Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Non-stationary Online Learning for Curved Losses: Improved Dynamic Regret via Mixability

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A fixed-share exponential-weight method achieves near-optimal dynamic regret for mixable losses, reducing the dimension dependence from $d^{10/3}$ to $d$.

desk verdict The d-improvement for mixable losses is real and the analysis is clean; Theorem 3 has a localized, likely repairable gap for very large path length. read the letter →

arxiv 2506.10616 v1 pith:7Q7EWRRW submitted 2025-06-12 cs.LG

Yu-Jie Zhang , Peng Zhao , Masashi Sugiyama This is my paper · ORCID
classification cs.LG MSC 68Q3290C25
keywords dynamicregretmixabilityexp-concavelossfixed-shareonlineconvexoptimizationnon-stationarylearningpathlengthexponentialweights
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Dynamic regret compares an online learner against a time-varying benchmark, and for convex losses the achievable rate is controlled by the benchmark's path length $P_T$. This paper shows that when the loss is stronger than convex — specifically $\eta$-mixable, a curvature property generalizing exp-concavity — a continuous exponential-weight method with fixed-share updates attains $\mathcal{O}(d \log T\,(1 + T^{1/3} P_T^{2/3}))$ dynamic regret for any comparator sequence. The key gain is in the dimension: the best prior bound for exp-concave losses carried a $d^{10/3}$ factor, while this one scales only linearly in $d$. The result applies directly to squared loss, least-squares regression, and logistic regression, and extends via a projection step to proper learning for general exp-concave functions.

What carries the argument

The analysis is carried by the mix loss $m_t(P) = -\frac{1}{\eta}\ln \mathbb{E}_{u\sim P}[e^{-\eta f_t(u)}]$, which converts the exponential-weight update into an online prediction game. The dynamic regret is split into three terms: the mixability gap (non-positive by construction of the prediction $z_t$), the mixability regret against a time-varying Gaussian comparator $Q_t = \mathcal{N}(u_t, \sigma^2 I_d)$, and the comparator gap, controlled by $\beta$-smoothness. The fixed-share update (mixing in a Gaussian prior with weight $\mu = 1/T$) makes the analysis work over the continuous space and is equivalent to a follow-the-leading-history ensemble. The variance $\sigma$ is only an analysis parameter and is tuned as $\sigma = P_T^{1/3} T^{-1/3}$ to balance the mixability regret and the comparator gap.

What would settle it

Compute, for a smooth $\eta$-mixable loss on $\mathbb{R}$, the exact dynamic regret of Algorithm 1 when the comparator alternates with $P_T \asymp T^{1/3}$; the claimed rate is $\mathcal{O}(T^{5/9}\log T)$ up to dimension factors, so exceeding that by a polynomial in $T$ would refute Theorem 1.

Watch

Extended reading notes

Core claim

The paper establishes that mixability, not exp-concavity per se, is the property that enables fast dynamic regret for curved losses. Its central claim is Theorem 1: under $\beta$-smoothness and $\eta$-mixability over the whole real line, Algorithm 1 — fixed-share exponential weights over Gaussian distributions on model parameters — achieves $\mathcal{O}(d \log T (1 + T^{1/3} P_T^{2/3}))$ for any sequence of comparators with path length $P_T$, with no prior knowledge of $P_T$. The same rate, Theorem 3, holds for proper learning of general exp-concave losses on a bounded convex domain, using a quadratic surrogate loss and a KL projection onto a set of Gaussian mixtures. For least-squares and logistic regression this replaces the previous $\mathcal{O}(d^{10/3} T^{1/3} P_T^{2/3})$ bound by a bound linear in $d$, which is the paper's headline improvement.

Load-bearing premise

The load-bearing premise is that the loss is $\eta$-mixable over every point of the real line (Assumption 3) and that the functions are uniformly smooth (Assumption 2); if mixability holds only on a bounded domain, the mixability gap in the regret decomposition can become positive and the bound collapses.

