arXiv subjects

Chulhee Yun

Publications and source records attributed to Chulhee Yun.

At least 37 records · Page 2Linked to original sources

arXiv2025

Convergence and Implicit Bias of Gradient Descent on Continual Linear Classification

We study continual learning on multiple linear classification tasks by sequentially running gradient descent (GD) for a fixed budget of iterations per task. When all tasks are jointly linearly separable and are presented in a cyclic/random order, we show the directional convergence of the trained linear classifier to the joint (offline) max-margin solution. This is surprising because GD training on a single task is implicitly biased towards the individual max-margin solution for the task, and the direction of the joint max-margin solution can be largely different from these individual solutions. Additionally, when tasks are given in a cyclic order, we present a non-asymptotic analysis on cycle-averaged forgetting, revealing that (1) alignment between tasks is indeed closely tied to catastrophic forgetting and backward knowledge transfer and (2) the amount of forgetting vanishes to zero as the cycle repeats. Lastly, we analyze the case where the tasks are no longer jointly separable and show that the model trained in a cyclic order converges to the unique minimum of the joint loss function.

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arXiv2025

Arithmetic Transformers Can Length-Generalize in Both Operand Length and Count

Transformers often struggle with length generalization, meaning they fail to generalize to sequences longer than those encountered during training. While arithmetic tasks are commonly used to study length generalization, certain tasks are considered notoriously difficult, e.g., multi-operand addition (requiring generalization over both the number of operands and their lengths) and multiplication (requiring generalization over both operand lengths). In this work, we achieve approximately 2-3x length generalization on both tasks, which is the first such achievement in arithmetic Transformers. We design task-specific scratchpads enabling the model to focus on a fixed number of tokens per each next-token prediction step, and apply multi-level versions of \Position Coupling (Cho et al., 2024; McLeish et al., 2024) to let Transformers know the right position to attend to. On the theory side, we prove that a 1-layer Transformer using our method can solve multi-operand addition, up to operand length and operand count that are exponential in embedding dimension.

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arXiv2025

Does SGD really happen in tiny subspaces?

Understanding the training dynamics of deep neural networks is challenging due to their high-dimensional nature and intricate loss landscapes. Recent studies have revealed that, along the training trajectory, the gradient approximately aligns with a low-rank top eigenspace of the training loss Hessian, referred to as the dominant subspace. Given this alignment, this paper explores whether neural networks can be trained within the dominant subspace, which, if feasible, could lead to more efficient training methods. Our primary observation is that when the SGD update is projected onto the dominant subspace, the training loss does not decrease further. This suggests that the observed alignment between the gradient and the dominant subspace is spurious. Surprisingly, projecting out the dominant subspace proves to be just as effective as the original update, despite removing the majority of the original update component. We observe similar behavior across practical setups, including the large learning rate regime (also known as Edge of Stability), Sharpness-Aware Minimization, momentum, and adaptive optimizers. We discuss the main causes and implications of this spurious alignment, shedding light on the dynamics of neural network training.

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arXiv2025

Parameter Expanded Stochastic Gradient Markov Chain Monte Carlo

Bayesian Neural Networks (BNNs) provide a promising framework for modeling predictive uncertainty and enhancing out-of-distribution robustness (OOD) by estimating the posterior distribution of network parameters. Stochastic Gradient Markov Chain Monte Carlo (SGMCMC) is one of the most powerful methods for scalable posterior sampling in BNNs, achieving efficiency by combining stochastic gradient descent with second-order Langevin dynamics. However, SGMCMC often suffers from limited sample diversity in practice, which affects uncertainty estimation and model performance. We propose a simple yet effective approach to enhance sample diversity in SGMCMC without the need for tempering or running multiple chains. Our approach reparameterizes the neural network by decomposing each of its weight matrices into a product of matrices, resulting in a sampling trajectory that better explores the target parameter space. This approach produces a more diverse set of samples, allowing faster mixing within the same computational budget. Notably, our sampler achieves these improvements without increasing the inference cost compared to the standard SGMCMC. Extensive experiments on image classification tasks, including OOD robustness, diversity, loss surface analyses, and a comparative study with Hamiltonian Monte Carlo, demonstrate the superiority of the proposed approach.

