Beyond Communication Overhead: A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning

Bassel Hamoud; Kfir Y. Levy; Ze'ev Zukerman

REVIEW 3 major objections 5 minor 13 references

Beyond Communication Overhead: A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A multilevel Monte Carlo estimator makes biased gradient compressors unbiased, restoring uncompressed SGD convergence rates.

desk verdict A promising MLMC compression idea undercut by an unproven uniform-variance assumption that the parallelization claim rests on. read the letter →

arxiv 2507.05508 v1 pith:RDVB27TD submitted 2025-07-07 cs.LG

Ze'ev Zukerman , Bassel Hamoud , Kfir Y. Levy This is my paper · ORCID

classification cs.LG

keywords distributedlearninggradientcompressionmultilevelMonteCarlounbiasedestimatorTop-ksparsificationbit-wisequantizationparallelSGDconvergenceguarantees

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

Distributed training is often limited by the cost of moving gradients to a central server, and compression introduces a dilemma: biased compressors such as Top-k work well in practice but break the unbiasedness on which standard parallel-SGD guarantees depend. This paper proposes wrapping any biased compressor in a multilevel Monte Carlo (MLMC) estimator $\tilde{g}_{t,i} = g^0_{t,i} + \frac{1}{p_l}(g^l_{t,i} - g^{l-1}_{t,i})$, with level $l$ sampled from probabilities $\{p_l\}$, so that the compressed gradient is a conditionally unbiased estimate of the true gradient. The paper claims that plugging this estimator into data-parallel SGD restores the asymptotic convergence rate of uncompressed SGD, $O(1/T + \sigma/\sqrt{MT})$, with only a slightly enlarged variance term, and that this permits scaling to $M = O(T)$ machines without performance loss. It derives variance-minimizing level probabilities for bit-wise and Top-k compression, including an adaptive per-sample variant, and validates the approach on BERT and CIFAR-10 training.

What carries the argument

The load-bearing object is the multilevel compressor $C^l$ of Definition 3.1, whose levels range from the zero map up to the identity $C^L(v) = v$. The estimator $\tilde{g}_{t,i} = g^0_{t,i} + \frac{1}{p_l}(g^l_{t,i} - g^{l-1}_{t,i})$, with $l \sim p_l$, is unbiased because in expectation the correction telescopes to $g^L = v$, transducing each level's bias into variance. The remaining design problem is the level distribution, which the paper chooses to minimize estimator variance: $p_l = 2^{-l}/(1-2^{-63})$ for fixed-point bit-wise compression, $p^l_{t,i} = \Delta^l_{t,i}/\sum_{l'}\Delta^{l'}_{t,i}$ for general compressors with $\Delta^l_{t,i} = \|g^l_{t,i} - g^{l-1}_{t,i}\|$, and $p^l_{t,i} \propto \sqrt{\alpha^l_{t,i} - \alpha^{l-1}_{t,i}}$ for s-Top-k, giving an adaptive per-sample, per-step scheme.

What would settle it

Train the adaptive MLMC-Top-k method on a problem whose gradients are dense and near-uniform, such as sorted gradient magnitudes that are nearly flat, measure the estimator variance $E\|\tilde{g} - v\|^2$ as the dimension $d$ grows, and compare the error curve to the predicted $O(1/T + \sigma/\sqrt{MT})$: variance growing linearly in $d$, or convergence degrading when $M$ approaches $T$, would falsify the uniform $\hat{\omega}$ bound behind Theorem 4.1.

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Extended reading notes

Core claim

The paper's central claim is that the MLMC gradient estimator $\tilde{g}_{t,i} = g^0_{t,i} + \frac{1}{p_l}(g^l_{t,i} - g^{l-1}_{t,i})$, where $l$ is sampled from a nonzero distribution $\{p_l\}$ and the highest compression level $L$ is the identity map, is a conditionally unbiased estimate of the true gradient $\nabla f_i(x_t)$ for any multilevel compressor (Lemma 3.2). Because the telescoping sum of residuals cancels the per-level bias, the estimator can replace the raw gradient inside ordinary parallel SGD without error feedback. Theorem 4.1 then gives, in the homogeneous setting, convex and nonconvex error bounds of the form $O(1/T + \hat{\omega}^2/(MT) + (\hat{\omega}+1)\sigma/\sqrt{MT})$, where the middle term is asymptotically negligible, so the rate is asymptotically identical to uncompressed SGD. The authors conclude that the method parallelizes over $M = O(T)$ machines, equivalently $O(\sqrt{N})$ workers for dataset size $N$, without degradation, whereas the error-feedback baseline EF21-SGDM is limited to $M = O(\sqrt{T})$, or $O(N^{1/3})$.

