Near-Linear Sample Complexity for L_p Polynomial Regression

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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.