The standard convergence analysis of mini-batch stochastic gradient descent (SGD) models gradient noise using a single variance term that treats all parameter directions equally, ignoring the fact that noise in high-curvature directions has less impact because learning rates are already constrained there. We introduce Curvature-Weighted Gradient Diversity (CWGD), a geometry-aware measure that weights per-sample gradient diversity by the inverse square root of the Hessian, providing a tighter proxy for the effective optimization noise. For strongly convex quadratic objectives with diagonal Hessians and isotropic noise, we prove that a CWGD-modulated cosine learning-rate schedule can reduce the asymptotic optimization error floor by up to a factor of two compared with standard cosine annealing. We implement this idea as CWGD-Cosine using a Hutchinson-based diagonal Hessian estimator that is exact for quadratic objectives. Across a range of condition numbers, batch sizes, and noise structures, CWGD-Cosine consistently achieves approximately 20% lower final optimization error than standard cosine annealing while incurring negligible overhead in the quadratic setting. We also identify and correct a degenerate curvature estimator, analyze the robustness of the proposed estimator, and explicitly discuss the limitations of the method, including Hessian staleness in non-convex optimization. These results establish CWGD as a principled geometry-aware measure of optimization noise and motivate future extensions to more general learning problems.

Air pollution was responsible for 2.6 million deaths across South Asia in 2021 alone, with brick manufacturing contributing significantly to this burden. In particular, the Indo-Gangetic Plain; a densely populated and highly polluted region spanning northern India, Pakistan, Bangladesh, and parts of Afghanistan sees brick kilns contributing 8-14% of ambient air pollution. Traditional monitoring approaches, such as field surveys and manual annotation using tools like Google Earth Pro, are time and labor-intensive. Prior ML-based efforts for automated detection have relied on costly high-resolution commercial imagery and non-public datasets, limiting reproducibility and scalability. In this work, we introduce SENTINELKILNDB, a publicly available, hand-validated benchmark of 62,671 brick kilns spanning three kiln types Fixed Chimney Bull's Trench Kiln (FCBK), Circular FCBK (CFCBK), and Zigzag kilns - annotated with oriented bounding boxes (OBBs) across 2.8 million km2 using free and globally accessible Sentinel-2 imagery. We benchmark state-of-the-art oriented object detection models and evaluate generalization across in-region, out-of-region, and super-resolution settings. SENTINELKILNDB enables rigorous evaluation of geospatial generalization and robustness for low-resolution object detection, and provides a new testbed for ML models addressing real-world environmental and remote sensing challenges at a continental scale. Datasets and code are available in SentinelKilnDB Dataset and SentinelKilnDB Benchmark, under the Creative Commons Attribution-NonCommercial 4.0 International License.

Designing effective black-box optimizers is hampered by limited problem-specific knowledge and manual control that spans months for almost every detail. In this paper, we present DesignX, the first automated algorithm design framework that generates an effective optimizer specific to a given black-box optimization problem within seconds. Rooted in the first principles, we identify two key sub-tasks: 1) algorithm structure generation and 2) hyperparameter control. To enable systematic construction, a comprehensive modular algorithmic space is first built, embracing hundreds of algorithm components collected from decades of research. We then introduce a dual-agent reinforcement learning system that collaborates on structural and parametric design through a novel cooperative training objective, enabling large-scale meta-training across 10k diverse instances. Remarkably, through days of autonomous learning, the DesignX-generated optimizers continuously surpass human-crafted optimizers by orders of magnitude, either on synthetic testbed or on realistic optimization scenarios such as Protein-docking, AutoML and UAV path planning. Further in-depth analysis reveals DesignX's capability to discover non-trivial algorithm patterns beyond expert intuition, which, conversely, provides valuable design insights for the optimization community.

Adaptive optimization is critical in federated learning, where enabling adaptivity on both the server and client sides has proven essential for achieving optimal performance. However, the scalability of such jointly adaptive systems is often hindered by resource limitations in communication and memory. In this paper, we introduce a class of efficient adaptive algorithms, named FedAda2 and its enhanced version FedAda2++, designed specifically for large-scale, cross-device federated environments.

