Softmax function is widely used in artificial neural networks for multiclass classification, multilabel classification, attention mechanisms, etc. However, its efficacy is often questioned in literature. The log-softmax loss has been shown to belong to a more generic class of loss functions, called spherical family, and its member log-Taylor softmax loss is arguably the best alternative in this class. In another approach which tries to enhance the discriminative nature of the softmax function, soft-margin softmax (SM-softmax) has been proposed to be the most suitable alternative. In this work, we investigate Taylor softmax, SM-softmax and our proposed SM-Taylor softmax, an amalgamation of the earlier two functions, as alternatives to softmax function. Furthermore, we explore the effect of expanding Taylor softmax up to ten terms (original work proposed expanding only to two terms) along with the ramifications of considering Taylor softmax to be a finite or infinite series during backpropagation. Our experiments for the image classification task on different datasets reveal that there is always a configuration of the SM-Taylor softmax function that outperforms the normal softmax function and its other alternatives.
Beijing Jiaotong University（北京交通大学）；China University of Petroleum (East China)（中国石油大学（华东））
Softmax is widely used in neural networks for multiclass classification, gate structure and attention mechanisms. The statistical assumption that the input is normal distributed supports the gradient stability of Softmax. However, when used in attention mechanisms such as transformers, since the correlation scores between embeddings are often not normally distributed, the gradient vanishing problem appears, and we prove this point through experimental confirmation. In this work, we suggest that replacing the exponential function by periodic functions, and we delve into some potential periodic …
Converting an n-dimensional vector to a probability distribution over n objects is a commonly used component in many machine learning tasks like multiclass classification, multilabel classification, attention mechanisms etc. For this, several probability mapping functions have been proposed and employed in literature such as softmax, sum-normalization, spherical softmax, and sparsemax, but there is very little understanding in terms how they relate with each other. Further, none of the above formulations offer an explicit control over the degree of sparsity. To address this, we develop a unifi…
The Implicit Bias of Depth: From Neural Collapse to Softmax Codes
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Department of Electrical and Computer Engineering, University of British Columbia（不列颠哥伦比亚大学电气与计算机工程系）
Statistical Advantage of Softmax Attention: Insights from Single-Location Regression
Institute of Mathematics, EPFL（EPFL数学研究所）；Statistical Physics of Computation Laboratory, École Polytechnique Fédérale de Lausanne（计算统计物理实验室，瑞士联邦理工学院）；Inria, École Normale Supérieure, PSL Research University（Inria，巴黎高等师范学院，PSL研究大学）；Laboratoire de Mathématiques d’Orsay, Université Paris Saclay and Institut Universitaire de France（奥赛数学实验室，巴黎萨克雷大学及法国国家科学院）；Departement d’Informatique, École Normale Supérieure, PSL & CNRS（信息科学系，巴黎高等师范学院，PSL与CNRS）
School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；ILLC, University of Amsterdam（阿姆斯特丹大学语言、逻辑与计算研究所）；Department of Computational Linguistics, University of Zurich（苏黎世大学计算语言学系）
Recently, it has been argued that encoder-decoder models can be made more interpretable by replacing the softmax function in the attention with its sparse variants. In this work, we introduce a novel, simple method for achieving sparsity in attention: we replace the softmax activation with a ReLU, and show that sparsity naturally emerges from such a formulation. Training stability is achieved with layer normalization with either a specialized initialization or an additional gating function. Our model, which we call Rectified Linear Attention (ReLA), is easy to implement and more efficient than…
Prominently used in support vector machines and logistic regressions, kernel functions (kernels) can implicitly map data points into high dimensional spaces and make it easier to learn complex decision boundaries. In this work, by replacing the inner product function in the softmax layer, we explore the use of kernels for contextual word classification. In order to compare the individual kernels, experiments are conducted on standard language modeling and machine translation tasks. We observe a wide range of performances across different kernel settings. Extending the results, we look at the g…
The softmax content-based attention mechanism has proven to be very beneficial in many applications of recurrent neural networks. Nevertheless it suffers from two major computational limitations. First, its computations for an attention lookup scale linearly in the size of the attended sequence. Second, it does not encode the sequence into a fixed-size representation but instead requires to memorize all the hidden states. These two limitations restrict the use of the softmax attention mechanism to relatively small-scale applications with short sequences and few lookups per sequence. In this wo…
Cardiff University（卡迪夫大学）；Osaka City University（大阪市立大学）；Beijing Jiaotong University（北京交通大学）；Shenzhen University（深圳大学）；Shenzhen Zhuiyi Technology Co., Ltd.（深圳追一科技有限公司）；Chinese Academy of Science（中国科学院）
