arXiv subjects

Taiji Suzuki

Publications and source records attributed to Taiji Suzuki.

At least 127 records · Page 7Linked to original sources

arXiv2021

MSR-DARTS: Minimum Stable Rank of Differentiable Architecture Search

In neural architecture search (NAS), differentiable architecture search (DARTS) has recently attracted much attention due to its high efficiency. It defines an over-parameterized network with mixed edges, each of which represents all operator candidates, and jointly optimizes the weights of the network and its architecture in an alternating manner. However, this method finds a model with the weights converging faster than the others, and such a model with fastest convergence often leads to overfitting. Accordingly, the resulting model cannot always be well-generalized. To overcome this problem, we propose a method called minimum stable rank DARTS (MSR-DARTS), for finding a model with the best generalization error by replacing architecture optimization with the selection process using the minimum stable rank criterion. Specifically, a convolution operator is represented by a matrix, and MSR-DARTS selects the one with the smallest stable rank. We evaluated MSR-DARTS on CIFAR-10 and ImageNet datasets. It achieves an error rate of 2.54% with 4.0M parameters within 0.3 GPU-days on CIFAR-10, and a top-1 error rate of 23.9% on ImageNet. The official code is available at https://github.com/mtaecchhi/msrdarts.git.

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arXiv2021

Graph Neural Networks Exponentially Lose Expressive Power for Node Classification

Graph Neural Networks (graph NNs) are a promising deep learning approach for analyzing graph-structured data. However, it is known that they do not improve (or sometimes worsen) their predictive performance as we pile up many layers and add non-lineality. To tackle this problem, we investigate the expressive power of graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our strategy is to generalize the forward propagation of a Graph Convolutional Network (GCN), which is a popular graph NN variant, as a specific dynamical system. In the case of a GCN, we show that when its weights satisfy the conditions determined by the spectra of the (augmented) normalized Laplacian, its output exponentially approaches the set of signals that carry information of the connected components and node degrees only for distinguishing nodes. Our theory enables us to relate the expressive power of GCNs with the topological information of the underlying graphs inherent in the graph spectra. To demonstrate this, we characterize the asymptotic behavior of GCNs on the Erdős -- Rényi graph. We show that when the Erdős -- Rényi graph is sufficiently dense and large, a broad range of GCNs on it suffers from the "information loss" in the limit of infinite layers with high probability. Based on the theory, we provide a principled guideline for weight normalization of graph NNs. We experimentally confirm that the proposed weight scaling enhances the predictive performance of GCNs in real data. Code is available at https://github.com/delta2323/gnn-asymptotics.

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arXiv2021

Optimization and Generalization Analysis of Transduction through Gradient Boosting and Application to Multi-scale Graph Neural Networks

It is known that the current graph neural networks (GNNs) are difficult to make themselves deep due to the problem known as over-smoothing. Multi-scale GNNs are a promising approach for mitigating the over-smoothing problem. However, there is little explanation of why it works empirically from the viewpoint of learning theory. In this study, we derive the optimization and generalization guarantees of transductive learning algorithms that include multi-scale GNNs. Using the boosting theory, we prove the convergence of the training error under weak learning-type conditions. By combining it with generalization gap bounds in terms of transductive Rademacher complexity, we show that a test error bound of a specific type of multi-scale GNNs that decreases corresponding to the number of node aggregations under some conditions. Our results offer theoretical explanations for the effectiveness of the multi-scale structure against the over-smoothing problem. We apply boosting algorithms to the training of multi-scale GNNs for real-world node prediction tasks. We confirm that its performance is comparable to existing GNNs, and the practical behaviors are consistent with theoretical observations. Code is available at https://github.com/delta2323/GB-GNN.

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arXiv2020

When Does Preconditioning Help or Hurt Generalization?

While second order optimizers such as natural gradient descent (NGD) often speed up optimization, their effect on generalization has been called into question. This work presents a more nuanced view on how the \textit{implicit bias} of first- and second-order methods affects the comparison of generalization properties. We provide an exact asymptotic bias-variance decomposition of the generalization error of overparameterized ridgeless regression under a general class of preconditioner $\boldsymbol{P}$, and consider the inverse population Fisher information matrix (used in NGD) as a particular example. We determine the optimal $\boldsymbol{P}$ for both the bias and variance, and find that the relative generalization performance of different optimizers depends on the label noise and the "shape" of the signal (true parameters): when the labels are noisy, the model is misspecified, or the signal is misaligned with the features, NGD can achieve lower risk; conversely, GD generalizes better than NGD under clean labels, a well-specified model, or aligned signal. Based on this analysis, we discuss several approaches to manage the bias-variance tradeoff, and the potential benefit of interpolating between GD and NGD. We then extend our analysis to regression in the reproducing kernel Hilbert space and demonstrate that preconditioned GD can decrease the population risk faster than GD. Lastly, we empirically compare the generalization error of first- and second-order optimizers in neural network experiments, and observe robust trends matching our theoretical analysis.

