Primal optimal transport is formulated as a solution to the Monge problem (MP) between measures \(\alpha\) and \(\beta\) and stated as: $$ \begin{aligned} \text{MP}(\alpha, \beta) = \inf_{T : \, T_\#\alpha =\beta}\int _\mathcal{X} c(x, T(x))\text{d}\alpha(x), \end{aligned} $$ However, for computational reasons, it is easier to solve dual problem, called Kantorovich problem (KP): $$ \begin{aligned} \text{KP}(\alpha, \beta) = \sup_{(f, g) \in L_1(\alpha) \times L_1(\beta)} \Bigr[\mathbb{E}_{\alpha} [f(x)] + \mathbb{E}_{\beta} [g(y)]\Bigr] + \inf_{\pi, \gamma > 0} \gamma \, \mathbb{E}_{\pi} [c(x, y) - f(x) - g(y)], \end{aligned} $$ Which is equivalent to: $$ \begin{aligned} \text{KP}(\alpha, \beta) &= \sup_{g \in L_1(\beta)} \Bigr[\mathbb{E}_{\alpha} [g^c(x)] + \mathbb{E}_{\beta} [g (y)]\Bigr] \\ &= \sup_{g \in L_1(\beta)} \inf_{T: \, \mathcal{X} \to \mathcal{Y}} \Bigr[\mathbb{E}_{\alpha}[ c(x, T(x)) ] + \mathbb{E}_{\beta} [g (y)] - \mathbb{E}_{\alpha} [g(T(x)) ]\Bigr]. \end{aligned} $$ The main computational burden lies in inner optimization problem over conjugate functions. Existing approaches either try to find an exact solution (which is computationaly heavy) or find rough approximation (which leads to instabilities). We propose to restrict set of possible conjugate functions by adding expectile regularization term \(\mathcal{R}_g\): $$ \mathbb{E}_{\alpha} [g^T(x)] + \mathbb{E}_{\beta} [g (y)] - \mathbb{E}_{\alpha, \beta} [\mathcal{R}_g(x, y)]. $$ Despite its simplicity, overall objective has several useful properties:

Expectiles are another way to capture statistics of some random variable \(X\). More formally, expectile \(\tau \in (0, 1)\) of r.v \(X\) is a solution to the assymetric least squares problem, defined as: $$ \begin{aligned} \arg \min _{m_{\tau}} \{\mathbb{E}\left[\mathcal{L}^{\tau}_2 (x - m_{\tau})\right]\} \\ \mathcal{L}^{\tau}_2 (u) = |\tau - \mathbb{1}(u < 0)|u^2 \end{aligned} $$ Intuitively, for low-valued expectiles, the model is penalized more for underestimation, while for large expectile values, we penalize for the overestimation, as shown in the graph above. By penalizing the overly optimistic approximation of the conjugate Kantorovich potential during the optimization, expectile regularization enforces "closeness" between the true conjugate and that approximated by a neural network.