Dennis Forster

Sebastian Salwig, Till Kahlke, Florian Hirschberger et al.

Gaussian Mixture Models (GMMs) range among the most frequently used machine learning models. However, training large, general GMMs becomes computationally prohibitive for datasets with many data points $N$ of high-dimensionality $D$. For GMMs with arbitrary covariances, we here derive a highly efficient variational approximation, which is integrated with mixtures of factor analyzers (MFAs). For GMMs with $C$ components, our proposed algorithm significantly reduces runtime complexity per iteration from $\mathcal{O}(NCD^2)$ to a complexity scaling linearly with $D$ and remaining constant w.r.t. $C$. Numerical validation of this theoretical complexity reduction then shows the following: the distance evaluations required for the entire GMM optimization process scale sublinearly with $NC$. On large-scale benchmarks, this sublinearity results in speed-ups of an order-of-magnitude compared to the state-of-the-art. As a proof of concept, we train GMMs with over 10 billion parameters on about 100 million images, and observe training times of approximately nine hours on a single state-of-the-art CPU.

Simon Damm, Dennis Forster, Dmytro Velychko et al.

The central objective function of a variational autoencoder (VAE) is its variational lower bound (the ELBO). Here we show that for standard (i.e., Gaussian) VAEs the ELBO converges to a value given by the sum of three entropies: the (negative) entropy of the prior distribution, the expected (negative) entropy of the observable distribution, and the average entropy of the variational distributions (the latter is already part of the ELBO). Our derived analytical results are exact and apply for small as well as for intricate deep networks for encoder and decoder. Furthermore, they apply for finitely and infinitely many data points and at any stationary point (including local maxima and saddle points). The result implies that the ELBO can for standard VAEs often be computed in closed-form at stationary points while the original ELBO requires numerical approximations of integrals. As a main contribution, we provide the proof that the ELBO for VAEs is at stationary points equal to entropy sums. Numerical experiments then show that the obtained analytical results are sufficiently precise also in those vicinities of stationary points that are reached in practice. Furthermore, we discuss how the novel entropy form of the ELBO can be used to analyze and understand learning behavior. More generally, we believe that our contributions can be useful for future theoretical and practical studies on VAE learning as they provide novel information on those points in parameters space that optimization of VAEs converges to.

Florian Hirschberger, Dennis Forster, Jörg Lücke

This paper represents a preliminary (pre-reviewing) version of a sublinear variational algorithm for isotropic Gaussian mixture models (GMMs). Further developments of the algorithm for GMMs with diagonal covariance matrices (instead of isotropic clusters) and their corresponding benchmarking results have been published by TPAMI (doi:10.1109/TPAMI.2021.3133763) in the paper "A Variational EM Acceleration for Efficient Clustering at Very Large Scales". We kindly refer the reader to the TPAMI paper instead of this much earlier arXiv version (the TPAMI paper is also open access). Publicly available source code accompanies the paper (see https://github.com/variational-sublinear-clustering). Please note that the TPAMI paper does not contain the benchmark on the 80 Million Tiny Images dataset anymore because we followed the call of the dataset creators to discontinue the use of that dataset. The aim of the project (which resulted in this arXiv version and the later TPAMI paper) is the exploration of the current efficiency and large-scale limits in fitting a parametric model for clustering to data distributions. To reduce computational complexity, we used a clustering objective based on truncated variational EM (which reduces complexity for many clusters) in combination with coreset objectives (which reduce complexity for many data points). We used efficient coreset construction and efficient seeding to translate the theoretical sublinear complexity gains into an efficient algorithm. In applications to standard large-scale benchmarks for clustering, we then observed substantial wall-clock speedups compared to already highly efficient clustering approaches. To demonstrate that the observed efficiency enables applications previously considered unfeasible, we clustered the entire and unscaled 80 Million Tiny Images dataset into up to 32,000 clusters.

Dennis Forster, Jörg Lücke

One iteration of standard $k$-means (i.e., Lloyd's algorithm) or standard EM for Gaussian mixture models (GMMs) scales linearly with the number of clusters $C$, data points $N$, and data dimensionality $D$. In this study, we explore whether one iteration of $k$-means or EM for GMMs can scale sublinearly with $C$ at run-time, while improving the clustering objective remains effective. The tool we apply for complexity reduction is variational EM, which is typically used to make training of generative models with exponentially many hidden states tractable. Here, we apply novel theoretical results on truncated variational EM to make tractable clustering algorithms more efficient. The basic idea is to use a partial variational E-step which reduces the linear complexity of $\mathcal{O}(NCD)$ required for a full E-step to a sublinear complexity. Our main observation is that the linear dependency on $C$ can be reduced to a dependency on a much smaller parameter $G$ which relates to cluster neighborhood relations. We focus on two versions of partial variational EM for clustering: variational GMM, scaling with $\mathcal{O}(NG^2D)$, and variational $k$-means, scaling with $\mathcal{O}(NGD)$ per iteration. Empirical results show that these algorithms still require comparable numbers of iterations to improve the clustering objective to same values as $k$-means. For data with many clusters, we consequently observe reductions of net computational demands between two and three orders of magnitude. More generally, our results provide substantial empirical evidence in favor of clustering to scale sublinearly with $C$.

Jörg Lücke, Dennis Forster

We show that $k$-means (Lloyd's algorithm) is obtained as a special case when truncated variational EM approximations are applied to Gaussian Mixture Models (GMM) with isotropic Gaussians. In contrast to the standard way to relate $k$-means and GMMs, the provided derivation shows that it is not required to consider Gaussians with small variances or the limit case of zero variances. There are a number of consequences that directly follow from our approach: (A) $k$-means can be shown to increase a free energy associated with truncated distributions and this free energy can directly be reformulated in terms of the $k$-means objective; (B) $k$-means generalizations can directly be derived by considering the 2nd closest, 3rd closest etc. cluster in addition to just the closest one; and (C) the embedding of $k$-means into a free energy framework allows for theoretical interpretations of other $k$-means generalizations in the literature. In general, truncated variational EM provides a natural and rigorous quantitative link between $k$-means-like clustering and GMM clustering algorithms which may be very relevant for future theoretical and empirical studies.

Dennis Forster, Jörg Lücke

Inference and learning for probabilistic generative networks is often very challenging and typically prevents scalability to as large networks as used for deep discriminative approaches. To obtain efficiently trainable, large-scale and well performing generative networks for semi-supervised learning, we here combine two recent developments: a neural network reformulation of hierarchical Poisson mixtures (Neural Simpletrons), and a novel truncated variational EM approach (TV-EM). TV-EM provides theoretical guarantees for learning in generative networks, and its application to Neural Simpletrons results in particularly compact, yet approximately optimal, modifications of learning equations. If applied to standard benchmarks, we empirically find, that learning converges in fewer EM iterations, that the complexity per EM iteration is reduced, and that final likelihood values are higher on average. For the task of classification on data sets with few labels, learning improvements result in consistently lower error rates if compared to applications without truncation. Experiments on the MNIST data set herein allow for comparison to standard and state-of-the-art models in the semi-supervised setting. Further experiments on the NIST SD19 data set show the scalability of the approach when a manifold of additional unlabeled data is available.

Dennis Forster, Abdul-Saboor Sheikh, Jörg Lücke

Classifiers for the semi-supervised setting often combine strong supervised models with additional learning objectives to make use of unlabeled data. This results in powerful though very complex models that are hard to train and that demand additional labels for optimal parameter tuning, which are often not given when labeled data is very sparse. We here study a minimalistic multi-layer generative neural network for semi-supervised learning in a form and setting as similar to standard discriminative networks as possible. Based on normalized Poisson mixtures, we derive compact and local learning and neural activation rules. Learning and inference in the network can be scaled using standard deep learning tools for parallelized GPU implementation. With the single objective of likelihood optimization, both labeled and unlabeled data are naturally incorporated into learning. Empirical evaluations on standard benchmarks show, that for datasets with few labels the derived minimalistic network improves on all classical deep learning approaches and is competitive with their recent variants without the need of additional labels for parameter tuning. Furthermore, we find that the studied network is the best performing monolithic (`non-hybrid') system for few labels, and that it can be applied in the limit of very few labels, where no other system has been reported to operate so far.