Bohang Zhang

Bohang Zhang, Du Jiang, Di He et al.

Designing neural networks with bounded Lipschitz constant is a promising way to obtain certifiably robust classifiers against adversarial examples. However, the relevant progress for the important $\ell_\infty$ perturbation setting is rather limited, and a principled understanding of how to design expressive $\ell_\infty$ Lipschitz networks is still lacking. In this paper, we bridge the gap by studying certified $\ell_\infty$ robustness from a novel perspective of representing Boolean functions. We derive two fundamental impossibility results that hold for any standard Lipschitz network: one for robust classification on finite datasets, and the other for Lipschitz function approximation. These results identify that networks built upon norm-bounded affine layers and Lipschitz activations intrinsically lose expressive power even in the two-dimensional case, and shed light on how recently proposed Lipschitz networks (e.g., GroupSort and $\ell_\infty$-distance nets) bypass these impossibilities by leveraging order statistic functions. Finally, based on these insights, we develop a unified Lipschitz network that generalizes prior works, and design a practical version that can be efficiently trained (making certified robust training free). Extensive experiments show that our approach is scalable, efficient, and consistently yields better certified robustness across multiple datasets and perturbation radii than prior Lipschitz networks. Our code is available at https://github.com/zbh2047/SortNet.

Bohang Zhang, Guhao Feng, Yiheng Du et al. · pku

Recently, subgraph GNNs have emerged as an important direction for developing expressive graph neural networks (GNNs). While numerous architectures have been proposed, so far there is still a limited understanding of how various design paradigms differ in terms of expressive power, nor is it clear what design principle achieves maximal expressiveness with minimal architectural complexity. To address these fundamental questions, this paper conducts a systematic study of general node-based subgraph GNNs through the lens of Subgraph Weisfeiler-Lehman Tests (SWL). Our central result is to build a complete hierarchy of SWL with strictly growing expressivity. Concretely, we prove that any node-based subgraph GNN falls into one of the six SWL equivalence classes, among which $\mathsf{SSWL}$ achieves the maximal expressive power. We also study how these equivalence classes differ in terms of their practical expressiveness such as encoding graph distance and biconnectivity. Furthermore, we give a tight expressivity upper bound of all SWL algorithms by establishing a close relation with localized versions of WL and Folklore WL (FWL) tests. Our results provide insights into the power of existing subgraph GNNs, guide the design of new architectures, and point out their limitations by revealing an inherent gap with the 2-FWL test. Finally, experiments demonstrate that $\mathsf{SSWL}$-inspired subgraph GNNs can significantly outperform prior architectures on multiple benchmarks despite great simplicity.

Bohang Zhang, Shengjie Luo, Liwei Wang et al.

Designing expressive Graph Neural Networks (GNNs) is a central topic in learning graph-structured data. While numerous approaches have been proposed to improve GNNs in terms of the Weisfeiler-Lehman (WL) test, generally there is still a lack of deep understanding of what additional power they can systematically and provably gain. In this paper, we take a fundamentally different perspective to study the expressive power of GNNs beyond the WL test. Specifically, we introduce a novel class of expressivity metrics via graph biconnectivity and highlight their importance in both theory and practice. As biconnectivity can be easily calculated using simple algorithms that have linear computational costs, it is natural to expect that popular GNNs can learn it easily as well. However, after a thorough review of prior GNN architectures, we surprisingly find that most of them are not expressive for any of these metrics. The only exception is the ESAN framework, for which we give a theoretical justification of its power. We proceed to introduce a principled and more efficient approach, called the Generalized Distance Weisfeiler-Lehman (GD-WL), which is provably expressive for all biconnectivity metrics. Practically, we show GD-WL can be implemented by a Transformer-like architecture that preserves expressiveness and enjoys full parallelizability. A set of experiments on both synthetic and real datasets demonstrates that our approach can consistently outperform prior GNN architectures.

Bohang Zhang, Du Jiang, Di He et al.

Recently, Zhang et al. (2021) developed a new neural network architecture based on $\ell_\infty$-distance functions, which naturally possesses certified $\ell_\infty$ robustness by its construction. Despite the novel design and theoretical foundation, so far the model only achieved comparable performance to conventional networks. In this paper, we make the following two contributions: $\mathrm{(i)}$ We demonstrate that $\ell_\infty$-distance nets enjoy a fundamental advantage in certified robustness over conventional networks (under typical certification approaches); $\mathrm{(ii)}$ With an improved training process we are able to significantly boost the certified accuracy of $\ell_\infty$-distance nets. Our training approach largely alleviates the optimization problem that arose in the previous training scheme, in particular, the unexpected large Lipschitz constant due to the use of a crucial trick called $\ell_p$-relaxation. The core of our training approach is a novel objective function that combines scaled cross-entropy loss and clipped hinge loss with a decaying mixing coefficient. Experiments show that using the proposed training strategy, the certified accuracy of $\ell_\infty$-distance net can be dramatically improved from 33.30% to 40.06% on CIFAR-10 ($ε=8/255$), meanwhile outperforming other approaches in this area by a large margin. Our results clearly demonstrate the effectiveness and potential of $\ell_\infty$-distance net for certified robustness. Codes are available at https://github.com/zbh2047/L_inf-dist-net-v2.

