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arxiv: 2606.28973 · v1 · pith:NNMBOB7Bnew · submitted 2026-06-27 · 🧮 math.OC

Sharp O(1/k) convergence rate for the Sinkhorn algorithm via a local analysis

Guillaume Wang This is my paper

Pith reviewed 2026-06-30 08:35 UTC · model grok-4.3

classification 🧮 math.OC
keywords Sinkhorn algorithmoptimal transportconvergence rateentropy regularizationbipartite graphmarginal errorrelative entropylocal analysis
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The pith

The Sinkhorn algorithm converges at the sharp rate of O(1/k) in ℓ1 marginal error and joint relative entropy.

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

The paper establishes that the Sinkhorn algorithm for entropy-regularized optimal transport converges at the rate O(1/k) in both the ℓ1-norm of the marginal error and the joint relative entropy. This rate is known to be optimal when the problem is asymptotically scalable. The proof proceeds by analyzing the bipartite graph of the problem and handling edges that carry positive mass in the optimal plan separately from those that do not, yielding a local bound that is then extended to a global one using a previous result showing an almost-sharp rate up to a logarithmic factor.

Core claim

We prove that the Sinkhorn algorithm converges at the rate of O(1/k) in ℓ1-norm marginal error and in joint relative entropy, which is known to be sharp in the asymptotically scalable case. The proof is based on examining the bipartite graph associated to the entropy-regularized optimal transport problem, and treating differently the edges that are assigned a positive mass in the optimal transport plan vs. those that are not. This yields a local convergence bound with the sharp rate, which is bootstrapped into a global bound using the author's previous result where we showed an almost-sharp rate up to a logarithmic factor.

What carries the argument

The bipartite graph of the entropy-regularized optimal transport problem, with separate contraction analysis on edges carrying positive mass in the optimal plan versus the remaining edges.

Load-bearing premise

The edges of the bipartite graph can be partitioned according to whether they carry positive mass in the optimal transport plan, allowing a separate local contraction that lifts to a global bound.

What would settle it

Run Sinkhorn iterations on an instance whose optimal plan support is known exactly and measure whether the ℓ1 marginal error decreases linearly in 1/k after the initial phase or stalls at a slower rate.

Figures

Figures reproduced from arXiv: 2606.28973 by Guillaume Wang.

Figure 1
Figure 1. Figure 1: An illustration of the DM decomposition Prop. 3.1. In the case represented here, p = 3 and the DM interaction DAG is the maximal one. In (a) and (b), instead of representing specific edges, for readability we used transparent bands to indicate the Ip, Jq for which E ∩ (Ip × Jq) ̸= ∅; so for example E does not contain any edge with one endpoint in I2 and the other in J1. Let us show by induction that for an… view at source ↗
read the original abstract

We prove that the Sinkhorn algorithm converges at the rate of $O(1/k)$ in $\ell_1$-norm marginal error and in joint relative entropy, which is known to be sharp in the asymptotically scalable case. The proof is based on examining the bipartite graph associated to the entropy-regularized optimal transport problem, and treating differently the edges that are assigned a positive mass in the optimal transport plan vs. those that are not. This yields a local convergence bound with the sharp rate, which is bootstrapped into a global bound using the author's previous result in arXiv:2604.26265 where we showed an almost-sharp rate up to a logarithmic factor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to prove a sharp O(1/k) convergence rate for the Sinkhorn algorithm in both ℓ1-norm marginal error and joint relative entropy for entropy-regularized optimal transport. The argument proceeds by partitioning the edges of the associated bipartite graph according to whether they receive positive mass in the optimal transport plan, establishing a local contraction at the sharp rate on this partition, and then bootstrapping the local bound into a global O(1/k) guarantee by invoking the author's earlier result (arXiv:2604.26265), which had obtained an almost-sharp rate up to a logarithmic factor.

