Daily Papers

In distributed deep learning with data parallelism, synchronizing gradients at each training step can cause a huge communication overhead, especially when many nodes work together to train large models. Local gradient methods, such as Local SGD, address this issue by allowing workers to compute locally for H steps without synchronizing with others, hence reducing communication frequency. While H has been viewed as a hyperparameter to trade optimization efficiency for communication cost, recent research indicates that setting a proper H value can lead to generalization improvement. Yet, selecting a proper H is elusive. This work proposes a theory-grounded method for determining H, named the Quadratic Synchronization Rule (QSR), which recommends dynamically setting H in proportion to 1{eta^2} as the learning rate eta decays over time. Extensive ImageNet experiments on ResNet and ViT show that local gradient methods with QSR consistently improve the test accuracy over other synchronization strategies. Compared with the standard data parallel training, QSR enables Local AdamW on ViT-B to cut the training time on 16 or 64 GPUs down from 26.7 to 20.2 hours or from 8.6 to 5.5 hours and, at the same time, achieves 1.16% or 0.84% higher top-1 validation accuracy.

In heterogeneous networks of coupled oscillators, phase frustration typically prevents the emergence of synchronization in the Sakaguchi--Kuramoto (SK) model. In this study, we propose an analytical framework to overcome this barrier and induce complete synchronization at a specified small coupling value in oscillators governed by phase-frustrated bi-harmonic coupling. We derive an optimal set of natural frequencies that is robust against added noise and correlated with the network degree heterogeneity, along with the parameters involved in the bi-harmonic coupling function that lead to complete synchronization (r = 1). In addition, we find complete synchronization transitions accompanied by hysteresis in scale-free networks, indicating a first-order (discontinuous) phase transition, whereas Erdős--Rényi networks exhibit complete synchronization through a second-order (continuous) phase transition. Furthermore, we use the mean-field approximation in the presence of optimal frequencies to determine the critical coupling strength associated with the synchronization transition in the pure second-harmonic Sakaguchi--Kuramoto model. Here, the obtained optimal natural frequencies scale linearly with the node degree, and the critical coupling strength for the onset of synchronization is derived analytically from the self-consistent equations. In this specific regime, we observe a perfectly ordered two-cluster synchronized state. These findings remain robust for higher-order harmonic coupling schemes, as well as across a diverse range of synthetic and empirical networks, including scale-free, Erdős--Rényi, Zachary Karate Club, and the C.~elegans neural network, demonstrating their general applicability.

We address the problem of model-free distributed stabilization of heterogeneous multi-agent systems using reinforcement learning (RL). Two algorithms are developed. The first algorithm solves a centralized linear quadratic regulator (LQR) problem without knowing any initial stabilizing gain in advance. The second algorithm builds upon the results of the first algorithm, and extends it to distributed stabilization of multi-agent systems with predefined interaction graphs. Rigorous proofs are provided to show that the proposed algorithms achieve guaranteed convergence if specific conditions hold. A simulation example is presented to demonstrate the theoretical results.

The angular synchronization problem aims to accurately estimate (up to a constant additive phase) a set of unknown angles theta_1, dots, theta_nin[0, 2pi) from m noisy measurements of their offsets theta_i-theta_j ;mod ; 2pi. Applications include, for example, sensor network localization, phase retrieval, and distributed clock synchronization. An extension of the problem to the heterogeneous setting (dubbed k-synchronization) is to estimate k groups of angles simultaneously, given noisy observations (with unknown group assignment) from each group. Existing methods for angular synchronization usually perform poorly in high-noise regimes, which are common in applications. In this paper, we leverage neural networks for the angular synchronization problem, and its heterogeneous extension, by proposing GNNSync, a theoretically-grounded end-to-end trainable framework using directed graph neural networks. In addition, new loss functions are devised to encode synchronization objectives. Experimental results on extensive data sets demonstrate that GNNSync attains competitive, and often superior, performance against a comprehensive set of baselines for the angular synchronization problem and its extension, validating the robustness of GNNSync even at high noise levels.

I introduce a novel mathematical framework integrating topological dynamics, operator algebras, and ergodic geometry to study lattices of asynchronous metric dynamical systems. Each node in the lattice carries an internal flow represented by a one-parameter family of operators, evolving on its own time scale. I formalize stratified state spaces capturing multiple levels of synchronized behavior, define an asynchronous evolution metric that quantifies phase-offset distances between subsystems, and characterize emergent coherent topologies arising when subsystems synchronize. Within this framework, I develop formal operators for the evolution of each subsystem and give precise conditions under which phase-aligned synchronization occurs across the lattice. The main results include: (1) the existence and uniqueness of coherent (synchronized) states under a contractive coupling condition, (2) stability of these coherent states and criteria for their emergence as a collective phase transition in a continuous operator topology, and (3) the influence of symmetries, with group-invariant coupling leading to flow-invariant synchrony subspaces and structured cluster dynamics. Proofs are given for each theorem, demonstrating full mathematical rigor. In a final section, I discuss hypothetical applications of this framework to symbolic lattice systems (e.g. subshifts), to invariant group actions on dynamical lattices, and to operator fields over stratified manifolds in the spirit of noncommutative geometry. Throughout, I write in the first person to emphasize the exploratory nature of this work. The paper avoids any reference to cosmology or observers, focusing instead on clean, formal mathematics suitable for a broad array of dynamical systems.

Synchronization in one dimension displays generic scale invariance with universal properties previously observed in surface kinetic roughening and the wider context of the Kardar-Parisi-Zhang (KPZ) universality class. This has been established for phase oscillators and also for some limit-cycle oscillators, both in the presence of columnar (quenched) disorder and of time-dependent noise, by extensive numerical simulations, and has been analytically motivated by continuum approximations in the strong oscillator coupling limit. The robustness and the precise boundaries in parameter space for such critical behavior remain unclear, however, which may preclude further developments, including the extension of these results to higher dimensions and the experimental observation of nonequilibrium criticality in synchronizing (e.g.~electronic or chemical) oscillators. We here present complete numerical phase diagrams of one-dimensional synchronization, including saturation times and values, but, most importantly, also dynamical features giving insight into the gradual emergence of synchronous dynamics, based on systems of phase oscillators with either type of randomness. In the absence of synchronization, the dynamics evolves as expected for random deposition (for time-dependent noise) or linear growth (for columnar disorder), while a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang behavior (with the corresponding type of randomness) is observed as the randomness strength, or the nonoddity of the coupling among oscillators, is increased in the synchronous region -- their combined effect being partially captured by the so-called KPZ coupling. The distortion of scaling due to phase slips near the desynchronization boundary, a feature that is likely to play a role in experimental contexts, is also discussed.

We propose the Kuramoto Graph Neural Network (KuramotoGNN), a novel class of continuous-depth graph neural networks (GNNs) that employs the Kuramoto model to mitigate the over-smoothing phenomenon, in which node features in GNNs become indistinguishable as the number of layers increases. The Kuramoto model captures the synchronization behavior of non-linear coupled oscillators. Under the view of coupled oscillators, we first show the connection between Kuramoto model and basic GNN and then over-smoothing phenomenon in GNNs can be interpreted as phase synchronization in Kuramoto model. The KuramotoGNN replaces this phase synchronization with frequency synchronization to prevent the node features from converging into each other while allowing the system to reach a stable synchronized state. We experimentally verify the advantages of the KuramotoGNN over the baseline GNNs and existing methods in reducing over-smoothing on various graph deep learning benchmark tasks.

We introduce for continual learning Autodiff Quadratic Consolidation (AQC), which approximates the previous loss function with a quadratic function, and Neural Consolidation (NC), which approximates the previous loss function with a neural network. Although they are not scalable to large neural networks, they can be used with a fixed pre-trained feature extractor. We empirically study these methods in class-incremental learning, for which regularization-based methods produce unsatisfactory results, unless combined with replay. We find that for small datasets, quadratic approximation of the previous loss function leads to poor results, even with full Hessian computation, and NC could significantly improve the predictive performance, while for large datasets, when used with a fixed pre-trained feature extractor, AQC provides superior predictive performance. We also find that using tanh-output features can improve the predictive performance of AQC. In particular, in class-incremental Split MNIST, when a Convolutional Neural Network (CNN) with tanh-output features is pre-trained on EMNIST Letters and used as a fixed pre-trained feature extractor, AQC can achieve predictive performance comparable to joint training.

Inspired by the diversity of biological neurons, quadratic artificial neurons can play an important role in deep learning models. The type of quadratic neurons of our interest replaces the inner-product operation in the conventional neuron with a quadratic function. Despite promising results so far achieved by networks of quadratic neurons, there are important issues not well addressed. Theoretically, the superior expressivity of a quadratic network over either a conventional network or a conventional network via quadratic activation is not fully elucidated, which makes the use of quadratic networks not well grounded. Practically, although a quadratic network can be trained via generic backpropagation, it can be subject to a higher risk of collapse than the conventional counterpart. To address these issues, we first apply the spline theory and a measure from algebraic geometry to give two theorems that demonstrate better model expressivity of a quadratic network than the conventional counterpart with or without quadratic activation. Then, we propose an effective training strategy referred to as ReLinear to stabilize the training process of a quadratic network, thereby unleashing the full potential in its associated machine learning tasks. Comprehensive experiments on popular datasets are performed to support our findings and confirm the performance of quadratic deep learning. We have shared our code in https://github.com/FengleiFan/ReLinear.

