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Hideaki Iiduka

Publications and source records attributed to Hideaki Iiduka.

At least 19 recordsLinked to original sources
arXiv2026

Vanilla SGD with Momentum Survives Heavy-Tailed Noise: Convergence Analysis without Gradient Clipping or Normalization

Stochastic gradient descent (SGD) is a cornerstone of modern optimization. While its performance under heavy-tailed noise is often addressed through specialized modifications such as gradient clipping or normalization, we investigate a more fundamental question: how does vanilla SGD, particularly with momentum, perform in the presence of heavy-tailed noise? In this paper, we refine existing convergence results for vanilla SGD and, more importantly, provide the first comprehensive convergence analysis of vanilla SGD with momentum for strongly convex, convex, and nonconvex objectives, without employing any gradient control mechanisms. Our results demonstrate that the obtained convergence rates are inferior to the optimal rates achieved by clipped or normalized variants of SGD, thereby revealing inherent limitations of vanilla methods under heavy-tailed noise. The theoretical findings are supported by experiments on synthetic functions.

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arXiv2026

Accelerating SGDM via Learning Rate and Batch Size Schedules: A Lyapunov-Based Analysis

We analyze the convergence behavior of stochastic gradient descent with momentum (SGDM) under dynamic learning-rate and batch-size schedules by introducing a novel and simpler Lyapunov function. We extend the existing theoretical framework to cover three practical scheduling strategies commonly used in deep learning: a constant batch size with a decaying learning rate, an increasing batch size with a decaying learning rate, and an increasing batch size with an increasing learning rate. Our results reveal a clear hierarchy in convergence: a constant batch size does not guarantee convergence of the expected gradient norm under our Lyapunov-based analysis, whereas an increasing batch size does, and simultaneously increasing both the batch size and learning rate achieves a provably faster decay. Empirical results validate our theory, showing that dynamically scheduled SGDM significantly outperforms its fixed-hyperparameter counterpart in convergence speed. We also evaluated a warmup schedule in experiments, which empirically outperformed all other strategies in convergence behavior.

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arXiv2026

Mini-Batch Stochastic Halpern Algorithm for Nonexpansive Fixed Point Problems

The Halpern algorithm is a powerful fixed point approximation method for finding the closest point in the fixed point set of a nonexpansive mapping to the initial point. However, in practice, it is not necessarily true that this algorithm can be applied to large-scale fixed point problems, since the computation of the nonexpansive mapping is expensive. In this paper, we present mini-batch stochastic Halpern algorithm to resolve the issue caused by the computational difficulty of the mapping. We preform a convergence analysis demonstrating that the algorithm with diminishing step sizes and increasing batch sizes converges in mean square to the closest point in the fixed point set to the initial point. We also perform a convergence rate analysis demonstrating that convergence speed of the algorithm depends on the settings of the diminishing step sizes. Furthermore, numerical experiments demonstrated the effectiveness of the proposed mini-batch stochastic Halpern algorithm.

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arXiv2026

Convergence Bound and Critical Batch Size of Muon Optimizer

Muon, a recently proposed optimizer that leverages the inherent matrix structure of neural network parameters, has demonstrated strong empirical performance, indicating its potential as a successor to standard optimizers such as AdamW. This paper presents theoretical analysis to support its practical success. We provide convergence proofs for Muon across four practical settings, systematically examining its behavior with and without the inclusion of Nesterov momentum and weight decay. We then demonstrate that the addition of weight decay ensures almost-sure boundedness of the parameter and gradient norms -- without relying on the commonly imposed bounded-gradient assumption -- and clarify the interplay between the weight decay coefficient and the learning rate. Finally, we derive a lower bound on the critical batch size for Muon -- the batch size that minimizes the stochastic first-order oracle (SFO) complexity of training. Because the resulting formula involves problem-dependent quantities that are not directly observable (gradient variance, target precision, effective rank), it does not predict the critical batch size in absolute terms; rather, it reveals how the hyperparameters $β$ (momentum) and $λ$ (weight decay) govern the qualitative scaling of this value. Our experiments validate these hyperparameter-dependent predictions across workloads including image classification and language modeling.

