arXiv subjects

Yuma Ichikawa

Publications and source records attributed to Yuma Ichikawa.

29 records · Page 2Linked to original sources

arXiv2025

Continuous Parallel Relaxation for Finding Diverse Solutions in Combinatorial Optimization Problems

Finding the optimal solution is often the primary goal in combinatorial optimization (CO). However, real-world applications frequently require diverse solutions rather than a single optimum, particularly in two key scenarios. The first scenario occurs in real-world applications where strictly enforcing every constraint is neither necessary nor desirable. Allowing minor constraint violations can often lead to more cost-effective solutions. This is typically achieved by incorporating the constraints as penalty terms in the objective function, which requires careful tuning of penalty parameters. The second scenario involves cases where CO formulations tend to oversimplify complex real-world factors, such as domain knowledge, implicit trade-offs, or ethical considerations. To address these challenges, generating (i) penalty-diversified solutions by varying penalty intensities and (ii) variation-diversified solutions with distinct structural characteristics provides valuable insights, enabling practitioners to post-select the most suitable solution for their specific needs. However, efficiently discovering these diverse solutions is more challenging than finding a single optimal one. This study introduces Continual Parallel Relaxation Annealing (CPRA), a computationally efficient framework for unsupervised-learning (UL)-based CO solvers that generates diverse solutions within a single training run. CPRA leverages representation learning and parallelization to automatically discover shared representations, substantially accelerating the search for these diverse solutions. Numerical experiments demonstrate that CPRA outperforms existing UL-based solvers in generating these diverse solutions while significantly reducing computational costs.

stat.ML↗

arXiv2025

High-dimensional Asymptotics of VAEs: Threshold of Posterior Collapse and Dataset-Size Dependence of Rate-Distortion Curve

In variational autoencoders (VAEs), the variational posterior often collapses to the prior, known as posterior collapse, which leads to poor representation learning quality. An adjustable hyperparameter beta has been introduced in VAEs to address this issue. This study sharply evaluates the conditions under which the posterior collapse occurs with respect to beta and dataset size by analyzing a minimal VAE in a high-dimensional limit. Additionally, this setting enables the evaluation of the rate-distortion curve of the VAE. Our results show that, unlike typical regularization parameters, VAEs face "inevitable posterior collapse" beyond a certain beta threshold, regardless of dataset size. Moreover, the dataset-size dependence of the derived rate-distortion curve suggests that relatively large datasets are required to achieve a rate-distortion curve with high rates. These findings robustly explain generalization behavior observed in various real datasets with highly non-linear VAEs.

stat.ML↗

arXiv2024

Controlling Continuous Relaxation for Combinatorial Optimization

Unsupervised learning (UL)-based solvers for combinatorial optimization (CO) train a neural network that generates a soft solution by directly optimizing the CO objective using a continuous relaxation strategy. These solvers offer several advantages over traditional methods and other learning-based methods, particularly for large-scale CO problems. However, UL-based solvers face two practical issues: (I) an optimization issue, where UL-based solvers are easily trapped at local optima, and (II) a rounding issue, where UL-based solvers require artificial post-learning rounding from the continuous space back to the original discrete space, undermining the robustness of the results. This study proposes a Continuous Relaxation Annealing (CRA) strategy, an effective rounding-free learning method for UL-based solvers. CRA introduces a penalty term that dynamically shifts from prioritizing continuous solutions, effectively smoothing the non-convexity of the objective function, to enforcing discreteness, eliminating artificial rounding. Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems. Additionally, CRA effectively eliminates artificial rounding and accelerates the learning process.

