The widely observed'benign overfitting phenomenon' in the neural network literature raises the challenge to the `bias-variance trade-off' doctrine in the statistical learning theory.Since the generalization ability of the'lazy trained' over-parametrized neural network can be well approximated by that of the neural tangent kernel regression,the curve of the excess risk (namely, the learning curve) of kernel ridge regression attracts increasing attention recently.However, most recent arguments on the learning curve are heuristic and are based on the'Gaussian design' assumption.In this paper, under mild and more realistic assumptions, we rigorously provide a full characterization of the learning curve in the asymptotic senseunder a power-law decay condition of the eigenvalues of the kernel and also the target function.The learning curve elaborates the effect and the interplay of the choice of the regularization parameter, the source condition and the noise.In particular, our results suggest that the'benign overfitting phenomenon' exists in over-parametrized neural networks only when the noise level is small.

Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner by having no failure probability and always returning the same results.

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.

The widely observed'benign overfitting phenomenon' in the neural network literature raises the challenge to the bias-variance trade-off' doctrine in the statistical learning theory.Since the generalization ability of the'lazy trained' over-parametrized neural network can be well approximated by that of the neural tangent kernel regression,the curve of the excess risk (namely, the learning curve) of kernel ridge regression attracts increasing attention recently.However, most recent arguments on the learning curve are heuristic and are based on the'Gaussian design' assumption.In this paper, under mild and more realistic assumptions, we rigorously provide a full characterization of the learning curve in the asymptotic senseunder a power-law decay condition of the eigenvalues of the kernel and also the target function.The learning curve elaborates the effect and the interplay of the choice of the regularization parameter, the source condition and the noise.In particular, our results suggest that the'benign overfitting phenomenon' exists in over-parametrized neural networks only when the noise level is small.

This paper explores the intricate behavior of deep neural networks (DNNs) through the lens of neuron activation dynamics. We propose a probabilistic framework that can analyze models' neuron activation patterns as a stochastic process, uncovering theoretical insights into neural scaling laws, such as over-parameterization and the power-law decay of loss with respect to dataset size. By deriving key mathematical relationships, we present that the number of activated neurons increases in the form of $N(1-(\frac{bN}{D+bN})^b)$, and the neuron activation should follows power-law distribution. Based on these two mathematical results, we demonstrate how DNNs maintain generalization capabilities even under over-parameterization, and we elucidate the phase transition phenomenon observed in loss curves as dataset size plotted in log-axis (i.e. the data magnitude increases linearly). Moreover, by combining the above two phenomenons and the power-law distribution of neuron activation, we derived the power-law decay of neural network's loss function as the data size scale increases. Furthermore, our analysis bridges the gap between empirical observations and theoretical underpinnings, offering experimentally testable predictions regarding parameter efficiency and model compressibility. These findings provide a foundation for understanding neural network scaling and present new directions for optimizing DNN performance.

Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner by having no failure probability and always returning the same results. We provide provable guarantees for deterministic column sampling using ridge leverage scores. The matrix sketch returned by our algorithm is a column subset of the original matrix, yielding additional interpretability. Like the randomized counterparts, the deterministic algorithm provides $(1+\epsilon)$ error column subset selection, $(1+\epsilon)$ error projection-cost preservation, and an additive-multiplicative spectral bound. We also show that under the assumption of power-law decay of ridge leverage scores, this deterministic algorithm is provably as accurate as randomized algorithms. Lastly, ridge regression is frequently used to regularize ill-posed linear least-squares problems. While ridge regression provides shrinkage for the regression coefficients, many of the coefficients remain small but non-zero. Performing ridge regression with the matrix sketch returned by our algorithm and a particular regularization parameter forces coefficients to zero and has a provable $(1+\epsilon)$ bound on the statistical risk. As such, it is an interesting alternative to elastic net regularization.

Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner by having no failure probability and always returning the same results. We provide provable guarantees for deterministic column sampling using ridge leverage scores. The matrix sketch returned by our algorithm is a column subset of the original matrix, yielding additional interpretability. Like the randomized counterparts, the deterministic algorithm provides (1 + {\epsilon}) error column subset selection, (1 + {\epsilon}) error projection-cost preservation, and an additive-multiplicative spectral bound. We also show that under the assumption of power-law decay of ridge leverage scores, this deterministic algorithm is provably as accurate as randomized algorithms. Lastly, ridge regression is frequently used to regularize ill-posed linear least- squares problems. While ridge regression provides shrinkage for the regression coefficients, many of the coefficients remain small but non-zero. Performing ridge regression with the matrix sketch returned by our algorithm and a particular regularization parameter forces coefficients to zero and has a provable (1 + {\epsilon}) bound on the statistical risk. As such, it is an interesting alternative to elastic net regularization.

Training classifiers on large databases is computationally demanding. It is desirable to develop efficient procedures for a reliable prediction of a classifier's suitability for implementing a given task, so that resources can be assigned to the most promising candidates or freed for exploring new classifier candidates. We propose such a practical and principled predictive method. Practical because it avoids the costly procedure of training poor classifiers on the whole training set, and principled because of its theoretical foundation. The effectiveness of the proposed procedure is demonstrated for both single-and multi-layer networks.

Training classifiers on large databases is computationally demanding. It is desirable to develop efficient procedures for a reliable prediction of a classifier's suitability for implementing a given task, so that resources can be assigned to the most promising candidates or freed for exploring new classifier candidates. We propose such a practical and principled predictive method. Practical because it avoids the costly procedure of training poor classifiers on the whole training set, and principled because of its theoretical foundation. The effectiveness of the proposed procedure is demonstrated for both single-and multi-layer networks.

Training classifiers on large databases is computationally demanding. Itis desirable to develop efficient procedures for a reliable prediction of a classifier's suitability for implementing a given task, so that resources can be assigned to the most promising candidates or freed for exploring new classifier candidates. We propose such a practical and principled predictive method. Practical because it avoids the costly procedure of training poor classifiers on the whole training set, and principled because of its theoretical foundation. The effectiveness of the proposed procedure is demonstrated for both single-and multi-layer networks.