Gradient Flow Dynamics and Implicit Bias of Diagonal Linear Networks under Infinitesimal Initialization

The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as $m \gtrsim \log(n)$, where $m$ is the network width and $n$ is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared $\ell_2$-norm scaling as $\sqrt{n}$, which is within a constant factor of the minimal $\ell_2$-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.

Training dynamics is central to understanding neural networks, yet its theoretical analysis remains difficult even for simple architectures and becomes substantially more challenging for general modern architectures. In this paper, we propose a convergence framework for analyzing gradient descent (GD) dynamics under a broad family of neural network architectures and datasets beyond the neural tangent kernel (NTK) regime. The framework is formulated at the level of network blocks and covers architectures including pre-normalized multi-layer transformers. More precisely, under mild assumptions, we prove that for almost all initializations, GD with regular learning rates converges to the neighbourhood of a stationary point. This is mainly proved by establishing an iterate-dependent PL-type inequality through analyticity and measure-zero arguments, and by proving Lipschitz smoothness along the GD trajectory through polynomial generalized smoothness and a local relaxed dissipative condition. We further interpret the theorem under Xavier initialization and practical architectural scaling, showing that the learning rate scale depends on the depth and effective bottleneck dimensions rather than the largest width. Finally, we derive structural nondegeneracy implications for residual connections and function composition, and provide a generic characterization of global minimizers within our framework.

Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and generalization properties of neural models. In this work, we show that in the high-dimensional regime, stochastic gradient descent on diagonal linear networks is well-approximated by continuous dynamics governed by a stochastic differential equation (SDE), which explicitly decouples the drift from the gradient noise. We further derive a deterministic partial differential equation whose solution propagates the relevant state of the iterates and characterizes the time evolution of a broad class of observable statistics, including the risk, curvature, and other metrics for optimality. Finally, we show that, under a suitable parametrization, the stochastic dynamics are globally well posed and converge exponentially fast to zero risk with high probability, yielding a fully explicit non-asymptotic description of their long-time behavior. Numerical simulations corroborate our theoretical findings.