We study the problem of efficiently estimating the coefficients of generalized linear models (GLMs) in the large-scale setting where the number of observations $n$ is much larger than the number of predictors $p$, i.e. $n\gg p \gg 1$. We show that in GLMs with random (not necessarily Gaussian) design, the GLM coefficients are approximately proportional to the corresponding ordinary least squares (OLS) coefficients. Using this relation, we design an algorithm that achieves the same accuracy as the maximum likelihood estimator (MLE) through iterations that attain up to a cubic convergence rate, and that are cheaper than any batch optimization algorithm by at least a factor of $\mathcal{O}(p)$. We provide theoretical guarantees for our algorithm, and analyze the convergence behavior in terms of data dimensions.

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for matrix completion has no spurious local minima --- all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve matrix completion with \textit{arbitrary} initialization in polynomial time.

We propose an online convex optimization algorithm (RescaledExp) that achieves optimal regret in the unconstrained setting without prior knowledge of any bounds on the loss functions. We prove a lower bound showing an exponential separation between the regret of existing algorithms that require a known bound on the loss functions and any algorithm that does not require such knowledge. RescaledExp matches this lower bound asymptotically in the number of iterations. RescaledExp is naturally hyperparameter-free and we demonstrate empirically that it matches prior optimization algorithms that require hyperparameter optimization.

We present a framework to train a structured prediction model by performing smoothing on the inference algorithm it builds upon. Smoothing overcomes the non-smoothness inherent to the maximum margin structured prediction objective, and paves the way for the use of fast primal gradient-based optimization algorithms. We illustrate the proposed framework by developing a novel primal incremental optimization algorithm for the structural support vector machine. The proposed algorithm blends an extrapolation scheme for acceleration and an adaptive smoothing scheme and builds upon the stochastic variance-reduced gradient algorithm. We establish its worst-case global complexity bound and study several practical variants. We present experimental results on two real-world problems, namely named entity recognition and visual object localization. The experimental results show that the proposed framework allows us to build upon efficient inference algorithms to develop large-scale optimization algorithms for structured prediction which can achieve competitive performance on the two real-world problems.

We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems. Our analysis is applicable to more general algorithms and optimization problems {\it \textbf{beyond}} convexity and strong convexity, e.g. Polyak-\L ojasiewicz and error bound conditions (possibly nonconvex).

As a significant direction in this line, hyperbolic representation learning has been shown to offer several advantages over conventional Euclidean geometry.

It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2% on ImageNet and saves up to 5x the pre-training compute on JFT.