Mixture-of-Experts (MoE) architectures have emerged as a powerful paradigm for scaling neural networks while maintaining computational efficiency. However, standard MoE implementations rely on two rigid design assumptions: (1) fixed Top-K routing where exactly K experts are activated per token, and (2) uniform expert allocation across all layers. This paper introduces DynaMoE, a novel MoE framework that relaxes both constraints through dynamic token-level expert activation and layer-wise adaptive capacity allocation. DynaMoE introduces a principled routing mechanism where the number of active experts per token varies based on input complexity. Concurrently, the framework implements six distinct scheduling strategies for distributing expert capacity across network depth, including descending, ascending, pyramid, and wave patterns. We theoretically analyze the expressivity gains of dynamic routing and derive bounds on computational efficiency. Through extensive experiments on MNIST, Fashion-MNIST, CIFAR-10 (image classification), and Recycling-the-Web (language modeling) across multiple model scales, we demonstrate that DynaMoE achieves superior parameter efficiency compared to static baselines. Our key finding is that optimal expert schedules are task- and scale-dependent: descending schedules (concentrating capacity in early layers) outperform uniform baselines on image classification. For language modeling, optimal schedules vary by model size, descending for Tiny, ascending for Small, and uniform for Medium. Furthermore, dynamic routing reduces gradient variance during training, leading to improved convergence stability. DynaMoE establishes a new framework for adaptive computation in neural networks, providing principled guidance for MoE architecture design.
National University of Singapore（新加坡国立大学）；RIKEN Center for Advanced Intelligence Project (AIP)（日本理化学研究所先进智能研究中心）；The University of Tokyo（东京大学）
We investigate the adversarial robustness of CNNs from the perspective of channel-wise activations. By comparing \textit{non-robust} (normally trained) and \textit{robustified} (adversarially trained) models, we observe that adversarial training (AT) robustifies CNNs by aligning the channel-wise activations of adversarial data with those of their natural counterparts. However, the channels that are \textit{negatively-relevant} (NR) to predictions are still over-activated when processing adversarial data. Besides, we also observe that AT does not result in similar robustness for all classes. Fo…
Recent works reveal that re-calibrating the intermediate activation of adversarial examples can improve the adversarial robustness of a CNN model. The state of the arts [Baiet al., 2021] and [Yanet al., 2021] explores this feature at the channel level, i.e. the activation of a channel is uniformly scaled by a factor. In this paper, we investigate the intermediate activation manipulation at a more fine-grained level. Instead of uniformly scaling the activation, we individually adjust each element within an activation and thus propose Element-Wise Activation Scaling, dubbed EWAS, to improve CNNs…
Electronics and Telecommunications Research Institute (ETRI)（电子与电信研究所（ETRI））；Ulsan National Institute of Science and Technology (UNIST)（蔚山科学技术院（UNIST））
This work presents deep asymmetric networks with a set of node-wise variant activation functions. The nodes' sensitivities are affected by activation function selections such that the nodes with smaller indices become increasingly more sensitive. As a result, features learned by the nodes are sorted by the node indices in the order of their importance. Asymmetric networks not only learn input features but also the importance of those features. Nodes of lesser importance in asymmetric networks can be pruned to reduce the complexity of the networks, and the pruned networks can be retrained witho…
Deep neural networks paved the way for significant improvements in image visual categorization during the last years. However, even though the tasks are highly varying, differing in complexity and difficulty, existing solutions mostly build on the same architectural decisions. This also applies to the selection of activation functions (AFs), where most approaches build on Rectified Linear Units (ReLUs). In this paper, however, we show that the choice of a proper AF has a significant impact on the classification accuracy, in particular, if fine, subtle details are of relevance. Therefore, we pr…
Element-wise activation functions play a critical role in deep neural networks via affecting the expressivity power and the learning dynamics. Learning-based activation functions have recently gained increasing attention and success. We propose a new perspective of learnable activation function through formulating them with element-wise attention mechanism. In each network layer, we devise an attention module which learns an element-wise, sign-based attention map for the pre-activation feature map. The attention map scales an element based on its sign. Adding the attention module with a rectif…
