On Parameter Estimation in Deviated Gaussian Mixture of Experts

• URL: http://arxiv.org/abs/2402.05220v2
• Date: Mon, 24 Jun 2024 05:13:06 GMT
• Title: On Parameter Estimation in Deviated Gaussian Mixture of Experts
• Authors: Huy Nguyen, Khai Nguyen, Nhat Ho,
• Abstract summary: We consider the parameter estimation problem in the deviated Gaussian mixture of experts. Data are generated from $g_0(Y|X)$ (null hypothesis) or they are generated from the whole mixture. We construct novel Voronoi-based loss functions to capture the convergence rates of maximum likelihood estimation.
• Score: 37.439768024583955
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the parameter estimation problem in the deviated Gaussian mixture of experts in which the data are generated from $(1 - \lambda^{\ast}) g_0(Y| X)+ \lambda^{\ast} \sum_{i = 1}^{k_{\ast}} p_{i}^{\ast} f(Y|(a_{i}^{\ast})^{\top}X+b_i^{\ast},\sigma_{i}^{\ast})$, where $X, Y$ are respectively a covariate vector and a response variable, $g_{0}(Y|X)$ is a known function, $\lambda^{\ast} \in [0, 1]$ is true but unknown mixing proportion, and $(p_{i}^{\ast}, a_{i}^{\ast}, b_{i}^{\ast}, \sigma_{i}^{\ast})$ for $1 \leq i \leq k^{\ast}$ are unknown parameters of the Gaussian mixture of experts. This problem arises from the goodness-of-fit test when we would like to test whether the data are generated from $g_{0}(Y|X)$ (null hypothesis) or they are generated from the whole mixture (alternative hypothesis). Based on the algebraic structure of the expert functions and the distinguishability between $g_0$ and the mixture part, we construct novel Voronoi-based loss functions to capture the convergence rates of maximum likelihood estimation (MLE) for our models. We further demonstrate that our proposed loss functions characterize the local convergence rates of parameter estimation more accurately than the generalized Wasserstein, a loss function being commonly used for estimating parameters in the Gaussian mixture of experts.

Related papers

• Variable Selection in Convex Piecewise Linear Regression [5.366354612549172]
  This paper presents Sparse Gradient as a solution for variable selection in convex piecewise linear regression. A non-asymptotic local convergence analysis is provided for SpGD under subGaussian noise.
  arXiv  Detail & Related papers  (2024-11-04T16:19:09Z)
• Dimension-free Private Mean Estimation for Anisotropic Distributions [55.86374912608193]
  Previous private estimators on distributions over $mathRd suffer from a curse of dimensionality. We present an algorithm whose sample complexity has improved dependence on dimension.
  arXiv  Detail & Related papers  (2024-11-01T17:59:53Z)
• Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes [5.00389879175348]
  We estimate the individual $beta$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $mathcal O(log(n) n-1/2)$.
  arXiv  Detail & Related papers  (2024-02-11T20:17:10Z)
• Statistical Learning under Heterogeneous Distribution Shift [71.8393170225794]
  Ground-truth predictor is additive $mathbbE[mathbfz mid mathbfx,mathbfy] = f_star(mathbfx) +g_star(mathbfy)$.
  arXiv  Detail & Related papers  (2023-02-27T16:34:21Z)
• Universality laws for Gaussian mixtures in generalized linear models [22.154969876570238]
  We investigate the joint statistics of the family of generalized linear estimators $(Theta_1, dots, Theta_M)$. This allow us to prove the universality of different quantities of interest, such as the training and generalization errors. We discuss the applications of our results to different machine learning tasks of interest, such as ensembling and uncertainty.
  arXiv  Detail & Related papers  (2023-02-17T15:16:06Z)
• Beyond Black Box Densities: Parameter Learning for the Deviated Components [15.501680326749515]
  A known density function estimate may have been previously obtained by a black box method. The increased complexity of the data set may result in the true density being deviated from the known estimate by a mixture distribution. We establish rates of convergence for the maximum likelihood estimates of $lambda*$ and $G*$ under Wasserstein metric.
  arXiv  Detail & Related papers  (2022-02-05T22:44:20Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• The Sample Complexity of Robust Covariance Testing [56.98280399449707]
  We are given i.i.d. samples from a distribution of the form $Z = (1-epsilon) X + epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $mathcalN(0, Sigma)$. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. We prove a sample complexity lower bound of $Omega(d2)$ for $epsilon$ an arbitrarily small constant and $gamma
  arXiv  Detail & Related papers  (2020-12-31T18:24:41Z)
• Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$ [5.457150493905064]
  We present a novel estimator with sub-Gaussian convergence. Our estimator does not require prior knowledge of the variance. Our estimator construction and analysis gives a framework generalizable to other problems.
  arXiv  Detail & Related papers  (2020-11-17T02:47:24Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)

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.