Estimating the Coefficients of a Mixture of Two Linear Regressions by Expectation Maximization
This work addresses the problem of parameter estimation in mixture models for researchers in statistics and machine learning, offering theoretical insights but is incremental as it builds on existing EM analysis tools.
The paper provides convergence guarantees for the EM algorithm when estimating coefficients in a symmetric mixture of two linear regressions, showing that with proper initialization, it converges to the true parameters at the parametric rate, and it performs well even under model misspecification.
We give convergence guarantees for estimating the coefficients of a symmetric mixture of two linear regressions by expectation maximization (EM). In particular, we show that the empirical EM iterates converge to the target parameter vector at the parametric rate, provided the algorithm is initialized in an unbounded cone. In particular, if the initial guess has a sufficiently large cosine angle with the target parameter vector, a sample-splitting version of the EM algorithm converges to the true coefficient vector with high probability. Interestingly, our analysis borrows from tools used in the problem of estimating the centers of a symmetric mixture of two Gaussians by EM. We also show that the population EM operator for mixtures of two regressions is anti-contractive from the target parameter vector if the cosine angle between the input vector and the target parameter vector is too small, thereby establishing the necessity of our conic condition. Finally, we give empirical evidence supporting this theoretical observation, which suggests that the sample based EM algorithm performs poorly when initial guesses are drawn accordingly. Our simulation study also suggests that the EM algorithm performs well even under model misspecification (i.e., when the covariate and error distributions violate the model assumptions).