Kai Yang, Jan Ackermann, Zhenyu He et al. · pku

As transformer-based language models are trained on increasingly large datasets and with vast numbers of parameters, finding more efficient alternatives to the standard Transformer has become very valuable. While many efficient Transformers and Transformer alternatives have been proposed, none provide theoretical guarantees that they are a suitable replacement for the standard Transformer. This makes it challenging to identify when to use a specific model and what directions to prioritize for further investigation. In this paper, we aim to understand the capabilities and limitations of efficient Transformers, specifically the Sparse Transformer and the Linear Transformer. We focus on their reasoning capability as exhibited by Chain-of-Thought (CoT) prompts and follow previous works to model them as Dynamic Programming (DP) problems. Our results show that while these models are expressive enough to solve general DP tasks, contrary to expectations, they require a model size that scales with the problem size. Nonetheless, we identify a class of DP problems for which these models can be more efficient than the standard Transformer. We confirm our theoretical results through experiments on representative DP tasks, adding to the understanding of efficient Transformers' practical strengths and weaknesses.

Xinnan Dai, Kai Yang, Jay Revolinsky et al.

Recent studies suggest that large language models (LLMs) possess the capability to solve graph reasoning tasks. Notably, even when graph structures are embedded within textual descriptions, LLMs can still effectively answer related questions. This raises a fundamental question: How can a decoder-only Transformer architecture understand underlying graph structures? To address this, we start with the substructure extraction task, interpreting the inner mechanisms inside the transformers and analyzing the impact of the input queries. Specifically, through both empirical results and theoretical analysis, we present Induced Substructure Filtration (ISF), a perspective that captures the substructure identification in the multi-layer transformers. We further validate the ISF process in LLMs, revealing consistent internal dynamics across layers. Building on these insights, we explore the broader capabilities of Transformers in handling diverse graph types. Specifically, we introduce the concept of thinking in substructures to efficiently extract complex composite patterns, and demonstrate that decoder-only Transformers can successfully extract substructures from attributed graphs, such as molecular graphs. Together, our findings offer a new insight on how sequence-based Transformers perform the substructure extraction task over graph data.

Jingchu Gai, Yiheng Du, Bohang Zhang et al. · nvidia

Graph spectra are an important class of structural features on graphs that have shown promising results in enhancing Graph Neural Networks (GNNs). Despite their widespread practical use, the theoretical understanding of the power of spectral invariants -- particularly their contribution to GNNs -- remains incomplete. In this paper, we address this fundamental question through the lens of homomorphism expressivity, providing a comprehensive and quantitative analysis of the expressive power of spectral invariants. Specifically, we prove that spectral invariant GNNs can homomorphism-count exactly a class of specific tree-like graphs which we refer to as parallel trees. We highlight the significance of this result in various contexts, including establishing a quantitative expressiveness hierarchy across different architectural variants, offering insights into the impact of GNN depth, and understanding the subgraph counting capabilities of spectral invariant GNNs. In particular, our results significantly extend Arvind et al. (2024) and settle their open questions. Finally, we generalize our analysis to higher-order GNNs and answer an open question raised by Zhang et al. (2024).

Shuqi Lu, Xiaohong Ji, Bohang Zhang et al.

Molecular pretrained representations (MPR) has emerged as a powerful approach for addressing the challenge of limited supervised data in applications such as drug discovery and material design. While early MPR methods relied on 1D sequences and 2D graphs, recent advancements have incorporated 3D conformational information to capture rich atomic interactions. However, these prior models treat molecules merely as discrete atom sets, overlooking the space surrounding them. We argue from a physical perspective that only modeling these discrete points is insufficient. We first present a simple yet insightful observation: naively adding randomly sampled virtual points beyond atoms can surprisingly enhance MPR performance. In light of this, we propose a principled framework that incorporates the entire 3D space spanned by molecules. We implement the framework via a novel Transformer-based architecture, dubbed SpaceFormer, with three key components: (1) grid-based space discretization; (2) grid sampling/merging; and (3) efficient 3D positional encoding. Extensive experiments show that SpaceFormer significantly outperforms previous 3D MPR models across various downstream tasks with limited data, validating the benefit of leveraging the additional 3D space beyond atoms in MPR models.

Bohang Zhang, Lingxiao Zhao, Haggai Maron

Incorporating spectral information to enhance Graph Neural Networks (GNNs) has shown promising results but raises a fundamental challenge due to the inherent ambiguity of eigenvectors. Various architectures have been proposed to address this ambiguity, referred to as spectral invariant architectures. Notable examples include GNNs and Graph Transformers that use spectral distances, spectral projection matrices, or other invariant spectral features. However, the potential expressive power of these spectral invariant architectures remains largely unclear. The goal of this work is to gain a deep theoretical understanding of the expressive power obtainable when using spectral features. We first introduce a unified message-passing framework for designing spectral invariant GNNs, called Eigenspace Projection GNN (EPNN). A comprehensive analysis shows that EPNN essentially unifies all prior spectral invariant architectures, in that they are either strictly less expressive or equivalent to EPNN. A fine-grained expressiveness hierarchy among different architectures is also established. On the other hand, we prove that EPNN itself is bounded by a recently proposed class of Subgraph GNNs, implying that all these spectral invariant architectures are strictly less expressive than 3-WL. Finally, we discuss whether using spectral features can gain additional expressiveness when combined with more expressive GNNs.

Bohang Zhang, Jingchu Gai, Yiheng Du et al.

Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a unified framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric.

Guhao Feng, Bohang Zhang, Yuntian Gu et al.

Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the expressivity of LLMs with CoT in solving fundamental mathematical and decision-making problems. By using circuit complexity theory, we first give impossibility results showing that bounded-depth Transformers are unable to directly produce correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of constant size suffice to solve both tasks by generating CoT derivations using a commonly used math language format. Moreover, we show LLMs with CoT can handle a general class of decision-making problems known as Dynamic Programming, thus justifying its power in tackling complex real-world tasks. Finally, an extensive set of experiments show that, while Transformers always fail to directly predict the answers, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.

Jikai Jin, Bohang Zhang, Haiyang Wang et al.

Distributionally robust optimization (DRO) is a widely-used approach to learn models that are robust against distribution shift. Compared with the standard optimization setting, the objective function in DRO is more difficult to optimize, and most of the existing theoretical results make strong assumptions on the loss function. In this work we bridge the gap by studying DRO algorithms for general smooth non-convex losses. By carefully exploiting the specific form of the DRO objective, we are able to provide non-asymptotic convergence guarantees even though the objective function is possibly non-convex, non-smooth and has unbounded gradient noise. In particular, we prove that a special algorithm called the mini-batch normalized gradient descent with momentum, can find an $ε$ first-order stationary point within $O( ε^{-4} )$ gradient complexity. We also discuss the conditional value-at-risk (CVaR) setting, where we propose a penalized DRO objective based on a smoothed version of the CVaR that allows us to obtain a similar convergence guarantee. We finally verify our theoretical results in a number of tasks and find that the proposed algorithm can consistently achieve prominent acceleration.

Bohang Zhang, Tianle Cai, Zhou Lu et al.

It is well-known that standard neural networks, even with a high classification accuracy, are vulnerable to small $\ell_\infty$-norm bounded adversarial perturbations. Although many attempts have been made, most previous works either can only provide empirical verification of the defense to a particular attack method, or can only develop a certified guarantee of the model robustness in limited scenarios. In this paper, we seek for a new approach to develop a theoretically principled neural network that inherently resists $\ell_\infty$ perturbations. In particular, we design a novel neuron that uses $\ell_\infty$-distance as its basic operation (which we call $\ell_\infty$-dist neuron), and show that any neural network constructed with $\ell_\infty$-dist neurons (called $\ell_{\infty}$-dist net) is naturally a 1-Lipschitz function with respect to $\ell_\infty$-norm. This directly provides a rigorous guarantee of the certified robustness based on the margin of prediction outputs. We then prove that such networks have enough expressive power to approximate any 1-Lipschitz function with robust generalization guarantee. We further provide a holistic training strategy that can greatly alleviate optimization difficulties. Experimental results show that using $\ell_{\infty}$-dist nets as basic building blocks, we consistently achieve state-of-the-art performance on commonly used datasets: 93.09% certified accuracy on MNIST ($ε=0.3$), 35.42% on CIFAR-10 ($ε=8/255$) and 16.31% on TinyImageNet ($ε=1/255$).

Bohang Zhang, Jikai Jin, Cong Fang et al.

Gradient clipping is commonly used in training deep neural networks partly due to its practicability in relieving the exploding gradient problem. Recently, \citet{zhang2019gradient} show that clipped (stochastic) Gradient Descent (GD) converges faster than vanilla GD/SGD via introducing a new assumption called $(L_0, L_1)$-smoothness, which characterizes the violent fluctuation of gradients typically encountered in deep neural networks. However, their iteration complexities on the problem-dependent parameters are rather pessimistic, and theoretical justification of clipping combined with other crucial techniques, e.g. momentum acceleration, are still lacking. In this paper, we bridge the gap by presenting a general framework to study the clipping algorithms, which also takes momentum methods into consideration. We provide convergence analysis of the framework in both deterministic and stochastic setting, and demonstrate the tightness of our results by comparing them with existing lower bounds. Our results imply that the efficiency of clipping methods will not degenerate even in highly non-smooth regions of the landscape. Experiments confirm the superiority of clipping-based methods in deep learning tasks.