Pith. sign in

REVIEW 4 major objections 4 minor 47 references

Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Gradient-free manifold federated learning matches first-order rate

desk verdict New estimator idea with a real rate-theorem flaw; worth a serious referee but needs major revision. read the letter →

arxiv 2507.22855 v2 pith:R6ZBRONW submitted 2025-07-30 math.OC cs.LG

Hongye Wang , Zhaoye Pan , Chang He , Jiaxiang Li , Bo Jiang This is my paper
classification math.OCcs.LG MSC 90C2690C1565K10
keywords federatedlearningRiemannianmanifoldszeroth-orderoptimizationprojectionoperatorgradient-freeestimationnonconvexlinearspeedupproximalsmoothness
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

This paper asks whether federated learning can run on constrained model spaces such as spheres, Stiefel manifolds, and low-rank matrix manifolds when clients cannot compute gradients and only receive noisy function values. It proposes a zeroth-order projection-based Riemannian federated algorithm whose local gradient estimates are formed from ordinary Euclidean random vectors followed by projection onto the manifold, avoiding tangent-space sampling, exponential maps, and parallel transport. The central theoretical claim is that, under smoothness and bounded-noise assumptions, the algorithm converges at rate $O(1/(\sqrt{n}\,\tau K) + 1/(n\tau m))$ in squared first-order optimality, matching the first-order projection-based Riemannian federated learning rate and exhibiting linear speedup in the number of clients $n$. If correct, this widens federated learning to black-box scenarios such as adversarial attacks on neural networks and low-rank network training, where only function evaluations are available but parameters must stay on a manifold.

What carries the argument

The central object is the projection-based zeroth-order Riemannian gradient estimator $G^P_\mu(x)$, whose perturbed point $x+\mu u_j$ leaves the manifold but is mapped back by $P_{\mathcal M}$. Its mechanism is proximal smoothness: within a $2\gamma$-tube around the manifold the projection is unique, its nonexpansiveness gives $\|P_{\mathcal M}(x)-P_{\mathcal M}(y)\|\le 2\|x-y\|$, and the normal-space component is controlled by $\langle v,y-x\rangle \le \frac{\|v\|}{4\gamma}\|y-x\|^2$ for $v\in N_x\mathcal M$ and $y\in\mathcal M$. These inequalities let Euclidean random perturbations stand in for tangent-space perturbations, eliminating tangent-vector sampling. The estimator feeds local updates with drift-correction terms, the server update $x^{k+1}=P_{\mathcal M}(x^k)+\eta_g\big((1/n)\sum_i \hat z_i^{k,\tau}-P_{\mathcal M}(x^k)\big)$ uses only projection, and the proof tracks a potential involving $f(P_{\mathcal M}(x^k))-f^*$ plus correction-term deviations, measuring optimality by $G_{\tilde\eta}(P_{\mathcal M}(x^k))$, the displacement of one centralized projected gradient step.

What would settle it

Inspect the sphere-constrained attack objective from Section 5.2: locate a ReLU decision boundary of the pretrained CNN inside the unit ball and evaluate the objective's Euclidean gradient on two sides of that boundary at distance $\delta$; if the gradient jump stays constant as $\delta\to 0$, then no finite $l_i$ satisfies Assumption 2.3, and Theorem 4.1's rate is not guaranteed for that experiment.

Watch

Extended reading notes

Core claim

The authors' central claim, stated on their own terms, is that the estimator $G^P_\mu(x) = \frac{\sqrt{pr}}{m}\sum_{j=1}^m \frac{F(P_{\mathcal M}(x+\mu u_j),\xi_j)-F(x,\xi_j)}{\mu}u_j$, with $u_j$ uniform on the unit sphere in the ambient Euclidean space and $P_{\mathcal M}$ the projection onto a compact smooth submanifold, is enough to replace tangent-space zeroth-order estimators. Using $2\gamma$-proximal smoothness of the manifold, Lemmas 3.1 through 3.4 establish that the estimator is bounded, has bias $O(\chi_f\mu)$, and has second moment $O(\chi_1\mu^2 + \chi_2/m)$. Algorithm 1 integrates these estimators with client-drift correction terms and a server-side projection average; Theorem 4.1 bounds the average squared optimality metric by $8\Omega_1/(\sqrt{n}\eta\tau K) + 64/(n\tau)(\chi_1\mu^2 + \chi_2/m) + 16(3+n)(\chi_G+\chi)\chi_f\mu/n$. With the smoothing parameter set to $\mu = O(1/(pr\,n\tau K))$, this is $O(1/(\sqrt{n}\,\tau K) + 1/(n\tau m))$, the same sublinear rate as the first-order counterpart, with linear speedup in the number of clients and faster convergence with more local updates.

