Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalization
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866929273448693760 |
|---|---|
| author | Wang, Yutong Sonthalia, Rishi Hu, Wei |
| author_facet | Wang, Yutong Sonthalia, Rishi Hu, Wei |
| contents | We study the generalization capability of nearly-interpolating linear regressors: $\boldsymbolβ$'s whose training error $τ$ is positive but small, i.e., below the noise floor. Under a random matrix theoretic assumption on the data distribution and an eigendecay assumption on the data covariance matrix $\boldsymbolΣ$, we demonstrate that any near-interpolator exhibits rapid norm growth: for $τ$ fixed, $\boldsymbolβ$ has squared $\ell_2$-norm $\mathbb{E}[\|{\boldsymbolβ}\|_{2}^{2}] = Ω(n^α)$ where $n$ is the number of samples and $α>1$ is the exponent of the eigendecay, i.e., $λ_i(\boldsymbolΣ) \sim i^{-α}$. This implies that existing data-independent norm-based bounds are necessarily loose. On the other hand, in the same regime we precisely characterize the asymptotic trade-off between interpolation and generalization. Our characterization reveals that larger norm scaling exponents $α$ correspond to worse trade-offs between interpolation and generalization. We verify empirically that a similar phenomenon holds for nearly-interpolating shallow neural networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalization Wang, Yutong Sonthalia, Rishi Hu, Wei Machine Learning We study the generalization capability of nearly-interpolating linear regressors: $\boldsymbolβ$'s whose training error $τ$ is positive but small, i.e., below the noise floor. Under a random matrix theoretic assumption on the data distribution and an eigendecay assumption on the data covariance matrix $\boldsymbolΣ$, we demonstrate that any near-interpolator exhibits rapid norm growth: for $τ$ fixed, $\boldsymbolβ$ has squared $\ell_2$-norm $\mathbb{E}[\|{\boldsymbolβ}\|_{2}^{2}] = Ω(n^α)$ where $n$ is the number of samples and $α>1$ is the exponent of the eigendecay, i.e., $λ_i(\boldsymbolΣ) \sim i^{-α}$. This implies that existing data-independent norm-based bounds are necessarily loose. On the other hand, in the same regime we precisely characterize the asymptotic trade-off between interpolation and generalization. Our characterization reveals that larger norm scaling exponents $α$ correspond to worse trade-offs between interpolation and generalization. We verify empirically that a similar phenomenon holds for nearly-interpolating shallow neural networks. |
| title | Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2403.07264 |