arXiv subjects

Taiji Suzuki

Publications and source records attributed to Taiji Suzuki.

At least 109 records · Page 6Linked to original sources

arXiv2022

Stochastic Gradient Descent with Exponential Convergence Rates of Expected Classification Errors

We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the squared loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.

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arXiv2022

Layer-wise Adaptive Graph Convolution Networks Using Generalized Pagerank

We investigate adaptive layer-wise graph convolution in deep GCN models. We propose AdaGPR to learn generalized Pageranks at each layer of a GCNII network to induce adaptive convolution. We show that the generalization bound for AdaGPR is bounded by a polynomial of the eigenvalue spectrum of the normalized adjacency matrix in the order of the number of generalized Pagerank coefficients. By analysing the generalization bounds we show that oversmoothing depends on both the convolutions by the higher orders of the normalized adjacency matrix and the depth of the model. We performed evaluations on node-classification using benchmark real data and show that AdaGPR provides improved accuracies compared to existing graph convolution networks while demonstrating robustness against oversmoothing. Further, we demonstrate that analysis of coefficients of layer-wise generalized Pageranks allows us to qualitatively understand convolution at each layer enabling model interpretations.

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arXiv2022

Excess Risk of Two-Layer ReLU Neural Networks in Teacher-Student Settings and its Superiority to Kernel Methods

While deep learning has outperformed other methods for various tasks, theoretical frameworks that explain its reason have not been fully established. To address this issue, we investigate the excess risk of two-layer ReLU neural networks in a teacher-student regression model, in which a student network learns an unknown teacher network through its outputs. Especially, we consider the student network that has the same width as the teacher network and is trained in two phases: first by noisy gradient descent and then by the vanilla gradient descent. Our result shows that the student network provably reaches a near-global optimal solution and outperforms any kernel methods estimator (more generally, linear estimators), including neural tangent kernel approach, random feature model, and other kernel methods, in a sense of the minimax optimal rate. The key concept inducing this superiority is the non-convexity of the neural network models. Even though the loss landscape is highly non-convex, the student network adaptively learns the teacher neurons.

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arXiv2022

Exponential Convergence Rates of Classification Errors on Learning with SGD and Random Features

Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive efficient large-scale learning algorithms. In this study, we consider solving a binary classification problem using random features and stochastic gradient descent. In recent research, an exponential convergence rate of the expected classification error under the strong low-noise condition has been shown. We extend these analyses to a random features setting, analyzing the error induced by the approximation of random features in terms of the distance between the generated hypothesis including population risk minimizers and empirical risk minimizers when using general Lipschitz loss functions, to show that an exponential convergence of the expected classification error is achieved even if random features approximation is applied. Additionally, we demonstrate that the convergence rate does not depend on the number of features and there is a significant computational benefit in using random features in classification problems because of the strong low-noise condition.

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arXiv2022

High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation

We study the first gradient descent step on the first-layer parameters $\boldsymbol{W}$ in a two-layer neural network: $f(\boldsymbol{x}) = \frac{1}{\sqrt{N}}\boldsymbol{a}^\topσ(\boldsymbol{W}^\top\boldsymbol{x})$, where $\boldsymbol{W}\in\mathbb{R}^{d\times N}, \boldsymbol{a}\in\mathbb{R}^{N}$ are randomly initialized, and the training objective is the empirical MSE loss: $\frac{1}{n}\sum_{i=1}^n (f(\boldsymbol{x}_i)-y_i)^2$. In the proportional asymptotic limit where $n,d,N\to\infty$ at the same rate, and an idealized student-teacher setting, we show that the first gradient update contains a rank-1 "spike", which results in an alignment between the first-layer weights and the linear component of the teacher model $f^*$. To characterize the impact of this alignment, we compute the prediction risk of ridge regression on the conjugate kernel after one gradient step on $\boldsymbol{W}$ with learning rate $η$, when $f^*$ is a single-index model. We consider two scalings of the first step learning rate $η$. For small $η$, we establish a Gaussian equivalence property for the trained feature map, and prove that the learned kernel improves upon the initial random features model, but cannot defeat the best linear model on the input. Whereas for sufficiently large $η$, we prove that for certain $f^*$, the same ridge estimator on trained features can go beyond this "linear regime" and outperform a wide range of random features and rotationally invariant kernels. Our results demonstrate that even one gradient step can lead to a considerable advantage over random features, and highlight the role of learning rate scaling in the initial phase of training.

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arXiv2022

Convex Analysis of the Mean Field Langevin Dynamics

As an example of the nonlinear Fokker-Planck equation, the mean field Langevin dynamics recently attracts attention due to its connection to (noisy) gradient descent on infinitely wide neural networks in the mean field regime, and hence the convergence property of the dynamics is of great theoretical interest. In this work, we give a concise and self-contained convergence rate analysis of the mean field Langevin dynamics with respect to the (regularized) objective function in both continuous and discrete time settings. The key ingredient of our proof is a proximal Gibbs distribution $p_q$ associated with the dynamics, which, in combination with techniques in [Vempala and Wibisono (2019)], allows us to develop a simple convergence theory parallel to classical results in convex optimization. Furthermore, we reveal that $p_q$ connects to the duality gap in the empirical risk minimization setting, which enables efficient empirical evaluation of the algorithm convergence.

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arXiv2022

Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis

We propose the particle dual averaging (PDA) method, which generalizes the dual averaging method in convex optimization to the optimization over probability distributions with quantitative runtime guarantee. The algorithm consists of an inner loop and outer loop: the inner loop utilizes the Langevin algorithm to approximately solve for a stationary distribution, which is then optimized in the outer loop. The method can thus be interpreted as an extension of the Langevin algorithm to naturally handle nonlinear functional on the probability space. An important application of the proposed method is the optimization of neural network in the mean field regime, which is theoretically attractive due to the presence of nonlinear feature learning, but quantitative convergence rate can be challenging to obtain. By adapting finite-dimensional convex optimization theory into the space of measures, we analyze PDA in regularized empirical / expected risk minimization, and establish quantitative global convergence in learning two-layer mean field neural networks under more general settings. Our theoretical results are supported by numerical simulations on neural networks with reasonable size.

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arXiv2021

Neural Network Module Decomposition and Recomposition

We propose a modularization method that decomposes a deep neural network (DNN) into small modules from a functionality perspective and recomposes them into a new model for some other task. Decomposed modules are expected to have the advantages of interpretability and verifiability due to their small size. In contrast to existing studies based on reusing models that involve retraining, such as a transfer learning model, the proposed method does not require retraining and has wide applicability as it can be easily combined with existing functional modules. The proposed method extracts modules using weight masks and can be applied to arbitrary DNNs. Unlike existing studies, it requires no assumption about the network architecture. To extract modules, we designed a learning method and a loss function to maximize shared weights among modules. As a result, the extracted modules can be recomposed without a large increase in the size. We demonstrate that the proposed method can decompose and recompose DNNs with high compression ratio and high accuracy and is superior to the existing method through sharing weights between modules.

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arXiv2021

A Scaling Law for Synthetic-to-Real Transfer: How Much Is Your Pre-training Effective?

Synthetic-to-real transfer learning is a framework in which a synthetically generated dataset is used to pre-train a model to improve its performance on real vision tasks. The most significant advantage of using synthetic images is that the ground-truth labels are automatically available, enabling unlimited expansion of the data size without human cost. However, synthetic data may have a huge domain gap, in which case increasing the data size does not improve the performance. How can we know that? In this study, we derive a simple scaling law that predicts the performance from the amount of pre-training data. By estimating the parameters of the law, we can judge whether we should increase the data or change the setting of image synthesis. Further, we analyze the theory of transfer learning by considering learning dynamics and confirm that the derived generalization bound is consistent with our empirical findings. We empirically validated our scaling law on various experimental settings of benchmark tasks, model sizes, and complexities of synthetic images.

