SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yuzheng Hu $^{1\dagger}$  Ruicheng Xian $^{1\star}$  Qilong Wu $^{1\star}$  Qiuling Fan $^{2}$  Lang Yin $^{1}$  Han Zhao $^{1\dagger}$
$^{1}$ Department of Computer Science  $^{2}$ Department of Mathematics
{yh46,rxian2,qilong3,qiuling2,langyin2,hanzhao}@illinois.edu
Linear scalarization, i.e., combining all loss functions by a weighted sum, has been the default choice in the literature of multi-task learning (MTL) since its inception. In recent years, there is a surge of interest in developing Specialized Multi-Task Optimizers (SMTOs) that treat MTL as a multi-objective optimization problem. However, it remains open whether there is a fundamental advantage of SMTOs over scalarization. In fact, heated debates exist in the community comparing these two types of algorithms, mostly from an empirical perspective. To approach the above question, in this paper, we revisit scalarization from a theoretical perspective. We focus on linear MTL models and study whether scalarization is capable of fully exploring the Pareto front. Our findings reveal that, in contrast to recent works that claimed empirical advantages of scalarization, scalarization is inherently incapable of full exploration, especially for those Pareto optimal solutions that strike the balanced trade-offs between multiple tasks. More concretely, when the model is under-parametrized, we reveal a multi-surface structure of the feasible region and identify necessary and sufficient conditions for full exploration. This leads to the conclusion that scalarization is in general incapable of tracing out the Pareto front. Our theoretical results partially answer the open questions in Xin et al. (2021), and provide a more intuitive explanation on why scalarization fails beyond non-convexity. We additionally perform experiments on a real-world dataset using both scalarization and state-of-the-art SMTOs. The experimental results not only corroborate our theoretical findings, but also unveil the potential of SMTOs in finding balanced solutions, which cannot be achieved by scalarization.
When labeled data from multiple tasks are available, multi-task learning (MTL) aims to learn a joint model for these tasks simultaneously in order to improve generalization (Caruana, 1997; Zhang and Yang, 2021; Ruder, 2017). One common approach towards this goal is learning a joint feature representation that is shared among multiple tasks, where each task then uses a separate task-specific head for its own prediction. The hope is that by sharing a joint feature representation, the effective sample size for feature learning will be larger than that for single-task learning, hence leading to better generalization across related tasks. This simple and natural idea has proven to be successful in many applications, including object segmentation (Misra et al., 2016), natural language processing (Collobert and Weston, 2008), autonomous driving (Yu et al., 2020a), to name a few.
One typical approach to learning the shared feature model from multiple tasks is called linear scalarization, where the learner provides a set of convex coefficients (in the form of a probability vector) to
linearly combine the loss from each task and reduces the learning problem into a scalar optimization problem. Since its inception, scalarization has been the default approach for MTL (Caruana, 1997; Ruder, 2017; Zhang and Yang, 2018; Crawshaw, 2020; Zhang and Yang, 2021), mainly due to its simplicity, scalability, as well as empirical success. However, in recent years, variants of the multiple gradient descent algorithm (MGDA) (Fliege and Svaiter, 2000) have been proposed and used in MTL (Sener and Koltun, 2018; Yu et al., 2020b; Chen et al., 2020; Liu et al., 2021a,b; Zhou et al., 2022). Referred to as specialized multi-task optimizers (SMTOs), the main idea behind them is to cast the learning problem explicitly as a multi-objective (multi-criteria) optimization problem, wherein the goal is to enforce convergence to a Pareto optimal solution across tasks. In the literature, a main driving force behind these studies has been the belief that they are better than linear scalarization when conflicting gradients among tasks are present (Yu et al., 2020b; Liu et al., 2021a). Nevertheless, recent empirical results tell an opposite story (Kurin et al., 2022; Xin et al., 2022; Lin et al., 2022)—with proper choices of hyperparameters and regularization techniques, scalarization matches or even surpasses SMTOs. Hence, it largely remains an open problem regarding whether there is an inherent advantage of using SMTOs for MTL over scalarization, at least from a theoretical perspective.
In this paper, we revisit scalarization from a theoretical perspective. We study whether scalarization is capable of fully exploring the Pareto front in linear MTL. Technically, we aim to understand the following question on full exploration:
For every Pareto optimum  $v$  on the Pareto front induced by multiple tasks, does there exist a weighted combination of task losses such that the optimal solution of the linearly scalarized objective corresponds to  $v$ ?
