Title: Optimistic Online-to-Batch Conversions for Accelerated Convergence and Universality
Venue: NeurIPS 2025
Accepted: yes
Keywords: offline optimization

convex optimization

online-to-batch conversion

acceleration
Abstract: In this work, we study offline convex optimization with smooth objectives, where the classical Nesterov's Accelerated Gradient (**NAG**) method achieves the optimal accelerated convergence. Extensive research has aimed to understand **NAG** from various perspectives, and a recent line of work approaches this from the viewpoint of online learning and online-to-batch conversion, emphasizing the role of *optimistic online algorithms* for acceleration. In this work, we contribute to this perspective by proposing novel *optimistic online-to-batch conversions* that incorporate optimism theoretically into the analysis, thereby significantly simplifying the online algorithm design while preserving the optimal convergence rates. Specifically, we demonstrate the effectiveness of our conversions through the following results: *(i)* when combined with simple online gradient descent, our optimistic conversion achieves the optimal accelerated convergence; *(ii)* our conversion also applies to strongly convex objectives, and by leveraging both optimistic online-to-batch conversion and optimistic online algorithms, we achieve the optimal accelerated convergence rate for strongly convex and smooth objectives, for the first time through the lens of online-to-batch conversion; *(iii)* our optimistic conversion can achieve universality to smoothness~---~applicable to both smooth and non-smooth objectives without requiring knowledge of the smoothness coefficient~---~and remains efficient as non-universal methods by using only one gradient query in each iteration. Finally, we highlight the effectiveness of our optimistic online-to-batch conversions by a precise correspondence with **NAG**.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes optimistic online-to-batch conversions, which incorporate the optimistic approach into the online-to-batch stage, enabling the use of non-optimistic methods for achieving the same rate as optimistic methods with stabilized online-to-batch conversions (Cutkosky 2019). This enables the authors to achieve acceleration, both in convex and strongly convex settings, and universality, with a single query per iteration, in contrast to previous work. Additional relations between online approaches, NAG, and the heavy-ball method are discussed.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. The presentation is well structured, helping the reader understand the difference between existing online-to-batch conversions and the new proposed conversions.\n2. The results cover several cases of interest, including convex Lipschitz and convex smooth, with and without stochastic noise, and strongly convex smooth case.\n\n**Weaknesses**\n\n3. My main concern is the novelty of the results. This work mainly offers a new perspective, while each of the results can be obtained from previous work, either directly or with a minor extension (e.g., universal algorithms, Nesterov 2015, Kavis et al. 2019, Rodomanov et al. 2024a,b, or using restarts to support the strongly convex case).\n\nQUESTIONS:\n1. Can the authors elaborate on what results, if any, were not established by previous work (and cannot be easily established from previous results)?\n2. Beyond elegance (and potentially practical use), is the single query per iteration of particular importance beyond saving a constant factor in the convergence rate? I do appreciate the simpler algorithm but missed if there is something beyond simplicity and constant factors.\n3. Does the approach enable guarantees in the strongly convex case with stochastic noise? I am aware that the $\\sigma / \\sqrt{T}$ term cannot be improved. (This is a question out of interest, it is ok if the authors simply do not know the answer.)\n\nOverall, this paper is interesting to read and provides some insights into acceleration and universality, with a somewhat simpler universal algorithm. That being said, articulating new guarantees or benefits achieved from this new perspective would be much appreciated.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper provides an overview and some new theory for online-to-batch conversions with a focus on universal and accelerated approaches.\n\nSTRENGTHS AND WEAKNESSES:\nOnline-to-batch conversions are not yet well understood in the optimization community and I think this paper does a good job of introducing existing approaches and expanding on them.\n\n- The proof techniques for the theory are a different and interesting take compared to the other approaches in the existing literature. Much of the overview content will be already familiar to researchers in the area, while for others it will be a good introduction and it's very useful to bring all these results together and provide a clarifying view.