The distribution of a neural network's latent representations has been successfully used to detect out-of-distribution (OOD) data. This work investigates whether this distribution moreover correlates with a model's epistemic uncertainty, thus indicates its ability to generalise to novel inputs. We first empirically verify that epistemic uncertainty can be identified with the surprise, thus the negative log-likelihood, of observing a particular latent representation. Moreover, we demonstrate that the output-conditional distribution of hidden representations also allows quantifying aleatoric uncertainty via the entropy of the predictive distribution. We analyse epistemic and aleatoric uncertainty inferred from the representations of different layers and conclude that deeper layers lead to uncertainty with similar behaviour as established - but computationally more expensive - methods (e.g. deep ensembles). While our approach does not require modifying the training process, we follow prior work and experiment with an additional regularising loss that increases the information in the latent representations. We find that this leads to improved OOD detection of epistemic uncertainty at the cost of ambiguous calibration close to the data distribution. We verify our findings on both classification and regression models.
Activation-Space Uncertainty Quantification for Pretrained Networks
Department of Engineering, University of Cambridge, Cambridge, UK（剑桥大学工程系）；Mathematical Institute（数学研究所）；Oxford-Man Institute, University of Oxford, Oxford, UK（牛津大学奥克斯曼研究所）
EPFL（洛桑联邦理工学院）；Idiap Research Institute（Idiap 研究所）；Singapore Management University（新加坡管理大学）；U.S. Army Research Laboratory（美国陆军研究实验室）
In this paper, we introduce the concept of Prior Activation Distribution (PAD) as a versatile and general technique to capture the typical activation patterns of hidden layer units of a Deep Neural Network used for classification tasks. We show that the combined neural activations of such a hidden layer have class-specific distributional properties, and then define multiple statistical measures to compute how far a test sample's activations deviate from such distributions. Using a variety of benchmark datasets (including MNIST, CIFAR10, Fashion-MNIST & notMNIST), we show how such PAD-based mea…
We introduce a new family of non-linear neural network activation functions that mimic the properties induced by the widely-used Matérn family of kernels in Gaussian process (GP) models. This class spans a range of locally stationary models of various degrees of mean-square differentiability. We show an explicit link to the corresponding GP models in the case that the network consists of one infinitely wide hidden layer. In the limit of infinite smoothness the Matérn family results in the RBF kernel, and in this case we recover RBF activations. Matérn activation functions result in similar app…
What do data tell us about physics-and what don't they tell us? There has been a surge of interest in using machine learning models to discover governing physical laws such as differential equations from data, but current methods lack uncertainty quantification to communicate their credibility. This work addresses this shortcoming from a Bayesian perspective. We introduce a novel model comprising "leaf" modules that learn to represent distinct experiments' spatiotemporal functional data as neural networks and a single "root" module that expresses a nonparametric distribution over their governi…
Although neural networks are capable of reaching astonishing performances on a wide variety of contexts, properly training networks on complicated tasks requires expertise and can be expensive from a computational perspective. In industrial applications, data coming from an open-world setting might widely differ from the benchmark datasets on which a network was trained. Being able to monitor the presence of such variations without retraining the network is of crucial importance. In this article, we develop a method to monitor trained neural networks based on the topological properties of thei…
Isik University（伊希克大学）；University of Stavanger（斯塔万格大学）
In standard Deep Neural Network (DNN) based classifiers, the general convention is to omit the activation function in the last (output) layer and directly apply the softmax function on the logits to get the probability scores of each class. In this type of architectures, the loss value of the classifier against any output class is directly proportional to the difference between the final probability score and the label value of the associated class. Standard White-box adversarial evasion attacks, whether targeted or untargeted, mainly try to exploit the gradient of the model loss function to c…
Resting Neurons, Active Insights: Robustifying Activation Sparsity in LLMs via Spontaneity
IBM Thomas J. Watson Research Center, Yorktown Heights, USA（IBM 托马斯·J·沃森研究中心，美国Yorktown Heights）；Halıcıoğlu Data Science Institute, UC San Diego, La Jolla, USA（哈利奇欧数据科学研究所，美国UC圣地亚哥La Jolla）；Stony Brook University, Stony Brook, USA（史泰文·布鲁克大学，美国Stony Brook）
Visualizing Uncertainty: Spatial Maps of Missing and Conflicting Evidence in Deep Learning
