Private Stochastic Convex Optimization with Optimal Rates (1908.09970v1)

Abstract: We study differentially private (DP) algorithms for stochastic convex optimization (SCO). In this problem the goal is to approximately minimize the population loss given i.i.d. samples from a distribution over convex and Lipschitz loss functions. A long line of existing work on private convex optimization focuses on the empirical loss and derives asymptotically tight bounds on the excess empirical loss. However a significant gap exists in the known bounds for the population loss. We show that, up to logarithmic factors, the optimal excess population loss for DP algorithms is equal to the larger of the optimal non-private excess population loss, and the optimal excess empirical loss of DP algorithms. This implies that, contrary to intuition based on private ERM, private SCO has asymptotically the same rate of $1/\sqrt{n}$ as non-private SCO in the parameter regime most common in practice. The best previous result in this setting gives rate of $1/n^{1/4}$. Our approach builds on existing differentially private algorithms and relies on the analysis of algorithmic stability to ensure generalization.

• The paper demonstrates that a differentially private algorithm using noisy mini-batch SGD achieves an optimal O(1/√n) excess loss, matching non-private rates.
• It leverages algorithmic stability and the Moreau-Yosida envelope to extend optimal performance from smooth to non-smooth loss functions.
• The study underlines practical implications for privacy-sensitive machine learning by ensuring high performance without significant computational overhead.

Overview of "Private Stochastic Convex Optimization with Optimal Rates"

The paper "Private Stochastic Convex Optimization with Optimal Rates," authored by Raef Bassily, Vitaly Feldman, Kunal Talwar, and Abhradeep Thakurta, addresses the task of achieving differential privacy in the context of stochastic convex optimization (SCO) while maintaining optimal performance metrics. This research explores the challenge of minimizing population loss with differential privacy constraints using independent and identically distributed (i.i.d.) samples drawn from a distribution over convex and Lipschitz loss functions.

Main Contributions

1. Optimal Excess Population Loss:
   • The authors bridge the existing gap in the bounds for excess population loss in differentially private settings compared to non-private settings. They demonstrate that, for practical parameter regimes, differential privacy incurs no additional asymptotic cost over non-private SCO. Specifically, the optimal excess population loss achievable with differentially private algorithms can reach a rate of $O(1/\sqrt{n})$, as in the non-private case, surpassing previous bounds of $O(1/n^{1/4})$.
2. Algorithmic Approach:
   • The paper presents a differential privacy (DP) algorithm based on noisy mini-batch stochastic gradient descent (SGD). This method leverages the notion of algorithmic stability to ensure generalization, achieving an excess population loss rate of $O(1/\sqrt{n})$ under smoothness assumptions common in the stochastic optimization literature.
3. Relaxation of Smoothness Assumptions:
   • The authors extend their method to handle non-smooth loss functions by utilizing the Moreau-Yosida envelope, a smoothing technique that retains the core benefits of convexity and smoothness. This generalization is crucial as it allows their algorithm to perform effectively across a broader class of optimization problems without losing the optimal population loss rate.
4. Objective Perturbation and Efficiency:
   • Building on classical results, the paper also discusses the use of objective perturbation, a method known to achieve DP under specific conditions such as low-rank Hessians. The authors enhance this approach to maintain optimal bounds on excess population loss while achieving near-linear gradient evaluation complexity, a significant improvement for large-scale data contexts.

Theoretical and Practical Implications

The theoretical advancements presented in this paper have profound implications for the deployment of machine learning models in privacy-sensitive applications. By achieving rates traditionally reserved for non-private algorithms, the research assures practitioners that privacy-preserving solutions do not necessitate significant performance compromises. This finding supports the practical use of differentially private approaches in real-world scenarios where data sensitivity is paramount.

Future Directions

The paper highlights various avenues for further exploration:

• Algorithmic Efficiency: Advancing algorithms to maintain the optimal excess population loss while further minimizing computational overhead remains a key focus area. This involves developing methods that are robust to a wider array of assumptions than those considered here.
• Complexity and Scalability: The application of these techniques to high-dimensional data and models with complex structures presents ongoing challenges. Improving the efficiency and scalability of private SCO algorithms is crucial for their applicability across various domains.

In conclusion, this paper makes significant strides in aligning differential privacy with the rigorous performance standards of stochastic optimization. Its insights contribute towards a deeper understanding of how privacy constraints influence algorithmic behavior and set the stage for robust, privacy-preserving optimizers in future machine learning pipelines.