-
Notifications
You must be signed in to change notification settings - Fork 1
/
info.json
16 lines (16 loc) · 1.96 KB
/
info.json
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
{
"abstract": " <p>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 prove that for every symmetric distribution $\\mathcal{D}$, there exists a set of $n^{O(\\log{(1/\\epsilon)})}$ functions $\\mathbb{S}$, such that for every disjunction $c$, there is function $p$, expressible as a linear combination of functions in $\\mathbb{S}$, 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 implies 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 in $\\ell_1$ 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.</p> <p>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.</p> </div",
"authors": [
"Vitaly Feldman",
"Pravesh Kothari"
],
"id": "feldman15a",
"issue": 106,
"pages": [
3455,
3467
],
"title": "Agnostic Learning of Disjunctions on Symmetric Distributions",
"volume": 16,
"year": 2015
}