Load-bearing premise

The convergence and parallelization guarantees rest on the MLMC estimator satisfying a uniform bound on its variance relative to the true gradient's norm, and for the adaptive Top-k variant this bound is proven only when a gradient's sorted entries decay exponentially; for dense, near-uniform gradients the variance can grow with the dimension, which would break the claimed parallelization limit.

Editorial extensions

If this is right

Biased compressors such as Top-k and fixed-point quantization can be dropped into plain data-parallel SGD without error feedback, because the MLMC estimate restores conditional unbiasedness.
The convergence rate is asymptotically the same as uncompressed SGD, $O(1/T + \sigma/\sqrt{MT})$, so compression no longer changes the dominant optimization cost.
The method permits $M = O(T)$ machines, equivalently $O(\sqrt{N})$ workers for a dataset of size $N$, before performance degrades, while EF21-SGDM is limited to $M = O(\sqrt{T})$, or $O(N^{1/3})$.
In the homogeneous setting, each machine sends about $2d$ bits per round with the fixed-point MLMC compressor rather than $64d$ bits for the uncompressed vector, a 32-fold communication saving.
Because the MLMC gradients are unbiased, the method can be composed with variance-reduced frameworks such as MARINA or DASHA, which the paper states would fully mitigate the heterogeneity term.
When sorted gradient magnitudes decay exponentially, the adaptive s-Top-k variant achieves $O(1/(r_{t,i} s))$ estimator variance, compared with $O(d/s)$ for Rand-k with the same budget, so it exploits gradient energy concentration.

Reading between the lines

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

Because Lemma 3.2 is distribution-free, the same MLMC wrapper should extend to any family of compressors forming a chain from the zero map to the identity, including structured quantizers where the residual has no coordinate-wise interpretation; the paper's RTN experiments hint at this, but a general variance analysis for such quantizers is not given.
The adaptive Top-k advantage relies on gradient energy being concentrated; for dense, near-uniform gradients, the estimator variance can scale with the dimension, so the $M=O(T)$ parallelization regime should be expected to shrink for such problems.
Sampling levels independently per machine and per step adds fresh randomness each round; coupling the level draws across machines, or reusing levels across steps, is a natural variance-reduction extension that the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Hi [Name],

You should know two things about this paper. First, the MLMC compression idea is real: Lemma 3.2's unbiasedness via telescoping is correct, and the variance-minimizing level probabilities in Lemmas 3.3-3.4 are derived cleanly. Second, the paper's flagship convergence claim—asymptotic equivalence to uncompressed SGD with M=O(T) parallelization—is not actually established for the adaptive Top-k method, and the empirical results, while suggestive, lack the details to confirm the advantage.

What's new: applying MLMC to biased gradient compressors is a neat generalization of importance sampling; the paper correctly notes that for standard Top-k the method reduces to IS, and the genuinely new part is the extension to structured compressors like s-Top-k, fixed-point, floating-point, and RTN. That generalization is useful and well motivated. The analysis of bit-wise compressors with exact optimal probabilities is solid. I also appreciate that the authors are explicit about the connection to IS.

Where it gets soft: Theorem 4.1 relies on a uniform compression coefficient ω̂ in E||g̃-v||² ≤ ω̂²||v||². For Alg. 3 with adaptive Top-k, Appendix D gives the exact second moment E||g̃||² = (Σ Δ_l)². For a dense uniform vector, every Δ_l = 1/√d, so this is d and the relative variance is (d-1)||v||². No dimension-independent ω̂ exists. The only variance control for Alg. 3 is Lemma 3.6, which assumes exponential decay of sorted entries (Assumption 3.5) and 'sufficiently large rd'—conditions that don't appear in Theorem 4.1. And the paper's justification that Gaussian-like gradients satisfy exponential decay is wrong: order statistics of a Gaussian sample decay like sqrt(log(d/j)), far slower than exponential. So the claimed M=O(T) parallelization limit, which follows from dropping ω̂ in the comparison, should really be M ≤ T/(ω̂+1)², and ω̂ can be √d for dense gradients. This is a load-bearing gap, not a cosmetic one.

