Neural Network Training Dynamics

Feature learning in shallow neural networks exhibits rich yet fragile dynamics, including prolonged plateaus, abrupt phase transitions, and sensitivity to optimization hyperparameters. While recent theoretical work has characterized these behaviors through the geometry of loss landscapes, saddle escape mechanisms, and emergent scaling laws, practical methods for actively shaping these dynamics remain limited. In this paper, we introduce AYLA, a principled loss reparameterization framework that dynamically modulates gradient magnitudes during training without altering the location of stationary points or optimal solutions. AYLA applies a smooth, sigmoid-controlled power-law transformation to empirical loss, yielding a state-dependent effective learning rate that accelerates descent in flat or saddle-dominated regions while stabilizing late-stage optimization. Crucially, AYLA preserves all critical points of the original objective, acting solely as a monotone transformation that reshapes optimization trajectories rather than objectives. We evaluate AYLA in controlled teacher student settings using two-layer tanh networks trained on synthetic Gaussian data. Across stochastic gradient descent and multiple loss-exponent schedules, AYLA consistently improves feature recovery. This evidence is observed in terms of weight alignment, per-neuron cosine similarity, hidden-activation correlation, and spectral properties of learned representations, while AYLA maintains competitive or faster loss convergence. Spectral analyses further demonstrate that AYLA mitigates rank collapse and promotes richer internal representations, signaling a transition from lazy to active feature-learning regimes. AYLA offers a lightweight, theoretically grounded way to improve shallow-network optimization, especially in resource-limited or noise-sensitive settings.

We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean loss in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations. We show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. We provide a detailed discussion of several types of solutions to the gradient flow equations. A main motivation for this work is to shed light on the interpretability question in supervised learning.

Continuous adaptive learning, the ability to adapt to the environment and keep improving performance, is a hallmark of natural intelligence. Biological organisms excel in acquiring, transferring, and retaining knowledge while adapting to volatile environments, making them a source of inspiration for artificial neural networks (ANNs). This study explores how neuromodulation, a building block of learning in biological systems, can help address catastrophic forgetting and enhance the robustness of ANNs in continual learning. Driven by neuromodulators including dopamine (DA), acetylcholine (ACh), serotonin (5-HT) and noradrenaline (NA), neuromodulatory processes in the brain operate at multiple scales, facilitating dynamic responses to environmental changes through mechanisms ranging from local synaptic plasticity to global network-wide adaptability. Importantly, the relationship between neuromodulators and their interplay in modulating sensory and cognitive processes is more complex than previously expected, demonstrating a "many-to-many" neuromodulator-to-task mapping. To inspire neuromodulation-aware learning rules, we highlight (i) how multi-neuromodulatory interactions enrich single-neuromodulator-driven learning, (ii) the impact of neuromodulators across multiple spatio-temporal scales, and correspondingly, (iii) strategies for approximating and integrating neuromodulated learning processes in ANNs, and (iv) an architectural-general formulation of multi-neuromodulatory dynamics. We also present a conceptual study to showcase how neuromodulation-inspired mechanisms, such as DA-driven reward processing and NA-based cognitive flexibility, can enhance ANN performance in a Go/No-Go task. Though multi-scale neuromodulation, we aim to bridge the gap between biological and artificial learning, paving the way for ANNs with greater flexibility, robustness, and adaptability.

Advancements in artificial intelligence call for a deeper understanding of the fundamental mechanisms underlying deep learning. In this work, we propose a theoretical framework to analyze learning dynamics through the lens of dynamical systems theory. We redefine the notions of linearity and nonlinearity in neural networks by introducing two fundamental transformation units at the neuron level: order-preserving transformations and non-order-preserving transformations. Different transformation modes lead to distinct collective behaviors in weight vector organization, different modes of information extraction, and the emergence of qualitatively different learning phases. Transitions between these phases may occur during training, accounting for key phenomena such as grokking. To further characterize generalization and structural stability, we introduce the concept of attraction basins in both sample and weight spaces. The distribution of neurons with different transformation modes across layers, along with the structural characteristics of the two types of attraction basins, forms a set of core metrics for analyzing the performance of learning models. Hyperparameters such as depth, width, learning rate, and batch size act as control variables for fine-tuning these metrics. Our framework not only sheds light on the intrinsic advantages of deep learning, but also provides a novel perspective for optimizing network architectures and training strategies.