However, most existing optimizers stop searching the parameter space once they reach a local minimum. Given the complex geometric properties of the loss landscape, it is difficult to guarantee that such a point is the lowest or provides the best generalization. To address this, we propose an adaptor "E" for gradient-based optimizers. The adapted optimizer tends to continue exploring along landscape 5.0 valleys (areas with low and nearly identical losses) in order to search for potentially1.0

Multi-task learning (MTL) is a widely explored paradigm that enables the simultaneous learning of multiple tasks using a single model. Despite numerous solutions, the key issues of optimization conflict and task imbalance remain under-addressed, limiting performance. Unlike existing optimization-based approaches that typically reweight task losses or gradients to mitigate conflicts or promote progress, we propose a novel approach based on Continual Optimization with Symmetry Teleportation (COST). During MTL optimization, when an optimization conflict arises, we seek an alternative loss-equivalent point on the loss landscape to reduce conflict. Specifically, we utilize a low-rank adapter (LoRA) to facilitate this practical teleportation by designing convergent, loss-invariant objectives. Additionally, we introduce a historical trajectory reuse strategy to continually leverage the benefits of advanced optimizers. Extensive experiments on multiple mainstream datasets demonstrate the effectiveness of our approach. COSTis a plug-and-play solution that enhances a wide range of existing MTL methods. When integrated with state-of-the-art methods, COSTachieves superior performance.

In modern optimization methods used in deep learning, each update depends on the history of previous iterations, often referred to as memory, and this dependence decays fast as the iterates go further into the past. For example, gradient descent with momentum has exponentially decaying memory through exponentially averaged past gradients. We introduce a general technique for identifying a memoryless algorithm that approximates an optimization algorithm with memory. It is obtained by replacing all past iterates in the update by the current one, and then adding a correction term arising from memory (also a function of the current iterate). This correction term can be interpreted as a perturbation of the loss, and the nature of this perturbation can inform how memory implicitly (anti-)regularizes the optimization dynamics. As an application of our theory, we find that Lion does not have the kind of implicit anti-regularization induced by memory that AdamW does, providing a theory-based explanation for Lion's better generalization performance recently documented [13]. Empirical evaluations confirm our theoretical findings.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Therefore, practitioners often treat gradients as inputs to task-specific algorithms, e.g. for pruning or optimization. Recent works explore learning algorithms that operate directly on gradients but use architectures that are not specifically designed for gradient processing, limiting their applicability. In this paper, we present a principled approach for designing architectures that process gradients. Our approach is guided by three principles: (1) equivariant design that preserves neuron permutation symmetries, (2) processing sets of gradients across multiple data points to capture curvature information, and (3) efficient gradient representation through rank-1 decomposition. Based on these principles, we introduce GradMetaNet, a novel architecture for learning on gradients, constructed from simple equivariant blocks. We prove universality results for GradMetaNet, and show that previous approaches cannot approximate natural gradient-based functions that GradMetaNet can. We then demonstrate GradMetaNet's effectiveness on a diverse set of gradient-based tasks on MLPs and transformers, such as learned optimization, INR editing, and estimating loss landscape curvature.

Large language models (LLMs) are increasingly used in learning algorithms, evaluations, and optimization tasks. Recent studies have shown that using LLM-based optimizers to automatically optimize model prompts, demonstrations, predictions themselves, or other components can significantly enhance the performance of AI systems, as demonstrated by frameworks such as DSPy and TextGrad. However, optimizers built on language models themselves are usually designed by humans with manual design choices; optimizers themselves are not optimized. Moreover, these optimizers are general purpose by design, to be useful to a broad audience, and are not tailored for specific tasks. To address these challenges, we propose metaTextGrad, which focuses on designing a meta-optimizer to further enhance existing optimizers and align them to be good optimizers for a given task. Our approach consists of two key components: a meta prompt optimizer and a meta structure optimizer. The combination of these two significantly improves performance across multiple benchmarks, achieving an average absolute performance improvement of up to 6% compared to the best baseline.