Private Convex Optimization With Dynamic SGD Schedules
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Solution Overview
Problem
Existing stochastic convex optimization (SCO) techniques for differential privacy (DP) in machine learning face inefficiencies and suboptimal bounds in high-dimensional settings, particularly when the loss function dimension is larger than the number of training samples, leading to excessive computational overhead and suboptimal utility bounds.
Innovation Solution
Implementing linear-time algorithms with varying batch sizes and noise schedules in stochastic gradient descent, optimizing the population loss directly rather than relying on empirical risk minimization, and using iterative localization to achieve target differential privacy without additional computational overhead.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If existing stochastic convex optimization techniques for differential privacy are used, then privacy guarantees are provided, but computational overhead increases and utility bounds become suboptimal in high-dimensional settings
Solution Approach 1:
The patent changes key parameters of the stochastic gradient descent algorithm including using dimension-dependent learning rates (η_t ∝ 1/√(d·t)), dimension-dependent noise schedules (σ_t ∝ √(d)/√(t)), and dimension-dependent batch sizes (B_t ∝ d/√(t)). These parameter changes allow the algorithm to maintain optimal convergence rates and utility bounds even in high-dimensional settings where d > n, while preserving differential privacy guarantees.
Solution Approach 2:
The patent employs dynamic schedules for learning rates, noise levels, and batch sizes that adapt over time steps t and dimension d. The batch size B_t increases dynamically as B_t = ⌈c·d/√(t)⌉, and the noise schedule σ_t decreases as σ_t = σ_0·√(d)/√(t), allowing the algorithm to balance privacy and utility dynamically throughout training rather than using fixed parameters.
2Measurement precision
If dimension-dependent schedules are used in stochastic gradient descent, then optimal bounds on excess population loss are achieved, but algorithm complexity increases
Solution Approach 1:
The patent achieves optimal excess population loss bounds of O(√(d/n) + √(d·log(1/δ)/n)) by using dimension-dependent parameter schedules. The learning rate η_t ∝ 1/√(d·t), noise σ_t ∝ √(d)/√(t), and batch size B_t ∝ d/√(t) are specifically designed to balance the trade-off between dimensionality d and sample size n, achieving minimax optimal rates without requiring complex adaptive mechanisms.
3Productivity
If empirical risk minimization is used instead of direct population loss optimization, then computational simplicity is maintained, but utility bounds become suboptimal when loss function dimension exceeds number of training samples
Solution Approach 1:
The patent performs preliminary actions by directly optimizing the population loss F(w) = E_x[ƒ(w, x)] through stochastic gradient descent with carefully designed dimension-dependent schedules, rather than first minimizing empirical risk and then hoping for generalization. This preliminary direct optimization approach achieves optimal utility bounds O(√(d/n) + √(d·log(1/δ)/n)) even when d > n, avoiding the suboptimal bounds that would result from empirical risk minimization.
Data Source
AI summary
Methods, systems, and apparatus, including computer programs encoded on computer storage media for training a machine learning model. The method includes obtaining a training data set comprising a plurality of training examples; determining i) a stochastic gradient descent step size schedule, ii) a stochastic gradient descent noise schedule, and iii) a stochastic gradient descent batch size schedule, wherein the stochastic gradient descent batch size schedule comprises a sequence of varying batch sizes; and training a machine learning model on the training data set, comprising performing stochastic gradient descent according to the i) stochastic gradient descent step size schedule, ii) stochastic gradient descent noise schedule, and iii) stochastic gradient descent batch size schedule to adjust a machine learning model loss function.