Recently, Rigollet and Weed (2019)^{1} introduced a consistent estimator for the problem of *uncoupled isotonic regression*. The authors also proposed a computationally efficient procedure for computing a relaxed version of the estimator, which achieves the same minimax rate.

Rigollet and Weed’s estimator relaxed estimator minimizes the Wasserstein-2 distance between the discrete, uniform measure on the observed data \({y_1, \dots, y_n}\) – named \(\hat{\pi}\) – and a quantization of the pushforward of the discrete uniform measure on the design points \({x_1, \dots, x_n}\) under the unknown regression function \(f\) convolved with a noise distribution \(\mathcal{D}\).

To implement Rigollet and Weed’s relaxed minimum-\(W_2\) estimator, one solves

\[\begin{aligned} \hat{\mu} \in \text{argmin}_{\mu \in \mathcal{M}_{\mathcal{A}, V}} W_2^2 \left(\Pi_{\mathcal{A}\sharp}(\mu \star \mathcal{D}), \hat{\pi} \right) \end{aligned}\]where \(\Pi_{\mathcal{A}\sharp}(\mu \star \mathcal{D})\) is the pushforward of the measure \(\mu \star \mathcal{D}\) by the projection operator defined in Rigollet and Weed (2019). This operator discretizes the continuous measure \(\mu \star \mathcal{D}\), concentrating its mass on an equi-spaced partition of the interval \([-V, V]\).

Because both measures considered are discrete, finding the Wasserstein distance between a given \(\Pi_{\mathcal{A}\sharp}(\mu \star \mathcal{D})\) and \(\hat{\pi}\) is equivalent to solving the linear program

\[\begin{aligned} \underset{\gamma_{ij}}{\text{minimize}}~~\sum_{ij} \gamma_{ij} &(y_i - \alpha_j)^2 \\ \text{subject to}~~\sum_{i} \gamma_{ij} &= a_j &j = 0, \dots, N \\ \sum_{j} \gamma_{ij} &= \hat{\pi}_i &i = 1, \dots, n \\ \gamma_{ij} &\geq 0 \end{aligned}\]where \(\alpha_j\) is the \(j^{\text{th}}\) element in the partition of \([-V, V]\) defined in Rigollet and Weed (2019), \(y_i\) is the \(i^{\text{th}}\) observed data point, and \(a_j = \Pi_{\mathcal{A}\sharp}(\mu \star \mathcal{D})_j\), the amount of probability mass placed at \(\alpha_j\).

From the dual problem to this linear program, we can obtain the subgradient of the \(W_2^2\) distance with respect to the measure \(a\). This subgradient is equal to the vector of optimal dual solutions corresponding to the first equality constraint in the linear program above. Therefore,

\[\begin{aligned} \frac{\partial W_2^2}{\partial a_j} = g_j^*~~~~~~~~j = 0, \dots, N \\ \end{aligned}\]where \(g_j^\ast\) is the \(j^{\text{th}}\) component of the optimal dual vector.

Define \(\mu_i := \mu (\alpha_i)\); in other words the measure weight *under \(\mu\)* given to \(\alpha_i\) (note that this is different than \(a_j\), which is the measure weight under \(\Pi_{\mathcal{A}\sharp}(\mu \star \mathcal{D})\)). Then based on the definition of the projection operator \(\Pi_{\mathcal{A}\sharp}\), \(a_j\) varies with \(\mu_i\) according to the relationship

where \(p\) is the density function of the noise distribution \(\mathcal{D}\). Hence, the partial derivatives of the \(a\)’s with respect to each \(\mu_i\) are

\[\begin{aligned} \frac{\partial a_j}{\partial \mu_i} = \int_{\alpha_j}^{\alpha_{j+1}} p(x ; \alpha_j, \sigma^2) dx \end{aligned}\]The Chain Rule then yields subgradients of the \(W_2^2\) objective with respect to the measure weights \(\mu_1, \dots, \mu_N\) according to

\[\begin{aligned} \frac{\partial W_2^2}{\partial \mu_i} = \sum_{j} \frac{\partial W_2^2}{\partial a_j} \frac{\partial a_j}{\partial \mu_i} \end{aligned}\]With these subgradients in hand, a projected descent algorithm of the form

\[\begin{aligned} \mu_{t+1} = \text{Proj}_\mathcal{S} \left( \mu_t - \lambda_t \frac{d W_2^2}{d \mu} \right) \end{aligned}\]may be used to arrive at the optimal value \(\mu^*\).

To speed convergence to the optimal \(\mu^*\), the CFM step-size rule^{2} can be used to determine the value of \(\lambda_t\) at each iteration. The operator \(\text{Proj}_\mathcal{S}\) is the projection operator onto the probability simplex.

An `R`

implementation of the minimum-Wasserstein estimator can be found at https://github.com/j-g-b/uncoupled.

^{1} Rigollet, P., Weed, J. (2019) *Uncoupled isotonic regression via minimum Wasserstein deconvolution*, Information and Inference: A Journal of the IMA.

^{2} Camerini P.M., Fratta L., Maffioli F. (1975) *On improving relaxation methods by modified gradient techniques*. In: Balinski M.L., Wolfe P. (eds) Nondifferentiable Optimization. Mathematical Programming Studies, vol 3. Springer, Berlin, Heidelberg.