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We propose a continuous optimization framework for discovering a latent directed acyclic graph (DAG) from observational data. Our approach optimizes over the polytope of permutation vectors, the so-called Permutahedron, to learn a topological ordering. Edges can be optimized jointly, or learned conditional on the ordering via a non-differentiable subroutine. Compared to existing continuous optimization approaches our formulation has a number of advantages including: 1. validity: optimizes over exact DAGs as opposed to other relaxations optimizing approximate DAGs; 2. modularity: accommodates any edge-optimization procedure, edge structural parameterization, and optimization loss; 3. end-to-end: either alternately iterates between node-ordering and edge-optimization, or optimizes them jointly. We demonstrate, on real-world data problems in protein-signaling and transcriptional network discovery, that our approach lies on the Pareto frontier of two key metrics, the SID and SHD., The Eleventh International Conference on Learning Representations

In this brief note, we formulate Principal Component Analysis (PCA) over datasets consisting not of points but of distributions, characterized by their location and covariance. Just like the usual PCA on points can be equivalently derived via a variance-maximization principle and via a minimization of reconstruction error, we derive a closed-form solution for distributional PCA from both of these perspectives., Comment: 4 pages, 1 figure

Many types of data from fields including natural language processing, computer vision, and bioinformatics, are well represented by discrete, compositional structures such as trees, sequences, or matchings. Latent structure models are a powerful tool for learning to extract such representations, offering a way to incorporate structural bias, discover insight about the data, and interpret decisions. However, effective training is challenging, as neural networks are typically designed for continuous computation. This text explores three broad strategies for learning with discrete latent structure: continuous relaxation, surrogate gradients, and probabilistic estimation. Our presentation relies on consistent notations for a wide range of models. As such, we reveal many new connections between latent structure learning strategies, showing how most consist of the same small set of fundamental building blocks, but use them differently, leading to substantially different applicability and properties.

International audience; We address the problem of learning binary decision trees that partition data for some downstream task. We propose to learn discrete parameters(i.e., for tree traversals and node pruning) and continuous parameters (i.e., for tree split functions and prediction functions) simultaneously usingargmin differentiation. We do so by sparsely relaxing a mixed-integer program for the discrete parameters, to allow gradients to pass throughthe program to continuous parameters. We derive customized algorithms to efficiently compute the forward and backward passes. This meansthat our tree learning procedure can be used as an (implicit) layer in arbitrary deep networks, and can be optimized with arbitrary loss functions. We demonstrate that our approach produces binary trees that are competitive with existing single tree and ensemble approaches, in both supervised and unsupervised settings. Further, apart from greedy approaches (which do not have competitive accuracies), our method is faster to train than all other tree-learning baselines we compare with. The code for reproducing the results is available at https://github.com/vzantedeschi/LatentTrees.

Neural networks and other machine learning models compute continuous representations, while humans communicate mostly through discrete symbols. Reconciling these two forms of communication is desirable for generating human-readable interpretations or learning discrete latent variable models, while maintaining end-to-end differentiability. Some existing approaches (such as the Gumbel-Softmax transformation) build continuous relaxations that are discrete approximations in the zero-temperature limit, while others (such as sparsemax transformations and the Hard Concrete distribution) produce discrete/continuous hybrids. In this paper, we build rigorous theoretical foundations for these hybrids, which we call "mixed random variables." Our starting point is a new "direct sum" base measure defined on the face lattice of the probability simplex. From this measure, we introduce new entropy and Kullback-Leibler divergence functions that subsume the discrete and differential cases and have interpretations in terms of code optimality. Our framework suggests two strategies for representing and sampling mixed random variables, an extrinsic ("sample-and-project") and an intrinsic one (based on face stratification). We experiment with both approaches on an emergent communication benchmark and on modeling MNIST and Fashion-MNIST data with variational auto-encoders with mixed latent variables., Comment: Accepted for oral presentation at ICLR 2022

