Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint

We consider forward-backward greedy algorithms for solving sparse feature selection problems with general convex smooth functions. A state-of-the-art greedy method, the Forward-Backward greedy algorithm (FoBa-obj) requires to solve a large number of optimization problems, thus it is not scalable for large-size problems. The FoBa-gdt algorithm, which uses the gradient information for feature selection at each forward iteration, significantly improves the efficiency of FoBa-obj. In this paper, we systematically analyze the theoretical properties of both forward-backward greedy algorithms. Our main contributions are: 1) We derive better theoretical bounds than existing analyses regarding FoBa-obj for general smooth convex functions; 2) We show that FoBa-gdt achieves the same theoretical performance as FoBa-obj under the same condition: restricted strong convexity condition. Our new bounds are consistent with the bounds of a special case (least squares) and fills a previously existing theoretical gap for general convex smooth functions; 3) We show that the restricted strong convexity condition is satisfied if the number of independent samples is more than $\bar{k}\log d$ where $\bar{k}$ is the sparsity number and $d$ is the dimension of the variable; 4) We apply FoBa-gdt (with the conditional random field objective) to the sensor selection problem for human indoor activity recognition and our results show that FoBa-gdt outperforms other methods (including the ones based on forward greedy selection and L1-regularization)

Distributed Bayesian Piecewise Sparse Linear Models

The importance of interpretability of machine learning models has been increasing due to emerging enterprise predictive analytics, threat of data privacy, accountability of artificial intelligence in society, and so on. Piecewise linear models have been actively studied to achieve both accuracy and interpretability. They often produce competitive accuracy against state-of-the-art non-linear methods. In addition, their representations (i.e., rule-based segmentation plus sparse linear formula) are often preferred by domain experts. A disadvantage of such models, however, is high computational cost for simultaneous determinations of the number of "pieces" and cardinality of each linear predictor, which has restricted their applicability to middle-scale data sets. This paper proposes a distributed factorized asymptotic Bayesian (FAB) inference of learning piece-wise sparse linear models on distributed memory architectures. The distributed FAB inference solves the simultaneous model selection issue without communicating $O(N)$ data where N is the number of training samples and achieves linear scale-out against the number of CPU cores. Experimental results demonstrate that the distributed FAB inference achieves high prediction accuracy and performance scalability with both synthetic and benchmark data.Comment: Short version of this paper will be published in IEEE BigData 201