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can exhibit the "catapult phase" [Lewkowycz et al. 2020] that arises when training such models with large learning rates. We then empirically show that the behaviour of neural quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models can be an effective tool for analysis of neural networks.

The enduring challenge in the field of artificial intelligence has been the control of systems to achieve desired behaviours. While for systems governed by straightforward dynamics equations, methods like Linear Quadratic Regulation (LQR) have historically proven highly effective, most real-world tasks, which require a general problem-solver, demand world models with dynamics that cannot be easily described by simple equations. Consequently, these models must be learned from data using neural networks. Most model predictive control (MPC) algorithms designed for visual world models have traditionally explored gradient-free population-based optimisation methods, such as Cross Entropy and Model Predictive Path Integral (MPPI) for planning. However, we present an exploration of a gradient-based alternative that fully leverages the differentiability of the world model. In our study, we conduct a comparative analysis between our method and other MPC-based alternatives, as well as policy-based algorithms. In a sample-efficient setting, our method achieves on par or superior performance compared to the alternative approaches in most tasks. Additionally, we introduce a hybrid model that combines policy networks and gradient-based MPC, which outperforms pure policy based methods thereby holding promise for Gradient-based planning with world models in complex real-world tasks.

Collaborative generation, which coordinates multiple diffusion trajectories to extend the capabilities of pretrained priors, has emerged as a powerful paradigm for extending the applicability of diffusion models. Among existing approaches, diffusion synchronization provides a scenario-agnostic solution by introducing general guidance mechanisms. However, current synchronization approaches rely heavily on heuristics and still require task-specific tailoring, which limits their generalizability and performance. In this work, we mathematically derive a synchronization framework based on optimal control, providing a principled explanation of diffusion synchronization. During sampling, we optimize control variables to guide multiple trajectories toward coherent solutions while remaining close to the underlying diffusion prior. Our method operates entirely at test-time without additional training, thereby enabling broad applicability across diverse generation scenarios when combined with strong pretrained priors. We demonstrate consistent improvements over baselines on three representative collaborative generation tasks, covering a wide range of modalities and applications. Beyond performance gains, our work establishes a novel foundation for collaborative generation, opening a principled path toward extending pretrained generative models to new collaborative generation settings.

This paper studies attack detection for discrete-time linear systems with stochastic process noise that produce both a vulnerable (i.e., attackable) linear measurement and a secured (i.e., unattackable) quadratic measurement. The motivating application of this model is a dynamic-game setting where the quadratic measurement is interpreted as a system-level utility or reward, and control inputs into the linear system are interpreted as control policies that, once applied, are known to all game participants and which steer the system towards a game-theoretic equilibrium (e.g., Nash equilibrium). To detect attacks on the linear channel, we develop a novel quadratic-utility-aware observer that leverages the secured quadratic output and enforces measurement consistency via a projection step. We establish three properties for this observer: feasibility of the true state, prox-regularity of the quadratic-constraint set, and a monotone error-reduction guarantee in the noise-free case. To detect adversarial manipulation, we compare linear and quadratic observer trajectories using a wild bootstrap maximum mean discrepancy (MMD) test that provides valid inference under temporal dependence. We validate our framework using numerical experiments of a pursuit-evasion game, where the quadratic observer preserves estimation accuracy under linear-sensor attacks, while the statistical test detects distributional divergence between the observers' trajectories.

Harmonic model predictive control (HMPC) is a recent model predictive control (MPC) formulation for tracking piece-wise constant references that includes a parameterized artificial harmonic reference as a decision variable, resulting in an increased performance and domain of attraction with respect to other MPC formulations. This article presents an extension of the HMPC formulation to track periodic harmonic/sinusoidal references and discusses its use for tracking arbitrary trajectories. The proposed formulation inherits the benefits of its predecessor, namely its good performance and large domain of attraction when using small prediction horizons, and that the complexity of its optimization problem does not depend on the period of the reference. We show closed-loop results discussing its performance and comparing it to other MPC formulations.

Time series play a fundamental role in many domains, capturing a plethora of information about the underlying data-generating processes. When a process generates multiple synchronized signals we are faced with multidimensional time series. In this context a fundamental problem is that of motif mining, where we seek patterns repeating twice with minor variations, spanning some of the dimensions. State of the art exact solutions for this problem run in time quadratic in the length of the input time series. We provide a scalable method to find the top-k motifs in multidimensional time series with probabilistic guarantees on the quality of the results. Our algorithm runs in time subquadratic in the length of the input, and returns the exact solution with probability at least 1-δ, where δ is a user-defined parameter. The algorithm is designed to be adaptive to the input distribution, self-tuning its parameters while respecting user-defined limits on the memory to use. Our theoretical analysis is complemented by an extensive experimental evaluation, showing that our algorithm is orders of magnitude faster than the state of the art.

Integral reinforcement learning (IntRL) demands the precise computation of the utility function's integral at its policy evaluation (PEV) stage. This is achieved through quadrature rules, which are weighted sums of utility functions evaluated from state samples obtained in discrete time. Our research reveals a critical yet underexplored phenomenon: the choice of the computational method -- in this case, the quadrature rule -- can significantly impact control performance. This impact is traced back to the fact that computational errors introduced in the PEV stage can affect the policy iteration's convergence behavior, which in turn affects the learned controller. To elucidate how computation impacts control, we draw a parallel between IntRL's policy iteration and Newton's method applied to the Hamilton-Jacobi-Bellman equation. In this light, computational error in PEV manifests as an extra error term in each iteration of Newton's method, with its upper bound proportional to the computational error. Further, we demonstrate that when the utility function resides in a reproducing kernel Hilbert space (RKHS), the optimal quadrature is achievable by employing Bayesian quadrature with the RKHS-inducing kernel function. We prove that the local convergence rates for IntRL using the trapezoidal rule and Bayesian quadrature with a Mat\'ern kernel to be O(N^{-2}) and O(N^{-b}), where N is the number of evenly-spaced samples and b is the Mat\'ern kernel's smoothness parameter. These theoretical findings are finally validated by two canonical control tasks.

Many modern Language Model (LM) pipelines return an averaged model, such as an exponential moving average of the training iterates, rather than the final iterate itself. This raises a fundamental question: given that we will return an iterate average, how should we change training to improve the performance of this average? We study this question by formulating optimizer design for the iterate-average estimator as an optimal-control problem. In a continuous-time stochastic quadratic model, we solve for the control strategy that minimizes the error of the returned average subject to a penalty on the size of the intervention. A practical approximation to this controller yields PACE, a lightweight wrapper around AdamW that pulls the live weights toward their exponential moving average with a clipped, per-coordinate control strength. We prove that a stylized version of PACE converges at the standard stochastic convex optimization rate, up to a factor depending on the averaging rule, while in the quadratic setting it can strictly improve the limiting squared error of the iterate-average estimator and can do so by an arbitrarily large factor on some instances. Empirically, our results suggest that PACE improves over AdamW and EMA-evaluated AdamW in supervised fine-tuning of 1-2B parameter LMs and in GPT-2 pretraining on FineWeb for a wide range of learning rates, decay schedules, and other hyperparameters.

Human cognition emerges from coordinated spiking dynamics in distributed neural circuits, where information is encoded via both firing rates and precise spike timing determined by brain rhythms. Inspired by this notion, we propose a brain-inspired learning primitive in which cognition-level neural synchrony emerges through iterative bottom-up and top-down interactions between micro-scale dynamics of spiking neurons and a macro-scale mechanism of oscillatory synchronization. Specifically, we model each parcel (e.g., a cortical region or an image pixel) in the target system as a spiking neuron embedded in a predefined connectivity scaffold. Low-level information is encoded in a spatiotemporal domain, where neurons are selectively grouped and fire spontaneously over time through self-organized dynamics. In the bottom-up route, oscillatory synchronization is formed from past spiking activity accumulated over a finite memory window. Since brain dynamics operate in a regime of partial and transient synchronization rather than global phase locking, we model oscillatory coordination using a time-delayed synchronization formulation, which enables a top-down modulation of heterogeneous neural spiking for a large-scale distributed system. Together, we devise a spiking-by-synchronization neural network (S2-Net) that uses rhythmic timing as a control mechanism for efficient information processing. Promising results have been achieved across a broad range of tasks, including neural activity decoding, energy-efficient signal processing, temporal binding and semantic reasoning.

We put forward the dynamical study of a novel higher-order small network of Chialvo neurons arranged in a ring-star topology, with the neurons interacting via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through higher-order multi-body interactions. We first analyze our model using the tools from nonlinear dynamics literature: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of chaotic attractors, and also an intriguing route to chaos starting from a fixed point, to period-doubling, to cyclic quasiperiodic closed invariant curves, to ultimately chaos. We numerically observe the existence of codimension-1 bifurcation patterns: saddle-node, period-doubling, and Neimark Sacker. We also qualitatively study the typical phase portraits of the system and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure respectively. Finally, we study the collective behavior of the neurons in terms of two synchronization measures: the cross-correlation coefficient, and the Kuramoto order parameter.

We propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark-Sacker, double Neimark-Sacker, flip- and fold-Neimark Sacker, and 1:1 and 1:2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.

Coordinated path following in multi-agent systems is a key challenge in robotics, with applications in automated logistics, surveillance, and collaborative exploration. Traditional formation control techniques often rely on time-parameterized trajectories and path integrals, which can result in synchronization issues and rigid behavior. In this work, we address the problem of sequential path following, where agents maintain fixed spatial separation along a common trajectory, guided by a leader under centralized control. We introduce Robot Conga, a leader-follower control strategy that updates each agent's desired state based on the leader's spatial displacement rather than time, assuming access to a global position reference, an assumption valid in indoor environments equipped with motion capture, vision-based tracking, or UWB localization systems. The algorithm was validated in simulation using both TurtleBot3 and quadruped (Laikago) robots. Results demonstrate accurate trajectory tracking, stable inter-agent spacing, and fast convergence, with all agents aligning within 250 time steps (approx. 0.25 seconds) in the quadruped case, and almost instantaneously in the TurtleBot3 implementation.

Training deep neural networks (DNNs) using traditional backpropagation (BP) presents challenges in terms of computational complexity and energy consumption, particularly for on-device learning where computational resources are limited. Various alternatives to BP, including random feedback alignment, forward-forward, and local classifiers, have been explored to address these challenges. These methods have their advantages, but they can encounter difficulties when dealing with intricate visual tasks or demand considerable computational resources. In this paper, we propose a novel Local Learning rule inspired by neural activity Synchronization phenomena (LLS) observed in the brain. LLS utilizes fixed periodic basis vectors to synchronize neuron activity within each layer, enabling efficient training without the need for additional trainable parameters. We demonstrate the effectiveness of LLS and its variations, LLS-M and LLS-MxM, on multiple image classification datasets, achieving accuracy comparable to BP with reduced computational complexity and minimal additional parameters. Furthermore, the performance of LLS on the Visual Wake Word (VWW) dataset highlights its suitability for on-device learning tasks, making it a promising candidate for edge hardware implementations.

A network of oscillators that synchronizes perfectly computes nothing further, so an attention architecture built from synchronization must locate its computation in structured departures from agreement. We introduce the Frustrated Synchronization Network (FSN), whose token states are phases on a torus and whose entire value pathway is one learned complex coupling kernel over harmonics and a one-step delay. Each component of the kernel is a frustration in the sense of the synchronization literature. The complex phases are static Kuramoto-Sakaguchi frustration angles, the signed harmonics are repulsive Daido components, and the delay term, which couples each token to the successors of the tokens it attends to, is algebraically identical to Kuramoto-Sakaguchi coupling whose frustration angle is the data's own transition, so next-token prediction is implemented as synchronization frustrated by the data. At matched one-million-parameter and training budgets on character-level text and code, the FSN's validation loss is below a tuned RoPE-SwiGLU transformer's at every epoch measured, and the comparison survives training the baseline to convergence: every thirty-epoch enwik8 seed finishes below the transformer's converged fifty-epoch loss of 1.611, and the FSN's completed fifty-epoch runs converge to 1.5953 +/- 0.0014. A variant with every feed-forward block replaced by mean-field coupling to learned collective modes, leaving no multilayer perceptron in the stack, tracks the transformer. On natural text the unfrustrated base layer falls behind the converged transformer at every copy depth, worst on long-range copy events; the kernel reverses the deficit at every depth of four and beyond. Headline comparisons are at the one-million-parameter scale; a scale ladder is complete through four million parameters with the advantage persisting, and remaining arms are marked as in progress.

A common pipeline in learning-based control is to iteratively estimate a model of system dynamics, and apply a trajectory optimization algorithm - e.g.~iLQR - on the learned model to minimize a target cost. This paper conducts a rigorous analysis of a simplified variant of this strategy for general nonlinear systems. We analyze an algorithm which iterates between estimating local linear models of nonlinear system dynamics and performing iLQR-like policy updates. We demonstrate that this algorithm attains sample complexity polynomial in relevant problem parameters, and, by synthesizing locally stabilizing gains, overcomes exponential dependence in problem horizon. Experimental results validate the performance of our algorithm, and compare to natural deep-learning baselines.

We first raise and tackle a ``time synchronization'' issue between the agent and the environment in non-stationary reinforcement learning (RL), a crucial factor hindering its real-world applications. In reality, environmental changes occur over wall-clock time (t) rather than episode progress (k), where wall-clock time signifies the actual elapsed time within the fixed duration t in [0, T]. In existing works, at episode k, the agent rolls a trajectory and trains a policy before transitioning to episode k+1. In the context of the time-desynchronized environment, however, the agent at time t_{k} allocates Δt for trajectory generation and training, subsequently moves to the next episode at t_{k+1}=t_{k}+Δt. Despite a fixed total number of episodes (K), the agent accumulates different trajectories influenced by the choice of interaction times (t_1,t_2,...,t_K), significantly impacting the suboptimality gap of the policy. We propose a Proactively Synchronizing Tempo (ProST) framework that computes a suboptimal sequence {t_1,t_2,...,t_K} (= { t_{1:K}}) by minimizing an upper bound on its performance measure, i.e., the dynamic regret. Our main contribution is that we show that a suboptimal {t_{1:K}} trades-off between the policy training time (agent tempo) and how fast the environment changes (environment tempo). Theoretically, this work develops a suboptimal {t_{1:K}} as a function of the degree of the environment's non-stationarity while also achieving a sublinear dynamic regret. Our experimental evaluation on various high-dimensional non-stationary environments shows that the ProST framework achieves a higher online return at suboptimal {t_{1:K}} than the existing methods.

We provide an algorithm for adaptive legged locomotion via online learning and model predictive control. The algorithm is composed of two interacting modules: model predictive control (MPC) and online learning of residual dynamics. The residual dynamics can represent modeling errors and external disturbances. We are motivated by the future of autonomy where quadrupeds will autonomously perform complex tasks despite real-world unknown uncertainty, such as unknown payload and uneven terrains. The algorithm uses random Fourier features to approximate the residual dynamics in reproducing kernel Hilbert spaces. Then, it employs MPC based on the current learned model of the residual dynamics. The model is updated online in a self-supervised manner using least squares based on the data collected while controlling the quadruped. The algorithm enjoys sublinear dynamic regret, defined as the suboptimality against an optimal clairvoyant controller that knows how the residual dynamics. We validate our algorithm in Gazebo and MuJoCo simulations, where the quadruped aims to track reference trajectories. The Gazebo simulations include constant unknown external forces up to 12g, where g is the gravity vector, in flat terrain, slope terrain with 20degree inclination, and rough terrain with 0.25m height variation. The MuJoCo simulations include time-varying unknown disturbances with payload up to 8~kg and time-varying ground friction coefficients in flat terrain.

Regulatory institutions (from content moderation platforms to financial supervisors) observe, deliberate, and intervene only after a characteristic delay. We ask whether this processing lag alone can destabilize a multi-agent system that would otherwise remain stable, without exogenous shocks, coordination among agents, or malicious actors. We study this question in two stages. First, we analyze a delayed replicator equation in which autonomous agents receive a benefit from radical behavior but face punishment based on a lagged institutional alarm signal. We derive a closed-form critical delay threshold beyond which the unique interior equilibrium loses stability through a Hopf bifurcation, and prove via center manifold reduction that the bifurcation is supercritical (producing bounded oscillations, not explosive growth) for the entire sigmoid response-function family. Second, we embed N=240 agents on a network and equip them with reinforcement learning (tabular Q-learning), comparing three decision architectures in a factorial design: non-reactive agents (fixed policy), reactive agents (threshold heuristic without memory), and Q-learning agents (adaptive with cumulative value estimates). The results reveal a hierarchy opposite to the naive expectation that learning amplifies instability: non-reactive agents are immune to delay (0% runaway across all tested values), reactive agents collapse catastrophically (96% runaway by delay geq 8 steps), and Q-learning agents achieve partial resilience (66% runaway at delay = 20). The destabilizing ingredient is reactivity to delayed signals: agents that immediately exploit low-alarm windows trigger oscillatory feedback loops. Learning buffers this through implicit punishment memory encoded in Q-values

Learning and decision making in the brain are key processes critical to survival, and yet are processes implemented by non-ideal biological building blocks which can impose significant error. We explore quantitatively how the brain might cope with this inherent source of error by taking advantage of two ubiquitous mechanisms, redundancy and synchronization. In particular we consider a neural process whose goal is to learn a decision function by implementing a nonlinear gradient dynamics. The dynamics, however, are assumed to be corrupted by perturbations modeling the error which might be incurred due to limitations of the biology, intrinsic neuronal noise, and imperfect measurements. We show that error, and the associated uncertainty surrounding a learned solution, can be controlled in large part by trading off synchronization strength among multiple redundant neural systems against the noise amplitude. The impact of the coupling between such redundant systems is quantified by the spectrum of the network Laplacian, and we discuss the role of network topology in synchronization and in reducing the effect of noise. A range of situations in which the mechanisms we model arise in brain science are discussed, and we draw attention to experimental evidence suggesting that cortical circuits capable of implementing the computations of interest here can be found on several scales. Finally, simulations comparing theoretical bounds to the relevant empirical quantities show that the theoretical estimates we derive can be tight.

