Title: Gradient-Variation Online Adaptivity for Accelerated Optimization with Hölder Smoothness
Venue: NeurIPS 2025
Accepted: yes
Keywords: online learning

regret

acceleration
Abstract: Smoothness is known to be crucial for acceleration in offline optimization, and for gradient-variation regret minimization in online learning. 
Interestingly, these two problems are actually closely connected --- accelerated optimization can be understood through the lens of gradient-variation online learning.
In this paper, we investigate online learning with *Hölder* functions, a general class encompassing both smooth and non-smooth (Lipschitz) functions, and explore its implications for offline optimization. 
For (strongly) convex online functions, we design the corresponding gradient-variation online learning algorithm whose regret smoothly interpolates between the optimal guarantees in smooth and non-smooth regimes. 
Notably, our algorithms do not require prior knowledge of the Hölder smoothness parameter, exhibiting strong adaptivity over existing methods.
Through online-to-batch conversion, this gradient-variation online adaptivity yields an optimal universal method for stochastic convex optimization under Hölder smoothness. 
However, achieving universality in offline strongly convex optimization is more challenging. 
We address this by integrating online adaptivity with a detection-based guess-and-check procedure, which, for the first time, yields a universal offline method that achieves accelerated convergence in the smooth regime while maintaining near-optimal convergence in the non-smooth one.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies regret bounds with gradient variation in the context of online learning.\n\nIt first extends existing regret bounds by incorporating gradient variation terms when the loss functions are Hölder smooth and Lipschitz continuous. The authors then provide an online-to-batch conversion technique that yields accelerated convergence rates of order $O(T^{-2})$ for smooth functions, thereby generalizing existing results. A key strength of the method is its universality—it does not require prior knowledge of the Lipschitz or smoothness constants.\n\nThe authors further extend their results to the strongly convex setting, deriving regret bounds and corresponding accelerated online-to-batch conversion rates. However, in this case, the method is not fully universal, as it requires prior knowledge of the strong convexity parameter. Additionally, the obtained rates are slightly suboptimal when the Lipschitz constant is unknown.\n\nFinally, the authors design a meta-algorithm that achieves dynamic regret bounds with gradient variation when the loss functions are Lipschitz and Hölder smooth. They also propose an algorithm that adapts to unknown strong convexity parameter.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper is well written and presents a clearly motivated framework, relevant to both online learning and convex optimization. It offers a substantial analysis of regret bounds incorporating gradient variation for Hölder smooth functions, providing a range of results that either improve upon existing work (by requiring fewer parameter assumptions) or generalize it (by considering Hölder smoothness with a generic parameter $\\nu$). These contributions add breadth and flexibility to the theory of learning with smooth losses.\n\nWeaknesses:\n\nIt is not entirely clear to what extent the results go beyond incremental generalizations of existing work—primarily extending known analyses from $\\nu = 1$ (standard smoothness) to a generic Hölder smoothness parameter $\\nu$. Could the authors clarify what technical challenges were encountered and how they were addressed in order to extend the results? For example, the analysis in Theorem 1 appears to follow standard Optimistic OMD techniques, and the online-to-batch conversion arguments seem to closely mirror those in [Cutkosky, 2019].\n\nQUESTIONS:\n1. Could the authors elaborate on the specific challenges involved in extending the online-to-batch conversion to the stochastic setting in the strongly convex case? Are there particular obstacles that prevent a direct extension of the techniques used in the smooth or convex cases?\n\n2. Is it possible to obtain an online-to-batch conversion for Theorem 6 in the case where the curvature parameter (i.e., the strong convexity constant) is unknown? If not, could the authors comment on what makes this technically difficult or whether a universal algorithm could adapt to unknown curvature in this setting?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper develops algorithms for online convex optimization and offline optimization under Hölder smoothness and convexity or strong convexity. Several algorithms are proposed, versions of optimistic gradient descent without and with round weights, and a meta-level learning algorithm. The algorithm does not have to know the class, and it yields optimal bounds in terms of the gradient variation.