Bayesian Inference with Deep Weakly Nonlinear Networks

• URL: http://arxiv.org/abs/2405.16630v1
• Date: Sun, 26 May 2024 17:08:04 GMT
• Title: Bayesian Inference with Deep Weakly Nonlinear Networks
• Authors: Boris Hanin, Alexander Zlokapa,
• Abstract summary: We show at a physics level of rigor that Bayesian inference with a fully connected neural network is solvable. We provide techniques to compute the model evidence and posterior to arbitrary order in $1/N$ and at arbitrary temperature.
• Score: 57.95116787699412
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show at a physics level of rigor that Bayesian inference with a fully connected neural network and a shaped nonlinearity of the form $\phi(t) = t + \psi t^3/L$ 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 $\psi$ 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 ($\psi=0$) 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.

Related papers

• Deep Neural Networks: Multi-Classification and Universal Approximation [0.0]
  We demonstrate that a ReLU deep neural network with a width of $2$ and a depth of $2N+4M-1$ layers can achieve finite sample memorization for any dataset comprising $N$ elements. We also provide depth estimates for approximating $W1,p$ functions and width estimates for approximating $Lp(Omega;mathbbRm)$ for $mgeq1$.
  arXiv  Detail & Related papers  (2024-09-10T14:31:21Z)
• Wide neural networks: From non-gaussian random fields at initialization to the NTK geometry of training [0.0]
  Recent developments in applications of artificial neural networks with over $n=1014$ parameters make it extremely important to study the large $n$ behaviour of such networks. Most works studying wide neural networks have focused on the infinite width $n to +infty$ limit of such networks. In this work we will study their behavior for large, but finite $n$.
  arXiv  Detail & Related papers  (2023-04-06T21:34:13Z)
• The Onset of Variance-Limited Behavior for Networks in the Lazy and Rich Regimes [75.59720049837459]
  We study the transition from infinite-width behavior to this variance limited regime as a function of sample size $P$ and network width $N$. We find that finite-size effects can become relevant for very small datasets on the order of $P* sim sqrtN$ for regression with ReLU networks.
  arXiv  Detail & Related papers  (2022-12-23T04:48:04Z)
• Understanding Deep Neural Function Approximation in Reinforcement Learning via $\epsilon$-Greedy Exploration [53.90873926758026]
  This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) We focus on the value based algorithm with the $epsilon$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces. Our analysis reformulates the temporal difference error in an $L2(mathrmdmu)$-integrable space over a certain averaged measure $mu$, and transforms it to a generalization problem under the non-iid setting.
  arXiv  Detail & Related papers  (2022-09-15T15:42:47Z)
• Neural Networks can Learn Representations with Gradient Descent [68.95262816363288]
  In specific regimes, neural networks trained by gradient descent behave like kernel methods. In practice, it is known that neural networks strongly outperform their associated kernels.
  arXiv  Detail & Related papers  (2022-06-30T09:24:02Z)
• The Rate of Convergence of Variation-Constrained Deep Neural Networks [35.393855471751756]
  We show that a class of variation-constrained neural networks can achieve near-parametric rate $n-1/2+delta$ for an arbitrarily small constant $delta$. The result indicates that the neural function space needed for approximating smooth functions may not be as large as what is often perceived.
  arXiv  Detail & Related papers  (2021-06-22T21:28:00Z)
• Locality defeats the curse of dimensionality in convolutional teacher-student scenarios [69.2027612631023]
  We show that locality is key in determining the learning curve exponent $beta$. We conclude by proving, using a natural assumption, that performing kernel regression with a ridge that decreases with the size of the training set leads to similar learning curve exponents to those we obtain in the ridgeless case.
  arXiv  Detail & Related papers  (2021-06-16T08:27:31Z)
• Fundamental tradeoffs between memorization and robustness in random features and neural tangent regimes [15.76663241036412]
  We prove for a large class of activation functions that, if the model memorizes even a fraction of the training, then its Sobolev-seminorm is lower-bounded. Experiments reveal for the first time, (iv) a multiple-descent phenomenon in the robustness of the min-norm interpolator.
  arXiv  Detail & Related papers  (2021-06-04T17:52:50Z)
• Towards Understanding Hierarchical Learning: Benefits of Neural Representations [160.33479656108926]
  In this work, we demonstrate that intermediate neural representations add more flexibility to neural networks. We show that neural representation can achieve improved sample complexities compared with the raw input. Our results characterize when neural representations are beneficial, and may provide a new perspective on why depth is important in deep learning.
  arXiv  Detail & Related papers  (2020-06-24T02:44:54Z)
• A Corrective View of Neural Networks: Representation, Memorization and Learning [26.87238691716307]
  We develop a corrective mechanism for neural network approximation. We show that two-layer neural networks in the random features regime (RF) can memorize arbitrary labels. We also consider three-layer neural networks and show that the corrective mechanism yields faster representation rates for smooth radial functions.
  arXiv  Detail & Related papers  (2020-02-01T20:51:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.