Instruction tuning aligns large language models, including multimodal ones, with diverse user intents, but scaling to heterogeneous mixtures is hindered by gradient interference and bandwidth-heavy synchronization. We ask whether these two bottlenecks can be addressed jointly by training parts of the mixture independently and reconciling them once in parameter space. We develop a local quadratic theory inside a shared flat basin that yields three results: weight merging produces a curvature-weighted variance reduction; PCA-aligned conflict splitting maximizes this gain along high-curvature directions; and merging additionally acts as spectral filtering with implicit norm regularization. These results directly motivate MERIT, a decentralized merge-ready instruction-tuning pipeline that estimates dataset-level gradient conflicts, partitions the mixture along the top PCA conflict axes, fine-tunes each partition independently with no inter-partition communication, and merges once via token-weighted averaging. On Qwen2.5-VL-3B with 136 Vision-FLAN tasks, MERIT improves the 8-benchmark average from 54.3 (joint training) to 57.0. The same recipe scales to a 7B model on a 1.6M-example, 176-source mixture -- matching or exceeding centralized joint training with minimal cost overhead -- and transfers to text-only FLAN. Our code is available at https://github.com/naver-ai/merit.

This work presents a novel online model-predictive trajectory planner for robotic manipulators called BoundMPC. This planner allows the collision-free following of Cartesian reference paths in the end-effector's position and orientation, including via-points, within desired asymmetric bounds of the orthogonal path error. The path parameter synchronizes the position and orientation reference paths. The decomposition of the path error into the tangential direction, describing the path progress, and the orthogonal direction, which represents the deviation from the path, is well known for the position from the path-following control in the literature. This paper extends this idea to the orientation by utilizing the Lie theory of rotations. Moreover, the orthogonal error plane is further decomposed into basis directions to define asymmetric Cartesian error bounds easily. Using piecewise linear position and orientation reference paths with via-points is computationally very efficient and allows replanning the pose trajectories during the robot's motion. This feature makes it possible to use this planner for dynamically changing environments and varying goals. The flexibility and performance of BoundMPC are experimentally demonstrated by two scenarios on a 7-DoF Kuka LBR iiwa 14 R820 robot. The first scenario shows the transfer of a larger object from a start to a goal pose through a confined space where the object must be tilted. The second scenario deals with grasping an object from a table where the grasping point changes during the robot's motion, and collisions with other obstacles in the scene must be avoided.

Reinforcement learning (RL) has shown promise in creating robust policies for robotics tasks. However, contemporary RL algorithms are data-hungry, often requiring billions of environment transitions to train successful policies. This necessitates the use of fast and highly-parallelizable simulators. In addition to speed, such simulators need to model the physics of the robots and their interaction with the environment to a level acceptable for transferring policies learned in simulation to reality. We present QuadSwarm, a fast, reliable simulator for research in single and multi-robot RL for quadrotors that addresses both issues. QuadSwarm, with fast forward-dynamics propagation decoupled from rendering, is designed to be highly parallelizable such that throughput scales linearly with additional compute. It provides multiple components tailored toward multi-robot RL, including diverse training scenarios, and provides domain randomization to facilitate the development and sim2real transfer of multi-quadrotor control policies. Initial experiments suggest that QuadSwarm achieves over 48,500 simulation samples per second (SPS) on a single quadrotor and over 62,000 SPS on eight quadrotors on a 16-core CPU. The code can be found in https://github.com/Zhehui-Huang/quad-swarm-rl.

Spatiotemporal neural dynamics and oscillatory synchronization are widely implicated in biological information processing and have been hypothesized to support flexible coordination such as feature binding. By contrast, most deep learning architectures represent and propagate information through activation values, neglecting the joint dynamics of rate and phase. In this work, we introduce Kuramoto oscillatory Phase Encoding (KoPE) as an additional, evolving phase state to Vision Transformers, incorporating a neuro-inspired synchronization mechanism to advance learning efficiency. We show that KoPE can improve training, parameter, and data efficiency of vision models through synchronization-enhanced structure learning. Moreover, KoPE benefits tasks requiring structured understanding, including semantic and panoptic segmentation, representation alignment with language, and few-shot abstract visual reasoning (ARC-AGI). Theoretical analysis and empirical verification further suggest that KoPE can accelerate attention concentration for learning efficiency. These results indicate that synchronization can serve as a scalable, neuro-inspired mechanism for advancing state-of-the-art neural network models.

Proper weight initialization prior to training has historically been one of the key factors that helped kick off the deep learning revolution. Initialization is even more crucial in "reservoir computing", where the weights of a readout layer are learned linearly while the reservoir weights are fixed and largely determine the richness, stability and memory of the resulting dynamics. In the infinite-width limit it has been shown that meaningful initializations are those sitting at an effective critical point of the randomly initialized model. The phase transition is controlled by the weight variance g^2 and separates an ordered phase from a chaotic one where information progressively degrades. Here we derive a simple criterion to estimate the critical g_c for a broad class of recurrent architectures and we show that it closely tracks the gain at which a gated-RNN reservoir achieves peak performance on a chaotic forecasting task. Finally, we argue that our criterion can serve as a design principle for future initialization schemes.

In this work, we consider the optimization process of minibatch stochastic gradient descent (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the d-dimensional Boolean hypercube labeled by the quadratic ``XOR'' function y = -x_ix_j, it is possible to train to a population error o(1) with d :polylog(d) samples. Our result considers simultaneously training both layers of the two-layer-neural network with ReLU activations via standard minibatch SGD on the logistic loss. To our knowledge, this work is the first to give a sample complexity of O(d) for efficiently learning the XOR function on isotropic data on a standard neural network with standard training. Our main technique is showing that the network evolves in two phases: a signal-finding phase where the network is small and many of the neurons evolve independently to find features, and a signal-heavy phase, where SGD maintains and balances the features. We leverage the simultaneous training of the layers to show that it is sufficient for only a small fraction of the neurons to learn features, since those neurons will be amplified by the simultaneous growth of their second layer weights.

Hybrid systems are prevalent in robotics. However, ensuring the stability of hybrid systems is challenging due to sophisticated continuous and discrete dynamics. A system with all its system modes stable can still be unstable. Hence special treatments are required at mode switchings to stabilize the system. In this work, we propose a hierarchical, neural network (NN)-based method to control general hybrid systems. For each system mode, we first learn an NN Lyapunov function and an NN controller to ensure the states within the region of attraction (RoA) can be stabilized. Then an RoA NN estimator is learned across different modes. Upon mode switching, we propose a differentiable planner to ensure the states after switching can land in next mode's RoA, hence stabilizing the hybrid system. We provide novel theoretical stability guarantees and conduct experiments in car tracking control, pogobot navigation, and bipedal walker locomotion. Our method only requires 0.25X of the training time as needed by other learning-based methods. With low running time (10-50X faster than model predictive control (MPC)), our controller achieves a higher stability/success rate over other baselines such as MPC, reinforcement learning (RL), common Lyapunov methods (CLF), linear quadratic regulator (LQR), quadratic programming (QP) and Hamilton-Jacobian-based methods (HJB). The project page is on https://mit-realm.github.io/hybrid-clf.

We study the hardmax limit of self-attention dynamics for token embeddings obtained in the zero-temperature (βto+infty) regime, and relate it to the finite-β setting. In this limit, the update rule can be viewed as a Frank-Wolfe step for a quadratic objective over the convex hull of the current token embeddings. When the key-query matrix is negative semidefinite, the method linearly contracts all tokens to a single cluster at the origin. When it is positive semidefinite, extending the hardmax rule to the entire convex hull induces a Voronoi diagram: vertices are stationary, interior points remain in their initial cells, and each token moves along a straight line toward its cell's vertex, yielding (super-)exponential convergence. As a byproduct, we also establish well-posedness of the associated ODE limit in this regime. Returning to the finite-β regime, we model self-attention dynamics as a Markov chain and prove dynamic metastability: with high probability, interior tokens reach near-vertex configurations in a constant number of steps and remain within a small neighborhood for times that grow exponentially in the inverse temperature β, before ultimately collapsing to the origin. Thus, the hardmax dynamics accurately approximate the finite-β process over exponentially long time horizons.

