Threshold activation functions are highly preferable in neural networks due to their efficiency in hardware implementations. Moreover, their mode of operation is more interpretable and resembles that of biological neurons. However, traditional gradient based algorithms such as Gradient Descent cannot be used to train the parameters of neural networks with threshold activations since the activation function has zero gradient except at a single non-differentiable point. To this end, we study weight decay regularized training problems of deep neural networks with threshold activations. We first show that regularized deep threshold network training problems can be equivalently formulated as a standard convex optimization problem, which parallels the LASSO method, provided that the last hidden layer width exceeds a certain threshold. We also derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network. We corroborate our theoretical results with various numerical experiments.
Overwhelming theoretical and empirical evidence shows that mildly overparametrized neural networks -- those with more connections than the size of the training data -- are often able to memorize the training data with $100\%$ accuracy. This was rigorously proved for networks with sigmoid activation functions and, very recently, for ReLU activations. Addressing a 1988 open question of Baum, we prove that this phenomenon holds for general multilayered perceptrons, i.e. neural networks with threshold activation functions, or with any mix of threshold and ReLU activations. Our construction is prob…
In past few years, linear rectified unit activation functions have shown its significance in the neural networks, surpassing the performance of sigmoid activations. RELU (Nair & Hinton, 2010), ELU (Clevert et al., 2015), PRELU (He et al., 2015), LRELU (Maas et al., 2013), SRELU (Jin et al., 2016), ThresholdedRELU, all these linear rectified activation functions have its own significance over others in some aspect. Most of the time these activation functions suffer from bias shift problem due to non-zero output mean, and high weight update problem in deep complex networks due to unit gradient, …
Universiti Tun Hussein Onn Malaysia（马来西亚敦胡先翁大学）；Institute of Engineering and Management（工程与管理学院）
Activation functions are essential for deep learning methods to learn and perform complex tasks such as image classification. Rectified Linear Unit (ReLU) has been widely used and become the default activation function across the deep learning community since 2012. Although ReLU has been popular, however, the hard zero property of the ReLU has heavily hindered the negative values from propagating through the network. Consequently, the deep neural network has not been benefited from the negative representations. In this work, an activation function called Flatten-T Swish (FTS) that leverage the…
University of Oxford（牛津大学）；The Alan Turing Institute（阿兰·图灵研究所）
Deep Convolutional Sparse Coding (D-CSC) is a framework reminiscent of deep convolutional neural networks (DCNNs), but by omitting the learning of the dictionaries one can more transparently analyse the role of the activation function and its ability to recover activation paths through the layers. Papyan, Romano, and Elad conducted an analysis of such an architecture, demonstrated the relationship with DCNNs and proved conditions under which the D-CSC is guaranteed to recover specific activation paths. A technical innovation of their work highlights that one can view the efficacy of the ReLU n…
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Binarization of neural network models is considered as one of the promising methods to deploy deep neural network models on resource-constrained environments such as mobile devices. However, Binary Neural Networks (BNNs) tend to suffer from severe accuracy degradation compared to the full-precision counterpart model. Several techniques were proposed to improve the accuracy of BNNs. One of the approaches is to balance the distribution of binary activations so that the amount of information in the binary activations becomes maximum. Based on extensive analysis, in stark contrast to previous work…
We address the problem of reconstructing sparse signals from noisy and compressive measurements using a feed-forward deep neural network (DNN) with an architecture motivated by the iterative shrinkage-thresholding algorithm (ISTA). We maintain the weights and biases of the network links as prescribed by ISTA and model the nonlinear activation function using a linear expansion of thresholds (LET), which has been very successful in image denoising and deconvolution. The optimal set of coefficients of the parametrized activation is learned over a training dataset containing measurement-sparse sig…
While the optimization problem behind deep neural networks is highly non-convex, it is frequently observed in practice that training deep networks seems possible without getting stuck in suboptimal points. It has been argued that this is the case as all local minima are close to being globally optimal. We show that this is (almost) true, in fact almost all local minima are globally optimal, for a fully connected network with squared loss and analytic activation function given that the number of hidden units of one layer of the network is larger than the number of training points and the networ…
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Google（谷歌）
Giving provable guarantees for learning neural networks is a core challenge of machine learning theory. Most prior work gives parameter recovery guarantees for one hidden layer networks, however, the networks used in practice have multiple non-linear layers. In this work, we show how we can strengthen such results to deeper networks -- we address the problem of uncovering the lowest layer in a deep neural network under the assumption that the lowest layer uses a high threshold before applying the activation, the upper network can be modeled as a well-behaved polynomial and the input distributi…
ReLU$^2$ Wins: Discovering Efficient Activation Functions for Sparse LLMs
Tsinghua University（清华大学）；Shanghai Jiao Tong University（上海交通大学）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；Renmin University of China（中国人民大学）
