Stochastic gradient-based descent (SGD), have long been central to training large language models (LLMs). However, their effectiveness is increasingly being questioned, particularly in large-scale applications where empirical evidence suggests potential performance limitations. In response, this paper proposes a stochastic conjugate subgradient method together with adaptive sampling tailored specifically for training LLMs. The method not only achieves faster convergence per iteration but also demonstrates improved scalability compared to traditional SGD techniques. It leverages sample complexity analysis to adaptively choose the sample size, employs a stochastic conjugate subgradient approach to determine search directions and utilizing an AdamW-like algorithm to adaptively adjust step sizes. This approach preserves the key advantages of first-order methods while effectively addressing the nonconvexity and non-smoothness inherent in LLMs training. Additionally, we provide a detailed analysis of the advantage of the algorithm. Experimental results show that the proposed method not only maintains, but in many cases surpasses, the scalability of traditional SGD techniques, significantly enhancing both the speed and accuracy of the optimization process.

While digital twins (DT) hold promise for providing real-time insights into complex energy assets, much of the current literature either does not offer a clear framework for information exchange between the model and the asset, lacks key features needed for real-time implementation, or gives limited attention to model uncertainty. Here, we aim to solve these gaps by proposing a variational digital twin (VDT) framework that augments standard neural architectures with a single Bayesian output layer. This lightweight addition, along with a novel VDT updating algorithm, lets a twin update in seconds on commodity GPUs while producing calibrated uncertainty bounds that can inform experiment design, control algorithms, and model reliability. The VDT is evaluated on four energy-sector problems. For critical-heat-flux prediction, uncertainty-driven active learning reaches R2 = 0.98 using 47 % fewer experiments and one-third the training time of random sampling. A three-year renewable-generation twin maintains R2 > 0.95 for solar output and curbs error growth for volatile wind forecasts via monthly updates that process only one month of data at a time. A nuclear reactor transient cooldown twin reconstructs thermocouple signals with R2 > 0.99 and preserves accuracy after 50 % sensor loss, demonstrating robustness to degraded instrumentation. Finally, a physics-informed Li-ion battery twin, retrained after every ten discharges, lowers voltage mean-squared error by an order of magnitude relative to the best static model while adapting its credible intervals as the cell approaches end-of-life. These results demonstrate that combining modest Bayesian augmentation with efficient update schemes turns conventional surrogates into uncertainty-aware, data-efficient, and computationally tractable DTs, paving the way for dependable models across industrial and scientific energy systems.

This work assesses both empirically and theoretically, using the performance estimation methodology, how robust different first-order optimization methods are when subject to relative inexactness in their gradient computations. Relative inexactness occurs, for example, when compressing the gradient using fewer bits of information, which happens when dealing with large-scale problems on GPUs. Three major families of methods are analyzed: constant step gradient descent, long-step methods, and accelerated methods. The latter two are first shown to be theoretically not robust to inexactness. Then, a semi-heuristic shortening factor is introduced to improve their theoretical guarantees. All methods are subsequently tested on a concrete inexact problem, with two different types of relative inexactness, and it is observed that both accelerated methods are much more robust than expected, and that the shortening factor significantly helps the long-step methods. In the end, all shortened methods appear to be promising, even in this inexact setting.

The Edge computing paradigm has gained prominence in both academic and industry circles in recent years. By implementing edge computing facilities and services in robotics, it becomes a key enabler in the deployment of artificial intelligence applications to robots. Time-sensitive robotics applications benefit from the reduced latency, mobility, and location awareness provided by the edge computing paradigm, which enables real-time data processing and intelligence at the network's edge. While the advantages of integrating edge computing into robotics are numerous, there has been no recent survey that comprehensively examines these benefits. This paper aims to bridge that gap by highlighting important work in the domain of edge robotics, examining recent advancements, and offering deeper insight into the challenges and motivations behind both current and emerging solutions. In particular, this article provides a comprehensive evaluation of recent developments in edge robotics, with an emphasis on fundamental applications, providing in-depth analysis of the key motivations, challenges, and future directions in this rapidly evolving domain. It also explores the importance of edge computing in real-world robotics scenarios where rapid response times are critical. Finally, the paper outlines various open research challenges in the field of edge robotics.

In this paper, we propose an improved numerical algorithm for solving minimax problems based on nonsmooth optimization, quadratic programming and iterative process. We also provide a rigorous proof of convergence for our algorithm under some mild assumptions, such as gradient continuity and boundedness. Such an algorithm can be widely applied in various fields such as robust optimization, imbalanced learning, etc.