Significance. If the local-to-global bootstrapping argument can be made rigorous without reintroducing a logarithmic factor, the result would establish the first sharp global rate for Sinkhorn iterations, matching the known lower bound in the asymptotically scalable regime. This would constitute a meaningful technical advance in the quantitative analysis of entropic optimal transport algorithms.

major comments (2)
  1. [Abstract] Abstract (final sentence): the claim that the local O(1/k) bound is bootstrapped into a global sharp O(1/k) bound via arXiv:2604.26265 is stated at a high level only. It is not shown whether the combination of the new local contraction with the prior almost-sharp result avoids reintroducing a log(k) factor during the transient phase before the iterates enter the local regime; this step is load-bearing for the headline global rate.
  2. The manuscript supplies only the high-level proof strategy; no explicit contraction mapping, error-term estimates, or analysis of the edge-partition under degeneracy (zero-mass edges or non-unique optimal plans) is visible. Consequently the local sharp-rate claim cannot be verified from the provided text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive feedback on our manuscript. Below we respond point-by-point to the major comments.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the claim that the local O(1/k) bound is bootstrapped into a global sharp O(1/k) bound via arXiv:2604.26265 is stated at a high level only. It is not shown whether the combination of the new local contraction with the prior almost-sharp result avoids reintroducing a log(k) factor during the transient phase before the iterates enter the local regime; this step is load-bearing for the headline global rate.

    Authors: The abstract is intentionally concise. In the body of the manuscript (Section 3), we show that the local contraction on the positive-mass subgraph is strict and applies once the marginal error falls below a fixed threshold δ > 0 that is independent of k (chosen from the asymptotic scalability assumption). The earlier global result (arXiv:2604.26265) is invoked only to guarantee that this threshold is reached after finitely many iterations whose number does not grow with k; the O(log k) factor therefore remains confined to a k-independent transient and does not pollute the subsequent 1/k tail. We will add a short clarifying sentence to the abstract and a remark after Theorem 3.1 to make this separation explicit. revision: yes

  2. Referee: [—] The manuscript supplies only the high-level proof strategy; no explicit contraction mapping, error-term estimates, or analysis of the edge-partition under degeneracy (zero-mass edges or non-unique optimal plans) is visible. Consequently the local sharp-rate claim cannot be verified from the provided text.

    Authors: The referee is correct that the excerpt supplied for review is high-level. The complete manuscript contains the explicit local contraction (Theorem 3.2), the error-term estimates obtained by restricting the KL divergence to the positive-mass edges (Lemmas 4.1–4.3), and the handling of degeneracy: zero-mass edges are shown to remain exactly zero after one iteration once the support is identified, while non-uniqueness is treated by a standard perturbation that preserves the O(1/k) rate. We will expand the introduction to signpost these sections more clearly so that the local analysis can be verified without reading the entire paper. revision: partial

Circularity Check

1 steps flagged

Global sharp O(1/k) rate obtained by bootstrapping local analysis onto self-authored prior result that only reached almost-sharp rate

specific steps
  1. self citation load bearing [Abstract]
    "This yields a local convergence bound with the sharp rate, which is bootstrapped into a global bound using the author's previous result in arXiv:2604.26265 where we showed an almost-sharp rate up to a logarithmic factor."

    The global sharp O(1/k) claim is justified solely by combining the new local analysis with the author's prior self-authored result (which stopped at almost-sharp); the load-bearing global step therefore reduces to that self-citation rather than being derived independently here.

full rationale

The paper's central claim of a sharp global O(1/k) convergence rate is achieved by deriving a local contraction on the edge partition and then explicitly bootstrapping it into a global bound via citation to the author's own earlier arXiv:2604.26265, which only established an almost-sharp rate (up to a logarithmic factor). This matches the self_citation_load_bearing pattern because the headline global sharpness depends on the prior self-work without an independent derivation of the global step in the present manuscript. No other patterns (self-definitional, fitted-input, etc.) are exhibited by the quoted text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard convex-analysis and graph-theoretic facts about entropy-regularized OT; no free parameters, ad-hoc constants, or new postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption The bipartite graph associated with the support of the entropy-regularized OT plan admits a well-defined partition into positive-mass and zero-mass edges.
    Invoked to obtain the local contraction that yields the sharp rate.

pith-pipeline@v0.9.1-grok · 5633 in / 1267 out tokens · 42484 ms · 2026-06-30T08:35:26.837745+00:00 · methodology

discussion (0)

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Reference graph

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