Despite the variety of robust regression methods that have been developed, current regression formulations are either NP-hard, or allow unbounded response to even a single leverage point. We present a general formulation for robust regression --Variational M-estimation--that unifies a number of robust regression methods while allowing a tractable approximation strategy. We develop an estimator that requires only polynomial-time, while achieving certain robustness and consistency guarantees. An experimental evaluation demonstrates the effectiveness of the new estimation approach compared to standard methods.

Subspace learning seeks a low dimensional representation of data that enables accurate reconstruction. However, in many applications, data is obtained from multiple sources rather than a single source (e.g. an object might be viewed by cameras at different angles, or a document might consist of text and images). The conditional independence of separate sources imposes constraints on their shared latent representation, which, if respected, can improve the quality of the learned low dimensional representation. In this paper, we present a convex formulation of multi-view subspace learning that enforces conditional independence while reducing dimensionality. For this formulation, we develop an efficient algorithm that recovers an optimal data reconstruction by exploiting an implicit convex regularizer, then recovers the corresponding latent representation and reconstruction model, jointly and optimally. Experiments illustrate that the proposed method produces high quality results.

Sparse learning models typically combine a smooth loss with a nonsmooth penalty, such as trace norm. Although recent developments in sparse approximation have offered promising solution methods, current approaches either apply only to matrix-norm constrained problems or provide suboptimal convergence rates. In this paper, we propose a boosting method for regularized learning that guarantees $\epsilon$ accuracy within $O(1/\epsilon)$ iterations. Performance is further accelerated by interlacing boosting with fixed-rank local optimization---exploiting a simpler local objective than previous work. The proposed method yields state-of-the-art performance on large-scale problems. We also demonstrate an application to latent multiview learning for which we provide the first efficient weak-oracle.

Robust regression and classification are often thought to require non-convex loss functions that prevent scalable, global training. However, such a view neglects the possibility of reformulated training methods that can yield practically solvable alternatives. A natural way to make a loss function more robust to outliers is to truncate loss values that exceed a maximum threshold. We demonstrate that a relaxation of this form of ``loss clipping'' can be made globally solvable and applicable to any standard loss while guaranteeing robustness against outliers. We present a generic procedure that can be applied to standard loss functions and demonstrate improved robustness in regression and classification problems.

We prove that linear projections between distribution families with fixed first and second moments are surjective, regardless of dimension. We further extend this result to families that respect additional constraints, such as symmetry, unimodality and log-concavity. By combining our results with classic univariate inequalities, we provide new worst-case analyses for natural risk criteria arising in different fields. One discovery is that portfolio selection under the worst-case value-at-risk and conditional value-at-risk criteria yields identical portfolios.