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For example, I was able to find that, using functionals, the squared mean error argmin is $[y]$:

$f^* = argmin_f E_{x,y \sim p*(x,y)}(y-f(x))^2$

gives us after deriving with respect to f:

$\int_y(-2y + 2f(x))p(y|x) = 0$, and thus:

$2E[Y] - 2f(x)*1 = 0$, which leads to $f^*(x) = E[Y]$.

However, I can't use the same logic to find the functional for:

$f^* = argmin_f E_{x,y \sim p*(x,y)}|y-f(x)|$

I know it's not differentiable at its minima, but is there any way to use functionals to motivate the minima as the median?

Thanks.

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As far as I know you can indeed use derivatives to justify the median as the minimizer of the absolute value loss. Considering $X\sim F$, where $F$ has density $f$ for simplicity, we are interested in \begin{align*} \min_{c \in \mathbb{R}} \mathbb{E}[|X-c|] = \min_{c \in \mathbb{R}} \left( -0.5\int_{-\infty}^{c} (x-c) f(x)dx + 0.5 \int_{c}^{\infty} (x-c) f(x)dx \right). \end{align*} Now we can try to get to our first order condition. When the variable (here $c$) is in the integral boundary this can be done via the Leibniz rule: https://en.wikipedia.org/wiki/Leibniz_integral_rule. Taking derivatives with respect to $c$ then leads to the first order condition \begin{align} 0 = 0.5 \int_{-\infty}^{c} f(x)dx - 0.5 \int_{c}^{\infty} f(x)dx = 0.5 F(c) - 0.5 (1-F(c)) = F(c) - 0.5, \end{align} where I assume that we can interchange limits and integrals. The last equation now is now equivalent to $c = F^{-1}(0.5)$ showing that the median is a stationary point. To show that the median is actually the minimum you can consider the function $g(c) = \mathbb{E}[|X-c|]$ and show that it is convex, which follows from the convexity of $|x|$.

While you put in the machine learning tag, this type of reasoning can be utilized in a couple of statistical contexts. It is the starting point for Quantile Regression, where the loss function is generalized to get arbitrary quantiles $F^{-1}(\alpha)$ for $\alpha \in (0,1)$. Statistical functionals (often defined via minimization of expected loss) are a standard starting point for statistics in general, but are specifically studied in Robust Statistics.


Edit: convexity in itself is not enough to guarantee the existence of a minimizer. As in real analysis coercivity (or a comparable condition) needs to be invoked. For the case discussed above Theorem 6.8 in E.L. Lehmann. "Theory of Point Estimation". John Wiley & Sons, 1983, is sufficient.

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  • $\begingroup$ Hi! Thank you for the very thorough answer. I just have a quick question: isn't $F^{-1}(0.5)$ supposed to be the mean? I've always intuitively thought that it was the center of mass, and in this case half of the mass is on the left of the mean equals that expression. I suppose in a continuous setting the median and the mean would be equivalent but not really sure how this exactly leads to the median. $\endgroup$ – OneRaynyDay Feb 13 '17 at 16:55
  • $\begingroup$ Hi @OneRaynyDay, the mean and median do not necessarily coincide. For non symmetric distributions the median $F^{-1}(0.5)$ and the mean $\int_{\mathbb{R}} x dF(x)$ are not the same. See for example the exponential distribution: en.wikipedia.org/wiki/Exponential_distribution $\endgroup$ – user3456032 Feb 13 '17 at 16:58
  • $\begingroup$ Ah. Thank you for that! Accepted :) have a great day. $\endgroup$ – OneRaynyDay Feb 13 '17 at 17:01
  • $\begingroup$ @OneRaynyDay you are welcome - I quickly added a reference closing the existence of a minimizer. I hope this helps and that you can have some fun with stats :-) $\endgroup$ – user3456032 Feb 13 '17 at 17:03
  • $\begingroup$ Haha, thank you! I have yet to take real analysis but I am really liking statistics/ML(coming from cs background) $\endgroup$ – OneRaynyDay Feb 13 '17 at 17:04

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