It seems that there is no universally accepted definition of chaotic behavior of a dynamical system.

Finally, we respectfully disagree that our paper is only tangentially related to NeurIPS.

In this paper, we contribute to the recent literature on the theoretical analysis of this phenomenon.

Mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional defined over the space of probability distributions. MFLD has gained attention due to its connection with noisy gradient descent for mean-field two-layer neural networks. Unlike standard Langevin dynamics, the nonlinearity of the objective functional induces particle interactions, necessitating multiple particles to approximate the dynamics in a finite-particle setting. Recent works (Chen et al., 2022; Suzuki et al., 2023b) have demonstrated the uniform-in-time propagation of chaos for MFLD, showing that the gap between the particle system and its mean-field limit uniformly shrinks over time as the number of particles increases. In this work, we improve the dependence on logarithmic Sobolev inequality (LSI) constants in their particle approximation errors, which can exponentially deteriorate with the regularization coefficient. Specifically, we establish an LSI-constant-free particle approximation error concerning the objective gap by leveraging the problem structure in risk minimization. As the application, we demonstrate improved convergence of MFLD, sampling guarantee for the mean-field stationary distribution, and uniform-in-time Wasserstein propagation of chaos in terms of particle complexity.

The success of contrastive learning is well known to be dependent on data augmentation.Although the degree of data augmentations has been well controlled by utilizing pre-defined techniques in some domains like vision, time-series data augmentation is less explored and remains a challenging problem due to the complexity of the data generation mechanism, such as the intricate mechanism involved in the cardiovascular system.Moreover, there is no widely recognized and general time-series augmentation method that can be applied across different tasks.In this paper, we propose a novel data augmentation method for time-series tasks that aims to connect intra-class samples together, and thereby find order in the latent space.Our method builds upon the well-known data augmentation technique of mixup by incorporating a novel approach that accounts for the non-stationary nature of time-series data.Also, by controlling the degree of chaos created by data augmentation, our method leads to improved feature representations and performance on downstream tasks.We evaluate our proposed method on three time-series tasks, including heart rate estimation, human activity recognition, and cardiovascular disease detection. Extensive experiments against the state-of-the-art methods show that the proposed method outperforms prior works on optimal data generation and known data augmentation techniques in three tasks, reflecting the effectiveness of the presented method. The source code is available at double-blind policy.

Recurrent Neural Networks (RNNs) frequently exhibit complicated dynamics, and their sensitivity to the initialization process often renders them notoriously hard to train. Recent works have shed light on such phenomena analyzing when exploding or vanishing gradients may occur, either of which is detrimental for training dynamics. In this paper, we point to a formal connection between RNNs and chaotic dynamical systems and prove a qualitatively stronger phenomenon about RNNs than what exploding gradients seem to suggest. Our main result proves that under standard initialization (e.g., He, Xavier etc.), RNNs will exhibit \textit{Li-Yorke chaos} with \textit{constant} probability \textit{independent} of the network's width. This explains the experimentally observed phenomenon of \textit{scrambling}, under which trajectories of nearby points may appear to be arbitrarily close during some timesteps, yet will be far away in future timesteps. In stark contrast to their feedforward counterparts, we show that chaotic behavior in RNNs is preserved under small perturbations and that their expressive power remains exponential in the number of feedback iterations. Our technical arguments rely on viewing RNNs as random walks under non-linear activations, and studying the existence of certain types of higher-order fixed points called \textit{periodic points} in order to establish phase transitions from order to chaos.

In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Stochastic Gradient Descent (SGD) algorithm applied to two-layer overparameterized neural networks, as the number or neurons (i.e., the size of the hidden layer) $N \to \plusinfty$. Following a probabilistic approach, we show `propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios, indicating that the statistical interaction between the particles asymptotically vanishes. In particular, we establish quantitative convergence with respect to $N$ of any particle to a solution of a mean-field McKean-Vlasov equation in the metric space endowed with the Wasserstein distance. In comparison to previous works on the subject, we consider settings in which the sequence of stepsizes in SGD can potentially depend on the number of neurons and the iterations. We then identify two regimes under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand. We perform various experiments on real datasets to validate our theoretical results, assessing the existence of these two regimes on classification problems and illustrating our convergence results.

Recurrent Neural Networks (RNNs) have found widespread applications in machine learning for time series prediction and dynamical systems reconstruction, and experienced a recent renaissance with improved training algorithms and architectural designs. Understanding why and how trained RNNs produce their behavior is important for scientific and medical applications, and explainable AI more generally. An RNN's dynamical repertoire depends on the topological and geometrical properties of its state space. Stable and unstable manifolds of periodic points play a particularly important role: They dissect a dynamical system's state space into different basins of attraction, and their intersections lead to chaotic dynamics with fractal geometry. Here we introduce a novel algorithm for detecting these manifolds, with a focus on piecewise-linear RNNs (PLRNNs) employing rectified linear units (ReLUs) as their activation function. We demonstrate how the algorithm can be used to trace the boundaries between different basins of attraction, and hence to characterize multistability, a computationally important property. We further show its utility in finding so-called homoclinic points, the intersections between stable and unstable manifolds, and thus establish the existence of chaos in PLRNNs. Finally we show for an empirical example, electrophysiological recordings from a cortical neuron, how insights into the underlying dynamics could be gained through our method.

As cities around the world aim to improve walkability and safety, understanding the irregular and unpredictable nature of pedestrian behavior has become increasingly important. This study introduces a data-driven framework for modeling chaotic pedestrian movement using empirically observed trajectory data and supervised learning. Videos were recorded during both daytime and nighttime conditions to capture pedestrian dynamics under varying ambient and traffic contexts. Pedestrian trajectories were extracted through computer vision techniques, and behavioral chaos was quantified using four chaos metrics: Approximate Entropy and Lyapunov Exponent, each computed for both velocity and direction change. A Principal Component Analysis (PCA) was then applied to consolidate these indicators into a unified chaos score. A comprehensive set of individual, group-level, and contextual traffic features was engineered and used to train Random Forest and CatBoost regression models. CatBoost models consistently achieved superior performance. The best daytime PCA-based CatBoost model reached an R^2 of 0.8319, while the nighttime PCA-based CatBoost model attained an R^2 of 0.8574. SHAP analysis highlighted that features such as distance travel, movement duration, and speed variability were robust contributors to chaotic behavior. The proposed framework enables practitioners to quantify and anticipate behavioral instability in real-world settings. Planners and engineers can use chaos scores to identify high-risk pedestrian zones, apprise infrastructure improvements, and calibrate realistic microsimulation models. The approach also supports adaptive risk assessment in automated vehicle systems by capturing short-term motion unpredictability grounded in observable, interpretable features.

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