\n\n- The exact correspondence between certain instantiations of optimism methods and Nesterov's acceleration is generally known to researchers in the area but is not clearly indicated in the literature, so the discussion here is welcome. \n\n- The use of the average iterate as the location of gradient evaluation was also separately discovered in the stochastic optimization literature separately from the online learning community, under the name “Primal Averaging” by Tao et al. [2020], and was further explored in “The Power of Factorial Powers” (Defazio and Gower, 2020). Some discussion of this related work would be welcome. A variant of these  online-to-batch conversions were also very recently shown to actually be usable for real-world deep learning training in \"The Road Less Scheduled\".\n\n- The UniXGrad paper (cited briefly already) also uses a similar difference-of-gradient's to define a universal method. Is there any relation between the methods? the UniXGrad paper uses general divergences which makes it not immediately clear to me.\n\nQUESTIONS:\nSee Above\n\nLIMITATIONS:\nSee Above", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies an online to batch conversion approach to convex optimization. The primary contribution of this work is demonstrating that online algorithms with optimistic hints are not necessary to obtain accelerated rates. The key change to the algorithm is evaluating the update gradient at a point which approximates the average of the iterates, without requiring access to (yet to be determined) next iterate. Due to the clever choice of evaluation point, a simple online to batch analysis results in an optimal rate with no requirement for direct hinting. The method can be extended to utilize adaptive step sizes in a manner robust to smoothness assumptions.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The explanation of the mechanism of the update mechanism, in comparison to previous anytime online to batch conversion is insightful and intuitive. Generally I think the discussion of previous anytime online to batch algorithms is good. \n- Most of the mathematical exposition is clear and correct (minus a few technicalities; see questions). Most of thee proofs are relatively easy to follow.\n\n### Weaknesses \n\n**Insufficient Differentiation/Discussion of Previous work**\nMy primary concern is that I am unsure whether this paper is a sufficiently novel/interesting contribution to the community. In particular, I don't think that you sufficiently highlight the connection to Wang and Abernethy (2018) on the following fronts:\n- Algorithm 2 in their paper looks to be essentially identical to your update. This fact should be highlighted in your remark/previous work\n- The analysis leading to your Theorem 2 is very similar to their Theorem 2 and Corollary 1.\n- Theorem 4 in your paper also appears to borrow heavily from their paper.\n- Since your step is the same as Algorithm 2 in their paper and Proposition 1 is lifted directly, the results from Section 4 don't feel that interesting. They are more like previous work. Furthermore, I don't think you can claim that you have established the equivalence between your method and NAG as you do in line 100.\n\nOverall I don't believe that the connection to Wang and Abernethy (2018) is sufficiently explained by the paper (in particular Remark 1 is certainly insufficient) which leaves me doubtful of exactly where the boundary between this contribution of this work and previous work is. Note that I don't have an issue building off of previous work per se, but I don't think this paper accurately summarises the relationship to previous work. Could you outline specifically how your method can be differentiated from this previous work/what your specific contribution is? So far as I can tell the adaptive step size is new but I am unsure how strong this result is.\n\n**Stochastic Case**\nOther works analyze the stochastic case in conjunction with the deterministic case, e.g, Cutosky (2019), Kavis (2019) and Joulani et al. (2022b). In particular, expectation bounds (under unbiased and bounded variance gradients assumptions) with some dependence on $\\sigma$ usually arise naturally from the online to batch framework. What were the barriers to taking this approach for, say, theorem 1-4? The stochastic analysis in this paper is focussed on one specific case and an almost sure boundedness (which is stronger than a bounded variance assumption).\n\n**Numerical Evaluation**\nA numerical comparison could significantly elevate the contribution of this paper. In particular, I would be interested to see if there is an advantage to integrating the optimism into the evaluation point, rather than though a hinting mechanism (even if there is a negative result).\n\n**Conclusion**\nOverall I am leaning towards a borderline reject pending your response to the weaknesses/technical questions. I could move to accept/reject depending on your responses. I am particularly interested if you can clearly explain 1. Your contribution over and above the results from Wang and Abernethy (2018) and 2. How you will adjust your introduction/summary of previous work to highlight how connected your method is with Wang and Abernethy (2018).\n\nQUESTIONS:\n- My reading is that Theorem 4 implicitly requires unconstrained problem to ensure $ x_t $ is feasible and $ \\nabla f( x^* ) = 0 $ for the final inequality in proof. Is this the case? If so you should make this explicit in the statement of the the theorem. Also the proof could be tightened by using the fact that $|| \\nabla f(\\tilde{x}_1) || = || \\nabla f(\\tilde{x}_1)- \\nabla f( x^* )|| \\leq L|| \\tilde{x}_1 - x^* ||$. \n- Could you discuss why the change in the algorithm is necessary in Theorem 4?  \n- You should make the $t$ values (e.g., $t\\geq0$?) that (16)-(17) are applicable explicit and prove the edge cases hold (particularly for (17) which includes the $x_{t-2}$ term). Similarly, you should define the edge cases for $\\tilde{x}_0$, $\\bar{x}_0$, and $A_0$ etc.\n- Could you elaborate on how (12) is obtained? What should $\\beta_{t-1}$ be?\n- line 566 why is the expectation conditioned on $x$?\n- I think that Section 1.3 could benefit from being cut down into a few dot points which highlight the main contributions.\n- Could you explain the working after line 594?\n\nTypos:\n- line 162: $\\ell_t$ is a scalar, so this statement doesn't make sense.\n- line 346: worth should be worthy?\n- In the working after line 476 I think the final equality should be an inequality. \n- line 527: \"the second step uses Lemma 3\" should this be Lemma 2?\n- line 557: should $2G^2$ be $4G^2$ after the last inequality?\n- line 585: filtration should be adapted to $\\tilde{x}_i$s instead of $x_i$s?\n\nLIMITATIONS:\nSee strengths/weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new online-to-batch conversion method and analysis technique to achieve acceleration from online gradient descent with gradients weighted differently. In contrast, previous online-to-batch conversion achieved accelerated rates when using some *optimistic* online learning algorithms. The idea of the analysis is to, in some sense, incorporate optimism only in the analysis, not as a requirement on the OL method itself. With that they show accelerated rates in the smooth and smooth strongly convex cases. Moreover, using their online-to-batch techniques they also show how to get acceleration without knowing the smoothness constant beforehand when the domain is bounded, what is known as an *universal algorithm*.\n\nSTRENGTHS AND WEAKNESSES:\nI believe this is a very strong submission: clearly written, with an elegant main idea and with very nice results. Of course, I may be missing some of the context from the recent work on online-to-batch conversion that happened more recently, but this feels like a simple method with a clean analysis and very strong results. I really appreciated the thorough discussion that the authors do with respect to related work. I was familiar with the use of the Fenchel game from Abernethy et al., so the connection to this work was quite interesting to me. Beyond acceleration in both smooth and smooth + stongly convex case, the authors also show how to get acceleration in bounded domains without know the smoothness constants, which was a very neat addition. I did not have the time to check the proofs for correctness properly, but the ideas in this paper are quite interesting.\n\nThe one main thing I was a bit confused was whether this optimistic online-to-batch conversion is restricted to online gradient descent. Algorithm 1 is describe with a general OL method and I know some of the results end up using O-t-B conversion with other algorithms, but in the paper the optimistic O-t-B conversion is described explicitly using OGD. I did not have the time to check whether Theorem 5 actually depends on the algorithm being OGD and I feel it was a bit confusing how dependent on it this method is.\n\nQUESTIONS:\nMy main question is whether Algorithm 2 (and Theorem 5) depend on the use of online gradient descent, of whether we could use (and if it makes sense) to use other online learning algorithms.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper proposes a new method called optimistic conversion to leverage online learning analysis for offline convex optimization. While several conversion techniques have been developed in the past, achieving an $O(1/T^2)$ convergence rate for offline smooth convex optimization typically required analysis of optimistic online methods. In contrast, the proposed optimistic conversion enables the derivation of an offline algorithm equivalent to Nesterov's accelerated gradient method using only a simple, non-optimistic online gradient descent, while still attaining the $O(1/T^2)$ convergence. In the strongly convex case, linear convergence can also be achieved. Moreover, this conversion facilitates the development of universal methods that do not rely on smoothness, as well as stochastic variants.

STRENGTHS AND WEAKNESSES:
**Strengths**  
- The paper addresses online-to-batch conversion, a timely and actively studied topic in the field.  
- The idea of optimistic conversion is intriguing and has the potential to aid the design of algorithms with improved theoretical guarantees.  
- The paper provides a thorough review of prior work, making it a valuable introduction to online-to-batch conversion. It also clearly positions the current work within the context of existing literature and describes their relationships well.  
- The paper is overall well-written and easy to follow.  

**Weakness**  
- While the paper presents a new framework for algorithm design and analysis, it does not appear to lead to new algorithms that improve upon existing theoretical results at this stage.

QUESTIONS:
Do the universal algorithms presented in Section 3.3 achieve linear convergence, similar to Theorem 3, when applied to strongly convex and smooth functions?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-03