Private Zeroth-Order Nonsmooth Nonconvex Optimization
Qinzi Zhang, Hoang Tran, Ashok Cutkosky
TL;DR
This paper presents a novel private zeroth-order algorithm for nonconvex, nonsmooth stochastic optimization that achieves optimal complexity and offers privacy guarantees without additional cost under certain conditions.
Contribution
It introduces a new private zeroth-order method for nonsmooth nonconvex optimization with optimal complexity matching non-private algorithms.
Findings
Achieves $(eta,eta ho^2/2)$-Rényi differential privacy.
Finds $( au, heta)$-stationary points with optimal complexity.
Provides privacy "for free" when $ ho \,\ge\, \sqrt{d}\epsilon$.
Abstract
We introduce a new zeroth-order algorithm for private stochastic optimization on nonconvex and nonsmooth objectives. Given a dataset of size , our algorithm ensures -R\'enyi differential privacy and finds a -stationary point so long as . This matches the optimal complexity of its non-private zeroth-order analog. Notably, although the objective is not smooth, we have privacy ``for free'' whenever .
Peer Reviews
Decision·ICLR 2024 poster
The paper is technically solid. The private setting is both important and interesting. This was done without sacrificing a good complexity; the sample complexity is optimal in some regimes $\rho > \sqrt{d} \epsilon$. In general, I think this is a nice paper with a clear idea and with good explanations that show the motivation for each statement or its proof. I really like the authors explained why the naive approach would give the sub-optimal rate in the appendix.
1. The result heavily based on the previous paper on the non-private version (Cutkosky et al., 2023.) makes it seem a bit incremental. 2. Understandably, there is no space in the current ICLR format for experimental evaluation, this is something that could be looked at in the future.
1. This work investigates the important problem of nonconvex and nonsmooth optimization, which is a frequent setting in modern machine learning. It provides an efficient algorithm that finds a stationary point while attaining differential privacy. The sample complexity required matches its non-private analog. 2. The paper is technically solid, clearly presented, and well-structured, with the key proof step contained in the main text, so that it is easy for the readers to follow the key proof id
1. As discussed in the paper, the need to sample $d$ iid estimators for each data point seems to be less natural. 2. It would be good to have some discussions on how differential privacy is attained in other similar (e.g., 1st order) optimization problems.
The paper studies a new topic that aims to obtain private zeroth-order algorithms for nonsmooth nonconvex optimization, while most existing works for private nonconvex optimization focus on first-order algorithms for smooth objective functions. The proposed method creatively combines existing results and leads to a non-trivial convergence analysis. The presentation of the paper is well-structured and clearly introduces different components of this new algorithm.
1. The reason to study differentially private (DP) zeroth-order methods for nonsmooth nonconvex optimization is not well motivated in this paper. I agree it is important to study DP nonconvex optimization, and there is indeed a rich literature that focuses on first-order methods. The paper mentions some applications of zeroth-order methods where gradients can be hard to obtain, including reinforcement learning. However, there is no notion of the dataset in these applications, then it is not imme
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Taxonomy
TopicsOptimization and Variational Analysis · Advanced Optimization Algorithms Research · Mathematical Inequalities and Applications