Computational Mathematics and Applications Seminar

Over the last decade, adversarial attack algorithms have revealed instabilities in artificial intelligence (AI) tools. These algorithms raise issues regarding safety, reliability and interpretability; especially in high risk settings. Mathematics is at the heart of this landscape, with ideas from  numerical analysis, optimization, and high dimensional stochastic analysis playing key roles. From a practical perspective, there has been a war of escalation between those developing attack and defence strategies. At a more theoretical level, researchers have also studied bigger picture questions concerning the existence and computability of successful attacks. I will present examples of attack algorithms for neural networks in image classification, for transformer models in optical character recognition and for large language models. I will also show how recent generative diffusion models can be used adversarially. From a more theoretical perspective, I will outline recent results on the overarching question of whether, under reasonable assumptions, it is inevitable that AI tools will be vulnerable to attack.

Optimal damping consists of identifying a viscosity vector that maximizes the decay rate of a mechanical system's response. This can be rephrased as minimizing the trace of the solution of a Lyapunov equation whose coefficient matrix, representing the system dynamics, depends on the dampers' viscosities. The latter must be nonnegative for a physically meaningful solution, and the system must be asymptotically stable at the solution.

In this talk, we present conditions under which the system is never stable or may not be stable for certain values of the viscosity vector, and, in the latter case, discuss how to modify the constraints so as to guarantee stability. We show that the KKT conditions of our nonlinear optimization problem are equivalent to a viscosity-dependent nonlinear residual function that is equal to zero at an optimal viscosity vector. To minimize this residual function, we propose a Barzilai-Borwein residual minimization algorithm (BBRMA) and a spectral projection gradient algorithm (SPG). The efficiency of both algorithms relies on a fast computation of the gradient for BBRMA, and both the objective function and its gradient for SPG. By fully exploiting the low-rank structure of the problem, we show how to compute these in $O(n^2)$ operations, $n$ being the size of the mechanical system.

This is joint work with Qingna Li (Beijing Institute of Technology).

Several interesting and emerging problems in statistics, machine learning and optimal transport can be cast as minimisation of (entropy-regularised) objective functions defined on an appropriate space of probability distributions.  Numerical methods have historically focused on linear objective functions, a setting in which one has access to an unnormalised density for the distributional target.  For nonlinear objectives, numerical methods are relatively under-developed; for example, mean-field Langevin dynamics is considered state-of-the-art.  In the nonlinear setting even basic questions, such as how to tell whether or not a sequence of numerical approximations has practically converged, remain unanswered.  Our main contribution is to present the first computable measure of sub-optimality for optimisation in this context.  

Joint work with Clémentine Chazal, Heishiro Kanagawa, Zheyang Shen and Anna Korba.

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