We study the cross-entropy method (CEM) for the non-convex optimization of a continuous and parameterized objective function and introduce a differentiable variant that enables us to differentiate the output of CEM with respect to the objective function's parameters. In the machine learning setting this brings CEM inside of the end-to-end learning pipeline where this has otherwise been impossible. We show applications in a synthetic energy-based structured prediction task and in non-convex continuous control. In the control setting we show how to embed optimal action sequences into a lower-dimensional space. DCEM enables us to fine-tune CEM-based controllers with policy optimization.
Aalto University（阿尔托大学）；University of Oxford（牛津大学）
Particle Filtering (PF) methods are an established class of procedures for performing inference in non-linear state-space models. Resampling is a key ingredient of PF, necessary to obtain low variance likelihood and states estimates. However, traditional resampling methods result in PF-based loss functions being non-differentiable with respect to model and PF parameters. In a variational inference context, resampling also yields high variance gradient estimates of the PF-based evidence lower bound. By leveraging optimal transport ideas, we introduce a principled differentiable particle filter …
We introduce an independence criterion based on entropy regularized optimal transport. Our criterion can be used to test for independence between two samples. We establish non-asymptotic bounds for our test statistic and study its statistical behavior under both the null hypothesis and the alternative hypothesis. The theoretical results involve tools from U-process theory and optimal transport theory. We also offer a random feature type approximation for large-scale problems, as well as a differentiable program implementation for deep learning applications. We present experimental results on e…
In this paper, we extend the recently proposed multivariate rank energy distance, based on the theory of optimal transport, for statistical testing of distributional similarity, to soft rank energy distance. Being differentiable, this in turn allows us to extend the rank energy to a subspace robust rank energy distance, dubbed Projected soft-Rank Energy distance, which can be computed via optimization over the Stiefel manifold. We show via experiments that using projected soft rank energy one can trade-off the detection power vs the false alarm via projections onto an appropriately selected lo…
We formulate the entropy of a quantized artificial neural network as a differentiable function that can be plugged as a regularization term into the cost function minimized by gradient descent. Our formulation scales efficiently beyond the first order and is agnostic of the quantization scheme. The network can then be trained to minimize the entropy of the quantized parameters, so that they can be optimally compressed via entropy coding. We experiment with our entropy formulation at quantizing and compressing well-known network architectures over multiple datasets. Our approach compares favora…
Ergodic LLC（Ergodic有限责任公司）；University of Michigan（密歇根大学）
Plasma supports collective modes and particle-wave interactions that leads to complex behavior in inertial fusion energy applications. While plasma can sometimes be modeled as a charged fluid, a kinetic description is useful towards the study of nonlinear effects in the higher dimensional momentum-position phase-space that describes the full complexity of plasma dynamics. We create a differentiable solver for the plasma kinetics 3D partial-differential-equation and introduce a domain-specific objective function. Using this framework, we perform gradient-based optimization of neural networks th…
Washington University in St. Louis（圣路易斯华盛顿大学）；Meta
Recent automated machine learning systems often use learning curves ranking models to inform decisions about when to stop unpromising trials and identify better model configurations. In this paper, we present a novel learning curve ranking model specifically tailored for ranking normalized entropy (NE) learning curves, which are commonly used in online advertising and recommendation systems. Our proposed model, self-Adaptive Curve Transformation augmented Relative curve Ranking (ACTR2), features an adaptive curve transformation layer that transforms raw lifetime NE curves into composite window…
FX-DARTS: Designing Topology-unconstrained Architectures with Differentiable Architecture Search and Entropy-based Super-network Shrinking
Beijing Normal University（北京师范大学）；Southern University of Science and Technology（南方科技大学）；University of Illinois Chicago（伊利诺伊大学香槟分校）；Politecnico di Milano（米兰理工大学）；Universita’ Della Svizzera Italiana（瑞士意大利大学）
Differentiable Entropy Regularization: A Complexity-Aware Approach for Neural Optimization
Department of Computer Science Iowa State University（计算机科学系爱荷华州立大学）；Department of Civil, Construction and Environmental Engineering Iowa State University（土木、建设与环境工程系爱荷华州立大学）
University of Amsterdam（阿姆斯特丹大学）；University of Edinburgh（爱丁堡大学）
Coreference evaluation metrics are hard to optimize directly as they are non-differentiable functions, not easily decomposable into elementary decisions. Consequently, most approaches optimize objectives only indirectly related to the end goal, resulting in suboptimal performance. Instead, we propose a differentiable relaxation that lends itself to gradient-based optimisation, thus bypassing the need for reinforcement learning or heuristic modification of cross-entropy. We show that by modifying the training objective of a competitive neural coreference system, we obtain a substantial gain in …