Machine learning models are increasingly being scaled in both sequence length and model dimension to reach longer contexts and better performance. However, existing architectures such as Transformers scale quadratically along both these axes. We ask: are there performant architectures that can scale sub-quadratically along sequence length and model dimension? We introduce Monarch Mixer (M2), a new architecture that uses the same sub-quadratic primitive along both sequence length and model dimension: Monarch matrices, a simple class of expressive structured matrices that captures many linear transforms, achieves high hardware efficiency on GPUs, and scales sub-quadratically. As a proof of concept, we explore the performance of M2 in three domains: non-causal BERT-style language modeling, ViT-style image classification, and causal GPT-style language modeling. For non-causal BERT-style modeling, M2 matches BERT-base and BERT-large in downstream GLUE quality with up to 27% fewer parameters, and achieves up to 9.1times higher throughput at sequence length 4K. On ImageNet, M2 outperforms ViT-b by 1% in accuracy, with only half the parameters. Causal GPT-style models introduce a technical challenge: enforcing causality via masking introduces a quadratic bottleneck. To alleviate this bottleneck, we develop a novel theoretical view of Monarch matrices based on multivariate polynomial evaluation and interpolation, which lets us parameterize M2 to be causal while remaining sub-quadratic. Using this parameterization, M2 matches GPT-style Transformers at 360M parameters in pretraining perplexity on The PILE--showing for the first time that it may be possible to match Transformer quality without attention or MLPs.

Classical convergence guarantees for gradient-based learning in games require the pseudo-gradient to be (strongly) monotone in Euclidean geometry as shown by rosen(1965), a condition that often fails even in simple games with strong cross-player couplings. We introduce Small-Gain Nash (SGN), a block small-gain condition in a custom block-weighted geometry. SGN converts local curvature and cross-player Lipschitz coupling bounds into a tractable certificate of contraction. It constructs a weighted block metric in which the pseudo-gradient becomes strongly monotone on any region where these bounds hold, even when it is non-monotone in the Euclidean sense. The continuous flow is exponentially contracting in this designed geometry, and projected Euler and RK4 discretizations converge under explicit step-size bounds derived from the SGN margin and a local Lipschitz constant. Our analysis reveals a certified ``timescale band'', a non-asymptotic, metric-based certificate that plays a TTUR-like role: rather than forcing asymptotic timescale separation via vanishing, unequal step sizes, SGN identifies a finite band of relative metric weights for which a single-step-size dynamics is provably contractive. We validate the framework on quadratic games where Euclidean monotonicity analysis fails to predict convergence, but SGN successfully certifies it, and extend the construction to mirror/Fisher geometries for entropy-regularized policy gradient in Markov games. The result is an offline certification pipeline that estimates curvature, coupling, and Lipschitz parameters on compact regions, optimizes block weights to enlarge the SGN margin, and returns a structural, computable convergence certificate consisting of a metric, contraction rate, and safe step-sizes for non-monotone games.

As is known, hybrid quadratic and subquadratic attention models in multi-head architectures have surpassed both Transformer and Linear RNN models , with these works primarily focusing on reducing KV complexity and improving efficiency. For further research on expressiveness, we introduce our series of models distilled from Qwen 2.5, based on pure native RWKV-7 attention, which aims to make RNN more expressive and demonstrates state tracking ability beyond transformers. We work with QRWK 32B based on RWKV-6 architecture, another approach that reduces the entire knowledge processing time to just 8 hours using 16 AMD MI300X GPUs while maintaining Qwen 2.5's performance. In fact, the distillation process can utilize any LLM, not just Qwen, and enables knowledge transfer from larger LLMs to smaller ones with more fewer tokens. We will explain the detailed process and share our insights on building more powerful foundation models. Please note that this is an ongoing work that will be updated continuously. The model checkpoints and source code are available at https://github.com/yynil/RWKVInside{https://github.com/yynil/RWKVInside}, https://huggingface.co/RWKV-Red-Team/ARWKV-7B-Preview-0.1{https://huggingface.co/RWKV-Red-Team/ARWKV-7B-Preview-0.1}.

This project focuses on a method to extract a frequency comb in mechanical means, for general interest and numerous practical applications in MEMS. The method of execution is the implementation of a beam that is exhibiting non-linear dynamics that is perturbed and analyzed for its transverse vibrations. The perturbation is an external harmonic driver with a chosen small amplitude and frequency (which is slightly detuned from the beam eigenfrequency), that when engaged with the unperturbed beam oscillations, causes it reach a state of "injection pulling" - an effect that occurs when one harmonic oscillator is coupled with a second one and causes it to oscillate in a frequency near its own. This causes the beam to reach SNIC bifurcation, rendering a frequency comb as desired. Theoretical analysis showed that the problem can be modelled using a non-linear equation of the beam, that translates to a form of the non-linear Duffing equation. While a solution to the dynamics function of the beam is hard to obtain in practice due to mathematical difficulties, a slow evolution model is suggested that is composed of functions of a amplitude and phase. Using several additional mathematical assumptions, the amplitude is seen to be related to the phase, while the phase equation solution is seen to be of the form of Adler's equation. These assumptions ultimately reduce the entire behaviour of the beam to a relatively simple solution to the Adler equation, which has a known analytical solution. Computerized numerical simulations are run on it to check the results and compare them to the theory and desired outcome. The results agreed with the theory and produce the expected frequency comb, showing the assumptions to be valid in extracting the comb.

Models of neural responses to stimuli with complex spatiotemporal correlation structure often assume that neurons are only selective for a small number of linear projections of a potentially high-dimensional input. Here we explore recent modeling approaches where the neural response depends on the quadratic form of the input rather than on its linear projection, that is, the neuron is sensitive to the local covariance structure of the signal preceding the spike. To infer this quadratic dependence in the presence of arbitrary (e.g. naturalistic) stimulus distribution, we review several inference methods, focussing in particular on two information-theory-based approaches (maximization of stimulus energy or of noise entropy) and a likelihood-based approach (Bayesian spike-triggered covariance, extensions of generalized linear models). We analyze the formal connection between the likelihood-based and information-based approaches to show how they lead to consistent inference. We demonstrate the practical feasibility of these procedures by using model neurons responding to a flickering variance stimulus.

Recent works in robotic manipulation through reinforcement learning (RL) or imitation learning (IL) have shown potential for tackling a range of tasks e.g., opening a drawer or a cupboard. However, these techniques generalize poorly to unseen objects. We conjecture that this is due to the high-dimensional action space for joint control. In this paper, we take an alternative approach and separate the task of learning 'what to do' from 'how to do it' i.e., whole-body control. We pose the RL problem as one of determining the skill dynamics for a disembodied virtual manipulator interacting with articulated objects. The whole-body robotic kinematic control is optimized to execute the high-dimensional joint motion to reach the goals in the workspace. It does so by solving a quadratic programming (QP) model with robotic singularity and kinematic constraints. Our experiments on manipulating complex articulated objects show that the proposed approach is more generalizable to unseen objects with large intra-class variations, outperforming previous approaches. The evaluation results indicate that our approach generates more compliant robotic motion and outperforms the pure RL and IL baselines in task success rates. Additional information and videos are available at https://kl-research.github.io/decoupskill

Many existing reinforcement learning (RL) methods employ stochastic gradient iteration on the back end, whose stability hinges upon a hypothesis that the data-generating process mixes exponentially fast with a rate parameter that appears in the step-size selection. Unfortunately, this assumption is violated for large state spaces or settings with sparse rewards, and the mixing time is unknown, making the step size inoperable. In this work, we propose an RL methodology attuned to the mixing time by employing a multi-level Monte Carlo estimator for the critic, the actor, and the average reward embedded within an actor-critic (AC) algorithm. This method, which we call Multi-level Actor-Critic (MAC), is developed especially for infinite-horizon average-reward settings and neither relies on oracle knowledge of the mixing time in its parameter selection nor assumes its exponential decay; it, therefore, is readily applicable to applications with slower mixing times. Nonetheless, it achieves a convergence rate comparable to the state-of-the-art AC algorithms. We experimentally show that these alleviated restrictions on the technical conditions required for stability translate to superior performance in practice for RL problems with sparse rewards.

We consider offline reinforcement learning (RL) methods in possibly nonstationary environments. Many existing RL algorithms in the literature rely on the stationarity assumption that requires the system transition and the reward function to be constant over time. However, the stationarity assumption is restrictive in practice and is likely to be violated in a number of applications, including traffic signal control, robotics and mobile health. In this paper, we develop a consistent procedure to test the nonstationarity of the optimal Q-function based on pre-collected historical data, without additional online data collection. Based on the proposed test, we further develop a sequential change point detection method that can be naturally coupled with existing state-of-the-art RL methods for policy optimization in nonstationary environments. The usefulness of our method is illustrated by theoretical results, simulation studies, and a real data example from the 2018 Intern Health Study. A Python implementation of the proposed procedure is available at https://github.com/limengbinggz/CUSUM-RL.

The two-body problem of variational electrodynamics possesses differential-delay equations of motion with state-dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets. From a periodic orbit possessing some mild properties at breaking points, we define a synchronization function in R x R3, which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points. The oscillatory functions are associated with a PDE identity in H2(R3), and we postulate ordering conditions for the PDE identity to define a Fredholm-Schroedinger operator with an O(1/r2) spin-orbit forcing term belonging to L2(R3). As an application, we introduce the Chemical Principle criterion to select orbits with asymptotically vanishing far-fields and estimate the Bohr radius parameter of the Fredholm-Schroedinger PDE using the boundary-layer of orbits chosen according to the Chemical Principle criterion. Last, working backward, we derive a Chemical Principle-like condition from the ordering conditions.