Also, the experiments are averaged over 5 seeds but show no error bars, no wall-clock time, and no code release. The communication-efficiency plots show MLMC ahead, but it's hard to tell whether the overhead of sending the level index and the sign bits is accounted for at the bits level.

Bottom line: the paper deserves a serious referee, and I'd send it to review with a request to fix the theorem statement or add the missing assumption, and to tighten the empirical reporting. The MLMC construction is a good idea; the current write-up overclaims its scope.

Referee Report

3 major / 5 minor

Summary. The paper proposes a multilevel Monte Carlo (MLMC) compression scheme for distributed SGD. A hierarchy of biased compressors C^l is used to build the estimator g̃ = g^0 + (1/p_l)(g^l - g^{l-1}), and Lemma 3.2 proves this estimator is conditionally unbiased. The authors derive variance-minimizing level probabilities for fixed-point and floating-point compressors and an adaptive per-sample distribution for Top-k/s-Top-k. They state convergence bounds in Theorem 4.1 that match parallel SGD up to a compression coefficient, argue that this enables M=O(T) parallelism, and present experiments on BERT fine-tuning and CIFAR-10. The main advertised claim is that MLMC 'transduces bias into variance' while preserving the empirical advantages of biased compressors.

Significance. The MLMC construction is elegant and the unbiasedness argument is clean. The closed-form variance-minimizing probability distributions for the fixed-point and adaptive Top-k cases are a genuine strength, and the experiments show consistent gains over Top-k, Rand-k, EF21-SGDM, and QSGD on the tested tasks. However, the theoretical significance as stated is larger than what is proven: the compression coefficient in Theorem 4.1 can depend on the dimension, and for adaptive Top-k its worst-case value is d-1, so the 'slightly increased variance' and the M=O(T) parallelization claim need to be restated with explicit constants. The non-uniformity assumption used to escape the worst case is also not empirically validated.

major comments (3)

[Theorem 4.1; App. D, Eq. (55); §F.3, Eq. (102)] The theorem and its proof in §F.1 apply Eq. (3) with a single coefficient ω̂. For Alg. 3 with adaptive Top-k, App. D gives the exact second moment E||g̃_{t,i}||² = (Σ_l Δ_{t,i}^l)². For v = (1/√d, ..., 1/√d) this equals d, so σ²_comp = (d-1)||v||². A uniform but dimension-dependent bound ω̂² = d-1 therefore does exist, so the statement of Theorem 4.1 is not false. However, the paper never states this bound, and the interpretation in the theorem's discussion and in §F.3 is not dimension-aware: the third term in Eq. (98) carries a factor (ω̂+1), which is √d+1 in the worst case. Calling this a 'slightly increased variance' and dropping ω̂ in the comparison leading to M ≤ T in Eq. (102) is misleading; the correct comparison is M ≤ T/(ω̂+1)², which for uniform dense gradients becomes M ≤ T/d. The claims in §F.3 should be restated with this dimension dependence made explicit.
[§3.3, Assumption 3.5, Lemma 3.6] The paper's only variance improvement over random sparsification for Alg. 3 rests on Assumption 3.5, which asserts exponential decay of sorted gradient entries. This assumption is not validated, and the motivation given for it is incorrect: for i.i.d. Gaussian entries, the j-th largest magnitude behaves like sqrt(log(d/j)), which is much slower than exponential. The inequality 'ae^{-x^2} ≤ be^{-x}' in §3.3 does not establish the required decay for order statistics. Since the practical advantage of adaptive MLMC-Top-k depends entirely on non-uniformity, the authors should either verify Assumption 3.5 empirically on the BERT/ResNet gradients used in the experiments or present a weaker, empirically grounded assumption.
[App. D, Eq. (55); comparison with unbiased compressors] In the worst case, App. D implies σ²_comp = ||v||_1² - ||v||², which equals the variance of Rand-k with a single randomly selected coordinate. Thus Theorem 4.1 does not by itself establish an advantage over unbiased compressors; the advantage is only guaranteed under non-uniformity assumptions. The paper should state this explicitly, because the current text contrasts 'slightly increased variance' with the variance of unbiased methods and suggests the MLMC estimator is uniformly better.

minor comments (5)

