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REVIEW 3 major objections 5 minor 1 cited by

Methods with Local Steps and Random Reshuffling for Generally Smooth Non-Convex Federated Optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Pith's one-line read The paper proves convergence of local-step federated methods under (L0,L1)-smoothness without data-homogeneity, bounded-variance, or cosine-relatedness assumptions.

desk verdict Solid theory, shaky bridge to experiments: the paper proves first convergence rates for local-step FL with reshuffling under (L0,L1)-smoothness, but the implemented pseudogradient stepsize is not covered by the proofs. read the letter →

arxiv 2412.02781 v3 pith:LWBP2ZUT submitted 2024-12-03 math.OC cs.LG

classification math.OCcs.LG MSC 90C2690C3068W15
keywords federatedlearninglocalstepsrandomreshufflinggeneralizedsmoothnessgradientclippingpartialparticipationPolyak-Lojasiewiczconditionnon-convexoptimization
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

Modern neural-network losses are often not $L$-smooth: the Hessian norm grows with the gradient norm. This paper claims that federated optimization with local steps can nevertheless be analyzed under the resulting generalized $(L_0,L_1)$-smoothness by combining small client stepsizes, larger server stepsizes, and gradient clipping applied once to the aggregated update. It proposes three methods—Clip-LocalGDJ, CLERR, and Clipped RR-CLI—and proves non-convex convergence rates for all three, plus linear rates under the Polyak–Łojasiewicz condition, without the data-homogeneity, bounded-variance, or cosine-relatedness assumptions that earlier federated analyses of generalized smoothness required. The rates reduce to the known standard-smooth rates when $L_1 = 0$, which is part of the paper's evidence that its conditions are the right generalization.

What carries the argument

The load-bearing object is the generalized-smoothness coefficient $\hat a_p = L_0 + L_1\|\nabla f(\hat x_{t_p})\|$ (with local variants $a_p$ and $\tilde a_t$), paired with a server stepsize constrained to the interval $\gamma_p \in [\zeta/\hat a_p, 1/(4\hat a_p)]$ and a pseudogradient $g_p = \frac{1}{\alpha_p(v-t_p)}(\hat x_{t_p} - \frac1M\sum_m x^m_{t_{p+1}-1})$ formed from the local iterates. This coupling acts as smoothed clipping: when the gradient is large, the server step is automatically short, while the small client stepsize $\alpha_p \le 1/(2H a_p)$ bounds client drift. The proofs use a descent lemma for $(L_0,L_1)$-smooth functions and a deviation bound showing the average squared distance of local iterates from the synchronization point is at most $O((H-1)^2 a_p^3 \alpha_p^2(\delta_p+\Delta^\star))$; the argument then splits each epoch according to whether $\|\nabla f\|$ is below or above $L_0/L_1$, which is what converts the stationarity measure into the stated rates. For symmetric smoothness, the proof additionally controls the exponential factor in the descent inequality by an induction that bounds all local gradient norms along the epoch.

What would settle it

On a two-client $(L_0,L_1)$-smooth problem where one client's local gradients are far larger than the other's, compute $a_p = L_0 + L_1 \max_m \|\nabla f_m(\hat x_{t_p})\|$ along the trajectory and check whether the tuned inner stepsize satisfies $\alpha_p \le 1/(2H a_p)$ at every epoch; because $a_p$ is not observable to the user, a run that converges while violating this condition would show the theorem's sufficient conditions are not the operative mechanism, while a run that diverges under the condition would refute the claimed guarantee for the implemented method.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that local-step federated methods can converge under $(L_0,L_1)$-smoothness without extra restrictive assumptions, as long as the server takes a clipped, larger step along the averaged pseudogradient while clients take small steps. Concretely, Theorem 1 bounds $\min_{0\le p\le P-1} \frac{\zeta}{8}\min\{\|\nabla f(\hat x_{t_p})\|^2/L_0, \|\nabla f(\hat x_{t_p})\|/L_1\}$ by a term of order $O(L_0\delta_0/(\zeta \varepsilon))$ for Clip-LocalGDJ, and Corollary 1 converts this into an iteration complexity matching standard LocalGD when $L_1=0$. Theorems 3 and 4 give the analogous guarantees for CLERR (local steps with random reshuffling) and Clipped RR-CLI (local steps, reshuffling, and partial participation); for each method the paper also proves a linear rate under the Polyak–Łojasiewicz condition, recovering the standard $O((L_0/\mu)\ln(2\delta_0/\varepsilon))$ rate when $L_1=0$. These are the first results of this kind for partial participation under generalized smoothness, according to the paper's literature survey.

