Agnostic Learning of Mixed Linear Regressions with EM and AM Algorithms (2406.01149v1)

Abstract: Mixed linear regression is a well-studied problem in parametric statistics and machine learning. Given a set of samples, tuples of covariates and labels, the task of mixed linear regression is to find a small list of linear relationships that best fit the samples. Usually it is assumed that the label is generated stochastically by randomly selecting one of two or more linear functions, applying this chosen function to the covariates, and potentially introducing noise to the result. In that situation, the objective is to estimate the ground-truth linear functions up to some parameter error. The popular expectation maximization (EM) and alternating minimization (AM) algorithms have been previously analyzed for this. In this paper, we consider the more general problem of agnostic learning of mixed linear regression from samples, without such generative models. In particular, we show that the AM and EM algorithms, under standard conditions of separability and good initialization, lead to agnostic learning in mixed linear regression by converging to the population loss minimizers, for suitably defined loss functions. In some sense, this shows the strength of AM and EM algorithms that converges to ``optimal solutions'' even in the absence of realizable generative models.

• The paper demonstrates that using loss minimization with EM and AM algorithms achieves robust convergence in agnostic mixed linear regression.
• The methodology improves initialization requirements, needing only a Θ(1) distance from true parameters compared to previous stricter bounds.
• The analysis provides detailed sample complexity and generalization error bounds, underscoring the approach's practical viability.

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

1. Loss Functions:
   • Min-Loss Function:

   $$ℓ_{\text{min}}(\theta_1,...,\theta_k;x,y) = \min_{j \in [k]} (y - x^\top \theta_j)^2.$$

• Soft-Min Loss Function:

  $$ℓ_{\text{softmin}}(\theta_1,...,\theta_k;x,y) = \sum_{j=1}^k p_{\theta_1,...,k}(x,y;\theta_j)(y - x^\top \theta_j)^2,$$

  where $p_{\theta_1,...,k}(x,y;\theta_j)$ is a soft assignment probability that converges to the min-loss for β → ∞.

1. 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.
2. 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/π^3 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:

1. 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.
2. 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.
3. 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.