Boosting is a very successful classification algorithm that produces a linear combination of "weak" classifiers (a.k.a. base learners) to obtain high quality classification models. In this paper we propose a new boosting algorithm where base learners have structure relationships in the functional space. Though such relationships are generic, our work is particularly motivated by the emerging topic of pattern based classification for semi-structured data including graphs. Towards an efficient incorporation of the structure information, we have designed a general model where we use an undirected graph to capture the relationship of subgraph-based base learners. In our method, we combine both L_1 norm and Laplacian based L_2 norm penalty with Logit loss function of Logit Boost. In this approach, we enforce model sparsity and smoothness in the functional space spanned by the basis functions. We have derived efficient optimization algorithms based on coordinate decent for the new boosting formulation and theoretically prove that it exhibits a natural grouping effect for nearby spatial or overlapping features. Using comprehensive experimental study, we have demonstrated the effectiveness of the proposed learning methods.