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arXiv2026

Mini-Batch Stochastic Krasnosel'ski\uı-Mann Algorithm for Nonexpansive Fixed Point Problems

The Krasnosel'ski\uı-Mann algorithm is a well-known method for finding fixed points of a nonexpansive mapping with strong theoretical guarantees. However, there are practical large-scale problems to which this algorithm cannot be applied. Here, to resolve the issue caused by the computational difficulty of the mapping, we define a computable mini-batch stochastic mapping, which is a unbiased estimator of the nonexpansive mapping, and implement it in the Krasnosel'ski\uı-Mann algorithm. We show that the algorithm with increasing batch sizes converges almost surely to a fixed point of the nonexpansive mapping. We also perform a convergence rate analysis on the algorithm.

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arXiv2026

Muon Converges under Heavy-Tailed Noise: Nonconvex Hölder-Smooth Empirical Risk Minimization

Muon is a recently proposed optimizer that enforces orthogonality in parameter updates by projecting gradients onto the Stiefel manifold, leading to stable and efficient training in large-scale deep neural networks. Meanwhile, the previously reported results indicated that stochastic noise in practical machine learning may exhibit heavy-tailed behavior, violating the bounded-variance assumption. In this paper, we consider the problem of minimizing a nonconvex Hölder-smooth empirical risk that works well with the heavy-tailed stochastic noise. We then show that Muon converges to a stationary point of the empirical risk under the boundedness condition accounting for heavy-tailed stochastic noise. In addition, we show that Muon converges faster than mini-batch SGD.

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arXiv2026

Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization

The Muon optimizer has recently attracted attention due to its orthogonalized first-order updates, and a deeper theoretical understanding of its convergence behavior is essential for guiding practical applications; however, existing convergence guarantees are either coarse or obtained under restrictive analytical settings. In this work, we establish sharper convergence guarantees for the Muon optimizer through a direct and simplified analysis that does not rely on restrictive assumptions on the update rule. Our results improve upon existing bounds by achieving faster convergence rates while covering a broader class of problem settings. These findings provide a more accurate theoretical characterization of Muon and offer insights applicable to a broader class of orthogonalized first-order methods.

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arXiv2026

Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach

Deep equilibrium models (DEQs) achieve infinitely deep network representations without stacking layers by exploring fixed points of layer transformations in neural networks. Such models constitute an innovative approach that achieves performance comparable to state-of-the-art methods in many large-scale numerical experiments, despite requiring significantly less memory. However, DEQs face the challenge of requiring vastly more computational time for training and inference than conventional methods, as they repeatedly perform fixed-point iterations with no convergence guarantee upon each input. Therefore, this study explored an approach to improve fixed-point convergence and consequently reduce computational time by restructuring the model architecture to guarantee fixed-point convergence. Our proposed approach for image classification, Lipschitz multiscale DEQ, has theoretically guaranteed fixed-point convergence for both forward and backward passes by hyperparameter adjustment, achieving up to a 4.75$\times$ speed-up in numerical experiments on CIFAR-10 at the cost of a minor drop in accuracy.

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arXiv2026

Using Stochastic Gradient Descent to Smooth Nonconvex Functions: Analysis of Implicit Graduated Optimization

The graduated optimization approach is a method for finding global optimal solutions for nonconvex functions by using a function smoothing operation with stochastic noise. This paper makes three contributions regarding graduated optimization. First, we extend the definition of function smoothing that is traditionally achieved through convolution with Gaussian noise and characterize for the first time function smoothing with heavy-tailed noise. Second, we show that light- or heavy-tailed stochastic noise in stochastic gradient descent (SGD) has the effect of smoothing the objective function, the degree of which is determined by the learning rate, batch size, and the moment of the stochastic noise. Using this finding, we propose and analyze a new graduated optimization algorithm that varies the degree of smoothing by varying the learning rate and batch size. Third, we relax the $σ$-nice property, a standard but restrictive condition in the analysis of graduated optimization. Our refinement enables convergence guarantees for a broader class of non-convex functions, thereby bridging the gap between theoretical assumptions and practical optimization landscapes.