Editorial extensions

If this is right

  • For least-squares and logistic regression, the new bound $\mathcal{O}(d\log T + d T^{1/3} P_T^{2/3}\log T)$ improves the dimension factor from $d^{10/3}$ to $d$, while matching the prior rate in $T$ and $P_T$ up to logs.
  • The method does not need advance knowledge of the path length $P_T$; the fixed-share parameter $\mu = 1/T$ and the analysis-only tuning of $\sigma$ are enough.
  • For the 1-dimensional squared loss, the result matches the best known bound under proper learning, with a closed-form implementation whose per-round cost is $\mathcal{O}(T)$.
  • Theorem 3 extends nearly optimal dynamic regret to proper learning for general exp-concave losses on arbitrary bounded convex domains, and it drops the smoothness assumption used in Theorem 1.
  • The mixability-based analysis avoids KKT-condition arguments, so the same decomposition can be reused for other choices of the distributions $P_t$ and reference comparators $Q_t$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition could likely be ported to other non-stationarity measures, such as squared path length or gradient-variation budgets, by choosing the reference distribution $Q_t$ to match the geometry; the paper hints at this flexibility but does not work it out.
  • The linear-$d$ dependence still leaves a gap to the lower bound's $d^{1/3}$ factor, so either the algorithm or the lower bound may be further improvable in dimension.
  • Because the logistic-loss implementation relies on sampling and the general OCO projection is only shown to exist, a natural testable extension is whether $\mathcal{O}(\log T)$ per-round updates, as achieved via geometric coverings in prior work, can be brought into the fixed-share framework.
  • The whole-line mixability assumption suggests the method may extend to unbounded domains and scale-free settings, but only if the fixed-share Gaussian prior is replaced by a heavier-tailed reset distribution.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a continuous fixed-share exponential-weights method for non-stationary online learning with mixable losses. Under smoothness and mixability, Theorem 1 gives dynamic regret O(d log T (1 + T^{1/3} P_T^{2/3})) for any comparator sequence, improving the prior O(d^{10/3} T^{1/3} P_T^{2/3}) dependence on dimension for least-squares and logistic regression. The proof decomposes dynamic regret into a mixability gap, a mixability regret, and a comparator gap, with the reference distributions Q_t = N(u_t, sigma^2 I_d) used only as an analysis device. The method is shown equivalent to a follow-the-leading-history ensemble, and closed-form updates are given for squared loss. Section 5 extends the result to general exp-concave OCO with proper learning by adding a KL projection onto a Gaussian-mixture set and a surrogate quadratic loss, yielding Theorem 3 with the same rate.

Significance. If the results are correct, the paper makes a substantial contribution: it removes the large O(d^{10/3}) dimension factor from the best-known dynamic regret for curved losses, matches the 1-dimensional squared-loss rate with a much simpler analysis, avoids KKT-based arguments, and is parameter-free with respect to the unknown path length P_T. The decomposition is self-contained, and the reference distributions Q_t are genuinely analysis devices rather than fitted parameters. The extension to proper learning for general exp-concave losses is conceptually interesting, though the projection step is existence-based and computationally non-trivial. The main caveat is that the proof of Theorem 3 has a gap when P_T > T, precisely in the regime where the claimed bound is weakest, and the proof of Theorem 1 contains an incorrect Gaussian-integral evaluation that needs repair.

