On the Power of Learning from $k$-Wise Queries

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 $Ex_{x \sim D}[q(x)]$ for any choice of the query function $q$ mapping $X$ to the reals, where $D$ is an unknown data distribution over $X$.) 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 $q$ mapping $X^k$ to the reals. Hence it is natural to ask whether algorithms using $k$-wise queries can solve learning problems more efficiently and by how much. Blum, Kalai and Wasserman (2003) 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 et al.. 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.