Our fundamental thesis is that, if the answer to the above question is affirmative, then there is no inherent advantage of SMTOs over scalarization on the representation level; otherwise, there might be cases where SMTOs can converge to Pareto optimal solutions that cannot be obtained via any scalarization objective. Our findings suggest that, in contrast to recent works that claimed empirical advantages of scalarization, it is inherently incapable of full exploration. Specifically, our contributions are as follows:
1. We reveal a multi-surface structure of the feasible region, and identify a common failure mode of scalarization, which we refer to as "gradient disagreement" (Section 3.1). This serves as an important first step towards understanding when and why scalarization fails to achieve a specific Pareto optimum.
2. We provide necessary and sufficient conditions for scalarization to fully explore the Pareto front, which lead to the conclusion that scalarization is incapable of tracing out the Pareto front in general (Section 3.2). To the best of our knowledge, we are the first to establish both necessary and sufficient conditions for full exploration in the non-convex setting.
3. We proceed to discuss implications of our theoretical results (Section 4.1), demonstrating how they answer the open questions in Xin et al. (2022), and provide a remedy solution via randomization to counteract the fundamental limitation of scalarization (Section 4.2).
4. We additionally perform experiments on a real-world dataset using both scalarization and state-of-the-art SMTOs (Section 5). The experimental results not only corroborate our theoretical findings, but also demonstrate the potential of SMTOs in finding balanced solutions.
We introduce the setup of multi-task learning (MTL) and relevant notations, along with essential background on the optimal value of the scalarization objective and the concept of Pareto optimality.
Problem setup. Following prior works, we consider multi-task learning with  $k$  regression tasks (Wu et al., 2020; Du et al., 2021). The training examples  $\{(x_{j},y_{1}^{j},\dots ,y_{k}^{j})\}_{j = 1}^{n}$  are drawn i.i.d. from some  $\mu \in \mathcal{P}(\mathcal{X}\times \mathcal{Y}_1\times \dots \times \mathcal{Y}_k)$ , where  $\mathcal{X}\subseteq \mathbb{R}^p$  is the shared input space and  $\mathcal{V}_i\subseteq \mathbb{R}$  is the output space for task  $i$ . We denote the label of task  $i$  as  $y_{i} = (y_{i}^{1},\dots ,y_{i}^{n})^{\top}\in \mathbb{R}^{n}$ , and concatenate the inputs and labels in matrix forms:  $X = [x_{1},\dots ,x_{n}]^{\top}\in \mathbb{R}^{n\times p}$  and  $Y = [y_{1},\dots ,y_{k}]\in \mathbb{R}^{n\times k}$ .
We focus on linear MTL, in which a linear neural network serves as the backbone for prediction. Despite its simplicity, this setting has attracted extensive research interest in the literature (Maurer, 2006; Maurer et al., 2016; Wu et al., 2020; Du et al., 2021; Tripuraneni et al., 2021). Without loss of
generality, the model that will be used for our study is a two-layer linear network $^3$ , which consists of a shared layer  $W \in \mathbb{R}^{p \times q}$  and a task-specific head  $a_i \in \mathbb{R}^q$  for each task. The prediction for task  $i$  on input  $x$  is given by  $f_i(x, W, a_i) \coloneqq x^\top Wa_i$ . We adopt Mean Squared Error (MSE) as the loss function, and the training loss for task  $i$  is given by  $L_i(W, a_i) \coloneqq \|XWa_i - y_i\|_2^2$ .
Notation. We use  $[k]$  to denote the index set  $\{1, \dots, k\}$ . For two square matrices  $M$  and  $N$  with the same dimension, we write  $M \succeq N$  if  $M - N$  is positive semi-definite, and  $M \succ N$  if  $M - N$  is positive definite. We denote  $\mathrm{diag}(D)$  as the diagonal matrix whose diagonal entries are given by  $D = \{d_i\}_{i \in [k]}$ , a set of real numbers; and  $\operatorname{span}(V)$  as the linear space spanned by  $V = \{v_i\}_{i \in [k]}$ , a set of vectors. For two vectors with the same size, we use  $\odot$  and  $\oslash$  to denote entry-wise multiplication and division. When there is no ambiguity, operations on vectors (e.g., absolute value, square, square root) are performed entry-wise. For any matrix  $M$ ,  $M^{\dagger}$  stands for its Moore-Penrose inverse. We use  $\mathrm{range}(M)$  and  $\mathrm{rank}(M)$  to denote its column space and rank, respectively. The expression  $\|\cdot\|$  denotes the  $\ell_2$  norm of a vector,  $\langle \cdot, \cdot \rangle$  the inner product of two vectors,  $\|\cdot\|_F$  the Frobenius norm of a matrix, and  $\rho(\cdot)$  the spectral radius of a square matrix.
