We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python.
EWHA Womans University（梨花女子大学）；Korea Advanced Institute of Science and Technology（韩国科学技术院）
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 generaliz…
Institute of High Performance Computing（高性能计算研究院）；Agency for Science, Technology and Research（科学、技术与研究局）；Beijing Institute of Big Data Research（北京大数据研究院）；Peking University（北京大学）；Princeton University（普林斯顿大学）
We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood mathematically as an weak approximation, which leads to a number of precise and useful results on the approximations of stochastic gradient descent (SGD), momentum SGD and stochastic Nesterov's accelerated gradient method in the general setting of stochastic objectives. We also demon…
Gradient boosting methods based on Structured Categorical Decision Trees (SCDT) have been demonstrated to outperform numerical and one-hot-encodings on problems where the categorical variable has a known underlying structure. However, the enumeration procedure in the SCDT is infeasible except for categorical variables with low or moderate cardinality. We propose and implement two methods to overcome the computational obstacles and efficiently perform Gradient Boosting on complex structured categorical variables. The resulting package, called StructureBoost, is shown to outperform established p…
In this paper we present CatBoost, a new open-sourced gradient boosting library that successfully handles categorical features and outperforms existing publicly available implementations of gradient boosting in terms of quality on a set of popular publicly available datasets. The library has a GPU implementation of learning algorithm and a CPU implementation of scoring algorithm, which are significantly faster than other gradient boosting libraries on ensembles of similar sizes.
QKCV Attention: Enhancing Time Series Forecasting with Static Categorical Embeddings for Both Lightweight and Pre-trained Foundation Models
Independent Researcher（独立研究者）；Key Laboratory of Brain-Machine Intelligence for Information Behavior (Ministry of Education and Shanghai)（信息行为脑机智能关键实验室）；School of Business and Management（商学院）
Accurately backpropagating the gradient through categorical variables is a challenging task that arises in various domains, such as training discrete latent variable models. To this end, we propose CARMS, an unbiased estimator for categorical random variables based on multiple mutually negatively correlated (jointly antithetic) samples. CARMS combines REINFORCE with copula based sampling to avoid duplicate samples and reduce its variance, while keeping the estimator unbiased using importance sampling. It generalizes both the ARMS antithetic estimator for binary variables, which is CARMS for tw…
To address the challenge of backpropagating the gradient through categorical variables, we propose the augment-REINFORCE-swap-merge (ARSM) gradient estimator that is unbiased and has low variance. ARSM first uses variable augmentation, REINFORCE, and Rao-Blackwellization to re-express the gradient as an expectation under the Dirichlet distribution, then uses variable swapping to construct differently expressed but equivalent expectations, and finally shares common random numbers between these expectations to achieve significant variance reduction. Experimental results show ARSM closely resembl…
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…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Microsoft Research（微软研究院）；Columbia University（哥伦比亚大学）
Sequence generation models are commonly refined with reinforcement learning over user-defined metrics. However, high gradient variance hinders the practical use of this method. To stabilize this method, we adapt to contextual generation of categorical sequences a policy gradient estimator, which evaluates a set of correlated Monte Carlo (MC) rollouts for variance control. Due to the correlation, the number of unique rollouts is random and adaptive to model uncertainty; those rollouts naturally become baselines for each other, and hence are combined to effectively reduce gradient variance. We a…
University of Science and Technology Beijing（北京科技大学）；University of Tennessee Knoxville（田纳西大学诺克斯维尔分校）
We propose NECA, a deep representation learning method for categorical data. Built upon the foundations of network embedding and deep unsupervised representation learning, NECA deeply embeds the intrinsic relationship among attribute values and explicitly expresses data objects with numeric vector representations. Designed specifically for categorical data, NECA can support important downstream data mining tasks, such as clustering. Extensive experimental analysis demonstrated the effectiveness of NECA.
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
Deep learning models suffer from a phenomenon called adversarial attacks: we can apply minor changes to the model input to fool a classifier for a particular example. The literature mostly considers adversarial attacks on models with images and other structured inputs. However, the adversarial attacks for categorical sequences can also be harmful. Successful attacks for inputs in the form of categorical sequences should address the following challenges: (1) non-differentiability of the target function, (2) constraints on transformations of initial sequences, and (3) diversity of possible probl…
Yandex（扬达克斯）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
This paper presents the key algorithmic techniques behind CatBoost, a new gradient boosting toolkit. Their combination leads to CatBoost outperforming other publicly available boosting implementations in terms of quality on a variety of datasets. Two critical algorithmic advances introduced in CatBoost are the implementation of ordered boosting, a permutation-driven alternative to the classic algorithm, and an innovative algorithm for processing categorical features. Both techniques were created to fight a prediction shift caused by a special kind of target leakage present in all currently exi…
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…
Training models with discrete latent variables is challenging due to the high variance of unbiased gradient estimators. While low-variance reparameterization gradients of a continuous relaxation can provide an effective solution, a continuous relaxation is not always available or tractable. Dong et al. (2020) and Yin et al. (2020) introduced a performant estimator that does not rely on continuous relaxations; however, it is limited to binary random variables. We introduce a novel derivation of their estimator based on importance sampling and statistical couplings, which we extend to the catego…
We propose a new model for supervised learning to rank. In our model, the relevance labels are assumed to follow a categorical distribution whose probabilities are constructed based on a scoring function. We optimize the training objective with respect to the multivariate categorical variables with an unbiased and low-variance gradient estimator. Learning-to-rank methods can generally be categorized into pointwise, pairwise, and listwise approaches. Although our scoring function is pointwise, the proposed framework permits flexibility over the choice of the loss function. In our new model, the…
Northeastern University（东北大学）；Australian National University（澳大利亚国立大学）
Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been pursued recently, and yet there is still a gap between abstract mathematical foundations and concrete applications to machine learning tasks. In this paper we introduce DisCoPyro as a categorical structure learning framework, which combines categorical structures (such as symmetric monoidal categories and operads) with amortized variational inference, and ca…
University College London（伦敦大学学院）；University of Southampton（南安普顿大学）
We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches.…
Exact Discrete Stochastic Simulation with Deep-Learning-Scale Gradient Optimization
Biofisika Institute (CSIC, UPV/EHU)（生物物理研究所（CSIC, UPV/EHU））；University of the Basque Country (UPV/EHU)（巴斯克大学（UPV/EHU））；IKERBASQUE, Basque Foundation for Science（IKERBASQUE，巴斯克科学基金会）；University of California（加利福尼亚大学）