Hyperparameter tuning remains a significant challenge in the training of deep neural networks (DNNs), requiring manual search or time-intensive grid searches that increase resource costs and limit the accessibility of machine learning. The global initial learning rate is among the most consequential of these hyperparameters. Adaptive and scheduling-based methods manage the learning rate during training but still require manual selection of an initial global value; learning-rate-free alternatives remove this selection at the cost of performance or stability on non-convex problems. We present ExpTest, an autonomous learning-rate controller that treats the training loss curve as an online signal and performs sequential statistical tests on theoretically motivated windows to detect convergent behavior and trigger learning-rate reductions. The framework combines a covariance-based initial learning-rate estimate, curvature-motivated window sizing, and two-phase test-driven decay, drawing on the approximately exponential decay behavior predicted under linearized network dynamics. We provide a mathematical motivation for ExpTest and evaluate it on regression, classification, forecasting, and natural-language tasks across fully-connected, convolutional, transformer-based, and pretrained architectures. Across these tasks, ExpTest achieves competitive performance relative to hand-tuned SGD-based baselines and recent learning-rate-free methods, without manual initial learning-rate selection or predefined scheduling.

Large-scale pretraining of transformers has been central to the success of foundation models. However, the scale of those models limits our understanding of the mechanisms at play during optimization. In this work, we study the training dynamics of transformers in a controlled and interpretable setting. On the sparse modular addition task, we demonstrate that specialized attention circuits, called clustering heads, can be implemented during gradient descent to solve the problem. Our experiments show that such pathways naturally emerge during training. By monitoring the evolution of tokens via a visual sandbox, we uncover a two-stage learning and the occurrences of loss spikes due to the high curvature of normalization layers. Our findings provide several insights into patterns observed in more practical settings, such as the pretraining of large language models.

Despite being widely adopted as a canonical framework for learning robust models, adversarial training suffers from robust overfitting. Existing empirical and theoretical explorations fail to provide a satisfactory mechanistic interpretation of the phenomenon. By modeling adversarial training with momentum SGD as a discrete-time dynamical system, we propose a PAC-Bayesian analytical framework that proves time-resolved robust generalization bounds. Specifically, our framework tracks the closed-form evolution of the posterior mean and covariance under both stationary and non-stationary transient regimes, connecting the model's robust generalization performance to learning rate, local loss geometry, and mini-batch stochastic gradients. By estimating the key quantities associated with the bound, we illustrate the underlying mechanism of robust overfitting. Our framework also shows how adversarial weight perturbation reduces robust generalization gaps by suppressing dominant loss-curvature modes, while suggesting that excessive penalization can be sub-optimal for optimization.

To understand the training dynamics of neural networks, prior studies have considered the mean-field limit of two-layer neural networks as the width tends to infinity, establishing theoretical guarantees for its convergence under gradient flow training as well as approximation and generalization capabilities. In this work, we study the infinite-width limit of a type of three-layer neural network where the first-layer weights are randomly sampled and untrained. To rigorously define the limiting model, we extend the mean-field theory by lifting the representation of neurons from Euclidean to functional spaces. This allows us to establish the mean-field training dynamics as a functional gradient flow with a time-varying kernel that remains positive-definite under suitable assumptions, thus proving a linear-rate convergence of its training loss. Furthermore, we define novel function spaces that contain the solutions obtained through the mean-field training dynamics and prove Rademacher complexity bounds for these spaces. Notably, our analysis applies to a range of scaling choices of the model, resulting in two distinct regimes of the mean-field limit that both exhibit feature learning through training.

Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward propagation of deep residual networks using continuity equations, in which the measure is conserved and infinite curves in the measure space connect the input distribution to the output one of a ResNet. We find ResNets with $L_2$ regularization attempt to learn the geodesic curve in the Wasserstein space, induced by the optimal transport map. Compared with plain networks, ResNets can better approximate the geodesic curve, which explains why ResNets can be optimized and generalize better. Numerical experiments show that the data tracks of a ResNet tend to be line-shaped in terms of the line-shape score, and the map learned by a ResNet is closer to the optimal transport map in terms of the optimal transport score. In a word, we conclude that ResNets learn the geodesic curve in the Wasserstein space and discretely engineer the data transformation in high-dimensional spaces.

Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems, recently, some empirical studies of DNNs reported a universal phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target function from low to high frequencies during the training. The F-Principle has been very useful in providing both qualitative and quantitative understandings of DNNs. In this paper, we rigorously investigate the F-Principle for the training dynamics of a general DNN at three stages: initial stage, intermediate stage, and final stage. For each stage, a theorem is provided in terms of proper quantities characterizing the F-Principle. Our results are general in the sense that they work for multilayer networks with general activation functions, population densities of data, and a large class of loss functions. Our work lays a theoretical foundation of the F-Principle for a better understanding of the training process of DNNs.