**
Learning mixtures of structured distributions over discrete domains
**

S. Chan and
I. Diakonikolas and
R. Servedio and
X. Sun.

In
* ACM-SIAM Symposium on Discrete Algorithms (SODA)*, 2013, pp. 1380--1394.

**Abstract:**
Let $\mathfrak{C}$ be a class of probability distributions over the discrete
domain $[n] = \{1,\dots,n\}.$
We show that if $\mathfrak{C}$ satisfies a rather general condition --
essentially, that each distribution in
$\mathfrak{C}$ can be well-approximated by a variable-width
histogram with few bins -- then there is a highly efficient (both in terms of
running time and sample complexity)
algorithm that can learn any mixture of $k$ unknown distributions from
$\mathfrak{C}.$

We analyze several natural types of distributions over $[n]$,
including log-concave, monotone hazard rate and unimodal distributions,
and show that they have the required structural property of being
well-approximated by a histogram with few bins.
Applying our general algorithm, we
obtain near-optimally efficient algorithms for all these mixture
learning problems as described below. More precisely,

**Log-concave distributions:** We learn any mixture of $k$
log-concave distributions over $[n]$ using $k \cdot
\tilde{O}(1/\eps^4)$ samples (independent of $n$) and running in time
$\tilde{O}(k \log(n) / \eps^4)$ bit-operations (note that reading a single
sample from $[n]$ takes $\Theta(\log n)$ bit operations).
For the special case $k=1$ we give an efficient
algorithm using $\tilde{O}(1/\eps^3)$
samples; this generalizes the main result of \cite{DDS12stoc} from the
class of Poisson Binomial Distributions to the much broader class of all
log-concave distributions. Our upper bounds are not far from
optimal since any algorithm for this learning problem requires
$\Omega(k/\eps^{5/2})$ samples.

**Monotone hazard rate (MHR) distributions:**
We learn any mixture of $k$ MHR distributions over $[n]$ using
$O(k \log (n/\eps)/\eps^4)$ samples and running in time $\tilde{O}(k
\log^2(n) / \eps^4)$ bit-operations. Any algorithm for this learning problem must use $\Omega(k \log(n)/\eps^3)$ samples.

** Unimodal distributions:**
We give an algorithm that learns any mixture of $k$ unimodal distributions
over $[n]$ using $O(k \log (n)/\eps^{4})$ samples and running in time
$\tilde{O}(k \log^2(n) / \eps^{4})$ bit-operations.
Any algorithm for this problem must use $\Omega(k \log(n)/\eps^3)$ samples.
pdf of conference version

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