University of the District of Columbia（哥伦比亚特区大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；Virginia Tech（弗吉尼亚理工大学）；U.S. Army DEVCOM Army Research Laboratory（美国陆军DEVCOM陆军研究实验室）；University of Oslo（奥斯陆大学）；Bowie State University（鲍伊州立大学）
Universidad de Cantabria（坎塔布里亚大学）；Instituto de Investigación Valdecilla (IDIVAL)（瓦尔迪西利亚研究所（IDIVAL））；CIBER-BBN（生物工程、生物材料与纳米医学生物医学研究网络中心）；Dartmouth College（达特茅斯学院）
The following work presents how autoencoding all the possible hidden activations of a network for a given problem can provide insight about its structure, behavior, and vulnerabilities. The method, termed self-introspection, can show that a trained model showcases similar activation patterns (albeit randomly distributed due to initialization) when shown data belonging to the same category, and classification errors occur in fringe areas where the activations are not as clearly defined, suggesting some form of random, slowly varying, implicit encoding occurring within deep networks, that can be…
In this paper, we propose a defence strategy to improve adversarial robustness by incorporating hidden layer representation. The key of this defence strategy aims to compress or filter input information including adversarial perturbation. And this defence strategy can be regarded as an activation function which can be applied to any kind of neural network. We also prove theoretically the effectiveness of this defense strategy under certain conditions. Besides, incorporating hidden layer representation we propose three types of adversarial attacks to generate three types of adversarial examples…
Seoul National University（首尔国立大学）；Hanyang University（汉阳大学）；NAVER Corp（NAVER公司）
An activation boundary for a neuron refers to a separating hyperplane that determines whether the neuron is activated or deactivated. It has been long considered in neural networks that the activations of neurons, rather than their exact output values, play the most important role in forming classification friendly partitions of the hidden feature space. However, as far as we know, this aspect of neural networks has not been considered in the literature of knowledge transfer. In this paper, we propose a knowledge transfer method via distillation of activation boundaries formed by hidden neuron…
Single layer feedforward networks with random weights are successful in a variety of classification and regression problems. These networks are known for their non-iterative and fast training algorithms. A major drawback of these networks is that they require a large number of hidden units. In this paper, we propose the use of multi-activation hidden units. Such units increase the number of tunable parameters and enable formation of complex decision surfaces, without increasing the number of hidden units. We experimentally show that multi-activation hidden units can be used either to improve t…
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence（伯努利数学、计算机科学与人工智能研究所）；University of Groningen（格罗宁根大学）
We study layered neural networks of rectified linear units (ReLU) in a modelling framework for stochastic training processes. The comparison with sigmoidal activation functions is in the center of interest. We compute typical learning curves for shallow networks with K hidden units in matching student teacher scenarios. The systems exhibit sudden changes of the generalization performance via the process of hidden unit specialization at critical sizes of the training set. Surprisingly, our results show that the training behavior of ReLU networks is qualitatively different from that of networks …
Deep neural networks are in the limelight of machine learning with their excellent performance in many data-driven applications. However, they can lead to inaccurate predictions when queried in out-of-distribution data points, which can have detrimental effects especially in sensitive domains, such as healthcare and transportation, where erroneous predictions can be very costly and/or dangerous. Subsequently, quantifying the uncertainty of the output of a neural network is often leveraged to evaluate the confidence of its predictions, and ensemble models have proved to be effective in measurin…
Kansas State University（堪萨斯州立大学）；University of Hartford（哈特福德大学）
One of the current key challenges in Explainable AI is in correctly interpreting activations of hidden neurons. It seems evident that accurate interpretations thereof would provide insights into the question what a deep learning system has internally \emph{detected} as relevant on the input, thus lifting some of the black box character of deep learning systems. The state of the art on this front indicates that hidden node activations appear to be interpretable in a way that makes sense to humans, at least in some cases. Yet, systematic automated methods that would be able to first hypothesize …
We analyze the joint probability distribution on the lengths of the vectors of hidden variables in different layers of a fully connected deep network, when the weights and biases are chosen randomly according to Gaussian distributions, and the input is in $\{ -1, 1\}^N$. We show that, if the activation function $ϕ$ satisfies a minimal set of assumptions, satisfied by all activation functions that we know that are used in practice, then, as the width of the network gets large, the `length process' converges in probability to a length map that is determined as a simple function of the variances …
