Isao Ono

Masahiro Nomura, Ryoki Hamano, Isao Ono

Symbolic regression (SR) with genetic programming (GP) aims to discover interpretable mathematical expressions directly from data. Despite its strong empirical success, the theoretical understanding of why GP-based SR generalizes beyond the training data remains limited. In this work, we provide a learning-theoretic analysis of SR models represented as expression trees. We derive a generalization bound for GP-style SR under constraints on tree size, depth, and learnable constants. Our result decomposes the generalization gap into two interpretable components: a structure-selection term, reflecting the combinatorial complexity of choosing an expression-tree structure, and a constant-fitting term, capturing the complexity of optimizing numerical constants within a fixed structure. This decomposition provides a theoretical perspective on several widely used practices in GP, including parsimony pressure, depth limits, numerically stable operators, and interval arithmetic. In particular, our analysis shows how structural restrictions reduce hypothesis-class growth while stability mechanisms control the sensitivity of predictions to parameter perturbations. By linking these practical design choices to explicit complexity terms in the generalization bound, our work offers a principled explanation for commonly observed empirical behaviors in GP-based SR and contributes towards a more rigorous understanding of its generalization properties.

Yukinari Hisaki, Isao Ono

In this paper, we propose an off-policy deep reinforcement learning (DRL) method utilizing the average reward criterion. While most existing DRL methods employ the discounted reward criterion, this can potentially lead to a discrepancy between the training objective and performance metrics in continuing tasks, making the average reward criterion a recommended alternative. We introduce RVI-SAC, an extension of the state-of-the-art off-policy DRL method, Soft Actor-Critic (SAC), to the average reward criterion. Our proposal consists of (1) Critic updates based on RVI Q-learning, (2) Actor updates introduced by the average reward soft policy improvement theorem, and (3) automatic adjustment of Reset Cost enabling the average reward reinforcement learning to be applied to tasks with termination. We apply our method to the Gymnasium's Mujoco tasks, a subset of locomotion tasks, and demonstrate that RVI-SAC shows competitive performance compared to existing methods.

Aoi Kato, Kenta Kojima, Masahiro Nomura et al.

Black-box discrete optimization (BB-DO) problems arise in many real-world applications, such as neural architecture search and mathematical model estimation. A key challenge in BB-DO is epistasis among parameters where multiple variables must be modified simultaneously to effectively improve the objective function. Estimation of Distribution Algorithms (EDAs) provide a powerful framework for tackling BB-DO problems. In particular, an EDA leveraging a Variational Autoencoder (VAE) has demonstrated strong performance on relatively low-dimensional problems with epistasis while reducing computational cost. Meanwhile, evolutionary algorithms such as DSMGA-II and P3, which integrate bit-flip-based local search with linkage learning, have shown excellent performance on high-dimensional problems. In this study, we propose a new memetic algorithm that combines VAE-based sampling with local search. The proposed method inherits the strengths of both VAE-based EDAs and local search-based approaches: it effectively handles high-dimensional problems with epistasis among parameters without incurring excessive computational overhead. Experiments on NK landscapes -- a challenging benchmark for BB-DO involving epistasis among parameters -- demonstrate that our method outperforms state-of-the-art VAE-based EDA methods, as well as leading approaches such as P3 and DSMGA-II.

Masahiro Nomura, Isao Ono

In this work, we propose a new variant of natural evolution strategies (NES) for high-dimensional black-box optimization problems. The proposed method, CR-FM-NES, extends a recently proposed state-of-the-art NES, Fast Moving Natural Evolution Strategy (FM-NES), in order to be applicable in high-dimensional problems. CR-FM-NES builds on an idea using a restricted representation of a covariance matrix instead of using a full covariance matrix, while inheriting an efficiency of FM-NES. The restricted representation of the covariance matrix enables CR-FM-NES to update parameters of a multivariate normal distribution in linear time and space complexity, which can be applied to high-dimensional problems. Our experimental results reveal that CR-FM-NES does not lose the efficiency of FM-NES, and on the contrary, CR-FM-NES has achieved significant speedup compared to FM-NES on some benchmark problems. Furthermore, our numerical experiments using 200, 600, and 1000-dimensional benchmark problems demonstrate that CR-FM-NES is effective over scalable baseline methods, VD-CMA and Sep-CMA.