Optimal value of scalarization (restated from Wu et al. (2020)). The objective function of scalarization is a weighted sum of losses:  $\sum_{i=1}^{k} \lambda_i L_i(W, a_i)$ . Here, the convex coefficients  $\{\lambda_i\}_{i \in [k]}$  satisfy  $\lambda_i \geq 0$  and  $\sum_{i} \lambda_i = 1$ , and are typically obtained via grid search (Vandenhende et al., 2021). For each task, we can decompose its loss based on the orthogonality principle (Kay, 1993)
$$
L _ {i} (W, a _ {i}) := \| X W a _ {i} - y _ {i} \| _ {2} ^ {2} = \| X W a _ {i} - \hat {y} _ {i} \| ^ {2} + \| \hat {y} _ {i} - y _ {i} \| ^ {2}, \tag {1}
$$
where  $\hat{y}_i = X(X^\top X)^\dagger X^\top y_i$  is the optimal predictor for task  $i$ . Denote  $\Lambda = \mathrm{diag}(\{\sqrt{\lambda_i}\}_{i=1}^k)$ ,  $A = [a_1, \dots, a_k] \in \mathbb{R}^{q \times k}$ ,  $\hat{Y} = [\hat{y}_1, \dots, \hat{y}_k] \in \mathbb{R}^{n \times k}$ . The scalarization objective can be compactly expressed as
$$
\left\| \left(X W A - Y\right) \Lambda \right\| _ {F} ^ {2} = \left\| \left(X W A - \hat {Y}\right) \Lambda \right\| _ {F} ^ {2} + \left\| (\hat {Y} - Y) \Lambda \right\| _ {F} ^ {2}. \tag {2}
$$
Throughout this paper, we assume the optimal linear predictions  $\{\hat{y}_i\}_{i\in [k]}$  are linearly independent. This leads to two distinct cases, depending on the width of the hidden layer  $q$ :
1. When  $q \geq k$ , the minimum value of the first term is zero, i.e., every task can simultaneously achieve its optimal loss  $\| \hat{y}_i - y_i \|^2$ .
2. When  $q < k$ , the minimum value of the first term is given by the rank- $q$  approximation. In particular, let  $\hat{Y}\Lambda = \sum_{i=1}^{k}\sigma_{i}u_{i}v_{i}^{\top}$  be the SVD of  $\hat{Y}\Lambda$ . According to the Eckart-Young-Mirsky Theorem (Eckart and Young, 1936), the best rank- $q$  approximation of  $\hat{Y}\Lambda$  under the Frobenius norm is given by the truncated SVD:  $\hat{Y}_{q,\Lambda}\Lambda \coloneqq \sum_{i=1}^{q}\sigma_{i}u_{i}v_{i}^{\top}$ . Hence, the minimum training risk is  $\|( \hat{Y}_{q,\Lambda} - \hat{Y})\Lambda \|_F^2 + \| (\hat{Y} - Y)\Lambda \|_F^2$ . This regime, referred to as the under-parameterized regime, constitutes the main focus of our study.
Pareto optimality. In the context of MTL, a point  $\theta^{*}$  (shared model parameters) is a Pareto optimum (PO) if there does not exist a  $\theta^{\prime}$  that achieves strictly better performance on at least one task and no worse performance on all other tasks compared to  $\theta^{*}$ . Formally, for a set of objectives  $\{L_i\}_{i\in [k]}$  that need to be minimized, if  $L_{i}(\theta^{\prime}) < L_{i}(\theta^{*})$  for some point  $\theta^{\prime}$ , then there must exist another task  $j$  such that  $L_{j}(\theta^{\prime}) > L_{j}(\theta^{*})$ . The Pareto front (a.k.a. Pareto frontier) is the collection of objective values  $(L_{1}(\theta),\dots ,L_{k}(\theta))$  corresponding to all PO.
