The Stochastic Gradient Boosted Distributed Decision Trees

by Jerry Ye, Jyh-Herng Chow, Jiang Chen and Zhaohui Zheng

This paper proposed two solutions forr implementing the exact stochastic GBDT, which was developed by the famous statician Friendman. I'd like to scan his two previous papers later as the first study of decision trees (others will be CART and C4.5... I guess).

The map/reduce implementation is based on the previously scanned paper, using horizontal splits while in the MPI implementation using vertical splits. The former is quite directly; the later requires communication using all-to-all broadcasting.

Maybe after studying the GBDT, I would have a better understanding of this paper.

A Framework for Learning from Dsitributed Data Using Sufficient Statistics and its Application to Learning Decision Trees

by Doina Caragea, Adrian Silvescu and Vasant Honavar

This paper addresses training a large-scale problem using two types of split of data: horizontal fragmentation (sample subsets) and vertical fragmentation (feature subsets). The orientation is determined when writing row-wise sample matrix.

The key idea behind this paper is to extract sufficient statistics from  splits of data so that we may aggregate the statistics in the last to get the exact model. This is quite direct in the case of MoE (when each  expert is an exponential family). This paper, however, has the emphasis in decision trees.

The decision trees are trained layer-wise, finding splits of data using features that maximized the information gain (or some similar criteria). The paper discussed two cases:

• if the data are horizontally splitted, each site compute the required statistics and they are all combine to find the best split; the split then is passed back to each site so that each site knows which node the samples are in and they could take a second collection of sufficient stats.
• if the data are vertically splitted, each site could figure out its own split and they will be compared and find the best; however, the split is then represented by the indices (so that each site may understand which samples are in which node);

This solution is accurate and can be applied to implementing the decision trees under map/reduce framework directly. And it should also apply to many hierarchical models too.

Collaborative Filtering on a Budget

by Alexandros Karatzoglou, Alex Smola and Markus Weimer

This paper talks about dealing with large scale collaborative filtering. The collaborative filtering can be formulated as a matrix factorization problem and we may try several loss functions with different regularizer. One typical example is the M3F paper previously scanned here. A convenient solver is stochastic gradient descent.

The core idea proposed is we may use two hash functions (one for user and another for items recommended) to aggregate the user matrix and item matrix to eliminate the computational cost in large-scale problems. These matrices are approximated with the help of Rademacher functions. But I have no idea why this is possible. Maybe I will take a look some day later.