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Bernoulli Variational Auto-Encoder in Torch

After formally introducing the concept of categorical variational auto-encoders in a previous article, this article presents a practical Torch implementation of variational auto-encoders with Bernoulli latent variables.

Update. The GitHub repository now contains several additional examples besides the code discussed in this article.

This is the third and probably final practical article in a series on variational auto-encoders and their implementation in Torch. Based on the Gaussian variational auto-encoder [] implemented in a previous article, this article discusses a simple implementation of a Bernoulli variational auto-encoder [][] where the latent variables are assumed to be Bernoulli distributed.

Previous articles:

Prerequisites. This article requires basic understanding of LUA and Torch; the code is also based on this article and the underlying mathematics are detailed in this article.

The code is available on GitHub:

Torch Denoising Variational Auto-Encoder on GitHub

Overview

The original variational auto-encoder as in [] is a continuous latent variable model. The model is intended to learn a latent space $\mathcal{Z} = \mathbb{R}^Q$ using a given set of samples $\{y_m\} \subseteq \mathcal{Y} = \mathbb{R}^R$ where $Q \ll R$. The model consists of the generative model $p(y | z)$ given a fixed prior $p(z)$, and the recognition (inference) model $q(z | y)$. The vanilla variational auto-encoder imposes a unit Gaussian prior

$p(z) = \mathcal{N}(z; 0, I_Q)$

such that the recognition model $q(z | y)$ also needs to be modeled as Gaussian distribution. The corresponding loss to be minimized can be written as:

$\mathcal{L}_{\text{VAE}} (w) = \text{KL}(q(z|y_m) | p(z)) - \frac{1}{L}\sum_{l = 1}^L \ln p(y_m | z_{l,m})$

where $y_m$ is a training sample and $z_{l,m} = g(\epsilon_{l,m}, y)$ with $\epsilon_{l,m} \sim \mathcal{N}(\epsilon ; 0, I_Q)$. Here, $g$ represents the so-called reparamterization trick:

$z_i = g_i(y, \epsilon_i) = \mu_i(y) + \epsilon_i \sigma_i^2(y)$

which ensures the differentiability of the model with respect to its input.

The latent variables can, however, also be modeled as discrete distributions. For example, the prior $p(z)$ and the recognition model $q(z|y)$ can both be modeled using Bernoulli distributions:

$p(z) = \prod_{i = 1}^Q \text{Ber}(z_i; 0.5)$

$q(z|x) = \prod_{i = 1}^Q \text{Ber}(z_i; \theta_i(y))$

where $\theta_i(y)$ is predicted by the encoder. The main difficulty of this latent space model is the reparameterization trick that allows to sample from $q(z|x)$ in a differentiable manner. In [][], this problem was solved using the following reparameterization trick, which has to be followed by a Sigmoid activation:

$z_i = g(y, \epsilon) = \sigma\left(\ln \epsilon - \ln (1 - \epsilon) + \ln \theta_i(y) - \ln (1 - \theta_i(y))\right)$(1)

where $\epsilon \sim U(0,1)$ is a uniformly distributed auxiliary variable. Finally, the loss to be minimized is, again,

$\mathcal{L}_{\text{VAE}} (w) = \text{KL}(q(z|y_m) | p(z)) - \frac{1}{L}\sum_{l = 1}^L \ln p(y_m | z_{l,m})$

where the Kullback-Leibler divergence is computed analytically as

$\text{KL}(q(z_i | y) | p(z_i)) = \text{KL}(\text{Ber}(z_i; \theta_i(y)) | \text{Ber}(z_i; 0.5))$

$=\sum_{k \in \{0,1\}} \ln \theta_i(y)^k (1 - \theta_i(y))^{1 - k} - \ln 0.5^k 0.5^{1-k}$.(2)

Implementation

The full implementation is discussed below.