Load-bearing premise

For the proved rates to apply, stepsizes must be set using the exact global gradient and the largest local gradient norm among all clients at every synchronization point, values a federated user would not have; the implemented algorithms instead use a pseudogradient heuristic that the theorems do not cover.

Editorial extensions

If this is right

  • For non-convex $(L_0,L_1)$-smooth distributed problems, Clip-LocalGDJ requires $O(L_0\delta_0/(\zeta\varepsilon))$ synchronization epochs to reach the generalized stationarity measure $\min\{\|\nabla f\|^2/L_0, \|\nabla f\|/L_1\}\le\varepsilon$, matching the standard LocalGD rate when $L_1=0$.
  • CLERR gives a random-reshuffling convergence rate for generalized smooth objectives that does not assume bounded variance or cosine relatedness, under the paper's claims.
  • Clipped RR-CLI extends the guarantee to partial participation of clients, with a rate that again becomes $O(L_0\delta_0/(\zeta\varepsilon))$ when the inner stepsizes are small.
  • Under the Polyak–Łojasiewicz condition, all three methods converge linearly with complexity $O(L_0/\mu \ln(2\delta_0/\varepsilon))$ in the $L_1=0$ limit, and the theorem shows a large-gradient phase of at most $64\delta_0 L_1^2/(\mu\zeta)$ epochs before the linear decay begins.
  • The results indicate that clipping once per epoch, rather than at every local step, is sufficient in theory for generalized smoothness, and the paper's synthetic experiments support that design choice.

Reading between the lines

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

  • If $(L_0,L_1)$-smoothness is the right model for deep-network losses, the paper's design principle suggests that federated systems should spend their clipping budget on the aggregated server step rather than on every local gradient; the theory proves this for the once-per-epoch variants, but a head-to-head comparison with per-step clipping under the same assumptions would test whether the design p
  • Because the theorems' stepsize conditions involve quantities unavailable to the user (the full gradient and the largest local gradient norm at each synchronization point), an adaptive scheme that estimates these quantities online, or a parameter-free version, is the natural next step before the guarantees can be applied directly to the implemented algorithms.
  • The same two-case 'small gradient / large gradient' proof technique could plausibly extend to local methods with communication compression or incomplete local epochs, which the paper itself lists as future work.
  • The experimental heuristic of replacing $\nabla f(x_t)$ by the pseudogradient $g_t$ in the server stepsize is covered only approximately by the appendix's perturbation argument; a full proof for that practical variant would close the gap between the implemented and the analyzed methods.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes three federated optimization algorithms for non-convex objectives satisfying the generalized (L0,L1)-smoothness condition: Clip-LocalGDJ (Algorithm 1), CLERR (Algorithm 2), and Clipped RR-CLI (Algorithm 3). The methods combine local steps, random reshuffling, partial client participation, and server-side gradient clipping. The main theorems (Theorems 1-4) give convergence rates for the non-convex setting, and additional theorems in the appendices cover the Polyak-Lojasiewicz case. The paper claims these are the first such results without restrictive assumptions such as data homogeneity, bounded variance, or cosine relatedness, and the rates recover standard smooth rates when L1=0. Experiments on a synthetic shifted-fourth-order problem, ResNet-18 on CIFAR-10, and logistic regression datasets are reported in support of the theory.

Significance. If the theorems are correct as stated with their full hypotheses, the paper fills a genuine gap: it extends federated learning methods with local steps, random reshuffling, and partial participation to (L0,L1)-smooth non-convex problems without imposing the homogeneity or variance assumptions used in prior work. The recovery of known L-smooth rates when L1=0 is a useful consistency check and is not a circular derivation from the target result. The analysis follows established proof templates and the appendix contains detailed derivations. However, the practical impact is currently limited by two gaps: the stepsize conditions in the theorems require oracle quantities that are not available to the server, and the experiments run a version of the algorithms whose analysis is explicitly omitted. These issues are fixable but need to be addressed before the claims can be accepted as stated.

