arXiv subjects

Shuche Wang

Publications and source records attributed to Shuche Wang.

13 recordsLinked to original sources

arXiv2026

Muon Learns More Robust and Transferable Features than Adam

Muon has recently emerged as a state-of-the-art optimizer for pretraining Large Language Models (LLMs) and vision classifiers. Despite its efficiency advantage over Adam and SGD, the feature-learning advantage of Muon remains unclear. This paper investigates Muon's feature-learning advantage through the lens of robustness and transferability. First, by evaluating pretrained models on corrupted images and texts, we show that features learned by Muon are consistently more robust than those learned by Adam and SGD across different architectures, including transformers and Convolutional Neural Networks (CNNs). Using trained layer-wise probes, we further show that this robustness advantage is reflected in larger logit margins across layers. Second, by training linear classifiers or fine-tuning full models from pretrained parameters on downstream tasks, we demonstrate that Muon-learned features transfer more effectively than those learned by Adam and SGD. This transferability advantage is further supported by the diversity of hidden states across layers, as measured by effective rank. Finally, in a representative classification problem with multi-component features, we prove that Muon attains larger margins and higher effective rank than Adam and SGD, providing theoretical support for our empirical findings.

cs.LG↗

arXiv2026

Why Muon Outperforms Adam: A Curvature Perspective

Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.

cs.LG↗

arXiv2026

Online Learning with Gradient-Variation Interval Regret

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

cs.LG↗

arXiv2026

List Reconstruction Problem with List Size Two

The problem of computing the cardinality of the intersection of multiple balls in the Hamming space has attracted a lot of attention recently due to their applications in the list reconstruction problem and information retrieval in Associative Memories. In previous work, most of the results are for the cases where the radii of each ball, $r$ and the distance between the centers of these balls, $k$ are fixed when the length $n$ of each codeword tend to infinity. In this work, we focus on the case where $r = \alpha n$ and $k=\beta n$ for some constants $\alpha$ and $\beta$ and compute the maximum asymptotic rate of the cardinality of the intersection of three balls. We provide the maximum asymptotic rate as a function of two parameters $\alpha$ and $\beta$. We also provide numerical results and compare these results with the intersection of two balls.

math.CO↗

arXiv2026

Bandit Convex Optimization with Gradient Prediction Adaptivity

Bandit convex optimization (BCO) is a fundamental online learning framework with partial feedback, where the learner observes only the loss incurred at the chosen decision point in each round. In this work, we investigate whether optimistic gradient predictions can improve worst-case regret guarantees in a prediction-adaptive manner. Specifically, given gradient predictions $m_t$, we seek regret bounds that scale with the cumulative prediction error $S_T=\sum_{t=1}^T \|\nabla f_t(x_t)-m_t\|^2.$ We first establish a negative result: under the single-point feedback protocol, an unavoidable $\Omega(\sqrt{T})$ regret lower bound persists even when $S_T=o(T)$, showing that the variance of gradient estimation fundamentally obscures the benefit of accurate predictions. To overcome this barrier, we propose \emph{Two-Point Variance-Reduced Optimistic Gradient Descent} (TP-VR-OPT) for the two-point feedback setting. The key idea is a novel variance-reduced gradient estimator whose variance scales with the prediction error rather than the gradient norm. This yields a regret bound of $O\big(\sqrt{d\,\mathbb{E}[S_T]}\big),$ where $d$ is the decision dimension. Complementing this result, we establish an information-theoretic lower bound that scales as $\Omega(\sqrt{\mathbb{E}[S_T]})$, providing a fundamental characterization of the best achievable prediction-adaptive regret and showing that TP-VR-OPT is optimal up to a factor of $\sqrt d$. We further develop adaptive variants that eliminate the need for prior knowledge of $\mathbb{E}[S_T]$ or the horizon $T$, and extend our framework to non-stationary environments, establishing dynamic regret guarantees that adapt simultaneously to the cumulative prediction error and the comparator path length.

