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• W2136499803 abstract "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." @default.
• W2136499803 created "2016-06-24" @default.
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• W2136499803 date "2014-11-01" @default.
• W2136499803 modified "2023-09-27" @default.
• W2136499803 title "Near-Optimal Density Estimation in Near-Linear Time Using Variable-Width Histograms" @default.
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