Multi-objective optimization aims to find a set of solutions that achieve the best trade-off among multiple conflicting objective functions. While various multi-objective optimization algorithms have been proposed so far, most of them aim to find finite solutions as an approximation of the Pareto set, which may not adequately capture the entire structure of the Pareto set, especially when the number of variables is large. To overcome this limitation, we propose a method to obtain a parametric hypersurface representing the entire Pareto set instead of a finite set of points. Since the Pareto set of an $M$-objective optimization problem typically forms an $(M-1)$-dimensional simplex, we use a B{é}zier simplex as a model to express the Pareto set. We then develop a stochastic gradient descent-based algorithm that updates the B{é}zier simplex model toward the Pareto set, introducing a preconditioning matrix to enhance convergence. Our convergence analysis demonstrated that the proposed algorithm outperforms naive stochastic gradient descent in terms of convergence rate. Furthermore, we validate the effectiveness of our method through various multi-objective optimization problem instances, including real-world problems.

Neural networks play a central role in the construction of learning models for artificial intelligence and machine learning. This is because neural networks are highly flexible and can approximate a wide variety of maps with high accuracy. The flexibility of neural networks is theoretically guaranteed by using universal approximation theorems. When the input and output spaces have finite dimensions, the universal approximation property of neural networks has been intensively investigated under various conditions of width, depth, and activation functions. However, these finite-dimensional results cannot be directly applied to settings in which an input or output space has infinite dimensions, such as functional data analysis and neural processes. This study provides a universal approximation theorem with neural networks for uniformly continuous maps between function spaces, whose dimensions can be infinite.

Various data exhibit symmetry, including permutations in graphs and point clouds. Machine learning methods that utilize this symmetry have achieved considerable success. In this study, we explore learning models for data exhibiting group symmetry. Our focus is on transforming deep neural networks using Reynolds operators, which average over the group to convert a function into an invariant or equivariant form. While learning methods based on Reynolds operators are well-established, they often face computational complexity challenges. To address this, we introduce two new methods that reduce the computational burden associated with the Reynolds operator: (i) Although the Reynolds operator traditionally averages over the entire group, we demonstrate that it can be effectively approximated by averaging over specific subsets of the group, termed the Reynolds design. (ii) We reveal that the pre-model does not require all input variables. Instead, using a select number of partial inputs (Reynolds dimension) is sufficient to achieve a universally applicable model. Employing these methods, which hinge on the Reynolds design and Reynolds dimension concepts, allows us to construct universally applicable models with manageable computational complexity. Our experiments on benchmark data indicate that our approach is more efficient than existing methods.

The classical approach to measure the expressive power of deep neural networks with piecewise linear activations is based on counting their maximum number of linear regions. This complexity measure is quite relevant to understand general properties of the expressivity of neural networks such as the benefit of depth over width. Nevertheless, it appears limited when it comes to comparing the expressivity of different network architectures. This lack becomes particularly prominent when considering permutation-invariant networks, due to the symmetrical redundancy among the linear regions. To tackle this, we propose a refined definition of piecewise linear function complexity: instead of counting the number of linear regions directly, we first introduce an equivalence relation among the linear functions composing a piecewise linear function and then count those linear functions relative to that equivalence relation. Our new complexity measure can clearly distinguish between the two aforementioned models, is consistent with the classical measure, and increases exponentially with depth.

Numerous invariant (or equivariant) neural networks have succeeded in handling the invariant data such as point clouds and graphs. However, a generalization theory for the neural networks has not been well developed, because several essential factors for the theory, such as network size and margin distribution, are not deeply connected to the invariance and equivariance. In this study, we develop a novel generalization error bound for invariant and equivariant deep neural networks. To describe the effect of invariance and equivariance on generalization, we develop a notion of a quotient feature space, which measures the effect of group actions for the properties. Our main result proves that the volume of quotient feature spaces can describe the generalization error. Furthermore, the bound shows that the invariance and equivariance significantly improves the leading term of the bound. We apply our result to a specific invariant and equivariant networks, such as DeepSets (Zaheer et al., NIPS 2017), and show that their generalization bound is considerably improved by $\sqrt{n!}$, where $n!$ is the number of permutations. We also discuss the expressive power of invariant DNNs and show that they can achieve an optimal approximation rate. Moreover, we conducted experiments to support our results.

The B'ezier simplex fitting is a novel data modeling technique which utilizes geometric structures of data to approximate the Pareto set of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from skeletons of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a single object. In this paper, we analyze the asymptotic risks of those B'ezier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a B'ezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a B'ezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a B'ezier simplex of degree three where the all-at-once fitting gets an advantage.

Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M-objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis.