Heejune Sheen

Siyu Chen, Heejune Sheen, Tianhao Wang et al.

In-context learning (ICL) is a cornerstone of large language model (LLM) functionality, yet its theoretical foundations remain elusive due to the complexity of transformer architectures. In particular, most existing work only theoretically explains how the attention mechanism facilitates ICL under certain data models. It remains unclear how the other building blocks of the transformer contribute to ICL. To address this question, we study how a two-attention-layer transformer is trained to perform ICL on $n$-gram Markov chain data, where each token in the Markov chain statistically depends on the previous $n$ tokens. We analyze a sophisticated transformer model featuring relative positional embedding, multi-head softmax attention, and a feed-forward layer with normalization. We prove that the gradient flow with respect to a cross-entropy ICL loss converges to a limiting model that performs a generalized version of the induction head mechanism with a learned feature, resulting from the congruous contribution of all the building blocks. In the limiting model, the first attention layer acts as a $\mathit{copier}$, copying past tokens within a given window to each position, and the feed-forward network with normalization acts as a $\mathit{selector}$ that generates a feature vector by only looking at informationally relevant parents from the window. Finally, the second attention layer is a $\mathit{classifier}$ that compares these features with the feature at the output position, and uses the resulting similarity scores to generate the desired output. Our theory is further validated by experiments.

Siyu Chen, Heejune Sheen, Tianhao Wang et al.

We study the dynamics of gradient flow for training a multi-head softmax attention model for in-context learning of multi-task linear regression. We establish the global convergence of gradient flow under suitable choices of initialization. In addition, we prove that an interesting "task allocation" phenomenon emerges during the gradient flow dynamics, where each attention head focuses on solving a single task of the multi-task model. Specifically, we prove that the gradient flow dynamics can be split into three phases -- a warm-up phase where the loss decreases rather slowly and the attention heads gradually build up their inclination towards individual tasks, an emergence phase where each head selects a single task and the loss rapidly decreases, and a convergence phase where the attention parameters converge to a limit. Furthermore, we prove the optimality of gradient flow in the sense that the limiting model learned by gradient flow is on par with the best possible multi-head softmax attention model up to a constant factor. Our analysis also delineates a strict separation in terms of the prediction accuracy of ICL between single-head and multi-head attention models. The key technique for our convergence analysis is to map the gradient flow dynamics in the parameter space to a set of ordinary differential equations in the spectral domain, where the relative magnitudes of the semi-singular values of the attention weights determines task allocation. To our best knowledge, our work provides the first convergence result for the multi-head softmax attention model.

Heejune Sheen, Siyu Chen, Tianhao Wang et al.

We study gradient flow on the exponential loss for a classification problem with a one-layer softmax attention model, where the key and query weight matrices are trained separately. Under a separability assumption on the data, we show that when gradient flow achieves the minimal loss value, it further implicitly minimizes the nuclear norm of the product of the key and query weight matrices. Such implicit regularization can be described by a Support Vector Machine (SVM) problem with respect to the attention weights. This finding contrasts with prior results showing that the gradient descent induces an implicit regularization on the Frobenius norm on the product weight matrix when the key and query matrices are combined into a single weight matrix for training. For diagonal key and query matrices, our analysis builds upon the reparameterization technique and exploits approximate KKT conditions of the SVM associated with the classification data. Moreover, the results are extended to general weights configurations given proper alignment of the weight matrices' singular spaces with the data features at initialization.

Siyu Chen, Heejune Sheen, Xuyuan Xiong et al.

We study the challenge of achieving theoretically grounded feature recovery using Sparse Autoencoders (SAEs) for the interpretation of Large Language Models. Existing SAE training algorithms often lack rigorous mathematical guarantees and suffer from practical limitations such as hyperparameter sensitivity and instability. To address these issues, we first propose a novel statistical framework for the feature recovery problem, which includes a new notion of feature identifiability by modeling polysemantic features as sparse mixtures of underlying monosemantic concepts. Building on this framework, we introduce a new SAE training algorithm based on ``bias adaptation'', a technique that adaptively adjusts neural network bias parameters to ensure appropriate activation sparsity. We theoretically \highlight{prove that this algorithm correctly recovers all monosemantic features} when input data is sampled from our proposed statistical model. Furthermore, we develop an improved empirical variant, Group Bias Adaptation (GBA), and \highlight{demonstrate its superior performance against benchmark methods when applied to LLMs with up to 1.5 billion parameters}. This work represents a foundational step in demystifying SAE training by providing the first SAE algorithm with theoretical recovery guarantees, thereby advancing the development of more transparent and trustworthy AI systems through enhanced mechanistic interpretability.

Heejune Sheen, Xiaonan Zhu, Yao Xie

We estimate the general influence functions for spatio-temporal Hawkes processes using a tensor recovery approach by formulating the location dependent influence function that captures the influence of historical events as a tensor kernel. We assume a low-rank structure for the tensor kernel and cast the estimation problem as a convex optimization problem using the Fourier transformed nuclear norm (TNN). We provide theoretical performance guarantees for our approach and present an algorithm to solve the optimization problem. Moreover, we demonstrate the efficiency of our estimation with numerical simulations.