introduction

• Variational inference applied to large datasets：Approximate intractable posterior distributions with a class of known probability distributions

• 发展历程：Large-scale topic model core from text（(Hoffman et al., 2013）-》Semi-supervised classification techniques》Drive the most like image model now,Default model for many physicochemical systems.But a posteriori approximate limit is more widely used
  • 平均场：The mean field is that the variational probability density function is considered to be each random variable.（参数）The product of the variational probability density functions of（i.e. factorization）
  • Deep Autoregressive Networks
  • 提出不足：Posterior variance underestimation、The limitations of the posterior approximation lead to theMAP（最大后验估计）Estimates are biased
  • Mixed Model Constraints Computational Power

amortized variational inference

• 观测变量x,潜在变量z,模型参数,Introduce the posterior likelihood distribution and follow the variational principle to obtainmarginal likelihood的边界,通常被作为F或者ELBO提到.（3）是目标函数

  

  • The current best practice to use small batch variational reasoning and stochastic gradient descent to perform the optimization to solve large data sets,主要需要解决
    • Efficient Computational Derivation
    • rich computable The focus of this article

•  Stochastic Backpropagation

  • Reparameterization:We use a known cardinality distribution and a differentiable transformation(Such as position scaling transformation or cumulative distribution function)to reparameterize latent variables e.g.

    

  • ###Backpropagation with Monte Carlo:Monte Carlo Fitting from Cardinal Extraction to Differentiate a Variable of a Variational Distribution

    

  • MCCV（Monte Carlo control variables）exists as an alternative to random backpropagation,Allow latent variables to be continuous/离散,But stochastic backpropagation for models with continuous latent variables,It has the smallest variance among competing estimators

• Inference Networks：A modulo that learns the inverse mapping from observations to latent variables

  • Represent approximate posterior distributions with recognition models or inference networksqφ(·),Based on this only need to compute a set of global variational parametersφ,Valid at both training time and test time
  • Take the Gaussian density function as an example

    

    
    where the mean functionμφ(x)and standard deviation functionσφ(x)is specified with a deep neural network

• Deep Latent Gaussian Models:由Llayer Gaussian latent variables at each layerZldeep directed graph model

  

  
  Each layer of latent variables depends on the layer above in a non-linear fashion,Nonlinear dependencies are specified by deep neural networks
  • 第LLayer Gaussian distribution does not depend on any other random variable
  • The prior of the latent variable is a unit Gaussian function as follows,observation likelihood distributionpθ (x|z)是以z1为条件,and any suitable distribution parameterized by a deep neural network
  • can be viewed as a variational encoder

Normalizing Flows

• By examining the border(3),we can see that allowIDKL[q‖p] = 0The optimal variational distribution of is aqφ(z|x) = pθ (z|x)的分布,即q为真实分布,但不可能,such as uncorrelated Gaussian functions or other mean-field approximations,So I want to find a flexible family of variational distributions that can contain real solutions,One way to achieve this ideal is based on the principle of non-generalized flows

• 有限流 finite flows

  • reversible smooth mapf,且f逆=g,组合g ◦ f (z) = z,用其将z变换为q(z),get random variablez ' = f (z)There is a distribution as follows,fHigh dimensions are reduced by Jacobian determinant

    

  • 将随机变量z0与分布q0通过Ktransform chainfkDensity obtained by successive transformationqK (z)为,即（6）for a continuousnormalizing流

    

  • 通过转换,without knowingqK的情况下计算qK,any expectationsEqK [h(z)]可以写成q0下的期望为: h(z)不依赖于qKYou do not need to calculate has nothing to dologdet-Jacobian项

    

• infinitesimal flow

  • Partial differential equations describing the initial densityq0(z)随时间演变： （Random parameters can change at any time）

    

  • Langevin Flow：Langevinstochastic differential equations given by

    