stat.ML↗

arXiv2024

Optimization by Parallel Quasi-Quantum Annealing with Gradient-Based Sampling

Learning-based methods have gained attention as general-purpose solvers due to their ability to automatically learn problem-specific heuristics, reducing the need for manually crafted heuristics. However, these methods often face scalability challenges. To address these issues, the improved Sampling algorithm for Combinatorial Optimization (iSCO), using discrete Langevin dynamics, has been proposed, demonstrating better performance than several learning-based solvers. This study proposes a different approach that integrates gradient-based update through continuous relaxation, combined with Quasi-Quantum Annealing (QQA). QQA smoothly transitions the objective function, starting from a simple convex function, minimized at half-integral values, to the original objective function, where the relaxed variables are minimized only in the discrete space. Furthermore, we incorporate parallel run communication leveraging GPUs to enhance exploration capabilities and accelerate convergence. Numerical experiments demonstrate that our method is a competitive general-purpose solver, achieving performance comparable to iSCO and learning-based solvers across various benchmark problems. Notably, our method exhibits superior speed-quality trade-offs for large-scale instances compared to iSCO, learning-based solvers, commercial solvers, and specialized algorithms.

cs.LG↗

arXiv2024

Ratio Divergence Learning Using Target Energy in Restricted Boltzmann Machines: Beyond Kullback--Leibler Divergence Learning

We propose ratio divergence (RD) learning for discrete energy-based models, a method that utilizes both training data and a tractable target energy function. We apply RD learning to restricted Boltzmann machines (RBMs), which are a minimal model that satisfies the universal approximation theorem for discrete distributions. RD learning combines the strength of both forward and reverse Kullback-Leibler divergence (KLD) learning, effectively addressing the "notorious" issues of underfitting with the forward KLD and mode-collapse with the reverse KLD. Since the summation of forward and reverse KLD seems to be sufficient to combine the strength of both approaches, we include this learning method as a direct baseline in numerical experiments to evaluate its effectiveness. Numerical experiments demonstrate that RD learning significantly outperforms other learning methods in terms of energy function fitting, mode-covering, and learning stability across various discrete energy-based models. Moreover, the performance gaps between RD learning and the other learning methods become more pronounced as the dimensions of target models increase.

stat.ML↗

arXiv2024

Statistical Mechanics of Min-Max Problems

Min-max optimization problems, also known as saddle point problems, have attracted significant attention due to their applications in various fields, such as fair beamforming, generative adversarial networks (GANs), and adversarial learning. However, understanding the properties of these min-max problems has remained a substantial challenge. This study introduces a statistical mechanical formalism for analyzing the equilibrium values of min-max problems in the high-dimensional limit, while appropriately addressing the order of operations for min and max. As a first step, we apply this formalism to bilinear min-max games and simple GANs, deriving the relationship between the amount of training data and generalization error and indicating the optimal ratio of fake to real data for effective learning. This formalism provides a groundwork for a deeper theoretical analysis of the equilibrium properties in various machine learning methods based on min-max problems and encourages the development of new algorithms and architectures.

cs.LG↗

arXiv2024

Training-Free Time-Series Anomaly Detection: Leveraging Image Foundation Models

Recent advancements in time-series anomaly detection have relied on deep learning models to handle the diverse behaviors of time-series data. However, these models often suffer from unstable training and require extensive hyperparameter tuning, leading to practical limitations. Although foundation models present a potential solution, their use in time series is limited. To overcome these issues, we propose an innovative image-based, training-free time-series anomaly detection (ITF-TAD) approach. ITF-TAD converts time-series data into images using wavelet transform and compresses them into a single representation, leveraging image foundation models for anomaly detection. This approach achieves high-performance anomaly detection without unstable neural network training or hyperparameter tuning. Furthermore, ITF-TAD identifies anomalies across different frequencies, providing users with a detailed visualization of anomalies and their corresponding frequencies. Comprehensive experiments on five benchmark datasets, including univariate and multivariate time series, demonstrate that ITF-TAD offers a practical and effective solution with performance exceeding or comparable to that of deep models.