Load-bearing premise

The load-bearing premise is that every noisy client function is smooth in the whole convex hull of the manifold in the ordinary Euclidean sense and has a uniformly bounded Euclidean gradient on the manifold; the projection estimator's bias, variance, and the whole convergence proof collapse without it.

Editorial extensions

If this is right

  • Federated clients that can only evaluate a noisy function, with no gradient information, can still train models on spheres, Stiefel manifolds, and low-rank manifolds at the same asymptotic sublinear rate as gradient-based Riemannian federated learning.
  • The server and clients need only a projection operator, so exponential maps, inverse exponential maps, and parallel transport are not required for the consensus step.
  • Because the estimator uses ordinary Euclidean random vectors, it avoids tangent-space sampling, which on the Stiefel manifold costs $O(np^2)$ per sample.
  • Convergence improves as the number of clients $n$ grows and as the number of local updates $\tau$ grows, so the algorithm inherits the linear-speedup property of first-order federated learning.
  • The method is demonstrated on black-box adversarial attacks with sphere-constrained perturbations and on low-rank neural network training with rank-constrained hidden layers, suggesting it supports communication-light constrained training.

Reading between the lines

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

  • The paper's assumptions are the main scope limit: Assumption 2.3 requires Euclidean smoothness on the whole convex hull of the manifold and uniformly bounded Euclidean gradients, which is not verified for ReLU-based networks; confirming or relaxing this requirement for nonsmooth activations is a natural next step.
  • The step-size condition depends inversely on $\chi_G = O(pr)$ and the bias is $O(\mu)$ with $\mu$ set to $O(1/(pr\,n\tau K))$, so high-dimensional ambient spaces shrink the allowed step size and smoothing radius; the big-O in $n,\tau,K$ hides this dimension cost in wall-clock time.
  • The estimator lemmas are client-independent, so the same projection-based estimator could be plugged into centralized or decentralized zeroth-order Riemannian optimization, not only federated setups.
  • The convergence measure $G_{\tilde\eta}(P_{\mathcal M}(x^k))$ vanishes exactly when the Riemannian gradient vanishes, so practitioners could monitor it directly during training as a proxy for first-order optimality.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a zeroth-order, projection-based federated learning algorithm on Riemannian submanifolds. The estimator (7) uses Euclidean random perturbations on the unit sphere followed by projection onto the manifold, avoiding tangent-space sampling and geometric operators such as exponential maps and parallel transport. The authors prove approximation properties of the estimator (Lemmas 3.2--3.4) and state a sublinear convergence bound for the full federated algorithm (Theorem 4.1), claiming a rate matching the first-order counterpart plus linear speedup in the number of clients. Experiments on kPCA, adversarial attacks on a CNN, and low-rank neural network training are presented as validation.

Significance. If established, the estimator is a clean and computationally attractive idea: it reduces zeroth-order Riemannian optimization to Euclidean perturbations plus projection, and the three lemmas in Section 3 provide a plausible toolkit for such estimators. The federated component with drift correction is also a natural extension of existing projection-based Riemannian FL. However, the central quantitative claim of the paper, the O(1/(√n τ K)) rate and the accompanying linear-speedup statement, is not supported by the theorem's own displayed bound, and the proof of Theorem 4.1 omits the decisive algebra. The paper therefore currently overstates its main deliverable.

