On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries
<|reference_start|>On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries: The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$--potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.<|reference_end|>
@article{joshi2024on,
    title={On the Complexity of Learning Sparse Functions with Statistical and
  Gradient Queries},
    author={Nirmit Joshi, Theodor Misiakiewicz, Nathan Srebro},
    journal={arXiv preprint arXiv:2407.05622},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.05622},
    primaryClass={cs.LG cs.DS}
}