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We consider forward-backward greedy algorithms for solving sparse featureselection problems with general convex smooth functions. A state-of-the-artgreedy method, the Forward-Backward greedy algorithm (FoBa-obj) requires tosolve a large number of optimization problems, thus it is not scalable forlarge-size problems. The FoBa-gdt algorithm, which uses the gradientinformation for feature selection at each forward iteration, significantlyimproves the efficiency of FoBa-obj. In this paper, we systematically analyzethe theoretical properties of both forward-backward greedy algorithms. Our maincontributions are: 1) We derive better theoretical bounds than existinganalyses regarding FoBa-obj for general smooth convex functions; 2) We showthat FoBa-gdt achieves the same theoretical performance as FoBa-obj under thesame condition: restricted strong convexity condition. Our new bounds areconsistent with the bounds of a special case (least squares) and fills apreviously existing theoretical gap for general convex smooth functions; 3) Weshow that the restricted strong convexity condition is satisfied if the numberof independent samples is more than $\bar{k}\log d$ where $\bar{k}$ is thesparsity number and $d$ is the dimension of the variable; 4) We apply FoBa-gdt(with the conditional random field objective) to the sensor selection problemfor human indoor activity recognition and our results show that FoBa-gdtoutperforms other methods (including the ones based on forward greedy selectionand L1-regularization).

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