Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as pseudorandom generators for halfspaces. In this work, we investigate the limits of these techniques by proving inapproximability results for the sign function. Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be learned with respect to log-concave distributions on $\mathbb{R}^n$ in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. We ask whether this technique can be extended beyond log-concave distributions, and establish a negative result. We show that polynomials of any degree cannot approximate the sign function to within arbitrarily low error for a large class of non-log-concave distributions on the real line, including those with densities proportional to $\exp(-|x|^{0.99})$. Secondly, we investigate the derandomization of Chernoff-type concentration inequalities. Chernoff-type tail bounds on sums of independent random variables have pervasive applications in theoretical computer science. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that these inequalities can be established for sums of random variables with only $O(\log(1/\delta))$-wise independence, for a tail probability of $\delta$. We show that their results are tight up to constant factors. These results rely on techniques from weighted approximation theory, which studies how well functions on the real line can be approximated by polynomials under various distributions. We believe that these techniques will have further applications in other areas of computer science.Comment: 22 page

Inductive queries for a drug designing robot scientist

It is increasingly clear that machine learning algorithms need to be integrated in an iterative scientific discovery loop, in which data is queried repeatedly by means of inductive queries and where the computer provides guidance to the experiments that are being performed. In this chapter, we summarise several key challenges in achieving this integration of machine learning and data mining algorithms in methods for the discovery of Quantitative Structure Activity Relationships (QSARs). We introduce the concept of a robot scientist, in which all steps of the discovery process are automated; we discuss the representation of molecular data such that knowledge discovery tools can analyse it, and we discuss the adaptation of machine learning and data mining algorithms to guide QSAR experiments

Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems

A new class of non-homogeneous state-affine systems is introduced for use in reservoir computing. Sufficient conditions are identified that guarantee first, that the associated reservoir computers with linear readouts are causal, time-invariant, and satisfy the fading memory property and second, that a subset of this class is universal in the category of fading memory filters with stochastic almost surely uniformly bounded inputs. This means that any discrete-time filter that satisfies the fading memory property with random inputs of that type can be uniformly approximated by elements in the non-homogeneous state-affine family.Comment: 41 page

Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the $N \times N$ adjacency matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value $\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed version of this matrix is at least $\Delta/\lambda$. We use this connection to prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran