Compared to point estimates calculated by standard neural networks, Bayesian neural networks (BNN) provide probability distributions over the output predictions and model parameters, i.e., the weights. Training the weight distribution of a BNN, however, is more involved due to the intractability of the underlying Bayesian inference problem and thus, requires efficient approximations. In this paper, we propose a novel approach for BNN learning via closed-form Bayesian inference. For this purpose, the calculation of the predictive distribution of the output and the update of the weight distribution are treated as Bayesian filtering and smoothing problems, where the weights are modeled as Gaussian random variables. This allows closed-form expressions for training the network's parameters in a sequential/online fashion without gradient descent. We demonstrate our method on several UCI datasets and compare it to the state of the art.

Large multilayer neural networks trained with backpropagation have recently achieved state-ofthe-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalable method for learning Bayesian neural networks, called probabilistic backpropagation (PBP). Similar to classical backpropagation, PBP works by computing a forward propagation of probabilities through the network and then doing a backward computation of gradients. A series of experiments on ten real-world datasets show that PBP is significantly faster than other techniques, while offering competitive predictive abilities. Our experiments also show that PBP provides accurate estimates of the posterior variance on the network weights.

在本文中，我们提出了一种在贝叶斯神经网络中执行近似高斯推理（Tagi）的分析方法。该方法使得后尺寸矢量和对角线协方差矩阵的分析高斯推断用于重量和偏差。提出的方法具有$ \ mathcal {o}（n）$的计算复杂性，与参数$ n $的数量，并且对回归和分类基准测试的测试确认，对于相同的网络架构，它匹配依赖于梯度背交的现有方法的性能。

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。

神经随机微分方程（NSDES）模拟随机过程作为神经网络的漂移和扩散函数。尽管已知NSDE可以进行准确的预测，但到目前为止，其不确定性定量属性仍未探索。我们报告了经验发现，即从NSDE获得良好的不确定性估计是计算上的过度估计。作为一种补救措施，我们开发了一种计算负担得起的确定性方案，该方案在动力学受NSD管辖时准确地近似过渡内核。我们的方法引入了匹配算法的二维力矩：沿着神经净层和沿时间方向水平的垂直力，这受益于有效近似的原始组合。我们对过渡内核的确定性近似适用于培训和预测。我们在多个实验中观察到，我们方法的不确定性校准质量只有在引入高计算成本后才通过蒙特卡洛采样来匹配。由于确定性培训的数值稳定性，我们的方法还提高了预测准确性。

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。

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。

We propose SWA-Gaussian (SWAG), a simple, scalable, and general purpose approach for uncertainty representation and calibration in deep learning. Stochastic Weight Averaging (SWA), which computes the first moment of stochastic gradient descent (SGD) iterates with a modified learning rate schedule, has recently been shown to improve generalization in deep learning. With SWAG, we fit a Gaussian using the SWA solution as the first moment and a low rank plus diagonal covariance also derived from the SGD iterates, forming an approximate posterior distribution over neural network weights; we then sample from this Gaussian distribution to perform Bayesian model averaging. We empirically find that SWAG approximates the shape of the true posterior, in accordance with results describing the stationary distribution of SGD iterates. Moreover, we demonstrate that SWAG performs well on a wide variety of tasks, including out of sample detection, calibration, and transfer learning, in comparison to many popular alternatives including MC dropout, KFAC Laplace, SGLD, and temperature scaling.

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.

Kullback-Leibler（KL）差异广泛用于贝叶斯神经网络（BNNS）的变异推理。然而，KL差异具有无限性和不对称性等局限性。我们检查了更通用，有限和对称的詹森 - 香农（JS）差异。我们根据几何JS差异为BNN制定新的损失函数，并表明基于KL差异的常规损失函数是其特殊情况。我们以封闭形式的高斯先验评估拟议损失函数的差异部分。对于任何其他一般的先验，都可以使用蒙特卡洛近似值。我们提供了实施这两种情况的算法。我们证明所提出的损失函数提供了一个可以调整的附加参数，以控制正则化程度。我们得出了所提出的损失函数在高斯先验和后代的基于KL差异的损失函数更好的条件。我们证明了基于嘈杂的CIFAR数据集和有偏见的组织病理学数据集的最新基于KL差异的BNN的性能提高。

Deep learning tools have gained tremendous attention in applied machine learning. However such tools for regression and classification do not capture model uncertainty. In comparison, Bayesian models offer a mathematically grounded framework to reason about model uncertainty, but usually come with a prohibitive computational cost. In this paper we develop a new theoretical framework casting dropout training in deep neural networks (NNs) as approximate Bayesian inference in deep Gaussian processes. A direct result of this theory gives us tools to model uncertainty with dropout NNsextracting information from existing models that has been thrown away so far. This mitigates the problem of representing uncertainty in deep learning without sacrificing either computational complexity or test accuracy. We perform an extensive study of the properties of dropout's uncertainty. Various network architectures and nonlinearities are assessed on tasks of regression and classification, using MNIST as an example. We show a considerable improvement in predictive log-likelihood and RMSE compared to existing state-of-the-art methods, and finish by using dropout's uncertainty in deep reinforcement learning.

We investigate the efficacy of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical evidence that this standard construction may be unnecessary. To this end, we prove that expressive predictive distributions require only small amounts of stochasticity. In particular, partially stochastic networks with only $n$ stochastic biases are universal probabilistic predictors for $n$-dimensional predictive problems. In empirical investigations, we find no systematic benefit of full stochasticity across four different inference modalities and eight datasets; partially stochastic networks can match and sometimes even outperform fully stochastic networks, despite their reduced memory costs.

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。

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.

