Agnostic Learning of Mixed Linear Regressions with EM and AM Algorithms

Agnostic Learning of Mixed Linear Regressions with EM and AM Algorithms

The paper "Agnostic Learning of Mixed Linear Regressions with EM and AM Algorithms" by Avishek Ghosh and Arya Mazumdar tackles the problem of mixed linear regression (MLR) through an agnostic learning perspective. Unlike traditional settings where generative models are assumed, this paper does not predefine such models and instead focuses on minimizing a suitable loss function using Expectation-Maximization (EM) and Alternating Minimization (AM) algorithms.

Overview

Mixed linear regression is a pivotal problem in both parametric statistics and machine learning. Given samples as tuples of covariates and labels, the objective of MLR is to identify a small set of linear functions that best fit these samples. Typically, the labels are generated by stochastically selecting one out of multiple linear functions (with possible noise). The primary aim has been to estimate these underlying linear functions.

The EM and AM algorithms are well-documented methodologies for solving MLR problems where generative models are presumed. This paper extends their application to an agnostic setup, where no such assumption is made about the generative nature of the samples. Specifically, this research demonstrates that, with the right initial conditions and separability, both EM and AM algorithms converge to minimizers of the population loss in an agnostic setting.

Theoretical and Practical Implications

Loss Functions:

Min-Loss Function:

[ ℓ{\text{min}}(\theta1,...,\thetak;x,y) = \min{j \in [k]} (y - x^\top \theta_j)^2. ]

Soft-Min Loss Function:

[ ℓ{\text{softmin}}(\theta1,...,\thetak;x,y) = \sum{j=1}^k p{\theta1,...,k}(x,y;\thetaj)(y - x^\top \thetaj)^2, ] where ( p{\theta1,...,k}(x,y;\theta_j) ) is a soft assignment probability that converges to the min-loss for β → ∞.

Algorithm Analysis:

• The paper analyzes both Gradient-EM and Gradient-AM methods. Under proper initialization and separability (denoted by ( \Delta ) and ( \lambda )), EM and AM algorithm estimates were shown to converge to the optimal parameters with an error floor proportional to ( \lambda ).
• The research presents improved initialization requirements over previous work, necessitating only a Θ(1) distance from the true parameters as opposed to an O(1/√d) requirement.

Sample Complexity and Generalization:

• The sample complexity bounds derived involve factors like the data dimension ( d ), number of functions ( k ), and minimum probability mass ( π_{\min} ). Though optimal in terms of ( d ), the analysis yields a dependency of 1/π³ in sample complexity, which the authors note as generally unavoidable for problems involving spectral properties of restricted Gaussians.

Numerical Results and Bold Claims

Theoretical guarantees and experiments illustrate that both EM and AM algorithms are effective in the absence of generative models. The strong claims about convergence rates and initialization conditions highlight the robustness of these algorithms. Moreover, the bounds on sample complexities and generalization errors add significant value to the algorithms' applicability in practical scenarios.

Future Directions

This research opens several future avenues:

Extensions Beyond Linear Regressions:

• The agnostic framework and the tools developed could be extended to other mixture models such as mixture of classifiers or experts, bringing broader applicability.

Relaxation of Gaussian Assumptions:

• While the current work focuses on Gaussian covariates, relaxing this assumption to include sub-Gaussian or other bounded distributions remains an interesting potential direction.

Refined Algorithmic Techniques:

• Investigating more refined techniques like Leave-One-Out (LOO) analyses to avoid resampling and improve on the computational complexity and practical efficiency of these algorithms.

Conclusion

This paper contributes a significant theoretical and practical advancement in the understanding and application of EM and AM algorithms for mixed linear regressions in an agnostic setting. The convergence guarantees, under reasonable conditions of separability and initialization, showcase the adaptability and robustness of these algorithms beyond traditional assumptions of generative models. The implications extend not only to theoretical machine learning but also to practical implementations in various domains requiring robust mixture model estimations.