[Author affiliations] 'Viterby Faculty' should be 'Viterbi Faculty'.
[§3.3, proof of Lemma 3.6] 'rank-k' should be 'Rand-k' in the comparison sentence.
[Figures 1-5] Error bars or confidence bands would strengthen the claims, since the comparisons are averaged over only 5 seeds.
[§1.1 and §5] The related work discusses Horváth & Richtárik (2021), which also constructs unbiased estimators from biased compressors; a few sentences in the method or experiments section explaining the difference in communication cost and variance would help position the contribution.
[Lemmas 3.3, 3.4, B.1] The Lagrangian derivations write the problem as min over p of an expression with max over λ, but the positivity of p_l is not mentioned; the optimal p_l is indeed positive, but this should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation chain; central MLMC unbiasedness and variance-minimizing probabilities are self-contained, with only non-load-bearing self-citations.

full rationale

Lemma 3.2 derives unbiasedness by telescoping: E[g̃_{t,i} | x_t] = E_z[Σ_{l=1}^L (g^l_{t,i} - g^{l-1}_{t,i})] = E_z[g^L_{t,i}] = ∇f_i(x_t); this is a genuine identity, not a restatement of the conclusion. Lemma 3.4 minimizes E[||g̃||²] = Σ_l (Δ_l)^2 / p_l over the probability simplex; the optimum p_l ∝ Δ_l follows from Lagrange multipliers, and the resulting variance (ΣΔ_l)^2 - ||v||² is a derived quantity, not a fitted parameter relabeled as a prediction. Appendix B and C similarly solve closed-form variance minimizations for bit-wise compressors. The paper's main convergence theorem is conditional on a compression coefficient ω̂ satisfying Eq. (3); exact ω̂ values are supplied for the non-adaptive compressors, while for adaptive Top-k the only variance control is Lemma 3.6 under Assumption 3.5 (exponential decay of sorted entries). For dense uniform gradients, Appendix D's exact expression gives dimension-dependent variance, so the uniform ω̂ bound and the M = O(T) parallelization claim in §F.3 are not established as stated. This is a correctness gap or false generalization, not circularity: Theorem 4.1 does not reduce to its assumptions by definition; it invokes a bound that is only partially proven. Self-citations to Dorfman et al. (2023, 2024) appear in Appendix F.1 for the standard SGD error bound, but Theorem 2.3 already states the same bound with Dekel et al. and Ghadimi & Lan as sources, so the self-citations are not load-bearing.

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

The central unbiasedness claim is a direct telescoping identity and needs no fitted parameters. The variance and parallelization claims rest on the uniform compression-coefficient bound and the exponential-decay assumption, which are domain assumptions about gradient structure rather than parameters fitted to data.

assumptions (4)

domain assumption The loss functions f_i are L-smooth and stochastic gradients have bounded variance σ² (Assumptions 2.1, 2.2).
Standard assumptions for SGD convergence used in Theorem 2.3 and Theorem 4.1.
domain assumption There exists a uniform compression coefficient ω̂ such that E[||g̃-v||²] ≤ ω̂²||v||² for the MLMC estimator (Eq. (3)).
Used throughout the proof of Theorem 4.1 in App. F.1; not established for the adaptive Top-k method outside the exponential-decay setting.
ad hoc to paper Sorted gradient magnitudes decay exponentially: |v(j)| = |v(0)| e^{-(r/2)j} (Assumption 3.5).
Introduced specifically to derive the O(1/(rs)) variance bound in Lemma 3.6, which supports the claimed advantage over Rand-k.
domain assumption Gradient entries can be normalized to |e| ≤ 1 and represented in 64-bit fixed point without significant error (Eq. (7)).
Needed for the fixed-point MLMC analysis in Section 3.1.

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Cite this review

Pith. "Pith review of Beyond Communication Overhead: A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning." pith.science (2026). https://pith.science/paper/RDVB27TD

@misc{pith2026250705508,
  author       = {Pith},
  title        = {Pith review of: Beyond Communication Overhead: A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDVB27TD}},
  note         = {Machine review of arXiv:2507.05508}
}

abstract

Distributed learning methods have gained substantial momentum in recent years, with communication overhead often emerging as a critical bottleneck. Gradient compression techniques alleviate communication costs but involve an inherent trade-off between the empirical efficiency of biased compressors and the theoretical guarantees of unbiased compressors. In this work, we introduce a novel Multilevel Monte Carlo (MLMC) compression scheme that leverages biased compressors to construct statistically unbiased estimates. This approach effectively bridges the gap between biased and unbiased methods, combining the strengths of both. To showcase the versatility of our method, we apply it to popular compressors, like Top-$k$ and bit-wise compressors, resulting in enhanced variants. Furthermore, we derive an adaptive version of our approach to further improve its performance. We validate our method empirically on distributed deep learning tasks.