\n\n\nLet me go into a bit more detail on how the results are achieved.\n\nTheorems 1 and 3 rely on \n(a) the optimistic OMD analysis, resulting in gradient variation appearing plus negative movement terms.\n(b) the Hölder-to-quadratic-plus-overhead inequality\n(c) decaying learning rate schedule, so that it eventually the negative terms from (a) start helping.\n\nThese are all well-known techniques. The contribution of this paper is to show that they can be chained together even in intermediate smoothness regimes i.e. nu in (0,1) and while retaining the gradient variation in the answer.\n\nThe results are then applied and extended for offline stochastic optimization, and for dynamic and universal regret.\n\nSTRENGTHS AND WEAKNESSES:\nQuality:\nThe achieved adaptivity result is an accomplishment.\n\nClarity:\nThe paper is well written and organized clearly. The results are compared to the existing literature.\n\nSignificance:\nI think this paper will be remembered for its Theorem 6. I find it positively amazing that the Hölder smoothness constants are not required for tuning the algorithm.\n\nOriginality:\nThe paper presents a tighter analysis of an existing algorithm.\n\nQUESTIONS:\nCan you relate to the techniques in Wang and Abernethy 2018, Acceleration through Optimistic No-Regret Dynamics?\n\nThe presentation of \"universality\" in online learning is not quite accurate. Going back to van Erven et al (2016), universality is adaptivity to the curvature parameter *without knowing it*. The paper under review instead adapts to a fixed given \\lambda or \\lambda=0. Wang et al (2019) show how to adapt to the strong convexity parameter (\\lambda in the paper under review) without knowing it up-front. Would their approach (of combining multiple learning rates) extend to your setting, and deliver adaptivity to any \\lambda without knowing it? That seems like a desirable and within-reach extension.\n\nThe universal regret result in Theorem 6 has a *better* exponent on log(T) than the \"just-strongly convex\" result in Theorem 3. This is strange. What is the catch? Why would we be satisfied with Theorem 3?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies online convex optimization under Holder smoothness, and shows that optimistic OGD can obtain $R_T\\le O(\\sqrt{\\sum_t\\sup_x\\\\|f_t(x)-f_{t-1}(x)\\\\|^2}+T^{\\frac{1-\\nu}{2}}+\\\\|\\nabla f_1(x_1)\\\\|)$ regret without any prior knowledge of the Holder smoothness parameters, and it is shown that this yields optimal rates in the stochastic optimization setting. The result is also extended to strongly-convex losses, to time-varying comparators (dynamic regret), and to universal regret.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\nThe main contribution in Theorem 1 is very nice and the analysis is very clean. The main tool is Lemma 1, which lets allows you to connect $\\\\|\\nabla f(x)-\\nabla f(y)\\\\|^2$ to $\\\\|x-y\\\\|^2$, similar to how you would with smooth losses, but with an additional $\\delta$ factor that appears only in the analysis and can therefore be tuned *a posteriori* at the end of the analysis. It's a very nice approach overall and leads to several extensions essentially \"for free\" after Theorem 1 is established.\n\n## Weaknesses\n\n**Universality**\n\nI think the insistence on defining methods as \"strongly universal\" will serve only to complicate the literature. For one, there are already other notions of universality, for instance Nesterov's Universal Algorithm, or in OCO algorithms which achieve guarantees on universal regret (as in Section 5.2) are sometimes referred to as being \"universal\".\n\nBut it is also more generally just a short-sighted naming convention. Intuitively, an algorithm should be deemed universal if it solves any problem in a problem class without manually adjusting any problem-dependent parameters; a practitioner should be able to just hit run and know the guarantee will hold. This is not the case here because the results depend on the diameter of the decision set. I think to really be able to call an algorithm universal here you'd have to develop a comparator-adaptive version. So let's suppose that you extend the approach here to achieve comparator adaptivity, what should we call such an algorithm? \"Extra Strongly Universal\"? \n\n**Loss of Adaptivity**\n\nThe key argument seems to unavoidably lead non-adaptive penalties (factors of $T$ instead of, say, $V_T$ or $\\sum_t\\\\|g_t\\\\|^2$) showing up in the bound. Prior works avoid such penalties from showing up, though they don't adapt to the Holder smoothness parameters; is it possible to improve the results here to achieve fully adaptive bounds? Or does adapting to the Holder smoothness parameters necessitate non-adaptive penalties? For example, the dynamic regret result in the Lipschitz losses setting ends up being $O(G\\sqrt{P_T T})$ instead of the usual $O(\\sqrt{P_T\\sum_t\\\\|g_t\\\\|^2})$.