Accelerating Sinkhorn for Entropy-Regularized Optimal Transport

Abstract: We propose Acc-Sinkhorn, a simple accelerated variant of Sinkhorn for entropy-regularized optimal transport (EOT). The method is derived from a bilevel optimization view: Sinkhorn row scaling solves the inner variable $u$ exactly and defines the reduced dual objective $f(v)=\min_u F(u,v)$, while the remaining column scaling is a unit-step dual mirror descent step in $v$. This structure yields a Hessian-driven Nesterov acceleration that keeps Sinkhorn's scaling form and per-iteration cost, using only extrapolated combinations of Sinkhorn iterates. We prove an $\mathcal{O}(1/k<sup>2)$ rate under a verifiable stability condition. For an $\varepsilon$-approximation of unregularized OT, the resulting complexity is $\widetilde{\mathcal{O}}(n<sup>2/\varepsilon)$, improved from $\widetilde{\mathcal{O}}(n<sup>2/\varepsilon<sup>2)$ for Sinkhorn. On synthetic problems, color transfer, and word alignment, Acc-Sinkhorn gives a $10\times$--$30\times$ speedup over Sinkhorn at small regularization.

• The paper introduces Acc-Sinkhorn, a Hessian-driven Nesterov acceleration that preserves Sinkhorn scaling and its O(nm) per-iteration cost while achieving an O(1/k²) rate under a stability condition.
• The method improves the estimated complexity of approximating unregularized optimal transport to Õ(n²/ε), compared with Õ(n²/ε²) for standard Sinkhorn, by balancing optimization error with entropic bias.
• Experiments on synthetic data, color transfer, and word alignment report roughly 10×–30× speedups at small regularization without meaningful loss in transport or alignment quality, although the practical step size lacks a complete theoretical guarantee.

Overview

This paper introduces Acc-Sinkhorn, an accelerated variant of the Sinkhorn algorithm for entropy-regularized optimal transport (EOT). The method is derived from a bilevel optimization perspective: Sinkhorn's row scaling exactly minimizes the dual objective over one block of variables, and the remaining column update is shown to be a unit-step dual mirror descent step on the reduced outer objective. This structure admits a Hessian-driven Nesterov acceleration that preserves Sinkhorn's scaling form and $\mathcal{O}(nm)$ per-iteration cost. The authors prove an $\mathcal{O}(1/k^2)$ convergence rate under a verifiable stability condition, yielding a total complexity of $\widetilde{\mathcal{O}}(n^2/\varepsilon)$ for approximating unregularized OT, improved from the $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$ complexity of standard Sinkhorn. Experiments report $10\times$–$30\times$ speedups at small regularization.

Bilevel and mirror descent structure of Sinkhorn

The paper works with the EOT dual objective $F(u,v)$, where $u$ and $v$ are Lagrange multipliers for the row and column marginal constraints. For fixed $v$, minimizing over $\mathcal{O}(1/k^2)$0 has a closed-form solution identical to Sinkhorn's row-scaling step. This defines the reduced objective

$\mathcal{O}(1/k^2)$1

whose gradient is the column-marginal residual after exact row matching. The key geometric observation is that with the mirror function $\mathcal{O}(1/k^2)$2, the Sinkhorn column update is exactly unit-step dual mirror descent:

$\mathcal{O}(1/k^2)$3

Moreover, using the Schur-complement form of the Hessian, $\mathcal{O}(1/k^2)$4, the paper establishes dual relative smoothness with constant $\mathcal{O}(1/k^2)$5, i.e., $\mathcal{O}(1/k^2)$6. This recovers the known $\mathcal{O}(1/k^2)$7 sublinear rate via Maddison et al.'s framework.

The shift invariance $\mathcal{O}(1/k^2)$8 means $\mathcal{O}(1/k^2)$9 is not strongly convex; the paper handles this by restricting to $\widetilde{\mathcal{O}}(n^2/\varepsilon)$0 via a normalized Sinkhorn iteration (which changes iterates only by multiples of $\widetilde{\mathcal{O}}(n^2/\varepsilon)$1 and leaves the primal plan unchanged), proving coercivity on $\widetilde{\mathcal{O}}(n^2/\varepsilon)$2, a Polyak–Łojasiewicz inequality on sublevel sets, and hence linear convergence.

