Self-supervised learning has become a cornerstone in computer vision, primarily divided into reconstruction-based methods like masked autoencoders (MAE) and discriminative methods such as contrastive learning (CL). Recent empirical observations reveal that MAE and CL capture different types of representations: CL tends to focus on global patterns, while MAE adeptly captures **both global and subtle local** information simultaneously. Despite a flurry of recent empirical investigations to shed light on this difference, theoretical understanding remains limited, especially on the dominant architecture **vision transformers** (ViTs). In this paper, to provide rigorous insights, we model the visual data distribution by considering two types of spatial features: dominant global features and comparatively minuscule local features, and study the impact of imbalance among these features. We analyze the training dynamics of one-layer softmax-based ViTs on both MAE and CL objectives using gradient descent. Our analysis shows that as the degree of feature imbalance varies, ViTs trained with the MAE objective effectively learn both global and local features to achieve near-optimal reconstruction, while the CL-trained ViTs favor predominantly global features, even under mild imbalance. These results provide a theoretical explanation for distinct behaviors of MAE and CL observed in empirical studies.

The ability to reason lies at the core of artificial intelligence (AI), and challenging problems usually call for deeper and longer reasoning to tackle. A crucial question about AI reasoning is whether models can extrapolate learned reasoning patterns to solve harder tasks with a longer chain-of-thought (CoT). In this work, we present a theoretical analysis of transformers learning on synthetic state-tracking tasks with gradient descent. Specifically: 1). We prove how the *algebraic structure* of state-tracking problems governs the length generalization of learned reasoning in transformers. In doing so, we formulate the **attention concentration** mechanism, linking the retrieval robustness of the attention layer to the task structure of long-context state tracking problems. 2). Moreover, we prove that a transformer can provably *self-improve* via a *recursive self-training* scheme that progressively extends the range of solvable problem lengths. We show that the model can achieve abilities outside the coverage of the base model in recursive training, different from prior theoretical works on self-improvement. To our knowledge, we provide the first *optimization guarantee* that constant-depth transformers provably learn $\text{NC}^1$-complete problems with CoT, significantly going beyond prior art confined in $\text{TC}^0$, unless the widely held conjecture $\text{TC}^0 \neq \text{NC}^1$ fails. Finally, we present a broad set of experiments supporting our theoretical results, confirming the length generalization behaviors and the mechanism of attention concentration.

Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a theoretical standpoint. This work investigates how transformers learn to solve symbolic multi-step reasoning problems through chain-of-thought processes, focusing on path-finding in trees. We analyze two intertwined tasks: a backward reasoning task, where the model outputs a path from a goal node to the root, and a more complex forward reasoning task, where the model implements two-stage reasoning by first identifying the goal-to-root path and then reversing it to produce the root-to-goal path. Our theoretical analysis, grounded in the dynamics of gradient descent, shows that trained one-layer transformers can provably solve both tasks with generalization guarantees to unseen trees. In particular, our multi-phase training dynamics for forward reasoning elucidate how different attention heads learn to specialize and coordinate autonomously to solve the two subtasks in a single autoregressive path. These results provide a mechanistic explanation of how trained transformers can implement sequential algorithmic procedures. Moreover, they offer insights into the emergence of reasoning abilities, suggesting that when tasks are structured to take intermediate chain-of-thought steps, even shallow multi-head transformers can effectively solve problems that would otherwise require deeper architectures.

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate $O(\frac{1}{{T}^2})$ with $T$ the number of steps, improving upon the $O(\frac{1}{T})$ rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate $O(\frac{1}{T})$, outperforming the rate $O(\frac{1}{\sqrt{T}})$ for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates $\ell_2$-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.

Transformers have recently revolutionized many domains in modern machine learning and one salient discovery is their remarkable in-context learning capability, where models can solve an unseen task by utilizing task-specific prompts without further parameters fine-tuning. This also inspired recent theoretical studies aiming to understand the in-context learning mechanism of transformers, which however focused only on *linear* transformers. In this work, we take the first step toward studying the learning dynamics of a one-layer transformer with *softmax* attention trained via gradient descent in order to in-context learn linear function classes. We consider a structured data model, where each token is randomly sampled from a set of feature vectors in either balanced or imbalanced fashion. For data with balanced features, we establish the finite-time convergence guarantee with near-zero prediction error by navigating our analysis over two phases of the training dynamics of the attention map. More notably, for data with imbalanced features, we show that the learning dynamics take a stage-wise convergence process, where the transformer first converges to a near-zero prediction error for the query tokens of dominant features, and then converges later to a near-zero error for query tokens of under-represented features, via one and four training phases. Our proof features new techniques for analyzing the competing strengths of two types of attention weights, the change of which determines different training phases.

