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 $X_i$, where $mathbb{E}[X_i] = p_i$. The $k$u0027th power of this distribution, for $k$ 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 learning all powers 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 PBDu0027s resembling the PBDu0027s 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 [COLTu002716] showed a lower bound of $Omega(2^{1/epsilon})$ samples to learn the $p_i$u0027s 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 $p_i$u0027s. To prove our two last lower bounds we extend the classical minimax risk definition from statistics to estimating functions of sequences of distributions.