Learning Orthogonal Multi-Index Models: A Fine-Grained Information Exponent Analysis
Keywords: multi-index model, information exponent, sample complexity, stochastic gradient descent
Abstract: The information exponent (Ben Arous et al. (2021)) --- which is equivalent to the lowest degree in the Hermite
expansion of the link function for Gaussian single-index models --- has played an important role in predicting the
sample complexity of online stochastic gradient descent (SGD) in various learning tasks. In this work, we
demonstrate that, for multi-index models, focusing solely on the lowest degree can miss key structural details
and result in suboptimal rates.
Specifically, we consider the task of learning target functions of form $f_*(\mathbf{x}) = \sum_{k=1}^{P} \phi({\mathbf{v_k}^*} \cdot \mathbf{x})$,
where $P \ll d$, the ground-truth directions $\\{ v_k^* \\}_{k=1}^P$ are orthonormal, and only the second and
$2L$-th Hermite coefficients of the link function $\phi$ can be nonzero.
Based on the theory of information exponent, when the lowest degree is $2L$, recovering the directions requires
$d^{2L-1}\mathrm{poly}(P)$ samples, and when the lowest degree is $2$, only the relevant subspace (not the exact
directions) can be recovered due to the rotational invariance of the second-order terms. In contrast, we show that
by considering both second- and higher-order terms, we can first learn the relevant space via the second-order
terms, and then the exact directions using the higher-order terms, and the overall sample complexity of online
SGD is $d \mathrm{poly}(P)$.
Primary Area: learning theory
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Submission Number: 7683
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