Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Data

In this work, we will establish that the Langevin Monte-Carlo algorithm can learn depth-2 neural nets of any size and for any data and we give non-asymptotic convergence rates for it. We achieve this via showing that under Total Variation distance and q-Renyi divergence, the iterates of Langevin Monte Carlo converge to the Gibbs distribution of Frobenius norm regularized losses for any of these nets, when using smooth activations and in both classification and regression settings. Most critically, the amount of regularization needed for our results is independent of the size of the net. The key observation of ours is that two layer neural loss functions can always be regularized by a constant amount such that they satisfy the Villani conditions, and thus their Gibbs measures satisfy a Poincare inequality.

Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the $\ell_2$-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

Realistic Data Augmentation Framework for Enhancing Tabular Reasoning

Existing approaches to constructing training data for Natural Language Inference (NLI) tasks, such as for semi-structured table reasoning, are either via crowdsourcing or fully automatic methods. However, the former is expensive and time-consuming and thus limits scale, and the latter often produces naive examples that may lack complex reasoning. This paper develops a realistic semi-automated framework for data augmentation for tabular inference. Instead of manually generating a hypothesis for each table, our methodology generates hypothesis templates transferable to similar tables. In addition, our framework entails the creation of rational counterfactual tables based on human written logical constraints and premise paraphrasing. For our case study, we use the InfoTabs, which is an entity-centric tabular inference dataset. We observed that our framework could generate human-like tabular inference examples, which could benefit training data augmentation, especially in the scenario with limited supervision.