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arXiv2020

Benefit of deep learning with non-convex noisy gradient descent: Provable excess risk bound and superiority to kernel methods

Establishing a theoretical analysis that explains why deep learning can outperform shallow learning such as kernel methods is one of the biggest issues in the deep learning literature. Towards answering this question, we evaluate excess risk of a deep learning estimator trained by a noisy gradient descent with ridge regularization on a mildly overparameterized neural network, and discuss its superiority to a class of linear estimators that includes neural tangent kernel approach, random feature model, other kernel methods, $k$-NN estimator and so on. We consider a teacher-student regression model, and eventually show that any linear estimator can be outperformed by deep learning in a sense of the minimax optimal rate especially for a high dimension setting. The obtained excess bounds are so-called fast learning rate which is faster than $O(1/\sqrt{n})$ that is obtained by usual Rademacher complexity analysis. This discrepancy is induced by the non-convex geometry of the model and the noisy gradient descent used for neural network training provably reaches a near global optimal solution even though the loss landscape is highly non-convex. Although the noisy gradient descent does not employ any explicit or implicit sparsity inducing regularization, it shows a preferable generalization performance that dominates linear estimators.

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arXiv2020

Generalization bound of globally optimal non-convex neural network training: Transportation map estimation by infinite dimensional Langevin dynamics

We introduce a new theoretical framework to analyze deep learning optimization with connection to its generalization error. Existing frameworks such as mean field theory and neural tangent kernel theory for neural network optimization analysis typically require taking limit of infinite width of the network to show its global convergence. This potentially makes it difficult to directly deal with finite width network; especially in the neural tangent kernel regime, we cannot reveal favorable properties of neural networks beyond kernel methods. To realize more natural analysis, we consider a completely different approach in which we formulate the parameter training as a transportation map estimation and show its global convergence via the theory of the infinite dimensional Langevin dynamics. This enables us to analyze narrow and wide networks in a unifying manner. Moreover, we give generalization gap and excess risk bounds for the solution obtained by the dynamics. The excess risk bound achieves the so-called fast learning rate. In particular, we show an exponential convergence for a classification problem and a minimax optimal rate for a regression problem.

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arXiv2020

Goodness-of-fit Test for Latent Block Models

Latent block models are used for probabilistic biclustering, which is shown to be an effective method for analyzing various relational data sets. However, there has been no statistical test method for determining the row and column cluster numbers of latent block models. Recent studies have constructed statistical-test-based methods for stochastic block models, which assume that the observed matrix is a square symmetric matrix and that the cluster assignments are the same for rows and columns. In this study, we developed a new goodness-of-fit test for latent block models to test whether an observed data matrix fits a given set of row and column cluster numbers, or it consists of more clusters in at least one direction of the row and the column. To construct the test method, we used a result from the random matrix theory for a sample covariance matrix. We experimentally demonstrated the effectiveness of the proposed method by showing the asymptotic behavior of the test statistic and measuring the test accuracy.

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arXiv2020

Domain Adaptation Regularization for Spectral Pruning

Deep Neural Networks (DNNs) have recently been achieving state-of-the-art performance on a variety of computer vision related tasks. However, their computational cost limits their ability to be implemented in embedded systems with restricted resources or strict latency constraints. Model compression has therefore been an active field of research to overcome this issue. Additionally, DNNs typically require massive amounts of labeled data to be trained. This represents a second limitation to their deployment. Domain Adaptation (DA) addresses this issue by allowing knowledge learned on one labeled source distribution to be transferred to a target distribution, possibly unlabeled. In this paper, we investigate on possible improvements of compression methods in DA setting. We focus on a compression method that was previously developed in the context of a single data distribution and show that, with a careful choice of data to use during compression and additional regularization terms directly related to DA objectives, it is possible to improve compression results. We also show that our method outperforms an existing compression method studied in the DA setting by a large margin for high compression rates. Although our work is based on one specific compression method, we also outline some general guidelines for improving compression in DA setting.

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arXiv2020

Spectral Pruning: Compressing Deep Neural Networks via Spectral Analysis and its Generalization Error

Compression techniques for deep neural network models are becoming very important for the efficient execution of high-performance deep learning systems on edge-computing devices. The concept of model compression is also important for analyzing the generalization error of deep learning, known as the compression-based error bound. However, there is still huge gap between a practically effective compression method and its rigorous background of statistical learning theory. To resolve this issue, we develop a new theoretical framework for model compression and propose a new pruning method called {\it spectral pruning} based on this framework. We define the ``degrees of freedom'' to quantify the intrinsic dimensionality of a model by using the eigenvalue distribution of the covariance matrix across the internal nodes and show that the compression ability is essentially controlled by this quantity. Moreover, we present a sharp generalization error bound of the compressed model and characterize the bias--variance tradeoff induced by the compression procedure. We apply our method to several datasets to justify our theoretical analyses and show the superiority of the the proposed method.