3 Linear scalarization through the lens of exploring the Pareto front
In this section, we provide a fine-grained analysis of the feasible region (formally defined in Section 3.1), revealing its multi-surface structure. We proceed to analyze whether scalarization is capable of tracing out the Pareto front, and identify necessary and sufficient conditions for full exploration.
Reformulation. Recall that minimizing Equation (1) is equivalent to minimizing the squared loss  $\| XW a_{i} - \hat{y}_{i}\|^{2}$ , where  $\hat{y}_i = X(X^\top X)^\dagger X^\top y_i$  is the optimal linear predictor for task  $i$ . Additionally, the task-specific heads can be optimized independently for each task. Thus, if we let  $Z = XW$  and
<sup>3</sup>Any deep linear networks for MTL can be reduced to the case of a two-layer linear network, with one layer of shared representations and one layer of task-specific heads.
As long as  $n \geq k$ , an arbitrarily small perturbation can make  $\{\hat{y}_i\}_{i \in [k]}$  linearly independent. We mainly adopt this assumption for ease of presentation.
Figure 1: Feasible MSEs of a three-task linear MTL problem with  $q = 1$ , formed by a union of multiple feasible MSEs surfaces defined in Theorem 3.1. Note that the balanced PO located at the intersection of the surfaces is not achievable by scalarization. See Appendix A.1 for implementation details.
Figure 2: When restricted to a single surface, each contains a (different) optimum point, and the gradients (w.r.t. minimizing total MSE) on each surface disagree at their intersection.
$a_{i} = (Z^{\top}Z)^{\dagger}Z^{\top}\hat{y}_{i}$  (the optimal task-specific head), then minimizing Equation (1) is equivalent to
$$
\min  _ {Z = X W} \| Z \left(Z ^ {\top} Z\right) ^ {\dagger} Z ^ {\top} \hat {y} _ {i} - \hat {y} _ {i} \| ^ {2}, \tag {3}
$$
which can be further transformed into
$$
\max _ {P _ {Z}} \hat {y} _ {i} ^ {\top} P _ {Z} \hat {y} _ {i}, \quad P _ {Z} = Z (Z ^ {\top} Z) ^ {\dagger} Z ^ {\top}.
$$
Therefore, the original MTL problem can be reformulated into the following multi-objective problem
$$
\max  _ {P _ {Z}} \left(\hat {y} _ {1} ^ {\top} P _ {Z} \hat {y} _ {1}, \dots , \hat {y} _ {k} ^ {\top} P _ {Z} \hat {y} _ {k}\right), \tag {4}
$$
where  $P_Z$  is a projection matrix with rank at most  $q$  (i.e., hidden layer width) and maps to the column space of  $X$ . Each quadratic term can be further expressed as  $\| P_Z \hat{y}_i \|^2$ . From here it is clear (also see Section 2) that when  $q \geq k$  (the over-parametrized regime), the  $k$  quadratic terms can be maximized simultaneously, meaning that the network has sufficient capacity to fit the target tasks. In this case, the Pareto front reduces to a singleton, and can be achieved by scalarization with any choice of convex coefficients. In fact, this holds true even for general non-linear networks as long as it is wide enough, i.e., over-parametrized, and we defer more discussions to Appendix A.2.
The main focus of our study, in contrast, lies in the under-parametrized regime  $q < k$ , where various tasks compete for the network's capacity for better prediction. We comment that this regime has been the focus of previous work and demonstrated to be beneficial in MTL (Wu et al., 2020; Wang et al., 2022). We study two extreme cases within the under-parameterized regime,  $q = 1$  and  $q = k - 1$ . By doing so, we anticipate that our results will hold for other  $k$ 's in between.
3.1 The multi-surface structure of the feasible region
The objective function of task  $i$  is  $\| P_Z \hat{y}_i \|^2$ , i.e., the square of the projection of  $\hat{y}_i$  on  $P_Z$ . Since we are interested in the Pareto front, we can assume w.l.o.g. that  $\mathrm{rank}(P_Z) = q$  and  $\mathrm{range}(P_Z) \subset \mathrm{span}(\{\hat{y}_i\}_{i \in [k]})$  (otherwise there is a loss of model capacity, and the corresponding point will no longer be a PO). Consequently, we define the feasible region as
$$
\mathcal {F} _ {q} := \left\{\left(\hat {y _ {1}} ^ {\top} P _ {Z} \hat {y _ {1}}, \dots , \hat {y _ {k}} ^ {\top} P _ {Z} \hat {y _ {k}}\right) ^ {\top} \Bigg | P _ {Z} ^ {2} = P _ {Z}, \operatorname {r a n k} (P _ {Z}) = q, \operatorname {r a n g e} (P _ {Z}) \subset \operatorname {s p a n} (\{\hat {y _ {i}} \} _ {i \in [ k ]}) \right\}. \tag {5}
$$
We now provide a fine-grained characterization of  $\mathcal{F}_1$  (corresponding to  $q = 1$ ) in the following theorem, which also serves as the basis for  $q = k - 1$ .