cs.LG↗

arXiv2024

Adaptive Flip Graph Algorithm for Matrix Multiplication

This study proposes the "adaptive flip graph algorithm", which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the inherent limitations of exploration and inefficient search encountered in the original flip graph algorithm, particularly when dealing with large matrix multiplication. For the limitation of exploration, the proposed algorithm adaptively transitions over the flip graph, introducing a flexibility that does not strictly reduce the number of multiplications. Concerning the issue of inefficient search in large instances, the proposed algorithm adaptively constraints the search range instead of relying on a completely random search, facilitating more effective exploration. Numerical experimental results demonstrate the effectiveness of the adaptive flip graph algorithm, showing a reduction in the number of multiplications for a $4\times 5$ matrix multiplied by a $5\times 5$ matrix from $76$ to $73$, and that from $95$ to $94$ for a $5 \times 5$ matrix multiplied by another $5\times 5$ matrix. These results are obtained in characteristic two.

cs.SC↗

arXiv2023

Learning Dynamics in Linear VAE: Posterior Collapse Threshold, Superfluous Latent Space Pitfalls, and Speedup with KL Annealing

Variational autoencoders (VAEs) face a notorious problem wherein the variational posterior often aligns closely with the prior, a phenomenon known as posterior collapse, which hinders the quality of representation learning. To mitigate this problem, an adjustable hyperparameter $β$ and a strategy for annealing this parameter, called KL annealing, are proposed. This study presents a theoretical analysis of the learning dynamics in a minimal VAE. It is rigorously proved that the dynamics converge to a deterministic process within the limit of large input dimensions, thereby enabling a detailed dynamical analysis of the generalization error. Furthermore, the analysis shows that the VAE initially learns entangled representations and gradually acquires disentangled representations. A fixed-point analysis of the deterministic process reveals that when $β$ exceeds a certain threshold, posterior collapse becomes inevitable regardless of the learning period. Additionally, the superfluous latent variables for the data-generative factors lead to overfitting of the background noise; this adversely affects both generalization and learning convergence. The analysis further unveiled that appropriately tuned KL annealing can accelerate convergence.

stat.ML↗

arXiv2023

Toward Unlimited Self-Learning MCMC with Parallel Adaptive Annealing

Self-learning Monte Carlo (SLMC) methods are recently proposed to accelerate Markov chain Monte Carlo (MCMC) methods using a machine learning model. With latent generative models, SLMC methods realize efficient Monte Carlo updates with less autocorrelation. However, SLMC methods are difficult to directly apply to multimodal distributions for which training data are difficult to obtain. To solve the limitation, we propose parallel adaptive annealing, which makes SLMC methods directly apply to multimodal distributions with a gradually trained proposal while annealing target distribution. Parallel adaptive annealing is based on (i) sequential learning with annealing to inherit and update the model parameters, (ii) adaptive annealing to automatically detect under-learning, and (iii) parallel annealing to mitigate mode collapse of proposal models. We also propose VAE-SLMC method which utilizes a variational autoencoder (VAE) as a proposal of SLMC to make efficient parallel proposals independent of any previous state using recently clarified quantitative properties of VAE. Experiments validate that our method can proficiently obtain accurate samples from multiple multimodal toy distributions and practical multimodal posterior distributions, which is difficult to achieve with the existing SLMC methods.

stat.ML↗

arXiv2022

Statistical-mechanical study of deep Boltzmann machine given weight parameters after training by singular value decomposition

Deep learning methods relying on multi-layered networks have been actively studied in a wide range of fields in recent years, and deep Boltzmann machines(DBMs) is one of them. In this study, a model of DBMs with some properites of weight parameters obtained by learning is studied theoretically by a statistical-mechanical approach based on the replica method. The phases characterizing the role of DBMs as a generator and their phase diagram are derived, depending meaningfully on the numbers of layers and units in each layer. It is found, in particular, that the correlation between the weight parameters in the hidden layers plays an essential role and that an increase in the layer number has a negative effect on DBM's performance as a generator when the correlation is smaller than a certain threshold value.

cond-mat.dis-nn↗