We investigate the landscape of the negative log-likelihood function of Gaussian Mixture Models (GMMs) with a general number of components in the population limit. As the objective function is non-convex, there can exist multiple spurious local minima that are not globally optimal, even for well-separated mixture models. Our study reveals that all local minima share a common structure that partially identifies the cluster centers (i.e., means of the Gaussian components) of the true location mixture. Specifically, each local minimum can be represented as a non-overlapping combination of two types of sub-configurations: (1) fitting a single mean estimate to multiple Gaussian components or (2) fitting multiple estimates to a single true component. These results apply to settings where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over- or under-specified. We also present a more fine-grained analysis for the setting of one-dimensional GMMs with three components, which provide sharper approximation error bounds with improved dependence on the separation parameter.

Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.

We consider the problem of sequential recommendation, where the current recommendation is made based on past interactions. This recommendation task requires efficient processing of the sequential data and aims to provide recommendations that maximize the long-term reward. To this end, we train a farsighted recommender by using an offline RL algorithm with the policy network in our model architecture that has been initialized from a pre-trained transformer model. The pre-trained model leverages the superb ability of the transformer to process sequential information. Compared to prior works that rely on online interaction via simulation, we focus on implementing a fully offline RL framework that is able to converge in a fast and stable way. Through extensive experiments on public datasets, we show that our method is robust across various recommendation regimes, including e-commerce and movie suggestions. Compared to state-of-the-art supervised learning algorithms, our algorithm yields recommendations of higher quality, demonstrating the clear advantage of combining RL and transformers.