Related papers: Lower bounds for uniform read-once threshold formu…
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically…
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give…
We propose a novel algorithm for optimizing multivariate linear threshold functions as split functions of decision trees to create improved Random Forest classifiers. Standard tree induction methods resort to sampling and exhaustive search…
This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable?…
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial…
This paper introduces a series of methods for traversing binary decision trees using arithmetic operations. We present a suite of binary tree traversal algorithms that leverage novel representation matrices to flatten the full binary tree…
Decision trees and their ensembles are popular in machine learning as easy-to-understand models. Several techniques have been proposed in the literature for learning tree-based classifiers, with different techniques working well for data…
This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an…
It is known that the size of the largest common subtree (i.e., the maximum agreement subtree) of two independent random binary trees with $n$ given labeled leaves is of order between $n^{0.366}$ and $n^{1/2}$. We improve the lower bound to…
In this paper, we study learning and testing decision tree of size and depth that are significantly smaller than the number of attributes $n$. Our main result addresses the problem of poly$(n,1/\epsilon)$ time algorithms with…
In this paper, we study the query complexity of Boolean functions in the presence of uncertainty, motivated by parallel computation with an unlimited number of processors where inputs are allowed to be unknown. We allow each query to…
We study a natural complexity measure of Boolean functions known as the rational degree. Denoted $\textrm{rdeg}(f)$, it is the minimal degree of a rational function that is equal to $f$ on the Boolean hypercube. For total functions $f$, it…
We investigate at decision trees that incorporate both traditional queries based on one attribute and queries based on hypotheses about the values of all attributes. Such decision trees are similar to ones studied in exact learning, where…
It is shown that the counting function of n Boolean variables can be implemented with the formulae of size O(n^3.06) over the basis of all 2-input Boolean functions and of size O(n^4.54) over the standard basis. The same bounds follow for…
Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$…
The use of machine learning algorithms in finance, medicine, and criminal justice can deeply impact human lives. As a consequence, research into interpretable machine learning has rapidly grown in an attempt to better control and fix…
The best current methods for exactly computing the number of satisfying assignments, or the satisfying probability, of Boolean formulas can be seen, either directly or indirectly, as building 'decision-DNNF' (decision decomposable negation…
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for…
We present a method for learning treewidth-bounded Bayesian networks from data sets containing thousands of variables. Bounding the treewidth of a Bayesian greatly reduces the complexity of inferences. Yet, being a global property of the…
Algorithms for learning decision trees often include heuristic local-search operations such as (1) adjusting the threshold of a cut or (2) also exchanging the feature of that cut. We study minimizing the number of classification errors by…