Academic

Optimal Rates for Pure {\varepsilon}-Differentially Private Stochastic Convex Optimization with Heavy Tails

arXiv:2604.06492v1 Announce Type: new Abstract: We study stochastic convex optimization (SCO) with heavy-tailed gradients under pure epsilon-differential privacy (DP). Instead of assuming a bound on the worst-case Lipschitz parameter of the loss, we assume only a bounded k-th moment. This assumption allows for unbounded, heavy-tailed stochastic gradient distributions, and can yield sharper excess risk bounds. The minimax optimal rate for approximate (epsilon, delta)-DP SCO is known in this setting, but the pure epsilon-DP case has remained open. We characterize the minimax optimal excess-risk rate for pure epsilon-DP heavy-tailed SCO up to logarithmic factors. Our algorithm achieves this rate in polynomial time with high probability. Moreover, it runs in polynomial time with probability 1 when the worst-case Lipschitz parameter is polynomially bounded. For important structured problem classes - including hinge/ReLU-type and absolute-value losses on Euclidean balls, ellipsoids, and pol

A
Andrew Lowy · · 1 min read · 24 views
#cs.LG #cs.CR #stat.ML

arXiv:2604.06492v1 Announce Type: new Abstract: We study stochastic convex optimization (SCO) with heavy-tailed gradients under pure epsilon-differential privacy (DP). Instead of assuming a bound on the worst-case Lipschitz parameter of the loss, we assume only a bounded k-th moment. This assumption allows for unbounded, heavy-tailed stochastic gradient distributions, and can yield sharper excess risk bounds. The minimax optimal rate for approximate (epsilon, delta)-DP SCO is known in this setting, but the pure epsilon-DP case has remained open. We characterize the minimax optimal excess-risk rate for pure epsilon-DP heavy-tailed SCO up to logarithmic factors. Our algorithm achieves this rate in polynomial time with high probability. Moreover, it runs in polynomial time with probability 1 when the worst-case Lipschitz parameter is polynomially bounded. For important structured problem classes - including hinge/ReLU-type and absolute-value losses on Euclidean balls, ellipsoids, and polytopes - we achieve the same excess-risk guarantee in polynomial time with probability 1 even when the worst-case Lipschitz parameter is infinite. Our approach is based on a novel framework for privately optimizing Lipschitz extensions of the empirical loss. We complement our excess risk upper bound with a novel high probability lower bound.

Executive Summary

This article addresses a significant gap in the literature on differentially private stochastic convex optimization (SCO) by establishing the minimax optimal excess-risk rate for pure epsilon-differential privacy (DP) in settings with heavy-tailed gradients. Moving beyond traditional Lipschitz assumptions, the authors consider a bounded k-th moment for gradients, which is more robust for heavy-tailed distributions. They propose a novel algorithm that achieves this optimal rate, up to logarithmic factors, in polynomial time. Crucially, their framework extends to scenarios where the worst-case Lipschitz parameter is infinite, particularly for structured loss functions. This work provides both an algorithmic upper bound and a high-probability lower bound, rigorously characterizing the performance limits.

Key Points

  • ▸ Establishes the minimax optimal excess-risk rate for pure epsilon-DP SCO with heavy-tailed gradients, a previously open problem.
  • ▸ Relaxes the standard Lipschitz assumption, instead assuming a bounded k-th moment for gradients, accommodating unbounded, heavy-tailed distributions.
  • ▸ Proposes a novel algorithm that achieves the optimal rate (up to log factors) in polynomial time, even with infinite worst-case Lipschitz parameters for certain problem classes.
  • ▸ Introduces a new framework for privately optimizing Lipschitz extensions of the empirical loss.
  • ▸ Provides both an algorithmic upper bound and a novel high-probability lower bound, ensuring a comprehensive characterization of the rate.

Merits

Addresses a Critical Open Problem

Successfully closes a significant theoretical gap by characterizing the minimax optimal rate for pure epsilon-DP in heavy-tailed SCO, extending prior work on approximate DP.

Robustness to Heavy Tails and Unbounded Gradients

The shift from Lipschitz bounds to bounded k-th moments is a crucial methodological advancement, making the framework applicable to more realistic and challenging datasets where gradients can be heavy-tailed and even unbounded.

Algorithmic Practicality

The proposed algorithm achieves the optimal rate in polynomial time, suggesting its potential for practical implementation, especially for structured problems where it guarantees polynomial time even with infinite Lipschitz parameters.

Rigorous Theoretical Contribution

The dual contribution of an algorithmic upper bound and a novel high-probability lower bound provides a complete and robust theoretical characterization of the problem's limits.

Novel Framework

The approach based on privately optimizing Lipschitz extensions of the empirical loss is an innovative technique that could have broader applicability in private optimization.

Demerits

Logarithmic Factors in Optimality

While optimal up to logarithmic factors, the exact constant factors and the impact of these logarithmic terms in real-world scenarios might warrant further investigation for tighter bounds.

Computational Complexity for General Cases

While polynomial time, the practical constant factors and exact degree of the polynomial for the general case (where Lipschitz parameter is not polynomially bounded) could still be high, impacting scalability for very large datasets.

Assumption on k-th Moment

While more flexible than Lipschitz, the assumption of a bounded k-th moment still requires some prior knowledge or estimation about the gradient distribution, which might not always be perfectly available in practice.

Expert Commentary

This paper represents a substantial theoretical advance in the intersection of differential privacy and robust optimization. The transition from worst-case Lipschitz assumptions to bounded k-th moments is not merely a technical tweak but a fundamental shift that significantly broadens the applicability of DP SCO to real-world, often messy, datasets. The elegance of achieving minimax optimality for pure epsilon-DP, a stricter privacy guarantee than approximate DP, is particularly noteworthy. The novel framework for private Lipschitz extensions is a powerful contribution, potentially serving as a building block for future research. While optimal up to log factors, the practical implications of these factors in specific applications warrant empirical validation. The work's strength lies in its comprehensive theoretical treatment, providing both algorithmic construction and tight lower bounds. This article will undoubtedly become a foundational reference for researchers and practitioners grappling with privacy-preserving machine learning in the presence of heavy-tailed data, pushing the boundaries of what is considered achievable under strong privacy constraints.

Recommendations

  • ✓ Future work could focus on tightening the logarithmic factors or exploring conditions under which they can be eliminated, potentially leading to even sharper bounds.
  • ✓ Empirical evaluation of the proposed algorithm on diverse real-world heavy-tailed datasets would be valuable to demonstrate its practical performance and scalability.
  • ✓ Investigate the applicability of the 'privately optimizing Lipschitz extensions' framework to other private learning settings, such as non-convex optimization or distributed learning.
  • ✓ Explore the sensitivity of the algorithm's performance to the choice or estimation of the k-th moment parameter, and develop methods for robust parameter selection.

Sources

Original: arXiv - cs.LG

Related Articles