We demonstrate that a continuous relaxation of the argmax operation can be used to create a differentiable approximation to greedy decoding for sequence-to-sequence (seq2seq) models. By incorporating this approximation into the scheduled sampling training procedure (Bengio et al., 2015)--a well-known technique for correcting exposure bias--we introduce a new training objective that is continuous and differentiable everywhere and that can provide informative gradients near points where previous decoding decisions change their value. In addition, by using a related approximation, we demonstrate …
MIT（麻省理工学院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
This work considers identifying parameters characterizing a physical system's dynamic motion directly from a video whose rendering configurations are inaccessible. Existing solutions require massive training data or lack generalizability to unknown rendering configurations. We propose a novel approach that marries domain randomization and differentiable rendering gradients to address this problem. Our core idea is to train a rendering-invariant state-prediction (RISP) network that transforms image differences into state differences independent of rendering configurations, e.g., lighting, shado…
Max Planck ETH Center for Learning Systems（马克斯·普朗克ETH学习系统中心）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Trajectory optimizers for model-based reinforcement learning, such as the Cross-Entropy Method (CEM), can yield compelling results even in high-dimensional control tasks and sparse-reward environments. However, their sampling inefficiency prevents them from being used for real-time planning and control. We propose an improved version of the CEM algorithm for fast planning, with novel additions including temporally-correlated actions and memory, requiring 2.7-22x less samples and yielding a performance increase of 1.2-10x in high-dimensional control problems.
Gathering the most information by picking the least amount of data is a common task in experimental design or when exploring an unknown environment in reinforcement learning and robotics. A widely used measure for quantifying the information contained in some distribution of interest is its entropy. Greedily minimizing the expected entropy is therefore a standard method for choosing samples in order to gain strong beliefs about the underlying random variables. We show that this approach is prone to temporally getting stuck in local optima corresponding to wrongly biased beliefs. We suggest ins…
Non-autoregressive machine translation models significantly speed up decoding by allowing for parallel prediction of the entire target sequence. However, modeling word order is more challenging due to the lack of autoregressive factors in the model. This difficultly is compounded during training with cross entropy loss, which can highly penalize small shifts in word order. In this paper, we propose aligned cross entropy (AXE) as an alternative loss function for training of non-autoregressive models. AXE uses a differentiable dynamic program to assign loss based on the best possible monotonic a…
ETH Zürich（苏黎世联邦理工学院）；IE University（IE大学）；HSE University（俄罗斯高等经济学院）
We propose an end-to-end trainable framework that processes large-scale visual data tensors by looking at a fraction of their entries only. Our method combines a neural network encoder with a tensor train decomposition to learn a low-rank latent encoding, coupled with cross-approximation (CA) to learn the representation through a subset of the original samples. CA is an adaptive sampling algorithm that is native to tensor decompositions and avoids working with the full high-resolution data explicitly. Instead, it actively selects local representative samples that we fetch out-of-core and on-de…
We present the Tamed Cross Entropy (TCE) loss function, a robust derivative of the standard Cross Entropy (CE) loss used in deep learning for classification tasks. However, unlike other robust losses, the TCE loss is designed to exhibit the same training properties than the CE loss in noiseless scenarios. Therefore, the TCE loss requires no modification on the training regime compared to the CE loss and, in consequence, can be applied in all applications where the CE loss is currently used. We evaluate the TCE loss using the ResNet architecture on four image datasets that we artificially conta…
Stanford University（斯坦福大学）；University of Maryland（马里兰大学）
Quantum machine learning is an emerging field at the intersection of machine learning and quantum computing. Classical cross entropy plays a central role in machine learning. We define its quantum generalization, the quantum cross entropy, prove its lower bounds, and investigate its relation to quantum fidelity. In the classical case, minimizing cross entropy is equivalent to maximizing likelihood. In the quantum case, when the quantum cross entropy is constructed from quantum data undisturbed by quantum measurements, this relation holds. Classical cross entropy is equal to negative log-likeli…
Designing DNA and protein sequences with improved function has the potential to greatly accelerate synthetic biology. Machine learning models that accurately predict biological fitness from sequence are becoming a powerful tool for molecular design. Activation maximization offers a simple design strategy for differentiable models: one-hot coded sequences are first approximated by a continuous representation which is then iteratively optimized with respect to the predictor oracle by gradient ascent. While elegant, this method suffers from vanishing gradients and may cause predictor pathologies …
The cross-entropy method is a simple but efficient method for global optimization. In this paper we provide two online variants of the basic CEM, together with a proof of convergence.
