Title: Comparator-Adaptive $\Phi$-Regret: Improved Bounds, Simpler Algorithms, and Applications to Games
Venue: NeurIPS 2025
Accepted: yes
Keywords: Swap Regret

$\Phi$-Regret

Learning in Games
Abstract: In the classic expert problem, $\Phi$-regret measures the gap between the learner's total loss and that achieved by applying the best action transformation $\phi \in \Phi$. A recent work by Lu et al., [2025] introduced an adaptive algorithm whose regret against a comparator $\phi$ depends on a certain sparsity-based complexity measure of $\phi$, recovering and interpolating optimal bounds for standard regret notions such as external, internal, and swap regret. In this work, we propose a general idea to achieve an even better comparator-adaptive $\Phi$-regret bound via much simpler algorithms compared to Lu et al., [2025]. Specifically, we discover a prior distribution over all possible binary transformations and show that it suffices to achieve prior-dependent regret against these transformations. Then, we propose two concrete and efficient algorithms to achieve so, where the first one combines multiple copies of the kernelized Hedge algorithm of Farina et al., [2022], and the second one combines multiple copies of a variant of the BM-reduction [Blum and Mansour, 2007]. To further showcase the power of our methods and the advantages over Lu et al., [2025] besides the simplicity and better regret bounds, we also show that our second approach can be extended to the game setting to achieve accelerated and adaptive convergence rate to $\Phi$-equilibria for a class of general-sum games. When specified to the special case of correlated equilibria, our bound improves over the existing ones from Anagnostides et al., [2022a,b].


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers minimizing the $\\Phi$ regret in the expert problem, and propose new algorithms achieving certain regret bounds that adapt to a sparsity based complexity measure of each comparing $\\phi$ instance, improving the results from a recent work (Lu et al., 2025). Different from the existing algorithm using wavelet inspired matrix features, the proposed algorithms are based on the arguably simpler idea of expert aggregation, with the twists being the incorporation of a carefully designed prior distribution over all possible $\\phi$s, as well as the idea of kernelization from (Farina et al., 2022). Finally, a variant of the algorithm is proposed to achieve an adaptive version of \"fast rate\" for learning in games, improving (Anagnostides et al., 2022).\n\nSTRENGTHS AND WEAKNESSES:\nOverall, I think the paper has several notable strengths making it well above the acceptance threshold. \n\n1. The results are strong as it removes several suboptimalities from the prior work and tightens its complexity description for the vast majority of transformation rules that are non-binary. \n\n2. The ideas behind the algorithms are substantially different from that of Lu et al, bringing a novel, complementary perspective to the problem. In general, adaptivity in online learning is typically achieved by either aggregating different nonadaptive subroutines or the \"parameter-free\" type of approaches. Their connections and differences have provided interesting insights to the field, and the present work contributes to this thread as well. \n\n3. The application to the fast rates in games is a nontrivial addition. \n\n4. Writing is exceptionally clear despite the complicated technicalities. \n\nOn the flip side, I would suggest adding an in-depth discussion on the computational complexity. I understand the key motivation of this adaptive $\\Phi$-regret problem as providing a beyond-the-worst improvement over the classical Blum-Mansour algorithm, therefore BM could be seen as the natural baseline. There, given the fixed point computational oracle which is matrix multiplication time ($d^3$ in practice but $d^\\omega$ in theory), the essential online learning occurs in a $d^2$ dimensional space, therefore the fixed point oracle is the only computational bottleneck and there has been some attempts accelerating it (such as power method, discussed in Appendix I.3 by Zhang et al. 2024). (Lu et al, 2025) is computationally similar, but in the present work the essential online learning appears to take $d^3\\log d$ time per iteration for both of the two proposed algorithms, dominating the fixed point computation. This makes the comparison to BM and Lu et al a bit more subtle, and some clarification would be nice. \n\nZhang, Brian, et al. \"Efficient $\\Phi $-Regret Minimization with Low-Degree Swap Deviations in Extensive-Form Games.\" Advances in Neural Information Processing Systems 37 (2024): 125192-125230.\n\nQUESTIONS:\nIn the fast kernelized algorithm (Algorithm 6), I wonder if the computation of $\\phi_t$ can be further reduced to matrix multiplication, thus $d^\\omega$ time.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-27"}

comments: SUMMARY:
This paper investigates instance-dependent \Phi-regret bounds. Building on techniques from prior work, the authors propose a refined complexity measure for the comparator, develop two efficient algorithms, and derive improved regret bounds.

STRENGTHS AND WEAKNESSES:
The paper is well written: the discussion of existing work is fair, the motivations behind the algorithmic development are clearly articulated, and the limitations are also acknowledged.

