Maybe an interesting observation sidenote i’ve made playing around with Jianlin’s Blogs L2 MEP. We can show that a single label L2 MEP secretly becomes a probability loss on a natural region of score space.
First let’s assume $m>0$.
We can define for $K$ classes
\[q_i = \frac{1}{K} + \frac{x_i-\bar{x}}{m}, \qquad \bar{x} = \frac{1}{K}\sum_{i=1}^{K} x_i.\]
Since $\sum_i q_i=1$ automatically $q$ is a probability vector exactly when $q_i\ge 0$ for every $i$ or equivalently when
\[x_i\ge \bar{x}-\frac{m}{K} \qquad \text{for every } i.\]
let
\[C_{m,y} = \left\{ z\in\mathbb{R}^K: z_y-z_j\ge m \text{ for every } j\ne y \right\}.\]
whenever $q$ is a probability vector the squared L2 MEP loss for label $y$ satisfies
\[ \operatorname{dist}_2^2\!\left(x, C_{m,y}\right) = m^2 \lVert q-e_y\rVert_2^2. \tag{1} \]
L2 MEP-Brier equivalence
To see this we can define
\[v_y = \bar{x}\mathbf{1} + m\left(e_y-\frac{1}{K}\mathbf{1}\right).\]
The vector $v_y$ lies in $C_{m,y}$. so for every $z\in C_{m,y}$,
\[\begin{aligned} \left\langle x-v_y,z-v_y\right\rangle &= m\sum_{j\ne y}q_j\left(z_j-z_y+m\right) \\ &\le 0. \end{aligned}\]
Euclidean projection characterization therefore gives $v_y=\operatorname{Proj}_{C_{m,y}}(x)$. Since $x-v_y=m(q-e_y)$ the identity (1) follows. For fixed $\bar{x}$ the vectors $v_y$ are also the vertices of the corresponding simplex in score space.
Thus on this region and after an affine reparameterization of the scores, single-label squared L2 MEP loss is exactly a scaled multiclass Brier score which is a classical quadratic loss often used in probability forecasts.
so consequently if the true conditional class probability vector is $p$ then
\[\begin{aligned} x^\star &= c\mathbf{1} + m\left(p-\frac{1}{K}\mathbf{1}\right). \end{aligned}\]
is a global population risk minimizer where $c\in\mathbb{R}$ is arbitrary because the MEP loss is invariant to adding the same value to every score.
so for the conditional population risk
\[R_p(x) = \sum_{y=1}^{K} p_y\operatorname{dist}_2^2\!\left(x,C_{m,y}\right),\]
each summand is convex and has gradient
\[2p_y\left(x-\operatorname{Proj}_{C_{m,y}}(x)\right).\]
At $x=x^\star$ this gives
\[\nabla R_p(x^\star) = 2m\sum_{y=1}^{K}p_y(p-e_y) = 0.\]
Convexity therefore proves that $x^\star$ is a global minimizer, including against scores outside the probability-vector region.
Hence a canonical centered population minimizer affinely encodes the conditional class probabilities with $m$ setting their scale. This gives L2 MEP a useful statistical interpretation and suggests it could be worthwile imo to explore the calibration of the transformed scores $q$ alongside classification accuracy. It does NOT imply that every finitely trained MEP model necessarily produces a valid or calibrated probability vector.