major comments (3)
  1. [Section 4 and Appendix F] The convergence theorems analyze Algorithms 1-3 with a global stepsize of the form 1/(c0+c1||grad f(x_t)||), where grad f is the full gradient. The experiments, however, replace this by the pseudogradient g_p (Algorithm 1) or g_t (Algorithms 2 and 3), as Appendix F explicitly states. Appendix F further says that the theoretical analysis of this practical version was deliberately not included. Appendix E only shows that ||g_p|| is close to ||grad f(hat x_{t_p})|| for small local stepsizes; it does not re-derive the descent lemmas used in Theorems 1-4. In particular, for CLERR and Clipped RR-CLI the stepsize becomes a random variable that is correlated with the random permutation through g_t, so the conditional-expectation step E[f(x_{t+1})|x_t] with a deterministic gamma_t is no longer justified. As written, the paper proves convergence for an algorithm that is not the one run in the experiments, and the experimental support for the theory is therefore incomplete.
  2. [Section 4, Theorems 1-4] The stepsize conditions in the theorems require quantities that are not available to the server in a federated setting. For example, Theorem 1 requires alpha_p to be bounded in terms of a_p = L0 + L1 max_m ||grad f_m(hat x_{t_p})||, and gamma_p to lie in an interval defined by hat a_p = L0 + L1 ||grad f(hat x_{t_p})||. The server would need to know L0, L1, and the maximum local gradient norm over all clients at every synchronization point, as well as delta0, Delta*, H, and P. The pseudocode only says 'choose small local stepsizes' without giving a procedure for computing or estimating these quantities. If these quantities are not available, the theorems do not apply to a practical implementation. The paper should state explicitly that the stepsizes are theoretical, or provide a procedure for setting them from computable quantities.
  3. [Section 4, Corollary 4] The statement of Corollary 4 says that if T >= 72 delta0 / (zeta epsilon) and gamma_t, eta_t are small enough, then the method reaches epsilon accuracy. The proof in Appendix D.1, however, requires T >= max{72 delta0 / (zeta epsilon), 12 Delta* / (zeta epsilon), 6 Delta* / (zeta epsilon)} and precise upper bounds on eta_t and gamma_t that depend on these quantities. As stated, the corollary omits the dependence on Delta* and Delta*, so if these constants are large the claimed iteration complexity is an underestimate. This is a load-bearing issue for the partial-participation result and should be corrected.
minor comments (5)
  1. [Section 4, Theorem 1] The main-text statement of Theorem 1 does not include the upper bound on alpha_p that is imposed in the appendix (alpha_p <= min{1/(2H a_p), 1/(c a_p) sqrt(hat a_p / a_p)} with c >= sqrt(P)). The main text says only 'choose small local stepsizes,' which is too vague for the theorem to be self-contained; please include the condition or provide an explicit reference to Appendix B.1.
  2. [Section 5.1.1 and Appendix F] The experiments are run over 3 or 10 random seeds, but Figures 2, 6, 7, and 8 show only mean curves without error bars or shaded regions. Adding variance information would make the comparisons between methods more convincing.
  3. [Section 5.3 and Appendix F.3] The abbreviation 'CRR-CLI' is used in Figure 4 and in Appendix F.3, while the method is named 'Clipped RR-CLI' in the main text. Please use a single consistent name for Algorithm 3.
  4. [Appendix F, first paragraph] The sentence 'Although, we decided not to include it to keep the presentation more concise' contains a typo; 'Although' should be 'However'.
  5. [Section 1.1] The contribution bullet says the results are 'tight,' but the paper only shows recovery of known rates in the special case L1=0; no lower bounds are established for the (L0,L1)-smooth setting. Consider softening the claim to avoid suggesting optimality in the broader class.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence rates are derived from the stated (L0,L1)-smoothness assumptions via standard descent lemmas, and the consistency checks with L-smooth rates are special-case verifications rather than inputs.