We consider a Bayesian diffusion control problem of expected terminal utility maximization. The controller imposes a prior distribution on the unknown drift of an underlying diffusion. The Bayesian optimal control, tracking the posterior distribution of the unknown drift, can be characterized explicitly. However, in practice, the prior will generally be incorrectly specified, and the degree of model misspecification can have a significant impact on policy performance. To mitigate this and reduce overpessimism, we introduce a distributionally robust Bayesian control (DRBC) formulation in which the controller plays a game against an adversary who selects a prior in divergence neighborhood of a baseline prior. The adversarial approach has been studied in economics and efficient algorithms have been proposed in static optimization settings. We develop a strong duality result for our DRBC formulation. Combining these results together with tools from stochastic analysis, we are able to derive a loss that can be efficiently trained (as we demonstrate in our numerical experiments) using a suitable neural network architecture. As a result, we obtain an effective algorithm for computing the DRBC optimal strategy. The methodology for computing the DRBC optimal strategy is greatly simplified, as we show, in the important case in which the adversary chooses a prior from a Kullback-Leibler distributional uncertainty set.

Optimizing high-dimensional black-box functions under black-box constraints is a pervasive task in a wide range of scientific and engineering problems. These problems are typically harder than unconstrained problems due to hard-to-find feasible regions. While Bayesian optimization (BO) methods have been developed to solve such problems, they often struggle with the curse of dimensionality. Recently, generative model-based approaches have emerged as a promising alternative for constrained optimization. However, they suffer from poor scalability and are vulnerable to mode collapse, particularly when the target distribution is highly multi-modal. In this paper, we propose a new framework to overcome these challenges. Our method iterates through two stages. First, we train flow-based models to capture the data distribution and surrogate models that predict both function values and constraint violations with uncertainty quantification. Second, we cast the candidate selection problem as a posterior inference problem to effectively search for promising candidates that have high objective values while not violating the constraints. During posterior inference, we find that the posterior distribution is highly multi-modal and has a large plateau due to constraints, especially when constraint feedback is given as binary indicators of feasibility. To mitigate this issue, we amortize the sampling from the posterior distribution in the latent space of flow-based models, which is much smoother than that in the data space. We empirically demonstrate that our method achieves superior performance on various synthetic and real-world constrained black-box optimization tasks. Our code is publicly available \href{https://github.com/umkiyoung/CiBO}{here}.

This thesis contributes to the theoretical understanding of local update algorithms, especially Local SGD, in distributed and federated optimization under realistic models of data heterogeneity. A central focus is on the bounded second-order heterogeneity assumption, which is shown to be both necessary and sufficient for local updates to outperform centralized or mini-batch methods in convex and non-convex settings. The thesis establishes tight upper and lower bounds in several regimes for various local update algorithms and characterizes the min-max complexity of multiple problem classes. At its core is a fine-grained consensus-error-based analysis framework that yields sharper finite-time convergence bounds under third-order smoothness and relaxed heterogeneity assumptions. The thesis also extends to online federated learning, providing fundamental regret bounds under both first-order and bandit feedback. Together, these results clarify when and why local updates offer provable advantages, and the thesis serves as a self-contained guide for analyzing Local SGD in heterogeneous environments.

HIV epidemiological data is increasingly complex, requiring advanced computation for accurate cluster detection and forecasting. We employed quantum-accelerated machine learning to analyze HIV prevalence at the ZIP-code level using AIDSVu and synthetic SDoH data for 2022. Our approach compared classical clustering (DBSCAN, HDBSCAN) with a quantum approximate optimization algorithm (QAOA), developed a hybrid quantum-classical neural network for HIV prevalence forecasting, and used quantum Bayesian networks to explore causal links between SDoH factors and HIV incidence. The QAOA-based method achieved 92% accuracy in cluster detection within 1.6 seconds, outperforming classical algorithms. Meanwhile, the hybrid quantum-classical neural network predicted HIV prevalence with 94% accuracy, surpassing a purely classical counterpart. Quantum Bayesian analysis identified housing instability as a key driver of HIV cluster emergence and expansion, with stigma exerting a geographically variable influence. These quantum-enhanced methods deliver greater precision and efficiency in HIV surveillance while illuminating critical causal pathways. This work can guide targeted interventions, optimize resource allocation for PrEP, and address structural inequities fueling HIV transmission.

Classifier guidance is intended to steer a diffusion process such that a given classifier reliably recognizes the generated data point as a certain class. However, most classifier guidance approaches are restricted to robust classifiers, which were specifically trained on the noise of the diffusion forward process. We extend classifier guidance to work with general, non-robust, classifiers that were trained without noise. We analyze the sensitivity of both non-robust and robust classifiers to noise of the diffusion process on the standard CelebA data set, the specialized SportBalls data set and the high-dimensional real-world CelebA-HQ data set. Our findings reveal that non-robust classifiers exhibit significant accuracy degradation under noisy conditions, leading to unstable guidance gradients. To mitigate these issues, we propose a method that utilizes one-step denoised image predictions and implements stabilization techniques inspired by stochastic optimization methods, such as exponential moving averages. Experimental results demonstrate that our approach improves the stability of classifier guidance while maintaining sample diversity and visual quality. This work contributes to advancing conditional sampling techniques in generative models, enabling a broader range of classifiers to be used as guidance classifiers.