The Lov\'asz Hinge: A Novel Convex Surrogate for Submodular Losses

Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the underlying loss function is increasing in the number of incorrect predictions. However, gradient or cutting-plane computation for these functions is NP-hard for non-supermodular loss functions. We propose instead a novel surrogate loss function for submodular losses, the Lov\'asz hinge, which leads to O(p log p) complexity with O(p) oracle accesses to the loss function to compute a gradient or cutting-plane. We prove that the Lov\'asz hinge is convex and yields an extension. As a result, we have developed the first tractable convex surrogates in the literature for submodular losses. We demonstrate the utility of this novel convex surrogate through several set prediction tasks, including on the PASCAL VOC and Microsoft COCO datasets

The Lov\'asz-Softmax loss: A tractable surrogate for the optimization of the intersection-over-union measure in neural networks

The Jaccard index, also referred to as the intersection-over-union score, is commonly employed in the evaluation of image segmentation results given its perceptual qualities, scale invariance - which lends appropriate relevance to small objects, and appropriate counting of false negatives, in comparison to per-pixel losses. We present a method for direct optimization of the mean intersection-over-union loss in neural networks, in the context of semantic image segmentation, based on the convex Lov\'asz extension of submodular losses. The loss is shown to perform better with respect to the Jaccard index measure than the traditionally used cross-entropy loss. We show quantitative and qualitative differences between optimizing the Jaccard index per image versus optimizing the Jaccard index taken over an entire dataset. We evaluate the impact of our method in a semantic segmentation pipeline and show substantially improved intersection-over-union segmentation scores on the Pascal VOC and Cityscapes datasets using state-of-the-art deep learning segmentation architectures.Comment: Accepted as a conference paper at CVPR 201

Learning to Discover Sparse Graphical Models

We consider structure discovery of undirected graphical models from observational data. Inferring likely structures from few examples is a complex task often requiring the formulation of priors and sophisticated inference procedures. Popular methods rely on estimating a penalized maximum likelihood of the precision matrix. However, in these approaches structure recovery is an indirect consequence of the data-fit term, the penalty can be difficult to adapt for domain-specific knowledge, and the inference is computationally demanding. By contrast, it may be easier to generate training samples of data that arise from graphs with the desired structure properties. We propose here to leverage this latter source of information as training data to learn a function, parametrized by a neural network that maps empirical covariance matrices to estimated graph structures. Learning this function brings two benefits: it implicitly models the desired structure or sparsity properties to form suitable priors, and it can be tailored to the specific problem of edge structure discovery, rather than maximizing data likelihood. Applying this framework, we find our learnable graph-discovery method trained on synthetic data generalizes well: identifying relevant edges in both synthetic and real data, completely unknown at training time. We find that on genetics, brain imaging, and simulation data we obtain performance generally superior to analytical methods

A low variance consistent test of relative dependency

We describe a novel non-parametric statistical hypothesis test of relative dependence between a source variable and two candidate target variables. Such a test enables us to determine whether one source variable is significantly more dependent on a first target variable or a second. Dependence is measured via the Hilbert-Schmidt Independence Criterion (HSIC), resulting in a pair of empirical dependence measures (source-target 1, source-target 2). We test whether the first dependence measure is significantly larger than the second. Modeling the covariance between these HSIC statistics leads to a provably more powerful test than the construction of independent HSIC statistics by sub-sampling. The resulting test is consistent and unbiased, and (being based on U-statistics) has favorable convergence properties. The test can be computed in quadratic time, matching the computational complexity of standard empirical HSIC estimators. The effectiveness of the test is demonstrated on several real-world problems: we identify language groups from a multilingual corpus, and we prove that tumor location is more dependent on gene expression than chromosomal imbalances. Source code is available for download at https://github.com/wbounliphone/reldep.Comment: International Conference on Machine Learning, Jul 2015, Lille, Franc