Demystifying Softmax Gating Function in Gaussian Mixture of Experts
- URL: http://arxiv.org/abs/2305.03288v2
- Date: Mon, 30 Oct 2023 01:13:03 GMT
- Title: Demystifying Softmax Gating Function in Gaussian Mixture of Experts
- Authors: Huy Nguyen and TrungTin Nguyen and Nhat Ho
- Abstract summary: We propose novel Voronoi loss functions among parameters and establish the convergence rates of maximum likelihood estimator (MLE) for solving parameter estimation.
Our findings show a connection between the convergence rate of the MLE and a solvability problem of a system of equations.
- Score: 34.53974702114644
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Understanding the parameter estimation of softmax gating Gaussian mixture of
experts has remained a long-standing open problem in the literature. It is
mainly due to three fundamental theoretical challenges associated with the
softmax gating function: (i) the identifiability only up to the translation of
parameters; (ii) the intrinsic interaction via partial differential equations
between the softmax gating and the expert functions in the Gaussian density;
(iii) the complex dependence between the numerator and denominator of the
conditional density of softmax gating Gaussian mixture of experts. We resolve
these challenges by proposing novel Voronoi loss functions among parameters and
establishing the convergence rates of maximum likelihood estimator (MLE) for
solving parameter estimation in these models. When the true number of experts
is unknown and over-specified, our findings show a connection between the
convergence rate of the MLE and a solvability problem of a system of polynomial
equations.
Related papers
- MG-Net: Learn to Customize QAOA with Circuit Depth Awareness [51.78425545377329]
Quantum Approximate Optimization Algorithm (QAOA) and its variants exhibit immense potential in tackling optimization challenges.
The requisite circuit depth for satisfactory performance is problem-specific and often exceeds the maximum capability of current quantum devices.
We introduce the Mixer Generator Network (MG-Net), a unified deep learning framework adept at dynamically formulating optimal mixer Hamiltonians.
arXiv Detail & Related papers (2024-09-27T12:28:18Z) - An Analysis of Quantum Annealing Algorithms for Solving the Maximum Clique Problem [49.1574468325115]
We analyse the ability of quantum D-Wave annealers to find the maximum clique on a graph, expressed as a QUBO problem.
We propose a decomposition algorithm for the complementary maximum independent set problem, and a graph generation algorithm to control the number of nodes, the number of cliques, the density, the connectivity indices and the ratio of the solution size to the number of other nodes.
arXiv Detail & Related papers (2024-06-11T04:40:05Z) - Sigmoid Gating is More Sample Efficient than Softmax Gating in Mixture of Experts [78.3687645289918]
We show that the sigmoid gating function enjoys a higher sample efficiency than the softmax gating for the statistical task of expert estimation.
We find that experts formulated as feed-forward networks with commonly used activation such as ReLU and GELU enjoy faster convergence rates under the sigmoid gating.
arXiv Detail & Related papers (2024-05-22T21:12:34Z) - Is Temperature Sample Efficient for Softmax Gaussian Mixture of Experts? [27.924615931679757]
We explore the impacts of a dense-to-sparse gating mixture of experts (MoE) on the maximum likelihood estimation under the MoE.
We propose using a novel activation dense-to-sparse gate, which routes the output of a linear layer to an activation function before delivering them to the softmax function.
arXiv Detail & Related papers (2024-01-25T01:09:09Z) - 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) - Statistical Perspective of Top-K Sparse Softmax Gating Mixture of
Experts [28.907764868329988]
We study the effects of the top-K sparse softmax gating function on both density and parameter estimations.
Our results hinge upon defining novel loss functions among parameters to capture different behaviors of the input regions.
Our findings suggest that the number of experts selected from the top-K sparse softmax gating function must exceed the total cardinality of a certain number of Voronoi cells.
arXiv Detail & Related papers (2023-09-25T03:28:01Z) - Towards Convergence Rates for Parameter Estimation in Gaussian-gated
Mixture of Experts [40.24720443257405]
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.
arXiv Detail & Related papers (2023-05-12T16:02:19Z) - Convex Bounds on the Softmax Function with Applications to Robustness
Verification [69.09991317119679]
The softmax function is a ubiquitous component at the output of neural networks and increasingly in intermediate layers as well.
This paper provides convex lower bounds and concave upper bounds on the softmax function, which are compatible with convex optimization formulations for characterizing neural networks and other ML models.
arXiv Detail & Related papers (2023-03-03T05:07:02Z) - Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions [73.33961743410876]
A soft-max function has two main efficiency measures: approximation and smoothness.
We identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness.
This leads to novel soft-max functions, each of which is optimal for a different application.
arXiv Detail & Related papers (2020-10-22T05:19:58Z)
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.