In-context learning (ICL) refers to a remarkable capability of pretrained large language models, which can learn a new task given a few examples during inference. However, theoretical understanding of ICL is largely under-explored, particularly whether transformers can be trained to generalize to unseen examples in a prompt, which will require the model to acquire contextual knowledge of the prompt for generalization. This paper investigates the training dynamics of transformers by gradient descent through the lens of non-linear regression tasks. The contextual generalization here can be attained via learning the template function for each task in-context, where all template functions lie in a linear space with $m$ basis functions. We analyze the training dynamics of one-layer multi-head transformers to {in-contextly} predict unlabeled inputs given partially labeled prompts, where the labels contain Gaussian noise and the number of examples in each prompt are not sufficient to determine the template. Under mild assumptions, we show that the training loss for a one-layer multi-head transformer converges linearly to a global minimum. Moreover, the transformer effectively learns to perform ridge regression over the basis functions. To our knowledge, this study is the first provable demonstration that transformers can learn contextual (i.e., template) information to generalize to both unseen examples and tasks when prompts contain only a small number of query-answer pairs.

Despite the remarkable success of deep multi-modal learning in practice, it has not been well-explained in theory. Recently, it has been observed that the best uni-modal network outperforms the jointly trained multi-modal network across different combinations of modalities on various tasks, which is counter-intuitive since multiple signals would bring more information (Wang et al., 2020). This work provides a theoretical explanation for the emergence of such performance gap in neural networks for the prevalent joint training framework. Based on a simplified data distribution that captures the realistic property of multi-modal data, we prove that for multi-modal late-fusion network with (smoothed) ReLU activation trained jointly by gradient descent, different modalities will compete with each other and only a subset of modalities will be learned by its corresponding encoder networks. We refer to this phenomenon as modality competition, and the losing modalities, which fail to be discovered, are the origins where the sub-optimality of joint training comes from. In contrast, for uni-modal networks with similar learning settings, we provably show that the networks will focus on learning modality-associated features. Experimentally, we illustrate that modality competition matches the intrinsic behavior of late-fusion joint training to supplement our theoretical results. To the best of our knowledge, our work is the first theoretical treatment towards the degenerating aspect of multi-modal learning in neural networks.

Despite the empirical success of deep meta-learning, theoretical understanding of overparameterized meta-learning is still limited. This paper studies the generalization of a widely used meta-learning approach, Model-Agnostic Meta-Learning (MAML), which aims to find a good initialization for fast adaptation to new tasks. Under a mixed linear regression model, we analyze the generalization properties of MAML trained with SGD in the overparameterized regime. We provide both upper and lower bounds for the excess risk of MAML, which captures how SGD dynamics affect these generalization bounds. With such sharp characterizations, we further explore how various learning parameters impact the generalization capability of overparameterized MAML, including explicitly identifying typical data and task distributions that can achieve diminishing generalization error with overparameterization, and characterizing the impact of adaptation learning rate on both excess risk and the early stopping time. Our theoretical findings are further validated by experiments.

The world provides us with data of multiple modalities. Intuitively, models fusing data from different modalities outperform their uni-modal counterparts, since more information is aggregated. Recently, joining the success of deep learning, there is an influential line of work on deep multi-modal learning, which has remarkable empirical results on various applications. However, theoretical justifications in this field are notably lacking. Can multi-modal learning provably perform better than uni-modal?In this paper, we answer this question under a most popular multi-modal fusion framework, which firstly encodes features from different modalities into a common latent space and seamlessly maps the latent representations into the task space. We prove that learning with multiple modalities achieves a smaller population risk than only using its subset of modalities. The main intuition is that the former has a more accurate estimate of the latent space representation. To the best of our knowledge, this is the first theoretical treatment to capture important qualitative phenomena observed in real multi-modal applications from the generalization perspective. Combining with experiment results, we show that multi-modal learning does possess an appealing formal guarantee.