An Automatic Inequality Prover and Instance Optimal Identity Testing

Gregory Valiant, Paul Valiant

Abstract: We consider the problem of verifying the identity of a distribution: Given the description of a distribution over a discrete finite or countably infinite support, $p=(p_1,p_2,\ldots)$, how many samples (independent draws) must one obtain from an unknown distribution, $q$, to distinguish, with high probability, the case that $p=q$ from the case that the total variation distance ($L_1$ distance) $\|p-q\|_1 \ge \epsilon$? We resolve this question, up to constant factors, on an instance by instance basis: there exist universal constants $c,c'$ and a function $f(p,\epsilon)$ on the known distribution $p$ and error parameter $\epsilon$, such that our tester distinguishes $p=q$ from $\|p-q\|_1\ge \epsilon$ using $f(p,\epsilon)$ samples with success probability $>2/3$, but no tester can distinguish $p=q$ from $\|p-q\|_1\ge c\cdot \epsilon$ when given $c'\cdot f(p,\epsilon)$ samples. The function $f(p,\epsilon)$ is upper-bounded by a multiple of $\frac{\|p\|_{2/3}}{\epsilon^2}$ but is more complicated. This result generalizes and tightens previous results: since distributions of support at most $n$ have $L_{2/3}$ norm bounded by $\sqrt{n},$ this result immediately shows that for such distributions, $O(\sqrt{n}/{\epsilon^2})$ samples suffice, tightening the previous bound of $O(\frac{\sqrt{n}\ {\rm polylog}\ n}{\epsilon^4})$ and matching the (tight) results for the case that $p$ is the uniform distribution of support $n$. The analysis of our very simple testing algorithm involves several hairy inequalities. To facilitate this analysis, we give a complete characterization of a general class of inequalities---generalizing Cauchy--Schwarz, Hölder's inequality, and the monotonicity of $L_p$ norms. Specifically, we characterize the set of sequences of triples $(a,b,c)_i=(a_1,b_1,c_1),\ldots,(a_r,b_r,c_r)$ for which it holds that for all finite sequences of positive numbers $(x)_j=x_1,\ldots$ and $(y)_j=y_1,\ldots,$ $\prod_{i=1}^r ( \sum_j x_j^{a_i} y_j^{b_i})^{c_i} \ge 1.$ For example, the standard Cauchy--Schwarz inequality corresponds to the triples $(a,b,c)_i=(1,0,\frac{1}{2}),$ $\,(0,1,\frac{1}{2}),$ $\,(\frac{1}{2},\frac{1}{2},-1)$. Our characterization is constructive and algorithmic, leveraging linear programming to prove or refute an inequality, which would otherwise have to be investigated, through trial and error, by hand. We hope the computational nature of our characterization will be useful to others and facilitate analyses like the one here.