SVRG and its variants are among the state of art optimization algorithms for large scale machine learning problems. It is well known that SVRG converges linearly when the objective function is strongly convex. However this setup can be restrictive, and does not include several important formulations such as Lasso, group Lasso, logistic regression, and some non-convex models including corrected Lasso and SCAD. In this paper, we prove that, for a class of statistical M-estimators covering examples mentioned above, SVRG solves the formulation with {\em a linear convergence rate} without strong convexity or even convexity. Our analysis makes use of {\em restricted strong convexity}, under which we show that SVRG converges linearly to the fundamental statistical precision of the model, i.e., the difference between true unknown parameter $\theta^*$ and the optimal solution $\hat{\theta}$ of the model.

Lozano, Aurélie C., Meinshausen, Nicolai

We propose a minimum distance estimation method for robust regression in sparse high-dimensional settings. The traditional likelihood-based estimators lack resilience against outliers, a critical issue when dealing with high-dimensional noisy data. Our method, Minimum Distance Lasso (MD-Lasso), combines minimum distance functionals, customarily used in nonparametric estimation for their robustness, with l1-regularization for high-dimensional regression. The geometry of MD-Lasso is key to its consistency and robustness. The estimator is governed by a scaling parameter that caps the influence of outliers: the loss per observation is locally convex and close to quadratic for small squared residuals, and flattens for squared residuals larger than the scaling parameter. As the parameter approaches infinity, the estimator becomes equivalent to least-squares Lasso. MD-Lasso enjoys fast convergence rates under mild conditions on the model error distribution, which hold for any of the solutions in a convexity region around the true parameter and in certain cases for every solution. Remarkably, a first-order optimization method is able to produce iterates very close to the consistent solutions, with geometric convergence and regardless of the initialization. A connection is established with re-weighted least-squares that intuitively explains MD-Lasso robustness. The merits of our method are demonstrated through simulation and eQTL data analysis.

Hara, Satoshi (National Institute of Informatics) | Maehara, Takanori (Shizuoka University)

We propose an algorithm for enumerating solutions to the Lasso regression problem.In ordinary Lasso regression, one global optimum is obtained and the resulting features are interpreted as task-relevant features.However, this can overlook possibly relevant features not selected by the Lasso.With the proposed method, we can enumerate many possible feature sets for human inspection, thus recording all the important features.We prove that by enumerating solutions, we can recover a true feature set exactly under less restrictive conditions compared with the ordinary Lasso.We confirm our theoretical results also in numerical simulations.Finally, in the gene expression and the text data, we demonstrate that the proposed method can enumerate a wide variety of meaningful feature sets, which are overlooked by the global optima.

Steinhardt, Jacob, Wager, Stefan, Liang, Percy

We present a sparse analogue to stochastic gradient descent that is guaranteed to perform well under similar conditions to the lasso. In the linear regression setup with irrepresentable noise features, our algorithm recovers the support set of the optimal parameter vector with high probability, and achieves a statistically quasi-optimal rate of convergence of Op(k log(d)/T), where k is the sparsity of the solution, d is the number of features, and T is the number of training examples. Meanwhile, our algorithm does not require any more computational resources than stochastic gradient descent. In our experiments, we find that our method substantially out-performs existing streaming algorithms on both real and simulated data.