major comments (3)
  1. [Appendix C.3, Theorem 3, and Lemma 8] The generalized Pythagorean inequality used to bound the mixability regret in Lemma 8 is valid only if the reference distribution Q_t belongs to the projection set M. For Q_t = N(u_t, sigma^2 I_d) this requires sigma^2 in [1/T, 1], since M constrains component covariances to have eigenvalues between 1/T and 1. Lemma 8 states no such restriction, and the proof of Theorem 3, Case 2, sets sigma = P_T^{1/3} T^{-1/3}; this satisfies sigma^2 <= 1 only when P_T <= T. Assumption 1 alone permits P_T > T (e.g., a comparator alternating between two points at distance D > 1). In that regime Q_t is not in M, so the inequality KL(Q_t || P') >= KL(Q_t || tilde P) used in Lemma 8 is not available and the stated for-all-P_T bound is unproved. The gap is likely repairable by choosing sigma = 1 when P_T > T and accepting a different O(d(PT + T) log T) rate, or by stating the theorem with the condition P_T <= T, but as written the proof is incomplete.
  2. [Appendix A.1, Lemma 2 and Eq. (21)] The Gaussian integral after Eq. (21) is evaluated as (2 pi / (eta beta t))^{d/2}, but the integrand has exponent -(1 + eta beta t)/2 * ||u - v*||^2, so the correct value is (2 pi / (1 + eta beta t))^{d/2}. Consequently, the stated bound ln P_t(u) <= (d/2) ln(eta beta t / (2 pi)) does not follow and can even be negative for small eta beta t, although the logarithm on the left is positive on the region W_t^(1) where it is used. This computation feeds directly into Eq. (32) and Eq. (34) for term (B-I) in the proof of Theorem 1, so it must be corrected; the correct factor with 1 + eta beta t preserves the argument up to constants and an additional lower-order condition on eta beta T.
  3. [Theorem 3, Algorithm 3 projection step] The projection step (16) is defined as an information projection onto the Gaussian-mixture set M, and Lemma 5 proves existence via total-variation closedness. However, no algorithm, finite-time convergence statement, or computational complexity is provided for computing this projection, while the corresponding update is essential to the definition of Algorithm 3. The paper states that the projection introduces 'non-trivial computational challenges' and leaves this open, which is acceptable as a caveat, but the claim that Algorithm 3 'attains' the regret bound should make clear that this is an existence-level algorithmic guarantee rather than a computationally implemented procedure.
minor comments (6)
  1. [Assumption 3] Assumption 3 states that the loss ell(z,y) is eta-mixable over z in R^d, but in the online prediction setting of Section 3 the prediction z_t is a scalar in R; this should be z in R (or the notation should be aligned with the per-round prediction z_t).
  2. [Appendix B.4, proof of Corollary 2] The final sentence of the proof says the method ensures an O((ln T)^{2/3} P_T^{2/3} T^{1/3}) bound, but the corollary states eO(d + d T^{1/3} P_T^{2/3}); the displayed bound omits the dimension factor and should be corrected.
  3. [Theorem 2 proof] In the display for eP_flh_{t+1}(u), the first occurrence of exp(-gamma f_t(u)) should be exp(-eta f_t(u)); the two following occurrences correctly use eta. This appears to be a typographical error in an algebraic identity.
  4. [Appendix B.2, Lemma 4 proof] The proof of mixability over R uses both Y and B for the same label bound, and the shifted distribution P' is described with Y in one place and B in the next; the notation should be made consistent.
  5. [Appendix A.1, Lemma 3] In the bound for TERM (B), the first inequality gives 2D^2/lambda^2 * ||P - Q||_1, but the text later writes D/lambda^2 * ||P - Q||_1; the D^2 dependence is the correct one and the constant should be adjusted consistently.
  6. [Section 5, surrogate loss] The constants in Lemma 11 and the coefficient gamma = min{1/(4GD), eta} should be checked for consistency: Lemma 11 is quoted with gamma <= min{1/(8GD), eta/2}, while the surrogate loss uses gamma/2 in the quadratic term, and the proof invokes Lemma 12 with the condition eta <= 1/(5GD). The current presentation mixes two different normalizations and would benefit from a single consistent statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dynamic regret bound is derived from a self-contained decomposition, the reference distributions are analysis-only devices, algorithm parameters are fixed, and prior self-citations are contextual rather than load-bearing.