Université de Montréal（蒙特利尔大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
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 …
SWAP-NAS: Sample-Wise Activation Patterns for Ultra-fast NAS
School of Computing Technologies, RMIT University（RMIT大学计算技术学院）；University of Technology Sydney（悉尼科技大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
Zero-Shot Neural Network Evaluation with Sample-Wise Activation Patterns
School of Computing Technologies, RMIT University（计算技术学院，皇家墨尔本理工大学）；Department of Electronic Engineering and Information Science, University of Science and Technology of China（电子工程与信息科学系，中国科学技术大学）
Deep neural networks (DNNs) are increasingly being deployed in safety-critical systems such as personal healthcare devices and self-driving cars. In such DNN-based systems, error resilience is a top priority since faults in DNN inference could lead to mispredictions and safety hazards. For latency-critical DNN inference on resource-constrained edge devices, it is nontrivial to apply conventional redundancy-based fault tolerance techniques. In this paper, we propose FitAct, a low-cost approach to enhance the error resilience of DNNs by deploying fine-grained post-trainable activation functions.…
Winterlight Labs（温特莱特实验室）；University of Toronto（多伦多大学）；Surgical Safety Technologies（手术安全技术公司）；Li Ka Shing Knowledge Institute, St Michael’s Hospital（圣迈克尔医院李嘉诚知识研究所）；Vector Institute for Artificial Intelligence（矢量人工智能研究所）
We seek to improve the data efficiency of neural networks and present novel implementations of parameterized piece-wise polynomial activation functions. The parameters are the y-coordinates of n+1 Chebyshev nodes per hidden unit and Lagrangian interpolation between the nodes produces the polynomial on [-1, 1]. We show results for different methods of handling inputs outside [-1, 1] on synthetic datasets, finding significant improvements in capacity of expression and accuracy of interpolation in models that compute some form of linear extrapolation from either ends. We demonstrate competitive o…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Lipschitz-constrained neural networks have many applications in machine learning. Since designing and training expressive Lipschitz-constrained networks is very challenging, there is a need for improved methods and a better theoretical understanding. Unfortunately, it turns out that ReLU networks have provable disadvantages in this setting. Hence, we propose to use learnable spline activation functions with at least 3 linear regions instead. We prove that this choice is optimal among all component-wise $1$-Lipschitz activation functions in the sense that no other weight constrained architectur…
Shanghai Jiao Tong University（上海交通大学）；Shanghai Qi Zhi Institute（上海期智研究院）；Alibaba Group（阿里巴巴集团）；Microsoft Research（微软研究院）；Institute for AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院（AIR））
An activation function is an element-wise mathematical function and plays a crucial role in deep neural networks (DNN). Many novel and sophisticated activation functions have been proposed to improve the DNN accuracy but also consume massive memory in the training process with back-propagation. In this study, we propose the nested forward automatic differentiation (Forward-AD), specifically for the element-wise activation function for memory-efficient DNN training. We deploy nested Forward-AD in two widely-used deep learning frameworks, TensorFlow and PyTorch, which support the static and dyna…
University of Manchester（曼彻斯特大学）；The Alan Turing Institute（艾伦·图灵研究所）
Standard neural network architectures are non-linear only by virtue of a simple element-wise activation function, making them both brittle and excessively large. In this paper, we consider methods for making the feed-forward layer more flexible while preserving its basic structure. We develop simple drop-in replacements that learn to adapt their parameterization conditional on the input, thereby increasing statistical efficiency significantly. We present an adaptive LSTM that advances the state of the art for the Penn Treebank and WikiText-2 word-modeling tasks while using fewer parameters and…
We present an approach for the verification of feed-forward neural networks in which all nodes have a piece-wise linear activation function. Such networks are often used in deep learning and have been shown to be hard to verify for modern satisfiability modulo theory (SMT) and integer linear programming (ILP) solvers. The starting point of our approach is the addition of a global linear approximation of the overall network behavior to the verification problem that helps with SMT-like reasoning over the network behavior. We present a specialized verification algorithm that employs this approxim…
