15 publications

Accelerated Rates between Stochastic and Adversarial Online Convex Optimization

Accelerated Rates between Stochastic and Adversarial Online Convex Optimization

Optimistic Online Mirror Descent for Bridging Stochastic and Adversarial Online Convex Optimization

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