Stochastic Inertial Krasnosel'skii-Mann Iteration Achieves Near-Optimal Sample Complexity
Tong Yang, Tao Jiang, Yuejie Chi, Ashok Cutkosky, Lin Xiao
stat.ML · cs.LG
Quick verdict
Stochastic KM + two inertial extrapolations attains Õ(ε⁻²) last-iterate sample complexity for nonexpansive fixed points; rigorous, no experiments shown.
Summary
The paper analyzes a stochastic inertial Krasnosel'skii–Mann (iKM) iteration for finding fixed points of nonexpansive operators in real Hilbert spaces. The method appends two inertial extrapolation steps to the stochastic KM iteration of Bravo and Cominetti (2024), using a single possibly biased stochastic oracle call per update: the oracle returns Tz_k + b_k + ξ_k with predictable bias and conditionally centered noise of variance at most σ². With a softplus-based parameter schedule governed by one tuning parameter a, Theorem 1 proves a last-iterate residual bound of order (1+1/a)D/K + (1+a)(σ log K/√K + B_K log K/K), where D = dist(x₁, Fix T) and B_K is the accumulated root-mean-square bias, valid simultaneously for x_K, y_K, z_K. When B_K = O(√K), this yields Õ(ε⁻²) sample complexity, matching the unbiased-oracle lower bound of Foster et al. (2019) up to a logarithmic factor and improving the O(ε⁻⁴) random-iterate rate known for stochastic KM; the authors state it is the first single-loop method achieving this for general nonexpansive maps without variance reduction or batching. With an exact oracle, the same scheme attains the worst-case-optimal O(1/K) deterministic last-iterate rate. Limitations: purely theoretical with no numerical experiments; guarantees are in expectation, not high probability; near-optimality holds only up to log factors and rests on a lower bound transferred from smooth convex optimization; the bias condition B_K = O(√K) is assumed rather than derived.
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Assessment at a glance
4
risk flags identified
● high 0
● medium 3
● low 1
5 strengths · 5 weaknesses · 5 author questions
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Key claims under review
- Stochastic iKM with the proposed softplus schedule (eqs. 10–12) satisfies a last-iterate residual bound O(1/K + σ log K/√K + B_K log K/K), simultaneously for x_K, y_K, and z_K (Theorem 1).
- When the accumulated RMS bias is O(√K), the method needs Õ(ε⁻²) oracle calls to reach residual ε, matching the unbiased-oracle lower bound of Foster et al. (2019) up to one log factor and improving the O(ε⁻⁴) random-iterate rate of stochastic KM (Bravo–Cominetti 2024, Cor. 5.4).
- To the authors' knowledge it is the first single-loop method for general nonexpansive fixed-point problems attaining this complexity without variance reduction or batching.
- With an exact oracle the same method attains the worst-case-optimal O(1/K) deterministic last-iterate rate, strictly improving classical KM's Θ(K⁻¹ᐟ²).
- Fast KM (Boț–Nguyen 2023) is a special case of deterministic iKM under a specific hyperparameter choice (Proposition 1, Appendix A).
Strengths
Dual optimality from one simple scheme
Theorem 1 (eq. 17) gives Õ(ε⁻²) stochastic complexity when B_K=O(√K) and, at σ=B_K=0, the worst-case-optimal O(1/K) deterministic last-iterate rate (Park–Ryu 2022, Thm 4.6) — one schedule interpolating both regimes, improving classical KM's Θ(K⁻¹ᐟ²) (Davis–Yin 2016).
Bias-robust, input-free algorithm
The oracle model T̂T_k(z_k) = Tz_k + b_k + ξ_k (eq. 6) with accumulated RMS bias B_K (eq. 7) requires no knowledge of the bias, σ, or D = dist(x₁, Fix T); the bound adapts automatically, unlike VR-GHAL which needs same-seed difference queries (§1.2).
Complete, self-contained schedule theory
Rigorous handling of measurability/integrability
Accurate positioning against prior art
Weaknesses
No empirical validation at all
Guarantee only in expectation
Assumed bias regime without examples
Log factors not shown necessary
Horizon-dependent schedule
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Scope & Claims
This is a pure optimization-theory paper (arXiv 2609.28543v1, stat.ML/cs.LG; authors from CMU, Meta FAIR, Yale, and BU). Setting: a nonexpansive T : H → H (eq. 3) with Fix T ≠ ∅, accessed only through stochastic responses T̂T_k(z_k) = Tz_k + b_k + ξ_k (eq. 6), where b_k is a predictable, history-dependent bias and ξ_k is conditionally centered noise with E‖ξ_k‖²|F_k ≤ σ² (eq. 8). The proposed stochastic inertial KM (iKM) iteration (eq. 4) adds two extrapolations (coefficients α_k, β_k) to the stochastic KM of Bravo–Cominetti (2024), keeping exactly one oracle call per update. Four claims structure the paper: (1) Theorem 1: the last-iterate bound C{(1+1/a)D/K + (1+a)(σ log K/√K + B_K log K/K)}, with D = dist(x₁, Fix T) and B_K the accumulated RMS bias (eq. 7), valid simultaneously for x_K, y_K, z_K; (2) under B_K = O(√K), horizon (eq. 18) O(D/ε + (σ+B)²ε⁻²log²((σ+B)/ε)) — Õ(ε⁻²) complexity within one log factor of the Foster et al. (2019, Thm 2) lower bound, improving the O(ε⁻⁴) random-iterate guarantee (Bravo–Cominetti 2024, Cor. 5.4); (3) the exact-oracle specialization attains the worst-case-optimal O(1/K) deterministic rate (Park–Ryu 2022, Thm 4.6), improving classical KM's Θ(K⁻¹ᐟ²) (Davis–Yin 2016); (4) Fast KM (Boț–Nguyen 2023) embeds as a deterministic iKM special case (Prop. 1, Appendix A).
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Frequently asked questions
What problem does this paper solve?
It determines how many noisy evaluations of a nonexpansive operator (for example, the update map of a first-order or operator-splitting method) are needed to find an approximate fixed point in a Hilbert space, when each evaluation may be both noisy and systematically biased.
What is the main result of the paper?
A single-loop algorithm making one oracle call per iteration reaches expected fixed-point residual ε with Õ(ε⁻²) total calls, provided the accumulated root-mean-square bias is O(√K). This matches a known lower bound up to a logarithmic factor and improves the previous best O(ε⁻⁴) rate for stochastic KM; with a noiseless oracle the same algorithm gets the optimal O(1/K) deterministic rate.
Does the method require variance reduction or mini-batching?
No. That is the central selling point: adding two inertial (momentum-like) extrapolation steps to stochastic KM suffices — one oracle call per iteration, no batches, no recursive variance reduction, and no need to know the bias or the distance to the solution set.
Does the paper include experiments or code?
No. It is a purely theoretical paper: theorems with complete proofs in the appendix plus a figure visualizing the parameter schedule. No numerical experiments, datasets, or code are reported.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Formula extraction flattened fractions and sub/superscripts (e.g., constraint (9) and schedule (10)–(12)); exact forms were cross-checked against derived relations such as (80) and (66), with minor residual uncertainty.
- Figure 1 is available only as extracted axis labels and legend text; visual shapes of the ω_k, α_k, β_k, λ_k curves could not be verified directly, and the β-panel tick values are ambiguous in extraction.
- Review is based on v1 full text (Sept 2026); the appendix appears complete through B.4.5, but page-level truncation beyond the provided text cannot be fully ruled out.
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