SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
r _ {i} \left(s _ {t}, a _ {t}\right) = - (1 - \xi) \left(s _ {t, i} ^ {2} + \sum_ {i \neq j} a _ {t, i} ^ {2}\right) - \xi \left(a _ {t, i} ^ {2} + \sum_ {i \neq j} s _ {t, i} ^ {2}\right). \tag {17}
$$
Here,  $\xi$  is the hyperparameter value that has been set to 0.1. The ultimate objective of the MO-LQR problem is to optimize the total reward while simultaneously taking into account the discount factor of  $\gamma = 0.9$ . The control matrix  $K_{\mathrm{LQR}}$  is assumed to be a diagonal matrix, and the diagonal elements of this matrix are treated as decision variables. The objectives are scaled with 0.01 for better illustration purposes.
Table 3 shows the number of decision variables and objectives used in this paper.
A.4 Results on All Problems
Results for all the examined problems are depicted in Figures 12-20, and combined with the results tabulated in Table 2 of the main paper, several conclusions can be made.
Behavior of LS-based PSL. The linear scalarization function can only learn the convex part of a Pareto front [39]. This fact is validated by Figure 13(e), where LS-based PSL can only learn several discrete solutions.
Table 3: Problem information for multiobjective synthetic, design, and LQR problems.
Problem	m	n
ZDT1	2	5
ZDT2	2	5
VLMOP1	2	5
VLMOP2	2	5
LQR2	2	2
Four Bar Truss Design	2	4
Hatch Cover Design	2	2
Rocket Injector Design	3	4
LQR3	3	3
(a)  $r = y^{\mathrm{radir}} + 0.1$
(b)  $r = y^{\mathrm{radir}} + 0.25$
(c)  $r = y^{\mathrm{nadir}} + 0.4$
(d)  $r = y^{\mathrm{radir}} + 0.55$
Figure 10: Hyperparameter study on ZDT1.
(e)  $r = y^{\mathrm{nadir}} + 0.70$
However, it is crucial to note that the connection between a solution and its corresponding preference vector,  $\lambda (\theta)$ , is non-uniform. This observation was rarely discussed in previous literature. Therefore, a uniform sampling of preferences will not result in a uniform sampling of Pareto objectives. This observation is supported by the results depicted in Figures 12(e), 15(e), and 17(e), where the learned Pareto objectives by LS-based PSL are not uniformly distributed. And as a result, the sparsity indicators are rather high, which indicates the learned front is sparse in some regions.
Time Consumption of EPO-based PSL. In comparison to our approach, the Exact Pareto Optimization [3] algorithm, which serves as the foundation for EPO-based PSL [12], exhibits low efficiency due to two factors.
1. To execute the Exact Pareto Optimization (EPO) algorithm, it is necessary to compute the gradients of all objectives,  $\nabla f_{i}(x)$ 's. This prerequisite entails performing  $m$  backpropagations in pytorch, resulting in higher computational costs. In contrast, our approach relies just one back-propagation operation, rendering it a more efficient option in comparison to EPO.
2. For each iteration, the Exact Pareto Optimization (EPO) algorithm entails solving a complicated optimization problem based on the specific value of  $f_{i}$ 's, utilizing the respective gradients of  $\nabla f_{i}(x)$ 's. In contrast, our method does not rely on solving optimization problems for each iteration.
Trade-off between Range and Sparsity. We find that the reference point  $r$  plays a crucial role in the performance of the final learned Pareto front for PSL-HV1. To demonstrate this effect, we consider different values of  $r$  from the set [0.1, 0.25, 0.4, 0.55, 0.7] in addition to the nadir point of each problem. The results are shown in the following two problems on the ZDT1 and ZDT2 problems.
Based on our empirical findings, emphasizing boundary Pareto objectives is crucial for recovering a complete Pareto set. From Figure 10 and 11, we observe that treating all coordinates  $\theta$  equally (small nadir point) typically results in a partial recovery of the Pareto set. When  $r$  exceeds a certain threshold (e.g., 0.4), raising its value directs Pareto objectives towards the boundary of the Pareto front, leading to poor uniformity of Pareto objective distribution.
(a)  $r = y^{\mathrm{adir}} + 0.1$
(b)  $r = y^{\mathrm{nadir}} + 0.25$
(d)  $r = y^{\mathrm{radir}} + 0.55$
(e)  $r = y^{\mathrm{radir}} + 0.70$
Figure 11: Hyperparameter study on ZDT2.
(c) Tche
Figure 12: ZDT1.
A.5 Pseudocode
For completeness, we provide the pseudocode of the proposed method. PSL-HV1 is selected as an example.  $\eta$  is a positive learning rate.
Algorithm 1: PSL-HV1.
Input: The multiobjective function  $f(\cdot)$  and initial Pareto neural model parameter  $\beta$ .
for  $k = 1$ : Niter do
Sample  $N$  preference angles  $\theta^{(1)},\ldots ,\theta^{(N)}$  from the positive unit sphere  $\mathbb{S}_+^{m - 1}$
Calculate the preference  $\lambda^{(1)},\ldots ,\lambda^{(N)}$  by Equation (26).
Calculate the empirical PSL gradient  $\nabla_{\beta}\overline{\mathcal{H}}_r$  by Equation (28) with  $N$  sampled preferences.
Update  $\beta$  by:  $\beta \gets \beta +\eta \nabla_{\beta}\hat{\overline{\mathcal{H}}} _r$
Output: The trained Pareto neural model  $x_{\beta}(\cdot)$ .
B Characters of Hypervolume Maximization
B.1 The Notation Table
To enhance the clarity of the paper, we have included a summary of the main notations in Table 4.
B.2 Hypervolume Calculation in the Polar Coordinate
The idea of calculating the hypervolume is similar to the Lemma 5 proved in Appendix of [32]. The difference is that, we use a distance function  $\rho_{\mathcal{X}}(\theta)$  as defined in Equation (6). For completeness, we provide the proof for Equation (5) in this section.
(c) Tche
Figure 13: ZDT2.
Figure 14: VLMOP1.
(c) Tche
Figure 15: VLMOP2.
(b) HV2
Figure 16: RE21.
(c) Tche
Figure 17: RE24.
(c) Tche
Figure 18: MO-LQR2.
(c) Tche
Figure 19: MO-LQR3.
(d) EPO
Figure 20: RE37.
Table 4: The notation table.
Variable	Definition
x	The decision variable.
X/Y	The decision/objective space.
n	The number of the decision variables.
N	The number of samples.
m	The number of objectives.
θ	The angular polar coordinate.
λ(θ),λ	An m-dimensional preference vector.
β	The model parameter.
ynadir/yideal	The nadir/ideal point of a given MOO problem.
T	The Pareto front.
Hr(A)	The hypervolume of set A w.r.t a reference .
Sm-1+	The (m-1)-D positive unit sphere.
Proof.  $\mathcal{H}_r(\mathcal{T})$  can be simplified by the following equations. Here  $\Omega$  denoted the dominated regions by the Pareto front, i.e.,  $\mathcal{H}_r(\mathcal{T}) = \Lambda (\Omega ,r)$
Figure 21: The hypervolume equals to the integral of dv.
$$
\mathcal {H} _ {r} (\mathcal {T}) = \int_ {\mathbb {R} ^ {m}} I _ {\Omega} d y _ {1} \dots d y _ {m}
$$
("dv" denoted the infinitesimal sector area.)
$$
= \underbrace {\int_ {0} ^ {\frac {\pi}{2}} \dots \int_ {0} ^ {\frac {\pi}{2}}} _ {m - 1} d v
$$
(dv equals the angle ratio multiplied by  $\rho^m$  multiplied by the unit for volume.)
$$
= \underbrace {\int_ {0} ^ {\frac {\pi}{2}} \dots \int_ {0} ^ {\frac {\pi}{2}}} _ {m - 1} \bar {c} _ {m} \cdot \frac {\rho \chi (\theta) ^ {m}}{2 \pi \cdot \pi^ {m - 2}} \underbrace {d \theta_ {1} \dots d \theta_ {m - 1}} _ {d \theta} \tag {18}
$$
(Re-arranging the terms.)
$$
\begin{array}{l} = \frac {\overline {{c}} _ {m}}{2 \pi^ {m - 1}} \underbrace {\int_ {0} ^ {\frac {\pi}{2}} \dots \int_ {0} ^ {\frac {\pi}{2}}} _ {m - 1} \rho_ {\chi} (\theta) ^ {m} d \theta \\ = \frac {\bar {c} _ {m}}{2 \pi^ {m - 1}} \cdot \left(\frac {\pi}{2}\right) ^ {m - 1} \cdot \mathbb {E} _ {\theta} [ \rho_ {\mathcal {X}} (\theta) ^ {m} ] \\ = c _ {m} \mathbb {E} _ {\theta} [ \rho_ {\mathcal {X}} (\theta) ^ {m} ]. \\ \end{array}
$$
We specify  $\theta \sim \mathrm{Unif}(\Theta) = \mathrm{Unif}([0, \frac{\pi}{2}]^{m-1})$  in Equation (18).
Line 2 holds since it represents the integral of  $\Omega$  expressed in polar coordinates, wherein the element  $dv$  corresponds to the volume associated with a segment obtained by varying  $d\theta$ .
Line 3 calculates the infinitesimal volume of  $dv$  by noticing the fact that the ratio of  $dv$  to  $\overline{c}_m$  is  $\frac{\rho_{\mathcal{X}}(\theta)^m}{2\pi\cdot\pi^{m-2}}$ . Line 4 is a simplification of Line 3. And Line 5 and 6 express the integral in its expectation form.
B.3 MOO Theory
In this section, we give the definitions and give proofs of the propositions used in the main paper.
We first formally define the term of aggregation function.
Definition 2 (Aggregation function (modified from the Definition 2.6.1 in [54])) A function  $g(\cdot, \lambda) : \mathbb{R}^m \mapsto \mathbb{R}$  representing the preferences of the decision maker among the objective is called an aggregation function.  $g(\cdot, \lambda) : \mathbb{R}^m \mapsto \mathbb{R}$  is decreasing, indicating that  $g(y^{(a)}, \lambda) < g(y^{(b)}, \lambda)$ , when  $y^{(a)} \prec y^{(b)}$ .
This definition is slightly different from the original definition in Definition 2.6.1 in [54]), as we modify the condition from "strictly decreasing" to "decreasing". The considered aggregation function  $\rho (x,\theta)$  does not satisfy the "strictly decreasing" condition in the original definition. When an aggregation function is only decreasing, the optimal objective can be weakly Pareto optimal. For this modified version of aggregation function, we have the following Lemma modified from Theorem 2.6.2 in [54].
Lemma 2. Let  $y^{*}$  belongs to the optimal solution set of  $g(\cdot, \lambda)$ , then  $y^{*}$  is weakly Pareto optimal or Pareto optimal. Further, if  $y^{*}$  be the only optimal solution of  $g(\cdot, \lambda)$ , then  $y^{*}$  is Pareto optimal.
We also provide the upper bound of the distance function  $\rho_{\mathcal{X}}(\theta)$  in this section by the following lemma.
Lemma 3 (The bound of in  $\rho_{\mathcal{X}}(\theta)$  Equation (6)).
$$
\rho_ {\mathcal {X}} (\theta) \leq B m ^ {1 / 2}, \tag {19}
$$
when  $b \leq r_i - f_i(x) \leq B$ ,  $\forall x \in \mathcal{X}, \forall i \in [m]$  and  $\|\lambda(\theta)\| = 1$ .
Proof. We show that the following inequalities hold,
$$
\begin{array}{l} \rho_ {\mathcal {X}} (\theta) \leq \max  _ {x \in \mathcal {X}, \| \lambda (\theta) \| = 1} \left(\min  _ {i \in [ m ]} \left\{\frac {r _ {i} - f _ {i} (x)}{\lambda_ {i} (\theta)} \right\}\right) \\ \left(r _ {i} - f _ {i} (x) \leq B\right) \\ \leq \max  _ {\| \lambda (\theta) \| = 1} \left(\min  _ {i \in [ m ]} \left\{\frac {B}{\lambda_ {i} (\theta)} \right\}\right) \tag {20} \\ \leq \frac {B}{m ^ {- 1 / 2}} = B m ^ {1 / 2}. \\ \end{array}
$$
The transition from line one to line two is due to the fact that the inequality  $r_i - f_i(x) \leq B$  holds for all  $x \in \mathcal{X}$  and for all  $i \in [m]$ . The transition from line two to line three is  $\max_{\| \lambda(\theta) \| = 1} \left( \min_{i \in [m]} \left\{ \frac{B}{\lambda_i(\theta)} \right\} \right)$  is an optimization problem under the constraint  $\| \lambda(\theta) \| = 1$ . The upper bound for this optimization is when  $\lambda_i = \ldots = \lambda_m = m^{-1/2}$ .
B.4  $\rho_{\chi}(\theta)$  as a Max-Min Problem
In this section, we first present the proof of Equation (6) from the main paper. To begin, we define the attainment surface  $A_{\mathcal{T}}$  [33] to handle disjointed PFs. Intuitively, the attainment surface is continuous, containing the weakly Pareto objectives and Pareto objectives (see Figure 22).
Definition 3 (Attainment surface [33]). The attainment surface  $A_{\mathcal{T}}$  induced by the Pareto front (PF)  $\mathcal{T}$  is defined as the boundary set of the PF.
$$
A _ {\mathcal {T}} = \{y \in \mathbb {R} ^ {m} | \exists y ^ {\prime} \in \mathcal {T}, y ^ {\prime} \preceq y, y ^ {\prime} \neq y \}. \tag {21}
$$
With this definition, the distance  $\rho_{\mathcal{X}}(\theta)$  between the reference point  $r$  and the point coordinated at  $\theta$  can be calculated using the following equation:
$$
\rho_ {\mathcal {X}} (\theta) = \max _ {x \in \mathcal {X}} \rho (x, \theta) = \max _ {x \in \mathcal {X}} \min _ {i \in [ m ]} \left\{\frac {r _ {i} - f _ {i} (x)}{\lambda_ {i}} \right\},
$$
where  $\lambda_{i}$  are shorthand's for  $\lambda_{i}(\theta)$ .
Proof. We first define the point  $P$ .
Definition 4.  $P$  is defined as the intersection point of the ray from  $r$  along angle  $\theta$  and the attainment surface.
There are two cases to consider:
①  $P$  is a Pareto objective, i.e.,  $P \in \mathcal{T}$ . ②  $P$  belongs to the attainment surface but not to the Pareto front, i.e.,  $P \in (A_{\mathcal{T}} \setminus \mathcal{T})$ .
Let  $x^{*}$  be one of the optimal solutions of Problem  $\max_{x\in \mathcal{X}}\rho (x,\theta)$ . Next, we will separately demonstrate the validity of the above-mentioned statements.
① When  $P$  is a Pareto objective, the according to Proposition 2, we have that  $y = P = f(x^{*})$  is the only objective of the problem  $\tilde{g}(y, \lambda) = \max_{y \in \mathcal{Y}} \min_{i \in [m]} \left\{ \frac{r_i - y_i}{\lambda_i} \right\}$ . And therefore,  $\rho_{\mathcal{X}}(\theta) = \frac{r_i - f_i(x^*)}{\lambda_i}, \forall i \in [m]$ .
② When  $y = P \in (A_{\mathcal{T}} \setminus \mathcal{T})$ . Then, according to Proposition 2,  $y$  is one of the optimal objective of problem  $\tilde{g}(y, \lambda) = \max_{y \in \mathcal{Y}} \min_{i \in [m]} \left\{ \frac{r_i - y_i}{\lambda_i} \right\}$ . In such a case, according to the definition of attainment surface (Equation (21)), we have that there exist at least one index  $j$ , where  $j = \arg \min_{\lambda_j} \frac{r_j - f_j(x^*)}{\lambda_j}$  such that  $\rho_{\mathcal{X}}(\theta) = \max_{x \in \mathcal{X}} \rho(x, \theta) = \max_{x \in \mathcal{X}} \min_{i \in [m]} \left\{ \frac{r_i - f_i(x)}{\lambda_i} \right\} = \min_{i \in [m]} \left\{ \frac{r_i - f_i(x^*)}{\lambda_i} \right\} = \frac{r_j - f_j(x^*)}{\lambda_j} = \frac{r_i - P_i}{\lambda_i}$ ,  $i \in [m]$ .
Figure 22: Illustration of the attainment surface on a case of a disjointed Pareto front.
The previous discussion explained that, in the case of a disjointed Pareto front, the distance between  $r$  and the attainment surface corresponding to angle  $\theta$  is denoted by  $\rho_{\mathcal{X}}(\theta)$ . In Figure 22, the black dot represents the solution for this scenario. It is worth noting that the integral of the distance function  $\rho_{\mathcal{X}}(\theta)$  yields the hypervolume of a disjointed Pareto front, i.e.,  $\mathcal{H}_r(\mathcal{T}) = \mathcal{H}_r(A_\mathcal{T})$ . This property aligns with the purpose of our paper.
In closing this section, we note that the absence of a (weakly) Pareto objective aligned with the preference angle  $\theta$ .
Remark 4. When  $P$  (in Definition 4) is not a (weakly) Pareto objective, The optimal solution  $y^{*}$  w.r.t the problem  $\tilde{g}(y, \lambda) = \max_{y \in \mathcal{Y}} \min_{i \in [m]} \left\{ \frac{r_i - y_i}{\lambda_i} \right\}$  belongs to the boundary of the Pareto front  $\partial \mathcal{T}$ . Further  $y^{*} = \arg \min_{y \in \mathcal{Y}} \max_{i \in [m]} \left\{ \frac{y_i - P_i}{\lambda_i} \right\}$ .
B.5 Pareto Front Hypervolume Calculation (Type2 and gradients)
In this section, we provide another form to calculate the Pareto front hypervolume. We first define region  $A$  as the set of points dominating the Pareto front.
$$
A = \{q \mid \exists p \in \mathcal {T}: p \preceq q \text {a n d} q \succeq p ^ {\mathrm {i d e a l}} \}. \tag {22}
$$
We use the notation  $\Lambda (\cdot)$  to represent the Lebesgue measure of a set. Geometrically, as illustrated in Figure 23, it can be observed that:
$$
\Lambda (A) + \mathcal {H} _ {r} (\beta) = \prod_ {i = 1} ^ {m} \left(r _ {i} - y _ {i} ^ {\text {i d e a l}}\right). \tag {23}
$$
The volume of  $A$  can be calculated in a polar coordinate as follows,
$$
\Lambda (A) = c _ {m} \int_ {(0, \frac {\pi}{2}) ^ {m - 1}} \bar {\rho} _ {\chi} (\theta) ^ {m} d \theta , \tag {24}
$$
Figure 23: The hypervolume calculation (Type2).
where  $c_{m}$  is a constant and  $\overline{\rho}_{\mathcal{X}}(\theta)$  represents the distance from the ideal point to the Pareto front at angle  $\theta$ . This distance function  $\overline{\rho}_{\mathcal{X}}(\theta)$  is obtained by solving the optimization problem assuming that any ray from  $\theta$  intersects with the Pareto front.
$$
\bar {\rho} _ {\mathcal {X}} (\theta) = \min  _ {x \in \mathcal {X}} \bar {\rho} _ {\mathcal {X}} (\theta , x) = \min  _ {x \in \mathcal {X}} \max  _ {i \in [ m ]} \left\{\frac {f _ {i} (x) - y _ {i} ^ {i d e a l}}{\lambda_ {i} (\theta)} \right\}, \quad \theta \in \left(0, \frac {\pi}{2}\right) ^ {m - 1}. \tag {25}
$$
The relationship between preference  $\lambda$  and the polar angle  $\theta$  is as follows:
$$
\left\{ \begin{array}{l} \lambda_ {1} (\theta) = \sin \theta_ {1} \sin \theta_ {2} \dots \sin \theta_ {m - 1} \\ \lambda_ {2} (\theta) = \sin \theta_ {1} \sin \theta_ {2} \dots \cos \theta_ {m - 1} \\ \dots \\ \lambda_ {m} (\theta) = \cos \theta_ {1}. \end{array} \right. \tag {26}
$$
Combining Equation (24) and Equation (25) implies that  $\overline{\mathcal{H}}_r(\beta)$  can be estimated as an expectation problem,
$$
\overline {{\mathcal {H}}} _ {r} (\beta) = \prod_ {i = 1} ^ {m} \left(r _ {i} - y _ {i} ^ {\text {i d e a l}}\right) - \frac {1}{m} c _ {m} \mathbb {E} _ {\theta} \left[ \bar {\rho} _ {\mathcal {X}} \left(x _ {\beta} (\theta), \theta\right) ^ {m} \right]. \tag {27}
$$
Remark 5. Compared to PSL-HV1, PSL-HV2 has weaker theoretical results due to its reliance on a stronger assumption. This assumption requires the presence of a (weakly) Pareto objective aligned on the ray originating from any coordinate  $\theta$ . An additional challenge with PSL-HV2 is that the Lipschitz constant cannot be bounded, making generalization theory for PSL-HV2 significantly more difficult (Equation (30)).