Hybrid private inference (PI) protocol, which synergistically utilizes both multi-party computation (MPC) and homomorphic encryption, is one of the most prominent techniques for PI. However, even the state-of-the-art PI protocols are bottlenecked by the non-linear layers, especially the activation functions. Although a standard non-linear activation function can generate higher model accuracy, it must be processed via a costly garbled-circuit MPC primitive. A polynomial activation can be processed via Beaver's multiplication triples MPC primitive but has been incurring severe accuracy drops so…
Purdue University（普渡大学）；Michigan State University（密歇根州立大学）；Texas Tech University（德克萨斯理工大学）
We propose reproducing activation functions (RAFs) to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. In NN training, RAFs can generate neural tangent kernels (NTKs) with a better condition …
Koç University（科奇大学）；Koç University TUPRAS Energy Center (KUTEM)（科奇大学TUPRAS能源中心）
Constructing first-principles models is usually a challenging and time-consuming task due to the complexity of the real-life processes. On the other hand, data-driven modeling, and in particular neural network models often suffer from issues such as overfitting and lack of useful and highquality data. At the same time, embedding trained machine learning models directly into the optimization problems has become an effective and state-of-the-art approach for surrogate optimization, whose performance can be improved by physics-informed training. In this study, it is proposed to upgrade piece-wise…
There has been an explosion of interest in designing high-performance Transformers. While Transformers have delivered significant performance improvements, training such networks is extremely memory intensive owing to storing all intermediate activations that are needed for gradient computation during backpropagation, especially for long sequences. To this end, we present Mesa, a memory-saving training framework for Transformers. Specifically, Mesa uses exact activations during forward pass while storing a low-precision version of activations to reduce memory consumption during training. The l…
University of California, Los Angeles（加州大学洛杉矶分校）；IBM Research（IBM研究院）；Massachusetts Institute of Technology（麻省理工学院）
Finding minimum distortion of adversarial examples and thus certifying robustness in neural network classifiers for given data points is known to be a challenging problem. Nevertheless, recently it has been shown to be possible to give a non-trivial certified lower bound of minimum adversarial distortion, and some recent progress has been made towards this direction by exploiting the piece-wise linear nature of ReLU activations. However, a generic robustness certification for general activation functions still remains largely unexplored. To address this issue, in this paper we introduce CROWN,…
Tsinghua Berkeley Shenzhen Institute, Tsinghua University（清华大学清华伯克利深圳学院）；Tsinghua Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际研究生院）；Key Lab. of Machine Perception (MoE), School of EECS, Peking University（北京大学信息科学技术学院机器感知与智能教育部重点实验室）；School of Information Technology, Deakin University（迪肯大学信息技术学院）；PCL Research Center of Networks and Communications, Peng Cheng Laboratory（鹏城实验室网络通信研究中心）
The study of adversarial examples and their activation has attracted significant attention for secure and robust learning with deep neural networks (DNNs). Different from existing works, in this paper, we highlight two new characteristics of adversarial examples from the channel-wise activation perspective: 1) the activation magnitudes of adversarial examples are higher than that of natural examples; and 2) the channels are activated more uniformly by adversarial examples than natural examples. We find that the state-of-the-art defense adversarial training has addressed the first issue of high…
Sapienza University of Rome（罗马第一大学）；Roma Tre University（罗马第三大学）
The design of activation functions is a growing research area in the field of neural networks. In particular, instead of using fixed point-wise functions (e.g., the rectified linear unit), several authors have proposed ways of learning these functions directly from the data in a non-parametric fashion. In this paper we focus on the kernel activation function (KAF), a recently proposed framework wherein each function is modeled as a one-dimensional kernel model, whose weights are adapted through standard backpropagation-based optimization. One drawback of KAFs is the need to select a single ker…