\n\n**Minor Weaknesses**\n\nThe main novelty is Theorem 1, and each of the extensions are generally very straight-forward afterwards. For example, the extension to stochastic optimization via anytime online-to-batch is well-known from Cutkosky 2019, and the result is basically immediate by combining the online-to-batch argument with Theorem 1. It is also well-known that gradient variation bounds imply improved regret under strong convexity, so these too are fairly easily obtained given Theorem 1, and obtaining the $\\sqrt{P_T T}$ dynamic regret bounds is also straight-forward once one has the static regret result.\n\nQUESTIONS:\n- Do you expect that the non-adaptive penalties can be improved to adaptive ones such as $V_T$ or $\\sum_t\\\\|g_t\\\\|^2$, or are the non-adaptive penalties unavoidable when adapting to the Holder smoothness parameters?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the challenge of achieving optimal regret bounds in online convex optimization (OCO) under gradient variation measures. Specifically, while prior works in OCO typically assume Lipschitz or first-order smooth conditions, the authors considered  $\\nu$-Holder conditions which interpolates these two regimes. The authors propose a parameter-free algorithm based on Online Gradient Descent (OGD) that achieves regret bounds of the order  $O(\\sqrt{V_T}+T^{\\frac{1-\\nu}{2}})$, where $V_T$ is gradient variation defined as the maximum total squared variation of the gradients through iterations. This matches the known lower bounds at the two corner points of Lipschitz and smooth functions. The authors further demonstrate that their method extends to various settings such as offline convex optimization, strongly convex optimization, and dynamic and universal regrets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:  The main conceptual contribution, adaptivity to unknown regularity under the gradient variation metric for Holder smooth functions, seem to be novel. The proposed algorithm provides a flexible tradeoff between the required sample complexity and the level of smoothness of the objective function. Rigorous analysis was provided and the achieved sample complexities are presented in clean forms. The paper is generally clear, with good use of tables, figures, and notation to present the main results. Section 2 provides an insightful summary of related works and positions the contribution well.\n\nWeakness: \n1. While the achievability results are clearly presented, it is not clear that whether the proposed algorithm achieve optimal regrets besides the two special cases of $\\nu=0,1$ as no lower bounds were formally presented as theorems or remarks. If no lower bound is provided, the authors should not claim in the abstract that the proposed algorithm is optimal. \n2.  According to the discussion provided by the author, the extended result to SCO seems to be already known in Rodomanov et al 2024.\n3. The formulation and assumptions in this work can be better clarified. For instance, while OCO is a well known framework, it is recommended to still provide a brief and concrete description before introducing the formulations for extensions.\n\nQUESTIONS:\nWe recommend the authors to address the issues in the weakness section, especially the optimality claim of the proposed results stated in the abstract needs to be appropriately justified. Besides, while the submitted work covers a wide range of extended settings, the explicit formulations in this work contain limitations that were not clearly stated. For instance, in the extension of SCO the authors focused on the settings of first-order oracles, while there has been extensive works on zeroth-order optimization, which has different fundamental sample complexities. The authors also focused on Holder conditions with $\\nu\\leq 1$, while there are works that explores higher-order smoothness conditions. To name a few, see the references at the end, which are by no means exclusive. We recommend the authors to clarify these limitations, and briefly mention if the proposed results can be extended to those settings. We would be happy to raise the score if the above issues are addressed adequately. \n\nBach and Perchet. \"Highly-smooth zero-th order online optimization\" \nAkhavan et al. \"Exploiting higher order smoothness in derivative-free optimization and continuous bandits\"\nNovitskii and Gasnikov. \"Improved exploiting higher order smoothness in derivative-free optimization and continuous bandit\"\nYu et al. \"Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity\"\nAkhavan and Tsybakov. \"Gradient-free stochastic optimization for additive models\"\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}