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arXiv2020

Compression based bound for non-compressed network: unified generalization error analysis of large compressible deep neural network

One of the biggest issues in deep learning theory is the generalization ability of networks with huge model size. The classical learning theory suggests that overparameterized models cause overfitting. However, practically used large deep models avoid overfitting, which is not well explained by the classical approaches. To resolve this issue, several attempts have been made. Among them, the compression based bound is one of the promising approaches. However, the compression based bound can be applied only to a compressed network, and it is not applicable to the non-compressed original network. In this paper, we give a unified frame-work that can convert compression based bounds to those for non-compressed original networks. The bound gives even better rate than the one for the compressed network by improving the bias term. By establishing the unified frame-work, we can obtain a data dependent generalization error bound which gives a tighter evaluation than the data independent ones.

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arXiv2020

Accelerated Sparsified SGD with Error Feedback

A stochastic gradient method for synchronous distributed optimization is studied. For reducing communication cost, we particularly focus on utilization of compression of communicated gradients. Several work has shown that {\it{sparsified}} stochastic gradient descent method (SGD) with {\it{error feedback}} asymptotically achieves the same rate as (non-sparsified) parallel SGD. However, from a viewpoint of non-asymptotic behavior, the compression error may cause slower convergence than non-sparsified SGD in early iterations. This is problematic in practical situations since early stopping is often adopted to maximize the generalization ability of learned models. For improving the previous results, we propose and theoretically analyse a sparsified stochastic gradient method with error feedback scheme combined with {\it{Nesterov's acceleration}}. It is shown that the necessary per iteration communication cost for maintaining the same rate as vanilla SGD can be smaller than non-accelerated methods in convex and even in nonconvex optimization problems. This indicates that our proposed method makes a better use of compressed information than previous methods. Numerical experiments are provided and empirically validates our theoretical findings.

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arXiv2020

Understanding Generalization in Deep Learning via Tensor Methods

Deep neural networks generalize well on unseen data though the number of parameters often far exceeds the number of training examples. Recently proposed complexity measures have provided insights to understanding the generalizability in neural networks from perspectives of PAC-Bayes, robustness, overparametrization, compression and so on. In this work, we advance the understanding of the relations between the network's architecture and its generalizability from the compression perspective. Using tensor analysis, we propose a series of intuitive, data-dependent and easily-measurable properties that tightly characterize the compressibility and generalizability of neural networks; thus, in practice, our generalization bound outperforms the previous compression-based ones, especially for neural networks using tensors as their weight kernels (e.g. CNNs). Moreover, these intuitive measurements provide further insights into designing neural network architectures with properties favorable for better/guaranteed generalizability. Our experimental results demonstrate that through the proposed measurable properties, our generalization error bound matches the trend of the test error well. Our theoretical analysis further provides justifications for the empirical success and limitations of some widely-used tensor-based compression approaches. We also discover the improvements to the compressibility and robustness of current neural networks when incorporating tensor operations via our proposed layer-wise structure.

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arXiv2020

Dimension-free convergence rates for gradient Langevin dynamics in RKHS

Gradient Langevin dynamics (GLD) and stochastic GLD (SGLD) have attracted considerable attention lately, as a way to provide convergence guarantees in a non-convex setting. However, the known rates grow exponentially with the dimension of the space. In this work, we provide a convergence analysis of GLD and SGLD when the optimization space is an infinite dimensional Hilbert space. More precisely, we derive non-asymptotic, dimension-free convergence rates for GLD/SGLD when performing regularized non-convex optimization in a reproducing kernel Hilbert space. Amongst others, the convergence analysis relies on the properties of a stochastic differential equation, its discrete time Galerkin approximation and the geometric ergodicity of the associated Markov chains.

math.PR↗

arXiv2020

Gradient Descent can Learn Less Over-parameterized Two-layer Neural Networks on Classification Problems

Recently, several studies have proven the global convergence and generalization abilities of the gradient descent method for two-layer ReLU networks. Most studies especially focused on the regression problems with the squared loss function, except for a few, and the importance of the positivity of the neural tangent kernel has been pointed out. On the other hand, the performance of gradient descent on classification problems using the logistic loss function has not been well studied, and further investigation of this problem structure is possible. In this work, we demonstrate that the separability assumption using a neural tangent model is more reasonable than the positivity condition of the neural tangent kernel and provide a refined convergence analysis of the gradient descent for two-layer networks with smooth activations. A remarkable point of our result is that our convergence and generalization bounds have much better dependence on the network width in comparison to related studies. Consequently, our theory provides a generalization guarantee for less over-parameterized two-layer networks, while most studies require much higher over-parameterization.