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arXiv2025

Convergence Analysis of SGD under Expected Smoothness

Stochastic gradient descent (SGD) is the workhorse of large-scale learning, yet classical analyses rely on assumptions that can be either too strong (bounded variance) or too coarse (uniform noise). The expected smoothness (ES) condition has emerged as a flexible alternative that ties the second moment of stochastic gradients to the objective value and the full gradient. This paper presents a self-contained convergence analysis of SGD under ES. We (i) refine ES with interpretations and sampling-dependent constants; (ii) derive bounds of the expectation of squared full gradient norm; and (iii) prove $O(1/K)$ rates with explicit residual errors for various step-size schedules. All proofs are given in full detail in the appendix. Our treatment unifies and extends recent threads (Khaled and Richtárik, 2020; Umeda and Iiduka, 2025).

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arXiv2025

Faster Convergence of Riemannian Stochastic Gradient Descent with Increasing Batch Size

We theoretically analyzed the convergence behavior of Riemannian stochastic gradient descent (RSGD) and found that using an increasing batch size leads to faster convergence than using a constant batch size, not only with a constant learning rate but also with a decaying learning rate, such as cosine annealing decay and polynomial decay. The convergence rate improves from $O(T^{-1}+C)$ with a constant batch size to $O(T^{-1})$ with an increasing batch size, where $T$ denotes the total number of iterations and $C$ is a constant. Using principal component analysis and low-rank matrix completion, we investigated, both theoretically and numerically, how an increasing batch size affects computational time as quantified by stochastic first-order oracle (SFO) complexity. An increasing batch size was found to reduce the SFO complexity of RSGD. Furthermore, an increasing batch size was found to offer the advantages of both small and large constant batch sizes.

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arXiv2025

Increasing Batch Size Improves Convergence of Stochastic Gradient Descent with Momentum

Stochastic gradient descent with momentum (SGDM), in which a momentum term is added to SGD, has been well studied in both theory and practice. The theoretical studies show that the settings of the learning rate and momentum weight affect the convergence of SGDM. Meanwhile, the practical studies have shown that the batch-size setting strongly affects the performance of SGDM. In this paper, we focus on mini-batch SGDM with a constant learning rate and constant momentum weight, which is frequently used to train deep neural networks. We show theoretically that using a constant batch size does not always minimize the expectation of the full gradient norm of the empirical loss in training a deep neural network, whereas using an increasing batch size definitely minimizes it; that is, an increasing batch size improves the convergence of mini-batch SGDM. We also provide numerical results supporting our analyses, indicating specifically that mini-batch SGDM with an increasing batch size converges to stationary points faster than with a constant batch size, while also reducing computational cost. Python implementations of the optimizers used in the numerical experiments are available at https://github.com/iiduka-researches/NSHB_increasing_batchsize_acml25/.

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arXiv2025

Both Asymptotic and Non-Asymptotic Convergence of Quasi-Hyperbolic Momentum using Increasing Batch Size

Momentum methods were originally introduced for their superiority to stochastic gradient descent (SGD) in deterministic settings with convex objective functions. However, despite their widespread application to deep neural networks -- a representative case of stochastic nonconvex optimization -- the theoretical justification for their effectiveness in such settings remains limited. Quasi-hyperbolic momentum (QHM) is an algorithm that generalizes various momentum methods and has been studied to better understand the class of momentum-based algorithms as a whole. In this paper, we provide both asymptotic and non-asymptotic convergence results for mini-batch QHM with an increasing batch size. We show that achieving asymptotic convergence requires either a decaying learning rate or an increasing batch size. Since a decaying learning rate adversely affects non-asymptotic convergence, we demonstrate that using mini-batch QHM with an increasing batch size -- without decaying the learning rate -- can be a more effective strategy. Our experiments show that even a finite increase in batch size can provide benefits for training neural networks.

