Emergence and scaling laws in SGD learning of shallow neural networks

• URL: http://arxiv.org/abs/2504.19983v1
• Date: Mon, 28 Apr 2025 16:58:55 GMT
• Title: Emergence and scaling laws in SGD learning of shallow neural networks
• Authors: Yunwei Ren, Eshaan Nichani, Denny Wu, Jason D. Lee,
• Abstract summary: We study the complexity of online gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data.<n>We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective.
• Score: 46.632052892298375
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the complexity of online stochastic gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data: $f_*(\boldsymbol{x}) = \sum_{p=1}^P a_p\cdot \sigma(\langle\boldsymbol{x},\boldsymbol{v}_p^*\rangle)$, $\boldsymbol{x} \sim \mathcal{N}(0,\boldsymbol{I}_d)$, where the activation $\sigma:\mathbb{R}\to\mathbb{R}$ is an even function with information exponent $k_*>2$ (defined as the lowest degree in the Hermite expansion), $\{\boldsymbol{v}^*_p\}_{p\in[P]}\subset \mathbb{R}^d$ are orthonormal signal directions, and the non-negative second-layer coefficients satisfy $\sum_{p} a_p^2=1$. We focus on the challenging ``extensive-width'' regime $P\gg 1$ and permit diverging condition number in the second-layer, covering as a special case the power-law scaling $a_p\asymp p^{-\beta}$ where $\beta\in\mathbb{R}_{\ge 0}$. We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective, and explicitly identify sharp transition times to recover each signal direction. In the power-law setting, we characterize scaling law exponents for the MSE loss with respect to the number of training samples and SGD steps, as well as the number of parameters in the student neural network. Our analysis entails that while the learning of individual teacher neurons exhibits abrupt transitions, the juxtaposition of $P\gg 1$ emergent learning curves at different timescales leads to a smooth scaling law in the cumulative objective.

Related papers

• Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit [75.4661041626338]
  We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$<n>We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ with a complexity that is not governed by information exponents.
  arXiv  Detail & Related papers  (2024-06-03T17:56:58Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• SGD Finds then Tunes Features in Two-Layer Neural Networks with near-Optimal Sample Complexity: A Case Study in the XOR problem [1.3597551064547502]
  We consider the optimization process of minibatch descent gradient (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the $d$-dimensional Boolean hypercube labeled by the quadratic XOR'' function $y = -x_ix_j$, it is possible to train to a population error $o(1)$ with $d :textpolylog(d)$ samples.
  arXiv  Detail & Related papers  (2023-09-26T17:57:44Z)
• Generalization and Stability of Interpolating Neural Networks with Minimal Width [37.908159361149835]
  We investigate the generalization and optimization of shallow neural-networks trained by gradient in the interpolating regime. We prove the training loss number minimizations $m=Omega(log4 (n))$ neurons and neurons $Tapprox n$. With $m=Omega(log4 (n))$ neurons and $Tapprox n$, we bound the test loss training by $tildeO (1/)$.
  arXiv  Detail & Related papers  (2023-02-18T05:06:15Z)
• Neural Networks Efficiently Learn Low-Dimensional Representations with SGD [22.703825902761405]
  We show that SGD-trained ReLU NNs can learn a single-index target of the form $y=f(langleboldsymbolu,boldsymbolxrangle) + epsilon$ by recovering the principal direction. We also provide compress guarantees for NNs using the approximate low-rank structure produced by SGD.
  arXiv  Detail & Related papers  (2022-09-29T15:29:10Z)
• High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation [89.21686761957383]
  We study the first gradient descent step on the first-layer parameters $boldsymbolW$ in a two-layer network. Our results demonstrate that even one step can lead to a considerable advantage over random features.
  arXiv  Detail & Related papers  (2022-05-03T12:09:59Z)
• Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK [58.5766737343951]
  We consider the dynamic of descent for learning a two-layer neural network. We show that an over-parametrized two-layer neural network can provably learn with gradient loss at most ground with Tangent samples.
  arXiv  Detail & Related papers  (2020-07-09T07:09:28Z)
• Agnostic Learning of a Single Neuron with Gradient Descent [92.7662890047311]
  We consider the problem of learning the best-fitting single neuron as measured by the expected square loss. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
  arXiv  Detail & Related papers  (2020-05-29T07:20:35Z)

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.