Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if allowed to grow large, they lose interpretability. In this paper, we present a novel mixed integer programming formulation to construct optimal decision trees of specified size. We take special structure of categorical features into account and allow combinatorial decisions (based on subsets of values of such a feature) at each node. We show that very good accuracy can be achieved with small trees using moderately-sized training sets. The optimization problems we solve are easily tractable with modern solvers.
We would now like to remark on other relevant uses of integer optimization in classification settings. In particular, @cite_7 considered the problem of learning optimal or's of and's'', which fits into the problem of learning optimal disjunctive normal forms (DNFs), where optimality is measured by a trade-off between the misclassification rate and the number of literals that appear in the or of and's''. The work in @cite_7 remarks on the relationship between this problem and learning optimal decision trees. In @cite_7 , for the sake of computational efficiency, the authors ultimately resort to optimally selecting from a subset of candidate suboptimal DNFs learned by heuristic means rather than solving their proposed mixed-integer optimization problem. Similarly, @cite_12 proposes learning DNF-like rules via integer optimization, and propose a formulation that can be viewed as boolean compressed sensing, lending theoretical credibility to solving a linear programming relaxation of their integer problem. Another integer model that minimizes misclassification error by choosing general partitions in feature space was proposed in @cite_13 , but when solving the model, global optimality certificates were not easily obtained on moderately-sized classification datasets, and the learned partition classifiers rarely outperformed CART, according to the overlapping author in @cite_9 .
@cite_9 @cite_13 @cite_12 @cite_7
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This paper proposes algorithms for learning two-level Boolean rules in Conjunctive Normal Form (CNF, i.e. AND-of-ORs) or Disjunctive Normal Form (DNF, i.e. OR-of-ANDs) as a type of human-interpretable classification model, aiming for a favorable trade-off between the classification accuracy and the simplicity of the rule. Two formulations are proposed. The first is an integer program whose objective function is a combination of the total number of errors and the total number of features used in the rule. We generalize a previously proposed linear programming (LP) relaxation from one-level to two-level rules. The second formulation replaces the 0-1 classification error with the Hamming distance from the current two-level rule to the closest rule that correctly classifies a sample. Based on this second formulation, block coordinate descent and alternating minimization algorithms are developed. Experiments show that the two-level rules can yield noticeably better performance than one-level rules due to their dramatically larger modeling capacity, and the two algorithms based on the Hamming distance formulation are generally superior to the other two-level rule learning methods in our comparison. A proposed approach to binarize any fractional values in the optimal solutions of LP relaxations is also shown to be effective. We analyze the problem of learning a single user's preferences in an active learning setting, sequentially and adaptively querying the user over a finite time horizon. Learning is conducted via choice-based queries, where the user selects her preferred option among a small subset of offered alternatives. These queries have been shown to be a robust and efficient way to learn an individual's preferences. We take a parametric approach and model the user's preferences through a linear classifier, using a Bayesian prior to encode our current knowledge of this classifier. The rate at which we learn depends on the alternatives offered at every time epoch. Under certain noise assumptions, we show that the Bayes-optimal policy for maximally reducing entropy of the posterior distribution of this linear classifier is a greedy policy, and that this policy achieves a linear lower bound when alternatives can be constructed from the continuum. Further, we analyze a different metric called misclassification error, proving that the performance of the optimal policy that minimizes misclassification error is bounded below by a linear function of differential entropy. Lastly, we numerically compare the greedy entropy reduction policy with a knowledge gradient policy under a number of scenarios, examining their performance under both differential entropy and misclassification error. A large number of compiler optimizations are nowadays available to users. These optimizations interact with each other and with the input code in several and complex ways. The sequence of application of optimization passes can have a significant impact on the performance achieved. The effect of the optimizations is both platform and application dependent. The exhaustive exploration of all viable sequences of compiler optimizations for a given code fragment is not feasible. As this exploration is a complex and time-consuming task, several researchers have focused on Design Space Exploration (DSE) strategies both to select optimization sequences to improve the performance of each function of the application and to reduce the exploration time. In this article, we present a DSE scheme based on a clustering approach for grouping functions with similarities and exploration of a reduced search space resulting from the combination of optimizations previously suggested for the functions in each group. The identification of similarities between functions uses a data mining method that is applied to a symbolic code representation. The data mining process combines three algorithms to generate clusters: the Normalized Compression Distance, the Neighbor Joining, and a new ambiguity-based clustering algorithm. Our experiments for evaluating the effectiveness of the proposed approach address the exploration of optimization sequences in the context of the ReflectC compiler, considering 49 compilation passes while targeting a Xilinx MicroBlaze processor, and aiming at performance improvements for 51 functions and four applications. Experimental results reveal that the use of our clustering-based DSE approach achieves a significant reduction in the total exploration time of the search space (20× over a Genetic Algorithm approach) at the same time that considerable performance speedups (41p over the baseline) were obtained using the optimized codes. Additional experiments were performed considering the LLVM compiler, considering 124 compilation passes, and targeting a LEON3 processor. The results show that our approach achieved geometric mean speedups of 1.49 × , 1.32 × , and 1.24 × for the best 10, 20, and 30 functions, respectively, and a global improvement of 7p over the performance obtained when compiling with -O2. We study binary classification in the presence of class-conditional random noise, where the learner gets to see labels that are flipped independently with some probability, and where the flip probability depends on the class. Our goal is to devise learning algorithms that are efficient and statistically consistent with respect to commonly used utility measures. In particular, we look at a family of measures motivated by their application in domains where cost-sensitive learning is necessary (for example, when there is class imbalance). In contrast to most of the existing literature on consistent classification that are limited to the classical 0-1 loss, our analysis includes more general utility measures such as the AM measure (arithmetic mean of True Positive Rate and True Negative Rate). For this problem of cost-sensitive learning under class-conditional random noise, we develop two approaches that are based on suitably modifying surrogate losses. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical utility maximization in the presence of i.i.d. data with noisy labels. If the loss function satis_es a simple symmetry condition, we show that using unbiased estimator leads to an efficient algorithm for empirical maximization. Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong utility bounds. This approach implies that methods already used in practice, such as biased SVM and weighted logistic regression, are provably noise-tolerant. For two practically important measures in our family, we show that the proposed methods are competitive with respect to recently proposed methods for dealing with label noise in several benchmark data sets.