The framework of multi-agent learning explores the dynamics of how an agent's

In this paper, we provide a simple, model-free algorithm for stabilizing fully observed dynamical systems.

Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transitions has remained limited, particularly in stochastic and data-driven settings. In this work, we establish a general, model-independent theorem providing sufficient conditions under which persistent entropy provably separates two phases. We show that persistent entropy exhibits an asymptotically non-vanishing gap across phases. The result relies only on continuity of persistent entropy along the convergent diagram sequence, or under mild regularization, and is therefore broadly applicable across data modalities, filtrations, and homological degrees. To connect asymptotic theory with finite-time computations, we introduce an operational framework based on topological stabilization, defining a topological transition time by stabilizing a chosen topological statistic over sliding windows, and a probability-based estimator of critical parameters within a finite observation horizon. We validate the framework on the Kuramoto synchronization transition, the Vicsek order-to-disorder transition in collective motion, and neural network training dynamics across multiple datasets and architectures. Across all experiments, stabilization of persistent entropy and collapse of variability across realizations provide robust numerical signatures consistent with the theoretical mechanism.

Control problems are always challenging since they arise from the real-world systems where stochasticity and randomness are of ubiquitous presence.

Curvature information -- particularly, the largest eigenvalue of the lossHessian, known as the sharpness -- often forms the basis for learning ratetuners. However, recent work has shown that the curvature information undergoescomplex dynamics during training, going from a phase of increasing sharpness toeventual stabilization. We analyze the closed-loop feedback effect betweenlearning rate tuning and curvature. We find that classical learning rate tunersmay yield greater one-step loss reduction, yet they ultimately underperform inthe long term when compared to constant learning rates in the full batch regime.These models break the stabilization of the sharpness, which we explain using asimplified model of the joint dynamics of the learning rate and the curvature.To further investigate these effects, we introduce a new learning rate tuningmethod, Curvature Dynamics Aware Tuning (CDAT), which prioritizes long termcurvature stabilization over instantaneous progress on the objective. In thefull batch regime, CDAT shows behavior akin to prefixed warm-up schedules on deeplearning objectives, outperforming tuned constant learning rates. In the minibatch regime, we observe that stochasticity introduces confounding effects thatexplain the previous success of some learning rate tuners at appropriate batchsizes. Our findings highlight the critical role of understanding the jointdynamics of the learning rate and curvature, beyond greedy minimization, todiagnose failures and design effective adaptive learning rate tuners.

Anderson mixing has been heuristically applied to reinforcement learning (RL) algorithms for accelerating convergence and improving the sampling efficiency of deep RL. Despite its heuristic improvement of convergence, a rigorous mathematical justification for the benefits of Anderson mixing in RL has not yet been put forward. In this paper, we provide deeper insights into a class of acceleration schemes built on Anderson mixing that improve the convergence of deep RL algorithms. Our main results establish a connection between Anderson mixing and quasi-Newton methods and prove that Anderson mixing increases the convergence radius of policy iteration schemes by an extra contraction factor.

The problem of orbital stabilization of periodic trajectories has been addressed in a series of publications: [1, 2, 3, 4, 5, 6, 7]. Many of these works, e.g., [1, 2, 4, 7], employ the transverse linearization approach, which approximates the dynamics near a reference periodic orbit by a linear time-varying (LTV) system with periodic coefficients. As shown in [2, 8], a feedback designed to stabilize the trivial solution of this auxiliary LTV system can be used to construct a control law that stabilizes the orbit of the original nonlinear system. Under the mild assumption of controllability of the LTV system over one period, the LQR approach can be used to design the feedback. The practical effectiveness of this method was demonstrated in experiments with real robotic systems in [9, 10, 11]. A substantially different stabilization method for the LTV system was proposed in [5], where the authors developed an alternative scheme combining Floquet theory with sliding-mode control. Following this line of work, we show that a specific feedback linearization of the transverse dynamics yields an LTV system endowed with a stable invariant subspace. In this setting, the control objective reduces to driving all trajectories into the stable subspace, which is achieved via sliding-mode-based control. This method does not require solving the computationally demanding periodic LQR problem.

Animals with foveated vision, including humans, experience microsaccades, small, rapid eye movements that they are not aware of. Inspired by this phenomenon, we develop a method for "Artificial Microsaccade Compensation". It can stabilize video captured by a tailless ornithopter that has resisted attempts to use camera-based sensing because it shakes at 12-20 Hz. Our approach minimizes changes in image intensity by optimizing over 3D rotation represented in SO(3). This results in a stabilized video, computed in real time, suitable for human viewing, and free from distortion. When adapted to hold a fixed viewing orientation, up to occasional saccades, it can dramatically reduce inter-frame motion while also benefiting from an efficient recursive update. When compared to Adobe Premier Pro's warp stabilizer, which is widely regarded as the best commercial video stabilization software available, our method achieves higher quality results while also running in real time.

Long-horizon multivariate time-series forecasting is challenging because realistic predictions must (i) denoise heterogeneous signals, (ii) track time-varying cross-series dependencies, and (iii) remain stable and physically plausible over long rollout horizons. We present PRISM, which couples a score-based diffusion preconditioner with a dynamic, correlation-thresholded graph encoder and a forecast head regularized by generic physics penalties. We prove contraction of the induced horizon dynamics under mild conditions and derive Lipschitz bounds for graph blocks, explaining the model's robustness. On six standard benchmarks , PRISM achieves consistent SOTA with strong MSE and MAE gains.