## Abstract

Convolutional neural networks often dominate fully-connected counterparts in generalization performance, especially on image classification tasks. This is often explained in terms of “better inductive bias.” However, this has not been made mathematically rigorous, and the hurdle is that the sufficiently wide fully-connected net can always simulate the convolutional net. Thus the training algorithm plays a role. The current work describes a natural task on which a provable sample complexity gap can be shown, for standard training algorithms. We construct a single natural distribution on R^{d} × {±1} on which any orthogonal-invariant algorithm (i.e. fully-connected networks trained with most gradient-based methods from gaussian initialization) requires Ω(d^{2}) samples to generalize while O(1) samples suffice for convolutional architectures. Furthermore, we demonstrate a single target function, learning which on all possible distributions leads to an O(1) vs Ω(d^{2}/ε) gap. The proof relies on the fact that SGD on fully-connected network is orthogonal equivariant. Similar results are achieved for ℓ_{2} regression and adaptive training algorithms, e.g. Adam and AdaGrad, which are only permutation equivariant.

Original language | English (US) |
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State | Published - 2021 |

Event | 9th International Conference on Learning Representations, ICLR 2021 - Virtual, Online Duration: May 3 2021 → May 7 2021 |

### Conference

Conference | 9th International Conference on Learning Representations, ICLR 2021 |
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City | Virtual, Online |

Period | 5/3/21 → 5/7/21 |

## All Science Journal Classification (ASJC) codes

- Language and Linguistics
- Computer Science Applications
- Education
- Linguistics and Language