Koji Hukushima

Yuma Ichikawa, Koji Hukushima

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.

Yuma Ichikawa, Koji Hukushima

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.

Yuma Ichikawa, Koji Hukushima

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.

Yuichi Ishida, Yuma Ichikawa, Aki Dote et al.

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.

Kai Nakaishi, Ryo Yoshida, Kohei Kajikawa et al.

Statistical analysis of corpora provides an approach to quantitatively investigate natural languages. This approach has revealed that several power laws consistently emerge across different corpora and languages, suggesting universal mechanisms underlying languages. Particularly, the power-law decay of correlation has been interpreted as evidence for underlying hierarchical structures in syntax, semantics, and discourse. This perspective has also been extended to child speeches and animal signals. However, the argument supporting this interpretation has not been empirically tested in natural languages. To address this problem, the present study examines the validity of the argument for syntactic structures. Specifically, we test whether the statistical properties of parse trees align with the assumptions in the argument. Using English and Japanese corpora, we analyze the mutual information, deviations from probabilistic context-free grammars (PCFGs), and other properties in natural language parse trees, as well as in the PCFG that approximates these parse trees. Our results indicate that the assumptions do not hold for syntactic structures and that it is difficult to apply the proposed argument to child speeches and animal signals, highlighting the need to reconsider the relationship between the power law and hierarchical structures.

Kai Nakaishi, Yoshihiko Nishikawa, Koji Hukushima

Large Language Models (LLMs) have demonstrated impressive performance. To understand their behaviors, we need to consider the fact that LLMs sometimes show qualitative changes. The natural world also presents such changes called phase transitions, which are defined by singular, divergent statistical quantities. Therefore, an intriguing question is whether qualitative changes in LLMs are phase transitions. In this work, we have conducted extensive analysis on texts generated by LLMs and suggested that a phase transition occurs in LLMs when varying the temperature parameter. Specifically, statistical quantities have divergent properties just at the point between the low-temperature regime, where LLMs generate sentences with clear repetitive structures, and the high-temperature regime, where generated sentences are often incomprehensible. In addition, critical behaviors near the phase transition point, such as a power-law decay of correlation and slow convergence toward the stationary state, are similar to those in natural languages. Our results suggest a meaningful analogy between LLMs and natural phenomena.

Ryo Tamura, Koji Hukushima, Akira Matsuo et al.

We propose a data-driven technique to estimate the spin Hamiltonian, including uncertainty, from multiple physical quantities. Using our technique, an effective model of KCu$_4$P$_3$O$_{12}$ is determined from the experimentally observed magnetic susceptibility and magnetization curves with various temperatures under high magnetic fields. An effective model, which is the quantum Heisenberg model on a zigzag chain with eight spins having $J_1= -8.54 \pm 0.51 \{\rm meV}$, $J_2 = -2.67 \pm 1.13 \{\rm meV}$, $J_3 = -3.90 \pm 0.15 \{\rm meV}$, and $J_4 = 6.24 \pm 0.95 \{\rm meV}$, describes these measured results well. These uncertainties are successfully determined by the noise estimation. The relations among the estimated magnetic interactions or physical quantities are also discussed. The obtained effective model is useful to predict hard-to-measure properties such as spin gap, spin configuration at the ground state, magnetic specific heat, and magnetic entropy.