  Representations of the world environment play a crucial role in artificial
intelligence. It is often inefficient to conduct reasoning and inference
directly in the space of raw sensory representations, such as pixel values of
images. Representation learning allows us to automatically discover suitable
representations from raw sensory data. For example, given raw sensory data, a
deep neural network learns nonlinear representations at its hidden layers,
which are subsequently used for classification (or regression) at its output
layer. This happens implicitly during training through minimizing a supervised
or unsupervised loss in common practical regimes of deep learning, unlike the
neural tangent kernel (NTK) regime. In this paper, we study the dynamics of
such implicit nonlinear representation learning, which is beyond the NTK
regime. We identify a pair of a new assumption and a novel condition, called
the common model structure assumption and the data-architecture alignment
condition. Under the common model structure assumption, the data-architecture
alignment condition is shown to be sufficient for the global convergence and
necessary for the global optimality. Moreover, our theory explains how and when
increasing the network size does and does not improve the training behaviors in
the practical regime. Our results provide practical guidance for designing a
model structure: e.g., the common model structure assumption can be used as a
justification for using a particular model structure instead of others. We also
derive a new training framework based on the theory. The proposed framework is
empirically shown to maintain competitive (practical) test performances while
providing global convergence guarantees for deep residual neural networks with
convolutions, skip connections, and batch normalization with standard benchmark
datasets, including CIFAR-10, CIFAR-100, and SVHN.
