Tomohiro Hayase

Ryo Karakida, Tomoumi Takase, Tomohiro Hayase et al.

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.

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.

Tomohiro Hayase, Ryo Karakida

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.

Tomohiro Hayase, Benoît Collins, Ryo Karakida

Self-attention layers have become fundamental building blocks of modern deep neural networks, yet their theoretical understanding remains limited, particularly from the perspective of random matrix theory. In this work, we provide a rigorous analysis of the singular value spectrum of the attention matrix and establish the first Gaussian equivalence result for attention. In a natural regime where the inverse temperature remains of constant order, we show that the singular value distribution of the attention matrix is asymptotically characterized by a tractable linear model. We further demonstrate that the distribution of squared singular values deviates from the Marchenko-Pastur law, which has been believed in previous work. Our proof relies on two key ingredients: precise control of fluctuations in the normalization term and a refined linearization that leverages favorable Taylor expansions of the exponential. This analysis also identifies a threshold for linearization and elucidates why attention, despite not being an entrywise operation, admits a rigorous Gaussian equivalence in this regime.

Tomohiro Hayase, Benoît Collins, Nakamasa Inoue

Hierarchical inductive biases are hypothesized to promote generalizable policies in reinforcement learning, as demonstrated by explicit hyperbolic latent representations and architectures. Therefore, a more flexible approach is to have these biases emerge naturally from the algorithm. We introduce Free Random Projection, an input mapping grounded in free probability theory that constructs random orthogonal matrices where hierarchical structure arises inherently. The free random projection integrates seamlessly into existing in-context reinforcement learning frameworks by encoding hierarchical organization within the input space without requiring explicit architectural modifications. Empirical results on multi-environment benchmarks show that free random projection consistently outperforms the standard random projection, leading to improvements in generalization. Furthermore, analyses within linearly solvable Markov decision processes and investigations of the spectrum of kernel random matrices reveal the theoretical underpinnings of free random projection's enhanced performance, highlighting its capacity for effective adaptation in hierarchically structured state spaces.

Benoit Collins, Tomohiro Hayase

Free Probability Theory (FPT) provides rich knowledge for handling mathematical difficulties caused by random matrices that appear in research related to deep neural networks (DNNs), such as the dynamical isometry, Fisher information matrix, and training dynamics. FPT suits these researches because the DNN's parameter-Jacobian and input-Jacobian are polynomials of layerwise Jacobians. However, the critical assumption of asymptotic freenss of the layerwise Jacobian has not been proven completely so far. The asymptotic freeness assumption plays a fundamental role when propagating spectral distributions through the layers. Haar distributed orthogonal matrices are essential for achieving dynamical isometry. In this work, we prove asymptotic freeness of layerwise Jacobians of multilayer perceptron (MLP) in this case. A key of the proof is an invariance of the MLP. Considering the orthogonal matrices that fix the hidden units in each layer, we replace each layer's parameter matrix with itself multiplied by the orthogonal matrix, and then the MLP does not change. Furthermore, if the original weights are Haar orthogonal, the Jacobian is also unchanged by this replacement. Lastly, we can replace each weight with a Haar orthogonal random matrix independent of the Jacobian of the activation function using this key fact.

Shohei Kubota, Hideaki Hayashi, Tomohiro Hayase et al.

The interpretability of neural networks (NNs) is a challenging but essential topic for transparency in the decision-making process using machine learning. One of the reasons for the lack of interpretability is random weight initialization, where the input is randomly embedded into a different feature space in each layer. In this paper, we propose an interpretation method for a deep multilayer perceptron, which is the most general architecture of NNs, based on identity initialization (namely, initialization using identity matrices). The proposed method allows us to analyze the contribution of each neuron to classification and class likelihood in each hidden layer. As a property of the identity-initialized perceptron, the weight matrices remain near the identity matrices even after learning. This property enables us to treat the change of features from the input to each hidden layer as the contribution to classification. Furthermore, we can separate the output of each hidden layer into a contribution map that depicts the contribution to classification and class likelihood, by adding extra dimensions to each layer according to the number of classes, thereby allowing the calculation of the recognition accuracy in each layer and thus revealing the roles of independent layers, such as feature extraction and classification.

Tomohiro Hayase, Suguru Yasutomi, Takashi Katoh

Selective forgetting or removing information from deep neural networks (DNNs) is essential for continual learning and is challenging in controlling the DNNs. Such forgetting is crucial also in a practical sense since the deployed DNNs may be trained on the data with outliers, poisoned by attackers, or with leaked/sensitive information. In this paper, we formulate selective forgetting for classification tasks at a finer level than the samples' level. We specify the finer level based on four datasets distinguished by two conditions: whether they contain information to be forgotten and whether they are available for the forgetting procedure. Additionally, we reveal the need for such formulation with the datasets by showing concrete and practical situations. Moreover, we introduce the forgetting procedure as an optimization problem on three criteria; the forgetting, the correction, and the remembering term. Experimental results show that the proposed methods can make the model forget to use specific information for classification. Notably, in specific cases, our methods improved the model's accuracy on the datasets, which contains information to be forgotten but is unavailable in the forgetting procedure. Such data are unexpectedly found and misclassified in actual situations.

Tomohiro Hayase, Ryo Karakida

The Fisher information matrix (FIM) is fundamental to understanding the trainability of deep neural nets (DNN), since it describes the parameter space's local metric. We investigate the spectral distribution of the conditional FIM, which is the FIM given a single sample, by focusing on fully-connected networks achieving dynamical isometry. Then, while dynamical isometry is known to keep specific backpropagated signals independent of the depth, we find that the parameter space's local metric linearly depends on the depth even under the dynamical isometry. More precisely, we reveal that the conditional FIM's spectrum concentrates around the maximum and the value grows linearly as the depth increases. To examine the spectrum, considering random initialization and the wide limit, we construct an algebraic methodology based on the free probability theory. As a byproduct, we provide an analysis of the solvable spectral distribution in two-hidden-layer cases. Lastly, experimental results verify that the appropriate learning rate for the online training of DNNs is in inverse proportional to depth, which is determined by the conditional FIM's spectrum.

Tomohiro Hayase

A well-conditioned Jacobian spectrum has a vital role in preventing exploding or vanishing gradients and speeding up learning of deep neural networks. Free probability theory helps us to understand and handle the Jacobian spectrum. We rigorously show almost sure asymptotic freeness of layer-wise Jacobians of deep neural networks as the wide limit. In particular, we treat the case that weights are initialized as Haar distributed orthogonal matrices.

Tomohiro Hayase

For random matrix models, the parameter estimation based on the traditional likelihood functions is not straightforward in particular when we have only one sample matrix. We introduce a new parameter optimization method for random matrix models which works even in such a case. The method is based on the spectral distribution instead of the traditional likelihood. In the method, the Cauchy noise has an essential role because the free deterministic equivalent, which is a tool in free probability theory, allows us to approximate the spectral distribution perturbed by Cauchy noises by a smooth and accessible density function. Moreover, we study an asymptotic property of determination gap, which has a similar role as generalization gap. Besides, we propose a new dimensionality recovery method for the signal-plus-noise model, and experimentally demonstrate that it recovers the rank of the signal part even if the true rank is not small. It is a simultaneous rank selection and parameter estimation procedure.