Choosing a Model by MSE

Choosing a Model by MSE

English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email [email protected].

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.]

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.
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