Optimal Differentially Private Optimization with Sparse Gradients

Core Concepts

Differentially private optimization algorithms with nearly optimal rates can be achieved by leveraging the sparsity of individual gradients, improving upon existing methods that do not exploit this structure.

Abstract

The content presents a theoretical framework for studying differentially private (DP) optimization under the assumption of sparse gradients. The key results are: New upper bounds for DP mean estimation with sparse data, improving upon existing algorithms particularly in the high-dimensional regime. The projection mechanism with a convex relaxation is shown to be nearly optimal, both for pure-DP and approximate-DP. Lower bounds that establish the near-optimality of the proposed algorithms. A novel block-diagonal construction is used to obtain lower bounds that exhibit the right low/high-dimensional transition. DP convex optimization algorithms for both empirical risk minimization (ERM) and stochastic convex optimization (SCO) that achieve nearly dimension-independent rates by leveraging gradient sparsity. These algorithms are based on regularized output perturbation with an additional ℓ∞ projection step. A bias-reduction technique for approximating stationary points in DP-ERM, which uses a random batch size schedule and a telescopic gradient estimator. This method achieves rates that depend on the sparsity rather than the dimension, modulo polylogarithmic factors. The results demonstrate that even for high-dimensional models, DP optimization is tractable under gradient sparsity, in contrast to the dense case where such poly-logarithmic rates are unattainable.

Stats

The ℓ2-sensitivity of the empirical mean is bounded by 2L√s/n. The ℓ2-sensitivity of the regularized ERM solution is bounded by 2√2sL/[λn].

Quotes

"Differentially private optimization algorithms with nearly optimal rates can be achieved by leveraging the sparsity of individual gradients, improving upon existing methods that do not exploit this structure." "The results demonstrate that even for high-dimensional models, DP optimization is tractable under gradient sparsity, in contrast to the dense case where such poly-logarithmic rates are unattainable."

Key Insights Distilled From

Differentially Private Optimization with Sparse Gradients

by Badi... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.10881.pdf

Deeper Inquiries

How can the proposed bias-reduction technique be extended to other DP optimization settings beyond ERM?

The proposed bias-reduction technique can be extended to other DP optimization settings by leveraging the concept of subsampling and variance reduction. In the context of stochastic optimization with sparse gradients, the bias-reduction method inspired by the simulation literature can be adapted to scenarios where the optimization problem involves different objectives or constraints. By incorporating random batch sizes in an exponentially increasing schedule and using a telescopic estimator of the gradient, the bias-reduction technique can help mitigate the regularization bias in various optimization settings. This approach allows for the amortization of errors within an iterative method, preventing their accumulation and improving the overall efficiency and accuracy of the optimization process. Additionally, the method can be tailored to specific optimization problems by adjusting the batch randomization and privacy accounting based on the requirements of the given scenario.

What are the computational and memory advantages of the approximately sparse solutions obtained by the ℓ1-minimization algorithm for DP mean estimation?

The approximately sparse solutions obtained by the ℓ1-minimization algorithm for DP mean estimation offer several computational and memory advantages. Efficient Computation: The ℓ1-minimization algorithm promotes sparse solutions by minimizing the ℓ1-norm of the output, leading to solutions with a reduced number of non-zero entries. This sparsity property allows for more efficient computation as operations involving sparse vectors are computationally less expensive compared to dense vectors. The algorithm can leverage this sparsity to speed up the optimization process and reduce computational complexity. Memory Efficiency: Sparse solutions require less memory to store and manipulate compared to dense solutions. By obtaining approximately sparse solutions through ℓ1-minimization, the memory requirements for storing the solution vectors are significantly reduced. This memory efficiency is particularly beneficial in scenarios where memory resources are limited or when dealing with large-scale datasets and high-dimensional feature spaces. Improved Interpretability: Sparse solutions are easier to interpret and analyze, as they highlight the most relevant features or components contributing to the solution. This can provide valuable insights into the underlying data patterns and help in making informed decisions based on the optimization results. Enhanced Scalability: The computational and memory advantages of sparse solutions make the ℓ1-minimization algorithm scalable to large datasets and high-dimensional spaces. The algorithm can efficiently handle complex optimization problems with sparse gradients, making it suitable for a wide range of applications in machine learning and optimization.

Can the block-diagonal lower bound construction be applied to other DP learning problems beyond mean estimation and optimization?

Yes, the block-diagonal lower bound construction can be applied to various other DP learning problems beyond mean estimation and optimization. The construction technique involves partitioning the dataset into blocks and leveraging the properties of these blocks to establish lower bounds on the optimization problem. This approach can be adapted to different DP learning scenarios that involve sparse data, convex optimization, or empirical risk minimization. Some examples of DP learning problems where the block-diagonal lower bound construction can be applied include: Sparse Regression: In scenarios where the regression problem involves sparse features or sparse gradients, the block-diagonal construction can be used to establish lower bounds on the regression error. By partitioning the dataset into blocks and analyzing the properties of these blocks, one can derive lower bounds that account for the sparsity of the data. Sparse Classification: For classification problems with sparse input features, the block-diagonal lower bound construction can help in proving lower bounds on the classification error. By considering the sparsity of the data and partitioning it into blocks, one can analyze the impact of the sparsity on the classification performance and derive meaningful lower bounds. Sparse Matrix Factorization: In matrix factorization problems involving sparse matrices, the block-diagonal construction can be utilized to establish lower bounds on the factorization error. By partitioning the matrix into blocks and examining the properties of these blocks, one can derive lower bounds that take into account the sparsity of the matrix and the factorization process. Overall, the block-diagonal lower bound construction is a versatile technique that can be adapted to various DP learning problems beyond mean estimation and optimization, providing valuable insights into the performance and complexity of these learning tasks.