Top-Down Induction of Decision Trees: Rigorous Guarantees and Inherent Limitations

Consider the following heuristic for building a decision tree for a function $f : \{0,1\}^n \to \{\pm 1\}$. Place the most influential variable $x_i$ of $f$ at the root, and recurse on the subfunctions $f_{x_i=0}$ and $f_{x_i=1}$ on the left and right subtrees respectively; terminate once the tree is an $\varepsilon$-approximation of $f$. We analyze the quality of this heuristic, obtaining near-matching upper and lower bounds: $\circ$ Upper bound: For every $f$ with decision tree size $s$ and every $\varepsilon \in (0,\frac1{2})$, this heuristic builds a decision tree of size at most $s^{O(\log(s/\varepsilon)\log(1/\varepsilon))}$. $\circ$ Lower bound: For every $\varepsilon \in (0,\frac1{2})$ and $s \le 2^{\tilde{O}(\sqrt{n})}$, there is an $f$ with decision tree size $s$ such that this heuristic builds a decision tree of size $s^{\tilde{\Omega}(\log s)}$. We also obtain upper and lower bounds for monotone functions: $s^{O(\sqrt{\log s}/\varepsilon)}$ and $s^{\tilde{\Omega}(\sqrt[4]{\log s } )}$ respectively. The lower bound disproves conjectures of Fiat and Pechyony (2004) and Lee (2009). Our upper bounds yield new algorithms for properly learning decision trees under the uniform distribution. We show that these algorithms---which are motivated by widely employed and empirically successful top-down decision tree learning heuristics such as ID3, C4.5, and CART---achieve provable guarantees that compare favorably with those of the current fastest algorithm (Ehrenfeucht and Haussler, 1989). Our lower bounds shed new light on the limitations of these heuristics. Finally, we revisit the classic work of Ehrenfeucht and Haussler. We extend it to give the first uniform-distribution proper learning algorithm that achieves polynomial sample and memory complexity, while matching its state-of-the-art quasipolynomial runtime

Parallelizing Deadlock Resolution in Symbolic Synthesis of Distributed Programs

Previous work has shown that there are two major complexity barriers in the synthesis of fault-tolerant distributed programs: (1) generation of fault-span, the set of states reachable in the presence of faults, and (2) resolving deadlock states, from where the program has no outgoing transitions. Of these, the former closely resembles with model checking and, hence, techniques for efficient verification are directly applicable to it. Hence, we focus on expediting the latter with the use of multi-core technology. We present two approaches for parallelization by considering different design choices. The first approach is based on the computation of equivalence classes of program transitions (called group computation) that are needed due to the issue of distribution (i.e., inability of processes to atomically read and write all program variables). We show that in most cases the speedup of this approach is close to the ideal speedup and in some cases it is superlinear. The second approach uses traditional technique of partitioning deadlock states among multiple threads. However, our experiments show that the speedup for this approach is small. Consequently, our analysis demonstrates that a simple approach of parallelizing the group computation is likely to be the effective method for using multi-core computing in the context of deadlock resolution

On the usage of the probability integral transform to reduce the complexity of multi-way fuzzy decision trees in Big Data classification problems

We present a new distributed fuzzy partitioning method to reduce the complexity of multi-way fuzzy decision trees in Big Data classification problems. The proposed algorithm builds a fixed number of fuzzy sets for all variables and adjusts their shape and position to the real distribution of training data. A two-step process is applied : 1) transformation of the original distribution into a standard uniform distribution by means of the probability integral transform. Since the original distribution is generally unknown, the cumulative distribution function is approximated by computing the q-quantiles of the training set; 2) construction of a Ruspini strong fuzzy partition in the transformed attribute space using a fixed number of equally distributed triangular membership functions. Despite the aforementioned transformation, the definition of every fuzzy set in the original space can be recovered by applying the inverse cumulative distribution function (also known as quantile function). The experimental results reveal that the proposed methodology allows the state-of-the-art multi-way fuzzy decision tree (FMDT) induction algorithm to maintain classification accuracy with up to 6 million fewer leaves.Comment: Appeared in 2018 IEEE International Congress on Big Data (BigData Congress). arXiv admin note: text overlap with arXiv:1902.0935

Affine arithmetic-based methodology for energy hub operation-scheduling in the presence of data uncertainty

In this study, the role of self-validated computing for solving the energy hub-scheduling problem in the presence of multiple and heterogeneous sources of data uncertainties is explored and a new solution paradigm based on affine arithmetic is conceptualised. The benefits deriving from the application of this methodology are analysed in details, and several numerical results are presented and discussed