Data augmentation (DA) is now a standard ingredient in modern machine learning pipelines, with extensive empirical evidence reporting improvements in generalization across modalities and tasks Mumuni and Mumuni (2022); Wang et al. (2025). It is often used to encode task-relevant symmetries directly into the training procedure, for instance by encouraging invariance to image rotations or other transformations of the input Shorten and Khoshgoftaar (2019); Chen et al. (2020). It has also been identified as one of the most effective regularization techniques across both supervised learning settings Bishop (1995); Cubuk et al. (2019); Mumuni and Mumuni (2022); Wang et al. (2025) and self-supervised/unsupervised learning Feng et al. (2021); Van Assel et al. (2025). Domain-specific augmentation pipelines have been central to progress in computer vision Shorten and Khoshgoftaar (2019); Kumar et al. (2024), natural language processing Feng et al. (2021); Shorten et al. (2021); Bayer et al. (2022), and time-series or audio applications Wen et al. (2020); Iwana and Uchida (2021); Iglesias et al. (2023). Despite these empirical successes, the benefits of DA remain highly task-and data-dependent, and augmentation schemes are often engineered in an ad hoc manner Fawzi et al. (2016); Cubuk et al. (2019); Lim et al. (2019); Hataya et al. (2020). In contrast with this rich empirical literature, comprehensive theoretical analyses of DA remain relatively scarce. Two classical starting points are, first, the interpretation of additive Gaussian noise as a form of explicit (ridge-like) regularization Bishop (1995); Lin et al. (2024), and second, the idea that leveraging distributional invariances and group structure in the learning objective helps decrease the variance of the model without increasing its bias Chen et al. (2020). Yet, when applied to modern and complex augmentation schemes, these works either provide only upper bounds on the generalization error Lin et al. (2024), or require very strong assumptions on the data distribution (e.g.

Self-supervised learning (SSL) -- a training paradigm in which models learn useful representations from unlabeled data by exploiting the data itself as a source of supervision -- has emerged as a foundational component of the recent success of transformer architectures. By avoiding the need for manual annotations, SSL retains many of the benefits traditionally associated with supervised learning while avoiding reliance on labeled data. Consequently, SSL is widely adopted as a pretraining paradigm for learning general-purpose representations that substantially accelerate the optimization of downstream tasks, especially in data-scarce settings. A canonical example of a self-supervised learning task is masked language modeling (MLM), in which a neural network is trained to predict masked tokens in text using the remaining tokens as contextual information (Devlin et al., 2019a; Howard and Ruder, 2018; Radford et al., 2018; Brown et al., 2020; OpenAI, 2024). For example, given the sentence "The capital of France is Paris", a typical MLM task would be to teach the model to infer that we are speaking about the capital of a country from the context "France" and "Paris" from the masked sentence "The [MASK] of France is Paris".

Partial Least Squares (PLS) is a widely used method for data integration, designed to extract latent components shared across paired high-dimensional datasets. Despite decades of practical success, a precise theoretical understanding of its behavior in high-dimensional regimes remains limited. In this paper, we study a data integration model in which two high-dimensional data matrices share a low-rank common latent structure while also containing individual-specific components. We analyze the singular vectors of the associated cross-covariance matrix using tools from random matrix theory and derive asymptotic characterizations of the alignment between estimated and true latent directions. These results provide a quantitative explanation of the reconstruction performance of the PLS variant based on Singular Value Decomposition (PLS-SVD) and identify regimes where the method exhibits counter-intuitive or limiting behavior. Building on this analysis, we compare PLS-SVD with principal component analysis applied separately to each dataset and show its asymptotic superiority in detecting the common latent subspace. Overall, our results offer a comprehensive theoretical understanding of high-dimensional PLS-SVD, clarifying both its advantages and fundamental limitations.

We prove a fundamental misalignment in transfer learning: the source regularization that minimizes source risk almost never coincides with the regularization maximizing transfer benefit. Through sharp phase boundaries for L2-SP ridge regression, we characterize the transfer-optimal source penalty $τ_0^*$ and show it diverges predictably from task-optimal values, requiring stronger regularization in high-SNR regimes and weaker regularization in low-SNR regimes. Additionally, in isotropic settings the decision to transfer is remarkably independent of target sample size and noise, depending only on task alignment and source characteristics. CIFAR-10 and MNIST experiments confirm this counterintuitive pattern persists in non-linear networks.

Fine-tuning has proven to be highly effective in adapting pre-trained models to perform better on new desired tasks with minimal data samples. Among the most widely used approaches are reparameterization methods, which update a target module by augmenting its frozen weight matrix with an additional trainable weight matrix. The most prominent example is Low Rank Adaption (LoRA), which gained significant attention in recent years. In this paper, we introduce a new class of reparameterization methods for transfer learning, designed to enhance the generalization ability of fine-tuned models. We establish the effectiveness of our approach in a high-dimensional binary classification setting using tools from Random Matrix Theory, and further validate our theoretical findings through more realistic experiments, such as fine-tuning LLMs.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count.

In-context learning (ICL) is a central capability of Transformer models, but the structures in data that enable its emergence and govern its robustness remain poorly understood. In this work, we study how the structure of pretraining tasks governs generalization in ICL. Using a solvable model for ICL of linear regression by linear attention, we derive an exact expression for ICL generalization error in high dimensions under arbitrary pretraining-testing task covariance mismatch. This leads to a new alignment measure that quantifies how much information about the pretraining task distribution is useful for inference at test time. We show that this measure directly predicts ICL performance not only in the solvable model but also in nonlinear Transformers. Our analysis further reveals a tradeoff between specialization and generalization in ICL: depending on task distribution alignment, increasing pretraining task diversity can either improve or harm test performance. Together, these results identify train-test task alignment as a key determinant of generalization in ICL.

Attention mechanisms have revolutionized machine learning (ML) by enabling efficient modeling of global dependencies across inputs. Their inherently parallelizable structures allow for efficient scaling with the exponentially increasing size of both pretrained data and model parameters. Yet, despite their central role as the computational backbone of modern large language models (LLMs), the theoretical understanding of Attentions, especially in the nonlinear setting, remains limited. In this paper, we provide a precise characterization of the \emph{in-context memorization error} of \emph{nonlinear Attention}, in the high-dimensional proportional regime where the number of input tokens $n$ and their embedding dimension $p$ are both large and comparable. Leveraging recent advances in the theory of large kernel random matrices, we show that nonlinear Attention typically incurs higher memorization error than linear ridge regression on random inputs. However, this gap vanishes, and can even be reversed, when the input exhibits statistical structure, particularly when the Attention weights align with the input signal direction. Our results reveal how nonlinearity and input structure interact with each other to govern the memorization performance of nonlinear Attention. The theoretical insights are supported by numerical experiments.