Probing RLVR training instability through the lens of objective-level hacking

Prolonged reinforcement learning with verifiable rewards (RLVR) has been shown to drive continuous improvements in the reasoning capabilities of large language models, but the training is often prone to instabilities, especially in Mixture-of-Experts (MoE) architectures. Training instability severely undermines model capability improvement, yet its underlying causes and mechanisms remain poorly understood. In this work, we introduce a principled framework for understanding RLVR instability through the lens of objective-level hacking. Unlike reward hacking, which arises from exploitable verifiers, objective-level hacking emerges from token-level credit misalignment and is manifested as system-level spurious signals in the optimization objective. Grounded in our framework, together with extensive experiments on a 30B MoE model, we trace the origin and formalize the mechanism behind a key pathological training dynamic in MoE models: the abnormal growth of the training-inference discrepancy, a phenomenon widely associated with instability but previously lacking a mechanistic explanation. These findings provide a concrete and causal account of the training dynamics underlying instabilities in MoE models, offering guidance for the design of stable RLVR algorithms.

Convergence Rate Analysis of the AdamW-Style Shampoo: Unifying One-sided and Two-Sided Preconditioning

This paper studies the AdamW-style Shampoo optimizer, an effective implementation of classical Shampoo that notably won the external tuning track of the AlgoPerf neural network training algorithm competition. Our analysis unifies one-sided and two-sided preconditioning and establishes the convergence rate $\frac{1}{K}\sum_{k=1}^K E\left[\|\nabla f(X_k)\|_*\right]\leq O(\frac{\sqrt{m+n}C}{K^{1/4}})$ measured by nuclear norm, where $K$ represents the iteration number, $(m,n)$ denotes the size of matrix parameters, and $C$ matches the constant in the optimal convergence rate of SGD. Theoretically, we have $\|\nabla f(X)\|_F\leq \|\nabla f(X)\|_*\leq \sqrt{m+n}\|\nabla f(X)\|_F$, supporting that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[\|\nabla f(X_k)\|_F\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case of $\|\nabla f(X)\|_*= Θ(\sqrt{m+n})\|\nabla f(X)\|_F$.

Lightweight posterior construction for gravitational-wave catalogs with the Kolmogorov-Arnold network

Neural density estimation has seen widespread applications in the gravitational-wave (GW) data analysis, which enables real-time parameter estimation for compact binary coalescences and enhances rapid inference for subsequent analysis such as population inference. In this work, we explore the application of using the Kolmogorov-Arnold network (KAN) to construct efficient and interpretable neural density estimators for lightweight posterior construction of GW catalogs. By replacing conventional activation functions with learnable splines, KAN achieves superior interpretability, higher accuracy, and greater parameter efficiency on related scientific tasks. Leveraging this feature, we propose a KAN-based neural density estimator, which ingests megabyte-scale GW posterior samples and compresses them into model weights of tens of kilobytes. Subsequently, analytic expressions requiring only several kilobytes can be further distilled from these neural network weights with minimal accuracy trade-off. In practice, GW posterior samples with fidelity can be regenerated rapidly using the model weights or analytic expressions for subsequent analysis. Our lightweight posterior construction strategy is expected to facilitate user-level data storage and transmission, paving a path for efficient analysis of numerous GW events in the next-generation GW detectors.

P/D-Device: Disaggregated Large Language Model between Cloud and Devices

Serving disaggregated large language models has been widely adopted in industrial practice for enhanced performance. However, too many tokens generated in decoding phase, i.e., occupying the resources for a long time, essentially hamper the cloud from achieving a higher throughput. Meanwhile, due to limited on-device resources, the time to first token (TTFT), i.e., the latency of prefill phase, increases dramatically with the growth on prompt length. In order to concur with such a bottleneck on resources, i.e., long occupation in cloud and limited on-device computing capacity, we propose to separate large language model between cloud and devices. That is, the cloud helps a portion of the content for each device, only in its prefill phase. Specifically, after receiving the first token from the cloud, decoupling with its own prefill, the device responds to the user immediately for a lower TTFT. Then, the following tokens from cloud are presented via a speed controller for smoothed TPOT (the time per output token), until the device catches up with the progress. On-device prefill is then amortized using received tokens while the resource usage in cloud is controlled. Moreover, during cloud prefill, the prompt can be refined, using those intermediate data already generated, to further speed up on-device inference. We implement such a scheme P/D-Device, and confirm its superiority over other alternatives. We further propose an algorithm to decide the best settings. Real-trace experiments show that TTFT decreases at least 60%, maximum TPOT is about tens of milliseconds, and cloud throughput increases by up to 15x.