-- Bernoulli variational auto-encoder.

require('math')
require('torch')
require('nn')
require('cunn')
require('optim')
require('image')

-- (1) The Kullback Leiber loss follows the Kullback Leibler loss of the Gaussian VAE.
-- The Kullback-Leibler divergence between two Bernoulli distribution can easily
-- be written down by summing over all possible states (i.e. 0 and 1).
--- @class KullbackLeiberDivergence
local KullbackLeiberDivergence, KullbackLeiberDivergenceParent = torch.class('nn.KullbackLeiberDivergence', 'nn.Module')

--- Initialize.
-- @param lambda weight of loss
function KullbackLeiberDivergence:__init(lambda, sizeAverage)
  self.lambda = lambda or 1
  self.prior = 0.5
  self.sizeAverage = sizeAverage or false
  self.loss = 0
end

--- Compute the Kullback-Leiber divergence; however, the input remains
-- unchanged - the divergence is saved in KullBackLeiblerDivergence.loss.
-- @param input probabilities
-- @return probabilities
function KullbackLeiberDivergence:updateOutput(input)

  -- (1.1) Forward pass of the KL divergence which is essentially
  -- an expectation over the log of the quotient of two Bernoulli distributions.
  -- Thus, considering all possible states (0, 1), this can be computed directly.
  self.loss = torch.cmul(input, torch.log(input + 1e-20) - torch.log(self.prior))
    + torch.cmul(1 - input, torch.log(1 - input + 1e-20) - torch.log(1 - self.prior))
  self.loss = self.lambda*torch.sum(self.loss)

  if self.sizeAverage then
    self.loss = self.loss/lib.utils.storageProd(#input)
  end

  self.output = input
  return self.output
end

--- Compute the backward pass of the Kullback-Leibler Divergence.
-- @param input probabilities
-- @param gradOutput gradients from top layer
-- @return gradients from top layer plus gradient of KL divergence with respect to probabilities
function KullbackLeiberDivergence:updateGradInput(input, gradOutput)

  -- (1.2) Backward pass, i.e. derivative of (1.1).
  local ones = input:clone():fill(1)
  self.gradInput = torch.log(input + 1e-20) + 1 - torch.log(self.prior) - torch.cdiv(ones, 1 - input + 1e-20)
    - torch.log(1 - input + 1e-20) + torch.cdiv(input, 1 - input + 1e-20) + torch.log(1 - self.prior)
  self.gradInput = self.lambda*self.gradInput
  --assert(not torch.any(self.gradInput:ne(self.gradInput)))

  if self.sizeAverage then
    self.gradInput = self.gradInput/lib.utils.storageProd(#input)
  end

  self.gradInput = self.gradInput + gradOutput
  --print(torch.mean(self.gradInput))
  return self.gradInput
end

-- (2) The reparameterization trick assumes that the next layer is a Sigmoid layer
-- in order to function correctly.
--- @class ReparameterizationSampler
local ReparameterizationSampler, ReparameterizationSamplerParent = torch.class('nn.ReparameterizationSampler', 'nn.Module')

--- Initialize.
-- @param temperature temperature of prediction
function ReparameterizationSampler:__init(temperature)
  self.temperature = temperature or 1
end

--- Sample from the provided mean and variance using the reparameterization trick.
-- @param input Bernoulli probabilities
-- @return sample
function ReparameterizationSampler:updateOutput(input)

  -- (2.1) Reparameterization:
  -- Let u be a uniform random variale in [0,1], p be the predicted probability (i.e. input),
  -- let l be the temperature.
  -- y = sigmoid((log(p) + log(u) - log(1 - u))/l)
  self.eps = torch.rand(input:size()):cuda()

  --self.output = (torch.log(input + 1e-20) + torch.log(self.eps) - torch.log(1 - self.eps))/self.temperature
  self.output = (torch.log(input + 1e-20) - torch.log(-torch.log(self.eps + 1e-20) + 1e-20))/self.temperature
  --print(torch.sigmoid(self.output))
  return self.output
end

--- Backward pass of the sampler.
-- @param input Bernoulli probabilities
-- @param gradOutput gradients of top layer
-- @return gradients with respect to input, table of two elements
function ReparameterizationSampler:updateGradInput(input, gradOutput)

