Recent advances in coreset methods have shown that a selection of representative datapoints can replace massive volumes of data for Bayesian inference, preserving the relevant statistical information and significantly accelerating subsequent downstream tasks. Existing variational coreset constructions rely on either selecting subsets of the observed datapoints, or jointly performing approximate inference and optimizing pseudodata in the observed space akin to inducing points methods in Gaussian Processes. So far, both approaches are limited by complexities in evaluating their objectives for general purpose models, and require generating samples from a typically intractable posterior over the coreset throughout inference and testing. In this work, we present a black-box variational inference framework for coresets that overcomes these constraints and enables principled application of variational coresets to intractable models, such as Bayesian neural networks. We apply our techniques to supervised learning problems, and compare them with existing approaches in the literature for data summarization and inference.

变异推理（VI）的核心原理是将计算复杂后概率密度计算的统计推断问题转换为可拖动的优化问题。该属性使VI比几种基于采样的技术更快。但是，传统的VI算法无法扩展到大型数据集，并且无法轻易推断出越野数据点，而无需重新运行优化过程。该领域的最新发展，例如随机，黑框和摊销VI，已帮助解决了这些问题。如今，生成的建模任务广泛利用摊销VI来实现其效率和可扩展性，因为它利用参数化函数来学习近似的后验密度参数。在本文中，我们回顾了各种VI技术的数学基础，以构成理解摊销VI的基础。此外，我们还概述了最近解决摊销VI问题的趋势，例如摊销差距，泛化问题，不一致的表示学习和后验崩溃。最后，我们分析了改善VI优化的替代差异度量。

We develop an optimization algorithm suitable for Bayesian learning in complex models. Our approach relies on natural gradient updates within a general black-box framework for efficient training with limited model-specific derivations. It applies within the class of exponential-family variational posterior distributions, for which we extensively discuss the Gaussian case for which the updates have a rather simple form. Our Quasi Black-box Variational Inference (QBVI) framework is readily applicable to a wide class of Bayesian inference problems and is of simple implementation as the updates of the variational posterior do not involve gradients with respect to the model parameters, nor the prescription of the Fisher information matrix. We develop QBVI under different hypotheses for the posterior covariance matrix, discuss details about its robust and feasible implementation, and provide a number of real-world applications to demonstrate its effectiveness.

隐式过程（IPS）代表一个灵活的框架，可用于描述各种模型，从贝叶斯神经网络，神经抽样器和数据生成器到许多其他模型。 IP还允许在功能空间上进行大致推断。公式的这种变化解决了参数空间的固有退化问题近似推断，即参数数量及其在大型模型中的强大依赖性。为此，文献中先前的作品试图采用IPS来设置先验并近似产生的后部。但是，这被证明是一项具有挑战性的任务。现有的方法可以调整先前的IP导致高斯预测分布，该分布未能捕获重要的数据模式。相比之下，通过使用另一个IP近似后验过程产生灵活预测分布的方法不能将先前的IP调整到观察到的数据中。我们在这里建议第一个可以实现这两个目标的方法。为此，我们依赖于先前IP的诱导点表示，就像在稀疏高斯过程中所做的那样。结果是一种可扩展的方法，用于与IP的近似推断，可以将先前的IP参数调整到数据中，并提供准确的非高斯预测分布。

One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this paper, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find the member of that family which is close to the target. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this paper is to catalyze statistical research on this class of algorithms.