major comments (4)
  1. [§4.2, Theorem 4.1] The displayed rate O(1/(√n τ K)) does not follow from the theorem's own bound. Since η_g = √n and η̃ = η_g η τ, the first error term is 8Ω_1/(√n η τ K) = 8Ω_1/(η̃ K). The theorem imposes η̃ ≤ min{1/(24ML), γ/(6 max{χ_G,χ}), 1/(χ L_P)}, a quantity independent of n and K, so this term is at least (8Ω_1/C)/K under the best allowed step-size choice. Substituting μ = O(1/(pr n τ K)) into the remaining terms gives a bound of order O(1/K + 1/(nτ m)), not O(1/(√n τ K) + 1/(nτ m)). This is load-bearing: the abstract, §1.2, and Remark 4.1 claim a first-order-matching rate and linear speedup, and these claims rest on the rate that the stated bound does not supply.
  2. [§C.1, Eq. (38)] The proof of Theorem 4.1 skips the exact step that produces the optimality measure: after combining equations (31) and (19), the paper states 'lengthy and nonessential algebraic manipulations are omitted' and jumps to (38). Equation (38) is where the coefficients of the potential, the ‖G_η̃(P_M(x^k))‖² term, and the 1/(n η̃) normalization must all match, so the omitted algebra is essential to the theorem. The full derivation and all constant constraints need to be supplied.
  3. [§C.2/Eq. (29)] Equation (29) appears to mishandle the n-dependence of the drift/Λ term. Summing the per-client inequality (28) over i should produce a sum over i of E‖Λ_i^k − Λ^k‖² (equivalently, an explicit factor n if the notation means the summed quantity). The displayed factor 9τ carries no n. Since Lemma C.2 and Lemma C.3 substitute (29) and the potential (15) divides the Λ-term by n, this factor propagates into the telescoping and could break the coefficient condition needed for the potential to decrease. This must be corrected and re-verified.
  4. [§5.2--5.3, Assumptions 2.1 and 2.3] The numerical applications fall outside the theorem's assumptions as stated. The low-rank manifold M = {x: rank(x) = R} in §5.3 is not compact, whereas Assumption 2.1 and the standing setup in §2 require a compact 2γ-proximally smooth submanifold. In addition, ReLU network objectives in §5.2 and §5.3 are not l-smooth over conv(M) with uniformly bounded Euclidean gradients, so Assumption 2.3 is not verified. If these experiments are meant to validate Theorem 4.1, the assumptions must be broadened with proof, or the experiments should be described as heuristic demonstrations outside the theorem's scope.
minor comments (4)
  1. [Lemma 3.4] In the proof of Lemma 3.4, χ_2 is defined to contain 2L²M⁴p²r²μ², so χ_2 depends on μ and is not a constant independent of the smoothing parameter. The statement should either exclude this μ-dependent term from χ_2 or state that the constants are allowed to depend on μ.
  2. [Algorithm 1 and Lemma C.1] The index ranges in Lemma C.1's equation (18) are inconsistent with the correction term update in Algorithm 1: the sums run from t=1 to τ−1 in (18), while the text immediately after includes t=0 and the algorithm's c^{k+1}_i averages over t=0, ..., τ−1. Please harmonize the indexing.
  3. [Table 1] The potential Ω^k in (15) is a scalar, but Table 1 lists its dimension as R^{np×r}; the entry should be R.
  4. [Throughout] There are several typographical errors, including 'rouds' and 'cilents' in Algorithm 1, 'CIF AR-10' instead of 'CIFAR-10' in §5.2, and a broken 'comp' fragment in the reader-facing text that should be corrected for publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.1's bound is derived from explicit assumptions via proved estimator and descent lemmas; the misstated rate in its closing line is a rigor gap, not a circular reduction.

