Estimating the gradients of stochastic nodes in stochastic computational graphs is one of the crucial research questions in the deep generative modeling community, which enables the gradient descent optimization on neural network parameters. Stochastic gradient estimators of discrete random variables are widely explored, for example, Gumbel-Softmax reparameterization trick for Bernoulli and categorical distributions. Meanwhile, other discrete distribution cases such as the Poisson, geometric, binomial, multinomial, negative binomial, etc. have not been explored. This paper proposes a generalized version of the Gumbel-Softmax estimator, which is able to reparameterize generic discrete distributions, not restricted to the Bernoulli and the categorical. The proposed estimator utilizes the truncation of discrete random variables, the Gumbel-Softmax trick, and a special form of linear transformation. Our experiments consist of (1) synthetic examples and applications on VAE, which show the efficacy of our methods; and (2) topic models, which demonstrate the value of the proposed estimation in practice.
Many problems in real life can be converted to combinatorial optimization problems (COPs) on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we…
Beijing Normal University（北京师范大学）；ColorfulClouds Tech（彩云科技）
In computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed…
Alan Turing Institute（艾伦·图灵研究所）；University of Warwick（华威大学）；University of Cambridge（剑桥大学）
Generative Adversarial Networks (GAN) have limitations when the goal is to generate sequences of discrete elements. The reason for this is that samples from a distribution on discrete objects such as the multinomial are not differentiable with respect to the distribution parameters. This problem can be avoided by using the Gumbel-softmax distribution, which is a continuous approximation to a multinomial distribution parameterized in terms of the softmax function. In this work, we evaluate the performance of GANs based on recurrent neural networks with Gumbel-softmax output distributions in the…
Google Brain（谷歌大脑）；University of Cambridge（剑桥大学）；MPI Tübingen（马克斯·普朗克智能系统研究所（蒂宾根））；Stanford University（斯坦福大学）
Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-o…
The Gumbel-Softmax is a continuous distribution over the simplex that is often used as a relaxation of discrete distributions. Because it can be readily interpreted and easily reparameterized, it enjoys widespread use. We propose a modular and more flexible family of reparameterizable distributions where Gaussian noise is transformed into a one-hot approximation through an invertible function. This invertible function is composed of a modified softmax and can incorporate diverse transformations that serve different specific purposes. For example, the stick-breaking procedure allows us to exten…
Many machine learning tasks require sampling a subset of items from a collection based on a parameterized distribution. The Gumbel-softmax trick can be used to sample a single item, and allows for low-variance reparameterized gradients with respect to the parameters of the underlying distribution. However, stochastic optimization involving subset sampling is typically not reparameterizable. To overcome this limitation, we define a continuous relaxation of subset sampling that provides reparameterization gradients by generalizing the Gumbel-max trick. We use this approach to sample subsets of f…
ETH Zürich（苏黎世联邦理工学院）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Institute for Advanced Study（高等研究院）
Gradient estimation in models with discrete latent variables is a challenging problem, because the simplest unbiased estimators tend to have high variance. To counteract this, modern estimators either introduce bias, rely on multiple function evaluations, or use learned, input-dependent baselines. Thus, there is a need for estimators that require minimal tuning, are computationally cheap, and have low mean squared error. In this paper, we show that the variance of the straight-through variant of the popular Gumbel-Softmax estimator can be reduced through Rao-Blackwellization without increasing…
When samples have internal structure, we often see a mismatch between the objective optimized during training and the model's goal during inference. For example, in sequence-to-sequence modeling we are interested in high-quality translated sentences, but training typically uses maximum likelihood at the word level. The natural training-time loss would involve a combinatorial problem -- dynamic programming-based global sequence alignment -- but solutions to combinatorial problems are not differentiable with respect to their input parameters, so surrogate, differentiable losses are used instead.…
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所模式识别国家重点实验室）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；Intel Labs China（英特尔中国研究院）
