Derives exact Frobenius norm imbalance identity for deep nonlinear networks, classifies activations into four classes, and obtains critical-depth escape time law τ★ = Θ(ε^{-(r-2)}) from reduction to scalar ODE on permutation-symmetric submanifold.
Gradient descent aligns the layers of deep linear networks
9 Pith papers cite this work, alongside 22 external citations. Polarity classification is still indexing.
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abstract
This paper establishes risk convergence and asymptotic weight matrix alignment --- a form of implicit regularization --- of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail, for gradient flow applied to strictly decreasing loss functions (with similar results for gradient descent with particular decreasing step sizes): (i) the risk converges to 0; (ii) the normalized i-th weight matrix asymptotically equals its rank-1 approximation $u_iv_i^{\top}$; (iii) these rank-1 matrices are aligned across layers, meaning $|v_{i+1}^{\top}u_i|\to1$. In the case of the logistic loss (binary cross entropy), more can be said: the linear function induced by the network --- the product of its weight matrices --- converges to the same direction as the maximum margin solution. This last property was identified in prior work, but only under assumptions on gradient descent which here are implied by the alignment phenomenon.
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Data symmetries generically do not induce conserved quantities in NN training for analytic non-polynomial losses, but can for MSE with tensorizable networks.
Gradient flow on deep diagonal linear LDA networks with balanced initialization converts additive updates to multiplicative updates, automatically conserving the (2/L) quasi-norm.
Establishes convergence guarantees for overparameterized 2-layer ReLU networks in flow matching, generalization bounds for the velocity-field objective, and Wasserstein guarantees for generated samples, using multi-task representation learning bounds.
Mini-batch noise reverses how Adam's β2 controls anti-regularization, making default momentum values suitable for small batches but requiring β1 closer to β2 for large batches to favor flatter minima.
Longer prediction horizons in predictive learning interact with model biases to recover the latent geometry of the task.
SPIN lets weak LLMs become strong by self-generating training data from previous model versions and training to prefer human-annotated responses over its own outputs, outperforming DPO even with extra GPT-4 data on benchmarks.
Gradient descent on wide shallow models with bounded nonlinearities converges globally in the mean-field limit as non-global critical points are unstable under the dynamics.
Unified framework characterizes conservation laws for gradient flow in feedforward networks with GELU/SiLU/SwiGLU, multihead attention with positional encodings, and MoE models under various gating.
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A Theory of Saddle Escape in Deep Nonlinear Networks
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Conservation Laws from Data Symmetry in Neural Networks
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Implicit Bias in Deep Linear Discriminant Analysis
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