  Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We
consider the problem of {\em density estimation}, in which a learning algorithm
is given i.i.d. draws from $p$ and must (with high probability) output a
hypothesis distribution that is close to $p$. The main contribution of this
paper is a highly efficient density estimation algorithm for learning using a
variable-width histogram, i.e., a hypothesis distribution with a piecewise
constant probability density function.
  In more detail, for any $k$ and $\epsilon$, we give an algorithm that makes
$\tilde{O}(k/\epsilon^2)$ draws from $p$, runs in $\tilde{O}(k/\epsilon^2)$
time, and outputs a hypothesis distribution $h$ that is piecewise constant with
$O(k \log^2(1/\epsilon))$ pieces. With high probability the hypothesis $h$
satisfies $d_{\mathrm{TV}}(p,h) \leq C \cdot \mathrm{opt}_k(p) + \epsilon$,
where $d_{\mathrm{TV}}$ denotes the total variation distance (statistical
distance), $C$ is a universal constant, and $\mathrm{opt}_k(p)$ is the smallest
total variation distance between $p$ and any $k$-piecewise constant
distribution. The sample size and running time of our algorithm are optimal up
to logarithmic factors. The "approximation factor" $C$ in our result is
inherent in the problem, as we prove that no algorithm with sample size bounded
in terms of $k$ and $\epsilon$ can achieve $C<2$ regardless of what kind of
hypothesis distribution it uses.
