Learning Powers of Poisson Binomial Distributions

We introduce the problem of simultaneously learning all powers of a Poisson Binomial Distribution (PBD). A PBD of order $n$ is the distribution of a sum of $n$ mutually independent Bernoulli random variables $Xi$, where $\mathbb{E}[Xi] = pi$. The $k$'th power of this distribution, for $k$ in a range $[m]$, is the distribution of $Pk = \sum{i=1}ⁿ Xi^{(k)}$, where each Bernoulli random variable $Xi^{(k)}$ has $\mathbb{E}[Xi^{(k)}] = (pi)^k$. The learning algorithm can query any power $Pk$ several times and succeeds in learning all powers in the range, if with probability at least $1- \delta$: given any $k \in [m]$, it returns a probability distribution $Qk$ with total variation distance from $Pk$ at most $\epsilon$. We provide almost matching lower and upper bounds on query complexity for this problem. We first show a lower bound on the query complexity on PBD powers instances with many distinct parameters $pi$ which are separated, and we almost match this lower bound by examining the query complexity of simultaneously learning all the powers of a special class of PBD's resembling the PBD's of our lower bound. We study the fundamental setting of a Binomial distribution, and provide an optimal algorithm which uses $O(1/\epsilon^2)$ samples. Diakonikolas, Kane and Stewart [COLT'16] showed a lower bound of $\Omega(2^{{1/\epsilon})$} samples to learn the $pi$'s within error $\epsilon$. The question whether sampling from powers of PBDs can reduce this sampling complexity, has a negative answer since we show that the exponential number of samples is inevitable. Having sampling access to the powers of a PBD we then give a nearly optimal algorithm that learns its $p_i$'s. To prove our two last lower bounds we extend the classical minimax risk definition from statistics to estimating functions of sequences of distributions.