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The rise of Mean Field Games (MFGs) has fostered a growing family of algorithms designed to solve large-scale multi-agent systems. However, the field currently lacks a standardized evaluation protocol, forcing researchers to rely on bespoke, isolated, and often simplistic environments. This fragmentation makes it difficult to assess the robustness, generalization, and failure modes of emerging methods. To address this gap, we propose a comprehensive benchmark suite for MFGs, focusing on the discrete-time, discrete-space, stationary setting. We introduce a taxonomy of problem classes, ranging from no-interaction and monotone games to potential and dynamics-coupled games, and provide prototypical environments for each. Furthermore, we present MF-Garnets, a method for generating random MFG instances to facilitate rigorous statistical testing. We benchmark a variety of learning algorithms across these environments, including a novel population-based approach (MF-PSO). To our knowledge, this is the first benchmark for MFGs.

Soft powered descent landings are becoming evermore important in the field of space exploration and scientific advancement, because of their ability to deliver mission critical payloads, both highly accurately and safely. The high efficiency restriction on planetary landings due to the severely limited fuel supply results in the need for optimal guidance trajectories and control. Model based reinforcement learning is proving to be an important tool in simplifying the highly complex process of generating optimal guidance trajectories and control. In this work, we present DreamLander, a state of the art model based reinforcement learning algorithm based on DreamerV3, to generate optimal guidance and control for a realistic 6 degree of freedom entry descent landing problem. The model based approach creates a sample efficient and highly generalizable model that can be applied to a variety of scenarios whilst maintaining a high degree of accuracy, landing within 2 meters of the designated landing zone, and with velocity norm less than 0.3 meters per second. DreamLander makes use of state sensor observations, making progress toward End-to-End solutions of the Entry Descent Landing problem.

Stochastic Gradient Descent (SGD) is a workhorse algorithm for continuous optimization. It is a folk belief among deep learning practitioners that learning rate decay in SGD substantially improves performance. However, it has remained unclear whether learning rate decay offers an asymptotic advantage for global nonconvex optimization. We answer this question in the affirmative by demonstrating a natural class of nonconvex functions for which, we prove that SGD with learning rate decay requires exponentially fewer gradient queries for global optimization than SGD with any fixed learning rate in the small-step size regime. To the best of our knowledge, this is the first exponential advantage that has been demonstrated for learning rate decay. The class of functions we consider includes many popular benchmark nonconvex functions. Our results provide a robust explanatory theory for the empirically observed benefits of learning rate decay for global optimization. We show that annealing performs the non-logconcave sampling task associated to the optimization problem in polynomial time, via a novel application of weak Poincaré inequalities.

Fundamental tasks in linear systems analysis and control require evaluating infinite-horizon output energies for autonomous systems and optimal value functions for controlled systems. These quantities are characterized by Lyapunov and algebraic Riccati equations, respectively. In this work, we study promise problems for their evaluation when the system matrices are sparse and represented succinctly by Boolean circuits specifying their nonzero entries, with input length polynomial in the logarithm of the system dimension. For dissipative autonomous systems with coordinate-vector initial states, we define the succinct Lyapunov value problem (SLV). Its extension to linear-quadratic optimal control is the succinct LQR value problem (SLQRV). We prove that SLV is $\BQP$-hard: every problem in bounded-error quantum polynomial time ($\BQP$) reduces to it in polynomial time. We then establish that the more general SLQRV problem admits a quantum algorithm running in time polynomial in the succinct input length. Since SLV is a special case of SLQRV, both problems are $\BQP$-complete. Consequently, unless bounded-error probabilistic polynomial time coincides with $\BQP$, no classical randomized algorithm can solve either problem on all promised instances in time polynomial in the succinct input length.

Exact score-based causal structure learning supports explicit structural constraints and optimisation certificates, but its branch-and-cut search can be computationally demanding. We augment exact causal structure learning methods with an upper bound on the maximum clique size of the learned undirected skeleton, enforced at integer candidates through lazy cycle and clique constraints. On static linear-Gaussian Erdős-Rényi and scale-free DAGs, we evaluate callback order, alternative clique-cut policies, incumbent repair, fractional separation, and Big-M bounds. We report causal-recovery metrics jointly with optimality gaps, primal and dual bounds, node throughput, MIPSOL events, and callback cost. Two variants receive particular attention: B3 selects the most violated minimal (k+1)-clique inequality at each callback, whereas C3 repairs an infeasible integer candidate by deleting violating edges and refitting affected regression columns using QR-based least squares. The results do not indicate that clique constraints uniformly accelerate optimisation. However, their benefit depends on graph density, sample size, and cut-management policy, while large collections of correlated constraints can reduce node throughput. Tighter valid Big-M bounds reduced the fixed-budget optimality gap in most tested settings, without necessarily changing SHD or F1 at a fixed time limit. These findings characterize the computational trade-offs of clique-constrained exact causal learning, and identify selective separation and incumbent repair as promising directions.