The Acc-Sinkhorn algorithm

Applying Hessian-driven Nesterov accelerated gradient flow to the reduced problem yields a two-sequence scheme in $\widetilde{\mathcal{O}}(n^2/\varepsilon)$3 with step size $\widetilde{\mathcal{O}}(n^2/\varepsilon)$4, where $\widetilde{\mathcal{O}}(n^2/\varepsilon)$5 is a strong-convexity parameter. In its practical form, each iteration requires only one normalized Sinkhorn step plus negligible vector operations: the Sinkhorn map computed for the $\widetilde{\mathcal{O}}(n^2/\varepsilon)$6-update is reused in the next $\widetilde{\mathcal{O}}(n^2/\varepsilon)$7-update. Because the true strong-convexity modulus is unknown (and degenerates along $\widetilde{\mathcal{O}}(n^2/\varepsilon)$8), the method embeds this inner scheme in a homotopy loop that halves $\widetilde{\mathcal{O}}(n^2/\varepsilon)$9 each round while growing the inner iteration count as $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$0. The result is a parameter-free outer procedure whose smoothness constant is fixed by geometry ($\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$1).

Convergence theory

The analysis uses the Lyapunov function

$\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$2

where $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$3 is a state-dependent diagonal metric satisfying $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$4. The continuous-time flow exhibits exponential Lyapunov decay up to perturbation terms; discretization introduces additional cross terms controlled by choosing $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$5 so that they become non-positive under dual relative smoothness.

The main theorem states that under a boundedness assumption on the iterates, the homotopy algorithm achieves

$\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$6

so reaching accuracy $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$7 requires $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$8 iterations, versus $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$9 for Sinkhorn. Combined with the entropic approximation error $10\times$0 and balancing optimization error against regularization bias ($10\times$1), this gives overall complexity $10\times$2 for an $10\times$3-approximation of the unregularized OT cost — matching the best accelerated methods (APDAGD achieves $10\times$4) while retaining Sinkhorn's per-iteration cost and requiring no line search.

Important caveat: the full guarantee holds under a verifiable stability condition (boundedness of iterates and control of the metric-variation term $10\times$5), which is enforced for sufficiently small $10\times$6 via line search. The practically used step size $10\times$7 lies at the boundary of what the proof covers; the authors observe stable behavior empirically but cannot yet justify it theoretically. Removing this condition is stated as open future work.

Numerical experiments

All experiments were run in MATLAB R2025b on an Apple M4 laptop, with log-domain implementations for numerical stability and stopping based on total $10\times$8 marginal violation.

Task	Setting	Speedup regime
Synthetic	$10\times$9, $30\times$0	$30\times$1 faster wall-clock
Color transfer	$30\times$2, RGB cost, $30\times$3	At $30\times$4: 239 iterations / 3.24 s vs. 3507 iterations / 49.69 s (~15×)
Word alignment	En–Fr, $30\times$5, cosine cost, top-1 ≈ 82%	At $30\times$6: 667 iterations / 1.49 s vs. 20320 iterations / 49.36 s (~33×)

In color transfer, both solvers produce comparable plans; Acc-Sinkhorn's advantage grows sharply as $30\times$7 decreases, precisely the regime where sharper plans are desired but Sinkhorn becomes slowest. In word alignment, accuracy is essentially unchanged between solvers (top-1 within about 1 point across $30\times$8 values), indicating the acceleration does not degrade solution quality. The synthetic results also indicate robustness to the homotopy hyperparameters $30\times$9 and $F(u,v)$0. No error bars or repeated-run statistics are reported; the authors note the experiments are illustrative rather than a comprehensive empirical evaluation.

Limitations and open questions

The central theoretical gap is the discrepancy between the proven guarantee and practical configuration: the accelerated rate is established only under the stability condition, which holds for sufficiently small $F(u,v)$1 (enforceable by backtracking line search), whereas experiments use the limiting choice $F(u,v)$2 without line search. The obstruction is bounding the metric-variation term uniformly; the authors show it is small in practice but leave its uniform control unresolved. A second open question is whether the boundedness lemma can be extended beyond $F(u,v)$3 to cover the observed regime directly. Additionally, the empirical evidence is limited to problems of dimension up to $F(u,v)$4 on CPU, and no statistical significance testing accompanies the timing comparisons.

Conclusion

The paper reframes Sinkhorn as exact block minimization followed by unit-step dual mirror descent with relative smoothness constant $F(u,v)$5, and exploits this structure to construct Acc-Sinkhorn, a Hessian-driven Nesterov acceleration that preserves Sinkhorn's scaling form and per-iteration cost. Under a verifiable stability condition, the method attains an $F(u,v)$6 rate and $F(u,v)$7 complexity for approximating unregularized OT, with $F(u,v)$8–$F(u,v)$9 empirical speedups at small regularization on synthetic data, color transfer, and word alignment. Closing the gap between the conservative provable step size and the empirically effective $u$0 remains the principal open problem.