cs.LG↗

arXiv2025

Parameter-free Algorithms for the Stochastically Extended Adversarial Model

We develop the first parameter-free algorithms for the Stochastically Extended Adversarial (SEA) model, a framework that bridges adversarial and stochastic online convex optimization. Existing approaches for the SEA model require prior knowledge of problem-specific parameters, such as the diameter of the domain $D$ and the Lipschitz constant of the loss functions $G$, which limits their practical applicability. Addressing this, we develop parameter-free methods by leveraging the Optimistic Online Newton Step (OONS) algorithm to eliminate the need for these parameters. We first establish a comparator-adaptive algorithm for the scenario with unknown domain diameter but known Lipschitz constant, achieving an expected regret bound of $\tilde{O}\big(\|u\|_2^2 + \|u\|_2(\sqrt{\sigma^2_{1:T}} + \sqrt{\Sigma^2_{1:T}})\big)$, where $u$ is the comparator vector and $\sigma^2_{1:T}$ and $\Sigma^2_{1:T}$ represent the cumulative stochastic variance and cumulative adversarial variation, respectively. We then extend this to the more general setting where both $D$ and $G$ are unknown, attaining the comparator- and Lipschitz-adaptive algorithm. Notably, the regret bound exhibits the same dependence on $\sigma^2_{1:T}$ and $\Sigma^2_{1:T}$, demonstrating the efficacy of our proposed methods even when both parameters are unknown in the SEA model.

cs.LG↗

arXiv2025

Muon Outperforms Adam in Tail-End Associative Memory Learning

The Muon optimizer is consistently faster than Adam in training Large Language Models (LLMs), yet the mechanism underlying its success remains unclear. This paper demystifies this mechanism through the lens of associative memory. By ablating the transformer components optimized by Muon, we reveal that the associative memory parameters of LLMs, namely the Value and Output (VO) attention weights and Feed-Forward Networks (FFNs), are the primary contributors to Muon's superiority. Motivated by this associative memory view, we then explain Muon's superiority on real-world corpora, which are intrinsically heavy-tailed: a few classes (tail classes) appear far less frequently than others. The superiority is explained through two key properties: (i) its update rule consistently yields a more isotropic singular spectrum than Adam; and as a result, (ii) on heavy-tailed data, it optimizes tail classes more effectively than Adam. Beyond empirical evidence, we theoretically confirm these findings by analyzing a one-layer associative memory model under class-imbalanced data. We prove that Muon consistently achieves balanced learning across classes regardless of feature embeddings, whereas Adam can induce large disparities in learning errors depending on embedding properties. In summary, our empirical observations and theoretical analyses reveal Muon's core advantage: its update rule aligns with the outer-product structure of linear associative memories, enabling more balanced and effective learning of tail classes in heavy-tailed distributions than Adam.

cs.LG↗

arXiv2024

A Mirror Descent-Based Algorithm for Corruption-Tolerant Distributed Gradient Descent

Distributed gradient descent algorithms have come to the fore in modern machine learning, especially in parallelizing the handling of large datasets that are distributed across several workers. However, scant attention has been paid to analyzing the behavior of distributed gradient descent algorithms in the presence of adversarial corruptions instead of random noise. In this paper, we formulate a novel problem in which adversarial corruptions are present in a distributed learning system. We show how to use ideas from (lazy) mirror descent to design a corruption-tolerant distributed optimization algorithm. Extensive convergence analysis for (strongly) convex loss functions is provided for different choices of the stepsize. We carefully optimize the stepsize schedule to accelerate the convergence of the algorithm, while at the same time amortizing the effect of the corruption over time. Experiments based on linear regression, support vector classification, and softmax classification on the MNIST dataset corroborate our theoretical findings.