Theorem 3.1. Let  $G = \hat{Y}^\top \hat{Y}$  and  $Q = G^{-1}$ . Define
$$
Q _ {i _ {1} \dots i _ {l}} := D _ {i _ {1} \dots i _ {l}} Q D _ {i _ {1} \dots i _ {l}}, \quad 0 \leq l \leq k, 1 \leq i _ {1} <   \dots <   i _ {l} \leq k, \tag {6}
$$
where  $D_{i_1\dots i_l}$  is a diagonal matrix with all diagonal entries equal to 1, except for the entries at positions  $i_1,\dots ,i_l$  which are equal to -1. Consider the following surface parameterized by  $v$ :
$$
E _ {i _ {1} \dots i _ {l}} := \left\{v: \sqrt {v} ^ {\top} Q _ {i _ {1} \dots i _ {l}} \sqrt {v} = 1 \right\}. \tag {7}
$$
$$
\mathcal {F} _ {1} = \bigcup_ {0 \leq l \leq k} \bigcup_ {1 \leq i _ {1} <   \dots <   i _ {l} \leq k} E _ {i _ {1} \dots i _ {l}}, \tag {8}
$$
i.e.,  $\mathcal{F}_1$  is the union of  $2^{k - 1}$  surfaces in the non-negative orthant (note  $E_{i_1\dots i_l} = E_{\overline{i_1\dots i_l}}$  by symmetry).
To help the readers gain some intuitions on the multi-surface structure demonstrated in Theorem 3.1, we visualize the feasible region of a three-task (i.e.,  $k = 3$ ) example in Figure 1 (more details at the end of the subsection). We also provide a proof sketch below (detailed proof in Appendix A.3).
Proof sketch of Theorem 3.1. A rank-1 projection matrix can be expressed as  $ss^{\top}$  with  $\| s \| = 1$ . The feasible region can therefore be equivalently characterized as
$$
\mathcal {F} _ {1} = \left\{\left(\langle \hat {y} _ {1}, s \rangle^ {2}, \dots , \langle \hat {y} _ {k}, s \rangle^ {2}\right) ^ {\top} \mid \| s \| = 1, s \in \operatorname {s p a n} \left(\left\{\hat {y} _ {i} \right\} _ {i \in [ k ]}\right) \right\}. \tag {9}
$$
Define  $v_{i} = \langle \hat{y}_{i}, s \rangle$  and  $v = (v_{1}, \dots, v_{k})^{\top}$ . We will show the set of  $v$  forms the boundary of an ellipsoid determined by  $Q$ . As a consequence, the set of  $|v|$  is the union of multiple surfaces, each corresponding to the boundary of an ellipsoid (reflection of  $Q$ ) in the non-negative orthant. The reflection is represented by the diagonal matrix  $D_{i_1\dots i_l}$  in the theorem statement. Finally,  $\mathcal{F}_1$ , which is the set of  $v^2$ , can be obtained by squaring the coordinates in the previous set. This finishes the proof of Theorem 3.1.
Since the orthogonal complement of a 1-dimensional subspace of  $\mathrm{span}(\{\hat{y}_i\}_{i\in [k]})$  is a  $(k - 1)$ -dimensional subspace, we immediately have the following result (proof deferred to Appendix A.4).
Theorem 3.2. Let  $t\coloneqq (\| \hat{y}_1\| ^2,\dots ,\| \hat{y}_k\| ^2)^\top$  . Consider the following surface parametrized by  $v$  ..
$$
I _ {i _ {1} \dots i _ {l}} = \left\{v: \sqrt {t - v} ^ {\top} Q _ {i _ {1} \dots i _ {l}} \sqrt {t - v} = 1 \right\}. \tag {10}
$$
$$
\mathcal {F} _ {k - 1} = \bigcup_ {0 \leq l \leq k} \bigcup_ {1 \leq i _ {1} <   \dots <   i _ {l} \leq k} I _ {i _ {1} \dots i _ {l}}, \tag {11}
$$
i.e.,  $\mathcal{F}_{k - 1}$  is the union of  $2^{k - 1}$  surfaces in the non-negative orthant ( $I_{i_1\dots i_l} = I_{\overline{i_1\dots i_l}}$  by symmetry).
