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Ryo Karakida

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10 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

Infinite-Width Limit of a Single Attention Layer: Analysis via Tensor Programs

Mana Sakai
Ryo Karakida
Masaaki Imaizumi

In modern theoretical analyses of neural networks, the infinite-width limit is often invoked to justify Gaussian approximations of neuron preactivations (e. g. , via neural network Gaussian processes or Tensor Programs). However, these Gaussian-based asymptotic theories have so far been unable to capture the behavior of attention layers, except under special regimes such as infinitely many heads or tailored scaling schemes. In this paper, leveraging the Tensor Programs framework, we rigorously identify the infinite-width limit distribution of variables within a single attention layer under realistic architectural dimensionality and standard $1/\sqrt{n}$-scaling with $n$ dimensionality. We derive the exact form of this limit law without resorting to infinite-head approximations or tailored scalings, demonstrating that it departs fundamentally from Gaussianity. This limiting distribution exhibits non-Gaussianity from a hierarchical structure, being Gaussian conditional on the random similarity scores. Numerical experiments validate our theoretical predictions, confirming the effectiveness of our theory at finite width and accurate description of finite-head attentions. Beyond characterizing a standalone attention layer, our findings lay the groundwork for developing a unified theory of deep Transformer architectures in the infinite-width regime.

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ICLR Conference 2025 Conference Paper

Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation

Satoki Ishikawa
Rio Yokota
Ryo Karakida

Local learning, which trains a network through layer-wise local targets and losses, has been studied as an alternative to backpropagation (BP) in neural computation. However, its algorithms often become more complex or require additional hyperparameters due to the locality, making it challenging to identify desirable settings where the algorithm progresses in a stable manner. To provide theoretical and quantitative insights, we introduce maximal update parameterization ($\mu$P) in the infinite-width limit for two representative designs of local targets: predictive coding (PC) and target propagation (TP). We verify that $\mu$P enables hyperparameter transfer across models of different widths. Furthermore, our analysis reveals unique and intriguing properties of $\mu$P that are not present in conventional BP. By analyzing deep linear networks, we find that PC's gradients interpolate between first-order and Gauss-Newton-like gradients, depending on the parameterization. We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient. For TP, even with the standard scaling of the last layer differing from classical $\mu$P, its local loss optimization favors the feature learning regime over the kernel regime.

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NeurIPS Conference 2025 Conference Paper

Recurrent Self-Attention Dynamics: An Energy-Agnostic Perspective from Jacobians

Akiyoshi Tomihari
Ryo Karakida

The theoretical understanding of self-attention (SA) has been steadily progressing. A prominent line of work studies a class of SA layers that admit an energy function decreased by state updates. While it provides valuable insights into inherent biases in signal propagation, it often relies on idealized assumptions or additional constraints not necessarily present in standard SA. Thus, to broaden our understanding, this work aims to relax these energy constraints and provide an energy-agnostic characterization of inference dynamics by dynamical systems analysis. In more detail, we first consider relaxing the symmetry and single-head constraints traditionally required in energy-based formulations. Next, we show that analyzing the Jacobian matrix of the state is highly valuable when investigating more general SA architectures without necessarily admitting an energy function. It reveals that the normalization layer plays an essential role in suppressing the Lipschitzness of SA and the Jacobian's complex eigenvalues, which correspond to the oscillatory components of the dynamics. In addition, the Lyapunov exponents computed from the Jacobians demonstrate that the normalized dynamics lie close to a critical state, and this criticality serves as a strong indicator of high inference performance. Furthermore, the Jacobian perspective also enables us to develop regularization methods for training and a pseudo-energy for monitoring inference dynamics.

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ICLR Conference 2024 Conference Paper

On the Parameterization of Second-Order Optimization Effective towards the Infinite Width

Satoki Ishikawa
Ryo Karakida

Second-order optimization has been developed to accelerate the training of deep neural networks and it is being applied to increasingly larger-scale models. In this study, towards training on further larger scales, we identify a specific parameterization for second-order optimization that promotes feature learning in a stable manner even if the network width increases significantly. Inspired by a maximal update parametrization, we consider a one-step update of the gradient and reveal the appropriate scales of hyperparameters including random initialization, learning rates, and damping terms. Our approach covers two major second-order optimization algorithms, K-FAC and Shampoo, and we demonstrate that our parametrization achieves higher generalization performance in feature learning. In particular, it enables us to transfer the hyperparameters across models with different widths.

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ICML Conference 2024 Conference Paper

Self-attention Networks Localize When QK-eigenspectrum Concentrates

Han Bao 0002
Ryuichiro Hataya
Ryo Karakida

The self-attention mechanism prevails in modern machine learning. It has an interesting functionality of adaptively selecting tokens from an input sequence by modulating the degree of attention localization, which many researchers speculate is the basis of the powerful model performance but complicates the underlying mechanism of the learning dynamics. In recent years, mainly two arguments have connected attention localization to the model performances. One is the rank collapse, where the embedded tokens by a self-attention block become very similar across different tokens, leading to a less expressive network. The other is the entropy collapse, where the attention probability approaches non-uniform and entails low entropy, making the learning dynamics more likely to be trapped in plateaus. These two failure modes may apparently contradict each other because the rank and entropy collapses are relevant to uniform and non-uniform attention, respectively. To this end, we characterize the notion of attention localization by the eigenspectrum of query-key parameter matrices and reveal that a small eigenspectrum variance leads attention to be localized. Interestingly, the small eigenspectrum variance prevents both rank and entropy collapse, leading to better model expressivity and trainability.

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ICML Conference 2024 Conference Paper

Understanding MLP-Mixer as a wide and sparse MLP

Tomohiro Hayase
Ryo Karakida

Multi-layer perceptron (MLP) is a fundamental component of deep learning, and recent MLP-based architectures, especially the MLP-Mixer, have achieved significant empirical success. Nevertheless, our understanding of why and how the MLP-Mixer outperforms conventional MLPs remains largely unexplored. In this work, we reveal that sparseness is a key mechanism underlying the MLP-Mixers. First, the Mixers have an effective expression as a wider MLP with Kronecker-product weights, clarifying that the Mixers efficiently embody several sparseness properties explored in deep learning. In the case of linear layers, the effective expression elucidates an implicit sparse regularization caused by the model architecture and a hidden relation to Monarch matrices, which is also known as another form of sparse parameterization. Next, for general cases, we empirically demonstrate quantitative similarities between the Mixer and the unstructured sparse-weight MLPs. Following a guiding principle proposed by Golubeva, Neyshabur and Gur-Ari (2021), which fixes the number of connections and increases the width and sparsity, the Mixers can demonstrate improved performance.

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ICML Conference 2023 Conference Paper

Understanding Gradient Regularization in Deep Learning: Efficient Finite-Difference Computation and Implicit Bias

Ryo Karakida
Tomoumi Takase
Tomohiro Hayase
Kazuki Osawa

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

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ICLR Conference 2022 Conference Paper

Learning curves for continual learning in neural networks: Self-knowledge transfer and forgetting

Ryo Karakida
Shotaro Akaho

Sequential training from task to task is becoming one of the major objects in deep learning applications such as continual learning and transfer learning. Nevertheless, it remains unclear under what conditions the trained model's performance improves or deteriorates. To deepen our understanding of sequential training, this study provides a theoretical analysis of generalization performance in a solvable case of continual learning. We consider neural networks in the neural tangent kernel (NTK) regime that continually learn target functions from task to task, and investigate the generalization by using an established statistical mechanical analysis of kernel ridge-less regression. We first show characteristic transitions from positive to negative transfer. More similar targets above a specific critical value can achieve positive knowledge transfer for the subsequent task while catastrophic forgetting occurs even with very similar targets. Next, we investigate a variant of continual learning which supposes the same target function in multiple tasks. Even for the same target, the trained model shows some transfer and forgetting depending on the sample size of each task. We can guarantee that the generalization error monotonically decreases from task to task for equal sample sizes while unbalanced sample sizes deteriorate the generalization. We respectively refer to these improvement and deterioration as self-knowledge transfer and forgetting, and empirically confirm them in realistic training of deep neural networks as well.

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NeurIPS Conference 2020 Conference Paper

Understanding Approximate Fisher Information for Fast Convergence of Natural Gradient Descent in Wide Neural Networks

Ryo Karakida
Kazuki Osawa

Natural Gradient Descent (NGD) helps to accelerate the convergence of gradient descent dynamics, but it requires approximations in large-scale deep neural networks because of its high computational cost. Empirical studies have confirmed that some NGD methods with approximate Fisher information converge sufficiently fast in practice. Nevertheless, it remains unclear from the theoretical perspective why and under what conditions such heuristic approximations work well. In this work, we reveal that, under specific conditions, NGD with approximate Fisher information achieves the same fast convergence to global minima as exact NGD. We consider deep neural networks in the infinite-width limit, and analyze the asymptotic training dynamics of NGD in function space via the neural tangent kernel. In the function space, the training dynamics with the approximate Fisher information are identical to those with the exact Fisher information, and they converge quickly. The fast convergence holds in layer-wise approximations; for instance, in block diagonal approximation where each block corresponds to a layer as well as in block tri-diagonal and K-FAC approximations. We also find that a unit-wise approximation achieves the same fast convergence under some assumptions. All of these different approximations have an isotropic gradient in the function space, and this plays a fundamental role in achieving the same convergence properties in training. Thus, the current study gives a novel and unified theoretical foundation with which to understand NGD methods in deep learning.

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NeurIPS Conference 2019 Conference Paper

The Normalization Method for Alleviating Pathological Sharpness in Wide Neural Networks

Ryo Karakida
Shotaro Akaho
Shun-ichi Amari

Normalization methods play an important role in enhancing the performance of deep learning while their theoretical understandings have been limited. To theoretically elucidate the effectiveness of normalization, we quantify the geometry of the parameter space determined by the Fisher information matrix (FIM), which also corresponds to the local shape of the loss landscape under certain conditions. We analyze deep neural networks with random initialization, which is known to suffer from a pathologically sharp shape of the landscape when the network becomes sufficiently wide. We reveal that batch normalization in the last layer contributes to drastically decreasing such pathological sharpness if the width and sample number satisfy a specific condition. In contrast, it is hard for batch normalization in the middle hidden layers to alleviate pathological sharpness in many settings. We also found that layer normalization cannot alleviate pathological sharpness either. Thus, we can conclude that batch normalization in the last layer significantly contributes to decreasing the sharpness induced by the FIM.

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