  -- (2.2) Derivative of reparameterization with respect to p.
  --local ones = input:clone():fill(1)
  --self.gradInput = torch.cmul(torch.cdiv(ones, input*self.temperature + 1e-20), gradOutput)
  self.gradInput = torch.cdiv(gradOutput, input + 1e-20)/self.temperature
  --assert(not torch.any(self.gradInput:ne(self.gradInput)))
  --print(torch.mean(self.gradInput))
  return self.gradInput
end

-- Data parameters.
H = 24
W = 24
rH = 8
rW = 8
N = 50000

-- Fix random seed.
torch.manualSeed(1)

inputs = torch.Tensor(N, 1, H, W):fill(0)
for i = 1, N do
  local h = torch.random(rH, rH)
  local w = torch.random(rW, rW)
  local aH = torch.random(1, H - h)
  local aW = torch.random(1, W - w)
  inputs[i][1]:sub(aH, aH + h, aW, aW + w):fill(1)
end

outputs = inputs:clone()

-- (3) The encoder consists of several linear layerReparameterizationSamplers followed by
-- the Kullback Leibler loss, the samples and the docoder; the decoder
-- mirrors the encoder.
-- (3.1) The encoder, as for vanilla VAE.
hidden = math.floor(2*H*W)
encoder = nn.Sequential()
encoder:add(nn.View(1*H*W))
encoder:add(nn.Linear(1*H*W, hidden))
--encoder:add(nn.BatchNormalization(hidden))
encoder:add(nn.ReLU(true))
encoder:add(nn.Linear(hidden, hidden))
--encoder:add(nn.BatchNormalization(hidden))
encoder:add(nn.ReLU(true))

code = 25
encoder:add(nn.Linear(hidden, code))

-- (3.2) As for vanilla VAEs.
decoder = nn.Sequential()
decoder:add(nn.Linear(code, hidden))
--decoder:add(nn.BatchNormalization(hidden))
decoder:add(nn.ReLU(true))
decoder:add(nn.Linear(hidden, hidden))
--decoder:add(nn.BatchNormalization(hidden))
decoder:add(nn.ReLU(true))
decoder:add(nn.Linear(hidden, 1*H*W))
decoder:add(nn.View(1, H, W))
decoder:add(nn.Sigmoid(true))

-- (3) The full model, i.e encoder followed by the Kullback Leibler
-- divergence and the reparameterization trick sampler.
-- The main difference to the Gaussian model is that a Sigmoid layer follows
-- the reparameterization sampler.
model = nn.Sequential()
model:add(encoder)
KLD = nn.KullbackLeiberDivergence()
model:add(nn.Sigmoid(true))
model:add(KLD)
model:add(nn.ReparameterizationSampler())
model:add(nn.Sigmoid(true))
model:add(decoder)
print(model)
model = model:cuda()

criterion = nn.BCECriterion()
criterion.sizeAverage = false
criterion = criterion:cuda()

parameters, gradParameters = model:getParameters()
parameters = parameters:cuda()
gradParameters = gradParameters:cuda()

batchSize = 16
learningRate = 0.001
epochs = 10
iterations = epochs*math.floor(N/batchSize)
lossIterations = 50 -- in which interval to report training
protocol = torch.Tensor(iterations, 2):fill(0)

for t = 1, iterations do

  -- Sample a random batch from the dataset.
  local shuffle = torch.randperm(N)
  shuffle = shuffle:narrow(1, 1, batchSize)
  shuffle = shuffle:long()

  local input = inputs:index(1, shuffle)
  local output = outputs:index(1, shuffle)

  input = input:cuda()
  output = output:cuda()

  --- Definition of the objective on the current mini-batch.
  -- This will be the objective fed to the optimization algorithm.
  -- @param x input parameters
  -- @return object value, gradients
  local feval = function(x)

    -- Get new parameters.
    if x ~= parameters then
      parameters:copy(x)
    end

    -- Reset gradients.
    gradParameters:zero()

