Towards Convergence Rates for Parameter Estimation in Gaussian-gated Mixture of Experts

• URL: http://arxiv.org/abs/2305.07572v2
• Date: Fri, 9 Feb 2024 14:51:16 GMT
• Title: Towards Convergence Rates for Parameter Estimation in Gaussian-gated Mixture of Experts
• Authors: Huy Nguyen, TrungTin Nguyen, Khai Nguyen, Nhat Ho
• Abstract summary: We provide a convergence analysis for maximum likelihood estimation (MLE) in the Gaussian-gated MoE model. Our findings reveal that the MLE has distinct behaviors under two complement settings of location parameters of the Gaussian gating functions. Notably, these behaviors can be characterized by the solvability of two different systems of equations.
• Score: 40.24720443257405
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Originally introduced as a neural network for ensemble learning, mixture of experts (MoE) has recently become a fundamental building block of highly successful modern deep neural networks for heterogeneous data analysis in several applications of machine learning and statistics. Despite its popularity in practice, a satisfactory level of theoretical understanding of the MoE model is far from complete. To shed new light on this problem, we provide a convergence analysis for maximum likelihood estimation (MLE) in the Gaussian-gated MoE model. The main challenge of that analysis comes from the inclusion of covariates in the Gaussian gating functions and expert networks, which leads to their intrinsic interaction via some partial differential equations with respect to their parameters. We tackle these issues by designing novel Voronoi loss functions among parameters to accurately capture the heterogeneity of parameter estimation rates. Our findings reveal that the MLE has distinct behaviors under two complement settings of location parameters of the Gaussian gating functions, namely when all these parameters are non-zero versus when at least one among them vanishes. Notably, these behaviors can be characterized by the solvability of two different systems of polynomial equations. Finally, we conduct a simulation study to empirically verify our theoretical results.

Related papers

• Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks [3.680127959836384]
  implicit gradient descent (IGD) outperforms the common gradient descent (GD) in handling certain multi-scale problems. We show that IGD converges a globally optimal solution at a linear convergence rate.
  arXiv  Detail & Related papers  (2024-07-03T06:10:41Z)
• Proximal Interacting Particle Langevin Algorithms [0.0]
  We introduce Proximal Interacting Particle Langevin Algorithms (PIPLA) for inference and learning in latent variable models. We propose several variants within the novel proximal IPLA family, tailored to the problem of estimating parameters in a non-differentiable statistical model. Our theory and experiments together show that PIPLA family can be the de facto choice for parameter estimation problems in latent variable models for non-differentiable models.
  arXiv  Detail & Related papers  (2024-06-20T13:16:41Z)
• A General Theory for Softmax Gating Multinomial Logistic Mixture of Experts [28.13187489224953]
  We propose a novel class of modified softmax gating functions which transform the input before delivering them to the gating functions. As a result, the previous interaction disappears and the parameter estimation rates are significantly improved.
  arXiv  Detail & Related papers  (2023-10-22T05:32:19Z)
• Heterogeneous Multi-Task Gaussian Cox Processes [61.67344039414193]
  We present a novel extension of multi-task Gaussian Cox processes for modeling heterogeneous correlated tasks jointly. A MOGP prior over the parameters of the dedicated likelihoods for classification, regression and point process tasks can facilitate sharing of information between heterogeneous tasks. We derive a mean-field approximation to realize closed-form iterative updates for estimating model parameters.
  arXiv  Detail & Related papers  (2023-08-29T15:01:01Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• On the Influence of Enforcing Model Identifiability on Learning dynamics of Gaussian Mixture Models [14.759688428864159]
  We propose a technique for extracting submodels from singular models. Our method enforces model identifiability during training. We show how the method can be applied to more complex models like deep neural networks.
  arXiv  Detail & Related papers  (2022-06-17T07:50:22Z)
• A Unified View of Stochastic Hamiltonian Sampling [18.300078015845262]
  This work revisits the theoretical properties of Hamiltonian differential equations (SDEs) for posterior sampling. We study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates.
  arXiv  Detail & Related papers  (2021-06-30T16:50:11Z)
• Post-mortem on a deep learning contest: a Simpson's paradox and the complementary roles of scale metrics versus shape metrics [61.49826776409194]
  We analyze a corpus of models made publicly-available for a contest to predict the generalization accuracy of neural network (NN) models. We identify what amounts to a Simpson's paradox: where "scale" metrics perform well overall but perform poorly on sub partitions of the data. We present two novel shape metrics, one data-independent, and the other data-dependent, which can predict trends in the test accuracy of a series of NNs.
  arXiv  Detail & Related papers  (2021-06-01T19:19:49Z)
• Understanding Overparameterization in Generative Adversarial Networks [56.57403335510056]
  Generative Adversarial Networks (GANs) are used to train non- concave mini-max optimization problems. A theory has shown the importance of the gradient descent (GD) to globally optimal solutions. We show that in an overized GAN with a $1$-layer neural network generator and a linear discriminator, the GDA converges to a global saddle point of the underlying non- concave min-max problem.
  arXiv  Detail & Related papers  (2021-04-12T16:23:37Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)

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.