Modulated differentiable STFT and balanced spectrum metric for freight train wheelset bearing cross-machine transfer monitoring under speed fluctuations
State Key Laboratory of Advanced Rail Autonomous Operation（先进轨道交通自主运行状态重点实验室）；Beijing Jiaotong University（北京交通大学）；School of Mechanical, Electronic and Control Engineering（机械电子控制工程学院）
Tokyo Institute of Technology（东京工业大学）；Gunma University（群马大学）；RIKEN AIP（理化学研究所先进智能项目中心）
In knowledge graph embedding, the theoretical relationship between the softmax cross-entropy and negative sampling loss functions has not been investigated. This makes it difficult to fairly compare the results of the two different loss functions. We attempted to solve this problem by using the Bregman divergence to provide a unified interpretation of the softmax cross-entropy and negative sampling loss functions. Under this interpretation, we can derive theoretical findings for fair comparison. Experimental results on the FB15k-237 and WN18RR datasets show that the theoretical findings are va…
A Layer Separation Optimization Framework for Cross-Entropy Training in Deep Learning
School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）；Department of Mathematics, Hong Kong Baptist University（香港 Baptist 大学数学系）
We propose a permutation-invariant loss function designed for the neural networks reconstructing a set of elements without considering the order within its vector representation. Unlike popular approaches for encoding and decoding a set, our work does not rely on a carefully engineered network topology nor by any additional sequential algorithm. The proposed method, Set Cross Entropy, has a natural information-theoretic interpretation and is related to the metrics defined for sets. We evaluate the proposed approach in two object reconstruction tasks and a rule learning task.
Jagiellonian University（雅盖隆大学）；AGH University of Science and Technology（AGH科技大学）
The R Package CEC performs clustering based on the cross-entropy clustering (CEC) method, which was recently developed with the use of information theory. The main advantage of CEC is that it combines the speed and simplicity of $k$-means with the ability to use various Gaussian mixture models and reduce unnecessary clusters. In this work we present a practical tutorial to CEC based on the R Package CEC. Functions are provided to encompass the whole process of clustering.
Many offline unsupervised change point detection algorithms rely on minimizing a penalized sum of segment-wise costs. We extend this framework by proposing to minimize a sum of discrepancies between segments. In particular, we propose to select the change points so as to maximize the cross-entropy between successive segments, balanced by a penalty for introducing new change points. We propose a dynamic programming algorithm to solve this problem and analyze its complexity. Experiments on two challenging datasets demonstrate the advantages of our method compared to three state-of-the-art approa…
Recently, substantial research efforts in Deep Metric Learning (DML) focused on designing complex pairwise-distance losses, which require convoluted schemes to ease optimization, such as sample mining or pair weighting. The standard cross-entropy loss for classification has been largely overlooked in DML. On the surface, the cross-entropy may seem unrelated and irrelevant to metric learning as it does not explicitly involve pairwise distances. However, we provide a theoretical analysis that links the cross-entropy to several well-known and recent pairwise losses. Our connections are drawn from…
The Rényi cross-entropy measure between two distributions, a generalization of the Shannon cross-entropy, was recently used as a loss function for the improved design of deep learning generative adversarial networks. In this work, we examine the properties of this measure and derive closed-form expressions for it when one of the distributions is fixed and when both distributions belong to the exponential family. We also analytically determine a formula for the cross-entropy rate for stationary Gaussian processes and for finite-alphabet Markov sources.
