Product distribution learning with imperfect advice

Given i.i.d.~samples from an unknown distribution $P$, the goal of distribution learning is to recover the parameters of a distribution that is close to $P$. When $P$ belongs to the class of product distributions on the Boolean hypercube $\{0,1{\}}^d$, it is known that $\mathrm{\Omega }(d\mathrm{/}{\epsilon }^2)$ samples are necessary to learn $P$ within total variation (TV) distance $\epsilon$. We revisit this problem when the learner is also given as advice the parameters of a product distribution $Q$. We show that there is an efficient algorithm to learn $P$ within TV distance $\epsilon$ that has sample complexity $\tilde{O}(d^{1-\eta }\mathrm{/}{\epsilon }^2)$, if $\mathrm{\parallel }\mathbf{p}-\mathbf{q}{\mathrm{\parallel }}_1<\epsilon d^{0.5-\mathrm{\Omega }(\eta )}$. Here, $\mathbf{p}$ and $\mathbf{q}$ are the mean vectors of $P$ and $Q$ respectively, and no bound on $\mathrm{\parallel }\mathbf{p}-\mathbf{q}{\mathrm{\parallel }}_1$ is known to the algorithm a priori.