Private Stochastic Convex Optimization: Optimal Rates in Linear Time (2005.04763v1)

Abstract: We study differentially private (DP) algorithms for stochastic convex optimization: the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. (2019) has established the optimal bound on the excess population loss achievable given $n$ samples. Unfortunately, their algorithm achieving this bound is relatively inefficient: it requires $O(\min{n^{3/2}, n^{5/2}/d})$ gradient computations, where $d$ is the dimension of the optimization problem. We describe two new techniques for deriving DP convex optimization algorithms both achieving the optimal bound on excess loss and using $O(\min{n, n^2/d})$ gradient computations. In particular, the algorithms match the running time of the optimal non-private algorithms. The first approach relies on the use of variable batch sizes and is analyzed using the privacy amplification by iteration technique of Feldman et al. (2018). The second approach is based on a general reduction to the problem of localizing an approximately optimal solution with differential privacy. Such localization, in turn, can be achieved using existing (non-private) uniformly stable optimization algorithms. As in the earlier work, our algorithms require a mild smoothness assumption. We also give a linear-time algorithm achieving the optimal bound on the excess loss for the strongly convex case, as well as a faster algorithm for the non-smooth case.

• The paper introduces two novel algorithms that achieve optimal population loss with significantly reduced gradient computations and enhanced privacy efficiency.
• It leverages variable batch sizes and iterative privacy amplification to attain optimal performance for the last iterate of SGD in private settings.
• The localization technique transforms the problem into simpler subproblems, offering scalable solutions for high-dimensional private convex optimization.

Analyzing Optimal Algorithms for Private Stochastic Convex Optimization

In the paper titled "Private Stochastic Convex Optimization: Optimal Rates in Linear Time," Feldman et al. explore the intersection of differential privacy (DP) and stochastic convex optimization (SCO), focusing on the development of efficient algorithms that maintain privacy while optimizing population loss based on i.i.d. samples of convex loss functions. The authors present significant advancements over existing methods by reducing computational complexity and refining privacy guarantees.

One major contribution of this work is addressing the inefficiencies found in the algorithm proposed by Bassily et al., which, though achieving optimal excess population loss bounds, incurred a substantial computational expense of up to $O(\min\{n^{3/2}, n^{5/2}/d\})$ gradient computations for $n$ samples in $d$-dimensional space. Feldman et al. offer two algorithms that match optimal non-private SCO algorithms with running times of $O(\min\{n, n^2/d\})$, hence significantly improving computational efficiency.

The first technique introduced relies on variable batch sizes and employs privacy amplification through iterative procedures. The Snowball-SGD algorithm, as coined, utilizes a batch size schedule that grows inversely with the number of steps remaining, effectively amplifying privacy guarantees by spreading noise addition across multiple iterations. This method leverages results from Jain et al. to ensure optimal bounds for the last iterate of SGD, which are shown to forego the traditional averaging strategies often used in gradient methods.

The second approach developed is termed localization. Instead of directly solving the optimization problem, this technique transforms it into a series of progressively easier localization problems. It employs differentially private versions of uniformly stable optimization procedures, ensuring that the final output remains both accurate and private with no computational overhead. Particularly noteworthy is how this method extends well to non-smooth optimization cases, demonstrating versatility across diverse problem settings.

Both of these approaches require assumptions about the smoothness of the loss functions, although the smoothness constraint is milder than previous methods. Notably, Feldman et al. illustrate how privatizing strongly convex optimization problems can also be achieved with minimal computational cost, maintaining optimal population loss bounds even when faced with high-dimensional data.

The authors provide theoretical validation and potential for application in machine learning tasks that require balancing privacy with high-dimensional data efficiency. Especially in the context of private data analysis where maintaining differential privacy often introduces complexity and computational constraints, these findings propose promising pathways for scalable, private machine learning models.

Looking forward, the paper opens doors to further exploration in bridging privacy with other optimization paradigms, particularly in multi-objective optimization domains or settings with intricate data dependencies. As differential privacy continues to be a critical concern in data-driven algorithm design, graphical models, and beyond, these insights promise significant implications for theoretical and practical advancements in AI and beyond. However, challenges remain, notably in optimizing algorithms for higher-order interaction between variables, or better accommodating domain-specific data structures without stringent smoothness assumptions.

Overall, Feldman et al.'s contribution to differentially private SCO not only enhances theoretical understanding but significantly impacts practical deployment of efficient private algorithms in real-world applications.