Efficient methods improve optimization with fewer transformations needed
Sparse Data Augmentation for Optimization with Provable Guarantees
Machine LearningArtificial Intelligence
Summary
When teaching computers to learn from geometric data, it helps to look at many versions of the data to understand its important features. However, checking every possible version can be very slow or impossible when there are too many. The authors show that using only a small set of these versions fixed in advance still lets the computer learn well while saving a lot of work. Their method requires fewer checks than previous approaches and comes with mathematical guarantees about its effectiveness. They achieve this by carefully analyzing how transformations combine using advanced math tools.
nonconvex optimizationdata augmentationgroup transformationsgradient descentstochastic gradient descentmachine learninggeometric learningrepresentation theoryspectral analysisoracle queries
Authors
Behrooz Tahmasebi, Melanie Weber
Abstract
In nonconvex optimization problems arising in geometric machine learning, data augmentation is commonly used to promote invariance by averaging empirical losses over transformations of the data. Computing the fully augmented objective, however, requires access to every element of the transformation group $G$, which may be prohibitively expensive when $G$ is large or accessible only through sampling. We study whether full augmentation can instead be approximated using a small, fixed sample of transformations acquired before optimization and reused thereafter. Under suitable regularity conditions, we show that, with probability at least $1-δ$, gradient descent (GD) on the resulting sparsely augmented objective returns an $\varepsilon$-stationary point of the fully augmented objective using $\mathcal{O}\bigl((\log |G|+\log(1/δ))/\varepsilon^2\bigr)$ group-transformation-oracle queries. By comparison, standard group stochastic gradient descent (group-SGD), which samples a fresh transformation at every iteration, uses $\mathcal{O}(1/\varepsilon^4)$ transformation queries. Therefore, gradient descent with fixed sparse augmentation requires fewer transformation queries than both GD applied to the fully augmented objective and group-SGD. Our proof techniques, which may be of independent interest, establish a uniform approximation of the full group-averaged gradient field by a random group average using spectral properties of group-induced operators and tools from representation theory.