我们提出了一种新的方法，可以在复杂模型（例如贝叶斯神经网络）中执行近似贝叶斯推断。该方法比马尔可夫链蒙特卡洛更可扩展到大数据，它具有比变异推断更具表现力的模型，并且不依赖于对抗训练（或密度比估计）。我们采用了构建两个模型的最新方法：（1）一个主要模型，负责执行回归或分类； （2）一个辅助，表达的（例如隐式）模型，该模型定义了主模型参数上的近似后验分布。但是，我们根据后验预测分布的蒙特卡洛估计值通过梯度下降来优化后验模型的参数 - 这是我们唯一的近似值（除后模型除外）。只需要指定一个可能性，可以采用各种形式，例如损失功能和合成可能性，从而提供无可能的方法的形式。此外，我们制定了该方法，使后样品可以独立于或有条件地取决于主要模型的输入。后一种方法被证明能够增加主要模型的明显复杂性。我们认为这在诸如替代和基于物理的模型之类的应用中很有用。为了促进贝叶斯范式如何提供不仅仅是不确定性量化的方式，我们证明了：不确定性量化，多模式以及具有最新预测的神经网络体系结构的应用。

表示学习已成为一种实用的方法，可以在重建方面成功地建立大量高维数据的丰富参数编码。在考虑具有测试训练分布变化的无监督任务时，概率的观点有助于解决预测过度自信和不良校准。但是，由于多种原因，即维度或顽固性问题的诅咒，直接引入贝叶斯推断仍然是一个艰难的问题。 Laplace近似（LA）在这里提供了一个解决方案，因为可以通过二阶Taylor膨胀在参数空间的某些位置通过二阶Taylor膨胀来建立重量的高斯近似值。在这项工作中，我们为洛杉矶启发的无监督表示学习提供了贝叶斯自动编码器。我们的方法实现了迭代的拉普拉斯更新，以获得新型自动编码器证据的新变化下限。二阶部分衍生物的巨大计算负担是通过Hessian矩阵的近似来跳过的。从经验上讲，我们通过为分布外检测提供了良好的不确定性，用于差异几何形状的大地测量和缺失数据归思的方法来证明拉普拉斯自动编码器的可伸缩性和性能。

We propose a simultaneous learning and pruning algorithm capable of identifying and eliminating irrelevant structures in a neural network during the early stages of training. Thus, the computational cost of subsequent training iterations, besides that of inference, is considerably reduced. Our method, based on variational inference principles using Gaussian scale mixture priors on neural network weights, learns the variational posterior distribution of Bernoulli random variables multiplying the units/filters similarly to adaptive dropout. Our algorithm, ensures that the Bernoulli parameters practically converge to either 0 or 1, establishing a deterministic final network. We analytically derive a novel hyper-prior distribution over the prior parameters that is crucial for their optimal selection and leads to consistent pruning levels and prediction accuracy regardless of weight initialization or the size of the starting network. We prove the convergence properties of our algorithm establishing theoretical and practical pruning conditions. We evaluate the proposed algorithm on the MNIST and CIFAR-10 data sets and the commonly used fully connected and convolutional LeNet and VGG16 architectures. The simulations show that our method achieves pruning levels on par with state-of the-art methods for structured pruning, while maintaining better test-accuracy and more importantly in a manner robust with respect to network initialization and initial size.

A large amount of recent research has the far-reaching goal of finding training methods for deep neural networks that can serve as alternatives to backpropagation (BP). A prominent example is predictive coding (PC), which is a neuroscience-inspired method that performs inference on hierarchical Gaussian generative models. These methods, however, fail to keep up with modern neural networks, as they are unable to replicate the dynamics of complex layers and activation functions. In this work, we solve this problem by generalizing PC to arbitrary probability distributions, enabling the training of architectures, such as transformers, that are hard to approximate with only Gaussian assumptions. We perform three experimental analyses. First, we study the gap between our method and the standard formulation of PC on multiple toy examples. Second, we test the reconstruction quality on variational autoencoders, where our method reaches the same reconstruction quality as BP. Third, we show that our method allows us to train transformer networks and achieve a performance comparable with BP on conditional language models. More broadly, this method allows neuroscience-inspired learning to be applied to multiple domains, since the internal distributions can be flexibly adapted to the data, tasks, and architectures used.

Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs, which learn a distribution over weights, are currently the state-of-the-art for estimating predictive uncertainty; however these require significant modifications to the training procedure and are computationally expensive compared to standard (non-Bayesian) NNs. We propose an alternative to Bayesian NNs that is simple to implement, readily parallelizable, requires very little hyperparameter tuning, and yields high quality predictive uncertainty estimates. Through a series of experiments on classification and regression benchmarks, we demonstrate that our method produces well-calibrated uncertainty estimates which are as good or better than approximate Bayesian NNs. To assess robustness to dataset shift, we evaluate the predictive uncertainty on test examples from known and unknown distributions, and show that our method is able to express higher uncertainty on out-of-distribution examples. We demonstrate the scalability of our method by evaluating predictive uncertainty estimates on ImageNet.

深度神经网络易于对异常值过度自信的预测。贝叶斯神经网络和深度融合都已显示在某种程度上减轻了这个问题。在这项工作中，我们的目标是通过提议预测由高斯混合模型的后续的高斯混合模型来结合这两种方法的益处，该高斯混合模型包括独立培训的深神经网络的LAPPALL近似的加权和。该方法可以与任何一组预先训练的网络一起使用，并且与常规合并相比，只需要小的计算和内存开销。理论上我们验证了我们的方法从训练数据中的培训数据和虚拟化的基本线上的标准不确定量级基准测试中的“远离”的过度控制。