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Inferring the most likely configuration for a subset of variables of a joint distribution given the remaining ones – which we refer to as co-generation – is an important challenge that is computationally demanding for all but the simplest settings.
This task has received a considerable amount of attention, particularly for classical ways of modeling distributions like structured prediction.
In contrast, almost nothing is known about this task when considering recently proposed techniques for modeling high-dimensional distributions, particularly generative adversarial nets (GANs).
Therefore, in this paper, we study the occurring challenges for co-generation with GANs.
To address those challenges we develop an annealed importance sampling (AIS) based Hamiltonian Monte Carlo (HMC) co-generation algorithm.
The presented approach significantly outperforms classical gradient-based methods on synthetic data and on CelebA.
While generative adversarial nets (GANs) [6] and variational auto-encoders (VAEs) [8] model a joint probability distribution which implicitly captures the correlations between multiple parts of the output, e.g., pixels in an image, and while those methods permit easy sampling from the entire output space domain, it remains an open question how to sample from part of the domain given the remainder?
We refer to this task as co-generation.
To enable co-generation for a domain unknown at training time, for GANs, optimization based algorithms have been proposed [15, 10] .
Intuitively, they aim at finding that latent sample which accurately matches the observed part.
However, successful training of the GAN leads to an increasingly ragged energy landscape, making the search for an appropriate latent variable via backpropagation through the generator harder and harder until it eventually fails.
To deal with this ragged energy landscape during co-generation, we develop a method using an annealed importance sampling (AIS) [11] based Hamiltonian Monte Carlo (HMC) algorithm [4, 12] , which is typically used to estimate (ratios of) the partition function [14, 13] .
Rather than focus on the partition function, the proposed approach leverages the benefits of AIS, i.e., gradually annealing a complex probability distribution, and HMC, i.e., avoiding a localized random walk.
We evaluate the proposed approach on synthetic data and imaging data (CelebA), showing compelling results via MSE and MSSIM metrics.
For more details and results please see our main conference paper [5] .
In the following we first motivate the problem of co-generation before we present an overview of our proposed approach and discuss the details of the employed Hamiltonian Monte Carlo method.
Assume we are given a well trained generatorx = G θ (z), parameterized by θ, which is able to produce samplesx from an implicitly modeled distribution p G (x|z) via a transformation of embeddings z [6, 1, 9, 2, 3] .
Further assume we are given partially observed data x o while the remaining part x h of the data x = (x o , x h ) is latent.
To reconstruct the latent parts of the data x h from available observations x o , a program can be formulated as follows:
where G θ (z) o denotes the restriction of the generated sample G θ (z) to the observed part.
Upon solving the program given in Eq. (1), we obtain an estimate for the missing datax
However, in practice, Eq. (1) turns out to be extremely hard to address, particularly if the generator G θ (z) is very well trained.
To see this, consider as an example a generator operating on a 2-dimensional latent space z = (z 1 , z 2 ) and 2-dimensional data x = (x 1 , x 2 ) (blue points in Fig. 2(a) ).
We use h = 1 and let x o = x 2 = 0.
In the first row of Fig. 1 we illustrate the loss surface of the objective given in Eq. (1) obtained when using a generator G θ (z) trained on the original 2-dimensional data for 500, 1.5k, 2.5k and 15k iterations (columns in Fig. 1 ).
We observe the latent space to become increasingly ragged, exhibiting folds that clearly separate different data regimes.
First (e.g., gradient descent (GD)) or second order optimization techniques cannot cope easily with such a loss landscape and likely get trapped in local optima.
We observe GD (red trajectory in Fig. 1 first row and loss in second row) to get stuck in a local optimum as the loss fails to decrease to zero once the generator better captures the data.
To prevent those local-optima issues for co-generation, we propose an annealed importance-sampling (AIS) based Hamiltonian Monte Carlo (HMC) method in the following (Alg.
1).
In order to reconstruct the hidden portion x h of the data
.
To obtain samplesẑ following the posterior distribution p(z|x o ), we use annealed importance sampling (AIS) [11] to gradually approach the complex and often high-dimensional posterior distribution p(z|x o ) by simulating a Markov Chain starting from the prior distribution p(z) = N (z|0, I), a standard normal distribution (zero mean and unit variance).
Formally, we define an annealing schedule for the parameter β t from β 0 = 0 to β T = 1.
At every time step t ∈ {1, . . . , T } we refine the samples drawn at the previous timestep t − 1 so as to represent the distributionp t (z|x o ) = p(z|x o ) βt p(z) 1−βt .
We use a sigmoid schedule for the parameter β t .
To successively refine the samples we use Hamilton Monte Carlo (HMC) sampling because a proposed update can be far from the current sample while still having a high acceptance probability.