Multi-agent large language model (LLM) systems often rely on verifier and critic agents to suppress hallucinations, but verification is delayed. During this delay, false claims can propagate through the agent network. We model this process as delayed consensus on a graph with grounded corrector nodes. Spectral decomposition by the grounded Laplacian yields a closed-form stability threshold for the verification dose: correction that is too strong or too delayed can turn consensus into oscillation. The most unstable regime occurs when the communication and verification delays coincide; for delay two, the threshold is the inverse golden ratio. The same framework gives a supermodular placement objective and a greedy (1-1/e)-approximation rule for assigning a limited corrector budget to influential nodes. Experiments across five open models confirm the predicted dose-delay oscillations. By contrast, grounded factual answering makes truth an absorbing boundary and eliminates the effect, suggesting that the instability is specific to signed-belief tasks while grounded verification remains stabilizing

The problem of robotic synchronisation and coordination is a long-standing one. Combining autonomous, computerised systems with unpredictable real-world conditions can have consequences ranging from poor performance to collisions and damage. This paper proposes using coupled oscillators to create a drone swarm that is decentralised and self organising. This allows for greater flexibility and adaptiveness than a hard-coded swarm, with more resilience and scalability than a centralised system. Our method allows for a variable number of drones to spontaneously form a swarm and react to changing swarm conditions. Additionally, this method includes provisions to prevent communication interference between drones, and signal processing techniques to ensure a smooth and cohesive swarm.

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide mathcal{O}(kappa) accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

Fast Weight Programmers (FWPs) encode temporal dependencies through dynamically updated parameters rather than recurrent hidden states. Quantum FWPs (QFWPs) extend this idea with variational quantum circuits (VQCs), but existing implementations rely on multi-qubit architectures that are difficult to scale on noisy intermediate-scale quantum (NISQ) devices and expensive to simulate classically. We propose gated QKAN-FWP, a fast-weight framework that integrates FWP with Quantum-inspired Kolmogorov-Arnold Network (QKAN) using single-qubit data re-uploading circuits as learnable nonlinear activation, known as DatA Re-Uploading ActivatioN (DARUAN). We further introduce a scalar-gated fast-weight update rule that stabilizes parameter evolution, supported by a theoretical analysis of its adaptive memory kernel, geometric boundedness, and parallelizable gradient paths. We evaluate the framework across time-series benchmarks, MiniGrid reinforcement learning, and highlight real-world solar cycle forecasting as our main practical result. In the long-horizon setting with 528-month input window and 132-month forecast horizon, our 12.5k-parameter model achieves lower scaled Mean Square Error (MSE), peak amplitude error, and peak timing error than a suite of classical recurrent baselines with up to 13x more parameters, including Long Short-Term Memory (LSTM) networks (25.9k-89.1k parameters), WaveNet-LSTM (167k), Vanilla recurrent neural network (11.5k), and a Modified Echo State Network (132k). To validate NISQ compatibility, we further deploy the trained fast programmer on IonQ and IBM Quantum processors, recovering forecasting accuracy within 0.1% relative MSE of the noiseless simulator at 1024 shots. These results position gated QKAN-FWP as a scalable, parameter-efficient, and NISQ-compatible approach to quantum-inspired sequence modeling.

Exact firing rate models, also known as next-generation neural mass models (NG-NMMs), provide a rigorous description of the dynamics of neural populations. While in its simplest form a single population only displays fixed-point activity, multi-population models may display a range of different behaviors. In this work, we study the dynamics of all-excitatory or all-inhibitory NG-NMMs coupled through sparse random networks with row-normalized network topology. Linear stability analysis of the homogeneous states of the system, representing asynchronous neural activity, provides a dispersion relation linking the emergence of spatiotemporal dynamics to the spectra of the connectivity matrix. Using bounds from random matrix theory, we identify the parameter regions where instabilities occur. In undirected networks, only inhibitory systems produce heterogeneous stationary patterns, corresponding to a winner-takes-all mechanism. In directed networks, exotic rhythmic states with high frequencies emerge in both, excitatory and inhibitory systems. Numerical simulations reveal that these hectic oscillatory states correspond to high-dimensional chaos with extensive properties.

Asynchronous reinforcement learning has become increasingly central to scaling LLM post-training, delivering major throughput gains by decoupling rollout generation from policy updates. However, widely used policy-gradient objectives such as REINFORCE and GRPO suffer under high asynchrony: stale rollouts produce heavy-tailed importance weights, so a small number of trajectories dominate updates and the policy-gradient estimator becomes markedly higher variance. Through systematic analysis on math, reasoning, and tool-use benchmarks, we find that this increasing variance is reliably predicted by collapsing effective sample size (ESS), which prior stabilization methods largely fail to address. Motivated by this diagnosis, we introduce Variance Controlled Policy Optimization (VCPO), a method that (i) dynamically scales the learning rate with ESS to dampen unreliable updates and (ii) applies a closed-form minimum-variance baseline for off-policy settings, without a critic model and adding minimal overhead. Empirically, across math and general reasoning benchmarks, this enables robustly stable asynchronous training compared to previous stabilization and algorithmic methods, even in highly off-policy regimes (128 steps off-policy). In a long-horizon, tool-use task, VCPO matches synchronous performance while delivering a 2.5times speedup in training time. Code is available at: https://github.com/mit-han-lab/vcpo

We verify that persistent observers in causally invariant hypergraph substrates satisfy the conditions of the Conant-Ashby Good Regulator Theorem. Building on Wolfram's hypergraph physics and Vanchurin's neural network cosmology, we formalize persistent observers as entities that minimize prediction error at their boundary with the environment. Applying a modern reformulation of the Conant-Ashby theorem, we demonstrate that hypergraph observers satisfy Good Regulator conditions, requiring them to maintain internal models. Once an internal model with loss function exists, the emergence of a Fisher information metric follows from standard information geometry. Invoking Amari's uniqueness theorem for reparameterization-invariant gradients, we show that natural gradient descent is the unique admissible learning rule. Under the ansatz M=F^2 for exponential family observers and one specific convergence time functional, we derive a closed-form formula for the regime parameter alpha in Vanchurin's Type II framework, with a quantum-classical threshold at kappa(F)=2. However, three alternative convergence models do not reproduce this result, so this prediction is strongly model-dependent. We further introduce the directional regime parameter alpha_{v_k} and the trace-free deviation tensor, showing that a single observer can simultaneously occupy different Vanchurin regimes along different eigendirections of the Fisher metric. This connects Wolfram and Vanchurin frameworks through established theorems, providing approximately 25-30% novel contribution.

To stabilize the periodic operation of a chemical reactor the oscillation period should be determined precisely in real time. The method discussed in the paper is based on adaptive sampling of the state variable with the use of chaotic mapping to itself. It enables precise determination of the oscillation period in real time and could be used for a proper control system, that can successfully control the process and maintain the oscillation period at a set level. The method was applied to a tank reactor and tubular reactor with recycle.

In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit. Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets.

In this paper, we introduce a new class of parameterized controllers, drawing inspiration from Model Predictive Control (MPC). The controller resembles a Quadratic Programming (QP) solver of a linear MPC problem, with the parameters of the controller being trained via Deep Reinforcement Learning (DRL) rather than derived from system models. This approach addresses the limitations of common controllers with Multi-Layer Perceptron (MLP) or other general neural network architecture used in DRL, in terms of verifiability and performance guarantees, and the learned controllers possess verifiable properties like persistent feasibility and asymptotic stability akin to MPC. On the other hand, numerical examples illustrate that the proposed controller empirically matches MPC and MLP controllers in terms of control performance and has superior robustness against modeling uncertainty and noises. Furthermore, the proposed controller is significantly more computationally efficient compared to MPC and requires fewer parameters to learn than MLP controllers. Real-world experiments on vehicle drift maneuvering task demonstrate the potential of these controllers for robotics and other demanding control tasks.

A modern challenge of Artificial Intelligence is learning multiple patterns at once (i.e.parallel learning). While this can not be accomplished by standard Hebbian associative neural networks, in this paper we show how the Multitasking Hebbian Network (a variation on theme of the Hopfield model working on sparse data-sets) is naturally able to perform this complex task. We focus on systems processing in parallel a finite (up to logarithmic growth in the size of the network) amount of patterns, mirroring the low-storage level of standard associative neural networks at work with pattern recognition. For mild dilution in the patterns, the network handles them hierarchically, distributing the amplitudes of their signals as power-laws w.r.t. their information content (hierarchical regime), while, for strong dilution, all the signals pertaining to all the patterns are raised with the same strength (parallel regime). Further, confined to the low-storage setting (i.e., far from the spin glass limit), the presence of a teacher neither alters the multitasking performances nor changes the thresholds for learning: the latter are the same whatever the training protocol is supervised or unsupervised. Results obtained through statistical mechanics, signal-to-noise technique and Monte Carlo simulations are overall in perfect agreement and carry interesting insights on multiple learning at once: for instance, whenever the cost-function of the model is minimized in parallel on several patterns (in its description via Statistical Mechanics), the same happens to the standard sum-squared error Loss function (typically used in Machine Learning).

