Understanding deep neural networks from the perspective of piecewise linear property

In recent years, deep learning models have been widely used and are behind major breakthroughs across many fields. Deep learning models are usually considered to be black boxes due to their large model structures and complicated hierarchical nonlinear transformations. As deep learning technology continues to develop, the understanding of deep learning models is raising concerns, such as the understanding of the training and prediction behaviors and the internal mechanism of models. In this thesis, we study the model understanding problem of deep neural networks from the perspective of piecewise linear property. First, we introduce the piecewise linear property. Next, we review the role and progress of deep learning understanding from the perspective of the piecewise linear property. The piecewise linear property reveals that deep neural networks with piecewise linear activation functions can generally divide the input space into a number of small disjointed regions that correspond to a local linear function within each region. Next, we investigate two typical understanding problems, namely model interpretation, and model complexity. In particular, we provide a series of derivations and analyses of the piecewise linear property of deep neural networks with piecewise linear activation functions. We propose an approach for interpreting the predictions given by such models based on the piecewise linear property. Next, we propose a method to provide local interpretation to a black box deep model by mimicking a piecewise linear approximation from the deep model. Then, we study deep neural networks with curve activation functions with the aim of providing piecewise linear approximations for these networks that would let them benefit from the piecewise linear property. After proposing a piecewise linear approximation framework, we investigate model complexity and model interpretation using the approximation. The thesis concludes by discussing future directions for understanding deep neural networks from the perspective of the piecewise linear property

Expected Complexity and Gradients of Deep Maxout Neural Networks and Implications to Parameter Initialization

Learning with neural networks depends on the particular parametrization of the functions represented by the network, that is, the assignment of parameters to functions. It also depends on the identity of the functions, which get assigned typical parameters at initialization, and, later, the parameters that arise during training. The choice of the activation function is a critical aspect of the network design that influences these function properties and requires investigation. This thesis focuses on analyzing the expected behavior of networks with maxout (multi-argument) activation functions. On top of enhancing the practical applicability of maxout networks, these findings add to the theoretical exploration of activation functions beyond the common choices. We believe this work can advance the study of activation functions and complicated neural network architectures. We begin by taking the number of activation regions as a complexity measure and showing that the practical complexity of deep networks with maxout activation functions is often far from the theoretical maximum. This analysis extends the previous results that were valid for deep neural networks with single-argument activation functions such as ReLU. Additionally, we demonstrate that a similar phenomenon occurs when considering the decision boundaries in classification tasks. We also show that the parameter space has a multitude of full-dimensional regions with widely different complexity and obtain nontrivial lower bounds on the expected complexity. Finally, we investigate different parameter initialization procedures and show that they can increase the speed of the gradient descent convergence in training. Further, continuing the investigation of the expected behavior, we study the gradients of a maxout network with respect to inputs and parameters and obtain bounds for the moments depending on the architecture and the parameter distribution. We observe that the distribution of the input-output Jacobian depends on the input, which complicates a stable parameter initialization. Based on the moments of the gradients, we formulate parameter initialization strategies that avoid vanishing and exploding gradients in wide networks. Experiments with deep fully-connected and convolutional networks show that this strategy improves SGD and Adam training of deep maxout networks. In addition, we obtain refined bounds on the expected number of linear regions, results on the expected curve length distortion, and results on the NTK. As the result of the research in this thesis, we develop multiple experiments and helpful components and make the code for them publicly available

Understanding the Evolution of Linear Regions in Deep Reinforcement Learning

Policies produced by deep reinforcement learning are typically characterised by their learning curves, but they remain poorly understood in many other respects. ReLU-based policies result in a partitioning of the input space into piecewise linear regions. We seek to understand how observed region counts and their densities evolve during deep reinforcement learning using empirical results that span a range of continuous control tasks and policy network dimensions. Intuitively, we may expect that during training, the region density increases in the areas that are frequently visited by the policy, thereby affording fine-grained control. We use recent theoretical and empirical results for the linear regions induced by neural networks in supervised learning settings for grounding and comparison of our results. Empirically, we find that the region density increases only moderately throughout training, as measured along fixed trajectories coming from the final policy. However, the trajectories themselves also increase in length during training, and thus the region densities decrease as seen from the perspective of the current trajectory. Our findings suggest that the complexity of deep reinforcement learning policies does not principally emerge from a significant growth in the complexity of functions observed on-and-around trajectories of the policy.Comment: NeurIPS 2022 camera read