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arXiv2024

Stochastic Extragradient with Flip-Flop Shuffling & Anchoring: Provable Improvements

In minimax optimization, the extragradient (EG) method has been extensively studied because it outperforms the gradient descent-ascent method in convex-concave (C-C) problems. Yet, stochastic EG (SEG) has seen limited success in C-C problems, especially for unconstrained cases. Motivated by the recent progress of shuffling-based stochastic methods, we investigate the convergence of shuffling-based SEG in unconstrained finite-sum minimax problems, in search of convergent shuffling-based SEG. Our analysis reveals that both random reshuffling and the recently proposed flip-flop shuffling alone can suffer divergence in C-C problems. However, with an additional simple trick called anchoring, we develop the SEG with flip-flop anchoring (SEG-FFA) method which successfully converges in C-C problems. We also show upper and lower bounds in the strongly-convex-strongly-concave setting, demonstrating that SEG-FFA has a provably faster convergence rate compared to other shuffling-based methods.

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arXiv2024

DASH: Warm-Starting Neural Network Training in Stationary Settings without Loss of Plasticity

Warm-starting neural network training by initializing networks with previously learned weights is appealing, as practical neural networks are often deployed under a continuous influx of new data. However, it often leads to loss of plasticity, where the network loses its ability to learn new information, resulting in worse generalization than training from scratch. This occurs even under stationary data distributions, and its underlying mechanism is poorly understood. We develop a framework emulating real-world neural network training and identify noise memorization as the primary cause of plasticity loss when warm-starting on stationary data. Motivated by this, we propose Direction-Aware SHrinking (DASH), a method aiming to mitigate plasticity loss by selectively forgetting memorized noise while preserving learned features. We validate our approach on vision tasks, demonstrating improvements in test accuracy and training efficiency.

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arXiv2024

Provable Benefit of Cutout and CutMix for Feature Learning

Patch-level data augmentation techniques such as Cutout and CutMix have demonstrated significant efficacy in enhancing the performance of vision tasks. However, a comprehensive theoretical understanding of these methods remains elusive. In this paper, we study two-layer neural networks trained using three distinct methods: vanilla training without augmentation, Cutout training, and CutMix training. Our analysis focuses on a feature-noise data model, which consists of several label-dependent features of varying rarity and label-independent noises of differing strengths. Our theorems demonstrate that Cutout training can learn low-frequency features that vanilla training cannot, while CutMix training can learn even rarer features that Cutout cannot capture. From this, we establish that CutMix yields the highest test accuracy among the three. Our novel analysis reveals that CutMix training makes the network learn all features and noise vectors "evenly" regardless of the rarity and strength, which provides an interesting insight into understanding patch-level augmentation.

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arXiv2024

Position Coupling: Improving Length Generalization of Arithmetic Transformers Using Task Structure

Even for simple arithmetic tasks like integer addition, it is challenging for Transformers to generalize to longer sequences than those encountered during training. To tackle this problem, we propose position coupling, a simple yet effective method that directly embeds the structure of the tasks into the positional encoding of a (decoder-only) Transformer. Taking a departure from the vanilla absolute position mechanism assigning unique position IDs to each of the tokens, we assign the same position IDs to two or more "relevant" tokens; for integer addition tasks, we regard digits of the same significance as in the same position. On the empirical side, we show that with the proposed position coupling, our models trained on 1 to 30-digit additions can generalize up to 200-digit additions (6.67x of the trained length). On the theoretical side, we prove that a 1-layer Transformer with coupled positions can solve the addition task involving exponentially many digits, whereas any 1-layer Transformer without positional information cannot entirely solve it. We also demonstrate that position coupling can be applied to other algorithmic tasks such as Nx2 multiplication and a two-dimensional task.

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arXiv2024

Fundamental Benefit of Alternating Updates in Minimax Optimization

The Gradient Descent-Ascent (GDA) algorithm, designed to solve minimax optimization problems, takes the descent and ascent steps either simultaneously (Sim-GDA) or alternately (Alt-GDA). While Alt-GDA is commonly observed to converge faster, the performance gap between the two is not yet well understood theoretically, especially in terms of global convergence rates. To address this theory-practice gap, we present fine-grained convergence analyses of both algorithms for strongly-convex-strongly-concave and Lipschitz-gradient objectives. Our new iteration complexity upper bound of Alt-GDA is strictly smaller than the lower bound of Sim-GDA; i.e., Alt-GDA is provably faster. Moreover, we propose Alternating-Extrapolation GDA (Alex-GDA), a general algorithmic framework that subsumes Sim-GDA and Alt-GDA, for which the main idea is to alternately take gradients from extrapolations of the iterates. We show that Alex-GDA satisfies a smaller iteration complexity bound, identical to that of the Extra-gradient method, while requiring less gradient computations. We also prove that Alex-GDA enjoys linear convergence for bilinear problems, for which both Sim-GDA and Alt-GDA fail to converge at all.