full rationale

The paper's derivation chain is self-contained, and no prediction reduces to its own input by construction. Theorem 1 is obtained from the three-term decomposition in Eq. (7)/(30): the mixability gap is non-positive by the defining property of the prediction rule (4), which is an algorithmic guarantee rather than a fitted value; the mixability regret is controlled by Lemma 1 via telescoping KL decompositions with the fixed-share update; the comparator gap uses beta-smoothness. The reference distributions Qt = N(ut, sigma^2 I_d) appear only in the analysis; sigma is tuned solely to minimize the final expression (sigma = T^{-1/2} for PT <= T^{-1/2}, sigma = PT^{1/3} T^{-1/3} otherwise). This is standard bound optimization, not estimating a parameter from a data subset and then 'predicting' a related quantity. The fixed-share parameter mu = 1/T is fixed, and no grid search over PT is converted into a claimed guarantee. Prior results are used as external benchmarks: the O(d^{10/3} T^{1/3} PT^{2/3}) upper bounds and the lower bound cited for comparison come from Baby & Wang (2021; 2022a;b) and Baby et al. (2023), while the mixability and exp-concavity machinery (Vovk 2001; Cesa-Bianchi & Lugosi 2006; Hazan 2016) is classical. The authors' own prior work (Zhang et al. 2023a) is cited only as context and is not load-bearing. Theorem 2 (equivalence of Algorithms 1 and 2) is proved by a direct recursion. No uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation: the Gaussian-mixture set M (Eq. 19) and the surrogate loss (Eq. 18) are introduced and analyzed in the paper, with existence of the projection supported by Csiszar (1975), an external source. One genuine concern in the proof of Theorem 3 is a correctness gap, not circularity: Lemma 8's Pythagorean inequality KL(Qt || P') >= KL(Qt || P~) requires Qt in M, which needs sigma^2 <= 1, but Case 2 of the proof sets sigma = PT^{1/3} T^{-1/3}, which can exceed 1 when PT > T under Assumption 1 (e.g., diameter D > 1). This leaves the stated for-all-PT bound unproved in that regime; the repair (choosing sigma = PT^{1/3}(DT)^{-1/3}, which is at most 1 since PT <= D(T-1)) is not written down. This is flagged under correctness risk, not circularity, per the separation rule. Verdict: no significant circularity; score 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No physical or empirical parameters are fitted; sigma is an analysis parameter and is not used by the algorithms. The main assumptions are smoothness and mixability for online prediction, and exp-concavity and bounded gradient for general OCO. The projection set M is a mathematical construction, not an invented entity.

free parameters (1)
  • sigma (reference Gaussian variance in proofs) = Theta(P_T^{1/3} T^{-1/3}) when P_T >= T^{-1/2}, else Theta(T^{-1/2})
    Chosen in the analysis, not in the algorithm, to balance the d P_T / sigma and d sigma^2 T terms; it never appears in Algorithm 1 or Algorithm 3 updates.
assumptions (5)
  • domain assumption The online loss f_t is beta-smooth over all of R^d (Assumption 2), and the base loss ell is eta-mixable over the prediction line (Assumption 3).
    Smoothness controls the comparator gap in Eq. (8); mixability makes the mixability gap non-positive. Both appear at Theorem 1, Section 3.3. The text writes 'over z in R^d' in Assumption 3 but the predictions z_t are scalar, so this is a typo to fix.
  • domain assumption For general OCO, f_t is eta-exp-concave over W and has gradients bounded by G (Assumptions 4 and 5).
    Used to construct the quadratic surrogate loss (18) so that the original regret is bounded by surrogate regret, citing Hazan Lemma 4.2, and to apply Gaussian exp-concavity in Lemma 7.
  • standard math The KL projection onto the Gaussian-mixture set M exists and satisfies the Pythagorean inequality for every Q_t in M (Csiszar 1975; Csiszar and Matus 2003).
    Invoked in Lemma 8, Appendix C.2; the set M is closed in total variation by Lemma 5, so the projection exists. Requires Q_t in M, which imposes 1/T <= sigma^2 <= 1.
  • standard math The greedy forecaster constructed from a mixable loss has non-positive mixability gap (Cesa-Bianchi and Lugosi 2006, Proposition 3.3).
    Used in Corollaries 1 and 2 to justify the closed-form predictions (12) and (13).
  • standard math Each component Gaussian satisfies the exp-concavity inequality of van Erven and Koolen (2016, Lemma 10) under gamma <= 1/(5 G D).
    Used in Lemma 7 for Gaussian mixtures; the paper cites this external lemma rather than proving it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-stationary Online Learning for Curved Losses: Improved Dynamic Regret via Mixability." pith.science (2026). https://pith.science/paper/7Q7EWRRW

@misc{pith2026250610616,
  author       = {Pith},
  title        = {Pith review of: Non-stationary Online Learning for Curved Losses: Improved Dynamic Regret via Mixability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Q7EWRRW}},
  note         = {Machine review of arXiv:2506.10616}
}
abstract

