Single Domain Generalization (SDG) remains a formidable challenge in the field of machine learning, particularly when models are deployed in environments that differ significantly from their training domains. In this paper, we propose a novel data augmentation approach, named as Model-aware Parametric Batch-wise Mixup (MPBM), to tackle the challenge of SDG. MPBM deploys adversarial queries generated with stochastic gradient Langevin dynamics, and produces model-aware augmenting instances with a parametric batch-wise mixup generator network that is carefully designed through an innovative attention mechanism. By exploiting inter-feature correlations, the parameterized mixup generator introduces additional versatility in combining features across a batch of instances, thereby enhancing the capacity to generate highly adaptive and informative synthetic instances for specific queries. The synthetic data produced by this adaptable generator network, guided by informative queries, is expected to significantly enrich the representation space covered by the original training dataset and subsequently enhance the prediction model's generalizability across diverse and previously unseen domains. To prevent excessive deviation from the training data, we further incorporate a real-data alignment-based adversarial loss into the learning process of MPBM, regularizing any tendencies toward undesirable expansions. We conduct extensive experiments on several benchmark datasets. The empirical results demonstrate that by augmenting the training set with informative synthesis data, our proposed MPBM method achieves the state-of-the-art performance for single domain generalization.

Deep networks often have an enormous amount of parameters an d are theoretically capable of overfitting the training data. However, in practice, deep networks trai ned via gradient descent (GD) or its variants often generalize well. This is commonly attributed to the implicit bias of GD, in which GD finds a certain solution that prevents overfitting ( Zhang et al., 2021; Neyshabur et al., 2017; Bartlett et al., 2021). Understanding the implicit bias of GD is one of the central topics in deep learni ng theory. The implicit bias of GD is relatively well-understood when t he network is homogeneous (see Soudry et al., 2018; Ji and Telgarsky, 2018; Lyu and Li, 2020; Ji and Telgarsky, 2020; Wu et al., 2023, and references therein). For linear networks trained on linearly separabl e data, GD diverges in norm while converging in direction to the maximum margin solution ( Soudry et al., 2018; Ji and Telgarsky, 2018; Wu et al., 2023). Similar results have been established for generic homogene ous networks that include a class of deep networks, assuming that the network at initialization can sepa rate the training data.

In today's world, machine learning is hard to imagine without large training datasets and models. This has led to the use of stochastic methods for training, such as stochastic gradient descent (SGD). SGD provides weak theoretical guarantees of convergence, but there are modifications, such as Stochastic Variance Reduced Gradient (SVRG) and StochAstic Recursive grAdient algoritHm (SARAH), that can reduce the variance. These methods require the computation of the full gradient occasionally, which can be time consuming. In this paper, we explore variants of variance reduction algorithms that eliminate the need for full gradient computations. To make our approach memory-efficient and avoid full gradient computations, we use two key techniques: the shuffling heuristic and idea of SAG/SAGA methods. As a result, we improve existing estimates for variance reduction algorithms without the full gradient computations. Additionally, for the non-convex objective function, our estimate matches that of classic shuffling methods, while for the strongly convex one, it is an improvement. We conduct comprehensive theoretical analysis and provide extensive experimental results to validate the efficiency and practicality of our methods for large-scale machine learning problems.

We solve a regularized weighted low-rank approximation problem by a stochastic gradient descent on a manifold. To guarantee the convergence of our stochastic gradient descent, we establish a convergence theorem on manifolds for retraction-based stochastic gradient descents admitting confinements. On sample data from the Netflix Prize training dataset, our algorithm outperforms the existing stochastic gradient descent on Euclidean spaces. We also compare the accelerated line search on this manifold to the existing accelerated line search on Euclidean spaces.

Understanding the advantages of deep neural networks trained by gradient descent (GD) compared to shallow models remains an open theoretical challenge. While the study of multi-index models with Gaussian data in high dimensions has provided analytical insights into the benefits of GD-trained neural networks over kernels, the role of depth in improving sample complexity and generalization in GD-trained networks remains poorly understood. In this paper, we introduce a class of target functions (single and multi-index Gaussian hierarchical targets) that incorporate a hierarchy of latent subspace dimensionalities. This framework enables us to analytically study the learning dynamics and generalization performance of deep networks compared to shallow ones in the high-dimensional limit. Specifically, our main theorem shows that feature learning with GD reduces the effective dimensionality, transforming a high-dimensional problem into a sequence of lower-dimensional ones. This enables learning the target function with drastically less samples than with shallow networks. While the results are proven in a controlled training setting, we also discuss more common training procedures and argue that they learn through the same mechanisms. These findings open the way to further quantitative studies of the crucial role of depth in learning hierarchical structures with deep networks.

In overparameterized logistic regression, gradient descent (GD) iterates diverge in norm while converging in direction to the maximum $\ell_2$-margin solution -- a phenomenon known as the implicit bias of GD. This work investigates additional regularization effects induced by early stopping in well-specified high-dimensional logistic regression. We first demonstrate that the excess logistic risk vanishes for early-stopped GD but diverges to infinity for GD iterates at convergence. This suggests that early-stopped GD is well-calibrated, whereas asymptotic GD is statistically inconsistent. Second, we show that to attain a small excess zero-one risk, polynomially many samples are sufficient for early-stopped GD, while exponentially many samples are necessary for any interpolating estimator, including asymptotic GD. This separation underscores the statistical benefits of early stopping in the overparameterized regime. Finally, we establish nonasymptotic bounds on the norm and angular differences between early-stopped GD and $\ell_2$-regularized empirical risk minimizer, thereby connecting the implicit regularization of GD with explicit $\ell_2$-regularization.

