Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization

We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an extension with weighted sampling probabilities on the dual variables, which has a better complexity than uniform sampling on unnormalized data. Both theoretically and empirically, we show that the SPDC method has comparable or better performance than several state-of-the-art optimization methods

Continuation of Nesterov's Smoothing for Regression with Structured Sparsity in High-Dimensional Neuroimaging

Predictive models can be used on high-dimensional brain images for diagnosis of a clinical condition. Spatial regularization through structured sparsity offers new perspectives in this context and reduces the risk of overfitting the model while providing interpretable neuroimaging signatures by forcing the solution to adhere to domain-specific constraints. Total Variation (TV) enforces spatial smoothness of the solution while segmenting predictive regions from the background. We consider the problem of minimizing the sum of a smooth convex loss, a non-smooth convex penalty (whose proximal operator is known) and a wide range of possible complex, non-smooth convex structured penalties such as TV or overlapping group Lasso. Existing solvers are either limited in the functions they can minimize or in their practical capacity to scale to high-dimensional imaging data. Nesterov's smoothing technique can be used to minimize a large number of non-smooth convex structured penalties but reasonable precision requires a small smoothing parameter, which slows down the convergence speed. To benefit from the versatility of Nesterov's smoothing technique, we propose a first order continuation algorithm, CONESTA, which automatically generates a sequence of decreasing smoothing parameters. The generated sequence maintains the optimal convergence speed towards any globally desired precision. Our main contributions are: To propose an expression of the duality gap to probe the current distance to the global optimum in order to adapt the smoothing parameter and the convergence speed. We provide a convergence rate, which is an improvement over classical proximal gradient smoothing methods. We demonstrate on both simulated and high-dimensional structural neuroimaging data that CONESTA significantly outperforms many state-of-the-art solvers in regard to convergence speed and precision.Comment: 11 pages, 6 figures, accepted in IEEE TMI, IEEE Transactions on Medical Imaging 201