# Private Stochastic Optimization With Large Worst-Case Lipschitz Parameter

arXiv: https://arxiv.org/abs/2209.07403 · HTML: https://mlsanity.com/paper/2209.07403/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 63 references that are in the map.
- First posted on arXiv 15 Sep 2022.

## Builds on (most cited first, 30 of 35)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Membership Inference Attacks against Machine Learning Models](https://mlsanity.com/paper/1610.05820/) (2016)
- [Extracting Training Data from Large Language Models](https://mlsanity.com/paper/2012.07805/) (2020)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Differentially Private Learning with Adaptive Clipping](https://mlsanity.com/paper/1905.03871/) (2019)
- [Private Stochastic Convex Optimization with Optimal Rates](https://mlsanity.com/paper/1908.09970/) (2019)
- [Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses](https://mlsanity.com/paper/2006.06914/) (2020)
- [Private Stochastic Convex Optimization: Optimal Rates in Linear Time](https://mlsanity.com/paper/2005.04763/) (2020)
- [Understanding Gradient Clipping in Private SGD: A Geometric Perspective](https://mlsanity.com/paper/2006.15429/) (2020)
- [Differentially Private Empirical Risk Minimization Revisited: Faster and More General](https://mlsanity.com/paper/1802.05251/) (2018)
- [Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08157/) (2020)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [Private Stochastic Convex Optimization: Optimal Rates in ℓ1 Geometry](https://mlsanity.com/paper/2103.01516/) (2021)
- [Unified Optimal Analysis of the (Stochastic) Gradient Method](https://mlsanity.com/paper/1907.04232/) (2019)
- [Private Mean Estimation of Heavy-Tailed Distributions](https://mlsanity.com/paper/2002.09464/) (2020)
- [Differentially Private Assouad, Fano, and Le Cam](https://mlsanity.com/paper/2004.06830/) (2020)
- [On Differentially Private Stochastic Convex Optimization with Heavy-tailed Data](https://mlsanity.com/paper/2010.11082/) (2020)
- [Non-Euclidean Differentially Private Stochastic Convex Optimization: Optimal Rates in Linear Time](https://mlsanity.com/paper/2103.01278/) (2021)
- [Improved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data](https://mlsanity.com/paper/2106.01336/) (2021)
- [Private Adaptive Gradient Methods for Convex Optimization](https://mlsanity.com/paper/2106.13756/) (2021)
- [Differentially Private Stochastic Optimization: New Results in Convex and Non-Convex Settings](https://mlsanity.com/paper/2107.05585/) (2021)
- [High Dimensional Differentially Private Stochastic Optimization with Heavy-tailed Data](https://mlsanity.com/paper/2107.11136/) (2021)
- [Private Federated Learning Without a Trusted Server: Optimal Algorithms for Convex Losses](https://mlsanity.com/paper/2106.09779/) (2021)
- [Private Non-Convex Federated Learning Without a Trusted Server](https://mlsanity.com/paper/2203.06735/) (2022)
- [Optimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits](https://mlsanity.com/paper/2106.02575/) (2021)
- [Private Non-smooth Empirical Risk Minimization and Stochastic Convex Optimization in Subquadratic Steps](https://mlsanity.com/paper/2103.15352/) (2021)
- [Beyond Uniform Lipschitz Condition in Differentially Private Optimization](https://mlsanity.com/paper/2206.10713/) (2022)
- [Private Multiplicative Weights Beyond Linear Queries](https://mlsanity.com/paper/1407.1571/) (2014)

## Built on by (most cited first, 3 of 3)

- [Optimal Differentially Private Model Training with Public Data](https://mlsanity.com/paper/2306.15056/) (2023)
- [Optimal Rates for Pure varepsilon-Differentially Private Stochastic Convex Optimization with Heavy Tails](https://mlsanity.com/paper/2604.06492/) (2026)
- [Near-Optimal Machine Unlearning Utility for Smooth Strongly Convex Losses](https://mlsanity.com/paper/2606.01527/) (2026)

## Abstract (arXiv)

We study differentially private (DP) stochastic optimization (SO) with loss functions whose worst-case Lipschitz parameter over all data may be extremely large or infinite. To date, the vast majority of work on DP SO assumes that the loss is uniformly Lipschitz continuous (i.e. stochastic gradients are uniformly bounded) over data. While this assumption is convenient, it often leads to pessimistic risk bounds. In many practical problems, the worst-case (uniform) Lipschitz parameter of the loss over all data may be huge due to outliers and/or heavy-tailed data. In such cases, the risk bounds for DP SO, which scale with the worst-case Lipschitz parameter, are vacuous. To address these limitations, we provide improved risk bounds that do not depend on the uniform Lipschitz parameter. Following a recent line of work [WXDX20, KLZ22], we assume that stochastic gradients have bounded k-th order moments for some k ≥ 2. Compared with works on uniformly Lipschitz DP SO, our risk bounds scale with the k-th moment instead of the uniform Lipschitz parameter of the loss, allowing for significantly faster rates in the presence of outliers and/or heavy-tailed data. For smooth convex loss functions, we provide linear-time algorithms with state-of-the-art excess risk. We complement our excess risk upper bounds with novel lower bounds. In certain parameter regimes, our linear-time excess risk bounds are minimax optimal. Second, we provide the first algorithm to handle non-smooth convex loss functions. To do so, we develop novel algorithmic and stability-based proof techniques, which we believe will be useful for future work in obtaining optimal excess risk. Finally, our work is the first to address non-convex non-uniformly Lipschitz loss functions satisfying the Proximal-PL inequality; this covers some practical machine learning models. Our Proximal-PL algorithm has near-optimal excess risk.