We use 0.01 as the leapfrog step size and employ 10 leapfrog updates per HMC loop for the synthetic 2D dataset and 20 leapfrog updates for real dataset at first.
The acceptance rate is 0.65, as recommended by Neal [12] .
Hamilton Monte Carlo (HMC) [4] is capable of traversing folds in an energy landscape.
For this, HMC methods trade potential energy U t (z) = − logp t (z|x o ) with kinetic energy K t (v).
∀z ∈ Z compute new proposal sample using leapfrog integration on Hamiltonian 8: ∀z ∈ Z use Metropolis Hastings to check whether to accept the proposal and update Z 9:
end for 10: end for 11: Return: Z HMC defines a Hamiltonian H(z, v) = U (z) + K(v) or conversely a joint probability distribution log p(z, v) ∝ −H(z, v) and proceeds by iterating three steps M times.
In a first step, the Hamiltonian is initialized by randomly sampling the momentum variable v, typically using a standard Gaussian.
In the second step, (z * , v * ) are proposed via leapfrog integration to move along a hypersurface of the Hamiltonian.
In the final third step we decide whether to accept the proposal (z * , v * ) computed via leapfrog integration.
Formally, we accept the proposal with probability min{1, exp (−H(z
If the proposed state (z * , v * ) is rejected, the m + 1-th iteration reuses z, otherwise z is replaced with z * in the m + 1-th iteration.
This process is shown in Alg.
1 line 6, 7 and 8.
Baselines: In the following, we evaluate the proposed approach on synthetic and imaging data.
We use two GD baselines, employing different initialization methods.
The first one samples a single z randomly.
The second picks that one sample z from 5000 points which best matches the objective given in Eq. (1) initially.
To illustrate the advantage of our proposed method over the common baseline, we first demonstrate our results on 2-dimensional synthetic data.
Specifically, the 2-dimensional data x = (x 1 , x 2 ) is drawn from a mixture of five equally weighted Gaussians each with a variance of 0.02, the means of In this experiment, we aim to reconstruct x = (x 1 , x 2 ), given x o = x 2 = 0.
The optimal solution for the reconstruction isx = (1, 0) , where the reconstruction error should be 0.
However, as discussed in reference to Fig. 1 earlier, we observe that energy barriers in the Z-space complicate optimization.
In contrast, our proposed AIS co-generation method only requires one initialization to achieve the desired result after 6, 000 AIS loops, as shown in Fig. 2 (15000 (d) ).
Specifically, reconstruction with generators trained for a different number of epochs (500, 1.5k and 15k) are shown in the rows.
The samples obtained from the generator for the data (blue points in column (a)) are illustrated in column (a) using black color.
Using the respective generator to solve the program given in Eq. (1) via GD yields results highlighted with yellow color in column (b).
The empirical reconstruction error frequency for this baseline is given in column (c).
The results and the reconstruction error frequency obtained with Alg.
1 are shown in columns (d, e).
We observe significantly better results and robustness to initialization.
In Fig. 3 we show for 100 samples that Alg.
1 moves them across the energy barriers during the annealing procedure, illustrating the benefits of AIS based HMC over GD.
To validate our method on real data, we evaluate on CelebA, using MSE and MSSIM metrics.
We use the progressive GAN architecture [7] .
The size of the input is 512 and the size of the output is 128 × 128.
We randomly mask blocks of width and height ranging from 30 to 60.
Then we use Alg.
1 for reconstruction with 500 HMC loops.
In Fig. 3 (a,b) , we observe that Alg.
1 outperforms over both baselines for all GAN training iterations on both MSSIM and MSE metrics.
In Fig. 4 we show results generated by both baselines and Alg.
1.
We propose a co-generation approach, i.e., we complete partially given input data, using annealed importance sampling (AIS) based on the Hamiltonian Monte Carlo (HMC).
Different from classical optimization based methods, specifically GD, which get easily trapped in local optima when solving this task, the proposed approach is much more robust.
Importantly, the method is able to traverse large energy barriers that occur when training generative adversarial nets.
Its robustness is due to AIS gradually annealing a probability distribution and HMC avoiding localized walks.
We show additional results for real data experiments.
We observe our proposed algorithm to recover masked images more accurately than baselines and to generate better high-resolution images given low-resolution images.
We show masked CelebA (Fig. 5) and LSUN (Fig. 6 ) recovery results for baselines and our method, given a Progressive GAN generator.
Note that our algorithm is pretty robust to the position of the z initialization, since the generated results are consistent in Fig. 5 .
(a) (b) (c)

assistant: text: Using annealed importance sampling on the co-generation problem.