Median trick

The median trick is a generic approach that increases the chances of a probabilistic algorithm to succeed. ^[1] Apparently first used in 1986 ^[2] by Jerrum et al. ^[3] for approximate counting algorithms, the technique was later applied to a broad selection of classification and regression problems. ^[2]

The idea of median trick is very simple: run the randomized algorithm with numeric output multiple times, and use the median of the obtained results as a final answer. For example, for sublinear in time algorithms the same algorithm can be run repeatedly (or in parallel) over random subsets of input data, and, per Chernoff inequality, the median of the results will converge to solution very fast. ^[4] For the algorithms that are sublinear in space (e.g., counting the distinct elements of a stream), different randomizations of the algorithm (say, with different hash functions) should be used for repeated runs over the same data. ^[5]

Statement

Given a set of independent random variables ${\textstyle X_{1},\dots ,X_{n}}$, and an unknown deterministic number ${\textstyle Y}$.

Suppose that each random variable ${\textstyle X_{i}}$ falls within ${\textstyle [Y\pm \epsilon ]}$ with probability ${\textstyle \geq p}$ where ${\textstyle p>1/2}$ is a constant, then the median trick states that ${\textstyle Med(X_{i})\in [Y\pm \epsilon ]}$ with probability ${\textstyle \geq 1-e^{-2n(p-1/2)^{2}}}$.

In other words, in order to ensure that ${\textstyle Y\in [Med(X_{i})\pm \epsilon ]}$ with probability ${\textstyle \geq 1-\delta }$, it suffices to use ${\textstyle {\frac {\ln {\frac {1}{\delta }}}{2(p-1/2)^{2}}}}$ samples.

Proof

Let ${\textstyle Z_{i}}$ be the indicator variable for the event that ${\textstyle X_{i}\in [Y\pm \epsilon ]}$. Then, the event ${\textstyle Med(X_{i})\in [Y\pm \epsilon ]}$ fails to occur only if at least half of ${\textstyle Z_{i}=1}$, that is, ${\textstyle {\frac {1}{n}}\sum _{i}Z_{i}\leq 1/2}$.

By Hoeffding's inequality, this event occurs with probability ${\textstyle \leq e^{-2n(p-1/2)^{2}}}$.

References

1. Kogler & Traxler 2017, p. 378.
2. Kogler & Traxler 2017, p. 380.
3. Jerrum, Valiant & Vazirani 1986, p. 182, Lemma 6.1.
4. Wang & Han 2015, p. 11.
5. Wang & Han 2015, pp. 17–18, Median Trick in Boosting Confidence.

Sources

• Kogler, Alexander; Traxler, Patrick (2017). "Parallel and Robust Empirical Risk Minimization via the Median Trick". Mathematical Aspects of Computer and Information Sciences. Cham: Springer International Publishing. doi:10.1007/978-3-319-72453-9_31. ISBN   978-3-319-72452-2. ISSN   0302-9743.
• Jerrum, Mark R.; Valiant, Leslie G.; Vazirani, Vijay V. (1986). "Random generation of combinatorial structures from a uniform distribution". Theoretical Computer Science. 43. Elsevier BV: 169–188. doi:10.1016/0304-3975(86)90174-x. ISSN   0304-3975.
• Wang, Dan; Han, Zhu (2015). "Basics for Sublinear Algorithms". Sublinear Algorithms for Big Data Applications. Cham: Springer International Publishing. doi:10.1007/978-3-319-20448-2_2. ISBN   978-3-319-20447-5. ISSN   2191-5768.