We give the first provably efficient algorithm for learning a one hidden layer convolutional network with respect to a general class of (potentially overlapping) patches under mild conditions on the underlying distribution. We prove that our framework captures commonly used schemes from computer vision, including one-dimensional and two-dimensional ``patch and stride'' convolutions. Our algorithm-- {\em Convotron}-- is inspired by recent work applying isotonic regression to learning neural networks. Convotron uses a simple, iterative update rule that is stochastic in nature and tolerant to noise (requires only that the conditional mean function is a one layer convolutional network, as opposed to the realizable setting). In contrast to gradient descent, Convotron requires no special initialization or learning-rate tuning to converge to the global optimum. We also point out that learning one hidden convolutional layer with respect to a Gaussian distribution and just {\em one} disjoint patch $P$ (the other patches may be arbitrary) is {\em easy} in the following sense: Convotron can efficiently recover the hidden weight vector by updating {\em only} in the direction of $P$.

We consider the problem of learning an one-hidden-layer neural network with non-overlapping convolutional layer and ReLU activation function, i.e., $f(Z; w, a) = \sum_j a_j\sigma(w^\top Z_j)$, in which both the convolutional weights $w$ and the output weights $a$ are parameters to be learned. We prove that with Gaussian input $\vZ$, there is a spurious local minimizer. Surprisingly, in the presence of the spurious local minimizer, starting from randomly initialized weights, gradient descent with weight normalization 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.

We study the loss surface of neural networks equipped with a hinge loss criterion and ReLU or leaky ReLU nonlinearities. Any such network defines a piecewise multilinear form in parameter space. By appealing to harmonic analysis we show that all local minima of such network are non-differentiable, except for those minima that occur in a region of parameter space where the loss surface is perfectly flat. Non-differentiable minima are therefore not technicalities or pathologies; they are heart of the problem when investigating the loss of ReLU networks. As a consequence, we must employ techniques from nonsmooth analysis to study these loss surfaces. We show how to apply these techniques in some illustrative cases.

It is widely conjectured that training algorithms for neural networks are successful because all local minima lead to similar performance; for example, see (LeCun et al., 2015; Choromanska et al., 2015; Dauphin et al., 2014). Performance is typically measured in terms of two metrics: training performance and generalization performance. Here we focus on the training performance of neural networks for binary classification, and provide conditions under which the training error is zero at all local minima of appropriately chosen surrogate loss functions. Our conditions are roughly in the following form: the neurons have to be increasing and strictly convex, the neural network should either be single-layered or is multi-layered with a shortcut-like connection, and the surrogate loss function should be a smooth version of hinge loss. We also provide counterexamples to show that, when these conditions are relaxed, the result may not hold.