full rationale

The derivation is self-contained: Assumptions 2.1–2.3 (proximal smoothness, unbiased bounded-variance oracle, Euclidean smoothness and bounded gradients on conv(M)) feed Lemma 3.1, whose proof converts them into Riemannian gradient Lipschitz and normal-space bounds using inequality (4), and Lemmas 3.2–3.4 prove (not assume) uniform boundedness, bias ‖E G^P_μ − gradf‖ ≤ χ_f μ, and variance ≤ χ_1μ² + χ_2/m. Lemma 4.1 is a descent lemma for projected Euclidean updates, Lemmas C.2–C.3 bound drift and one-round descent using only Assumptions 2.1–2.3 and Lemmas 3.1–3.4, and Theorem 4.1 telescopes the potential Ω^k of (15) via (38)–(39). No fitted parameter is renamed as a prediction, and no load-bearing premise rests on a self-citation: the algorithm skeleton, the optimality metric G̃_η, and the validity claim ‖G̃_η(P_M(x^k))‖ = 0 iff ‖gradf(P_M(x^k))‖ = 0 are credited to external Zhang et al. (2024a,b), and the only author-overlapping citation (He et al. 2024) appears in related-work comparisons about tangent-space sampling cost, not in the proof. Flagged for correctness, not circularity: the printed rate O(1/(√nτK)) in Theorem 4.1 does not follow from the displayed bound, because η̃ = √nητ ≤ min{1/(24ML), γ/(6 max{χ_G, χ}), 1/(χL_P)} is a constant, so 8Ω_1/(√nητK) = 8Ω_1/(η̃K) ≥ (8Ω_1/C)/K, i.e., the first term is Θ(1/K), yielding O(1/K + 1/(nτm)) after μ = O(1/(pr nτK)); this internal inconsistency, together with two omitted passages (Lemma C.1 deferred to Zhang et al. (2024a), and the “lengthy and nonessential algebraic manipulations” after eq. (38) in Section C.1), weakens the strongest quantitative claim as a matter of rigor, but the target rate is nowhere assumed as an input, so the derivation is not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard manifold and proximal-smoothness assumptions, an oracle noise model, and a smoothness assumption on conv(M). No free parameter is fitted to data; algorithmic hyperparameters such as mu, m, tau, eta, and eta_g are tunable but are not used to force the target rate. No invented entities are introduced.

assumptions (4)
  • domain assumption The manifold M is 2-gamma-proximally smooth (Assumption 2.1), so projection is unique in the 2-gamma tube and satisfies the Lipschitz and normal-residual inequalities (3) through (5).
    Used throughout to replace retractions by projections; cited from Davis et al. (2025) and Zhang et al. (2024b). Standard for compact embedded submanifolds but depends on an unspecified constant gamma.
  • domain assumption The zeroth-order oracle is unbiased and has bounded gradient variance: E F = f, E gradF = gradf, and E ||gradF - gradf||^2 <= sigma^2 (Assumption 2.2).
    Defines the black-box setting and is required for Lemmas 3.3 and 3.4.
  • domain assumption Each noisy function F_i is l_i-smooth on the convex hull of M in the Euclidean sense with uniformly bounded Euclidean gradient on M (Assumption 2.3).
    Basis of the Riemannian Lipschitz inequality Lemma 3.1; strong and unverified for deep-network objectives.
  • standard math For any point in the proximal tube, x - P_M(x) lies in the normal space at P_M(x) (Lemma A.2).
    Used in the bias proof of Lemma 3.3 and in Lemma 4.1; follows from first-order optimality of the projection.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach." pith.science (2026). https://pith.science/paper/R6ZBRONW

@misc{pith2026250722855,
  author       = {Pith},
  title        = {Pith review of: Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6ZBRONW}},
  note         = {Machine review of arXiv:2507.22855}
}
read the original abstract

Federated learning (FL) has emerged as a powerful paradigm for collaborative model training across distributed clients while preserving data privacy. However, existing FL algorithms predominantly focus on unconstrained optimization problems with exact gradient information, limiting its applicability in scenarios where only noisy function evaluations are accessible or where model parameters are constrained. To address these challenges, we propose a novel zeroth-order projection-based algorithm on Riemannian manifolds for FL. By leveraging the projection operator, we introduce a computationally efficient zeroth-order Riemannian gradient estimator. Unlike existing estimators, ours requires only a simple Euclidean random perturbation, eliminating the need to sample random vectors in the tangent space, thus reducing computational cost. Theoretically, we first prove the approximation properties of the estimator and then establish the sublinear convergence of the proposed algorithm, matching the rate of its first-order counterpart. Numerically, we first assess the efficiency of our estimator using kernel principal component analysis. Furthermore, we apply the proposed algorithm to two real-world scenarios: zeroth-order attacks on deep neural networks and low-rank neural network training to validate the theoretical findings.