From Macro to Micro: Probing Dataset Diversity in Language Model Fine-Tuning

Dataset diversity plays a pivotal role for the successful training of many machine learning models, particularly in the supervised fine-tuning (SFT) stage of large language model (LLM) development. Despite increasing recognition of its importance, systematic analyses of dataset diversity still remain underexplored. To address this gap, this work presents a systematic taxonomy of existing diversity-control strategies, which primarily focus on the instruction component, operating at either macroscopic (entire instruction semantics) or mesoscopic levels (instruction units), and furthermore introduces a novel analysis of microscopic diversity within the response component, specifically analyzing the statistical distribution of tokens in SFT training samples. In the experimental evaluation, we construct fixed-size datasets (e.g., 10,000 samples each) from a corpus of 117,000 open-source SFT samples, incorporating six distinct diversity-control strategies spanning macro-, meso-, and microscopic levels applied to both instructions and responses. We then fine-tune LLMs on these datasets to assess the six diversity-control strategies. Results reveal that while macroscopic and mesoscopic strategies lead to higher performance with increasing diversity, the microscopic strategy in responses exhibits both a stronger correlation between model performance and the degree of diversity and superior performance with maximum diversity across all strategies. These findings offer actionable insights for constructing high-performance SFT datasets.

Stepsize anything: A unified learning rate schedule for budgeted-iteration training

The expanding computational costs and limited resources underscore the critical need for budgeted-iteration training, which aims to achieve optimal learning within predetermined iteration budgets.While learning rate schedules fundamentally govern the performance of different networks and tasks, particularly in budgeted-iteration scenarios, their design remains largely heuristic, lacking theoretical foundations.In addition, the optimal learning rate schedule requires extensive trial-and-error selection, making the training process inefficient.In this work, we propose the Unified Budget-Aware (UBA) schedule, a theoretically grounded learning rate schedule that consistently outperforms commonly-used schedules among diverse architectures and tasks under different constrained training budgets.First, we bridge the gap by constructing a novel training budget-aware optimization framework, which explicitly accounts for the robustness to landscape curvature variations.From this framework, we derive the UBA schedule, controlled by a single hyper-parameter $\varphi$ that provides a trade-off between flexibility and simplicity, eliminating the need for per-network numerical optimization. Moreover, we establish a theoretical connection between $\varphi$ and the condition number, adding interpretation and justification to our approach. Besides, we prove the convergence for different values of $\varphi$.We offer practical guidelines for its selection via theoretical analysis and empirical results.xtensive experimental results show that UBA \textit{consistently surpasses} the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets.

On the $O(\frac{\sqrt{d}}{K^{1/4}})$ Convergence Rate of AdamW Measured by $\ell_1$ Norm

As the default optimizer for training large language models, AdamW has achieved remarkable success in deep learning. However, its convergence behavior is not theoretically well-understood. This paper establishes the convergence rate $\frac{1}{K}\sum_{k=1}^KE\left[\|\nabla f(x^k)\|_1\right]\leq O(\frac{\sqrt{d}C}{K^{1/4}})$ for AdamW measured by $\ell_1$ norm, where $K$ represents the iteration number, $d$ denotes the model dimension, and $C$ matches the constant in the optimal convergence rate of SGD. Theoretically, we have $E\left[\|\nabla f(x)\|_1\right]\geq\sqrt{\frac{2d}{\pi}}E\left[\|\nabla f(x)\|_2\right]$ when each element of $\nabla f(x)$ is generated from Gaussian distribution $\mathcal N(0,1)$. Empirically, our experimental results on real-world deep learning tasks reveal $\|\nabla f(x)\|_1=\varTheta(\sqrt{d})\|\nabla f(x)\|_2$. Both support that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[\|\nabla f(x^k)\|_2\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD.

Convergence Rate Analysis of LION

The LION (evoLved sIgn mOmeNtum) optimizer for deep neural network training was found by Google via program search, with the simple sign update yet showing impressive performance in training large scale networks. Although previous studies have investigated its convergence properties, a comprehensive analysis, especially the convergence rate, is still desirable. Recognizing that LION can be regarded as solving a specific constrained problem, this paper focuses on demonstrating its convergence to the Karush-Kuhn-Tucker (KKT) point at the rate of $\cal O(\sqrt{d}K^{-1/4})$ measured by gradient $\ell_1$ norm, where $d$ is the problem dimension and $K$ is the number of iteration steps. Step further, we remove the constraint and establish that LION converges to the critical point of the general unconstrained problem at the same rate. This rate not only delivers the currently optimal dependence on the problem dimension $d$ but also tightly matches the theoretical lower bound for nonconvex stochastic optimization algorithms, which is typically measured using the gradient $\ell_2$ norm, with respect to the number of iterations $K$. Through extensive experiments, we not only demonstrate that LION achieves lower loss and higher performance compared to standard SGD, but also empirically confirm that the gradient $\ell_1/\ell_2$ norm ratio aligns with $\Theta(\sqrt{d})$, thus proving that our convergence rate matches the theoretical lower bound with respect to $d$ in the empirical sense.