Learning low-degree functions from a logarithmic number of random queries

Abstract: We prove that every bounded function $f:{-1,1}^n\to[-1,1]$ of degree at most $d$ can be learned with $L_2$-accuracy $\varepsilon$ and confidence $1-\delta$ from $\log(\tfrac{n}{\delta})\,\varepsilon^{-d-1} C^{d^{3/2}\sqrt{\log d}}$ random queries, where $C>1$ is a universal finite constant.