Tight Sample Bounds for Renyi and Min-Entropy Estimation

Estimating entropy from samples is fundamental in information theory and property testing. Shannon entropy measures average uncertainty and can be estimated to constant additive accuracy over a $k$-symbol alphabet using $Θ(k/\log k)$ samples. Min-entropy depends only on the most likely symbol. Both are special cases of order-$α$ R'{e}nyi entropy, $H_α$. We characterize the sample complexity of estimating min-entropy and R'{e}nyi entropy for $k$ and integer $α>1$; our lower bounds also hold for noninteger $α\ge1.001$. We prove that min-entropy estimation to constant additive accuracy has sample complexity $Θ(k\log k)$. The upper bound uses the largest empirical frequency and concentration via dyadic grouping. The matching lower bound hides a slightly heavier symbol at a uniformly random location. Thus, min-entropy requires $Θ(\log^2 k)$ more samples than Shannon entropy and corrects a previously stated $Θ(k/\log k)$ characterization. For every integer $2\leα\le c_0\log k$, we prove the matching fixed-accuracy bound $Θ_{c_0}(αk^{1-1/α})$. Previous results gave $Ω_α(k^{1-1/α})$ for fixed integer $α>1$ and $O_{c_0}(α^2k^{1-1/α})$ for all integer $α>1$. Our upper bound analyzes an unbiased falling-factorial estimator based on $α$-way collisions, while a hidden-heavy-coordinate construction gives the matching lower bound and shows that the factor $α$ is unavoidable. For every real $1.001\leα\le c_0\log k$, we prove the uniform lower bound $Ω_{c_0}(αk^{1-1/α})$. Finally, since $0\le H_α(p)-H_\infty(p)\le\log k/(α-1)$, min-entropy uniformly approximates $H_α$ when $α$ is a sufficiently large multiple of $\log k$. Combining this reduction with our min-entropy bounds gives $Θ_\varepsilon(k\log k)$ sample complexity in the high-order regime.

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 $k$'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.