We establish a theoretical link between evolutionary algorithms and variational parameter optimization of probabilistic generative models with binary hidden variables.
 While the novel approach is independent of the actual generative model, here we use two such models to investigate its applicability and scalability: a noisy-OR Bayes Net (as a standard example of binary data) and Binary Sparse Coding (as a model for continuous data).

 Learning of probabilistic generative models is first formulated as approximate maximum likelihood optimization using variational expectation maximization (EM).
 We choose truncated posteriors as variational distributions in which discrete latent states serve as variational parameters. In the variational E-step,
 the latent states are then  
optimized according to a tractable free-energy objective . Given a data point, we can show that evolutionary algorithms can be used for the variational optimization loop by (A)~considering the bit-vectors of the latent states as genomes of individuals, and by (B)~defining the fitness of the
 individuals as the (log) joint probabilities given by the used generative model.

 As a proof of concept, we apply the novel evolutionary EM approach to the optimization of the parameters of noisy-OR Bayes nets and binary sparse coding on artificial and real data (natural image patches). Using point mutations and single-point cross-over for the evolutionary algorithm, we find that scalable variational EM algorithms are obtained which efficiently improve the data likelihood. In general we believe that, with the link established here, standard as well as recent results in the field of evolutionary optimization can be leveraged to address the difficult problem of parameter optimization in generative models. Evolutionary algorithms (EA) have been introduced (e.g. BID3 BID23 as a technique for function optimization using methods inspired by biological evolutionary processes such as mutation, recombination, and selection. As such EAs are of interest as tools to solve Machine Learning problems, and they have been frequently applied to a number of tasks such as clustering BID21 BID9 , reinforcement learning BID25 , and hierarchical unsupervised BID16 or deep supervised learning (e.g., BID32 BID32 BID33 BID22 for recent examples). In some of these tasks EAs have been investigated as alternatives to standard procedures BID9 ), but most frequently EAs are used to solve specific sub-problems. For example, for classification with Deep Neural Networks (DNNs LeCun et al., 2015; BID27 , EAs are frequently applied to solve the sub-problem of selecting the best DNN architectures for a given task (e.g. BID32 BID33 or more generally to find the best hyper-parameters of a DNN (e.g. BID12 BID22 .Inspired by these previous contributions, we here ask if EAs and learning algorithms can be linked more tightly. To address this question we make use of the theoretical framework of probabilistic generative models and expectation maximization (EM Dempster et al., 1977) approaches for parameter optimization. The probabilistic approach in combination with EM is appealing as it establishes a very general unifying framework able to encompass diverse algorithms from clustering and dimensionality reduction BID24 BID34 over feature learning and sparse coding BID18 to deep learning approaches BID20 . However, for most generative data models, EM is computationally intractable and requires approximations. Variational EM is a very prominent such approximation and is continuously further developed to become more efficient, more accurate and more autonomously applicable. Variational EM seeks to approximately solve optimization problems of functions with potentially many local optima in potentially very high dimensional spaces. The key observation exploited in this study is that a variational EM algorithm can be formulated such that latent states serve as variational parameters. If the latent states are then considered as genomes of individuals, EAs emerge as a very natural choice for optimization in the variational loop of EM. The training of generative models is a very intensively studied branch of Machine Learning. If EM is applied for training, most non-elementary models require approximations. For this reason, sophisticated and mathematically grounded approaches such as sampling or variational EM have been developed in order to derive sufficiently precise and efficient learning algorithms.Evolutionary algorithms (EAs) have also been applied in conjunction with EM. BID21 , for instance, have used EAs for clustering with Gaussian mixture models (GMMs). However, the GMM parameters are updated by their approach relatively conventionally using EM, while EAs are used to select the best GMM models for the clustering problem (using a min. description length criterion). Such a use of EAs is similar to DNN optimization where EAs optimize DNN hyperparameters in an outer optimization loop BID32 BID12 BID22 Suganuma et al., 2017, etc) , while the DNNs themselves are optimized using standard error-minimization algorithms. Still other approaches have used EAs to directly optimize, e.g., a clustering objective. But in these cases EAs replace EM approaches for optimization (compare Hruschka et al., 2009) . In contrast to all such previous applications, we have here shown that EAs and EM can be combined directly and intimately: Alg. 1 defines EAs as an integral part of EM, and as such EAs address the key optimization problem arising in the training of generative models.We see the main contribution of our study in the establishment of this close theoretical link between EAs and EM. This novel link will make it possible to leverage an extensive body of knowledge and experience from the community of evolutionary approaches for learning algorithms. Our numerical experiments are a proof of concept which shows that EAs are indeed able to train generative models with large hidden spaces and local optima. For this purpose we used very basic EAs with elementary selection, mutation, cross-over operators.EAs more specialized to the specific optimization problems arising in the training of generative models have great potentials in future improvements of accuracy and scalability, we believe. In our experiments, we have only just started to exploit the abilities of EAs for learning algorithms. Still, our results represent, to the knowledge of the authors, the first examples of noisy-OR or sparse coding models trained with EAs (although both models have been studied very extensively before). Most importantly, we have pointed out a novel mathematically grounded way how EAs can be used for generative models with binary latents in general. The approach here established is, moreover, not only very generically formulated using the models' joint probabilities but it is also very straightforward to apply.
We present Evolutionary EM as a novel algorithm for unsupervised training of generative models with binary latent variables that intimately connects variational EM with evolutionary optimization The paper presents a combination of evolutionary computation and variational EM for models with binary latent variables represented via a particle-based approximation The paper makes an attempt to tightly integrate expectation-maximization training algorithms with evolutionary algorithms.