math.OC↗

arXiv2024

Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults

Although gradient descent with Polyak's momentum is widely used in modern machine and deep learning, a concrete understanding of its effects on the training trajectory remains elusive. In this work, we empirically show that for linear diagonal networks and nonlinear neural networks, momentum gradient descent with a large learning rate displays large catapults, driving the iterates towards much flatter minima than those found by gradient descent. We hypothesize that the large catapult is caused by momentum "prolonging" the self-stabilization effect (Damian et al., 2023). We provide theoretical and empirical support for our hypothesis in a simple toy example and empirical evidence supporting our hypothesis for linear diagonal networks.

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arXiv2024

Linear attention is (maybe) all you need (to understand transformer optimization)

Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet canonical linearized shallow Transformer model. Specifically, we train linear Transformers to solve regression tasks, inspired by J.~von Oswald et al.~(ICML 2023), and K.~Ahn et al.~(NeurIPS 2023). Most importantly, we observe that our proposed linearized models can reproduce several prominent aspects of Transformer training dynamics. Consequently, the results obtained in this paper suggest that a simple linearized Transformer model could actually be a valuable, realistic abstraction for understanding Transformer optimization.

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arXiv2023

PLASTIC: Improving Input and Label Plasticity for Sample Efficient Reinforcement Learning

In Reinforcement Learning (RL), enhancing sample efficiency is crucial, particularly in scenarios when data acquisition is costly and risky. In principle, off-policy RL algorithms can improve sample efficiency by allowing multiple updates per environment interaction. However, these multiple updates often lead the model to overfit to earlier interactions, which is referred to as the loss of plasticity. Our study investigates the underlying causes of this phenomenon by dividing plasticity into two aspects. Input plasticity, which denotes the model's adaptability to changing input data, and label plasticity, which denotes the model's adaptability to evolving input-output relationships. Synthetic experiments on the CIFAR-10 dataset reveal that finding smoother minima of loss landscape enhances input plasticity, whereas refined gradient propagation improves label plasticity. Leveraging these findings, we introduce the PLASTIC algorithm, which harmoniously combines techniques to address both concerns. With minimal architectural modifications, PLASTIC achieves competitive performance on benchmarks including Atari-100k and Deepmind Control Suite. This result emphasizes the importance of preserving the model's plasticity to elevate the sample efficiency in RL. The code is available at https://github.com/dojeon-ai/plastic.

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arXiv2023

Fair Streaming Principal Component Analysis: Statistical and Algorithmic Viewpoint

Fair Principal Component Analysis (PCA) is a problem setting where we aim to perform PCA while making the resulting representation fair in that the projected distributions, conditional on the sensitive attributes, match one another. However, existing approaches to fair PCA have two main problems: theoretically, there has been no statistical foundation of fair PCA in terms of learnability; practically, limited memory prevents us from using existing approaches, as they explicitly rely on full access to the entire data. On the theoretical side, we rigorously formulate fair PCA using a new notion called \emph{probably approximately fair and optimal} (PAFO) learnability. On the practical side, motivated by recent advances in streaming algorithms for addressing memory limitation, we propose a new setting called \emph{fair streaming PCA} along with a memory-efficient algorithm, fair noisy power method (FNPM). We then provide its {\it statistical} guarantee in terms of PAFO-learnability, which is the first of its kind in fair PCA literature. Lastly, we verify the efficacy and memory efficiency of our algorithm on real-world datasets.

