On the worst-case error of least squares algorithms for $L_2$-approximation with high probability

It was recently shown in [4] that, for $L_2$-approximation of functions from a Hilbert space, function values are almost as powerful as arbitrary linear information, if the approximation numbers are square-summable. That is, we showed that $$e_n \,\lesssim\, \sqrt{\frac{1}{k_n} \sum_{j\geq k_n} a_j^2} \qquad \text{ with }\quad k_n \asymp \frac{n}{\ln(n)},$$ where $e_n$ are the sampling numbers and $a_k$ are the approximation numbers. In particular, if $(a_k)\in\ell_2$, then $e_n$ and $a_n$ are of the same polynomial order. For this, we presented an explicit (weighted least squares) algorithm based on i.i.d. random points and proved that this works with positive probability. This implies the existence of a good deterministic sampling algorithm. Here, we present a modification of the proof in [4] that shows that the same algorithm works with probability at least $1-{n^{-c}}$ for all $c>0$.