The softmax function is widely used in artificial neural networks for the multiclass classification problems, where the softmax transformation enforces the output to be positive and sum to one, and the corresponding loss function allows to use maximum likelihood principle to optimize the model. However, softmax leaves a large margin for loss function to conduct optimizing operation when it comes to high-dimensional classification, which results in low-performance to some extent. In this paper, we provide an empirical study on a simple and concise softmax variant, namely sparse-softmax, to alle…
In a multi-class classification problem, it is standard to model the output of a neural network as a categorical distribution conditioned on the inputs. The output must therefore be positive and sum to one, which is traditionally enforced by a softmax. This probabilistic mapping allows to use the maximum likelihood principle, which leads to the well-known log-softmax loss. However the choice of the softmax function seems somehow arbitrary as there are many other possible normalizing functions. It is thus unclear why the log-softmax loss would perform better than other loss alternatives. In par…
A softmax operator applied to a set of values acts somewhat like the maximization function and somewhat like an average. In sequential decision making, softmax is often used in settings where it is necessary to maximize utility but also to hedge against problems that arise from putting all of one's weight behind a single maximum utility decision. The Boltzmann softmax operator is the most commonly used softmax operator in this setting, but we show that this operator is prone to misbehavior. In this work, we study a differentiable softmax operator that, among other properties, is a non-expansio…
We empirically investigate the (negative) expected accuracy as an alternative loss function to cross entropy (negative log likelihood) for classification tasks. Coupled with softmax activation, it has small derivatives over most of its domain, and is therefore hard to optimize. A modified, leaky version is evaluated on a variety of classification tasks, including digit recognition, image classification, sequence tagging and tree tagging, using a variety of neural architectures such as logistic regression, multilayer perceptron, CNN, LSTM and Tree-LSTM. We show that it yields comparable or bett…
The softmax function is a ubiquitous component at the output of neural networks and increasingly in intermediate layers as well. This paper provides convex lower bounds and concave upper bounds on the softmax function, which are compatible with convex optimization formulations for characterizing neural networks and other ML models. We derive bounds using both a natural exponential-reciprocal decomposition of the softmax as well as an alternative decomposition in terms of the log-sum-exp function. The new bounds are provably and/or numerically tighter than linear bounds obtained in previous wor…
We formally prove the connection between k-means clustering and the predictions of neural networks based on the softmax activation layer. In existing work, this connection has been analyzed empirically, but it has never before been mathematically derived. The softmax function partitions the transformed input space into cones, each of which encompasses a class. This is equivalent to putting a number of centroids in this transformed space at equal distance from the origin, and k-means clustering the data points by proximity to these centroids. Softmax only cares in which cone a data point falls,…
Function approximation is a powerful approach for structuring large decision problems that has facilitated great achievements in the areas of reinforcement learning and game playing. Regression counterfactual regret minimization (RCFR) is a simple algorithm for approximately solving imperfect information games with normalized rectified linear unit (ReLU) parameterized policies. In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and yields a regret bound with a better dependence on the number of actions. We derive approximation error-a…
Neural networks utilize the softmax as a building block in classification tasks, which contains an overconfidence problem and lacks an uncertainty representation ability. As a Bayesian alternative to the softmax, we consider a random variable of a categorical probability over class labels. In this framework, the prior distribution explicitly models the presumed noise inherent in the observed label, which provides consistent gains in generalization performance in multiple challenging tasks. The proposed method inherits advantages of Bayesian approaches that achieve better uncertainty estimation…
Many applications of generative models rely on the marginalization of their high-dimensional output probability distributions. Normalization functions that yield sparse probability distributions can make exact marginalization more computationally tractable. However, sparse normalization functions usually require alternative loss functions for training since the log-likelihood is undefined for sparse probability distributions. Furthermore, many sparse normalization functions often collapse the multimodality of distributions. In this work, we present $\textit{ev-softmax}$, a sparse normalization…