Figures

Figures reproduced from arXiv: 2507.22855 by the authors.

Figure 1
Figure 1. Euclidean perturbation Remark 3.1. Previous Riemannian zeroth-order methods Li et al. (2023a,b); Maass et al. (2022) rely on retractions to construct gradient estimators, which requires sampling random vectors in the tangent space and leads to additional overhead. By leveraging the fact that projections can operate on arbitrary ambient vectors, we avoid the sampling procedure in the tangent space. For example, on th… view at source ↗
Figure 2
Figure 2. presents the objective function value gap over runtime. The results align with our theoretical analysis: leveraging the projection operator eliminates the need for sampling tangent random vectors, significantly reducing computation time. (a) Iris: N = 150, r = 2, µ = 1e − 4, m = 100 (b) Digits: N = 200, r = 4, µ = 1e − 4, m = 50 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Example of the proposed attack algorithm on a deer picture. Each subfigure demonstrates [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Impact of number of local updates. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Impact of number of participated clients. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Impact of the batch size in zeroth-order Riemannian gradient estimator. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Impact of number of participated clients. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Impact of the batch size in zeroth-order Riemannian gradient estimator. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Impact of number of rank. 6 Conclusion In the paper, we introduce a zeroth-order projection-based Riemannian algorithm for FL, overcoming the limitations of existing Riemannian FL frameworks that rely on exact gradient information. By leveraging the projection operator…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

47 extracted references · 35 canonical work pages

  1. [1]

    Absil and J

    P.-A. Absil and J. Malick. Projection-like retractions on matrix manifolds. SIAM Journal on Optimization, 22 0 (1): 0 135--158, 2012

    work page 2012

  2. [2]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization algorithms on matrix manifolds. Princeton University Press, 2008

    work page 2008

  3. [3]

    Balasubramanian and S

    K. Balasubramanian and S. Ghadimi. Zeroth-order nonconvex stochastic optimization: Handling constraints, high dimensionality, and saddle points. Foundations of Computational Mathematics, 22 0 (1): 0 35--76, 2022

    work page 2022

  4. [4]

    N. Boumal. An introduction to optimization on smooth manifolds. Cambridge University Press, 2023

    work page 2023

  5. [5]

    Carlini and D

    N. Carlini and D. Wagner. Towards evaluating the robustness of neural networks. In 2017 ieee symposium on security and privacy (sp), pages 39--57. Ieee, 2017

    2017

  6. [6]

    P.-Y. Chen, H. Zhang, Y. Sharma, J. Yi, and C.-J. Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM workshop on artificial intelligence and security, pages 15--26, 2017

    work page 2017

  7. [7]

    S. Chen, A. Garcia, M. Hong, and S. Shahrampour. Decentralized riemannian gradient descent on the stiefel manifold. In International Conference on Machine Learning, pages 1594--1605. PMLR, 2021

    work page 2021

  8. [8]

    Davis, D

    D. Davis, D. Drusvyatskiy, and Z. Shi. Stochastic optimization over proximally smooth sets. SIAM Journal on Optimization, 35 0 (1): 0 157--179, 2025

    work page 2025

Show all 47 references
  1. [9]

    Deng and J

    K. Deng and J. Hu. Decentralized projected riemannian gradient method for smooth optimization on compact submanifolds. arXiv preprint arXiv:2304.08241, 2023

    2023 arXiv

  2. [10]

    L. Deng. The mnist database of handwritten digit images for machine learning research [best of the web]. IEEE signal processing magazine, 29 0 (6): 0 141--142, 2012

    2012

  3. [11]

    X. Fan, Z. Gao, Y. Wu, Y. Jia, and M. Harandi. Learning a gradient-free riemannian optimizer on tangent spaces. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 7377--7384, 2021

    2021

  4. [12]

    W. Fang, Z. Yu, Y. Jiang, Y. Shi, C. N. Jones, and Y. Zhou. Communication-efficient stochastic zeroth-order optimization for federated learning. IEEE Transactions on Signal Processing, 70: 0 5058--5073, 2022

