Learning general sparse additive models from point queries in high dimensions

We consider the problem of learning a $d$-variate function $f$ defined on the cube $[-1,1{]}^d\subset {\mathbb{R}}^d$, where the algorithm is assumed to have black box access to samples of $f$ within this domain. Denote ${\mathcal{S}}_r\subset \left(\genfrac{}{}{0px}{}{[d]}{r}\right);r=1,\dots ,r_0$ to be sets consisting of unknown $r$-wise interactions amongst the coordinate variables. We then focus on the setting where $f$ has an additive structure, i.e., it can be represented as $$f=\sum _{\mathbf{j}\in {\mathcal{S}}_1}{\varphi }_{\mathbf{j}}+\sum _{\mathbf{j}\in {\mathcal{S}}_2}{\varphi }_{\mathbf{j}}+\cdots +\sum _{\mathbf{j}\in {\mathcal{S}}_{r_0}}{\varphi }_{\mathbf{j}},$$ where each ${\varphi }_{\mathbf{j}}$; $\mathbf{j}\in {\mathcal{S}}_r$ is at most $r$-variate for $1\le r\le r_0$. We derive randomized algorithms that query $f$ at carefully constructed set of points, and exactly recover each ${\mathcal{S}}_r$ with high probability. In contrary to the previous work, our analysis does not rely on numerical approximation of derivatives by finite order differences.