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the sample complexity of learning one-hidden-layer convolutional neural networks (CNNs) with non-overlapping filters. We propose a novel algorithm called approximate gradient descent for training CNNs, and show that, with high probability, the proposed algorithm with random initialization grants a linear convergence to the ground-truth parameters up to statistical precision. Compared with existing work, our result applies to general non-trivial, monotonic and Lipschitz continuous activation functions including ReLU, Leaky ReLU, Sigmod and Softplus etc. Moreover, our sample complexity …
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
Recurrent neural networks (RNNs) provide state-of-the-art performance in processing sequential data but are memory intensive to train, limiting the flexibility of RNN models which can be trained. Reversible RNNs---RNNs for which the hidden-to-hidden transition can be reversed---offer a path to reduce the memory requirements of training, as hidden states need not be stored and instead can be recomputed during backpropagation. We first show that perfectly reversible RNNs, which require no storage of the hidden activations, are fundamentally limited because they cannot forget information from the…
Dropout has been commonly used to quantify prediction uncertainty, i.e, the variations of model predictions on a given input example. However, using dropout in practice can be expensive as it requires running dropout inferences many times. In this paper, we study how to estimate dropout prediction uncertainty in a resource-efficient manner. We demonstrate that we can use neuron activation strengths to estimate dropout prediction uncertainty under different dropout settings and on a variety of tasks using three large datasets, MovieLens, Criteo, and EMNIST. Our approach provides an inference-on…
Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）
Here is presented an analysis of an autoencoder with binary activations $\{0, 1\}$ and binary $\{0, 1\}$ random weights. Such set up puts this model at the intersection of different fields: neuroscience, information theory, sparse coding, and machine learning. It is shown that the sparse activation of the hidden layer arises naturally in order to preserve information between layers. Furthermore, with a large enough hidden layer, it is possible to get zero reconstruction error for any input just by varying the thresholds of neurons. The model preserves the similarity of inputs at the hidden lay…
In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show $\mathit{local~linear~convergence}$ guarantees of gradient descent under a resampling rule. For homogeneou…
The Chinese University of Hong Kong（香港中文大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Does over-parameterization eliminate sub-optimal local minima for neural networks? An affirmative answer was given by a classical result in [59] for 1-hidden-layer wide neural networks. A few recent works have extended the setting to multi-layer neural networks, but none of them has proved every local minimum is global. Why is this result never extended to deep networks? In this paper, we show that the task is impossible because the original result for 1-hidden-layer network in [59] can not hold. More specifically, we prove that for any multi-layer network with generic input data and non-linea…
MILA, Université de Montréal（蒙特利尔大学 MILA）；École Polytechnique de Montréal（蒙特利尔理工学院）
We propose zoneout, a novel method for regularizing RNNs. At each timestep, zoneout stochastically forces some hidden units to maintain their previous values. Like dropout, zoneout uses random noise to train a pseudo-ensemble, improving generalization. But by preserving instead of dropping hidden units, gradient information and state information are more readily propagated through time, as in feedforward stochastic depth networks. We perform an empirical investigation of various RNN regularizers, and find that zoneout gives significant performance improvements across tasks. We achieve competit…
We analyze the hidden activations of neural network policies of deep reinforcement learning (RL) agents and show, empirically, that it's possible to know a priori if a state representation will lend itself to fast learning. RL agents in high-dimensional states have two main learning burdens: (1) to learn an action-selection policy and (2) to learn to discern between useful and non-useful information in a given state. By learning a latent representation of these high-dimensional states with an auxiliary model, the latter burden is effectively removed, thereby leading to accelerated training pro…
For one-hidden-layer ReLU networks, we prove that all differentiable local minima are global inside differentiable regions. We give the locations and losses of differentiable local minima, and show that these local minima can be isolated points or continuous hyperplanes, depending on an interplay between data, activation pattern of hidden neurons and network size. Furthermore, we give necessary and sufficient conditions for the existence of saddle points as well as non-differentiable local minima, and their locations if they exist.
