Forward Thinking: Building and Training Neural Networks One Layer at a Time

Forward Thinking: Building Deep Random Forests

The success of deep neural networks has inspired many to wonder whether other learners could benefit from deep, layered architectures. We present a general framework called forward thinking for deep learning that generalizes the architectural flexibility and sophistication of deep neural networks while also allowing for (i) different types of learning functions in the network, other than neurons, and (ii) the ability to adaptively deepen the network as needed to improve results. This is done by training one layer at a time, and once a layer is trained, the input data are mapped forward through…

Efficient Forward Architecture Search

We propose a neural architecture search (NAS) algorithm, Petridish, to iteratively add shortcut connections to existing network layers. The added shortcut connections effectively perform gradient boosting on the augmented layers. The proposed algorithm is motivated by the feature selection algorithm forward stage-wise linear regression, since we consider NAS as a generalization of feature selection for regression, where NAS selects shortcuts among layers instead of selecting features. In order to reduce the number of trials of possible connection combinations, we train jointly all possible con…

Using activation histograms to bound the number of affine regions in ReLU feed-forward neural networks

Several current bounds on the maximal number of affine regions of a ReLU feed-forward neural network are special cases of the framework [1] which relies on layer-wise activation histogram bounds. We analyze and partially solve a problem in algebraic topology the solution of which would fully exploit this framework. Our partial solution already induces slightly tighter bounds and suggests insight in how parameter initialization methods can affect the number of regions. Furthermore, we extend the framework to allow the composition of subnetwork instead of layer-wise activation histogram bounds t…

A Framework for the construction of upper bounds on the number of affine linear regions of ReLU feed-forward neural networks

We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width …

Differentiable Forward and Backward Fixed-Point Iteration Layers

Recently, several studies proposed methods to utilize some classes of optimization problems in designing deep neural networks to encode constraints that conventional layers cannot capture. However, these methods are still in their infancy and require special treatments, such as analyzing the KKT condition, for deriving the backpropagation formula. In this paper, we propose a new layer formulation called the fixed-point iteration (FPI) layer that facilitates the use of more complicated operations in deep networks. The backward FPI layer is also proposed for backpropagation, which is motivated b…

分布关联与上下文内推理：前馈层与注意力层的研究

Distributional Associations vs In-Context Reasoning: A Study of Feed-forward and Attention Layers

本文在受控合成环境中研究Transformer前馈层与注意力层的分工，发现前馈层偏向学习二元词组等分布关联，注意力层专注上下文内推理，并揭示梯度噪声是差异关键，最后在Pythia模型上验证了类似现象。