full rationale

The paper's central derivation chain is not circular. Theorems 1-4 all follow the same structure: apply the generalized-smooth descent inequality (Lemma 1, cited to Zhang et al. 2020b and Gorbunov et al. 2024) to bound f(x_{t+1}), control the client-drift or reshuffling-bias terms through Lemmas 4, 7, and the V_t bound in Appendix D, and then invoke the standard recurrence lemma of Mishchenko et al. (2020, Lemma 6). The final rates are consequences of explicit choices such as P >= 32*delta_0/(zeta*epsilon) and small enough inner stepsizes; none of the quantities defining the target bound are inserted into the assumptions or fitted from the data. The claim that the rates recover O(L0*delta_0/epsilon) for LocalGD, RR, and clipped GD when L1 = 0 is a consistency check, not an assumption: it is obtained by specializing the same bound, not by assuming the final rate. The self-citations are auxiliary rather than load-bearing: Lemma 1's second statement is a short proof from the descent inequality, the RR-CLI base algorithm is cited as prior work being extended, and the RR variance lemma from Malinovsky et al. (2022) is a standard sampling-without-replacement fact. None of these citations assume the paper's conclusion. The genuine limitations are non-circular correctness/implementability gaps: the stepsize conditions in Theorems 1-4 require oracle quantities such as hat_a_p = L0 + L1||grad f(hat_x_{t_p})|| and a_p = L0 + L1 max_m ||grad f_m(hat_x_{t_p})||, and Appendix F explicitly states that the theoretical analysis of the 'practical' pseudogradient stepsize used in the experiments was deliberately omitted, while Appendix E provides only a norm-closeness heuristic. These are gaps between the proved algorithm and the implemented one, not circular reasoning.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The theory relies on accepted smoothness and PL assumptions and on prior lemmas; no new postulates or entities are introduced. The only hand-chosen constant in the rates is zeta, which is not fitted to data.

free parameters (1)
  • zeta
    User-chosen constant with 0 < zeta <= 1/4 that constrains the server stepsize; it appears in the convergence rates but is not fitted to data.
assumptions (6)
  • domain assumption Assumption 1: f, f_m, f_mj are bounded below
    Used in Lemmas 1-2 to relate gradient norms to function suboptimality; standard in non-convex optimization.
  • domain assumption Assumption 2 (asymmetric (L0,L1)-smoothness) applied to f, each f_m, and each f_mj
    Central relaxation of smoothness introduced by Zhang et al. (2020b); proofs rely on the descent lemma derived from it in Lemma 1.
  • domain assumption Assumption 3 (symmetric (L0,L1)-smoothness) used for some theorems
    Stronger variant with sup over gradient norms; needed for the exponential descent lemma in Lemma 2, used in Theorems 5-9.
  • domain assumption Assumption 4 (Polyak-Lojasiewicz condition) for linear-rate results
    Used in Theorems 2, 6, 7, 9, 10 to obtain linear convergence after the large-gradient phase; standard strong-convexity substitute.
  • standard math Lemma 1 and Lemma 2 from earlier papers (Zhang et al. 2020b; Gorbunov et al. 2024; Chen et al. 2023)
    Provides the descent lemma and the gradient-to-suboptimality bound for (L0,L1)-smooth functions; cited rather than reproved.
  • standard math Lemma 6 from Mishchenko et al. (2020) for unrolling recurrences
    Used to convert per-step inequalities into min-rate bounds; a standard technical tool in this literature.

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Cite this review

Pith. "Pith review of Methods with Local Steps and Random Reshuffling for Generally Smooth Non-Convex Federated Optimization." pith.science (2026). https://pith.science/paper/LWBP2ZUT

@misc{pith2026241202781,
  author       = {Pith},
  title        = {Pith review of: Methods with Local Steps and Random Reshuffling for Generally Smooth Non-Convex Federated Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWBP2ZUT}},
  note         = {Machine review of arXiv:2412.02781}
}
abstract

Non-convex Machine Learning problems typically do not adhere to the standard smoothness assumption. Based on empirical findings, Zhang et al. (2020b) proposed a more realistic generalized $(L_0, L_1)$-smoothness assumption, though it remains largely unexplored. Many existing algorithms designed for standard smooth problems need to be revised. However, in the context of Federated Learning, only a few works address this problem but rely on additional limiting assumptions. In this paper, we address this gap in the literature: we propose and analyze new methods with local steps, partial participation of clients, and Random Reshuffling without extra restrictive assumptions beyond generalized smoothness. The proposed methods are based on the proper interplay between clients' and server's stepsizes and gradient clipping. Furthermore, we perform the first analysis of these methods under the Polyak-{\L} ojasiewicz condition. Our theory is consistent with the known results for standard smooth problems, and our experimental results support the theoretical insights.

