Challenges in Training PINNs: A Loss Landscape Perspective

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.

PROMISE: Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates

This paper introduces PROMISE ($\textbf{Pr}$econditioned Stochastic $\textbf{O}$ptimization $\textbf{M}$ethods by $\textbf{I}$ncorporating $\textbf{S}$calable Curvature $\textbf{E}$stimates), a suite of sketching-based preconditioned stochastic gradient algorithms for solving 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. In contrast, traditional stochastic gradient methods require careful hyperparameter tuning to succeed, and degrade in the presence of ill-conditioning, a ubiquitous phenomenon in machine learning. 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.

Robust, randomized preconditioning for kernel ridge regression

This paper introduces two randomized preconditioning techniques for robustly solving kernel ridge regression (KRR) problems with a medium to large number of data points ($10^4 \leq N \leq 10^7$). The first method, RPCholesky preconditioning, is capable of accurately solving the full-data KRR problem in $O(N^2)$ arithmetic operations, assuming sufficiently rapid polynomial decay of the kernel matrix eigenvalues. The second method, KRILL preconditioning, offers an accurate solution to a restricted version of the KRR problem involving $k \ll N$ selected data centers at a cost of $O((N + k^2) k \log k)$ operations. The proposed methods solve a broad range of KRR problems and overcome the failure modes of previous KRR preconditioners, making them ideal for practical applications.

SketchySGD: Reliable Stochastic Optimization via Robust Curvature Estimates

We introduce SketchySGD, a stochastic quasi-Newton method that uses sketching to approximate the curvature of the loss function. Quasi-Newton methods are among the most effective algorithms in traditional optimization, where they converge much faster than first-order methods such as SGD. However, for contemporary deep learning, quasi-Newton methods are considered inferior to first-order methods like SGD and Adam owing to higher per-iteration complexity and fragility due to inexact gradients. SketchySGD circumvents these issues by a novel combination of subsampling, randomized low-rank approximation, and dynamic regularization. In the convex case, we show SketchySGD with a fixed stepsize converges to a small ball around the optimum at a faster rate than SGD for ill-conditioned problems. In the non-convex case, SketchySGD converges linearly under two additional assumptions, interpolation and the Polyak-Lojaciewicz condition, the latter of which holds with high probability for wide neural networks. Numerical experiments on image and tabular data demonstrate the improved reliability and speed of SketchySGD for deep learning, compared to standard optimizers such as SGD and Adam and existing quasi-Newton methods.

Can we globally optimize cross-validation loss? Quasiconvexity in ridge regression

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 optimization 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.