For network architecture search (NAS), it is crucial but challenging to simultaneously guarantee both effectiveness and efficiency. Towards achieving this goal, we develop a differentiable NAS solution, where the search space includes arbitrary feed-forward network consisting of the predefined number of connections. Benefiting from a proposed ensemble Gumbel-Softmax estimator, our method optimizes both the architecture of a deep network and its parameters in the same round of backward propagation, yielding an end-to-end mechanism of searching network architectures. Extensive experiments on a v…
ETH Zürich（苏黎世联邦理工学院）；University of Toronto（多伦多大学）；Google Research, Brain Team（谷歌研究院大脑团队）；DeepMind（深度思维公司）；Institute for Advanced Study（高等研究院）
The Gumbel-Max trick is the basis of many relaxed gradient estimators. These estimators are easy to implement and low variance, but the goal of scaling them comprehensively to large combinatorial distributions is still outstanding. Working within the perturbation model framework, we introduce stochastic softmax tricks, which generalize the Gumbel-Softmax trick to combinatorial spaces. Our framework is a unified perspective on existing relaxed estimators for perturbation models, and it contains many novel relaxations. We design structured relaxations for subset selection, spanning trees, arbore…
Permutations and matchings are core building blocks in a variety of latent variable models, as they allow us to align, canonicalize, and sort data. Learning in such models is difficult, however, because exact marginalization over these combinatorial objects is intractable. In response, this paper introduces a collection of new methods for end-to-end learning in such models that approximate discrete maximum-weight matching using the continuous Sinkhorn operator. Sinkhorn iteration is attractive because it functions as a simple, easy-to-implement analog of the softmax operator. With this, we can…
Borealis AI（Borealis人工智能公司）；Vector Institute（向量研究所）；University of Guelph（圭尔夫大学）
ML models often operate within the context of a larger system that can adapt its response when the ML model is uncertain, such as falling back on safe defaults or a human in the loop. This commonly encountered operational context calls for principled techniques for training ML models with the option to abstain from predicting when uncertain. Selective neural networks are trained with an integrated option to abstain, allowing them to learn to recognize and optimize for the subset of the data distribution for which confident predictions can be made. However, optimizing selective networks is chal…
Recently, in many systems such as speech recognition and visual processing, deep learning has been widely implemented. In this research, we are exploring the possibility of using deep learning in community detection among the graph datasets. Graphs have gained growing traction in different fields, including social networks, information graphs, the recommender system, and also life sciences. In this paper, we propose a method of community detection clustering the nodes of various graph datasets. We cluster different category datasets that belong to Affiliation networks, Animal networks, Human c…
Gumbel-Softmax Flow Matching with Straight-Through Guidance for Controllable Biological Sequence Generation
Duke University（杜克大学）；University of Pennsylvania（宾夕法尼亚大学）；Duke-NUS Medical School（杜克-新加坡国立大学医学院）；Mila, Quebec AI Institute（米拉魁北克人工智能研究所）；Université de Montréal（蒙特利尔大学）
Computer Science 5000 Technology Dr NW Huntsville Alabama Usa 35805
Graph is a ubiquitous data structure in data science that is widely applied in social networks, knowledge representation graphs, recommendation systems, etc. When given a graph dataset consisting of one graph or more graphs, where the graphs are weighted in general, the first step is often to find clusters in the graphs. In this paper, we present some ongoing research results on graph clustering algorithms for clustering weighted graph datasets, which we name as Weighted Graph Node Clustering via Gumbel Softmax (WGCGS for short). We apply WGCGS on the Karate club weighted network dataset. Our …
University of Amsterdam（阿姆斯特丹大学）；Inception Institute of Artificial Intelligence（起源人工智能研究院）
Multi-task learning aims to explore task relatedness to improve individual tasks, which is of particular significance in the challenging scenario that only limited data is available for each task. To tackle this challenge, we propose variational multi-task learning (VMTL), a general probabilistic inference framework for learning multiple related tasks. We cast multi-task learning as a variational Bayesian inference problem, in which task relatedness is explored in a unified manner by specifying priors. To incorporate shared knowledge into each task, we design the prior of a task to be a learna…
