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Minimax Private Estimation of Smooth Optimal-Transport Maps

NeurIPS 2026 · Federated learning and privacy › Differential privacy · approach: Local-DP and private estimation · 0 citations (OpenAlex)

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Proposes differentially private plug-in estimators for smooth optimal transport maps based on private wavelet density estimation (central and local DP) and a private-quantile reduction for d=1, claiming near-minimax rates with matching lower bounds.

Problem

Estimating smooth optimal transport maps between two distributions from samples under central and local differential privacy constraints, with minimax-optimal rates.

Approach

Privatize boundary-corrected Daubechies wavelet coefficients of the source and target densities via Laplace noise, then compute the OT map between the resulting proper density surrogates (post-processing); in d=1 reduce OT to matching private quantiles.

Key results

• Theoretical: private plug-in estimator achieves rate max(n^{-2(s+1)/(2s+d)}, (pi_{n,eps})^{-2(s+1)/(s+d)}) for d>=3, matching the derived lower bound up to log factors for d>=2 under central and local DP.
• d=1 central DP quantile-based estimator achieves near-minimax rate PolyLog(n,eps) * max(1/n, 1/(n^2 eps^2)).
• Experiments on synthetic unif-unif benchmarks (d=2,3) show private smooth estimators can outperform the non-private empirical OT baseline in d=2 at larger n; the resampling step is the main computational bottleneck.

Surprising findings

• Under privacy the parametric-rate threshold shifts: privacy makes dimension matter already at d=2 rather than d=3 as in the non-private setting.
• Private smooth estimators can outperform the naive non-private empirical OT baseline in d=2 in experiments.

Limitations

• Computational bottleneck from resampling synthetic samples from surrogate densities before OT computation.
• Local-DP d=1 rate not shown to be minimax optimal; upper bound gap remains.
• Requires density lower/upper bounds and Hölder smoothness assumptions; epsilon = Omega(n^{-1+gamma}) condition for d=1 quantile result.

Experimental setup

Datasets: synthetic source-target distribution pairs (unif-unif, etc.)

Benchmarks: synthetic OT benchmarks with known transport maps

Metrics: mean squared error (L2(P_X))

Baselines: EOT baseline (entropically regularized OT map) empirical OT barycentric projection private semi-dual OT estimator (central DP)

Builds on: Brenier-potential semi-dual estimator (non-private minimax OT) stability bounds for smooth OT maps entropic OT barycentric projection Laplace mechanism

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