Brain-like intelligent systems need brain-like learning methods. Equilibrium Propagation (EP) is a biologically plausible learning framework with strong potential for brain-inspired computing hardware. However, existing im-plementations of EP suffer from instability and prohibi-tively high computational costs. Inspired by the structure and dynamics of the brain, we propose a biologically plau-sible Feedback-regulated REsidual recurrent neural network (FRE-RNN) and study its learning performance in EP framework. Feedback regulation enables rapid convergence by reducing the spectral radius. The improvement in con-vergence property reduces the computational cost and train-ing time of EP by orders of magnitude, delivering perfor-mance on par with backpropagation (BP) in benchmark tasks. Meanwhile, residual connections with brain-inspired topologies help alleviate the vanishing gradient problem that arises when feedback pathways are weak in deep RNNs. Our approach substantially enhances the applicabil-ity and practicality of EP in large-scale networks that un-derpin artificial intelligence. The techniques developed here also offer guidance to implementing in-situ learning in physical neural networks.

Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers (SCS) showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos, establishing a paradigmatic link between random connectivity, dynamical instability, and internally generated fluctuations in neural circuits. Here, we extend this framework to two-population firing-rate networks with segregated excitatory and inhibitory neurons and target-specific inhibitory couplings that break excitation--inhibition balance. Using dynamical mean-field theory, we derive self-consistent equations for the macroscopic mean activities and autocorrelations, together with stability criteria distinguishing mean-driven and fluctuation-driven instabilities. We show that target-specific inhibition organizes the phase diagram into three qualitative classes: inhibition-dominated or strictly balanced networks display only quiescent activity and asynchronous chaos; excitation-dominated networks display persistent activity together with either synchronous chaos with non-vanishing mean activity or coherent oscillations, depending on the stability-matrix eigenvalues. Crucially, coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory; rather, their onset suppresses the chaotic component, reminiscent of input-induced suppression of chaos. These results generalize SCS theory to recurrent networks with explicit excitatory--inhibitory structure and identify target-specific inhibition as a key control parameter for large-scale neural dynamics.

We develop a predictor-feedback cooperative adaptive cruise control (CACC) design relying on a multiple-predecessor-following (MPF) topology-based nominal delay-free CACC law. We consider vehicular platoons with heterogeneous vehicles, whose dynamics are described by a third-order linear system subject to actuation delay, along with vehicle-to-vehicle (V2V) communication delay. The design achieves individual vehicle stability, string stability, and zero, steady-state speed/spacing tracking errors, for any value of the actuation delay. The proofs of individual vehicle stability, string stability, and regulation rely on employment of an input-output approach on the frequency domain, capitalizing on the delay-compensating property of the design, which enables as to derive explicit string stability conditions on control and vehicle models parameters. The theoretical guarantees of string stability and the respective conditions on parameters are illustrated also numerically. We present consistent simulation results, for a ten-vehicle platoon, illustrating the potential of the design in traffic throughput improvement, as compared with a predictor-feedback CACC design in which, each ego vehicle's controller utilizes information only from a single preceding vehicle. We also present simulation results in a realistic scenario in which the leading vehicle's trajectory is obtained from NGSIM data.

We consider an improper reinforcement learning setting where a learner is given M base controllers for an unknown Markov decision process, and wishes to combine them optimally to produce a potentially new controller that can outperform each of the base ones. This can be useful in tuning across controllers, learnt possibly in mismatched or simulated environments, to obtain a good controller for a given target environment with relatively few trials. Towards this, we propose two algorithms: (1) a Policy Gradient-based approach; and (2) an algorithm that can switch between a simple Actor-Critic (AC) based scheme and a Natural Actor-Critic (NAC) scheme depending on the available information. Both algorithms operate over a class of improper mixtures of the given controllers. For the first case, we derive convergence rate guarantees assuming access to a gradient oracle. For the AC-based approach we provide convergence rate guarantees to a stationary point in the basic AC case and to a global optimum in the NAC case. Numerical results on (i) the standard control theoretic benchmark of stabilizing an cartpole; and (ii) a constrained queueing task show that our improper policy optimization algorithm can stabilize the system even when the base policies at its disposal are unstable.

Distributed training of foundation models via DDP is limited by interconnect bandwidth. While infrequent communication strategies reduce synchronization frequency, they remain bottlenecked by the memory and communication requirements of optimizer states. Low-rank optimizers can alleviate these constraints; however, in the local-update regime, workers lack access to the full-batch gradients required to compute low-rank projections, which degrades performance. We propose LoRDO, a principled framework unifying low-rank optimization with infrequent synchronization. We first demonstrate that, while global projections based on pseudo-gradients are theoretically superior, they permanently restrict the optimization trajectory to a low-rank subspace. To restore subspace exploration, we introduce a full-rank quasi-hyperbolic update. LoRDO achieves near-parity with low-rank DDP in language modeling and downstream tasks at model scales of 125M--720M, while reducing communication by approx 10 times. Finally, we show that LoRDO improves performance even more in very low-memory settings with small rank/batch size.

From a viewpoint of stochastic thermodynamics, we derive equations that describe the collective dynamics near the order-disorder transition in the globally coupled XY model and near the synchronization-desynchronization transition in the Kuramoto model. A new way of thinking is to interpret the deterministic time evolution of a macroscopic variable as an external operation to a thermodynamic system. We then find that the irreversible work determines the equation for the collective dynamics. When analyzing the Kuramoto model, we employ a generalized concept of irreversible work which originates from a non-equilibrium identity associated with steady state thermodynamics.

We present DPBench, a benchmark for evaluating coordination in multi-agent systems built from large language models. Existing benchmarks measure task-level success under a fixed protocol; the structural conditions under which coordination succeeds or fails at all have not been characterised. DPBench adapts the Dining Philosophers problem into a controlled testbed where the action protocol, the communication structure, and the group size each vary independently. We evaluate six agents: GPT-5.2, Claude Opus 4.5, Grok 4.1, Gemini 2.5 Flash, Llama 4 Maverick, and a uniform-random baseline. Under simultaneous action at N=5 with the default prompt, deadlock ranges from 25.0% (95% Wilson CI [11.2, 46.9]) for GPT-5.2 to 90.0% [74.4, 96.5] for Gemini 2.5 Flash; sequential action is solved by four of the six. Holding the model fixed at Gemini 2.5 Flash, three protocol variables drive deadlock from 90% to within CI of zero: three rounds of pre-commitment communication (0.0% vs. single-round 86.7%), a prompt encoding a classical concurrency primitive (0.0% for resource-ordering and symmetry-breaking, against 100% for the minimal prompt), or doubling the group from N=5 to N=10 (90.0% to 10.0%). Single-round messaging and memory of past timesteps do not change the rate at the sample size we ran. Whether the same model coordinates or deadlocks is determined by the protocol, not by the model's capability.

The integration of event and tracking data has become essential for advanced analysis in soccer. However, synchronizing these two modalities remains a significant challenge due to temporal and spatial inaccuracies in manually recorded event timestamps. Existing synchronizers typically rely on annotated event locations, which themselves are prone to spatial errors and thus can distort synchronization results. To address this issue, we propose ELASTIC (Event-Location-AgnoSTIC synchronizer), a synchronization framework that only uses features derived from tracking data. ELASTIC also explicitly detects the end times of pass-like events and separates the detection of major and minor events, which improves the completeness of the synchronized output and reduces error cascade across events. We annotated the ground truth timestamps of 2,134 events from three Eredivisie matches to measure the synchronization accuracy, and the experimental results demonstrate that ELASTIC outperforms existing synchronizers by a large margin.

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

Digital quadruplets aiming to improve road safety, traffic efficiency, and driving cooperation for future connected automated vehicles are proposed with the enlightenment of ACP based parallel driving. The ACP method denotes Artificial societies, Computational experiments, and Parallel execution modules for cyber-physical-social systems. Four agents are designed in the framework of digital quadruplets: descriptive vehicles, predictive vehicles, prescriptive vehicles, and real vehicles. The three virtual vehicles (descriptive, predictive, and prescriptive) dynamically interact with the real one in order to enhance the safety and performance of the real vehicle. The details of the three virtual vehicles in the digital quadruplets are described. Then, the interactions between the virtual and real vehicles are presented. The experimental results of the digital quadruplets demonstrate the effectiveness of the proposed framework.

Model merging combines several independently fine-tuned experts into a single multi-task model without any training data, reducing the storage, serving, and decentralized-development costs of large foundation models. State-of-the-art merging methods formulate merging as a layer-wise quadratic interference minimization problem. Although this problem admits an exact closed-form pseudoinverse solution, that solution underperforms hundreds of iterations of gradient descent in practice. The iterative loop dominates the cost of the pipeline, yet its effectiveness has remained unexplained. We revisit this regime and show that the iterative solver does not primarily act as an optimizer; rather, it serves as an implicit spectral regularizer for an ill-posed normal equation, where small-eigenvalue directions of the per-layer interference operator amplify proxy noise. Building on this finding, we formalize multi-task model merging as a noisy linear inverse problem and propose a spectral filtering estimator parameterized by a per-direction filter. We instantiate this estimator with SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent, with a hard top-K truncation that suppresses noise-amplifying small-eigenvalue directions. Furthermore, we propose SWUDI-A, an adaptive variant that replaces the global rank hyperparameter with per-layer rank rules, further improving robustness across architectures. Both variants share a single symmetric eigendecomposition per linear layer and require no training data or optimizer state. Across four general benchmarks and a multimodal merging benchmark spanning VQA, Geometry, Chart, OCR, Grounding, and modality merging, our proposed spectral solvers match or outperform state-of-the-art merging methods. Crucially, they reduce wall-clock time by 28-72x and peak GPU memory by up to 50%.