Non-stationary online learning has drawn much attention in recent years. Despite considerable progress, dynamic regret minimization has primarily focused on convex functions, leaving the functions with stronger curvature (e.g., squared or logistic loss) underexplored. In this work, we address this gap by showing that the regret can be substantially improved by leveraging the concept of mixability, a property that generalizes exp-concavity to effectively capture loss curvature. Let $d$ denote the dimensionality and $P_T$ the path length of comparators that reflects the environmental non-stationarity. We demonstrate that an exponential-weight method with fixed-share updates achieves an $\mathcal{O}(d T^{1/3} P_T^{2/3} \log T)$ dynamic regret for mixable losses, improving upon the best-known $\mathcal{O}(d^{10/3} T^{1/3} P_T^{2/3} \log T)$ result (Baby and Wang, 2021) in $d$. More importantly, this improvement arises from a simple yet powerful analytical framework that exploits the mixability, which avoids the Karush-Kuhn-Tucker-based analysis required by existing work.

Figures

Figures reproduced from arXiv: 2506.10616 by the authors.

Figure 1
Figure 1. An illustration of Algorithm 2, an alternative implemen￾tation of the fixed share method. The following theorem establishes the equivalence between Algorithm 1 and Algorithm 2. Theorem 2. The sequence of distributions {Pt} T t=1 re￾turned by the fixed-share algorithm (Algorithm 1) is iden￾tical to that of the FLH-type algorithm (Algorithm 2) with the same sequence of input loss functions {ft} T t=1 and the same para… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Adaptive Bayesian Online Learning via Expert Aggregation

    stat.ML 2026-07 conditional novelty 6.0 of 10

    Aggregating Bayesian experts by predictive loss gives fast regret bounds, long-run randomized conformal coverage, and minimax-adaptive Gaussian process regression.

Reference graph

Works this paper leans on

41 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    M., Chernov, A

    Adamskiy, D., Koolen, W. M., Chernov, A. V., and Vovk, V. A closer look at adaptive regret. Journal of Machine Learning Research, 17: 0 23:1--23:21, 2016

    work page 2016

  3. [3]

    and Wang, Y.-X

    Baby, D. and Wang, Y.-X. Optimal dynamic regret in exp-concave online learning. In Proceedings of the 34th Conference on Learning Theory (COLT), pp.\ 359--409, 2021

    work page 2021

  4. [4]

    and Wang, Y.-X

    Baby, D. and Wang, Y.-X. Optimal dynamic regret in proper online learning with strongly convex losses and beyond. In Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS), pp.\ 1805--1845, 2022 a

    work page 2022

  5. [5]

    and Wang, Y.-X

    Baby, D. and Wang, Y.-X. Optimal dynamic regret in LQR control. In Advances in Neural Information Processing Systems 35 (NeurIPS), pp.\ 24879--24892, 2022 b

    work page 2022

  6. [6]

    Non-stationary contextual pricing with safety constraints

    Baby, D., Xu, J., and Wang, Y.-X. Non-stationary contextual pricing with safety constraints. Transactions on Machine Learning Research, 2023, 2023

    work page 2023

  7. [7]

    Besbes, O., Gur, Y., and Zeevi, A. J. Non-stationary stochastic optimization. Operations Research, 63 0 (5): 0 1227--1244, 2015

    work page 2015

  8. [8]

    Bubeck, S., Eldan, R., and Lee, Y. T. Kernel-based methods for bandit convex optimization. Journal of the ACM, 68 0 (4), 2021

    work page 2021

Show all 41 references
  1. [9]

    and Lugosi, G

    Cesa-Bianchi, N. and Lugosi, G. Prediction, L earning, and G ames . Cambridge U niversity P ress, 2006

    2006

  2. [10]

    A new look at shifting regret

    Cesa - Bianchi, N., Gaillard, P., Lugosi, G., and Stoltz, G. A new look at shifting regret. ArXiv preprint, arXiv:1202.3323, 2012 a

    2012 arXiv

  3. [11]

    Mirror descent meets fixed share (and feels no regret)