Google Research（谷歌研究院）；National Institutes of Health（美国国立卫生研究院）
Listwise learning-to-rank methods form a powerful class of ranking algorithms that are widely adopted in applications such as information retrieval. These algorithms learn to rank a set of items by optimizing a loss that is a function of the entire set -- as a surrogate to a typically non-differentiable ranking metric. Despite their empirical success, existing listwise methods are based on heuristics and remain theoretically ill-understood. In particular, none of the empirically successful loss functions are related to ranking metrics. In this work, we propose a cross entropy-based learning-to…
University of Oxford（牛津大学）；University of Cambridge（剑桥大学）；KAIST（韩国科学技术院）；University of British Columbia（英属哥伦比亚大学）
We develop a new Low-level, First-order Probabilistic Programming Language (LF-PPL) suited for models containing a mix of continuous, discrete, and/or piecewise-continuous variables. The key success of this language and its compilation scheme is in its ability to automatically distinguish parameters the density function is discontinuous with respect to, while further providing runtime checks for boundary crossings. This enables the introduction of new inference engines that are able to exploit gradient information, while remaining efficient for models which are not everywhere differentiable. W…
University of Hildesheim（希尔德斯海姆大学）；University of Freiburg（弗莱堡大学）
Metafeatures, or dataset characteristics, have been shown to improve the performance of hyperparameter optimization (HPO). Conventionally, metafeatures are precomputed and used to measure the similarity between datasets, leading to a better initialization of HPO models. In this paper, we propose a cross dataset surrogate model called Differentiable Metafeature-based Surrogate (DMFBS), that predicts the hyperparameter response, i.e. validation loss, of a model trained on the dataset at hand. In contrast to existing models, DMFBS i) integrates a differentiable metafeature extractor and ii) is op…
Duke University（杜克大学）；University of Washington（华盛顿大学）
We consider the variational problem of cross-entropy loss with $n$ feature vectors on a unit hypersphere in $\mathbb{R}^d$. We prove that when $d \geq n - 1$, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that as $n \rightarrow \infty$ with fixed $d$, the minimizing points will distribute uniformly on the hypersphere and show a connection with the frame potential of Benedetto & Fickus.
National Library of Medicine, National Institutes of Health（美国国立医学图书馆，美国国立卫生研究院）
Minimizing cross-entropy is a widely used method for training artificial neural networks. Many training procedures based on backpropagation use cross-entropy directly as their loss function. Instead, this theoretical essay investigates a dual process model with two processes, in which one process minimizes the Kullback-Leibler divergence while its dual counterpart minimizes the Shannon entropy. Postulating that learning consists of two dual processes complementing each other, the model defines an equilibrium state for both processes in which the loss function assumes its minimum. An advantage …
Artificial Intelligence Graduate School, Pohang University of Science and Technology（浦项科技大学人工智能研究生院）；Department of Mathematics, Pohang University of Science and Technology（浦项科技大学数学系）
Singapore Management University（新加坡管理大学）；Tencent AI Lab（腾讯AI实验室）
We propose a new training objective named order-agnostic cross entropy (OaXE) for fully non-autoregressive translation (NAT) models. OaXE improves the standard cross-entropy loss to ameliorate the effect of word reordering, which is a common source of the critical multimodality problem in NAT. Concretely, OaXE removes the penalty for word order errors, and computes the cross entropy loss based on the best possible alignment between model predictions and target tokens. Since the log loss is very sensitive to invalid references, we leverage cross entropy initialization and loss truncation to ens…
IBM Research AI, IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心IBM研究人工智能部门）；IBM Watson（IBM沃森）
We introduce Multi-Frame Cross-Entropy training (MFCE) for convolutional neural network acoustic models. Recognizing that similar to RNNs, CNNs are in nature sequence models that take variable length inputs, we propose to take as input to the CNN a part of an utterance long enough that multiple labels are predicted at once, therefore getting cross-entropy loss signal from multiple adjacent frames. This increases the amount of label information drastically for small marginal computational cost. We show large WER improvements on hub5 and rt02 after training on the 2000-hour Switchboard benchmark…
Shell Global Solutions International B.V.（壳牌全球解决方案国际公司）；Delft University of Technology (TU Delft)（代尔夫特理工大学）
Traditional empirical risk minimization (ERM) for semantic segmentation can disproportionately advantage or disadvantage certain target classes in favor of an (unfair but) improved overall performance. Inspired by the recently introduced tilted ERM (TERM), we propose tilted cross-entropy (TCE) loss and adapt it to the semantic segmentation setting to minimize performance disparity among target classes and promote fairness. Through quantitative and qualitative performance analyses, we demonstrate that the proposed Stochastic TCE for semantic segmentation can efficiently improve the low-performi…
Enhancing Cross Entropy with a Linearly Adaptive Loss Function for Optimized Classification Performance
Extreme Materials Research Center, Korea Institute of Science and Technology（极端材料研究中心，韩国科学技术院）；Climate and Environmental Research Institute, Korea Institute of Science and Technology（气候与环境研究所，韩国科学技术院）；Division of AI-Robotics, KIST Campus, University of Science and Technology（人工智能机器人系，韩国科学技术院校园）
