This problem is particularly challenging when using gradient-based Markov Chain Monte Carlo (MCMC) algorithms due to diminishing gradients, which occurs when the tails of the target density decay at a slow (e.g.

Mean field games (MFGs) are a promising framework for modeling the behavior of large-population systems. However, solving MFGs can be challenging due to the coupling of forward population evolution and backward agent dynamics. Typically, obtaining mean field Nash equilibria (MFNE) involves an iterative approach where the forward and backward processes are solved alternately, known as fixed-point iteration (FPI). This method requires fully observed population propagation and agent dynamics over the entire spatial domain, which could be impractical in some real-world scenarios. To overcome this limitation, this paper introduces a novel online single-agent model-free learning scheme, which enables a single agent to learn MFNE using online samples, without prior knowledge of the state-action space, reward function, or transition dynamics. Specifically, the agent updates its policy through the value function (Q), while simultaneously evaluating the mean field state (M), using the same batch of observations. We develop two variants of this learning scheme: off-policy and on-policy QM iteration. We prove that they efficiently approximate FPI, and a sample complexity guarantee is provided. The efficacy of our methods is confirmed by numerical experiments.

We study the complexity of heavy-tailed sampling and present a separation result in terms of obtaining high-accuracy versus low-accuracy guarantees i.e., samplers that require only $O(\log(1/\varepsilon))$ versus $\Omega(\text{poly}(1/\varepsilon))$ iterations to output a sample which is $\varepsilon$-close to the target in $\chi^2$-divergence. Our results are presented for proximal samplers that are based on Gaussian versus stable oracles. We show that proximal samplers based on the Gaussian oracle have a fundamental barrier in that they necessarily achieve only low-accuracy guarantees when sampling from a class of heavy-tailed targets. In contrast, proximal samplers based on the stable oracle exhibit high-accuracy guarantees, thereby overcoming the aforementioned limitation. We also prove lower bounds for samplers under the stable oracle and show that our upper bounds cannot be fundamentally improved.

In target tracking and sensor fusion contexts it is not unusual to deal with a large number of Gaussian densities that encode the available information (multiple hypotheses), as in applications where many sensors, affected by clutter or multimodal noise, take measurements on the same scene. In such cases reduction procedures must be implemented, with the purpose of limiting the computational load. In some situations it is required to fuse all available information into a single hypothesis, and this is usually done by computing the barycenter of the set. However, such computation strongly depends on the chosen dissimilarity measure, and most often it must be performed making use of numerical methods, since in very few cases the barycenter can be computed analytically. Some issues, like the constraint on the covariance, that must be symmetric and positive definite, make it hard the numerical computation of the barycenter of a set of Gaussians. In this work, Fixed-Point Iterations (FPI) are presented for the computation of barycenters according to several dissimilarity measures, making up a useful toolbox for fusion/reduction of Gaussian sets in applications where specific dissimilarity measures are required.

As a former professor in artificial intelligence, one of my favorite – and surely one of the oldest – technological myths is found in the masterpiece, the Iliad. In Homer's poem narrating the Trojan War, the God of metalworking, Hephaestus, engineers one of the first robots known to history, a handmaiden designed to assist him in his forge. Not happy with limiting himself to manufacturing, Hephaestus steps it up by designing Talos, an automated bronze giant whose purpose was to protect ancient Crete from pirates and invaders. While thousands of years have passed since Hephaestus' mythical robots came to life, today's intelligent machines – strong with skillful AI – are making headway in our own workplaces. Take the factories and warehouses adversely affected by the pandemic as an example. With fewer and fewer workers willing and able to assist our manufacturers and fulfilment centers, many are embracing AI and machine learning to automate tasks such as quality control which are traditionally reliant on scores of human workers.

This piece explores myths about Artificial Intelligence, such as "I need to go to a university and hire an AI PhD" and "I need to collect millions of images to even know if using AI is possible." As a former professor in artificial intelligence, one of my favorite–and surely one of the oldest–technological myths is found in the masterpiece, the Iliad. In Homer's poem narrating the Trojan War, the God of metalworking, Hephaestus, engineers one of the first robots known to history, a handmaiden designed to assist him in his forge. Not happy with limiting himself to manufacturing, Hephaestus steps it up by designing Talos, an automated bronze giant whose purpose was to protect ancient Crete from pirates and invaders. While thousands of years have passed since Hephaestus' mythical robots came to life, today's intelligent machines–strong of skillful AI–are making headway in our own workplaces.