The construction of the prior is interesting, but the paper lacks a clear motivation for introducing it, and it is unclear whether this prior (and the resulting c_\phi) is the best possible choice. While the results presented are fairly comprehensive, the paper would benefit from a more thorough justification of the complexity measure c_\phi. It indeed interpolates between external and swap regret, but it is unclear what precise middle ground it captures. Moreover, c_\phi is defined as the minimum of two complexities, which seems somewhat ad hoc. It would strengthen the paper to provide a justification showing that c_\phi is inherent: for example, demonstrating that for some reasonable class of comparators (lying between \Phi-external/internal and \Phi-best), c_\phi is in fact the best complexity measure achievable.

QUESTIONS:
See above

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-28

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel framework for comparator-adaptive $\\Phi$-regret, significantly improving upon the work of Lu et al. (2025). The authors propose two efficient algorithms—based on kernelized Multiplicative Weight Update (MWU) and a prior-aware BM-reduction—to achieve tighter regret bounds, removing unnecessary terms from the prior work.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- The paper achieves tighter regret bounds, matching the minimax-optimal bounds for standard regret notions (external, internal, swap).\n- The proposed kernelized MWU and BM-reduction approaches are computationally more efficient than prior methods.\n- The extension to games is notable, providing the first adaptive and accelerated $\\Phi$-equilibrium guarantee.\n\n**Weaknesses:**\n\n- Although the work is purely theoretical, lacking experimental results to demonstrate practical performance. Empirical evaluation on real-world datasets or game instances would strengthen the claims.\n- The $\\Phi$-equilibrium bound includes an $O(N^2\\log d)$ term (Theorem 6.4), which scales quadratically with the number of players $N$. This may limit applicability to large multi-agent systems.\n\nQUESTIONS:\n- Given the focus on upper bounds for comparator-adaptive $\\Phi$-regret, is it possible to provide a lower bound analysis to establish the optimality of the proposed bounds, especially in relation to the complexity measure $c\\_{\\phi}$?\n- Regarding the kernelized MWU algorithm (Section 4), the paper mentions an $O(d^3)$ complexity per iteration. How does this complexity impact the algorithm’s scalability for large $d$? Are there practical optimizations considered to handle high-dimensional scenarios?\n- Can the framework be extended to handle heterogeneous agents with different action spaces and loss functions, as encountered in real-world games, and if so, how would the current analysis need to be adapted to accommodate such heterogeneity?\n\nLIMITATIONS:\n- Game Assumptions: The nonnegative-social-external-regret assumption restricts the applicability to specific game classes, limiting generality.\n- Lack of Experiments: The absence of empirical results prevents validation of the algorithms’ efficiency and practical utility, especially compared to the method in Lu et al. (2025).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper builds upon the recent work of Lu et al. (2025), providing $\\Phi$-regret bounds that are based on a sparsity complexity measure $c_\\phi$ that recovers the almost optimal no-external, no-internal, and no-swap regret guarantees. $\\Phi$-regret is a well-established regret notion, where given a sequence of probabilities $p_1,\\ldots,p_T \\in \\Delta_d$, it measures the difference between the experienced cost and the cost of the best transformation $\\phi: \\Delta_d \\mapsto \\Delta_d$,\n$\\mathcal{R}(\\phi):=\\sum_{t=1}^T \\ell_t^\\top \\cdot p_t - \\ell_t^\\top \\cdot \\phi(p_t)$. Lu et al. (2025) provide an algorithm with $\\mathcal{R}(\\phi) := O\\left(\\sqrt{c_\\phi(T + d) \\log^3 d}\\right)$, while the authors provide an improved bound of $\\mathcal{R}(\\phi) := O\\left(\\sqrt{c_\\phi T \\log d}\\right)$ with a significantly simpler algorithm.\n\nThe basic technical idea of the authors is to introduce a prior probability distribution on the set of $0$–$1$ row-stochastic matrices and to establish $\\Phi$-regret bounds by running the famous MWU on this set. The authors then use MWU as a meta-algorithm in order to select the optimal step size. The authors additionally show that, despite dealing with an exponentially large action set, their algorithm can be efficiently implemented using kernelization ideas.\n\nIn Section $5$, the authors introduce an alternative approach based on the Blum–Mansour reduction. Finally, in Section 6, the authors show how to extend their algorithm so as to achieve $\\tilde{\\mathcal{O}}(1/T)$ for approximate $\\Phi$-equilibrium in a specific class of normal-form games.\n\nSTRENGTHS AND WEAKNESSES:\nI think the paper presents solid theoretical results that will be of interest to the online learning community at NeurIPS. Beyond the improved regret guarantees over those of Lu et al., the proposed algorithms are both simple and elegant. The presentation of the paper is very clear, and the technical ideas are well-explained.\n\nPerhaps the only result I found somewhat expected was the $\\tilde{O}(1/T)$ convergence rate to $\\Phi$-equilibrium, given the extensive recent literature on the subject. Despite the considered class has been previously considered to the literature, the assumption feels a bit artificial. At the same time, I believe that a similar result for general normal-form games would add value to the paper.\n\nQUESTIONS:\n1. What are the limitations of extending your results for general sum normal form games?\n\n2. Can you elaborate more on the class of games satisfying Assumption~6.3?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}