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

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 $1-δ$ from $\log(\tfrac{n}δ)\,\varepsilon^{-d-1} C^{d^{3/2}\sqrt{\log d}}$ random queries, where $C>1$ is a universal finite constant.