We propose and analyze a new family of algorithms for training neural networks with ReLU activations. Our algorithms are based on the technique of alternating minimization: estimating the activation patterns of each ReLU for all given samples, interleaved with weight updates via a least-squares step. The main focus of our paper are 1-hidden layer networks with $k$ hidden neurons and ReLU activation. We show that under standard distributional assumptions on the $d-$dimensional input data, our algorithm provably recovers the true `ground truth' parameters in a linearly convergent fashion. This h…
The University of Sydney（悉尼大学）；University of Science and Technology of China（中国科学技术大学）
Understanding the loss surface of a neural network is fundamentally important to the understanding of deep learning. This paper presents how piecewise linear activation functions substantially shape the loss surfaces of neural networks. We first prove that {\it the loss surfaces of many neural networks have infinite spurious local minima} which are defined as the local minima with higher empirical risks than the global minima. Our result demonstrates that the networks with piecewise linear activations possess substantial differences to the well-studied linear neural networks. This result holds…
Georgia Institute of Technology（佐治亚理工学院）；JD.COM American Technologies Corporation（京东美国技术公司）
Many neural networks use the tanh activation function, however when given a probability distribution as input, the problem of computing the output distribution in neural networks with tanh activation has not yet been addressed. One important example is the initialization of the echo state network in reservoir computing, where random initialization of the reservoir requires time to wash out the initial conditions, thereby wasting precious data and computational resources. Motivated by this problem, we propose a novel solution utilizing a moment based approach to propagate uncertainty through an…
Indian Institute of Technology, Bombay（印度理工学院孟买分校）
Current research suggests that the key factors in designing neural network architectures involve choosing number of filters for every convolution layer, number of hidden neurons for every fully connected layer, dropout and pruning. The default activation function in most cases is the ReLU, as it has empirically shown faster training convergence. We explore whether ReLU is the best choice if one is aiming to desire better modularity structure within a neural network.
PRISM: Recovering Instruction Sets from Language Model Activations
Center for Cybersecurity Systems & Networks, Amrita Vishwa Vidyapeetham（阿姆里塔·维什瓦·维迪亚佩瑟姆网络安全系统与网络中心）；Microsoft（微软）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）
Kavli Institute for Systems Neuroscience and Centre for Neural Computation, Norwegian University of Science and Technology（挪威科技大学 Kavli 系统神经科学研究所与神经计算中心）；SISSA—Cognitive Neuroscience（SISSA 认知神经科学）
We study the type of distributions that Restricted Boltzmann Machines (RBMs) with different activation functions can express by investigating the effect of the activation function of the hidden nodes on the marginal distribution they impose on observed binary nodes. We report an exact expression for these marginals in the form of a model of interacting binary variables with the explicit form of the interactions depending on the hidden node activation function. We study the properties of these interactions in detail and evaluate how the accuracy with which the RBM approximates distributions ove…
Neural network models are known to reinforce hidden data biases, making them unreliable and difficult to interpret. We seek to build models that `know what they do not know' by introducing inductive biases in the function space. We show that periodic activation functions in Bayesian neural networks establish a connection between the prior on the network weights and translation-invariant, stationary Gaussian process priors. Furthermore, we show that this link goes beyond sinusoidal (Fourier) activations by also covering triangular wave and periodic ReLU activation functions. In a series of expe…
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…
In a function approximation with a neural network, an input dataset is mapped to an output index by optimizing the parameters of each hidden-layer unit. For a unary function, we present constraints on the parameters and its second derivative by constructing a continuum version of a one-hidden-layer neural network with the rectified linear unit (ReLU) activation function. The network is accurately implemented because the constraints decrease the degrees of freedom of the parameters. We also explain the existence of a smooth binary function that cannot be precisely represented by any such neural…