    -- Evaluate function on mini-batch.
    local pred = model:forward(input)
    local f = criterion:forward(pred, output)

    protocol[t][1] = f
    protocol[t][2] = KLD.loss

    -- Estimate df/dW.
    local df_do = criterion:backward(pred, output)
    model:backward(input, df_do)

    -- return f and df/dX
    return f, gradParameters
  end

  adamState = adamState or {
      learningRate = learningRate,
      momentum = 0,
      learningRateDecay = 5e-7
  }

  -- Returns the new parameters and the objective evaluated
  -- before the update.
  p, f = optim.adam(feval, parameters, adamState)

  if t%lossIterations == 0 then
    local loss = torch.mean(protocol:narrow(2, 1, 1):narrow(1, t - lossIterations + 1, lossIterations))
    local KLDLoss = torch.mean(protocol:narrow(2, 2, 1):narrow(1, t - lossIterations + 1, lossIterations))
    print('[Training] ' .. t .. '/' .. iterations .. ': ' .. loss .. ' | ' .. KLDLoss)
  end
end

randoms = torch.Tensor(20 * H, 20 * W)

-- Sample 20 x 20 points
for i = 1, 20  do
  for j = 1, 20 do
    local sample = torch.rand(1, code)
    sample[sample:gt(0.5)] = 1
    sample[sample:lt(1)] = 0
    local random = decoder:forward(sample:cuda())
    random = random:float()
    randoms[{{(i - 1) * H + 1, i * H}, {(j - 1) * W + 1, j * W}}] = random
  end
end

image.save('random.png', randoms)

The implementation mostly follows our implementation of the original variational auto-encoder, except for some minor changes. Also note that the size of the latent space $\mathcal{Z}$ needs to be increased significantly. While $2$ dimensions were sufficient before, significantly more binary dimensions are needed — which also influences training time.

  1. The Kullback-Leibler loss can be calculated analytically, as both the prior $p(z_i)$ and the recognition model $q(z_i|x)$ are Bernoulli distributed.
    1. The forward pass essentially implemented Equation (2).
    2. In the backward pass, Equation (2) is differentiated with respect to the predicted probabilities $\theta_i(y)$.
  2. The reparameterization trick which can be found in Equation (1) is also implemented as separate nn module.
    1. In the forward pass, a $\epsilon \sim U(0,1)$ is sampled and Equation (1) is applied. Additionally, the module implements a temperature parameters; the higher the temperature, the closer the approximation comes to true sampling from Bernoulli variables. The implementation also illustrates that a Sigmoid activation module needs to follow the reparameterization module.
    2. The backward pass implements the derivative of Equation (1) with respect to the sampled variables; which is made possible by the random auxiliary variable $\epsilon$.
  3. The overall model changes only slightly; specifically, after the encoder, the Kullback-Leibler divergence and the reparameterization layers are added. Finally, a sigmoid activation layer needs to be added before the decoder.

Figure 1 (click to enlarge): Random samples from the learned latent space for a $2$-dimensional space (left) and a $50$-dimensional space (right).

Qualitative results are shown in Figure 1. The results also illustrate the implications of binary latent variables as — for the same dimensionality — expressiveness of the generative model is lost. For a $2$-dimensional latent space, in particular, we see that the model can only capture binary steps in both directions. For a $50$-dimensional latent space, the model is able to interpolate better, but the reconstructions are still inferior to continuous latent variables.

  • [] D. P. Kingma and M. Welling. Auto-encoding variational bayes. CoRR, abs/1312.6114, 2013.
  • [] D. J. Im, S. Ahn, R. Memisevic, and Y. Bengio. Denoising criterion for variational auto-encoding framework. In AAAI Conference on Artificial Intelligence, pages 2059-2065, 2017.
  • [] E. Jang, S. Gu, and B. Poole. Categorical reparameterization with gumbel-softmax. CoRR, abs/1611.01144, 2016.
  • [] C. J. Maddison, A. Mnih, and Y. W. Teh. The concrete distribution: A continuous relaxation of discrete random variables. CoRR, abs/1611.00712, 2016.

What is your opinion on this article? Did you find it interesting or useful? Let me know your thoughts in the comments below: