**4. Analysis and Extensions**

#### *4.1. Dual Problem: A Numerical Solution*

To simplify presentation of the numerical algorithm of problem (28)'s solution, we suppose that the uncertainty set <sup>F</sup>*<sup>L</sup>* takes the form <sup>F</sup>*<sup>L</sup>* <sup>=</sup> {*<sup>F</sup>* <sup>∈</sup> <sup>F</sup> : <sup>L</sup> - *L*}, i.e., it is restricted by the conformity constraint only.

Let us consider the case <sup>C</sup> {*qj*}*j*=1,*<sup>M</sup>* <sup>⊂</sup> <sup>R</sup>*m*, which corresponds to the practical problem of Bayesian classification [10,38]. Here the dual problem (28) has the form **<sup>w</sup>** (*y*) = Argmax *<sup>w</sup>*∈conv(W(C|*y*)) **<sup>J</sup>**∗(*w*). Its solution can be represented as **<sup>w</sup>** (*y*) = <sup>∑</sup>*<sup>M</sup> <sup>j</sup>*=<sup>1</sup> <sup>P</sup>*j*(*y*)*w*(*qj*|*y*),

where <sup>P</sup>(*y*) row(P1(*y*), ... , <sup>P</sup>*M*(*y*)) is a solution to the standard quadratic programming problem (QP problem):

$$\hat{\boldsymbol{P}}(\boldsymbol{y}) \in \operatorname\*{Argmin}\_{\substack{p\_1, \dots, p\_M > 0 \\ \sum\_{j=1}^M p\_j = 1}} \left( \sum\_{j=1}^M p\_j w\_1(q\_j|\boldsymbol{y}) - \sum\_{j, j'=1}^M p\_j p\_{j'} \langle w\_2(q\_j|\boldsymbol{y}), w\_2(q\_{j'}|\boldsymbol{y}) \rangle \right). \tag{30}$$

Consequently, in the case of finite C the minimax estimation problem can be reduced to the standard QP problem with well-investigated properties and advanced numerical procedures.

Utilization of the finite subsets C(·) instead of the original domain C allows to calculate the "mesh" approximations for the solution to (8).

Let:

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$$\begin{array}{ll}\omega\_1^{\mathfrak{e}}(\boldsymbol{y}) \triangleq \max\_{\substack{q\_1, q\_2 \in \mathcal{C}:\\ \|q\_1 - q\_2\| < \epsilon\\ \|q\_1 - q\_2\| < \epsilon}} \left| w\_1(q\_1|\boldsymbol{y}) - w\_1(q\_2|\boldsymbol{y}) \right|,\\ \omega\_2^{\mathfrak{e}}(\boldsymbol{y}) \triangleq \max\_{\substack{q\_1, q\_2 \in \mathcal{C}:\\ \|q\_1 - q\_2\| < \epsilon}} \left\| w\_2(q\_1|\boldsymbol{y}) - w\_2(q\_2|\boldsymbol{y}) \right\|\end{array} \tag{31}$$

be modulus of continuity for *w*1(*y*|*q*) and *w*2(*y*|*q*).

The assertion below characterizes the divergence rate of the approximating solutions to the initial minimax estimate.

**Lemma 2.** *If* {**w** (*n*|*y*)}*n*∈<sup>N</sup> *are corresponding solutions to the problems:*

$$
\hat{\mathbf{w}}^n(y) = \underset{w \in \text{conv}(W(\mathcal{C}(c\_n)|y))}{\text{Arg}\max} \mathbf{J}\_\*(w)
$$

*then the following sequences are convergent as <sup>n</sup>* ↓ 0*:*

$$\begin{array}{l}\hat{\mathbf{w}}^{n}(\boldsymbol{y}) \to \hat{\mathbf{w}}(\boldsymbol{y}),\\\mathcal{I}(\boldsymbol{y}) - \max\_{\boldsymbol{w} \in \text{conv}(\mathcal{W}(\mathcal{C}(\boldsymbol{c}\_{n})|\boldsymbol{y}))} \mathbf{J}\_{\*}(\boldsymbol{w}) \lessapprox \omega\_{1}^{\epsilon\_{n}}(\boldsymbol{y}) + \mathsf{K}[\omega\_{2}^{\epsilon\_{n}}(\boldsymbol{y})]^{2} \downarrow 0\tag{32}$$

*with some constant* 0 < *K* < ∞*.*

The proof of Lemma 2 is given in Appendix D.