已知生物制剂在他们的生活过程中学习许多不同的任务，并且能够重新审视以前的任务和行为，而没有表现不损失。相比之下，人工代理容易出于“灾难性遗忘”，在以前任务上的性能随着所获取的新的任务而恶化。最近使用该方法通过鼓励参数保持接近以前任务的方法来解决此缺点。这可以通过（i）使用特定的参数正常数来完成，该参数正常数是在参数空间中映射合适的目的地，或（ii）通过将渐变投影到不会干扰先前任务的子空间来指导优化旅程。然而，这些方法通常在前馈和经常性神经网络中表现出子分子表现，并且经常性网络对支持生物持续学习的神经动力学研究感兴趣。在这项工作中，我们提出了自然的持续学习（NCL），一种统一重量正则化和预测梯度下降的新方法。 NCL使用贝叶斯重量正常化来鼓励在收敛的所有任务上进行良好的性能，并将其与梯度投影结合使用先前的精度，这可以防止在优化期间陷入灾难性遗忘。当应用于前馈和经常性网络中的连续学习问题时，我们的方法占据了标准重量正则化技术和投影的方法。最后，训练有素的网络演变了特定于任务特定的动态，这些动态被认为是学习的新任务，类似于生物电路中的实验结果。

现代深度学习方法构成了令人难以置信的强大工具，以解决无数的挑战问题。然而，由于深度学习方法作为黑匣子运作，因此与其预测相关的不确定性往往是挑战量化。贝叶斯统计数据提供了一种形式主义来理解和量化与深度神经网络预测相关的不确定性。本教程概述了相关文献和完整的工具集，用于设计，实施，列车，使用和评估贝叶斯神经网络，即使用贝叶斯方法培训的随机人工神经网络。

变分推理是一种强大的范例，用于近似贝叶斯推论，具有许多吸引人的属性，包括支持模型学习和数据分配。通过对比MCMC方法，如Hamiltonian Monte Carlo不共享这些属性，但由于与参数方法相反，因此仍然有吸引力，因此MCMC是无偏见的。由于这些原因，研究人员试图将两类算法的优势结合起来，最近的方法更接近在实践中实现这一愿景。然而，支持这些混合方法中的数据分配可能是一个挑战，通过引入可以与其他变分参数共同学习的替代可能性来解决的缺点。从理论上，理论上我们认为所产生的算法允许用户在推理保真度和计算成本之间进行直观的折衷。在一个广泛的经验比较中，我们表明我们的方法在实践中表现良好，并且它非常适合在概率编程框架中的黑匣子推断。

We present the GPry algorithm for fast Bayesian inference of general (non-Gaussian) posteriors with a moderate number of parameters. GPry does not need any pre-training, special hardware such as GPUs, and is intended as a drop-in replacement for traditional Monte Carlo methods for Bayesian inference. Our algorithm is based on generating a Gaussian Process surrogate model of the log-posterior, aided by a Support Vector Machine classifier that excludes extreme or non-finite values. An active learning scheme allows us to reduce the number of required posterior evaluations by two orders of magnitude compared to traditional Monte Carlo inference. Our algorithm allows for parallel evaluations of the posterior at optimal locations, further reducing wall-clock times. We significantly improve performance using properties of the posterior in our active learning scheme and for the definition of the GP prior. In particular we account for the expected dynamical range of the posterior in different dimensionalities. We test our model against a number of synthetic and cosmological examples. GPry outperforms traditional Monte Carlo methods when the evaluation time of the likelihood (or the calculation of theoretical observables) is of the order of seconds; for evaluation times of over a minute it can perform inference in days that would take months using traditional methods. GPry is distributed as an open source Python package (pip install gpry) and can also be found at https://github.com/jonaselgammal/GPry.

How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions is two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, we show that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. Theoretical advantages are reflected in experimental results.

