[Paper Review] Private Stochastic Convex Optimization with Optimal Rates

Raef Bassily, Vitaly Feldman+2 more|arXiv (Cornell University)|Jan 1, 2019

Stochastic Gradient Optimization Techniques56 citations

TL;DR

This paper establishes the optimal excess population loss rate for differentially private stochastic convex optimization (SCO), showing it matches the non-private rate of $1/\sqrt{n}$ up to logarithmic factors. By leveraging algorithmic stability and refining existing private algorithms, the authors close a long-standing gap between private and non-private SCO performance.

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.

Motivation & Objective

• To close the gap between known bounds for excess population loss in differentially private stochastic convex optimization (SCO) and non-private SCO.
• To determine whether private SCO inherits the same convergence rate as non-private SCO in practical parameter regimes.
• To establish tight bounds on the optimal excess population loss for DP algorithms, resolving an open question in private optimization.

Proposed method

• The authors analyze the generalization properties of differentially private algorithms using algorithmic stability, a key tool for bounding population loss.
• They build on existing private optimization algorithms and refine their analysis to achieve tighter generalization guarantees.
• The approach involves relating the excess population loss to both the optimal non-private rate and the optimal excess empirical loss of private algorithms.
• By combining stability analysis with known results on private empirical risk minimization, they derive a tight characterization of the optimal population loss rate.
• They show that the optimal excess population loss is the maximum of two terms: the non-private optimal rate and the private empirical loss rate.
• The analysis is conducted under standard assumptions: convex and Lipschitz loss functions, i.i.d. samples from a distribution.

Experimental results

Research questions

• RQ1 What is the optimal excess population loss rate for differentially private stochastic convex optimization?
• RQ2 Does private SCO inherit the same $1/\sqrt{n}$ convergence rate as non-private SCO in practical settings?
• RQ3 How does the optimal excess population loss for private SCO compare to the optimal excess empirical loss of private algorithms?
• RQ4 Can algorithmic stability be used to tightly characterize the generalization error in private SCO?
• RQ5 Is there a fundamental gap between private and non-private SCO rates, or can private algorithms achieve the same asymptotic rate?

Key findings

• The optimal excess population loss for differentially private stochastic convex optimization is, up to logarithmic factors, equal to the maximum of the optimal non-private excess population loss and the optimal excess empirical loss of private algorithms.
• The excess population loss rate for private SCO matches the non-private rate of $1/\sqrt{n}$, resolving a long-standing open question.
• This result implies that private SCO does not suffer from a worse asymptotic rate than non-private SCO in the standard $n$-sample regime.
• The previous best-known rate for private SCO was $1/n^{1/4}$, which is significantly slower than $1/\sqrt{n}$.
• The analysis demonstrates that the gap between private and non-private SCO rates is not inherent, but rather a consequence of prior algorithmic and analytical limitations.
• The work establishes that algorithmic stability is sufficient to achieve tight generalization bounds in private SCO, enabling the derivation of optimal rates.