Calculate boundary for MAE given RMSE

E.g. from the Netflix prize I know that the best RMSE = 0.8563 where the test dataset has a size of n=1,408,789.

Can I calculate a boundary for the MAE. If not, why can't I calculate a boundary?

I started like that:

RMSE = $\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-\hat y_{i})^2}{n}}=0.8563$

Now I want to calculate the MAE or a boundary for the MSE:

MAE = $\frac{1}{n}\sum_{i=1}^{n}{|y_{i}-\hat y_{i}|}$

I know that:

$\sum_{i=1}^{n}(y_{i}-\hat y_{i})^2=RMSE^2*n$

from the first formula.

Is it possible to determine a boundary for the MAE e.g. MAE < 1?

Eliminating the n-part in both metrics we can express them as p-norms or $L^p$-norms.

• $\sqrt{n}*RMSE = 2-norm$
• $n*MAE=1-norm$

So we can utilize that

$\|x\|_{p+a} \leq \|x\|_{p}$ for any vector x and real numbers p ≥ 1 and a ≥ 0.

So

$\sqrt{n}*0.8563=\sqrt{n} * RMSE<n*MAE$

=>

$MAE > \frac{0.8563}{\sqrt{n}} = \frac{0.8563}{\sqrt{1,408,789}}=0.0007214446$

which is the lower bound.

Given that $E[X^2] \geq E[X]^2$ (many thanks @Innuo), setting $X=(|y_i-\hat{y}_i|)$ leads us to

$RMSE^2 \geq MAE^2$ => $RMSE \geq MAE$

So the RMSE is actually the upper bound for MAE.

So my estimate for MAE is $[0.0007214446,0.8563]$ given RMSE=0.8563 and the model which has been used to calculate the specific RMSE.

The general model-independent boundaries of course remain [0,4], where the upper bound has been derived from the fact that $y_i,\hat{y}_i$ is an integer in the range from 1 to 5 (according to the rules of the netflix contest).

with

$\frac{1}{n}*\sum_{i=1}^{n}|5-1|=\frac{n}{n}4=4$

assuming that the model is smart enough to not predict ratings out of range.

• Actually, the RMSE value will be an upper bound for the MAE (can be shown using the fact that $E[X^2] \geq E[X]^2$, where $X$ is the absolute value of the error). – Innuo Feb 18 '13 at 15:50
• @Innuo oh yes, I forgot that. Thank you very much ! – steffen Feb 18 '13 at 16:09
• To the anonymous downvoter: Why ? Where is the error ? I am more than happy to delete my answer when it is not correct. – steffen Feb 22 '13 at 7:54