Theorems 3.1 and 3.2 not only offer us a glimpse into the shape of the feasible region but also provide valuable insights into the fundamental limitation of linear scalarization. The key observation is that, scalarization fails to reach the intersection points of two (or more) surfaces when the gradients with respect to the two (or more) surfaces disagree (i.e., they lie in different directions). We refer to this phenomenon as gradient disagreement, and provide an illustrating example in Figure 2. Essentially, the multi-surface structure that we have revealed implies the abundance of intersection points on the feasible region  $\mathcal{F}_q$ . Moreover, if any of these points were to be an PO, scalarization would not be able to trace out the Pareto front (see Appendix A.5 for a detailed explanation).
We illustrate this observation using a concrete example with three tasks in Figure 1. The feasible region consists of four surfaces, among which  $E_{\emptyset}$  is dominated by other surfaces. The intersection of  $E_1, E_2, E_3$  (the red point) is a PO that achieves balanced performance across tasks, yet scalarization cannot reach this point since it lies in a valley.
3.2 Necessary and sufficient conditions for full exploration
In this section, we provide an in-depth analysis of the Pareto front, which is a subset of the feasible region. Concretely, we develop necessary and sufficient conditions for scalarization to fully explore the Pareto front.
We first review some important concepts that will be frequently used in this section. Denote  $\mathcal{Q} \coloneqq \{Q_{i_1\dots i_l}\mid 0\leq l\leq k, 1\leq i_1 < \dots < i_l\leq k\}$  as the collection of matrices which define the surfaces in the previous section. Denote  $\mathcal{G} \coloneqq \{G_{i_1\dots i_l} = Q_{i_1\dots i_l}^{-1}\mid 0\leq l\leq k, 1\leq i_1 < \dots < i_l\leq k\}$ .
Note  $G_{i_1\dots i_l} = D_{i_1\dots i_l}\hat{Y}^\top \hat{Y}D_{i_1\dots i_l} = (\hat{Y}D_{i_1\dots i_l})^\top (\hat{Y}D_{i_1\dots i_l})$ . Therefore,  $\mathcal{G}$  is a collection of gram matrices which measure the task similarity, and each element is obtained by flipping the signs of some optimal predictors. We use  $\mathcal{E}$  and  $\mathcal{I}$  to denote the collection of surfaces defined in Equations (7) and (10), respectively. We also formally define a class of matrix as follows.
Definition 3.3 (Doubly non-negative matrix). A doubly non-negative matrix  $M$  is a real positive semi-definite (PSD) matrix with non-negative entries.
Our analysis begins with an observation that the distribution of PO across different surfaces plays a vital role in the success of scalarization. Specifically, we want to avoid the "bad" case like the one demonstrated in Figure 1, where a PO lies on the intersection of multiple surfaces. A natural solution to circumvent this issue is to force the Pareto front to lie on a single surface. We therefore draw our attention to this particular scenario, and provide a necessary and sufficient condition for such configuration in the following lemma.
Lemma 3.4. When  $q = 1$ , the Pareto front of the feasible region belongs to a single surface  $E^{*} \in \mathcal{E}$ , if and only if the corresponding  $G^{*} = (Q^{*})^{-1} \in \mathcal{G}$  is doubly non-negative. We refer to the existence of such a doubly non-negative  $G^{*}$  in  $\mathcal{G}$  as C1.
We outline the proof sketch below. The detailed proof can be found in Appendix B.1.
Proof sketch of Lemma 3.4. For the  $if$  part, we employ a geometric argument. Specifically, denote the acute cone formed by the optimal predictors as  $\mathcal{A}$ . We will show that, if  $s$  (a unit vector in Equation (9)) does not belong to the dual cone  $\mathcal{A}^*$ , then its reflection w.r.t. the dual cone  $s'$  is a better point than  $s$ . As a consequence, we can remove the absolute value from the proof sketch of Theorem 3.1. This further implies that the Pareto front belongs to a single surface.