Many electroencephalography (EEG) applications rely on channel selection methods to remove the least informative channels, e.g., to reduce the amount of electrodes to be mounted, to decrease the computational load, or to reduce overfitting effects and improve performance. Wrapper-based channel selection methods aim to match the channel selection step to the target model, yet they require to re-train the model multiple times on different candidate channel subsets, which often leads to an unacceptably high computational cost, especially when said model is a (deep) neural network. To alleviate th…
Federated Variational Preference Alignment with Gumbel-Softmax Prior for Personalized User Preferences
Graduate School of AI, POSTECH, Pohang, Republic of Korea（POSTECH人工智能研究生院）；Department of CSE, POSTECH, Pohang, Republic of Korea（POSTECH计算机科学与工程系）；National AI Research Lab, Seoul, Republic of Korea（首尔国家人工智能研究实验室）
The drivers of compositionality in artificial languages that emerge when two (or more) agents play a non-visual referential game has been previously investigated using approaches based on the REINFORCE algorithm and the (Neural) Iterated Learning Model. Following the more recent introduction of the \textit{Straight-Through Gumbel-Softmax} (ST-GS) approach, this paper investigates to what extent the drivers of compositionality identified so far in the field apply in the ST-GS context and to what extent do they translate into (emergent) systematic generalisation abilities, when playing a visual …
University of Amsterdam（阿姆斯特丹大学）；ORTEC；CIFAR（加拿大高等研究院）
The well-known Gumbel-Max trick for sampling from a categorical distribution can be extended to sample $k$ elements without replacement. We show how to implicitly apply this 'Gumbel-Top-$k$' trick on a factorized distribution over sequences, allowing to draw exact samples without replacement using a Stochastic Beam Search. Even for exponentially large domains, the number of model evaluations grows only linear in $k$ and the maximum sampled sequence length. The algorithm creates a theoretical connection between sampling and (deterministic) beam search and can be used as a principled intermediat…
Network quantization, which aims to reduce the bit-lengths of the network weights and activations, has emerged as one of the key ingredients to reduce the size of neural networks for their deployments to resource-limited devices. In order to overcome the nature of transforming continuous activations and weights to discrete ones, recent study called Relaxed Quantization (RQ) [Louizos et al. 2019] successfully employ the popular Gumbel-Softmax that allows this transformation with efficient gradient-based optimization. However, RQ with this Gumbel-Softmax relaxation still suffers from bias-varian…
Discrete and especially binary random variables occur in many machine learning models, notably in variational autoencoders with binary latent states and in stochastic binary networks. When learning such models, a key tool is an estimator of the gradient of the expected loss with respect to the probabilities of binary variables. The straight-through (ST) estimator gained popularity due to its simplicity and efficiency, in particular in deep networks where unbiased estimators are impractical. Several techniques were proposed to improve over ST while keeping the same low computational complexity:…
Australian National University（澳大利亚国立大学）；Pohang University of Science and Technology（浦项科技大学）
Mutual information is widely applied to learn latent representations of observations, whilst its implication in classification neural networks remain to be better explained. We show that optimising the parameters of classification neural networks with softmax cross-entropy is equivalent to maximising the mutual information between inputs and labels under the balanced data assumption. Through experiments on synthetic and real datasets, we show that softmax cross-entropy can estimate mutual information approximately. When applied to image classification, this relation helps approximate the point…
University of Pennsylvania（宾夕法尼亚大学）；Princeton University（普林斯顿大学）
With an eye towards human-centered automation, we contribute to the development of a systematic means to infer features of human decision-making from behavioral data. Motivated by the common use of softmax selection in models of human decision-making, we study the maximum likelihood parameter estimation problem for softmax decision-making models with linear objective functions. We present conditions under which the likelihood function is convex. These allow us to provide sufficient conditions for convergence of the resulting maximum likelihood estimator and to construct its asymptotic distribu…
University of Alberta（阿尔伯塔大学）；Noah’s Ark Lab, Huawei（华为诺亚方舟实验室）；Alberta Machine Intelligence Institute (Amii)（阿尔伯塔机器智能研究所（Amii））