在使用多模式贝叶斯后部分布时，马尔可夫链蒙特卡罗（MCMC）算法难以在模式之间移动，并且默认变分或基于模式的近似推动将低估后不确定性。并且，即使找到最重要的模式，难以评估后部的相对重量。在这里，我们提出了一种使用MCMC，变分或基于模式的模式的并行运行的方法，以便尽可能多地击中多种模式或分离的区域，然后使用贝叶斯堆叠来组合这些用于构建分布的加权平均值的可扩展方法。通过堆叠从多模式后分布的堆叠，最小化交叉验证预测误差的结果，并且代表了比变分推断更好的不确定度，但它不一定是相当于渐近的，以完全贝叶斯推断。我们呈现理论一致性，其中堆叠推断逼近来自未衰退的模型和非混合采样器的真实数据生成过程，预测性能优于完全贝叶斯推断，因此可以被视为祝福而不是模型拼写下的诅咒。我们展示了几个模型家庭的实际实施：潜在的Dirichlet分配，高斯过程回归，分层回归，马蹄素变量选择和神经网络。

The kernel function and its hyperparameters are the central model selection choice in a Gaussian proces (Rasmussen and Williams, 2006). Typically, the hyperparameters of the kernel are chosen by maximising the marginal likelihood, an approach known as Type-II maximum likelihood (ML-II). However, ML-II does not account for hyperparameter uncertainty, and it is well-known that this can lead to severely biased estimates and an underestimation of predictive uncertainty. While there are several works which employ a fully Bayesian characterisation of GPs, relatively few propose such approaches for the sparse GPs paradigm. In this work we propose an algorithm for sparse Gaussian process regression which leverages MCMC to sample from the hyperparameter posterior within the variational inducing point framework of Titsias (2009). This work is closely related to Hensman et al. (2015b) but side-steps the need to sample the inducing points, thereby significantly improving sampling efficiency in the Gaussian likelihood case. We compare this scheme against natural baselines in literature along with stochastic variational GPs (SVGPs) along with an extensive computational analysis.

随机梯度马尔可夫链蒙特卡洛（SGMCMC）被认为是大型模型（例如贝叶斯神经网络）中贝叶斯推断的金标准。由于从业人员在这些模型中面临速度与准确性权衡，因此变异推理（VI）通常是可取的选择。不幸的是，VI对后部的分解和功能形式做出了有力的假设。在这项工作中，我们提出了一个新的非参数变分近似，该近似没有对后验功能形式进行假设，并允许从业者指定算法应尊重或断裂的确切依赖性。该方法依赖于在修改的能量函数上运行的新的langevin型算法，其中潜在变量的一部分是在马尔可夫链的早期迭代中平均的。这样，统计依赖性可以以受控的方式破裂，从而使链条混合更快。可以以“辍学”方式进一步修改该方案，从而导致更大的可扩展性。我们在CIFAR-10，SVHN和FMNIST上测试RESNET-20的计划。在所有情况下，与SG-MCMC和VI相比，我们都会发现收敛速度和/或最终精度的提高。

近似贝叶斯深度学习方法对于解决在智能系统中部署深度学习组件时，包括在智能系统中部署深度学习组件的几个问题，包括减轻过度自信的错误并提供增强的鲁棒性，从而超出分发示例。但是，现有近似贝叶斯推理方法的计算要求可以使它们不适合部署包括低功耗边缘设备的智能IOT系统。在本文中，我们为监督深度学习提供了一系列近似贝叶斯推理方法，并在应用这些方法对当前边缘硬件上的挑战和机遇。我们突出了几种潜在的解决方案来降低模型存储要求，提高计算可扩展性，包括模型修剪和蒸馏方法。

Variational inference has become a widely used method to approximate posteriors in complex latent variables models. However, deriving a variational inference algorithm generally requires significant model-specific analysis, and these efforts can hinder and deter us from quickly developing and exploring a variety of models for a problem at hand. In this paper, we present a "black box" variational inference algorithm, one that can be quickly applied to many models with little additional derivation. Our method is based on a stochastic optimization of the variational objective where the noisy gradient is computed from Monte Carlo samples from the variational distribution. We develop a number of methods to reduce the variance of the gradient, always maintaining the criterion that we want to avoid difficult model-based derivations. We evaluate our method against the corresponding black box sampling based methods. We find that our method reaches better predictive likelihoods much faster than sampling methods. Finally, we demonstrate that Black Box Variational Inference lets us easily explore a wide space of models by quickly constructing and evaluating several models of longitudinal healthcare data.

