Generative Adversarial Networks (GAN) (, 2014) are an effective method for training generative models of complex data such as natural images. However, they are notoriously hard to train and can suffer from the problem of missing modes where the model is not able to produce examples in certain regions of the space. We propose an iterative procedure, called AdaGAN, where at every step we add a new component into a mixture model by running a GAN algorithm on a reweighted sample. This is inspired by boosting algorithms, where many potentially weak individual predictors are greedily aggregated to form a strong composite predictor. We prove that such an incremental procedure leads to convergence to the true distribution in a finite number of steps if each step is optimal, and convergence at an exponential rate otherwise. We also illustrate experimentally that this procedure addresses the problem of missing modes.
Similarly, Barron and Li @cite_1 studied the construction of mixtures minimizing the Kullback divergence and proposed a greedy procedure for doing so. They also proved that under certain conditions, finite mixtures can approximate arbitrary mixtures at a rate @math where @math is the number of components in the mixture when the weight of each newly added component is @math . These results are specific to the Kullback divergence but are consistent with our more general results.
@cite_1
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The authors propose a procedure for determining the unknown number of components in mixtures by generalizing a Bayesian testing method proposed by Mengersen & Robert (1996). The testing criterion they propose involves a Kullback-Leibler distance, which may be weighted or not. They give explicit formulas for the weighted distance for a number of mixture distributions and propose a stepwise testing procedure to select the minimum number of components adequate for the data. Their procedure, which is implemented using the BUGS software, exploits a fast collapsing approach which accelerates the search for the minimum number of components by avoiding full refitting at each step. The performance of their method is compared, using both distances, to the Bayes factor approach. We analyze mixture density approximation and estimation. We form a convex set of density functions by taking the convex hull of a parametric family, e.g. mixtures of the Gaussian location family. A sequence of finite mixture densities is formulated to provide a parsimonious approximation for the target density. If the target density itself is in the convex hull, we show that the approximation error goes to zero with a rate of 1 k, where k is the number of components in the approximation. If the target density is outside of the convex hull, the approximation error is equal to the best achievable error plus a term that goes to zero with a rate of 1 k. A greedy algorithm that introduces one component at each step is shown to achieve such an error rate. Similarly, a greedy estimation algorithm is provided to find such approximation for data from an arbitrary density. This algorithm estimates one mixture component at one time. We prove that such an algorithm achieves a likelihood nearly as good as the MLE (maximum likelihood estimate) over the whole convex hull. And we identify the difference as being bounded by order O(1 k), where k is the number of components in the estimate. Risks of such estimators are shown to be bounded by a sum of approximation error and estimation error. The error terms are identified. An optimal choice of k can be derived by minimizing the risk bound. Acting as a similar role as the bandwidth in non-parainetric density estimation, k controls two error terms in opposite directions. A large k reduces approximation error and increases estimation error. A MDL (minimum description length) principle is derived to provide an estimation method for k. And the estimated k is shown to achieve the risk bound as if we know the best k in advance. A new information projection theory is derived to expand the approximating class to include its information closure. We prove the existence and uniqueness of a f* in the closure of the convex hull C (in a sense we identify), such that D ( fpf* ) = infg∈CD fpg , where Dfpg is the Kullback-Leibler divergence. And log(fk) → log(f*) in L1 (f) for any sequence fk in C with Dfpfk→ infg∈CD fpg. Other characterizing properties of f* are also given. This article concerns the greedy learning of gaussian mixtures. In the greedy approach, mixture components are inserted into the mixture one after the other. We propose a heuristic for searching for the optimal component to insert. In a randomized manner, a set of candidate new components is generated. For each of these candidates, we find the locally optimal new component and insert it into the existing mixture. The resulting algorithm resolves the sensitivity to initialization of state-of-the-art methods, like expectation maximization, and has running time linear in the number of data points and quadratic in the (final) number of mixture components. Due to its greedy nature, the algorithm can be particularly useful when the optimal number of mixture components is unknown. Experimental results comparing the proposed algorithm to other methods on density estimation and texture segmentation are provided. In recent years analysis of complexity of learning Gaussian mixture models from sampled data has received significant attention in computational machine learning and theory communities. In this paper we present the first result showing that polynomial time learning of multidimensional Gaussian Mixture distributions is possible when the separation between the component means is arbitrarily small. Specifically, we present an algorithm for learning the parameters of a mixture of k identical spherical Gaussians in n-dimensional space with an arbitrarily small separation between the components, which is polynomial in dimension, inverse component separation and other input parameters for a fixed number of components k. The algorithm uses a projection to k dimensions and then a reduction to the 1-dimensional case. It relies on a theoretical analysis showing that two 1-dimensional mixtures whose densities are close in the L norm must have similar means and mixing coefficients. To produce the necessary lower bound for the L norm in terms of the distances between the corresponding means, we analyze the behavior of the Fourier transform of a mixture of Gaussians in one dimension around the origin, which turns out to be closely related to the properties of the Vandermonde matrix obtained from the component means. Analysis of minors of the Vandermonde matrix together with basic function approximation results allows us to provide a lower bound for the norm of the mixture in the Fourier domain and hence a bound in the original space. Additionally, we present a separate argument for reconstructing variance.