Bayesian Inference with Deep Weakly Nonlinear Networks
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929359359574016 |
|---|---|
| author | Hanin, Boris Zlokapa, Alexander |
| author_facet | Hanin, Boris Zlokapa, Alexander |
| contents | We show at a physics level of rigor that Bayesian inference with a fully connected neural network and a shaped nonlinearity of the form $ϕ(t) = t + ψt^3/L$ is (perturbatively) solvable in the regime where the number of training datapoints $P$ , the input dimension $N_0$, the network layer widths $N$, and the network depth $L$ are simultaneously large. Our results hold with weak assumptions on the data; the main constraint is that $P < N_0$. We provide techniques to compute the model evidence and posterior to arbitrary order in $1/N$ and at arbitrary temperature. We report the following results from the first-order computation:
1. When the width $N$ is much larger than the depth $L$ and training set size $P$, neural network Bayesian inference coincides with Bayesian inference using a kernel. The value of $ψ$ determines the curvature of a sphere, hyperbola, or plane into which the training data is implicitly embedded under the feature map.
2. When $LP/N$ is a small constant, neural network Bayesian inference departs from the kernel regime. At zero temperature, neural network Bayesian inference is equivalent to Bayesian inference using a data-dependent kernel, and $LP/N$ serves as an effective depth that controls the extent of feature learning.
3. In the restricted case of deep linear networks ($ψ=0$) and noisy data, we show a simple data model for which evidence and generalization error are optimal at zero temperature. As $LP/N$ increases, both evidence and generalization further improve, demonstrating the benefit of depth in benign overfitting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bayesian Inference with Deep Weakly Nonlinear Networks Hanin, Boris Zlokapa, Alexander Machine Learning Artificial Intelligence Probability Data Analysis, Statistics and Probability We show at a physics level of rigor that Bayesian inference with a fully connected neural network and a shaped nonlinearity of the form $ϕ(t) = t + ψt^3/L$ is (perturbatively) solvable in the regime where the number of training datapoints $P$ , the input dimension $N_0$, the network layer widths $N$, and the network depth $L$ are simultaneously large. Our results hold with weak assumptions on the data; the main constraint is that $P < N_0$. We provide techniques to compute the model evidence and posterior to arbitrary order in $1/N$ and at arbitrary temperature. We report the following results from the first-order computation: 1. When the width $N$ is much larger than the depth $L$ and training set size $P$, neural network Bayesian inference coincides with Bayesian inference using a kernel. The value of $ψ$ determines the curvature of a sphere, hyperbola, or plane into which the training data is implicitly embedded under the feature map. 2. When $LP/N$ is a small constant, neural network Bayesian inference departs from the kernel regime. At zero temperature, neural network Bayesian inference is equivalent to Bayesian inference using a data-dependent kernel, and $LP/N$ serves as an effective depth that controls the extent of feature learning. 3. In the restricted case of deep linear networks ($ψ=0$) and noisy data, we show a simple data model for which evidence and generalization error are optimal at zero temperature. As $LP/N$ increases, both evidence and generalization further improve, demonstrating the benefit of depth in benign overfitting. |
| title | Bayesian Inference with Deep Weakly Nonlinear Networks |
| topic | Machine Learning Artificial Intelligence Probability Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2405.16630 |