Figures

Figures reproduced from arXiv: 2412.02781 by the authors.

Figure 1
Figure 1. Function residual for (4), αt = 10−7 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. CLERR-h chooses a rather big outer stepsize, while the outer clipping level is very tiny. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Function residual for (4), starting from different [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Function residual for (4), starting from x0 = (1, ..., 1) with batch size 16. In the final experiment, we consider methods with partial participation. The goal of this experiment is to investigate how clipping, local steps, partial participation and random reshuffling …
Figure 5
Figure 5. Figure 5: Function residual for (4), αt = 10−7 . The best parameters are provided in the legend. In these experiments we compare methods with random reshuffling, that shuffle data once at the start of training process. The main idea is to show the positive impact of random reshu…
Figure 6
Figure 6. Figure 6: Loss, gradient norm and accuracy on train and test dataset for ResNet-18 on CIFAR-10. [PITH_FULL_IMAGE:figures/full_fig_p062_6.png]
Figure 7
Figure 7. Figure 7: Function residual for (4), starting from different [PITH_FULL_IMAGE:figures/full_fig_p063_7.png]
Figure 8
Figure 8. Figure 8: Function residual for (4), starting from [PITH_FULL_IMAGE:figures/full_fig_p064_8.png]
Figure 9
Figure 9. Figure 9: Algorithm 2 with different step sizes on ResNet-18 on CIFAR-10. [PITH_FULL_IMAGE:figures/full_fig_p065_9.png]
Figure 10
Figure 10. Figure 10: Gradient norm for logistic regression problem on gisette and realsim datasets. The best [PITH_FULL_IMAGE:figures/full_fig_p066_10.png]

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Reference graph

Works this paper leans on

20 extracted references · 13 canonical work pages · cited by 1 Pith paper

  1. [1]

    For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L1 ∇f ˆxtp

    Let ˜P be the number of steps p, so that ∇f ˆxtp ≥ L0 L1 . For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L1 ∇f ˆxtp . Therefore, we get µζ f (ˆxtp ) − f ⋆ 4L1 ∇f ˆxtp ≤ f ˆxtp − f ˆxtp+1 + 3(H − 1)2a3 pα2 p f (ˆxtp ) − f ⋆ + ∆⋆ 2ˆap . Notice that the relation ˆap ≤ 2L1 ∇f ˆxtp and Lemma 1 together imply ∇f ˆxtp 4L1 ≤ ∇f ˆxtp 2 2ˆap ≤ f ˆxtp − f ⋆. Hence, w...

  2. [2]

    For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L0

    Suppose now that ∇f ˆxtp ≤ L0 L1 . For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L0. Hence, µζ f (ˆxtp ) − f ⋆ 4L0 ≤ f ˆxtp − f ˆxtp+1 + 3(H − 1)2a3 pα2 p f (ˆxtp ) − f ⋆ + ∆⋆ 2ˆap . Subtracting f ⋆ on both sides and introducing δp def = f ˆxtp − f ⋆, we obtain δp+1 ≤ δpρ + 3(H − 1)2a3 pα2 p f (ˆxtp ) − f ⋆ + ∆⋆ 2ˆap , where ρ def = 1 − µζ 4L0 . 25 Publish...

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    16 Published as a conference paper at ICLR 2025 Quoc Tran-Dinh, Nhan H

    ISBN 978-1-57735-876-3. 16 Published as a conference paper at ICLR 2025 Quoc Tran-Dinh, Nhan H. Pham, D. Phan, and Lam M. Nguyen. Feddr - randomized douglas- rachford splitting algorithms for nonconvex federated composite optimization. In Neural Information Processing Systems , 2021. URL https://api.semanticscholar.org/ CorpusID:235376727. Bohan Wang, Yus...

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    Yang You, Jing Li, Sashank J

    doi: 10.1109/TSP.2018.2878551. Yang You, Jing Li, Sashank J. Reddi, Jonathan Hseu, Sanjiv Kumar, Srinadh Bhojanapalli, Xiaodan Song, James Demmel, Kurt Keutzer, and Cho-Jui Hsieh. Large batch optimization for deep learning: Training bert in 76 minutes. arXiv: Learning, 2019. Bohang Zhang, Jikai Jin, Cong Fang, and Liwei Wang. Improved analysis of clipping...