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arXiv2021

Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse of dimensionality can be avoided if the smoothness of the target function is highly anisotropic. Unlike existing studies, our analysis does not require a low-dimensional structure of the input data. We also investigate the minimax optimality of deep learning and compare its performance with that of the kernel method (more generally, linear estimators). The results show that deep learning has better dependence on the input dimensionality if the target function possesses anisotropic smoothness, and it achieves an adaptive rate for functions with spatially inhomogeneous smoothness.

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arXiv2021

Decomposable-Net: Scalable Low-Rank Compression for Neural Networks

Compressing DNNs is important for the real-world applications operating on resource-constrained devices. However, we typically observe drastic performance deterioration when changing model size after training is completed. Therefore, retraining is required to resume the performance of the compressed models suitable for different devices. In this paper, we propose Decomposable-Net (the network decomposable in any size), which allows flexible changes to model size without retraining. We decompose weight matrices in the DNNs via singular value decomposition and adjust ranks according to the target model size. Unlike the existing low-rank compression methods that specialize the model to a fixed size, we propose a novel backpropagation scheme that jointly minimizes losses for both of full- and low-rank networks. This enables not only to maintain the performance of a full-rank network {\it without retraining} but also to improve low-rank networks in multiple sizes. Additionally, we introduce a simple criterion for rank selection that effectively suppresses approximation error. In experiments on the ImageNet classification task, Decomposable-Net yields superior accuracy in a wide range of model sizes. In particular, Decomposable-Net achieves the top-1 accuracy of $73.2\%$ with $0.27\times$MACs with ResNet-50, compared to Tucker decomposition ($67.4\% / 0.30\times$), Trained Rank Pruning ($70.6\% / 0.28\times$), and universally slimmable networks ($71.4\% / 0.26\times$).

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arXiv2021

A Goodness-of-fit Test on the Number of Biclusters in a Relational Data Matrix

Biclustering is a method for detecting homogeneous submatrices in a given observed matrix, and it is an effective tool for relational data analysis. Although there are many studies that estimate the underlying bicluster structure of a matrix, few have enabled us to determine the appropriate number of biclusters in an observed matrix. Recently, a statistical test on the number of biclusters has been proposed for a regular-grid bicluster structure, where we assume that the latent bicluster structure can be represented by row-column clustering. However, when the latent bicluster structure does not satisfy such regular-grid assumption, the previous test requires a larger number of biclusters than necessary (i.e., a finer bicluster structure than necessary) for the null hypothesis to be accepted, which is not desirable in terms of interpreting the accepted bicluster structure. In this study, we propose a new statistical test on the number of biclusters that does not require the regular-grid assumption and derive the asymptotic behavior of the proposed test statistic in both null and alternative cases. We illustrate the effectiveness of the proposed method by applying it to both synthetic and practical relational data matrices.

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arXiv2021

On Learnability via Gradient Method for Two-Layer ReLU Neural Networks in Teacher-Student Setting

Deep learning empirically achieves high performance in many applications, but its training dynamics has not been fully understood theoretically. In this paper, we explore theoretical analysis on training two-layer ReLU neural networks in a teacher-student regression model, in which a student network learns an unknown teacher network through its outputs. We show that with a specific regularization and sufficient over-parameterization, the student network can identify the parameters of the teacher network with high probability via gradient descent with a norm dependent stepsize even though the objective function is highly non-convex. The key theoretical tool is the measure representation of the neural networks and a novel application of a dual certificate argument for sparse estimation on a measure space. We analyze the global minima and global convergence property in the measure space.

