Natural Reweighted Wake-Sleep.

Helmholtz Machines (HMs) are a class of generative models composed of two Sigmoid Belief Networks (SBNs), acting respectively as an encoder and a decoder. These models are commonly trained using a two-step optimization algorithm called Wake-Sleep (WS) and more recently by improved versions, such as Reweighted Wake-Sleep (RWS) and Bidirectional Helmholtz Machines (BiHM). The locality of the connections in an SBN induces sparsity in the Fisher Information Matrices associated to the probabilistic models, in the form of a finely-grained block-diagonal structure. In this paper we exploit this property to efficiently train SBNs and HMs using the natural gradient. We present a novel algorithm, called Natural Reweighted Wake-Sleep (NRWS), that corresponds to the geometric adaptation of its standard version. In a similar manner, we also introduce Natural Bidirectional Helmholtz Machine (NBiHM). Differently from previous work, we will show how for HMs the natural gradient can be efficiently computed without the need of introducing any approximation in the structure of the Fisher information matrix. The experiments performed on standard datasets from the literature show a consistent improvement of NRWS and NBiHM not only with respect to their non-geometric baselines but also with respect to state-of-the-art training algorithms for HMs. The improvement is quantified both in terms of speed of convergence as well as value of the log-likelihood reached after training.

Changing the Geometry of Representations: α-Embeddings for NLP Tasks.

Word embeddings based on a conditional model are commonly used in Natural Language Processing (NLP) tasks to embed the words of a dictionary in a low dimensional linear space. Their computation is based on the maximization of the likelihood of a conditional probability distribution for each word of the dictionary. These distributions form a Riemannian statistical manifold, where word embeddings can be interpreted as vectors in the tangent space of a specific reference measure on the manifold. A novel family of word embeddings, called α-embeddings have been recently introduced as deriving from the geometrical deformation of the simplex of probabilities through a parameter α, using notions from Information Geometry. After introducing the α-embeddings, we show how the deformation of the simplex, controlled by α, provides an extra handle to increase the performances of several intrinsic and extrinsic tasks in NLP. We test the α-embeddings on different tasks with models of increasing complexity, showing that the advantages associated with the use of α-embeddings are present also for models with a large number of parameters. Finally, we show that tuning α allows for higher performances compared to the use of larger models in which additionally a transformation of the embeddings is learned during training, as experimentally verified in attention models.