Figures

Figures reproduced from arXiv: 2507.05508 by the authors.

**Figure 1.** Finetuning BERT on GLUE SST2 communication efficiency comparison of the Adaptive MLMC-Top-k (Alg. 3), Top-k, EF21-SGDM, Rand-k, and uncompressed SGD for sparsification levels k ∈ {0.01n, 0.05n, 0.1n, 0.5n}, M = 4, 32 machines, and a batch size of 16 samples, averaged over 5 different seeds. 6. Conclusions We presented a novel method that bridges the gap between unbiased and biased compression approaches typically u… view at source ↗

**Figure 4.** CIFAR-10 image classification using ResNet18, communication efficiency comparison of our MLMC-Top-k Compressor (Alg. 3),Top-k, Rand-k, EF21-SGDM, and uncompressed SGD, for M = 4 machines and a batch size of 128 and for M = 32 machines and a batch size of 64, averaged over 5 different seeds. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗

**Figure 5.** CIFAR-10 image classification using ResNet18, iteration efficiency comparison of our MLMC-Top-k Compressor (Alg. 3),Topk, Rand-k, EF21-SGDM, and uncompressed SGD, for M = 4 machines and a batch size of 128 and for M = 32 machines and a batch size of 64, averaged over 5 different seeds. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

Figures from the paper (1 more)

**Figure 6.** Figure 6: Finetuning BERT on GLUE SST2 communication efficiency (top row) and iteration efficiency (bottom row) comparison of the Adaptive MLMC-RTN (Alg. 3), RTN with l ∈ {2, 4, 8, 16}, and uncompressed SGD, for M = 4 and M = 32 machines and a batch size of 16 samples. The resul…

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Works this paper leans on

13 extracted references · 7 canonical work pages

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This implies that the number of the elements we transmit is less or equal to 1 rt,i . Thus, we obtain the following 18 A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning approximation to the variance: σ2 t,comp ≈ ∥vt,i∥2   1 − e−rt,i·s 1 − e− rt,i 2 s 2 − 1   (73) ≈ ∥vt,i∥2 rt,i · s rt,i 2 s 2 − 1 ! (74) = ∥vt,...

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URL https://arxiv.org/abs/2107.06917. 12 A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning A. Proof of Lemma 3.2 Lemma 3.2 For any multilevel compressor C l, l∈ [L], any non-zero level probabilities {pl}L l=1, the MLMC gradient estimator ˜gt,i ≜ g0 t,i + 1 pl (gl t,i − gl−1 t,i ) is a conditionally unbiased estimate...

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(2023), nonconvex case): 1 T TX t=1 E ∥∇f (xt)∥2 ∈ O ∆1L αT + ∆1Lσ1/2 α1/2T 3/4 + ∆1Lσ√ M T (101) Let us consider our bounds in Eq

guarantees the following bound (Corollary 3 in Fatkhullin et al. (2023), nonconvex case): 1 T TX t=1 E ∥∇f (xt)∥2 ∈ O ∆1L αT + ∆1Lσ1/2 α1/2T 3/4 + ∆1Lσ√ M T (101) Let us consider our bounds in Eq. (98)-(99). Note that asymptotically, the third term is dominant. Therefore, M can be as large as o(T ), or equivalently o( √ N ), where N is the size of the who...

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The proof follows very similarly to the one for the convex case

Heterogeneous Nonconvex Case. The proof follows very similarly to the one for the convex case. Here, we have for η ≤ 1 L ((Dorfman et al., 2024)): 1 T TX t=1 E[∥∇f (xt)∥2] ≤ 2∆1 T η + ηL T TX t=1 EV 2 t . (120) where ∆1 = f (x1) − f (x∗) and V 2 t = E[∥˜gt − ∇f (xt)∥2 |xt]. Plugging in the expression for V 2 t in Eq. (113) yields: 1 T TX t=1 E[∥∇f (xt)∥2]...