    2022

  5. [13]

    Ghadimi and G

    S. Ghadimi and G. Lan. Stochastic first-and zeroth-order methods for nonconvex stochastic programming. SIAM journal on optimization, 23 0 (4): 0 2341--2368, 2013

    2013

  6. [14]

    C. He, Z. Pan, X. Wang, and B. Jiang. Riemannian accelerated zeroth-order algorithm: Improved robustness and lower query complexity. In Forty-first International Conference on Machine Learning, 2024

    2024

  7. [15]

    Huang, W

    Z. Huang, W. Huang, P. Jawanpuria, and B. Mishra. Federated learning on riemannian manifolds with differential privacy. arXiv preprint arXiv:2404.10029, 2024 a

    2024 arXiv

  8. [16]

    Huang, W

    Z. Huang, W. Huang, P. Jawanpuria, and B. Mishra. Riemannian federated learning via averaging gradient stream. arXiv preprint arXiv:2409.07223, 2024 b

    2024

  9. [17]

    Kairouz, H

    P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings, et al. Advances and open problems in federated learning. Foundations and trends in machine learning , 14 0 (1--2): 0 1--210, 2021

    2021

  10. [18]

    S. P. Karimireddy, M. Jaggi, S. Kale, M. Mohri, S. J. Reddi, S. U. Stich, and A. T. Suresh. Mime: Mimicking centralized stochastic algorithms in federated learning. arXiv preprint arXiv:2008.03606, 2020 a

    2008 arXiv

  11. [19]

    S. P. Karimireddy, S. Kale, M. Mohri, S. Reddi, S. Stich, and A. T. Suresh. Scaffold: Stochastic controlled averaging for federated learning. In International conference on machine learning, pages 5132--5143. PMLR, 2020 b

    2020

  12. [20]

    Khanduri, P

    P. Khanduri, P. Sharma, H. Yang, M. Hong, J. Liu, K. Rajawat, and P. Varshney. Stem: A stochastic two-sided momentum algorithm achieving near-optimal sample and communication complexities for federated learning. Advances in Neural Information Processing Systems, 34: 0 6050--6061, 2021

    2021

  13. [21]

    Kone c n \`y

    J. Kone c n \`y . Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492, 2016

    2016 arXiv

  14. [22]

    K. B. Letaief, Y. Shi, J. Lu, and J. Lu. Edge artificial intelligence for 6g: Vision, enabling technologies, and applications. IEEE Journal on Selected Areas in Communications, 40 0 (1): 0 5--36, 2021

    2021

  15. [23]

    Li and S

    J. Li and S. Ma. Federated learning on riemannian manifolds. arXiv preprint arXiv:2206.05668, 2022

    2022 arXiv

  16. [24]

    J. Li, K. Balasubramanian, and S. Ma. Stochastic zeroth-order riemannian derivative estimation and optimization. Mathematics of Operations Research, 48 0 (2): 0 1183--1211, 2023 a

    2023

  17. [25]

    J. Li, K. Balasubramanian, and S. Ma. Zeroth-order riemannian averaging stochastic approximation algorithms. arXiv preprint arXiv:2309.14506, 2023 b

    2023 arXiv

  18. [26]

    J. Li, X. Chen, S. Ma, and M. Hong. Problem-parameter-free decentralized nonconvex stochastic optimization. arXiv preprint arXiv:2402.08821, 2024

    2024 arXiv

  19. [27]

    X. Lian, C. Zhang, H. Zhang, C.-J. Hsieh, W. Zhang, and J. Liu. Can decentralized algorithms outperform centralized algorithms? a case study for decentralized parallel stochastic gradient descent. Advances in neural information processing systems, 30, 2017

    2017

  20. [28]

    A. I. Maass, C. Manzie, D. Nesic, J. H. Manton, and I. Shames. Tracking and regret bounds for online zeroth-order euclidean and riemannian optimization. SIAM Journal on Optimization, 32 0 (2): 0 445--469, 2022

    2022

  21. [29]

