On the number of response regions of deep feed forward networks with piece-wise linear activations (1312.6098v5)

Abstract: This paper explores the complexity of deep feedforward networks with linear pre-synaptic couplings and rectified linear activations. This is a contribution to the growing body of work contrasting the representational power of deep and shallow network architectures. In particular, we offer a framework for comparing deep and shallow models that belong to the family of piecewise linear functions based on computational geometry. We look at a deep rectifier multi-layer perceptron (MLP) with linear outputs units and compare it with a single layer version of the model. In the asymptotic regime, when the number of inputs stays constant, if the shallow model has $kn$ hidden units and $n_0$ inputs, then the number of linear regions is $O(k^{n_0}n^{n_0})$. For a $k$ layer model with $n$ hidden units on each layer it is $\Omega(\left\lfloor {n}/{n_0}\right\rfloor^{k-1}n^{n_0})$. The number $\left\lfloor{n}/{n_0}\right\rfloor^{k-1}$ grows faster than $k^{n_0}$ when $n$ tends to infinity or when $k$ tends to infinity and $n \geq 2n_0$. Additionally, even when $k$ is small, if we restrict $n$ to be $2n_0$, we can show that a deep model has considerably more linear regions that a shallow one. We consider this as a first step towards understanding the complexity of these models and specifically towards providing suitable mathematical tools for future analysis.

• The paper introduces a computational-geometry framework to bound the number of linear response regions in deep ReLU networks.
• It demonstrates that deep networks exponentially outperform shallow ones in representational capacity given equivalent hidden units.
• The analysis implies that increased network depth and width enhance non-linear decision boundary complexity, guiding future neural architecture design.

Analyzing the Representational Complexity of Deep Feedforward Networks with Piecewise Linear Activations

The paper "On the number of response regions of deep feedforward networks with piecewise linear activations" presents a detailed examination of the complexity and representational power of deep feedforward networks, focusing specifically on networks that utilize rectified linear unit (ReLU) activations. This paper contributes to the broader ongoing discourse contrasting the depth of neural network architectures, seeking to quantify the potential benefits of deploying deep over shallow architectures.

Summary of Results

The paper introduces a thorough framework grounded in computational geometry to quantify the representational capabilities of deep rectifier multi-layer perceptrons (MLPs). A central focus of the paper is on elucidating how the number of linear regions formed by a deep network compares to those formed by a shallow equivalent, especially given identical numbers of hidden units. Notably, the authors demonstrate that the number of response regions grows substantially faster in deep models as opposed to shallow ones. This increase is more significant in scenarios where the model has a fixed input dimensionality but varies in depth or layer width.

Mathematically, for a shallow model with $kn$ hidden units and $n_0$ inputs, the number of linear regions is bounded by $O(k^{n_0}n^{n_0})$, while for a deep model with $k$ layers and $n$ units per layer, it is bound by $$\Omega(\left\lfloor {n}/{n_0}\right\rfloor^{k-1}n^{n_0})$$. The paper underscores that as either the layer width ($n$) or number of layers ($k$) increases, the number of linear regions proliferates significantly, inherently indicating a higher complexity and potential for expressiveness in deeper configurations.

Methodological Insights

The authors employ theoretical tools from computational geometry to compute these upper bounds, highlighting hyperplane arrangements and vector space partitioning as key underpinning concepts. This geometric perspective facilitates evaluating complex network behavior in terms of linear decision boundaries, akin to evaluating hyperplane arrangements in linear algebra and Geometry. Moreover, this analytical scaffold allows the extrapolation of results to wider classes of piecewise linear functions beyond strict ReLU MLPs, suggesting applications for other architectural frameworks like maxout networks.

Implications and Speculations on Future AI Developments

The results presented bear significant implications for both theoretical understanding and practical applications of deep learning. On a theoretical level, these findings substantiate claims regarding the superior expressiveness of deep networks when compared to their shallow counterparts, especially emphasizing the efficiency in parameter usage—an essential consideration for neural network design and optimization.

Practically, these insights can drive developments in neural architecture design, particularly in optimizing depth-oriented architectures for tasks with complex, highly non-linear decision boundaries. With the findings suggesting that deeper networks can better approximate certain classes of functions, this holds potential for advancing applications in computer vision, natural language processing, and beyond.

In prognosticating the research landscape, this paper lays foundational groundwork for exploring novel piecewise linear units and non-standard architectures that may better exploit this expressiveness. Future research could delve into alternative activation schemes that maintain or enhance linear region proliferation while optimizing for other performance metrics, such as robustness or interpretability.

In conclusion, the paper contributes significantly to the understanding of deep networks' representational dynamics, combining rigorous geometric perspective with practical insights that enhance our broader grasp of neural computation models.