Path-conditioned training: a principled way to rescale ReLU neural networks

Stable and efficient training of ReLU networks with large depth is highly sensitive to weight initialization. Improper initialization can cause permanent neuron inactivation dying ReLU and exacerbate gradient instability as network depth increases. Methods such as He, Xavier, and orthogonal initialization preserve variance or promote approximate isometry. However, they do not necessarily regulate the pre-activation mean or control activation sparsity, and their effectiveness often diminishes in very deep architectures. This work introduces a semi-orthogonal initialization specifically optimized for ReLU by solving an optimization problem on the Stiefel manifold, thereby preserving scale and calibrating the pre-activation statistics from the outset. A family of closed-form solutions and an efficient sampling scheme are derived. Theoretical analysis at initialization shows that the proposed initialization mitigates dying ReLU, slows the decay of activation variance, and alleviates gradient vanishing, thereby stabilizing signal and gradient flow in deep architectures. Empirically, across MNIST, Fashion-MNIST, multiple tabular datasets, few-shot settings, and ReLU-family activations, the proposed initialization maintains stable performance as network depth increases, with particularly clear advantages in few-shot learning and large-depth settings.

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.

Given a labeled dataset $\{(x_i, y_i)\}_{i = 1}^n$, consider drawing a parameter vector $\hatθ$ uniformly at random from those that interpolate the data within a width-$W$ depth-$D$ ReLU architecture with continuous weights bounded within $[-M, M]$. We study the generalization properties of $\hatθ$ in the case where the dataset can, in fact, be explained by a smaller $w$-by-$d$ ReLU network. Despite the obliviousness of the selection rule for $\hatθ$, it adaptively generalizes at the rate $\tilde O\left(\frac{wdWD}{n}\right)$, rather than being governed by the VC dimension of the full $W$-by-$D$ hypothesis class. The reason is due to a particular inductive bias in ReLU networks: the $w$-by-$d$ parameter witnesses an exponential volume of $W$-by-$D$ interpolating parameters, which dwarfs the volume of poorly generalizing parameters (conditioned on interpolation). We explicitly construct this set of near-equivalent $W$-by-$D$ parameters by a proof technique called deep lifting, and then tie the achievable volume to stability properties of the $w$-by-$d$ witness. Examples show that these techniques allow us to convert approximation-theoretic results into generalization guarantees for nonparametric classes. We further show that generalization of uniform-random interpolators can survive label corruptions.