Agnostic Learning of Disjunctions on Symmetric Distributions

We consider the problem of approximating and learning disjunctions (or equivalently, conjunctions) on symmetric distributions over $\{0,1\}^n$. Symmetric distributions are distributions whose PDF is invariant under any permutation of the variables. We give a simple proof that for every symmetric distribution $\mathcal{D}$, there exists a set of $n^{O(\log{(1/\epsilon)})}$ functions $\mathcal{S}$, such that for every disjunction $c$, there is function $p$, expressible as a linear combination of functions in $\mathcal{S}$, such that $p$ $\epsilon$-approximates $c$ in $\ell_1$ distance on $\mathcal{D}$ or $\mathbf{E}_{x \sim \mathcal{D}}[ |c(x)-p(x)|] \leq \epsilon$. This directly gives an agnostic learning algorithm for disjunctions on symmetric distributions that runs in time $n^{O( \log{(1/\epsilon)})}$. The best known previous bound is $n^{O(1/\epsilon^4)}$ and follows from approximation of the more general class of halfspaces (Wimmer, 2010). We also show that there exists a symmetric distribution $\mathcal{D}$, such that the minimum degree of a polynomial that $1/3$-approximates the disjunction of all $n$ variables is $\ell_1$ distance on $\mathcal{D}$ is $\Omega( \sqrt{n})$. Therefore the learning result above cannot be achieved via $\ell_1$-regression with a polynomial basis used in most other agnostic learning algorithms. Our technique also gives a simple proof that for any product distribution $\mathcal{D}$ and every disjunction $c$, there exists a polynomial $p$ of degree $O(\log{(1/\epsilon)})$ such that $p$ $\epsilon$-approximates $c$ in $\ell_1$ distance on $\mathcal{D}$. This was first proved by Blais et al. (2008) via a more involved argument

Robust classification via MOM minimization

We present an extension of Vapnik's classical empirical risk minimizer (ERM) where the empirical risk is replaced by a median-of-means (MOM) estimator, the new estimators are called MOM minimizers. While ERM is sensitive to corruption of the dataset for many classical loss functions used in classification, we show that MOM minimizers behave well in theory, in the sense that it achieves Vapnik's (slow) rates of convergence under weak assumptions: data are only required to have a finite second moment and some outliers may also have corrupted the dataset. We propose an algorithm inspired by MOM minimizers. These algorithms can be analyzed using arguments quite similar to those used for Stochastic Block Gradient descent. As a proof of concept, we show how to modify a proof of consistency for a descent algorithm to prove consistency of its MOM version. As MOM algorithms perform a smart subsampling, our procedure can also help to reduce substantially time computations and memory ressources when applied to non linear algorithms. These empirical performances are illustrated on both simulated and real datasets

On the hardness of learning intersections of two halfspaces

AbstractWe show that unless NP=RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in Rn using a hypothesis which is a function of up to ℓ halfspaces (linear threshold functions) for any integer ℓ. Specifically, we show that for every integer ℓ and an arbitrarily small constant ε>0, unless NP=RP, no polynomial time algorithm can distinguish whether there is an intersection of two halfspaces that correctly classifies a given set of labeled points in Rn, or whether any function of ℓ halfspaces can correctly classify at most 12+ε fraction of the points