A composition theorem for parity kill number

In this work, we study the parity complexity measures ${\mathsf{C}^{\oplus}_{\min}}[f]$ and ${\mathsf{DT^{\oplus}}}[f]$. ${\mathsf{C}^{\oplus}_{\min}}[f]$ is the \emph{parity kill number} of $f$, the fewest number of parities on the input variables one has to fix in order to "kill" $f$, i.e. to make it constant. ${\mathsf{DT^{\oplus}}}[f]$ is the depth of the shortest \emph{parity decision tree} which computes $f$. These complexity measures have in recent years become increasingly important in the fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and pseudorandomness \cite{BK12, Sha11, CT13}. Our main result is a composition theorem for ${\mathsf{C}^{\oplus}_{\min}}$. The $k$-th power of $f$, denoted $f^{\circ k}$, is the function which results from composing $f$ with itself $k$ times. We prove that if $f$ is not a parity function, then ${\mathsf{C}^{\oplus}_{\min}}[f^{\circ k}] \geq \Omega({\mathsf{C}_{\min}}[f]^{k}).$ In other words, the parity kill number of $f$ is essentially supermultiplicative in the \emph{normal} kill number of $f$ (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of $\mathsf{Sort}^{\circ k}$ and $\mathsf{HI}^{\circ k}$. Here $\mathsf{Sort}$ is the sort function due to Ambainis \cite{Amb06}, and $\mathsf{HI}$ is Kushilevitz's hemi-icosahedron function \cite{NW95}. In doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and \cite{TWXZ13}

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

On the Computational Power of Radio Channels

Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

On the parity complexity measures of Boolean functions

The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any Boolean function is polynomially related to the non-deterministic complexity of the function or its complement. We also show that they are polynomially related to an analogue of the block sensitivity. We further study parity decision trees in their relations with an intermediate variant of the decision trees, as well as with communication complexity.Comment: submitted to TCS on 16-MAR-200

Optimal Direct Sum Results for Deterministic and Randomized Decision Tree Complexity

A Direct Sum Theorem holds in a model of computation, when solving some k input instances together is k times as expensive as solving one. We show that Direct Sum Theorems hold in the models of deterministic and randomized decision trees for all relations. We also note that a near optimal Direct Sum Theorem holds for quantum decision trees for boolean functions.Comment: 7 page