Alternating Regret for Online Convex Optimization

• URL: http://arxiv.org/abs/2502.12529v1
• Date: Tue, 18 Feb 2025 04:32:52 GMT
• Title: Alternating Regret for Online Convex Optimization
• Authors: Soumita Hait, Ping Li, Haipeng Luo, Mengxiao Zhang,
• Abstract summary: We show that the continuous Hedge algorithm achieves $tildemathcalO(dfrac23Tfrac13)$ alternating regret for any adversariald$-dimensional OCO problems. We show some algorithm-specific alternating regret lower bounds, including a somewhat surprising $Omega(sqrtT)$ lower bound for a Regret Matching variant.
• Score: 43.12477663757647
• License:
• Abstract: Motivated by alternating learning dynamics in two-player games, a recent work by Cevher et al.(2024) shows that $o(\sqrt{T})$ alternating regret is possible for any $T$-round adversarial Online Linear Optimization (OLO) problem, and left as an open question whether the same is true for general Online Convex Optimization (OCO). We answer this question in the affirmative by showing that the continuous Hedge algorithm achieves $\tilde{\mathcal{O}}(d^{\frac{2}{3}}T^{\frac{1}{3}})$ alternating regret for any adversarial $d$-dimensional OCO problems. We show that this implies an alternating learning dynamic that finds a Nash equilibrium for any convex-concave zero-sum games or a coarse correlated equilibrium for any convex two-player general-sum games at a rate of $\tilde{\mathcal{O}}(d^{\frac{2}{3}}/T^{\frac{2}{3}})$. To further improve the time complexity and/or the dimension dependence, we propose another simple algorithm, Follow-the-Regularized-Leader with a regularizer whose convex conjugate is 3rd-order smooth, for OCO with smooth and self-concordant loss functions (such as linear or quadratic losses). We instantiate our algorithm with different regularizers and show that, for example, when the decision set is the $\ell_2$ ball, our algorithm achieves $\tilde{\mathcal{O}}(T^{\frac{2}{5}})$ alternating regret with no dimension dependence (and a better $\tilde{\mathcal{O}}(T^{\frac{1}{3}})$ bound for quadratic losses). We complement our results by showing some algorithm-specific alternating regret lower bounds, including a somewhat surprising $\Omega(\sqrt{T})$ lower bound for a Regret Matching variant that is widely used in alternating learning dynamics.

Related papers

• Towards Optimal Regret in Adversarial Linear MDPs with Bandit Feedback [30.337826346496385]
  We study online reinforcement learning in linear Markov decision processes with adversarial losses and bandit feedback. We introduce two algorithms that achieve improved regret performance compared to existing approaches.
  arXiv  Detail & Related papers  (2023-10-17T19:43:37Z)
• Projection-free Online Exp-concave Optimization [21.30065439295409]
  We present an LOO-based ONS-style algorithm, which using overall $O(T)$ calls to a LOO, guarantees in worst case regret bounded by $widetildeO(n2/3T2/3)$. Our algorithm is most interesting in an important and plausible low-dimensional data scenario.
  arXiv  Detail & Related papers  (2023-02-09T18:58:05Z)
• Optimal Dynamic Regret in LQR Control [23.91519151164528]
  We consider the problem of nonstochastic control with a sequence of quadratic losses, i.e., LQR control. We provide an online algorithm that achieves an optimal dynamic (policy) regret of $tildeO(textmaxn1/3 mathcalTV(M_1:n)2/3, 1)$.
  arXiv  Detail & Related papers  (2022-06-18T18:00:21Z)
• Near-Optimal Learning of Extensive-Form Games with Imperfect Information [54.55092907312749]
  We present the first line of algorithms that require only $widetildemathcalO((XA+YB)/varepsilon2)$ episodes of play to find an $varepsilon$-approximate Nash equilibrium in two-player zero-sum games. This improves upon the best known sample complexity of $widetildemathcalO((X2A+Y2B)/varepsilon2)$ by a factor of $widetildemathcalO(maxX,
  arXiv  Detail & Related papers  (2022-02-03T18:18:28Z)
• Optimal Regret Algorithm for Pseudo-1d Bandit Convex Optimization [51.23789922123412]
  We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost/reward functions admit a "pseudo-1d" structure. We show a lower bound of $min(sqrtdT, T3/4)$ for the regret of any algorithm, where $T$ is the number of rounds. We propose a new algorithm sbcalg that combines randomized online gradient descent with a kernelized exponential weights method to exploit the pseudo-1d structure effectively.
  arXiv  Detail & Related papers  (2021-02-15T08:16:51Z)
• Revisiting Smoothed Online Learning [70.09792747315323]
  We investigate the problem of smoothed online learning, in which the online learner suffers both a hitting cost and a switching cost. To bound the competitive ratio, we assume the hitting cost is known to the learner in each round, and investigate the greedy algorithm which simply minimizes the weighted sum of the hitting cost and the switching cost.
  arXiv  Detail & Related papers  (2021-02-13T14:15:55Z)
• Near-Optimal Reinforcement Learning with Self-Play [50.29853537456737]
  We focus on self-play algorithms which learn the optimal policy by playing against itself without any direct supervision. We propose an optimistic variant of the emphNash Q-learning algorithm with sample complexity $tildemathcalO(SAB)$, and a new emphNash V-learning algorithm with sample complexity $tildemathcalO(S(A+B))$.
  arXiv  Detail & Related papers  (2020-06-22T05:00:13Z)
• Taking a hint: How to leverage loss predictors in contextual bandits? [63.546913998407405]
  We study learning in contextual bandits with the help of loss predictors. We show that the optimal regret is $mathcalO(minsqrtT, sqrtmathcalETfrac13)$ when $mathcalE$ is known.
  arXiv  Detail & Related papers  (2020-03-04T07:36:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.