Learning neural networks using only a small amount of data is an important ongoing research topic with tremendous potential for applications. We introduce a regularizer for the variational modeling of inverse problems in imaging based on normalizing flows, called patchNR. It involves a normalizing flow learned on patches of very few images. The subsequent reconstruction method is completely unsupervised and the same regularizer can be used for different forward operators acting on the same class of images.
By investigating the distribution of patches versus those of the whole image class, we prove that our variational model is indeed a MAP approach. Numerical examples for low-dose CT, limited-angle CT and superresolution of material images demonstrate that our method provides high quality results among unsupervised methods, but requires only very few data. Further, the appoach also works if only the low resolution image is available.
In the second part of the talk I will generalize normalizing flows to stochastic normalizing flows to improve their expressivity.Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. A unified framework to handle these approaches appear to be Markov chains. We consider stochastic normalizing flows as a pair of Markov chains fulfilling some properties and show how many state-of-the-art models for data generation fit into this framework. Indeed including stochastic layers improves the expressivity of the network and allows for generating multimodal distributions from unimodal ones. The Markov chains point of view enables us to couple both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers, variational autoencoders and diffusion normalizing flows in a mathematically sound way. Our framework establishes a useful mathematical tool to combine the various approaches.
Joint work with F. Altekrüger, A. Denker, P. Hagemann, J. Hertrich, P. Maass[-]

First-order non-convex optimization is at the heart of neural networks training. Recent analyses showed that the Polyak-Lojasiewicz condition is particularly well-suited to analyze the convergence of the training error for these architectures. In this short presentation, I will propose extensions of this condition that allows for more flexibility and application scenarios, and show how stochastic gradient descent converges under these conditions. Then, I will show how to use these conditions to prove the convergence of the test error for simple deep learning architectures in an online setting.[-]

Many of us use smartphones and rely on tools like auto-complete and spelling auto-correct to make using these devices more pleasant, but building these tools presents a challenge. On the one hand, the machine-learning algorithms used to provide these features require data to learn from, but on the other hand, who among us is willing to send a carbon copy of all our text messages to device manufacturers to provide that data? 'Local differential privacy' and related concepts have emerged as the gold standard model in which to analyze tradeoffs between losses in utility and privacy for solutions to such problems. In this talk, we give a new state-of-the-art algorithm for estimating histograms of user data, making use of projective geometry over finite fields coupled with a reconstruction algorithm based on dynamic programming.
This talk is based on joint work with Vitaly Feldman (Apple), Huy Le Nguyen (Northeastern), and Kunal Talwar (Apple).[-]

We present a list of counterexamples to conjectures in smooth convex coercive optimization. We will detail two extensions of the gradient descent method, of interest in machine learning: gradient descent with exact line search, and Bregman descent (also known as mirror descent). We show that both are non convergent in general. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of convex compact sets in the plane, whose boundaries are Ck curves (k ¿ 1, arbitrary) with positive curvature, there exists a Ck convex function for which each set of the sequence is a sublevel set. The talk will provide proof arguments for this results and detail how it can be used to construct the anounced counterexamples.[-]