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    CLERR clips outer gradients at the level of 10, so this also does not help method to converge to a better area

    Despite the fact that both Nastya and CLERR choose bigger outer stepsizes compared to inner stepsize, jumping does not have any impact on this problem. CLERR clips outer gradients at the level of 10, so this also does not help method to converge to a better area. Moreover, we provide results of heuristically modified Algorithm 2, where we fix clipping lev...

  6. [11]

    For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L1 ∇f ˆxtp

    Let ˜P be the number of steps p, so that ∇f ˆxtp ≥ L0 L1 . For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L1 ∇f ˆxtp . Therefore, we get µζ f (ˆxtp ) − f ⋆ 4L1 ∇f ˆxtp ≤ f ˆxtp − f ˆxtp+1 + 7(H − 1)2a3 pα2 p f (ˆxtp ) − f ⋆ + ∆⋆ ˆap . Notice that the relation ˆap ≤ 2L1 ∇f ˆxtp and Lemma 1 together imply ∇f ˆxtp 4L1 ≤ ∇f ˆxtp 2 2ˆap ≤ f ˆxtp − f ⋆. Hence, we...

  7. [12]

    For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L0

    Suppose now that ∇f ˆxtp ≤ L0 L1 . For such p, we have L0 + L1 ∇f ˆxtp = ˆap ≤ 2L0. Hence, µζ f (ˆxtp ) − f ⋆ 4L0 ≤ f ˆxtp − f ˆxtp+1 + 7(H − 1)2a3 pα2 p f (ˆxtp ) − f ⋆ + ∆⋆ ˆap . Subtracting f ⋆ on both sides and introducing δp def = f ˆxtp − f ⋆, we obtain δp+1 ≤ δpρ + 7(H − 1)2a3 pα2 p f (ˆxtp ) − f ⋆ + ∆⋆ ˆap , where ρ def = 1 − µζ 4L0 . Let αp def =...

  8. [13]

    For such t, we haveL0+L1 ∥∇f (xt)∥ = ˆat ≤ 2L1 ∥∇f (xt)∥

    Let ˜T be the number of stepst, so that ∥∇f (ˆxt)∥ ≥L0 L1 . For such t, we haveL0+L1 ∥∇f (xt)∥ = ˆat ≤ 2L1 ∥∇f (xt)∥ . Therefore, we get µζ (f (xt) − f ⋆) 4L1 ∥∇f (xt)∥ ≤ f (xt) − f (xt+1) + 3α2 t ˜at 3 8ˆat δt((N − 1)(2N − 1) + 2(N + 1)) + 2(N + 1)∆ ⋆ . Notice that the relation ˆat ≤ 2L1 ∥∇f (xt)∥ and Lemma 1 together imply ∥∇f (xt)∥ 4L1 ≤ ∥∇f (xt)∥2 2ˆa...

Show all 20 references
  1. [14]

    For such t, we have L0 +L1 ∥∇f (xt)∥ = ˆap ≤ 2L0

    Suppose now that ∥∇f (xt)∥ ≤L0 L1 . For such t, we have L0 +L1 ∥∇f (xt)∥ = ˆap ≤ 2L0. Hence, µζ (f (xt) − f ⋆) 4L0 ≤ f (xt) − f (xt+1) + 3α2 t ˜at 3 8ˆat δt((N − 1)(2N − 1) + 2(N + 1)) + 2(N + 1)∆ ⋆ . Subtracting f ⋆ on both sides and introducing δt def = f (xt) − f ⋆, we obta...

  2. [15]

    For such t, we haveL0+L1 ∥∇f (xt)∥ = ˆat ≤ 2L1 ∥∇f (xt)∥

    Let ˜T be the number of stepst, so that ∥∇f (ˆxt)∥ ≥L0 L1 . For such t, we haveL0+L1 ∥∇f (xt)∥ = ˆat ≤ 2L1 ∥∇f (xt)∥ . Therefore, we get µζ (f (xt) − f ⋆) 4L1 ∥∇f (xt)∥ ≤ f (xt) − f (xt+1) + α2 t ˜a3 t 2ˆat δt((N − 1)(2N − 1) + 2(N + 1)) + 2(N + 1)∆ ⋆ . Notice that the relatio...