Learned Index Structures (LIS) have significantly advanced data management by leveraging machine learning models to optimize data indexing. However, designing these structures often involves critical trade-offs, making it challenging for both designers and end-users to find an optimal balance tailored to specific workloads and scenarios. While some indexes offer adjustable parameters that demand intensive manual tuning, others rely on fixed configurations based on heuristic auto-tuners or expert knowledge, which may not consistently deliver optimal performance. This paper introduces LITune, a novel framework for end-to-end automatic tuning of Learned Index Structures. LITune employs an adaptive training pipeline equipped with a tailor-made Deep Reinforcement Learning (DRL) approach to ensure stable and efficient tuning. To accommodate long-term dynamics arising from online tuning, we further enhance LITune with an on-the-fly updating mechanism termed the O2 system. These innovations allow LITune to effectively capture state transitions in online tuning scenarios and dynamically adjust to changing data distributions and workloads, marking a significant improvement over other tuning methods. Our experimental results demonstrate that LITune achieves up to a 98% reduction in runtime and a 17-fold increase in throughput compared to default parameter settings given a selected Learned Index instance. These findings highlight LITune's effectiveness and its potential to facilitate broader adoption of LIS in real-world applications.

Language Models (LLMs) are often quantized to lower precision to reduce the memory cost and latency in inference. However, quantization often degrades model performance, thus fine-tuning is required for various down-stream tasks. Traditional fine-tuning methods such as stochastic gradient descent and Adam optimization require backpropagation, which are error-prone in the low-precision settings. To overcome these limitations, we propose the Quantized Zeroth-Order (QuZO) framework, specifically designed for fine-tuning LLMs through low-precision (e.g., 4- or 8-bit) forward passes. Our method can avoid the error-prone low-precision straight-through estimator, and utilizes optimized stochastic rounding to mitigate the increased bias. QuZO simplifies the training process, while achieving results comparable to first-order methods in ${\rm FP}8$ and superior accuracy in ${\rm INT}8$ and ${\rm INT}4$ training. Experiments demonstrate that low-bit training QuZO achieves performance comparable to MeZO optimization on GLUE, Multi-Choice, and Generation tasks, while reducing memory cost by $2.94 \times$ in LLaMA2-7B fine-tuning compared to quantized first-order methods.

Understanding the quantum control landscape (QCL) is important for designing effective quantum control strategies. In this study, we analyze the QCL for a single two-level quantum system (qubit) using various control strategies. We employ Principal Component Analysis (PCA), to visualize and analyze the QCL for higher dimensional control parameters. Our results indicate that dimensionality reduction techniques such as PCA, can play an important role in understanding the complex nature of quantum control in higher dimensions. Evaluations of traditional control techniques and machine learning algorithms reveal that Genetic Algorithms (GA) outperform Stochastic Gradient Descent (SGD), while Q-learning (QL) shows great promise compared to Deep Q-Networks (DQN) and Proximal Policy Optimization (PPO). Additionally, our experiments highlight the importance of reward function design in DQN and PPO demonstrating that using immediate reward results in improved performance rather than delayed rewards for systems with short time steps. A study of solution space complexity was conducted by using Cluster Density Index (CDI) as a key metric for analyzing the density of optimal solutions in the landscape. The CDI reflects cluster quality and helps determine whether a given algorithm generates regions of high fidelity or not. Our results provide insights into effective quantum control strategies, emphasizing the significance of parameter selection and algorithm optimization.

Recent MLLMs have shown emerging visual understanding and reasoning abilities after being pre-trained on large-scale multimodal datasets. Unlike pre-training, where MLLMs receive rich visual-text alignment, instruction-tuning is often text-driven with weaker visual supervision, leading to the degradation of pre-trained visual understanding and causing visual forgetting. Existing approaches, such as direct fine-tuning and continual learning methods, fail to explicitly address this issue, often compressing visual representations and prioritizing task alignment over visual retention, which further worsens visual forgetting. To overcome this limitation, we introduce a novel perspective leveraging effective rank to quantify the degradation of visual representation richness, interpreting this degradation through the information bottleneck principle as excessive compression that leads to the degradation of crucial pre-trained visual knowledge. Building on this view, we propose a modality-decoupled gradient descent (MDGD) method that regulates gradient updates to maintain the effective rank of visual representations while mitigating the over-compression effects described by the information bottleneck. By explicitly disentangling the optimization of visual understanding from task-specific alignment, MDGD preserves pre-trained visual knowledge while enabling efficient task adaptation. To enable lightweight instruction-tuning, we further develop a memory-efficient fine-tuning approach using gradient masking, which selectively updates a subset of model parameters to enable parameter-efficient fine-tuning (PEFT), reducing computational overhead while preserving rich visual representations. Extensive experiments across various downstream tasks and backbone MLLMs demonstrate that MDGD effectively mitigates visual forgetting from pre-trained tasks while enabling strong adaptation to new tasks.