We conclude this section by presenting the analytical expression for  $\nabla_{\beta}\mathcal{H}_r(\beta)$ . The gradient for PSL-HV1 can be computed using the chain rule, which yields:
$$
\nabla_ {\beta} \overline {{\mathcal {H}}} _ {r} = \left\{ \begin{array}{l l} m c _ {m} \mathbb {E} _ {\theta} [ \rho (x _ {\beta} (\theta), \theta) ^ {m - 1} \underbrace {\frac {\partial \rho (x _ {\beta} (\theta) , \theta)}{\partial x _ {\beta} (\theta)}} _ {1 \times n} \underbrace {\frac {\partial x _ {\beta} (\theta)}{\partial \beta}} _ {n \times d} ], & \rho (x _ {\beta} (\theta), \theta) \geq 0. \\ c _ {m} \mathbb {E} _ {\theta} [ \underbrace {\frac {\partial \rho (x _ {\beta} (\theta) , \theta)}{\partial x _ {\beta} (\theta)}} _ {1 \times n} \underbrace {\frac {\partial x _ {\beta} (\theta)}{\partial \beta}} _ {n \times d} ], & \text {O t h e r w i s e .} \end{array} \right. \tag {28}
$$
where we specify  $\theta$  to satisfy the law of  $\mathrm{Unif}(\Theta)$ . The gradient of PSL-HV2 can be calculated by,
$$
\nabla_ {\beta} \overline {{\mathcal {H}}} _ {r} = - m c _ {m} \mathbb {E} _ {\theta} \left[ \bar {\rho} _ {\mathcal {X}} \left(x _ {\beta} (\theta), \theta\right) ^ {m - 1} \right] \underbrace {\frac {\partial \bar {\rho} _ {\mathcal {X}} \left(x _ {\beta} (\theta) , \theta\right)}{\partial x _ {\beta} (\theta)}} _ {1 \times n} \underbrace {\frac {\partial x _ {\beta} (\theta)}{\partial \beta}} _ {n \times d}. \tag {29}
$$
B.6 Proof of Theorem 1
Definitions and preliminaries. The proof will heavily utilize the existing results on Rademacher complexity of MLPs. We will first provide some useful definitions and facts. We start with the definition of Rademacher complexity as follows:
Definition 5 (Rademacher complexity, Definition 13.1 in [55]). Given a set of vectors  $V \subseteq \mathbb{R}^n$ , we define the (unnormized) Rademacher complexity as
$$
\operatorname {U R a d} (V) := \underset {u \in V} {\mathbb {E}} \sup  \langle \epsilon , u \rangle ,
$$
where each coordinate  $\epsilon_{i}$  is an i.i.d. Rademacher random variable, meaning  $\operatorname*{Pr}[\epsilon_i = +1] = \frac{1}{2} = \operatorname*{Pr}[\epsilon_i = -1]$ . Furthermore, we can accordingly discuss the behavior of a function class  $\mathcal{G}$  on  $S = \{z_{i}\}_{i=1}^{N}$  by using the following set:
$$
\mathcal {G} _ {| S} := \left\{\left(g \left(z _ {1}\right), \dots , g \left(z _ {N}\right)\right): g \in \mathcal {G} \right\} \subseteq \mathbb {R} ^ {N},
$$
and its Rademacher complexity is
$$
\operatorname {U R a d} \left(\mathcal {G} _ {| S}\right) = \underset {\epsilon} {\mathbb {E}} \sup  _ {u \in \mathcal {G} _ {| S}} \langle \epsilon , u \rangle = \underset {\epsilon} {\mathbb {E}} \sup  _ {g \in \mathcal {G}} \sum_ {i} \epsilon_ {i} g \left(z _ {i}\right).
$$
Utilizing Rademacher complexity, we can conveniently bound the generalization error via the following theorem:
Theorem 2 (Uniform Generalization Error, Theorem 13.1 and Corollary 13.1 in [55]). Let  $\mathcal{G}$  be given with  $g(z)\in [a,b]$  a.s.  $\forall g\in \mathcal{G}$ . We collect i.i.d. samples  $S = \{z_{i}\}_{i = 1}^{N}$  from the law of random variable  $Z$ . With probability  $\geq 1 - \delta$
$$
\sup _ {g \in \mathcal {G}} \mathbb {E} g (Z) - \frac {1}{N} \sum_ {i} g \left(z _ {i}\right) \leq \frac {2}{N} \mathrm {U R a d} \left(\mathcal {G} _ {| S}\right) + 3 (b - a) \sqrt {\frac {\ln (2 / \delta)}{2 N}}.
$$
Specifically, the Rademacher complexity in using MLP is provided by the following theorem:
Theorem 3 (Rademacher complexity of MLP, Theorem 1 in [56]). Let 1-Lipschitz positive homogeneous activation  $\sigma_{i}$  be given, and
$$
\mathcal {G} ^ {M L P} := \left\{\theta \mapsto \sigma_ {L} \left(W _ {L} \sigma_ {L - 1} \left(\dots \sigma_ {1} \left(W _ {1} \theta\right) \dots\right)\right): \| W _ {i} \| _ {\mathrm {F}} \leq B _ {w} \right\}
$$
$$
\operatorname {U R a d} \left(\mathcal {G} _ {| S} ^ {M L P}\right) \leq B _ {w} ^ {L} \| X _ {\theta} \| _ {F} (1 + \sqrt {2 L \ln (2)}).
$$
We can then utilize the following composition character of Rademacher complexity, to help induce the final Rademacher complexity of hypervolume.
Lemma 4 (Rademacher complexity of compositional function class, adapted from Lemma 13.3 in [55]). Let  $g: \Theta \to \mathbb{R}^n$  be a vector of  $n$  multivariate functions  $g^{(1)}, g^{(2)}, \ldots, g^{(n)}$ ,  $\mathcal{G}$  denote the function class of  $g$ , and further  $\mathcal{G}^{(j)}$  be the function class of  $g^{(j)}, \forall j$ . We have a "partially Lipschitz continuous" function  $\ell(g(\theta), \theta)$  so that  $|\ell(g_1(\theta), \theta) - \ell(g_2(\theta), \theta)| \leq L_\ell \|g_1(\theta) - g_2(\theta)\|$  for all  $g_1, g_2 \in \mathcal{G}$  and a certain  $L_\ell > 0$ ; the associated function class of  $\ell$  is denoted as  $\mathcal{G}^\ell$ . We then have
$$
\mathrm {U R a d} \left(\mathcal {G} _ {| S} ^ {\ell}\right) \leq \sqrt {2} L _ {\ell} \sum_ {j = 1} ^ {n} \mathrm {U R a d} \left(\mathcal {G} _ {| S} ^ {(j)}\right).
$$
Proof. This proof extends Lemma 13.3 in [55] for vector-valued  $g$  and "partially Lipschitz continuous"  $\ell$ . We first similarly have
$$
\begin{array}{l} \operatorname {U R a d} \left(\mathcal {G} _ {| S} ^ {\ell}\right) = \mathbb {E} \sup  _ {g \in \mathcal {G}} \sum_ {i} \epsilon_ {i} \ell \left(g \left(\theta_ {i}\right), \theta_ {i}\right) \\ = \frac {1}{2} \underset {\epsilon_ {2: N}} {\mathbb {E}} \sup  _ {f, h \in \mathcal {G}} \left(\ell \left(f \left(\theta_ {1}\right), \theta_ {1}\right) - \ell \left(h \left(\theta_ {1}\right), \theta_ {1}\right) + \sum_ {i = 2} ^ {N} \epsilon_ {i} \left(\ell \left(f \left(\theta_ {i}\right), \theta_ {i}\right) + \ell \left(h \left(\theta_ {i}\right), \theta_ {i}\right)\right)\right) \\ \leq \frac {1}{2} \mathbb {E} _ {\epsilon_ {2: N}} \sup  _ {f, h \in \mathcal {G}} \left(L _ {\ell} \| f (\theta_ {1}) - h (\theta_ {1}) \| + \sum_ {i = 2} ^ {N} \epsilon_ {i} \left(\ell \left(f \left(\theta_ {i}\right), \theta_ {i}\right) + \ell \left(h \left(\theta_ {i}\right), \theta_ {i}\right)\right)\right) \\ \leq \frac {1}{2} \mathbb {E} _ {\epsilon} \sup  _ {f ^ {(j)}, h ^ {(j)} \in \mathcal {G} ^ {(j)}} \left(L _ {\ell} \sqrt {2} \left| \sum_ {j = 1} ^ {n} \epsilon_ {1} ^ {(j)} \left(f ^ {(j)} \left(\theta_ {1}\right) - h ^ {(j)} \left(\theta_ {1}\right)\right) \right| + \right. \\ \left. \sum_ {i = 2} ^ {N} \epsilon_ {i} \left(\ell \left(f \left(\theta_ {i}\right), \theta_ {i}\right) + \ell \left(h \left(\theta_ {i}\right), \theta_ {i}\right)\right)\right), \\ \end{array}
$$