We propose to view the outcomes of the agents' dynamics as inducing a "meta-game" between the users.

Importance sampling is a rare event simulation technique used in Monte Carlo simulations to bias the sampling distribution towards the rare event of interest. By assigning appropriate weights to sampled points, importance sampling allows for more efficient estimation of rare events or tails of distributions. However, importance sampling can fail when the proposal distribution does not effectively cover the target distribution. In this work, we propose a method for more efficient sampling by updating the proposal distribution in the latent space of a normalizing flow. Normalizing flows learn an invertible mapping from a target distribution to a simpler latent distribution. The latent space can be more easily explored during the search for a proposal distribution, and samples from the proposal distribution are recovered in the space of the target distribution via the invertible mapping. We empirically validate our methodology on simulated robotics applications such as autonomous racing and aircraft ground collision avoidance.

The long-run behavior of multi-agent learning - and, in particular, no-regret learning - is relatively well-understood in potential games, where players have aligned interests. By contrast, in harmonic games - the strategic counterpart of potential games, where players have conflicting interests - very little is known outside the narrow subclass of 2-player zero-sum games with a fully-mixed equilibrium. Our paper seeks to partially fill this gap by focusing on the full class of (generalized) harmonic games and examining the convergence properties of follow-the-regularized-leader (FTRL), the most widely studied class of no-regret learning schemes. As a first result, we show that the continuous-time dynamics of FTRL are Poincar\'e recurrent, that is, they return arbitrarily close to their starting point infinitely often, and hence fail to converge. In discrete time, the standard, "vanilla" implementation of FTRL may lead to even worse outcomes, eventually trapping the players in a perpetual cycle of best-responses. However, if FTRL is augmented with a suitable extrapolation step - which includes as special cases the optimistic and mirror-prox variants of FTRL - we show that learning converges to a Nash equilibrium from any initial condition, and all players are guaranteed at most O(1) regret. These results provide an in-depth understanding of no-regret learning in harmonic games, nesting prior work on 2-player zero-sum games, and showing at a high level that harmonic games are the canonical complement of potential games, not only from a strategic, but also from a dynamic viewpoint.

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Contemporary applications of machine learning in two-team e-sports and the superior expressivity of multi-agent generative adversarial networks raise important and overlooked theoretical questions regarding optimization in two-team games. Formally, two-team zero-sum games are defined as multi-player games where players are split into two competing sets of agents, each experiencing a utility identical to that of their teammates and opposite to that of the opposing team. We focus on the solution concept of Nash equilibria (NE). We first show that computing NE for this class of games is $\textit{hard}$ for the complexity class ${\mathrm{CLS}}$. To further examine the capabilities of online learning algorithms in games with full-information feedback, we propose a benchmark of a simple -- yet nontrivial -- family of such games. These games do not enjoy the properties used to prove convergence for relevant algorithms. In particular, we use a dynamical systems perspective to demonstrate that gradient descent-ascent, its optimistic variant, optimistic multiplicative weights update, and extra gradient fail to converge (even locally) to a Nash equilibrium. On a brighter note, we propose a first-order method that leverages control theory techniques and under some conditions enjoys last-iterate local convergence to a Nash equilibrium. We also believe our proposed method is of independent interest for general min-max optimization.

Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of black - box m odels is that they underperform under blind conditions since no physical knowledge is incorporated. Physics - based ML aims to address this problem by retaining the mathematical flexibility of ML techniques while incorporating physics. In accord, this paper proposes to embed mechanics - based models into the mean function of a Gaussian Process (GP) model and characterize potential discrepancies through kernel machines. A specific class of kernel function is promoted, which has a connection with the gradient of the physics - based model with respect to the input and parameters and shares similarity with the exact Auto - covariance function of linear dynamical systems. The spectral properties of the kernel function enable considering dominant periodic processes origin ating from physics misspecification. Nevertheless, the stationarity of the kernel function is a difficult hurdle in the sequential processing of long data sets, resolved through hierarchical Bayesian techniques. This implementation is also advantageous to mitigate computational costs, alleviating the scalability of GPs when dealing with sequential data. Using numerical and experimental examples, potential applications of the proposed method to structural dynamics inverse problems are demonstrated. Postdoctoral Fellow, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, Email: osedehi@connect.ust.hk Ph.D. Student, Department of Civil and Environmental Engineering, The Hong Kong Universi ty of Science and Technology, Hong Kong, Email: akosikova@connect.ust.hk

Last week, Google Research held an online workshop on the conceptual understanding of deep learning. The workshop, which featured presentations by award-winning computer scientists and neuroscientists, discussed how new findings in deep learning and neuroscience can help create better artificial intelligence systems. While all the presentations and discussions were worth watching (and I might revisit them again in the coming weeks), one in particular stood out for me: A talk on word representations in the brain by Christos Papadimitriou, professor of computer science at Columbia University. In his presentation, Papadimitriou, a recipient of the Gödel Prize and Knuth Prize, discussed how our growing understanding of information-processing mechanisms in the brain might help create algorithms that are more robust in understanding and engaging in conversations. Papadimitriou presented a simple and efficient model that explains how different areas of the brain inter-communicate to solve cognitive problems.

Last week, Google Research held an online workshop on the conceptual understanding of deep learning. The workshop, which featured presentations by award-winning computer scientists and neuroscientists, discussed how new findings in deep learning and neuroscience can help create better artificial intelligence systems. While all the presentations and discussions were worth watching (and I might revisit them again in the coming weeks), one, in particular, stood out for me: A talk on word representations in the brain by Christos Papadimitriou, professor of computer science at the University of Columbia. In his presentation, Papadimitriou, a recipient of the Gödel Prize and Knuth Prize, discussed how our growing understanding of information-processing mechanisms in the brain might help create algorithms that are more robust in understanding and engaging in conversations. Papadimitriou presented a simple and efficient model that explains how different areas of the brain inter-communicate to solve cognitive problems.