stat.ML↗

arXiv2023

Practical Sharpness-Aware Minimization Cannot Converge All the Way to Optima

Sharpness-Aware Minimization (SAM) is an optimizer that takes a descent step based on the gradient at a perturbation $y_t = x_t + ρ\frac{\nabla f(x_t)}{\lVert \nabla f(x_t) \rVert}$ of the current point $x_t$. Existing studies prove convergence of SAM for smooth functions, but they do so by assuming decaying perturbation size $ρ$ and/or no gradient normalization in $y_t$, which is detached from practice. To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant $ρ$ and gradient normalization in $y_t$) and explore their convergence properties on smooth functions with (non)convexity assumptions. Perhaps surprisingly, in many scenarios, we find out that SAM has limited capability to converge to global minima or stationary points. For smooth strongly convex functions, we show that while deterministic SAM enjoys tight global convergence rates of $\tilde Θ(\frac{1}{T^2})$, the convergence bound of stochastic SAM suffers an inevitable additive term $O(ρ^2)$, indicating convergence only up to neighborhoods of optima. In fact, such $O(ρ^2)$ factors arise for stochastic SAM in all the settings we consider, and also for deterministic SAM in nonconvex cases; importantly, we prove by examples that such terms are unavoidable. Our results highlight vastly different characteristics of SAM with vs. without decaying perturbation size or gradient normalization, and suggest that the intuitions gained from one version may not apply to the other.

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arXiv2023

Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory

Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon. The sharpness increases at the early phase of training (referred to as progressive sharpening), and eventually saturates close to the threshold of $2 / \text{(step size)}$. In this paper, we start by demonstrating through empirical studies that when the EoS phenomenon occurs, different GD trajectories (after a proper reparameterization) align on a specific bifurcation diagram independent of initialization. We then rigorously prove this trajectory alignment phenomenon for a two-layer fully-connected linear network and a single-neuron nonlinear network trained with a single data point. Our trajectory alignment analysis establishes both progressive sharpening and EoS phenomena, encompassing and extending recent findings in the literature.

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arXiv2023

On the Training Instability of Shuffling SGD with Batch Normalization

We uncover how SGD interacts with batch normalization and can exhibit undesirable training dynamics such as divergence. More precisely, we study how Single Shuffle (SS) and Random Reshuffle (RR) -- two widely used variants of SGD -- interact surprisingly differently in the presence of batch normalization: RR leads to much more stable evolution of training loss than SS. As a concrete example, for regression using a linear network with batch normalization, we prove that SS and RR converge to distinct global optima that are "distorted" away from gradient descent. Thereafter, for classification we characterize conditions under which training divergence for SS and RR can, and cannot occur. We present explicit constructions to show how SS leads to distorted optima in regression and divergence for classification, whereas RR avoids both distortion and divergence. We validate our results by confirming them empirically in realistic settings, and conclude that the separation between SS and RR used with batch normalization is relevant in practice.

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arXiv2023

Tighter Lower Bounds for Shuffling SGD: Random Permutations and Beyond

We study convergence lower bounds of without-replacement stochastic gradient descent (SGD) for solving smooth (strongly-)convex finite-sum minimization problems. Unlike most existing results focusing on final iterate lower bounds in terms of the number of components $n$ and the number of epochs $K$, we seek bounds for arbitrary weighted average iterates that are tight in all factors including the condition number $κ$. For SGD with Random Reshuffling, we present lower bounds that have tighter $κ$ dependencies than existing bounds. Our results are the first to perfectly close the gap between lower and upper bounds for weighted average iterates in both strongly-convex and convex cases. We also prove weighted average iterate lower bounds for arbitrary permutation-based SGD, which apply to all variants that carefully choose the best permutation. Our bounds improve the existing bounds in factors of $n$ and $κ$ and thereby match the upper bounds shown for a recently proposed algorithm called GraB.

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arXiv2023

Provable Benefit of Mixup for Finding Optimal Decision Boundaries

We investigate how pair-wise data augmentation techniques like Mixup affect the sample complexity of finding optimal decision boundaries in a binary linear classification problem. For a family of data distributions with a separability constant $κ$, we analyze how well the optimal classifier in terms of training loss aligns with the optimal one in test accuracy (i.e., Bayes optimal classifier). For vanilla training without augmentation, we uncover an interesting phenomenon named the curse of separability. As we increase $κ$ to make the data distribution more separable, the sample complexity of vanilla training increases exponentially in $κ$; perhaps surprisingly, the task of finding optimal decision boundaries becomes harder for more separable distributions. For Mixup training, we show that Mixup mitigates this problem by significantly reducing the sample complexity. To this end, we develop new concentration results applicable to $n^2$ pair-wise augmented data points constructed from $n$ independent data, by carefully dealing with dependencies between overlapping pairs. Lastly, we study other masking-based Mixup-style techniques and show that they can distort the training loss and make its minimizer converge to a suboptimal classifier in terms of test accuracy.

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