Deep learning models are often successfully trained using gradient descent, despite the worst case hardness of the underlying non-convex optimization problem. The key question is then under what conditions can one prove that optimization will succeed. Here we provide a strong result of this kind. We consider a neural net with one hidden layer and a convolutional structure with no overlap and a ReLU activation function. For this architecture we show that learning is NP-complete in the general case, but that when the input distribution is Gaussian, gradient descent converges to the global optimu…
We propose a novel Shapley value approach to help address neural networks' interpretability and "vanishing gradient" problems. Our method is based on an accurate analytical approximation to the Shapley value of a neuron with ReLU activation. This analytical approximation admits a linear propagation of relevance across neural network layers, resulting in a simple, fast and sensible interpretation of neural networks' decision making process. We then derived a globally continuous and non-vanishing Shapley gradient, which can replace the conventional gradient in training neural network layers with…
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）
We provide new theoretical insights on why over-parametrization is effective in learning neural networks. For a $k$ hidden node shallow network with quadratic activation and $n$ training data points, we show as long as $ k \ge \sqrt{2n}$, over-parametrization enables local search algorithms to find a \emph{globally} optimal solution for general smooth and convex loss functions. Further, despite that the number of parameters may exceed the sample size, using theory of Rademacher complexity, we show with weight decay, the solution also generalizes well if the data is sampled from a regular distr…
fortiss - Research Institute of the Free State of Bavaria（fortiss——巴伐利亚自由州研究院）；DENSO CORPORATION（电装公司）
For using neural networks in safety critical domains, it is important to know if a decision made by a neural network is supported by prior similarities in training. We propose runtime neuron activation pattern monitoring - after the standard training process, one creates a monitor by feeding the training data to the network again in order to store the neuron activation patterns in abstract form. In operation, a classification decision over an input is further supplemented by examining if a pattern similar (measured by Hamming distance) to the generated pattern is contained in the monitor. If t…
University of Oulu（奥卢大学）；National University of Defense Technology（国防科技大学）
Binary neural networks (BNNs) constrain weights and activations to +1 or -1 with limited storage and computational cost, which is hardware-friendly for portable devices. Recently, BNNs have achieved remarkable progress and been adopted into various fields. However, the performance of BNNs is sensitive to activation distribution. The existing BNNs utilized the Sign function with predefined or learned static thresholds to binarize activations. This process limits representation capacity of BNNs since different samples may adapt to unequal thresholds. To address this problem, we propose a dynamic…
In this paper we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the number of observations are fewer than the number of parameters in the model. We show that with quadratic activations the optimization landscape of training such shallow neural networks has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. This result holds for an arbitrary training data of input/output pairs. For differentiable…
The training of two-layer neural networks with nonlinear activation functions is an important non-convex optimization problem with numerous applications and promising performance in layerwise deep learning. In this paper, we develop exact convex optimization formulations for two-layer neural networks with second degree polynomial activations based on semidefinite programming. Remarkably, we show that semidefinite lifting is always exact and therefore computational complexity for global optimization is polynomial in the input dimension and sample size for all input data. The developed convex fo…
Bio-inspired spiking neural networks (SNNs), operating with asynchronous binary signals (or spikes) distributed over time, can potentially lead to greater computational efficiency on event-driven hardware. The state-of-the-art SNNs suffer from high inference latency, resulting from inefficient input encoding, and sub-optimal settings of the neuron parameters (firing threshold, and membrane leak). We propose DIET-SNN, a low-latency deep spiking network that is trained with gradient descent to optimize the membrane leak and the firing threshold along with other network parameters (weights). The …
Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We prove that if an activation function satisfies some mild conditions and number of neurons in a two-layered fully connected neural network with this activation function is beyond a certain threshold, then gradient descent on quadratic loss function finds the optimal weights of input layer for global minima in linear time. This threshold value is an improvement over previously obtained values. We hypothesise that this bound cannot be improved by the method we are using in this work.
Robotics Institute, Carnegie Mellon University（卡内基梅隆大学机器人研究所）
Stochastic Gradient Descent (SGD) is the central workhorse for training modern CNNs. Although giving impressive empirical performance it can be slow to converge. In this paper we explore a novel strategy for training a CNN using an alternation strategy that offers substantial speedups during training. We make the following contributions: (i) replace the ReLU non-linearity within a CNN with positive hard-thresholding, (ii) reinterpret this non-linearity as a binary state vector making the entire CNN linear if the multi-layer support is known, and (iii) demonstrate that under certain conditions …
Adacc: An Adaptive Framework Unifying Compression and Activation Recomputation for LLM Training
The State Key Laboratory of Blockchain and Data Security（区块链与数据安全国家重点实验室）；Hangzhou High-Tech Zone (Binjiang) Institute of Blockchain and Data Security（杭州高新技术区（滨江）区块链与数据安全研究院）；Shanghai Jiao Tong University（上海交通大学）；Huawei Cloud（华为云）；Zhejiang Lab（浙江实验室）
