Takashi Takahashi

Takashi Takahashi

Self-training (ST) is a simple yet effective semi-supervised learning method. However, why and how ST improves generalization performance by using potentially erroneous pseudo-labels is still not well understood. To deepen the understanding of ST, we derive and analyze a sharp characterization of the behavior of iterative ST when training a linear classifier by minimizing the ridge-regularized convex loss on binary Gaussian mixtures, in the asymptotic limit where input dimension and data size diverge proportionally. The results show that ST improves generalization in different ways depending on the number of iterations. When the number of iterations is small, ST improves generalization performance by fitting the model to relatively reliable pseudo-labels and updating the model parameters by a large amount at each iteration. This suggests that ST works intuitively. On the other hand, with many iterations, ST can gradually improve the direction of the classification plane by updating the model parameters incrementally, using soft labels and small regularization. It is argued that this is because the small update of ST can extract information from the data in an almost noiseless way. However, in the presence of label imbalance, the generalization performance of ST underperforms supervised learning with true labels. To overcome this, two heuristics are proposed to enable ST to achieve nearly compatible performance with supervised learning even with significant label imbalance.

Takashi Takahashi

Under-bagging (UB), which combines under-sampling and bagging, is a popular ensemble learning method for training classifiers on an imbalanced data. Using bagging to reduce the increased variance caused by the reduction in sample size due to under-sampling is a natural approach. However, it has recently been pointed out that in generalized linear models, naive bagging, which does not consider the class imbalance structure, and ridge regularization can produce the same results. Therefore, it is not obvious whether it is better to use UB, which requires an increased computational cost proportional to the number of under-sampled data sets, when training linear models. Given such a situation, in this study, we heuristically derive a sharp asymptotics of UB and use it to compare with several other popular methods for learning from imbalanced data, in the scenario where a linear classifier is trained from a two-component mixture data. The methods compared include the under-sampling (US) method, which trains a model using a single realization of the under-sampled data, and the simple weighting (SW) method, which trains a model with a weighted loss on the entire data. It is shown that the performance of UB is improved by increasing the size of the majority class while keeping the size of the minority fixed, even though the class imbalance can be large, especially when the size of the minority class is small. This is in contrast to US, whose performance is almost independent of the majority class size. In this sense, bagging and simple regularization differ as methods to reduce the variance increased by under-sampling. On the other hand, the performance of SW with the optimal weighting coefficients is almost equal to UB, indicating that the combination of reweighting and regularization may be similar to UB.

Kaito Takanami, Takashi Takahashi, Ayaka Sakata

Self-distillation (SD), a technique where a model improves itself using its own predictions, has attracted attention as a simple yet powerful approach in machine learning. Despite its widespread use, the mechanisms underlying its effectiveness remain unclear. In this study, we investigate the efficacy of hyperparameter-tuned multi-stage SD with a linear classifier for binary classification on noisy Gaussian mixture data. For the analysis, we employ the replica method from statistical physics. Our findings reveal that the primary driver of SD's performance improvement is denoising through hard pseudo-labels, with the most notable gains observed in moderately sized datasets. We also identify two practical heuristics to enhance SD: early stopping that limits the number of stages, which is broadly effective, and bias parameter fixing, which helps under label imbalance. To empirically validate our theoretical findings derived from our toy model, we conduct additional experiments on CIFAR-10 classification using pretrained ResNet backbone. These results provide both theoretical and practical insights, advancing our understanding and application of SD in noisy settings.

Kaito Takanami, Takashi Takahashi, Yoshiyuki Kabashima

In-context learning (ICL) is a key building block of modern large language models, yet its theoretical mechanisms remain poorly understood. It is particularly mysterious how ICL operates in real-world applications where tasks have a common structure. In this work, we address this problem by analyzing a linear attention model trained on low-rank regression tasks. Within this setting, we precisely characterize the distribution of predictions and the generalization error in the high-dimensional limit. Moreover, we find that statistical fluctuations in finite pre-training data induce an implicit regularization. Finally, we identify a sharp phase transition of the generalization error governed by task structure. These results provide a framework for understanding how transformers learn to learn the task structure.

Takashi Takahashi, Yoshiyuki Kabashima

We consider the variable selection problem of generalized linear models (GLMs). Stability selection (SS) is a promising method proposed for solving this problem. Although SS provides practical variable selection criteria, it is computationally demanding because it needs to fit GLMs to many re-sampled datasets. We propose a novel approximate inference algorithm that can conduct SS without the repeated fitting. The algorithm is based on the replica method of statistical mechanics and vector approximate message passing of information theory. For datasets characterized by rotation-invariant matrix ensembles, we derive state evolution equations that macroscopically describe the dynamics of the proposed algorithm. We also show that their fixed points are consistent with the replica symmetric solution obtained by the replica method. Numerical experiments indicate that the algorithm exhibits fast convergence and high approximation accuracy for both synthetic and real-world data.

Takashi Takahashi, Yoshiyuki Kabashima

Resampling techniques are widely used in statistical inference and ensemble learning, in which estimators' statistical properties are essential. However, existing methods are computationally demanding, because repetitions of estimation/learning via numerical optimization/integral for each resampled data are required. In this study, we introduce a computationally efficient method to resolve such problem: replicated vector approximate message passing. This is based on a combination of the replica method of statistical physics and an accurate approximate inference algorithm, namely the vector approximate message passing of information theory. The method provides tractable densities without repeating estimation/learning, and the densities approximately offer an arbitrary degree of the estimators' moment in practical time. In the experiment, we apply the proposed method to the stability selection method, which is commonly used in variable selection problems. The numerical results show its fast convergence and high approximation accuracy for problems involving both synthetic and real-world datasets.