  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 $X_i$, where
$\mathbb{E}[X_i] = p_i$. The $k$'th power of this distribution, for $k$ in a
range $[m]$, is the distribution of $P_k = \sum_{i=1}^n X_i^{(k)}$, where each
Bernoulli random variable $X_i^{(k)}$ has $\mathbb{E}[X_i^{(k)}] = (p_i)^k$.
The learning algorithm can query any power $P_k$ 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 $Q_k$ with total
variation distance from $P_k$ 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 $p_i$ 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
$p_i$'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.