    Cesa - Bianchi, N., Gaillard, P., Lugosi, G., and Stoltz, G. Mirror descent meets fixed share (and feels no regret). In Advances in Neural Information Processing Systems 25 (NIPS), pp.\ 989--997, 2012 b

    2012

  4. [12]

    I-divergence geometry of probability distributions and minimization problems

    Csisz \'a r, I. I-divergence geometry of probability distributions and minimization problems. The Annals of Probability, pp.\ 146--158, 1975

    1975

  5. [13]

    and Matus, F

    Csisz \'a r, I. and Matus, F. Information projections revisited. IEEE Transactions on Information Theory, 49 0 (6): 0 1474--1490, 2003

    2003

  6. [14]

    Parameter-free, dynamic, and strongly-adaptive online learning

    Cutkosky, A. Parameter-free, dynamic, and strongly-adaptive online learning. In Proceedings of the 37th International Conference on Machine Learning (ICML), pp.\ 2250--2259, 2020

    2020

  7. [15]

    J., Kale, S., Luo, H., Mohri, M., and Sridharan, K

    Foster, D. J., Kale, S., Luo, H., Mohri, M., and Sridharan, K. Logistic regression: The importance of being improper. In Proceedings of the 31st Conference on Learning Theory (COLT), pp.\ 167--208, 2018

    2018

  8. [16]

    Introduction to O nline C onvex O ptimization

    Hazan, E. Introduction to O nline C onvex O ptimization. Foundations and Trends in Optimization, 2 0 (3-4): 0 157--325, 2016

    2016

  9. [17]

    and Seshadhri, C

    Hazan, E. and Seshadhri, C. Efficient learning algorithms for changing environments. In Proceedings of the 26th International Conference on Machine Learning (ICML), pp.\ 393--400, 2009

    2009

  10. [18]

    Logarithmic regret algorithms for online convex optimization

    Hazan, E., Agarwal, A., and Kale, S. Logarithmic regret algorithms for online convex optimization. Machine Learning, 69 0 (2-3): 0 169--192, 2007

    2007

  11. [19]

    Information Theory for Continuous Systems

    Ihara, S. Information Theory for Continuous Systems. World Scientific, 1993

    1993

  12. [20]

    An optimal algorithm for bandit convex optimization with strongly-convex and smooth loss

    Ito, S. An optimal algorithm for bandit convex optimization with strongly-convex and smooth loss. In Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), pp.\ 2229--2239, 2020

    2020

  13. [21]

    and Cutkosky, A

    Jacobsen, A. and Cutkosky, A. Unconstrained online learning with unbounded losses. In Proceedings of the 40th International Conference on Machine Learning (ICML), pp.\ 14590--14630, 2023

    2023

  14. [22]

    Mixability made efficient: Fast online multiclass logistic regression

    J \' e z \' e quel, R., Gaillard, P., and Rudi, A. Mixability made efficient: Fast online multiclass logistic regression. In Advances in Neural Information Processing Systems 34 (NeurIPS), pp.\ 23692--23702, 2021

    2021

  15. [23]

    Near-optimal dynamic regret for adversarial linear mixture mdps

    Li, L.-F., Zhao, P., and Zhou, Z.-H. Near-optimal dynamic regret for adversarial linear mixture mdps. In Advances in Neural Information Processing Systems 37 (NeurIPS), pp.\ 55858--55883, 2024

    2024

  16. [24]

    J., Hadiji, H., and van Erven, T

    Mayo, J. J., Hadiji, H., and van Erven, T. Scale-free unconstrained online learning for curved losses. In Proceedings of the 35th Conference on Learning Theory (COLT), pp.\ 4464--4497, 2022

    2022

  17. [25]

    Online learning via sequential complexities

    Rakhlin, A., Sridharan, K., and Tewari, A. Online learning via sequential complexities. Journal of Machine Learning Research, 16: 0 155--186, 2015

    2015

  18. [26]

    and Ben-David, S

    Shalev-Shwartz, S. and Ben-David, S. Understanding M achine L earning: F rom T heory to A lgorithms . Cambridge U niversity P ress, 2014

    2014

  19. [27]

