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We study the problem of estimating the value of sums of the form Sp≜∑(xip) when one has the ability to sample xi≥0 with probability proportional to its magnitude. When p=2, this problem is equivalent to estimating the selectivity of a self-join query in database systems when one can sample rows randomly. We also study the special case when {xi} is the degree sequence of a graph, which corresponds to counting the number of p-stars in a graph when one has the ability to sample edges randomly. Our algorithm for a (1±ε)-multiplicative approximation of Sp has query and time complexities O(mloglognϵ2S1/pp). Here, m=∑xi/2 is the number of edges in the graph, or equivalently, half the number of records in the database table. Similarly, n is the number of vertices in the graph and the number of unique values in the database table. We also provide tight lower bounds (up to polylogarithmic factors) in almost all cases, even when {xi} is a degree sequence and one is allowed to use the structure of the graph to try to get a better estimate. We are not aware of any prior lower bounds on the problem of join selectivity estimation. For the graph problem, prior work which assumed the ability to sample only vertices uniformly gave algorithms with matching lower bounds (Gonen et al. in SIAM J Comput 25:1365–1411, 2011). With the ability to sample edges randomly, we show that one can achieve faster algorithms for approximating the number of star subgraphs, bypassing the lower bounds in this prior work. For example, in the regime where Sp≤n, and p=2, our upper bound is O~(n/S1/2p), in contrast to their Ω(n/S1/3p) lower bound when no random edge queries are available. In addition, we consider the problem of counting the number of directed paths of length two when the graph is directed. This problem is equivalent to estimating the selectivity of a join query between two distinct tables. We prove that the general version of this problem cannot be solved in sublinear time. However, when the ratio between in-degree and out-degree is bounded—or equivalently, when the ratio between the number of occurrences of values in the two columns being joined is bounded—we give a sublinear time algorithm via a reduction to the undirected case. 

Several well-studied models of access to data samples, including statistical queries, local differential privacy and low-communication algorithms rely on queries that provide information about a function of a single sample. (For example, a statistical query (SQ) gives an estimate of $\E_{x\sim D}[q(x)]$ for any choice of the query function $q:X\rightarrow \R$, where $D$ is an unknown data distribution.) Yet some data analysis algorithms rely on properties of functions that depend on multiple samples. Such algorithms would be naturally implemented using $k$-wise queries each of which is specified by a function $q:X^k\rightarrow \R$. Hence it is natural to ask whether algorithms using $k$-wise queries can solve learning problems more efficiently and by how much. Blum, Kalai, Wasserman~\cite{blum2003noise} showed that for any weak PAC learning problem over a fixed distribution, the complexity of learning with $k$-wise SQs is smaller than the (unary) SQ complexity by a factor of at most $2^k$. We show that for more general problems over distributions the picture is substantially richer. For every $k$, the complexity of distribution-independent PAC learning with $k$-wise queries can be exponentially larger than learning with $(k+1)$-wise queries. We then give two approaches for simulating a $k$-wise query using unary queries. The first approach exploits the structure of the problem that needs to be solved. It generalizes and strengthens (exponentially) the results of Blum \etal \cite{blum2003noise}. It allows us to derive strong lower bounds for learning DNF formulas and stochastic constraint satisfaction problems that hold against algorithms using $k$-wise queries. The second approach exploits the $k$-party communication complexity of the $k$-wise query function.