Neural network verifiers based on linear bound propagation scale impressively to massive models but can be surprisingly loose when neuron coupling is crucial. Conversely, semidefinite programming (SDP) verifiers capture inter-neuron coupling naturally, but their cubic complexity restricts them to only small models. In this paper, we propose SDP-CROWN, a novel hybrid verification framework that combines the tightness of SDP relaxations with the scalability of bound-propagation verifiers. At the core of SDP-CROWN is a new linear bound—derived via SDP principles—that explicitly captures $\ell_{2}$-norm-based inter-neuron coupling while adding only one extra parameter per layer. This bound can be integrated seamlessly into any linear bound-propagation pipeline, preserving the inherent scalability of such methods yet significantly improving tightness. In theory, we prove that our inter-neuron bound can be up to a factor of $\sqrt{n}$ tighter than traditional per-neuron bounds. In practice, when incorporated into the state-of-the-art $\alpha$-CROWN verifier, we observe markedly improved verification performance on large models with up to 65 thousand neurons and 2.47 million parameters, achieving tightness that approaches that of costly SDP-based methods.

Non-convex gradient descent is a common approach for estimating a low-rank $n\times n$ ground truth matrix from noisy measurements, because it has per-iteration costs as low as $O(n)$ time, and is in theory capable of converging to a minimax optimal estimate. However, the practitioner is often constrained to just tens to hundreds of iterations, and the slow and/or inconsistent convergence of non-convex gradient descent can prevent a high-quality estimate from being obtained. Recently, the technique of \emph{preconditioning} was shown to be highly effective at accelerating the local convergence of non-convex gradient descent when the measurements are noiseless. In this paper, we describe how preconditioning should be done for noisy measurements to accelerate local convergence to minimax optimality. For the symmetric matrix sensing problem, our proposed preconditioned method is guaranteed to locally converge to minimax error at a linear rate that is immune to ill-conditioning and/or over-parameterization. Using our proposed preconditioned method, we perform a 60 megapixel medical image denoising task, and observe significantly reduced noise levels compared to previous approaches.

Adversarial training is well-known to produce high-quality neural network models that are empirically robust against adversarial perturbations. Nevertheless, once a model has been adversarially trained, one often desires a certification that the model is truly robust against all future attacks. Unfortunately, when faced with adversarially trained models, all existing approaches have significant trouble making certifications that are strong enough to be practically useful. Linear programming (LP) techniques in particular face a “convex relaxation barrier” that prevent them from making high-quality certifications, even after refinement with mixed-integer linear programming (MILP) and branch-and-bound (BnB) techniques. In this paper, we propose a nonconvex certification technique, based on a low-rank restriction of a semidefinite programming (SDP) relaxation. The nonconvex relaxation makes strong certifications comparable to much more expensive SDP methods, while optimizing over dramatically fewer variables comparable to much weaker LP methods. Despite nonconvexity, we show how off-the-shelf local optimization algorithms can be used to achieve and to certify global optimality in polynomial time. Our experiments find that the nonconvex relaxation almost completely closes the gap towards exact certification of adversarially trained models.

The matrix completion problem seeks to recover a $d\times d$ ground truth matrix of low rank $r\ll d$ from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with $d$ so large that even the simplest full-dimension vector operations with $O(d)$ time complexity become prohibitively expensive. Stochastic gradient descent (SGD) is one of the few algorithms capable of solving matrix completion on a huge scale, and can also naturally handle streaming data over an evolving ground truth. Unfortunately, SGD experiences a dramatic slow-down when the underlying ground truth is ill-conditioned; it requires at least $O(\kappa\log(1/\epsilon))$ iterations to get $\epsilon$-close to ground truth matrix with condition number $\kappa$. In this paper, we propose a preconditioned version of SGD that preserves all the favorable practical qualities of SGD for huge-scale online optimization while also making it agnostic to $\kappa$. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to $\epsilon$-accuracy in $O(\log(1/\epsilon))$ iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with $\kappa=1$. In our numerical experiments, we observe a similar acceleration forill-conditioned matrix completion under the root mean square error (RMSE) loss, Euclidean distance matrix (EDM) completion under pairwise square loss, and collaborative filtering under the Bayesian Personalized Ranking (BPR) loss.