Delft University of Technology（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）
We illustrate the detrimental effect, such as overconfident decisions, that exponential behavior can have in methods like classical LDA and logistic regression. We then show how polynomiality can remedy the situation. This, among others, leads purposefully to random-level performance in the tails, away from the bulk of the training data. A directly related, simple, yet important technical novelty we subsequently present is softRmax: a reasoned alternative to the standard softmax function employed in contemporary (deep) neural networks. It is derived through linking the standard softmax to Gaus…
In this work, we propose a new loss to improve feature discriminability and classification performance. Motivated by the adaptive cosine/coherence estimator (ACE), our proposed method incorporates angular information that is inherently learned by artificial neural networks. Our learnable ACE (LACE) transforms the data into a new "whitened" space that improves the inter-class separability and intra-class compactness. We compare our LACE to alternative state-of-the art softmax-based and feature regularization approaches. Our results show that the proposed method can serve as a viable alternative…
Duke Kunshan University（昆山杜克大学）；Sun Yat-sen University（中山大学）
In this paper, we explore the encoding/pooling layer and loss function in the end-to-end speaker and language recognition system. First, a unified and interpretable end-to-end system for both speaker and language recognition is developed. It accepts variable-length input and produces an utterance level result. In the end-to-end system, the encoding layer plays a role in aggregating the variable-length input sequence into an utterance level representation. Besides the basic temporal average pooling, we introduce a self-attentive pooling layer and a learnable dictionary encoding layer to get the…
MILA（蒙特利尔学习算法研究所）；University of Montréal（蒙特利尔大学）；CIFAR（加拿大高等研究院）
Despite being the standard loss function to train multi-class neural networks, the log-softmax has two potential limitations. First, it involves computations that scale linearly with the number of output classes, which can restrict the size of problems we are able to tackle with current hardware. Second, it remains unclear how close it matches the task loss such as the top-k error rate or other non-differentiable evaluation metrics which we aim to optimize ultimately. In this paper, we introduce an alternative classification loss function, the Z-loss, which is designed to address these two iss…
ST Engineering Electronics-SUTD Cyber Security Laboratory（新科工程电子-新加坡科技设计大学网络安全实验室）；Singapore University of Technology and Design（新加坡科技设计大学）；Ben-Gurion University of the Negev（本-古里安大学）；Deutsche Telekom Innovation Laboratories at Ben-Gurion University of the Negev（本-古里安大学德国电信创新实验室）
We investigate to what extent alternative variants of Artificial Neural Networks (ANNs) are susceptible to adversarial attacks. We analyse the adversarial robustness of conventional, stochastic ANNs and Spiking Neural Networks (SNNs) in the raw image space, across three different datasets. Our experiments reveal that stochastic ANN variants are almost equally as susceptible as conventional ANNs when faced with simple iterative gradient-based attacks in the white-box setting. However we observe, that in black-box settings, stochastic ANNs are more robust than conventional ANNs, when faced with …
University of Tokyo（东京大学）；University of Cambridge（剑桥大学）；National Chiao Tung University（国立交通大学）；Japan Science and Technology Agency（日本科学技术振兴机构）
In this paper, we present contemporary techniques for visualising the feature space of a deep learning image classification neural network. These techniques are viewed in the context of a feed-forward network trained to classify low resolution fluorescence images of white blood cells captured using optofluidic imaging. The model has two output classes corresponding to two different cell types, which are often difficult to distinguish by eye. This paper has two major sections. The first looks to develop the information space presented by dimension reduction techniques, such as t-SNE, used to em…
Modern deep learning models have achieved great success in predictive accuracy for many data modalities. However, their application to many real-world tasks is restricted by poor uncertainty estimates, such as overconfidence on out-of-distribution (OOD) data and ungraceful failing under distributional shift. Previous benchmarks have found that ensembles of neural networks (NNs) are typically the best calibrated models on OOD data. Inspired by this, we leverage recent theoretical advances that characterize the function-space prior of an ensemble of infinitely-wide NNs as a Gaussian process, ter…
The advent of larger machine learning (ML) models have improved state-of-the-art (SOTA) performance in various modeling tasks, ranging from computer vision to natural language. As ML models continue increasing in size, so does their respective energy consumption and computational requirements. However, the methods for tracking, reporting, and comparing energy consumption remain limited. We presentEnergyVis, an interactive energy consumption tracker for ML models. Consisting of multiple coordinated views, EnergyVis enables researchers to interactively track, visualize and compare model energy c…