eess.SP↗

arXiv2024

Permutation and Multi-permutation Codes Correcting Multiple Deletions

Permutation codes in the Ulam metric, which can correct multiple deletions, have been investigated extensively recently. In this work, we are interested in the maximum size of permutation codes in the Ulam metric and aim to design permutation codes that can correct multiple deletions with efficient decoding algorithms. We first present an improvement on the Gilbert--Varshamov bound of the maximum size of these permutation codes by analyzing the independence number of the auxiliary graph. The idea is widely used in various cases and our contribution in this section is enumerating the number of triangles in the auxiliary graph and showing that it is small enough. Next, we design permutation codes correcting multiple deletions with a decoding algorithm. In particular, the constructed permutation codes can correct $t$ deletions with at most $(3t-1) \log n+o(\log n)$ bits of redundancy where $n$ is the length of the code. Our construction is based on a new mapping which yields a new connection between permutation codes in the Hamming metric and permutation codes in various metrics. Furthermore, we construct permutation codes that correct multiple bursts of deletions using this new mapping. Finally, we extend the new mapping for multi-permutations and construct the best-known multi-permutation codes in Ulam metric.

cs.IT↗

arXiv2023

Codes for Correcting Asymmetric Adjacent Transpositions and Deletions

Codes in the Damerau--Levenshtein metric have been extensively studied recently owing to their applications in DNA-based data storage. In particular, Gabrys, Yaakobi, and Milenkovic (2017) designed a length-$n$ code correcting a single deletion and $s$ adjacent transpositions with at most $(1+2s)\log n$ bits of redundancy. In this work, we consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions may occur. We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting a single deletion, $s^+$ right-shift, and $s^-$ left-shift errors with at most $(1+s)\log (n+s+1)+1$ bits of redundancy where $s=s^{+}+s^{-}$. In addition, we investigate codes correcting $t$ $0$-deletions, $s^+$ right-shift, and $s^-$ left-shift errors with both uniquely-decoding and list-decoding algorithms. Our main contribution here is the construction of a list-decodable code with list size $O(n^{\min\{s+1,t\}})$ and with at most $(\max \{t,s+1\}) \log n+O(1)$ bits of redundancy, where $s=s^{+}+s^{-}$. Finally, we construct both non-systematic and systematic codes for correcting blocks of $0$-deletions with $\ell$-limited-magnitude and $s$ adjacent transpositions.

cs.IT↗

arXiv2023

Codes for Correcting $t$ Limited-Magnitude Sticky Deletions

Codes for correcting sticky insertions/deletions and limited-magnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$-sticky deletions with $\ell$-limited-magnitude and propose a non-systematic code for correcting this type of error with redundancy $2t(1-1/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\frac{\lceil2t(1-1/p)\rceil\cdot\lceil\log p\rceil}{\log p} \log(n+1)+O(\log\log n)$, where $p$ is the smallest prime larger than $\ell+1$.

cs.IT↗

arXiv2022

Non-binary Codes for Correcting a Burst of at Most t Deletions

The problem of correcting deletions has received significant attention, partly because of the prevalence of these errors in DNA data storage. In this paper, we study the problem of correcting a consecutive burst of at most $t$ deletions in non-binary sequences. We first propose a non-binary code correcting a burst of at most 2 deletions for $q$-ary alphabets. Afterwards, we extend this result to the case where the length of the burst can be at most $t$ where $t$ is a constant. Finally, we consider the setup where the sequences that are transmitted are permutations. The proposed codes are the largest known for their respective parameter regimes.

cs.IT↗

arXiv2022

Low-redundancy codes for correcting multiple short-duplication and edit errors

Due to its higher data density, longevity, energy efficiency, and ease of generating copies, DNA is considered a promising storage technology for satisfying future needs. However, a diverse set of errors including deletions, insertions, duplications, and substitutions may arise in DNA at different stages of data storage and retrieval. The current paper constructs error-correcting codes for simultaneously correcting short (tandem) duplications and at most $p$ edits, where a short duplication generates a copy of a substring with length $\leq 3$ and inserts the copy following the original substring, and an edit is a substitution, deletion, or insertion. Compared to the state-of-the-art codes for duplications only, the proposed codes correct up to $p$ edits (in addition to duplications) at the additional cost of roughly $8p(\log_q n)(1+o(1))$ symbols of redundancy, thus achieving the same asymptotic rate, where $q\ge 4$ is the alphabet size and $p$ is a constant. Furthermore, the time complexities of both the encoding and decoding processes are polynomial when $p$ is a constant with respect to the code length.

cs.IT↗