    McMahan, E

    B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial intelligence and statistics, pages 1273--1282. PMLR, 2017

    2017

  22. [30]

    Nesterov and V

    Y. Nesterov and V. Spokoiny. Random gradient-free minimization of convex functions. Foundations of Computational Mathematics, 17 0 (2): 0 527--566, 2017

    2017

  23. [31]

    K. K. Patel, L. Wang, B. E. Woodworth, B. Bullins, and N. Srebro. Towards optimal communication complexity in distributed non-convex optimization. Advances in Neural Information Processing Systems, 35: 0 13316--13328, 2022

    2022

  24. [32]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, et al. Scikit-learn: Machine learning in python. the Journal of machine Learning research, 12: 0 2825--2830, 2011

    2011

  25. [33]

    Rabad \'a n and A

    R. Rabad \'a n and A. J. Blumberg. Topological data analysis for genomics and evolution: topology in biology. Cambridge University Press, 2019

    2019

  26. [34]

    S. U. Stich. Local sgd converges fast and communicates little. arXiv preprint arXiv:1805.09767, 2018

    2018 arXiv

  27. [35]

    H. Tang, X. Lian, M. Yan, C. Zhang, and J. Liu. d^2 : Decentralized training over decentralized data. In International Conference on Machine Learning, pages 4848--4856. PMLR, 2018

    2018

  28. [36]

    Townsend, N

    J. Townsend, N. Koep, and S. Weichwald. Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation. Journal of Machine Learning Research, 17 0 (137): 0 1--5, 2016

    2016

  29. [37]

    C.-C. Tu, P. Ting, P.-Y. Chen, S. Liu, H. Zhang, J. Yi, C.-J. Hsieh, and S.-M. Cheng. Autozoom: Autoencoder-based zeroth order optimization method for attacking black-box neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pages 742--749, 2019

    2019

  30. [38]

    Utpala, A

    S. Utpala, A. Han, P. Jawanpuria, and B. Mishra. Improved differentially private riemannian optimization: Fast sampling and variance reduction. Transactions on Machine Learning Research, 2023

    2023

  31. [39]

    T. Wang, Y. Huang, and D. Li. From the greene--wu convolution to gradient estimation over riemannian manifolds. arXiv preprint arXiv:2108.07406, 2021

    2021 arXiv

  32. [40]

    K. Q. Weinberger and L. K. Saul. Unsupervised learning of image manifolds by semidefinite programming. International journal of computer vision, 70: 0 77--90, 2006

    2006

  33. [41]

    H. Xiao, T. Yan, and S. Zhao. Riemannian svrg with barzilai-borwein scheme for federated learning. Journal of Industrial and Management Optimization, 21 0 (2): 0 1546--1567, 2025

    2025

  34. [42]

    N. Xiao, X. Liu, and K.-C. Toh. Dissolving constraints for riemannian optimization. Mathematics of Operations Research, 49 0 (1): 0 366--397, 2024

    2024

  35. [43]

    Xue and V

    Y. Xue and V. Lau. Riemannian low-rank model compression for federated learning with over-the-air aggregation. IEEE Transactions on Signal Processing, 71: 0 2172--2187, 2023

    2023

  36. [44]

    X. Yi, S. Zhang, T. Yang, and K. H. Johansson. Zeroth-order algorithms for stochastic distributed nonconvex optimization. Automatica, 142: 0 110353, 2022

    2022

  37. [45]

    K. Yuan, I. Chatzinikolaidis, and Z. Li. Bayesian optimization for whole-body control of high-degree-of-freedom robots through reduction of dimensionality. IEEE Robotics and Automation Letters, 4 0 (3): 0 2268--2275, 2019

    2019

  38. [46]

    Zhang, J

    J. Zhang, J. Hu, and M. Johansson. Composite federated learning with heterogeneous data. In ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 8946--8950. IEEE, 2024 a

    2024

  39. [47]

    Zhang, J

    J. Zhang, J. Hu, A. M.-C. So, and M. Johansson. Nonconvex federated learning on compact smooth submanifolds with heterogeneous data. arXiv preprint arXiv:2406.08465, 2024 b

    2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.