Gradient Descent Learns One-hidden-layer CNN: Don't be Afraid of Spurious Local Minima
This addresses the challenge of non-convex optimization in neural networks for researchers, showing that gradient descent can avoid spurious minima in a specific architecture, though it is incremental as it focuses on a simplified model.
The paper tackles the problem of learning a one-hidden-layer convolutional neural network with ReLU activation, proving that gradient descent can recover true parameters from a teacher network despite the presence of spurious local minima, achieving recovery with constant probability that can be boosted to probability 1 with multiple restarts.
We consider the problem of learning a one-hidden-layer neural network with non-overlapping convolutional layer and ReLU activation, i.e., $f(\mathbf{Z}, \mathbf{w}, \mathbf{a}) = \sum_j a_jσ(\mathbf{w}^T\mathbf{Z}_j)$, in which both the convolutional weights $\mathbf{w}$ and the output weights $\mathbf{a}$ are parameters to be learned. When the labels are the outputs from a teacher network of the same architecture with fixed weights $(\mathbf{w}^*, \mathbf{a}^*)$, we prove that with Gaussian input $\mathbf{Z}$, there is a spurious local minimizer. Surprisingly, in the presence of the spurious local minimizer, gradient descent with weight normalization from randomly initialized weights can still be proven to recover the true parameters with constant probability, which can be boosted to probability $1$ with multiple restarts. We also show that with constant probability, the same procedure could also converge to the spurious local minimum, showing that the local minimum plays a non-trivial role in the dynamics of gradient descent. Furthermore, a quantitative analysis shows that the gradient descent dynamics has two phases: it starts off slow, but converges much faster after several iterations.