Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm ($\texttt{ADASAP}$) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. $\texttt{ADASAP}$ outperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, $\texttt{ADASAP}$ scales to a dataset with $> 3 \cdot 10^8$ samples, a feat which has not been accomplished in the literature.

This paper explores challenges in training Physics-Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill-conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-BFGS, showing the superiority of Adam+L-BFGS, and introduce a novel second-order optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.

We introduce the CRONOS algorithm for convex optimization of two-layer neural networks. CRONOS is the first algorithm capable of scaling to high-dimensional datasets such as ImageNet, which are ubiquitous in modern deep learning. This significantly improves upon prior work, which has been restricted to downsampled versions of MNIST and CIFAR-10. Taking CRONOS as a primitive, we then develop a new algorithm called CRONOS-AM, which combines CRONOS with alternating minimization, to obtain an algorithm capable of training multi-layer networks with arbitrary architectures. Our theoretical analysis proves that CRONOS converges to the global minimum of the convex reformulation under mild assumptions. In addition, we validate the efficacy of CRONOS and CRONOS-AM through extensive large-scale numerical experiments with GPU acceleration in JAX. Our results show that CRONOS-AM can obtain comparable or better validation accuracy than predominant tuned deep learning optimizers on vision and language tasks with benchmark datasets such as ImageNet and IMDb. To the best of our knowledge, CRONOS is the first algorithm which utilizes the convex reformulation to enhance performance on large-scale learning tasks.

Ill-conditioned problems are ubiquitous in large-scale machine learning: as a data set grows to include more and more features correlated with the labels, the condition number increases. Yet traditional stochastic gradient methods converge slowly on these ill-conditioned problems, even with careful hyperparameter tuning. This paper introduces PROMISE (Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates), a suite of sketching-based preconditioned stochastic gradient algorithms that deliver fast convergence on ill-conditioned large-scale convex optimization problems arising in machine learning. PROMISE includes preconditioned versions of SVRG, SAGA, and Katyusha; each algorithm comes with a strong theoretical analysis and effective default hyperparameter values. Empirically, we verify the superiority of the proposed algorithms by showing that, using default hyperparameter values, they outperform or match popular tuned stochastic gradient optimizers on a test bed of 51 ridge and logistic regression problems assembled from benchmark machine learning repositories. On the theoretical side, this paper introduces the notion of quadratic regularity in order to establish linear convergence of all proposed methods even when the preconditioner is updated infrequently. The speed of linear convergence is determined by the quadratic regularity ratio, which often provides a tighter bound on the convergence rate compared to the condition number, both in theory and in practice, and explains the fast global linear convergence of the proposed methods.

This paper develops a scalable new algorithm, called NysADMM, to minimize a smooth convex loss function with a convex regularizer. NysADMM accelerates the inexact Alternating Direction Method of Multipliers (ADMM) by constructing a preconditioner for the ADMM subproblem from a randomized low-rank Nyström approximation. NysADMM comes with strong theoretical guarantees: it solves the ADMM subproblem in a constant number of iterations when the rank of the Nyström approximation is the effective dimension of the subproblem regularized Gram matrix. In practice, ranks much smaller than the effective dimension can succeed, so NysADMM uses an adaptive strategy to choose the rank that enjoys analogous guarantees. Numerical experiments on real-world datasets demonstrate that NysADMM can solve important applications, such as the lasso, logistic regression, and support vector machines, in half the time (or less) required by standard solvers. The breadth of problems on which NysADMM beats standard solvers is a surprise: it suggests that ADMM is a dominant paradigm for numerical optimization across a wide range of statistical learning problems that are usually solved with bespoke methods.

Models like LASSO and ridge regression are extensively used in practice due to their interpretability, ease of use, and strong theoretical guarantees. Cross-validation (CV) is widely used for hyperparameter tuning in these models, but do practical methods minimize the true out-of-sample loss? A recent line of research promises to show that the optimum of the CV loss matches the optimum of the out-of-sample loss (possibly after simple corrections). It remains to show how tractable it is to minimize the CV loss.In the present paper, we show that, in the case of ridge regression, the CV loss may fail to be quasiconvex and thus may have multiple local optima. We can guarantee that the CV loss is quasiconvex in at least one case: when the spectrum of the covariate matrix is nearly flat and the noise in the observed responses is not too high. More generally, we show that quasiconvexity status is independent of many properties of the observed data (response norm, covariate-matrix right singular vectors and singular-value scaling) and has a complex dependence on the few that remain. We empirically confirm our theory using simulated experiments.