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arXiv2025

Optimal Growth Schedules for Batch Size and Learning Rate in SGD that Reduce SFO Complexity

The unprecedented growth of deep learning models has enabled remarkable advances but introduced substantial computational bottlenecks. A key factor contributing to training efficiency is batch-size and learning-rate scheduling in stochastic gradient methods. However, naive scheduling of these hyperparameters can degrade optimization efficiency and compromise generalization. Motivated by recent theoretical insights, we investigated how the batch size and learning rate should be increased during training to balance efficiency and convergence. We analyzed this problem on the basis of stochastic first-order oracle (SFO) complexity, defined as the expected number of gradient evaluations needed to reach an $ε$-approximate stationary point of the empirical loss. We theoretically derived optimal growth schedules for the batch size and learning rate that reduce SFO complexity and validated them through extensive experiments. Our results offer both theoretical insights and practical guidelines for scalable and efficient large-batch training in deep learning.

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arXiv2025

Adaptive Batch Size and Learning Rate Scheduler for Stochastic Gradient Descent Based on Minimization of Stochastic First-order Oracle Complexity

The convergence behavior of mini-batch stochastic gradient descent (SGD) is highly sensitive to the batch size and learning rate settings. Recent theoretical studies have identified the existence of a critical batch size that minimizes stochastic first-order oracle (SFO) complexity, defined as the expected number of gradient evaluations required to reach a stationary point of the empirical loss function in a deep neural network. An adaptive scheduling strategy is introduced to accelerate SGD that leverages theoretical findings on the critical batch size. The batch size and learning rate are adjusted on the basis of the observed decay in the full gradient norm during training. Experiments using an adaptive joint scheduler based on this strategy demonstrated improved convergence speed compared with that of existing schedulers.

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arXiv2025

Momentum Does Not Reduce Stochastic Noise in Stochastic Gradient Descent

For nonconvex objective functions, including those found in training deep neural networks, stochastic gradient descent (SGD) with momentum is said to converge faster and have better generalizability than SGD without momentum. In particular, adding momentum is thought to reduce stochastic noise. To verify this, we estimated the magnitude of gradient noise by using convergence analysis and an optimal batch size estimation formula and found that momentum does not reduce gradient noise. We also analyzed the effect of search direction noise, which is stochastic noise defined as the error between the search direction of the optimizer and the steepest descent direction, and found that it inherently smooths the objective function and that momentum does not reduce search direction noise either. Finally, an analysis of the degree of smoothing introduced by search direction noise revealed that adding momentum offers limited advantage to SGD.

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arXiv2025

Increasing Both Batch Size and Learning Rate Accelerates Stochastic Gradient Descent

The performance of mini-batch stochastic gradient descent (SGD) strongly depends on setting the batch size and learning rate to minimize the empirical loss in training the deep neural network. In this paper, we present theoretical analyses of mini-batch SGD with four schedulers: (i) constant batch size and decaying learning rate scheduler, (ii) increasing batch size and decaying learning rate scheduler, (iii) increasing batch size and increasing learning rate scheduler, and (iv) increasing batch size and warm-up decaying learning rate scheduler. We show that mini-batch SGD using scheduler (i) does not always minimize the expectation of the full gradient norm of the empirical loss, whereas it does using any of schedulers (ii), (iii), and (iv). Furthermore, schedulers (iii) and (iv) accelerate mini-batch SGD. The paper also provides numerical results of supporting analyses showing that using scheduler (iii) or (iv) minimizes the full gradient norm of the empirical loss faster than using scheduler (i) or (ii).

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arXiv2025

A general framework of Riemannian adaptive optimization methods with a convergence analysis

This paper proposes a general framework of Riemannian adaptive optimization methods. The framework encapsulates several stochastic optimization algorithms on Riemannian manifolds and incorporates the mini-batch strategy that is often used in deep learning. Within this framework, we also propose AMSGrad on embedded submanifolds of Euclidean space. Moreover, we give convergence analyses valid for both a constant and a diminishing step size. Our analyses also reveal the relationship between the convergence rate and mini-batch size. In numerical experiments, we applied the proposed algorithm to principal component analysis and the low-rank matrix completion problem, which can be considered to be Riemannian optimization problems. Python implementations of the methods used in the numerical experiments are available at https://github.com/iiduka-researches/202408-adaptive.

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