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1 (pl t,i)2 (∆l t,i)2 # (47) amd by writing the expectation w.r.t pl explicitly, we have: E[∥˜gt,i∥2] = LX l=1

Since the Exponent is E = ⌊log2(e)⌋ + 1023, the Floating-Point biased compressor satisfies Eq. (4) with α = 1 − 2−l. We apply the MLMC scheme with the Floating-Point Compressor and thus only need to transmit the residual gl t,i − gl−1 t,i . The residual has the same Exponent and Sign bits as the original entry, but contains only one information bit of the...

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Pith tools

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

[1] [1]

This implies that the number of the elements we transmit is less or equal to 1 rt,i . Thus, we obtain the following 18 A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning approximation to the variance: σ2 t,comp ≈ ∥vt,i∥2   1 − e−rt,i·s 1 − e− rt,i 2 s 2 − 1   (73) ≈ ∥vt,i∥2 rt,i · s rt,i 2 s 2 − 1 ! (74) = ∥vt,...

work page 2024

[2] [6]

B., Ramage, D., and Richt´arik, P

Konecn´y, J., McMahan, H. B., Ramage, D., and Richt´arik, P. Federated optimization: Distributed machine learning for on-device intelligence. ArXiv, abs/1610.02527,

arXiv

[3] [7]

Recht, B., R´e, C., Wright, S

URL https://arxiv.org/abs/ 1901.09269. Recht, B., R´e, C., Wright, S. J., and Niu, F. Hogwild: A lock- free approach to parallelizing stochastic gradient descent. In Neural Information Processing Systems,

arXiv 1901

[4] [8]

11 A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning Wang, A., Singh, A., Michael, J., Hill, F., Levy, O., and Bowman, S. R. GLUE: A multi-task benchmark and anal- ysis platform for natural language understanding. CoRR, abs/1804.07461,

arXiv

[5] [9]

12 A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning A

URL https://arxiv.org/abs/2107.06917. 12 A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning A. Proof of Lemma 3.2 Lemma 3.2 For any multilevel compressor C l, l∈ [L], any non-zero level probabilities {pl}L l=1, the MLMC gradient estimator ˜gt,i ≜ g0 t,i + 1 pl (gl t,i − gl−1 t,i ) is a conditionally unbiased estimate...

arXiv

[6] [12]

(2023), nonconvex case): 1 T TX t=1 E ∥∇f (xt)∥2 ∈ O ∆1L αT + ∆1Lσ1/2 α1/2T 3/4 + ∆1Lσ√ M T (101) Let us consider our bounds in Eq

guarantees the following bound (Corollary 3 in Fatkhullin et al. (2023), nonconvex case): 1 T TX t=1 E ∥∇f (xt)∥2 ∈ O ∆1L αT + ∆1Lσ1/2 α1/2T 3/4 + ∆1Lσ√ M T (101) Let us consider our bounds in Eq. (98)-(99). Note that asymptotically, the third term is dominant. Therefore, M can be as large as o(T ), or equivalently o( √ N ), where N is the size of the who...

work page 2023

[7] [13]

The proof follows very similarly to the one for the convex case

Heterogeneous Nonconvex Case. The proof follows very similarly to the one for the convex case. Here, we have for η ≤ 1 L ((Dorfman et al., 2024)): 1 T TX t=1 E[∥∇f (xt)∥2] ≤ 2∆1 T η + ηL T TX t=1 EV 2 t . (120) where ∆1 = f (x1) − f (x∗) and V 2 t = E[∥˜gt − ∇f (xt)∥2 |xt]. Plugging in the expression for V 2 t in Eq. (113) yields: 1 T TX t=1 E[∥∇f (xt)∥2]...

work page 2024

[8] [52]

1 (pl t,i)2 (∆l t,i)2 # (47) amd by writing the expectation w.r.t pl explicitly, we have: E[∥˜gt,i∥2] = LX l=1

Since the Exponent is E = ⌊log2(e)⌋ + 1023, the Floating-Point biased compressor satisfies Eq. (4) with α = 1 − 2−l. We apply the MLMC scheme with the Floating-Point Compressor and thus only need to transmit the residual gl t,i − gl−1 t,i . The residual has the same Exponent and Sign bits as the original entry, but contains only one information bit of the...

work page 2026