    The many faces of exponential weights in online learning

    van der Hoeven, D., van Erven, T., and Kotlowski, W. The many faces of exponential weights in online learning. In Proceedings of the 31st Conference on Learning Theory (COLT), pp.\ 2067--2092, 2018

    2018

  20. [28]

    and Koolen, W

    van Erven, T. and Koolen, W. M. Metagrad: Multiple learning rates in online learning. In Advances in Neural Information Processing Systems 29 (NIPS), pp.\ 3666--3674, 2016

    2016

  21. [29]

    D., and Williamson, R

    van Erven, T., Gr \" u nwald, P., Reid, M. D., and Williamson, R. C. Mixability in statistical learning. In Advances in Neural Information Processing Systems 25 (NeurIPS), pp.\ 1691--1699, 2012

    2012

  22. [30]

    D., Mehta, N

    van Erven, T., Gr \" u nwald, P. D., Mehta, N. A., Reid, M. D., and Williamson, R. C. Fast rates in statistical and online learning. Journal of Machine Learning Research, 16: 0 1793--1861, 2015

    2015

  23. [31]

    A game of prediction with expert advice

    Vovk, V. A game of prediction with expert advice. Journal of Computer and System Sciences, 56 0 (2): 0 153--173, 1998

    1998

  24. [32]

    Competitive on-line statistics

    Vovk, V. Competitive on-line statistics. International Statistical Review, 69 0 (2): 0 213--248, 2001

    2001

  25. [33]

    and Zhdanov, F

    Vovk, V. and Zhdanov, F. Prediction with expert advice for the brier game. In Proceedings of the 25th International Conference on Machine Learning (ICML), pp.\ 1104--1111, 2008

    2008

  26. [34]

    and Luo, H

    Wei, C.-Y. and Luo, H. Non-stationary reinforcement learning without prior knowledge: An optimal black-box approach. In Proceedings of the 34th Conference on Learning Theory (COLT), pp.\ 4300--4354, 2021

    2021

  27. [35]

    Adaptive online learning in dynamic environments

    Zhang, L., Lu, S., and Zhou, Z.-H. Adaptive online learning in dynamic environments. In Advances in Neural Information Processing Systems 31 (NeurIPS), pp.\ 1330--1340, 2018

    2018

  28. [36]

    Adapting to continuous covariate shift via online density ratio estimation

    Zhang, Y.-J., Zhang, Z.-Y., Zhao, P., and Sugiyama, M. Adapting to continuous covariate shift via online density ratio estimation. In Advances in Neural Information Processing Systems 36 (NeurIPS), pp.\ 29074--29113, 2023 a

    2023

  29. [37]

    Unconstrained dynamic regret via sparse coding

    Zhang, Z., Cutkosky, A., and Paschalidis, Y. Unconstrained dynamic regret via sparse coding. In Advances in Neural Information Processing Systems 36 (NeurIPS), pp.\ 3477--3486, 2023 b

    2023

  30. [38]

    Dynamic regret of convex and smooth functions

    Zhao, P., Zhang, Y.-J., Zhang, L., and Zhou, Z.-H. Dynamic regret of convex and smooth functions. In Advances in Neural Information Processing Systems 33 (NeurIPS), pp.\ 12510--12520, 2020

    2020

  31. [39]

    Efficient methods for non-stationary online learning

    Zhao, P., Xie, Y.-F., Zhang, L., and Zhou, Z.-H. Efficient methods for non-stationary online learning. In Advances in Neural Information Processing Systems 35 (NeurIPS), pp.\ 11573--11585, 2022

    2022

  32. [40]

    Adaptivity and non-stationarity: Problem-dependent dynamic regret for online convex optimization

    Zhao, P., Zhang, Y.-J., Zhang, L., and Zhou, Z.-H. Adaptivity and non-stationarity: Problem-dependent dynamic regret for online convex optimization. Journal of Machine Learning Research, 25 0 (98): 0 1 -- 52, 2024

    2024

  33. [41]

    Online convex programming and generalized infinitesimal gradient ascent

    Zinkevich, M. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on Machine Learning (ICML), pp.\ 928--936, 2003

    2003

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.