Research in computational deep learning has directed considerable efforts towards hardware-oriented optimisations for deep neural networks, via the simplification of the activation functions, or the quantization of both activations and weights. The resulting non-differentiability (or even discontinuity) of the networks poses some challenging problems, especially in connection with the learning process. In this paper, we address several questions regarding both the expressivity of quantized neural networks and approximation techniques for non-differentiable networks. First, we answer in the aff…
Layer-wise Quantization for Quantized Optimistic Dual Averaging
National University of Singapore (NUS)（新加坡国立大学）；EPFL（洛桑联邦理工学院）；Neural Magic；University of Oslo (UiO)（奥斯陆大学）；Integreat, Norwegian Centre for Knowledge-driven Machine Learning（挪威知识驱动机器学习中心）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）
The un-rectifying technique expresses a non-linear point-wise activation function as a data-dependent variable, which means that the activation variable along with its input and output can all be employed in optimization. The ReLU network in this study was un-rectified means that the activation functions could be replaced with data-dependent activation variables in the form of equations and constraints. The discrete nature of activation variables associated with un-rectifying ReLUs allows the reformulation of deep learning problems as problems of combinatorial optimization. However, we demonst…
Sapienza University of Rome（罗马第一大学）；University of Cantabria（坎塔布里亚大学）
Gated recurrent neural networks have achieved remarkable results in the analysis of sequential data. Inside these networks, gates are used to control the flow of information, allowing to model even very long-term dependencies in the data. In this paper, we investigate whether the original gate equation (a linear projection followed by an element-wise sigmoid) can be improved. In particular, we design a more flexible architecture, with a small number of adaptable parameters, which is able to model a wider range of gating functions than the classical one. To this end, we replace the sigmoid func…
University of Science and Technology Beijing（北京科技大学）；Hua Xia General Processor Technologies（华夏通用处理器技术公司）；General Processor Technologies（通用处理器技术公司）
Deep neural networks have been applied in many applications exhibiting extraordinary abilities in the field of computer vision. However, complex network architectures challenge efficient real-time deployment and require significant computation resources and energy costs. These challenges can be overcome through optimizations such as network compression. Network compression can often be realized with little loss of accuracy. In some cases accuracy may even improve. This paper provides a survey on two types of network compression: pruning and quantization. Pruning can be categorized as static if…
ExpertWeaver: Unlocking the Inherent MoE in Dense LLMs with GLU Activation Patterns
Zhejiang University（浙江大学）；Shanghai AI Laboratory（上海人工智能实验室）；Shanghai Innovation Institute（上海创新研究院）；Fudan University（复旦大学）；Chinese University of Hong Kong（中国香港大学）
We propose a new optimization method for training feed-forward neural networks. By rewriting the activation function as an equivalent proximal operator, we approximate a feed-forward neural network by adding the proximal operators to the objective function as penalties, hence we call the lifted proximal operator machine (LPOM). LPOM is block multi-convex in all layer-wise weights and activations. This allows us to use block coordinate descent to update the layer-wise weights and activations in parallel. Most notably, we only use the mapping of the activation function itself, rather than its de…
Université Paris Cité（巴黎西岱大学）；EDF R&D（法国电力集团研发中心）；IUF（法国大学研究院）
Data series classification is an important and challenging problem in data science. Explaining the classification decisions by finding the discriminant parts of the input that led the algorithm to some decisions is a real need in many applications. Convolutional neural networks perform well for the data series classification task; though, the explanations provided by this type of algorithm are poor for the specific case of multivariate data series. Addressing this important limitation is a significant challenge. In this paper, we propose a novel method that solves this problem by highlighting …
Brown University（布朗大学）；Massachusetts Institute of Technology（麻省理工学院）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
We propose two approaches of locally adaptive activation functions namely, layer-wise and neuron-wise locally adaptive activation functions, which improve the performance of deep and physics-informed neural networks. The local adaptation of activation function is achieved by introducing a scalable parameter in each layer (layer-wise) and for every neuron (neuron-wise) separately, and then optimizing it using a variant of stochastic gradient descent algorithm. In order to further increase the training speed, an activation slope based slope recovery term is added in the loss function, which furt…