  3. [16]

    For such t, we have L0 +L1 ∥∇f (xt)∥ = ˆap ≤ 2L0

    Suppose now that ∥∇f (xt)∥ ≤L0 L1 . For such t, we have L0 +L1 ∥∇f (xt)∥ = ˆap ≤ 2L0. Hence, µζ (f (xt) − f ⋆) 4L0 ≤ f (xt) − f (xt+1) + α2 t ˜a3 t 2ˆat δt((N − 1)(2N − 1) + 2(N + 1)) + 2(N + 1)∆ ⋆ . Subtracting f ⋆ on both sides and introducing δt def = f (xt) − f ⋆, we obtai...

  4. [17]

    For such t, we haveL0+L1 ∥∇f (xt)∥ = ˆat ≤ 2L1 ∥∇f (xt)∥

    Let ˜T be the number of stepst, so that ∥∇f (ˆxt)∥ ≥L0 L1 . For such t, we haveL0+L1 ∥∇f (xt)∥ = ˆat ≤ 2L1 ∥∇f (xt)∥ . Therefore, we get µζ (f (xt) − f ⋆) 4L1 ∥∇f (xt)∥ ≤ f (xt) − f (xt+1) + 2ˆat˜a2 t + ˆa3 t 4ˆa2 t (f (xt) − f ⋆) η2 t at + η2 t R2ˆat + γ2 t N ˜at + η2 t Rat +...

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    practical

    Suppose now that ∥∇f (xt)∥ ≤L0 L1 . For such t, we have L0 +L1 ∥∇f (xt)∥ = ˆap ≤ 2L0. Hence, µζ (f (xt) − f ⋆) 4L0 ≤ f (xt) − f (xt+1) + 2ˆat˜a2 t + ˆa3 t 4ˆa2 t (f (xt) − f ⋆) η2 t at + η2 t R2ˆat + γ2 t N ˜at + η2 t Rat + 2ˆat˜a2 t + ˆa3 t 4ˆa2 t η2 t at∆⋆ + γ2 t N ˜at∆ ⋆ + ...

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    Speaking of outer steps, Algorithm 3 chooses global stepsize equal to 5 · 10−7 with clipping level 1016. And CE-FedAvg-PP has server stepsize equal to 10. The grids of hyperparameters are provided in Table 5. 64 Published as a conference paper at ICLR 2025 Method Cl. Stepsize ...

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    ISBN 9781450312851

    Omnipress. ISBN 9781450312851. Benjamin Recht and Christopher R ´e. Parallel stochastic gradient algorithms for large-scale matrix completion. Mathematical Programming Computation , 5:201 – 226, 2013. URL https:// api.semanticscholar.org/CorpusID:17109415. Herbert Robbins and ...

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    12 Published as a conference paper at ICLR 2025 Priya Goyal, Piotr Doll ´ar, Ross B

    URL https://proceedings.mlr.press/v97/qian19b.html. 12 Published as a conference paper at ICLR 2025 Priya Goyal, Piotr Doll ´ar, Ross B. Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, Andrew Tulloch, Yangqing Jia, and Kaiming He. Accurate, large minibatch sgd: Tra...

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    URL https://proceedings.neurips.cc/paper_files/paper/2020/ file/1457c0d6bfcb4967418bfb8ac142f64a-Paper.pdf. Xufeng Cai, Cheuk Yin Lin, and Jelena Diakonikolas. Empirical risk minimization with shuffled SGD: A primal-dual perspective and improved bounds. CoRR, abs/2306.12498, 2...

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    Wenlin Chen, Samuel Horv ´ath, and Peter Richt ´arik

    URL https://proceedings.neurips.cc/paper_files/paper/2021/ file/ab9ebd57177b5106ad7879f0896685d4-Paper.pdf. Wenlin Chen, Samuel Horv ´ath, and Peter Richt ´arik. Optimal client sampling for federated learn- ing. Trans. Mach. Learn. Res., 2022, 2020. URL https://api.semanticsch...

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    Dan Alistarh, Torsten Hoefler, Mikael Johansson, Sarit Khirirat, Nikola Konstantinov, and C ´edric Renggli

    URL https://api.semanticscholar.org/CorpusID:263605847. Dan Alistarh, Torsten Hoefler, Mikael Johansson, Sarit Khirirat, Nikola Konstantinov, and C ´edric Renggli. The convergence of sparsified gradient methods. In Proceedings of the 32nd Inter- national Conference on Neural I...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.