Fully-connected deep neural networks with weights initialized from
independent Gaussian distributions can be tuned to criticality, which prevents
the exponential growth or decay of signals propagating through the network.
However, such networks still exhibit fluctuations that grow linearly with the
depth of the network, which may impair the training of networks with width
comparable to depth. We show analytically that rectangular networks with tanh
activations and weights initialized from the ensemble of orthogonal matrices
have corresponding preactivation fluctuations which are independent of depth,
to leading order in inverse width. Moreover, we demonstrate numerically that,
at initialization, all correlators involving the neural tangent kernel (NTK)
and its descendants at leading order in inverse width -- which govern the
evolution of observables during training -- saturate at a depth of $\sim 20$,
rather than growing without bound as in the case of Gaussian initializations.
We speculate that this structure preserves finite-width feature learning while
reducing overall noise, thus improving both generalization and training speed.
We provide some experimental justification by relating empirical measurements
of the NTK to the superior performance of deep nonlinear orthogonal networks
trained under full-batch gradient descent on the MNIST and CIFAR-10
classification tasks.

完全连接的深度神经网络，其权重从独立的高斯分布初始化，可以调整到临界状态，阻止信号在网络中传播时呈指数增长或衰减。然而，这种网络仍然会表现出与网络深度线性增长的波动，这可能会削弱与深度相当的宽度网络的训练。我们在理论上证明了矩形网络与双曲正切激活函数以及从正交矩阵集合初始化权重，其相应的预激活波动与深度无关，以逆宽度为导数阶主导。此外，我们通过数值实验表明，初始化时在逆宽度方向上控制可观测量的演变的神经切向核（NTK）及其后代涉及的相关者的饱和深度约为 20，而不像高斯初始化的情况下无限增长。我们推测这种结构保留了有限宽度特征学习的同时，降低了整体噪声，从而改善了泛化能力和训练速度。通过将 NTK 的经验测量与深度非线性正交网络在 MNIST 和 CIFAR-10 分类任务上的卓越性能联系起来，我们提供了一些实验上的证明。

深度网络中的特征学习与泛化及正交权重

Feature Learning and Generalization in Deep Networks with Orthogonal  Weights

Artificial neural networks have revolutionized machine learning in recent
years, but a complete theoretical framework for their learning process is still
lacking. Substantial progress has been made for infinitely wide networks. In
this regime, two disparate theoretical frameworks have been used, in which the
network's output is described using kernels: one framework is based on the
Neural Tangent Kernel (NTK) which assumes linearized gradient descent dynamics,
while the Neural Network Gaussian Process (NNGP) kernel assumes a Bayesian
framework. However, the relation between these two frameworks has remained
elusive. This work unifies these two distinct theories using a Markov proximal
learning model for learning dynamics in an ensemble of randomly initialized
infinitely wide deep networks. We derive an exact analytical expression for the
network input-output function during and after learning, and introduce a new
time-dependent Neural Dynamical Kernel (NDK) from which both NTK and NNGP
kernels can be derived. We identify two learning phases characterized by
different time scales: gradient-driven and diffusive learning. In the initial
gradient-driven learning phase, the dynamics is dominated by deterministic
gradient descent, and is described by the NTK theory. This phase is followed by
the diffusive learning stage, during which the network parameters sample the
solution space, ultimately approaching the equilibrium distribution
corresponding to NNGP. Combined with numerical evaluations on synthetic and
benchmark datasets, we provide novel insights into the different roles of
initialization, regularization, and network depth, as well as phenomena such as
early stopping and representational drift. This work closes the gap between the
NTK and NNGP theories, providing a comprehensive framework for understanding
the learning process of deep neural networks in the infinite width limit.

通过使用具有随机初始化的无限宽度深度网络集合的马尔可夫接近学习模型，结合数值评估来合并和统一神经切向核（NTK）和神经网络高斯过程（NNGP）理论，并提供对机器学习中深度神经网络学习过程的全面理解。