Scaling laws predict loss from compute but not how capabilities interact. We measure the coupling between reasoning and truthfulness across 63 base models from 16 families and find a regime change invisible to loss curves: below a family-dependent critical scale N_c, capabilities anticorrelate; above it, they cooperate. N_c approx 3.5B parameters [2.9B, 13.4B] (bootstrap 95% CI), but model size is not the only variable that determines phase. Architecture, data curation, and training recipe each shift N_c independently: curated training eliminated the coupling dip between Qwen generations (0.025 to 0.830 at matched scale), Gemma-4 at 4B achieves coupling 0.871, characteristic of 13B+ standard-trained models, through distillation and architectural innovation, and Phi at 1B matches web-trained coupling at 10B through data curation alone. Width normalization eliminates the anticorrelation across all tested families, supporting an output-projection bottleneck. Internally, 38 of 40 models show zero competing attention heads. A sparse-regression ODE cross-predicts held-out Llama-2 at 5.6% error. The diagnostic requires no model internals -- only public benchmark scores across a model family. The cooperative regime extends to the frontier (r = +0.72, 34 models, 10 labs). Code, data, and an open-source activation-steering tool for any open-weight model are released alongside an interactive dashboard that diagnoses any model's coupling phase, suggests concrete interventions (data curation, width, benchmark rotation), and provides ODE scaling predictions, frontier diagnostics, and eigenstructure analysis: https://zehenlabs.com/cape/.

Oscillations and synchronization are widely believed to play a fundamental role in representation and computation. However, existing machine learning approaches based on synchronization dynamics have largely been confined to specialized settings such as object discovery, with limited evidence of scalability to standard vision benchmarks or logic reasoning tasks. We propose the Winfree Oscillatory Neural Network (WONN), a dynamical neural architecture based on generalized Winfree dynamics. WONN evolves representations on the torus (S^1)^d through structured oscillatory interactions, combining phase-based inductive biases with flexible and hierarchical interaction mechanisms instantiated as either fixed trigonometric mappings or learnable neural networks. We evaluate WONN on image recognition and complex reasoning tasks, including CIFAR, ImageNet, Maze-hard, and Sudoku. Across these domains, WONN achieves competitive or superior performance with strong parameter efficiency. In particular, WONN is, to our knowledge, the first synchronization-based oscillatory architecture to scale competitively to ImageNet-1K. Furthermore, on Maze-hard, WONN achieves 80.1% accuracy using only 1% of the parameters of prior state-of-the-art models. These results suggest that structured oscillatory dynamics provide a scalable and parameter-efficient alternative to conventional neural architectures.

Non-cooperative game theory provides a robust framework for analyzing distributed resource allocation in multi-user wireless networks, with Iterative Water-Filling (IWF) emerging as a canonical solution for power control problems. Although classical fixed-point theorems guarantee the existence of a Nash Equilibrium (NE) under mild concavity and compactness conditions, the convergence of practical iterative algorithms to that equilibrium remains a challenging endeavor. This challenge intensifies under varying update schedules, interference regimes, and imperfections such as channel estimation errors or feedback delay. In this paper, we present an in-depth examination of IWF in multi-user systems under three different update schemes: (1) synchronous sequential updates, (2) synchronous simultaneous updates, and (3) totally asynchronous updates. We first formulate the water-filling operator in a multi-carrier environment, then recast the iterative process as a fixed-point problem. Using contraction mapping principles, we demonstrate sufficient conditions under which IWF converges to a unique NE and highlight how spectral radius constraints, diagonal dominance, and careful step-size selection are pivotal for guaranteeing convergence. We further discuss robustness to measurement noise, partial updates, and network scaling to emphasize the practical viability of these schemes. This comprehensive analysis unifies diverse threads in the literature while offering novel insights into asynchronous implementations. Our findings enable network designers to ascertain system parameters that foster both stable convergence and efficient spectrum usage.

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.

Continual learning, the ability of a model to adapt to an ongoing sequence of tasks without forgetting earlier ones, is a central goal of artificial intelligence. To better understand its underlying mechanisms, we study the limitations of continual learning in a tractable yet representative setting. Specifically, we analyze one-hidden-layer quadratic neural networks trained by gradient descent on a sequence of XOR-cluster datasets with Gaussian noise, where different tasks correspond to clusters with orthogonal means. Our analysis is based on a tight characterization of gradient descent dynamics for the training loss, which yields explicit bounds on the rate of train-time forgetting as functions of the number of iterations, sample size, number of tasks, and hidden-layer width. We then leverage an algorithmic stability framework to bound the generalization gap, leading to corresponding guarantees on test-time forgetting. Together, our results provide the first closed-form guarantees for forgetting in continual learning with neural networks and show how key problem parameters jointly govern forgetting dynamics. Numerical experiments corroborate our theoretical results.

Multi-agent systems (MASs) can autonomously learn to solve previously unknown tasks by means of each agent's individual intelligence as well as by collaborating and exploiting collective intelligence. This article considers a group of autonomous agents learning to track the same given reference trajectory in a possibly small number of trials. We propose a novel collective learning control method that combines iterative learning control (ILC) with a collective update strategy. We derive conditions for desirable convergence properties of such systems. We show that the proposed method allows the collective to combine the advantages of the agents' individual learning strategies and thereby overcomes trade-offs and limitations of single-agent ILC. This benefit is achieved by designing a heterogeneous collective, i.e., a different learning law is assigned to each agent. All theoretical results are confirmed in simulations and experiments with two-wheeled-inverted-pendulum robots (TWIPRs) that jointly learn to perform the desired maneuver.

Achieving robust coordination and cooperation is a central challenge in multi-agent reinforcement learning (MARL). Uncovering the mechanisms underlying such emergent behaviors calls for a dynamical understanding of learn processes. In this work, we investigate the dynamics of actor-critic agents in stochastic games, focusing on the impact of entropy regularization. By leveraging time-scale separation, we derive the system's evolution equations, which are then formally analyzed using dynamical systems theory. We find that in the constant-sum game of Matching Pennies, the system exhibits chaotic behavior. Entropy regularization mitigates this chaos and drives the dynamics toward convergence to fair cooperation. In contrast, in the general-sum game of the Prisoner's Dilemma, the system displays multistability. Interestingly, the three stable equilibria of the system correspond to the well-known ALLC (Always Cooperate), ALLD (Always Defect), and GRIM (Grim Trigger) strategies from evolutionary game theory (EGT). Entropy regularization strengthens system resilience by enlarging the basin of attraction of the cooperative equilibrium. Our findings reveal a close link between the mechanism of direct reciprocity in EGT and how cooperation emerges in MARL, offering insights for designing more robust and collaborative multi-agent systems.

The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a Runge-Kutta collocation method at Radau points) and Dassl, based on backward differentiation formulas, among the others. When solving fractional ordinary differential equations (ODEs), the derivative operator is replaced by a non-local one and the fractional ODE is reformulated as a Volterra integral equation, to which these codes cannot be directly applied. This article is a follow-up of the article by the authors (Guglielmi and Hairer, SISC, 2025) for differential equations with distributed delays. The main idea is to approximate the fractional kernel t^{α-1}/ Γ(α) (α>0) by a sum of exponential functions or by a sum of exponential functions multiplied by a monomial, and then to transform the fractional integral (of convolution type) into a set of ordinary differential equations. The augmented system is typically stiff and thus requires the use of an implicit method. It can have a very large dimension and requires a special treatment of the arising linear systems. The present work presents an algorithm for the construction of an approximation of the fractional kernel by a sum of exponential functions, and it shows how the arising linear systems in a stiff time integrator can be solved efficiently. It is explained how the code Radau5 can be used for solving fractional differential equations. Numerical experiments illustrate the accuracy and the efficiency of the proposed method. Driver examples are publicly available from the homepages of the authors.

We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task. Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics. While prior work has identified that individual neurons learn single-frequency Fourier features and phase alignment, it does not fully explain how these features combine into a global solution. We bridge this gap by formalizing a diversification condition that emerges during training when overparametrized, consisting of two parts: phase symmetry and frequency diversification. We prove that these properties allow the network to collectively approximate a flawed indicator function on the correct logic for the modular addition task. While individual neurons produce noisy signals, the phase symmetry enables a majority-voting scheme that cancels out noise, allowing the network to robustly identify the correct sum. Furthermore, we explain the emergence of these features under random initialization via a lottery ticket mechanism. Our gradient flow analysis proves that frequencies compete within each neuron, with the "winner" determined by its initial spectral magnitude and phase alignment. From a technical standpoint, we provide a rigorous characterization of the layer-wise phase coupling dynamics and formalize the competitive landscape using the ODE comparison lemma. Finally, we use these insights to demystify grokking, characterizing it as a three-stage process involving memorization followed by two generalization phases, driven by the competition between loss minimization and weight decay.

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.