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arXiv2020

Meta Cyclical Annealing Schedule: A Simple Approach to Avoiding Meta-Amortization Error

The ability to learn new concepts with small amounts of data is a crucial aspect of intelligence that has proven challenging for deep learning methods. Meta-learning for few-shot learning offers a potential solution to this problem: by learning to learn across data from many previous tasks, few-shot learning algorithms can discover the structure among tasks to enable fast learning of new tasks. However, a critical challenge in few-shot learning is task ambiguity: even when a powerful prior can be meta-learned from a large number of prior tasks, a small dataset for a new task can simply be very ambiguous to acquire a single model for that task. The Bayesian meta-learning models can naturally resolve this problem by putting a sophisticated prior distribution and let the posterior well regularized through Bayesian decision theory. However, currently known Bayesian meta-learning procedures such as VERSA suffer from the so-called {\it information preference problem}, that is, the posterior distribution is degenerated to one point and is far from the exact one. To address this challenge, we design a novel meta-regularization objective using {\it cyclical annealing schedule} and {\it maximum mean discrepancy} (MMD) criterion. The cyclical annealing schedule is quite effective at avoiding such degenerate solutions. This procedure includes a difficult KL-divergence estimation, but we resolve the issue by employing MMD instead of KL-divergence. The experimental results show that our approach substantially outperforms standard meta-learning algorithms.

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arXiv2019

On the minimax optimality and superiority of deep neural network learning over sparse parameter spaces

Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we discuss the performance of deep learning and other methods on a nonparametric regression problem with a Gaussian noise. Whereas existing theoretical studies of deep learning have been based mainly on mathematical theories of well-known function classes such as Hölder and Besov classes, we focus on function classes with discontinuity and sparsity, which are those naturally assumed in practice. To highlight the effectiveness of deep learning, we compare deep learning with a class of linear estimators representative of a class of shallow estimators. It is shown that the minimax risk of a linear estimator on the convex hull of a target function class does not differ from that of the original target function class. This results in the suboptimality of linear methods over a simple but non-convex function class, on which deep learning can attain nearly the minimax-optimal rate. In addition to this extreme case, we consider function classes with sparse wavelet coefficients. On these function classes, deep learning also attains the minimax rate up to log factors of the sample size, and linear methods are still suboptimal if the assumed sparsity is strong. We also point out that the parameter sharing of deep neural networks can remarkably reduce the complexity of the model in our setting.

stat.ML↗

arXiv2019

Understanding the Effects of Pre-Training for Object Detectors via Eigenspectrum

ImageNet pre-training has been regarded as essential for training accurate object detectors for a long time. Recently, it has been shown that object detectors trained from randomly initialized weights can be on par with those fine-tuned from ImageNet pre-trained models. However, the effects of pre-training and the differences caused by pre-training are still not fully understood. In this paper, we analyze the eigenspectrum dynamics of the covariance matrix of each feature map in object detectors. Based on our analysis on ResNet-50, Faster R-CNN with FPN, and Mask R-CNN, we show that object detectors trained from ImageNet pre-trained models and those trained from scratch behave differently from each other even if both object detectors have similar accuracy. Furthermore, we propose a method for automatically determining the widths (the numbers of channels) of object detectors based on the eigenspectrum. We train Faster R-CNN with FPN from randomly initialized weights, and show that our method can reduce ~27% of the parameters of ResNet-50 without increasing Multiply-Accumulate operations and losing accuracy. Our results indicate that we should develop more appropriate methods for transferring knowledge from image classification to object detection (or other tasks).

cs.CV↗

arXiv2019

Gradient Noise Convolution (GNC): Smoothing Loss Function for Distributed Large-Batch SGD

Large-batch stochastic gradient descent (SGD) is widely used for training in distributed deep learning because of its training-time efficiency, however, extremely large-batch SGD leads to poor generalization and easily converges to sharp minima, which prevents naive large-scale data-parallel SGD (DP-SGD) from converging to good minima. To overcome this difficulty, we propose gradient noise convolution (GNC), which effectively smooths sharper minima of the loss function. For DP-SGD, GNC utilizes so-called gradient noise, which is induced by stochastic gradient variation and convolved to the loss function as a smoothing effect. GNC computation can be performed by simply computing the stochastic gradient on each parallel worker and merging them, and is therefore extremely easy to implement. Due to convolving with the gradient noise, which tends to spread along a sharper direction of the loss function, GNC can effectively smooth sharp minima and achieve better generalization, whereas isotropic random noise cannot. We empirically show this effect by comparing GNC with isotropic random noise, and show that it achieves state-of-the-art generalization performance for large-scale deep neural network optimization.

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