Dataset Distillation Efficiently Encodes Low-Dimensional Representations from Gradient-Based Learning of Non-Linear Tasks

TL;DR

This paper provides a theoretical analysis of dataset distillation, demonstrating how it encodes low-dimensional task-relevant information into synthetic data for neural networks, with implications for model generalization and memory efficiency.

Contribution

It offers the first theoretical framework linking task structure, intrinsic dimensionality, and dataset distillation efficiency in gradient-based neural network training.

Findings

Low-dimensional structure is efficiently encoded into distilled data.

The distilled dataset achieves high generalization with memory complexity $ ilde{ heta}(r^2d+L)$.

The work is among the first to connect task structure with dataset compression in theory.

Abstract

Dataset distillation, a training-aware data compression technique, has recently attracted increasing attention as an effective tool for mitigating costs of optimization and data storage. However, progress remains largely empirical. Mechanisms underlying the extraction of task-relevant information from the training process and the efficient encoding of such information into synthetic data points remain elusive. In this paper, we theoretically analyze practical algorithms of dataset distillation applied to the gradient-based training of two-layer neural networks with width $L$. By focusing on a non-linear task structure called multi-index model, we prove that the low-dimensional structure of the problem is efficiently encoded into the resulting distilled data. This dataset reproduces a model with high generalization ability for a required memory complexity of $\tilde{\Theta}$$(r^2d+L)$, where $d$ and $r$ are the input and intrinsic dimensions of the task. To the best of our knowledge, this is one of the first theoretical works that include a specific task structure, leverage its intrinsic dimensionality to quantify the compression rate and study dataset distillation implemented solely via gradient-based algorithms.