Neural networks have been used prominently in several machine learning and statistics applications. In general, the underlying optimization of neural networks is non-convex which makes their performance analysis challenging. In this paper, we take a novel approach to this problem by asking whether one can constrain neural network weights to make its optimization landscape have good theoretical properties while at the same time, be a good approximation for the unconstrained one. For two-layer neural networks, we provide affirmative answers to these questions by introducing Porcupine Neural Networks (PNNs) whose weight vectors are constrained to lie over a finite set of lines. We show that most local optima of PNN optimizations are global while we have a characterization of regions where bad local optimizers may exist. Moreover, our theoretical and empirical results suggest that an unconstrained neural network can be approximated using a polynomially-large PNN.
References @cite_11 @cite_8 @cite_14 @cite_34 have studied the convergence of the local search algorithms such as gradient descent methods to the global optimum of the neural network optimization with zero hidden neurons and a single output. In this case, the loss function of the neural network optimization has a single local optimizer which is the same as the global optimum. However, for neural networks with hidden neurons, the landscape of the loss function is more complicated than the case with no hidden neurons.
@cite_8 @cite_14 @cite_34 @cite_11
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Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine the convergence performance of optimization algorithms. In this paper, we provide full (necessary and sufficient) characterization of the analytical forms for the critical points (as well as global minimizers) of the square loss functions for various neural networks. We show that the analytical forms of the critical points characterize the values of the corresponding loss functions as well as the necessary and sufficient conditions to achieve global minimum. Furthermore, we exploit the analytical forms of the critical points to characterize the landscape properties for the loss functions of these neural networks. One particular conclusion is that: The loss function of linear networks has no spurious local minimum, while the loss function of one-hidden-layer nonlinear networks with ReLU activation function does have local minimum that is not global minimum. We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic gradient descent (SGD), we show that it converges in expectation to either the global or the local max-margin direction if SGD converges. We further explore the implicit bias of these algorithms in learning a multi-neuron network under certain stationary conditions, and show that the learned classifier maximizes the margins of each sample pattern partition under the ReLU activation. One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural networks. For an @math hidden node shallow neural network with ReLU activation and @math training data, we show as long as @math is large enough and no two inputs are parallel, randomly initialized gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. Our analysis relies on the following observation: over-parameterization and random initialization jointly restrict every weight vector to be close to its initialization for all iterations, which allows us to exploit a strong convexity-like property to show that gradient descent converges at a global linear rate to the global optimum. We believe these insights are also useful in analyzing deep models and other first order methods. One of the main difficulties in analyzing neural networks is the non-convexity of the loss function which may have many bad local minima. In this paper, we study the landscape of neural networks for binary classification tasks. Under mild assumptions, we prove that after adding one special neuron with a skip connection to the output, or one special neuron per layer, every local minimum is a global minimum.