Policy gradient (PG) estimators are ineffective in dealing with softmax policies that are sub-optimally saturated, which refers to the situation when the policy concentrates its probability mass on sub-optimal actions. Sub-optimal policy saturation may arise from bad policy initialization or sudden changes in the environment that occur after the policy has already converged. Current softmax PG estimators require a large number of updates to overcome policy saturation, which causes low sample efficiency and poor adaptability to new situations. To mitigate this problem, we propose a novel PG est…
In many applications we seek to maximize an expectation with respect to a distribution over discrete variables. Estimating gradients of such objectives with respect to the distribution parameters is a challenging problem. We analyze existing solutions including finite-difference (FD) estimators and continuous relaxation (CR) estimators in terms of bias and variance. We show that the commonly used Gumbel-Softmax estimator is biased and propose a simple method to reduce it. We also derive a simpler piece-wise linear continuous relaxation that also possesses reduced bias. We demonstrate empirical…
Tsinghua University（清华大学）；Microsoft Research Asia（微软亚洲研究院）
Value function estimation is an important task in reinforcement learning, i.e., prediction. The Boltzmann softmax operator is a natural value estimator and can provide several benefits. However, it does not satisfy the non-expansion property, and its direct use may fail to converge even in value iteration. In this paper, we propose to update the value function with dynamic Boltzmann softmax (DBS) operator, which has good convergence property in the setting of planning and learning. Experimental results on GridWorld show that the DBS operator enables better estimation of the value function, whi…
Reparameterization of variational auto-encoders with continuous random variables is an effective method for reducing the variance of their gradient estimates. In the discrete case, one can perform reparametrization using the Gumbel-Max trick, but the resulting objective relies on an $\arg \max$ operation and is non-differentiable. In contrast to previous works which resort to softmax-based relaxations, we propose to optimize it directly by applying the direct loss minimization approach. Our proposal extends naturally to structured discrete latent variable models when evaluating the $\arg \max$…
Training large-scale mixture of experts models efficiently on modern hardware requires assigning datapoints in a batch to different experts, each with a limited capacity. Recently proposed assignment procedures lack a probabilistic interpretation and use biased estimators for training. As an alternative, we propose two unbiased estimators based on principled stochastic assignment procedures: one that skips datapoints which exceed expert capacity, and one that samples perfectly balanced assignments using an extension of the Gumbel-Matching distribution [29]. Both estimators are unbiased, as the…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；University of Helsinki（赫尔辛基大学）
Variational approximations are increasingly based on gradient-based optimization of expectations estimated by sampling. Handling discrete latent variables is then challenging because the sampling process is not differentiable. Continuous relaxations, such as the Gumbel-Softmax for categorical distribution, enable gradient-based optimization, but do not define a valid probability mass for discrete observations. In practice, selecting the amount of relaxation is difficult and one needs to optimize an objective that does not align with the desired one, causing problems especially with models havi…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；Airtificial Intelligence Research Institute（人工智能研究所）；Moscow Institute of Physics and Technology（莫斯科物理与技术学院）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术学院）；AIRI（人工智能研究所）；Sber AI Lab；MIPT（莫斯科物理与技术学院）
Structured latent variables allow incorporating meaningful prior knowledge into deep learning models. However, learning with such variables remains challenging because of their discrete nature. Nowadays, the standard learning approach is to define a latent variable as a perturbed algorithm output and to use a differentiable surrogate for training. In general, the surrogate puts additional constraints on the model and inevitably leads to biased gradients. To alleviate these shortcomings, we extend the Gumbel-Max trick to define distributions over structured domains. We avoid the differentiable …
Training of discrete latent variable models remains challenging because passing gradient information through discrete units is difficult. We propose a new class of smoothing transformations based on a mixture of two overlapping distributions, and show that the proposed transformation can be used for training binary latent models with either directed or undirected priors. We derive a new variational bound to efficiently train with Boltzmann machine priors. Using this bound, we develop DVAE++, a generative model with a global discrete prior and a hierarchy of convolutional continuous variables. …