Recent Progress has shown that exploitation of hidden layer neurons in convolution neural networks incorporating with a carefully designed activation function can yield better classification results in the field of computer vision. The paper firstly introduces a novel deep learning architecture aiming to mitigate the gradient-vanishing problem, in which the earlier hidden layer neurons could be directly connected with the last hidden layer and feed into the last layer for classification. We then design a generalized linear rectifier function as the activation function that can approximate arbi…
Many neural network architectures rely on the choice of the activation function for each hidden layer. Given the activation function, the neural network is trained over the bias and the weight parameters. The bias catches the center of the activation, and the weights capture the scale. Here we propose to train the network over a shape parameter as well. This view allows each neuron to tune its own activation function and adapt the neuron curvature towards a better prediction. This modification only adds one further equation to the back-propagation for each neuron. Re-formalizing activation fun…
Université Paris-Saclay（巴黎-萨克雷大学）；CNRS（法国国家科学研究中心）；CEA（法国原子能和替代能源委员会）；Institut de physique théorique（理论物理研究所）；Courant Institute, New York University（纽约大学柯朗研究所）
We study the dynamics of optimization and the generalization properties of one-hidden layer neural networks with quadratic activation function in the over-parametrized regime where the layer width $m$ is larger than the input dimension $d$. We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width $m^*\le m$. We describe how the empirical loss landscape is affected by the number $n$ of data samples and the width $m^*$ of the teacher network. In particular we determine how the probability that there be no spurious minima …
We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because of the identity mapping inherent to ResNets, our network has alternating layers of dimension one and $d$. This stands in sharp contrast to fully connected networks, which are not universal approximators if their width is the input dimension $d$ [Lu et al, 2017; Hanin and Sellke, 2017]. Hence, our result implies an increase in representational power for narrow …
We prove a negative result for the approximation of functions defined on compact subsets of $\mathbb{R}^d$ (where $d \geq 2$) using feedforward neural networks with one hidden layer and arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions that are as difficult to approximate using these neural networks as one may want. We also demonstrate an analogous result (for general $d \in \mathbb{N}$) for neural networks with an \emph{arbitrary} number of hidden layers, for activation functions that are either rational functions or continuous splin…
Neural networks with random hidden nodes have gained increasing interest from researchers and practical applications. This is due to their unique features such as very fast training and universal approximation property. In these networks the weights and biases of hidden nodes determining the nonlinear feature mapping are set randomly and are not learned. Appropriate selection of the intervals from which weights and biases are selected is extremely important. This topic has not yet been sufficiently explored in the literature. In this work a method of generating random weights and biases is pro…
In this work, we propose the marginal structured SVM (MSSVM) for structured prediction with hidden variables. MSSVM properly accounts for the uncertainty of hidden variables, and can significantly outperform the previously proposed latent structured SVM (LSSVM; Yu & Joachims (2009)) and other state-of-art methods, especially when that uncertainty is large. Our method also results in a smoother objective function, making gradient-based optimization of MSSVMs converge significantly faster than for LSSVMs. We also show that our method consistently outperforms hidden conditional random fields (HCR…
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 s…
Princeton University（普林斯顿大学）；Facebook AI Research（脸书人工智能研究院）；Stanford University（斯坦福大学）
We show that the gradient descent algorithm provides an implicit regularization effect in the learning of over-parameterized matrix factorization models and one-hidden-layer neural networks with quadratic activations. Concretely, we show that given $\tilde{O}(dr^{2})$ random linear measurements of a rank $r$ positive semidefinite matrix $X^{\star}$, we can recover $X^{\star}$ by parameterizing it by $UU^\top$ with $U\in \mathbb R^{d\times d}$ and minimizing the squared loss, even if $r \ll d$. We prove that starting from a small initialization, gradient descent recovers $X^{\star}$ in $\tilde{…
From Associations to Activations: Comparing Behavioral and Hidden-State Semantic Geometry in LLMs
Computational Modelling Lab, Humboldt-Universität zu Berlin, Berlin, Germany（洪堡-柏林大学计算建模实验室）；Department of Education and Psychology, Freie Universität Berlin, Berlin, Germany（柏林自由大学教育与心理学系）；Department for AI in Society, Science, and Technology, Zuse Institute Berlin, Germany（柏林祖克研究所人工智能与社会、科学与技术部门）；Institute of Mathematics, Technische Universitat Berlin, Germany（柏林技术大学数学研究所）