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arXiv2021

Bias-Variance Reduced Local SGD for Less Heterogeneous Federated Learning

Recently, local SGD has got much attention and been extensively studied in the distributed learning community to overcome the communication bottleneck problem. However, the superiority of local SGD to minibatch SGD only holds in quite limited situations. In this paper, we study a new local algorithm called Bias-Variance Reduced Local SGD (BVR-L-SGD) for nonconvex distributed optimization. Algorithmically, our proposed bias and variance reduced local gradient estimator fully utilizes small second-order heterogeneity of local objectives and suggests randomly picking up one of the local models instead of taking the average of them when workers are synchronized. Theoretically, under small heterogeneity of local objectives, we show that BVR-L-SGD achieves better communication complexity than both the previous non-local and local methods under mild conditions, and particularly BVR-L-SGD is the first method that breaks the barrier of communication complexity $Θ(1/\varepsilon)$ for general nonconvex smooth objectives when the heterogeneity is small and the local computation budget is large. Numerical results are given to verify the theoretical findings and give empirical evidence of the superiority of our method.

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arXiv2021

Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regime

We analyze the convergence of the averaged stochastic gradient descent for overparameterized two-layer neural networks for regression problems. It was recently found that a neural tangent kernel (NTK) plays an important role in showing the global convergence of gradient-based methods under the NTK regime, where the learning dynamics for overparameterized neural networks can be almost characterized by that for the associated reproducing kernel Hilbert space (RKHS). However, there is still room for a convergence rate analysis in the NTK regime. In this study, we show that the averaged stochastic gradient descent can achieve the minimax optimal convergence rate, with the global convergence guarantee, by exploiting the complexities of the target function and the RKHS associated with the NTK. Moreover, we show that the target function specified by the NTK of a ReLU network can be learned at the optimal convergence rate through a smooth approximation of a ReLU network under certain conditions.

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arXiv2021

Selective Inference for Latent Block Models

Model selection in latent block models has been a challenging but important task in the field of statistics. Specifically, a major challenge is encountered when constructing a test on a block structure obtained by applying a specific clustering algorithm to a finite size matrix. In this case, it becomes crucial to consider the selective bias in the block structure, that is, the block structure is selected from all the possible cluster memberships based on some criterion by the clustering algorithm. To cope with this problem, this study provides a selective inference method for latent block models. Specifically, we construct a statistical test on a set of row and column cluster memberships of a latent block model, which is given by a squared residue minimization algorithm. The proposed test, by its nature, includes and thus can also be used as the test on the set of row and column cluster numbers. We also propose an approximated version of the test based on simulated annealing to avoid combinatorial explosion in searching the optimal block structure. The results show that the proposed exact and approximated tests work effectively, compared to the naive test that did not take the selective bias into account.

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arXiv2021

Gradient Descent in RKHS with Importance Labeling

Labeling cost is often expensive and is a fundamental limitation of supervised learning. In this paper, we study importance labeling problem, in which we are given many unlabeled data and select a limited number of data to be labeled from the unlabeled data, and then a learning algorithm is executed on the selected one. We propose a new importance labeling scheme that can effectively select an informative subset of unlabeled data in least squares regression in Reproducing Kernel Hilbert Spaces (RKHS). We analyze the generalization error of gradient descent combined with our labeling scheme and show that the proposed algorithm achieves the optimal rate of convergence in much wider settings and especially gives much better generalization ability in a small label noise setting than the usual uniform sampling scheme. Numerical experiments verify our theoretical findings.

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arXiv2021

Estimation error analysis of deep learning on the regression problem on the variable exponent Besov space

Deep learning has achieved notable success in various fields, including image and speech recognition. One of the factors in the successful performance of deep learning is its high feature extraction ability. In this study, we focus on the adaptivity of deep learning; consequently, we treat the variable exponent Besov space, which has a different smoothness depending on the input location $x$. In other words, the difficulty of the estimation is not uniform within the domain. We analyze the general approximation error of the variable exponent Besov space and the approximation and estimation errors of deep learning. We note that the improvement based on adaptivity is remarkable when the region upon which the target function has less smoothness is small and the dimension is large. Moreover, the superiority to linear estimators is shown with respect to the convergence rate of the estimation error.

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