我们提出了一种新的非参数混合物模型，用于多变量回归问题，灵感来自概率K-Nearthimest邻居算法。使用有条件指定的模型，对样本外输入的预测基于与每个观察到的数据点的相似性，从而产生高斯混合物表示的预测分布。在混合物组件的参数以及距离度量标准的参数上，使用平均场变化贝叶斯算法进行后推断，并具有基于随机梯度的优化过程。在与数据大小相比，输入 - 输出关系很复杂，预测分布可能偏向或多模式的情况下，输入相对较高的尺寸，该方法尤其有利。对五个数据集进行的计算研究，其中两个是合成生成的，这说明了我们的高维输入的专家混合物方法的明显优势，在验证指标和视觉检查方面都优于竞争者模型。

统计模型是机器学习的核心，具有广泛适用性，跨各种下游任务。模型通常由通过最大似然估计从数据估计的自由参数控制。但是，当面对现实世界数据集时，许多模型运行到一个关键问题：它们是在完全观察到的数据方面配制的，而在实践中，数据集会困扰缺失数据。来自不完整数据的统计模型估计理论在概念上类似于潜在变量模型的估计，其中存在强大的工具，例如变分推理（VI）。然而，与标准潜在变量模型相比，具有不完整数据的参数估计通常需要估计缺失变量的指数 - 许多条件分布，因此使标准的VI方法是棘手的。通过引入变分Gibbs推理（VGI），是一种新的通用方法来解决这个差距，以估计来自不完整数据的统计模型参数。我们在一组合成和实际估算任务上验证VGI，从不完整的数据中估算重要的机器学习模型，VAE和标准化流程。拟议的方法，同时通用，实现比现有的特定模型特定估计方法竞争或更好的性能。

贝叶斯范式有可能解决深度神经网络的核心问题，如校准和数据效率低差。唉，缩放贝叶斯推理到大量的空间通常需要限制近似。在这项工作中，我们表明它足以通过模型权重的小子集进行推动，以便获得准确的预测后断。另一个权重被保存为点估计。该子网推断框架使我们能够在这些子集上使用表现力，否则难以相容的后近近似。特别是，我们将子网线性化LAPLACE作为一种简单，可扩展的贝叶斯深度学习方法：我们首先使用线性化的拉普拉斯近似来获得所有重量的地图估计，然后在子网上推断出全协方差高斯后面。我们提出了一个子网选择策略，旨在最大限度地保护模型的预测性不确定性。经验上，我们的方法对整个网络的集合和较少的表达后近似进行了比较。

We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the minibatch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gaussian dropout objectives correspond to SGVB with local reparameterization, a scale-invariant prior and proportionally fixed posterior variance. Our method allows inference of more flexibly parameterized posteriors; specifically, we propose variational dropout, a generalization of Gaussian dropout where the dropout rates are learned, often leading to better models. The method is demonstrated through several experiments.

我们提出了使用多级蒙特卡罗（MLMC）方法的变分推理的差异减少框架。我们的框架是基于Reparameterized梯度估计的梯度估计，并在优化中从过去更新历史记录获得的“回收”参数。此外，我们的框架还提供了一种基于随机梯度下降（SGD）的新优化算法，其自适应地估计根据梯度方差的比率用于梯度估计的样本大小。理论上，通过我们的方法，梯度估计器的方差随着优化进行而降低，并且学习率调度器函数有助于提高收敛。我们还表明，就\ Texit {信噪比}比率而言，我们的方法可以通过提高初始样本大小来提高学习速率调度器功能的梯度估计的质量。最后，我们确认我们的方法通过使用多个基准数据集的基线方法的实验比较来实现更快的收敛性并降低梯度估计器的方差，并降低了与其他方法相比的其他方法。