# Choosing a Model by MSE

*English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email sota.ai.community@gmail.com.*

*Task 3 of the school stage of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group II), grades 9–11. The official answer and solution are in a separate file.*

Two sets of points are given (we will call the first one the training set and the second one the validation set).

$$\text{Training: } (-0.5, -0.75), (0, -0.5), (0.5, 1.25), (1, 1.5).$$

$$\text{Validation: } (2, 5.0), (3, 11.5), (4, 14.0).$$

*[Figure: see the [original statement](https://siriusolymp.ru/2025/ai).]*

*Translator's transcription of the figure: the seven points above plotted on a grid ($x$ from −6 to 9, $y$ from −1 to 14), with the legend “Training points” (orange circles) and “Validation points” (blue squares). In the PDF the figure is printed at the top of the next page, inside the solution.*

Four models are considered (that is, rules that specify how $y$ depends on $x$):

- $y = 2x$,
- $y = x + 0.5$,
- $y = \frac{1}{2}(x + 2)^2 - 2$,
- $y = (x + 1)^2 - 1$.

The quality of each model on a set of points is assessed by the formula:

$$\frac{\left(y_1 - \hat{y}_1\right)^2 + \left(y_2 - \hat{y}_2\right)^2 + \ldots + \left(y_n - \hat{y}_n\right)^2}{n},$$

where $y_i$ is the actual value and $\hat{y}_i$ is the value given by the model's formula at the same point $x_i$. The smaller the error, the better the model describes the data of this set.

Choose the model with the smallest mean squared error on the training set.<br>
Choose the model with the smallest mean squared error on the combined set of all seven points (training and validation together).

**Scoring criterion:** 6 points for each correct answer

**Maximum score for the task — 12**