Testing for Differences in Gaussian Graphical Models: Applications to Brain Connectivity

Functional brain networks are well described and estimated from data with Gaussian Graphical Models (GGMs), e.g. using sparse inverse covariance estimators. Comparing functional connectivity of subjects in two populations calls for comparing these estimated GGMs. Our goal is to identify differences in GGMs known to have similar structure. We characterize the uncertainty of differences with confidence intervals obtained using a parametric distribution on parameters of a sparse estimator. Sparse penalties enable statistical guarantees and interpretable models even in high-dimensional and low-sample settings. Characterizing the distributions of sparse models is inherently challenging as the penalties produce a biased estimator. Recent work invokes the sparsity assumptions to effectively remove the bias from a sparse estimator such as the lasso. These distributions can be used to give confidence intervals on edges in GGMs, and by extension their differences. However, in the case of comparing GGMs, these estimators do not make use of any assumed joint structure among the GGMs. Inspired by priors from brain functional connectivity we derive the distribution of parameter differences under a joint penalty when parameters are known to be sparse in the difference. This leads us to introduce the debiased multi-task fused lasso, whose distribution can be characterized in an efficient manner. We then show how the debiased lasso and multi-task fused lasso can be used to obtain confidence intervals on edge differences in GGMs. We validate the techniques proposed on a set of synthetic examples as well as neuro-imaging dataset created for the study of autism

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

Re-Weighted Softmax Cross-Entropy to Control Forgetting in Federated Learning

In Federated Learning, a global model is learned by aggregating model updates computed at a set of independent client nodes, to reduce communication costs multiple gradient steps are performed at each node prior to aggregation. A key challenge in this setting is data heterogeneity across clients resulting in differing local objectives which can lead clients to overly minimize their own local objective, diverging from the global solution. We demonstrate that individual client models experience a catastrophic forgetting with respect to data from other clients and propose an efficient approach that modifies the cross-entropy objective on a per-client basis by re-weighting the softmax logits prior to computing the loss. This approach shields classes outside a client's label set from abrupt representation change and we empirically demonstrate it can alleviate client forgetting and provide consistent improvements to standard federated learning algorithms. Our method is particularly beneficial under the most challenging federated learning settings where data heterogeneity is high and client participation in each round is low

Convex relaxations of penalties for sparse correlated variables with bounded total variation

International audienceWe study the problem of statistical estimation with a signal known to be sparse, spatially contiguous, and containing many highly correlated variables. We take inspiration from the recently introduced k-support norm, which has been successfully applied to sparse prediction problems with correlated features, but lacks any explicit structural constraints commonly found in machine learning and image processing. We address this problem by incorporating a total variation penalty in the k-support framework. We introduce the (k, s) support total variation norm as the tightest convex relaxation of the intersection of a set of sparsity and total variation constraints. We show that this norm leads to an intractable combinatorial graph optimization problem, which we prove to be NP-hard. We then introduce a tractable relaxation with approximation guarantees that scale well for grid structured graphs. We devise several first-order optimization strategies for statistical parameter estimation with the described penalty. We demonstrate the effectiveness of this penalty on classification in the low-sample regime, classification with M/EEG neuroimaging data, and image recovery with synthetic and real data background subtracted image recovery tasks. We extensively analyse the application of our penalty on the complex task of identifying predictive regions from low-sample high-dimensional fMRI brain data, we show that our method is particularly useful compared to existing methods in terms of accuracy, interpretability, and stability