Masahiro Nomura, Isao Ono

Natural Evolution Strategies (NES) is a promising framework for black-box continuous optimization problems. NES optimizes the parameters of a probability distribution based on the estimated natural gradient, and one of the key parameters affecting the performance is the learning rate. We argue that from the viewpoint of the natural gradient method, the learning rate should be determined according to the estimation accuracy of the natural gradient. To do so, we propose a new learning rate adaptation mechanism for NES. The proposed mechanism makes it possible to set a high learning rate for problems that are relatively easy to optimize, which results in speeding up the search. On the other hand, in problems that are difficult to optimize (e.g., multimodal functions), the proposed mechanism makes it possible to set a conservative learning rate when the estimation accuracy of the natural gradient seems to be low, which results in the robust and stable search. The experimental evaluations on unimodal and multimodal functions demonstrate that the proposed mechanism works properly depending on a search situation and is effective over the existing method, i.e., using the fixed learning rate.

Masahiro Nomura, Isao Ono

In this paper, we propose a new natural evolution strategy for unconstrained black-box function optimization (BBFO) problems and implicitly constrained BBFO problems. BBFO problems are known to be difficult because explicit representations of objective functions are not available. Implicit constraints make the problems more difficult because whether or not a solution is feasible is revealed when the solution is evaluated with the objective function. DX-NES-IC is one of the promising methods for implicitly constrained BBFO problems. DX-NES-IC has shown better performance than conventional methods on implicitly constrained benchmark problems. However, DX-NES-IC has a problem in that the moving speed of the probability distribution is slow on ridge structure. To address the problem, we propose the Fast Moving Natural Evolution Strategy (FM-NES) that accelerates the movement of the probability distribution on ridge structure by introducing the rank-one update into DX-NES-IC. The rank-one update is utilized in CMA-ES. Since naively introducing the rank-one update makes the search performance deteriorate on implicitly constrained problems, we propose a condition of performing the rank-one update. We also propose to reset the shape of the probability distribution when an infeasible solution is sampled at the first time. In numerical experiments using unconstrained and implicitly constrained benchmark problems, FM-NES showed better performance than DX-NES-IC on problems with ridge structure and almost the same performance as DX-NES-IC on the others. Furthermore, FM-NES outperformed xNES, CMA-ES, xNES with the resampling technique, and CMA-ES with the resampling technique.

Youhei Akimoto, Yuichi Nagata, Isao Ono et al.

This paper explores the theoretical basis of the covariance matrix adaptation evolution strategy (CMA-ES) from the information geometry viewpoint. To establish a theoretical foundation for the CMA-ES, we focus on a geometric structure of a Riemannian manifold of probability distributions equipped with the Fisher metric. We define a function on the manifold which is the expectation of fitness over the sampling distribution, and regard the goal of update of the parameters of sampling distribution in the CMA-ES as maximization of the expected fitness. We investigate the steepest ascent learning for the expected fitness maximization, where the steepest ascent direction is given by the natural gradient, which is the product of the inverse of the Fisher information matrix and the conventional gradient of the function. Our first result is that we can obtain under some types of parameterization of multivariate normal distribution the natural gradient of the expected fitness without the need for inversion of the Fisher information matrix. We find that the update of the distribution parameters in the CMA-ES is the same as natural gradient learning for expected fitness maximization. Our second result is that we derive the range of learning rates such that a step in the direction of the exact natural gradient improves the parameters in the expected fitness. We see from the close relation between the CMA-ES and natural gradient learning that the default setting of learning rates in the CMA-ES seems suitable in terms of monotone improvement in expected fitness. Then, we discuss the relation to the expectation-maximization framework and provide an information geometric interpretation of the CMA-ES.