Near-Linear Sample Complexity for $L_p$ Polynomial Regression

TL;DR

This paper establishes near-linear sample complexity bounds for $L_p$ polynomial regression across all $p$, using random sampling from the Chebyshev measure, with novel bounds on the $L_p$ Lewis weight function.

Contribution

It introduces the first near-optimal sample complexity results for $L_p$ polynomial regression for all $p$, extending beyond the well-understood $L_2$ and $L_______$ cases.

Findings

Sample complexity is linear in polynomial degree $d$ for all $p$.

Sampling from the Chebyshev measure is near-optimal for $L_p$ regression.

New bounds on the $L_p$ Lewis weight function for $p \u2264 2$.

Abstract

We study $L_p$ polynomial regression. Given query access to a function $f:[-1,1] \rightarrow \mathbb{R}$, the goal is to find a degree $d$ polynomial $\hat{q}$ such that, for a given parameter $\varepsilon > 0$, $$ \|\hat{q}-f\|_p\le (1+\varepsilon) \cdot \min_{q:\text{deg}(q)\le d}\|q-f\|_p. $$ Here $\|\cdot\|_p$ is the $L_p$ norm, $\|g\|_p = (\int_{-1}^1 |g(t)|^p dt)^{1/p}$. We show that querying $f$ at points randomly drawn from the Chebyshev measure on $[-1,1]$ is a near-optimal strategy for polynomial regression in all $L_p$ norms. In particular, to find $\hat q$, it suffices to sample $O(d\, \frac{\text{polylog}\,d}{\text{poly}\,\varepsilon})$ points from $[-1,1]$ with probabilities proportional to this measure. While the optimal sample complexity for polynomial regression was well understood for $L_2$ and $L_\infty$, our result is the first that achieves sample complexity linear in $d$ and error $(1+\varepsilon)$ for other values of $p$ without any assumptions. Our result requires two main technical contributions. The first concerns $p\leq 2$, for which we provide explicit bounds on the $L_p$ Lewis weight function of the infinite linear operator underlying polynomial regression. Using tools from the orthogonal polynomial literature, we show that this function is bounded by the Chebyshev density. Our second key contribution is to take advantage of the structure of polynomials to reduce the $p>2$ case to the $p\leq 2$ case. By doing so, we obtain a better sample complexity than what is possible for general $p$-norm linear regression problems, for which $\Omega(d^{p/2})$ samples are required.