Training neural network models with discrete (categorical or structured) latent variables can be computationally challenging, due to the need for marginalization over large or combinatorial sets. To circumvent this issue, one typically resorts to sampling-based approximations of the true marginal, requiring noisy gradient estimators (e.g., score function estimator) or continuous relaxations with lower-variance reparameterized gradients (e.g., Gumbel-Softmax). In this paper, we propose a new training strategy which replaces these estimators by an exact yet efficient marginalization. To achieve this, we parameterize discrete distributions over latent assignments using differentiable sparse mappings: sparsemax and its structured counterparts. In effect, the support of these distributions is greatly reduced, which enables efficient marginalization. We report successful results in three tasks covering a range of latent variable modeling applications: a semisupervised deep generative model, a latent communication game, and a generative model with a bit-vector latent representation. In all cases, we obtain good performance while still achieving the practicality of sampling-based approximations., Accepted for spotlight presentation at NeurIPS 2020

Structured prediction requires manipulating a large number of combinatorial structures, e.g., dependency trees or alignments, either as latent or output variables. Recently, the SparseMAP method has been proposed as a differentiable, sparse alternative to maximum a posteriori (MAP) and marginal inference. SparseMAP returns a combination of a small number of structures, a desirable property in some downstream applications. However, SparseMAP requires a tractable MAP inference oracle. This excludes, e.g., loopy graphical models or factor graphs with logic constraints, which generally require approximate inference. In this paper, we introduce LP-SparseMAP, an extension of SparseMAP that addresses this limitation via a local polytope relaxation. LP-SparseMAP uses the flexible and powerful domain specific language of factor graphs for defining and backpropagating through arbitrary hidden structure, supporting coarse decompositions, hard logic constraints, and higher-order correlations. We derive the forward and backward algorithms needed for using LP-SparseMAP as a hidden or output layer. Experiments in three structured prediction tasks show benefits compared to SparseMAP and Structured SVM., Comment: 34 pages, 5 tables, 4 figures. ICML 2020

These notes aim to shed light on the recently proposed structured projected intermediate gradient optimization technique (SPIGOT, Peng et al., 2018). SPIGOT is a variant of the straight-through estimator (Bengio et al., 2013) which bypasses gradients of the argmax function by back-propagating a surrogate "gradient." We provide a new interpretation to the proposed gradient and put this technique into perspective, linking it to other methods for training neural networks with discrete latent variables. As a by-product, we suggest alternate variants of SPIGOT which will be further explored in future work., Comment: 7 pages

Modern neural networks are often augmented with an attention mechanism, which tells the network where to focus within the input. We propose in this paper a new framework for sparse and structured attention, building upon a smoothed max operator. We show that the gradient of this operator defines a mapping from real values to probabilities, suitable as an attention mechanism. Our framework includes softmax and a slight generalization of the recently-proposed sparsemax as special cases. However, we also show how our framework can incorporate modern structured penalties, resulting in more interpretable attention mechanisms, that focus on entire segments or groups of an input. We derive efficient algorithms to compute the forward and backward passes of our attention mechanisms, enabling their use in a neural network trained with backpropagation. To showcase their potential as a drop-in replacement for existing ones, we evaluate our attention mechanisms on three large-scale tasks: textual entailment, machine translation, and sentence summarization. Our attention mechanisms improve interpretability without sacrificing performance; notably, on textual entailment and summarization, we outperform the standard attention mechanisms based on softmax and sparsemax., In proceedings of NeurIPS 2017; added errata

Factorization machines and polynomial networks are supervised polynomial models based on an efficient low-rank decomposition. We extend these models to the multi-output setting, i.e., for learning vector-valued functions, with application to multi-class or multi-task problems. We cast this as the problem of learning a 3-way tensor whose slices share a common basis and propose a convex formulation of that problem. We then develop an efficient conditional gradient algorithm and prove its global convergence, despite the fact that it involves a non-convex basis selection step. On classification tasks, we show that our algorithm achieves excellent accuracy with much sparser models than existing methods. On recommendation system tasks, we show how to combine our algorithm with a reduction from ordinal regression to multi-output classification and show that the resulting algorithm outperforms simple baselines in terms of ranking accuracy., Comment: Published at NIPS 2017. 17 pages, including appendix

International audience; Scikit-learn is an increasingly popular machine learning li- brary. Written in Python, it is designed to be simple and efficient, accessible to non-experts, and reusable in various contexts. In this paper, we present and discuss our design choices for the application programming interface (API) of the project. In particular, we describe the simple and elegant interface shared by all learning and processing units in the library and then discuss its advantages in terms of composition and reusability. The paper also comments on implementation details specific to the Python ecosystem and analyzes obstacles faced by users and developers of the library.