To tackle these problems, four main strategies will be studied. A first direction of the research program (Risk Analysis) will be to prove risk bounds on the error (in expectation or with high probability) of such algorithms, for various deep learning tasks such as regression or classification. Proving that the generalization error is small, as well as the optimization error and the empirical risk, would enable us to show that the risk of the learning algorithm is small. This would be a strong robustness guarantee. An intern is currently working on the landscape of the objective function in deep learning. He will most probably start a PhD in September. A second direction of the research program (Features Stability) will be to design a neural network with stable features which is a required condition to achieve interpretable features [13]. In a first step, it will be demonstrated that specific neural networks architectures based on dropout and ReLU have this stability capability. The stability condition should lead to conditions on the number of samples and the Dropout parameter to ensure the stability. The second step will be to demonstrate on some hypothesis that the stability leads to the generalization. The third study, from a numerical and theoretical point of view, how stability can be enforced by enforcing parsimony in the intermediate layers. We will adopt for that: – an empirical point of view, by implementing networks in which the intermediate layers must be sparse and by testing these networks on concrete problems; – a theoretical point of view by adapting co-sparsity conditions (see, for example, in [16]) in the case of deep networks. As mentioned in the Villani report, game theory might carry the seeds of the next AI revolution. A third direction of the research program (Game theory and artificial intelligence) is to investigate the possible applications of game theory concepts and methods to AI in particular new game-theoretic algorithms for GAN training. We would like to investigate how some algorithms that have been originally designed for games could prove useful for GAN training. These algorithms may come from non-cooperative game theory (beyond fictitious play) or from adversarial online learning (no-regret algorithms), and will need to be largely adapted to the GAN context. Convergence bounds may be combined with approximation and generalization guarantees for neural networks. In addition, payoffs in GANs may be modified, e.g., they could be defined through new probability metrics designed for partially observable Markov decision processes and dynamic games with incomplete information. More generally, it is worthwhile to try to go beyond the 2-player zero-sum game setup and design new games to learn and mimic distributions with high-dimensional data, with improved empirical and theoretical properties. Multi-player games could prove useful and might become an important topic. The fourth direction of the research program (Certifying robustness of deep networks) focuses on the robustness of neural networks thanks to the Lasserre’s Hierarchy which can be used to solve global polynomial optimization problems over semi-algebraic sets [22]. It has many applications among which providing numerical certificates for formal proofs and certification [23]. In a first step we would like to investigate how this approach could be used to extend the approaches of [26] [24] to provide a hierarchy of adversarial robustness bounds for deep networks. In a second step we will investigate how the specific structure of neural network robustness certification problems could be used to devise efficient numerical algorithms to scale the approach up to real world networks. A second axe (Christoffel-Darboux kernels with applications in deep learning explainability) is to reformulate the initial problem as an optimization problem over probability measures. Recent research [27], investigated the ability of Christoffel-Darboux kernels to capture information about the support of an unknown probability measure. A distinguishing feature of this approach is to allow one to infer support characteristics, based on the knowledge of finitely many moments of the underlying measure.