School of Computing, National University of Singapore, Singapore（新加坡国立大学计算机学院）；Department of Statistics, University of Oxford, United Kingdom（英国牛津大学统计系）；School of Intelligence Science（智能科学学院）；Technology, Nanjing University, China（技术学院，南京大学，中国）
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Alibaba Group（阿里巴巴集团）
A widely-used actor-critic reinforcement learning algorithm for continuous control, Deep Deterministic Policy Gradients (DDPG), suffers from the overestimation problem, which can negatively affect the performance. Although the state-of-the-art Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm mitigates the overestimation issue, it can lead to a large underestimation bias. In this paper, we propose to use the Boltzmann softmax operator for value function estimation in continuous control. We first theoretically analyze the softmax operator in continuous action space. Then, we uncov…
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…
Instituto de Telecomunicações（电信研究所）；University of Amsterdam（阿姆斯特丹大学）；Instituto Superior Técnico（高等技术学院）；Unbabel（Unbabel公司）
Training neural network models with discrete (categorical or structured) latent variables can be computationally challenging, due to the need for marginalization over large or combinatorial sets. To circumvent this issue, one typically resorts to sampling-based approximations of the true marginal, requiring noisy gradient estimators (e.g., score function estimator) or continuous relaxations with lower-variance reparameterized gradients (e.g., Gumbel-Softmax). In this paper, we propose a new training strategy which replaces these estimators by an exact yet efficient marginalization. To achieve …
University of Maryland Baltimore County（马里兰大学巴尔的摩县分校）
Within the context of event modeling and understanding, we propose a new method for neural sequence modeling that takes partially-observed sequences of discrete, external knowledge into account. We construct a sequential neural variational autoencoder, which uses Gumbel-Softmax reparametrization within a carefully defined encoder, to allow for successful backpropagation during training. The core idea is to allow semi-supervised external discrete knowledge to guide, but not restrict, the variational latent parameters during training. Our experiments indicate that our approach not only outperfor…
While deep generative models have succeeded in image processing, natural language processing, and reinforcement learning, training that involves discrete random variables remains challenging due to the high variance of its gradient estimation process. Monte Carlo is a common solution used in most variance reduction approaches. However, this involves time-consuming resampling and multiple function evaluations. We propose a Gapped Straight-Through (GST) estimator to reduce the variance without incurring resampling overhead. This estimator is inspired by the essential properties of Straight-Throu…
Replicated Softmax model, a well-known undirected topic model, is powerful in extracting semantic representations of documents. Traditional learning strategies such as Contrastive Divergence are very inefficient. This paper provides a novel estimator to speed up the learning based on Noise Contrastive Estimate, extended for documents of variant lengths and weighted inputs. Experiments on two benchmarks show that the new estimator achieves great learning efficiency and high accuracy on document retrieval and classification.
Improving Discrete Optimisation Via Decoupled Straight-Through Estimator
College of Design and Engineering, National University of Singapore, Singapore（设计与工程学院，新加坡国立大学）；Google Deepmind, California, USA（DeepMind，美国加州）；New York University, USA（纽约大学）
The University of Hong Kong（香港大学）；ByteDance Inc.（字节跳动公司）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）
Recently, random feature attentions (RFAs) are proposed to approximate the softmax attention in linear time and space complexity by linearizing the exponential kernel. In this paper, we first propose a novel perspective to understand the bias in such approximation by recasting RFAs as self-normalized importance samplers. This perspective further sheds light on an \emph{unbiased} estimator for the whole softmax attention, called randomized attention (RA). RA constructs positive random features via query-specific distributions and enjoys greatly improved approximation fidelity, albeit exhibiting…
Propensity Straight-Through Gradients for Discrete Stochastic Systems
Biofisika Institute (CSIC, UPV/EHU)（生物物理研究所（西班牙国家研究委员会、巴斯克大学））；University of the Basque Country（巴斯克大学）；IKERBASQUE, Basque Foundation for Science（巴斯克科学基金会IKERBASQUE）；University of California（加利福尼亚大学）