    
    If we take a random variablezWith the initial densityq0(z)通过Langevin流(9),Then the rule for density transformation is given byFokker-Planck方程(Or the probability ofKolmogorov方程
    • dξ(t)是一个E[ξi(t)] = 0and as shown belowWiener过程
      • (Wiener过程：A typical Markov random process)
    • F是drift vector ,D=GG^T是diffusion matrix
    • The transformed samples are intdensity of timeqt(z)变化为:
    • In machine learning, this is usually brought into：
      • If we start from the initial densityq0(z)开始,并通过Langevin SDEEvolution of the samplez0,the result pointz∞将按照q∞(z)∝e- L(z)分布
  • Hamiltonian Flow：Hamiltonian Monte Carlo Dynamically generate results from the graph below,in augmentation space ̃z = (z, ω)can also be seen asnormalizing flow

    

Inference with Normalizing Flows

• The current scale is calculated as follows,D是隐藏层维度,L是隐藏层数,In addition, computing the gradient of the Jacobian involves several alsoO(LD3)的附加运算,contains a numerically unstable matrix inverse.Therefore to seek low cost

  

• Invertible Linear-time Transformations：

  • 在O(D)时间内计算logdet-Jacobian项(Using matrix determinant lemma)

    

    • λ是自由变量,h(·)a smooth nonlinear element
    • 在映射时
    • 此时（7）改写成：
      • Modify the initial density by applying a series of contractions and expansions in the direction normal to the hyperplaneq0,Hence it is called flat flow
  • 替换：参考点z0nearby changing initial densityq0family of transformations

    

    • Allows linear time computation of determinants,For radial contraction and expansion around the reference point,so called radial flow
  • Influence of expansion and contraction on uniform Gaussian initial density,The Gaussian transformation of this sphere is converted into a bimodal distribution

    

  • 输入输出模型

    

• Flow-Based Free Energy Bound：长度为kflow about（3）的展开

  

• Algorithm Summary and Complexity：Jointly sample and compute the logarithm of the inference model-det-The algorithmic complexity of the Jacobian term isO(LN 2) + O(KD)

  

  
  Lis the number of deterministic layers used to map data to flow parameters,Nis the average hidden layer size,Kis the flow length,Dis the dimension of the latent variable.
  • 更新

Alternative Flow-based Posteriors

• a finite volume-preserving flow ,Neural Network Transformation Example NICE

  

  • 其中
  • The upper triangular part of the Jacobian matrix obtained in this form is zero,so that the determinant is1.

• The resulting density of the forward and inverse transformation is given by

  

Results:评估在DLGMused in inferencenormalizing flow-based posterior approximations 效果

• Simulated annealing by following a gradient,对模型参数θand variational parametersφTraining with random backpropagation.

  

  
  βt∈[0,1],After ten thousand iterations from0.01到1
  • Simple example calculation of hidden variables for each data point and variable update,4variable windowMaxout Nonlinear Hidden Layer400个
    • 输入向量
    • ∆：MaxoutNonlinear window size
  • 使用100A mini-batch of data points andRMSprop优化（rate = 1 × 10−5 and momentum = 0.9）,结果是在50Updated collected ten thousand parameters.Each experiment is repeated with a different random seed100次,We report mean scores and standard errors

• Representative Power of Normalizing Flows

  • 参数确定：unnormalized 2D densities p(z)∝exp[−U (z)]

    

  • 结果：Line chart for summary before3between true and approximate densitieskl -divergence的结果,K流长度,Mapping adopts（10）的

    

    • 原始是 a diagonal Gaussian q0(z)=
    • 我们发现NICEand plane flow(13)The same asymptotic performance can be achieved,we grow stream length,But the plane flow(13)The parameters of the need
      • probably because of the flow(13)But is learningNICEWhat is needed is an additional mechanism to mix components that have been randomly initialized but not learned

• MNIST and CIFAR-10 Images：6Wan training map10The handwritten figure
  • 对于50Updated parameters40hidden variableDLGMs训练
    • at different flow lengthsKcomparative analysisDLGM+NF和DLGM+NICEdifferent formula changes of

      

    • NF效果更好

      

    • log-likelihoods表现更好