Representation learning, particularly multi-task representation learning, has gained widespread popularity in various deep learning applications, ranging from computer vision to natural language processing, due to its remarkable generalization performance. Despite its growing use, our understanding of the underlying mechanisms remains limited. In this paper, we provide a theoretical analysis elucidating why multi-task representation learning outperforms its single-task counterpart in scenarios involving over-parameterized two-layer convolutional neural networks trained by gradient descent. Our analysis is based on a data model that encompasses both task-shared and task-specific features, a setting commonly encountered in real-world applications. We also present experiments on synthetic and real-world data to illustrate and validate our theoretical findings.

Agnostic learning of Boolean halfspaces is a fundamental problem in computational learning theory, but it is known to be computationally hard even for weak learning. Recent work \citep{chandrasekaran2024smoothed} proposed smoothed analysis as a way to bypass such hardness, but existing frameworks rely on additive Gaussian perturbations, making them unsuitable for discrete domains. We introduce a new smoothed agnostic learning framework for Boolean inputs, where perturbations are modeled via random bit flips. This defines a natural discrete analogue of smoothed optimality generalizing the Gaussian case. Under strictly subexponential assumptions on the input distribution, we give an efficient algorithm for learning halfspaces in this model, with runtime and sample complexity $\tilde{O}(n^{\mathrm{poly}(\frac{1}{\sigma\epsilon})})$. Previously, such algorithms were known only with strong structural assumptions for the discrete hypercube—for example, independent coordinates or symmetric distributions. Our result provides the first computationally efficient guarantee for smoothed agnostic learning of halfspaces over the Boolean hypercube, bridging the gap between worst-case intractability and practical learnability in discrete settings.

Discrete diffusion models have emerged as powerful tools for high-quality data generation. Despite their success in discrete spaces, such as text generation tasks, the acceleration of discrete diffusion models remains under-explored. In this paper, we propose discrete non-Markov diffusion models (DNDM), which naturally induce the predetermined transition time set. This enables a training-free sampling algorithm that significantly reduces the number of function evaluations (i.e., calls to the neural network), making the sampling process much faster. Furthermore, we study the transition from finite to infinite step sampling, offering new insights into bridging the gap between discrete and continuous-time processes for discrete diffusion models. Extensive experiments on natural language generation and machine translation tasks demonstrate the superior performance of our method in terms of both generation speed and sample quality compared to existing methods for discrete diffusion models. Codes are available at \url{https://github.com/uclaml/DNDM}.

The rapid evolution of large language models (LLMs) has expanded their capabilities across various data modalities, extending from well-established image data to increasingly popular graph data. Given the limitation of LLMs in hallucinations and inaccuracies in recalling factual knowledge, Knowledge Graph (KG) has emerged as a crucial data modality to support more accurate reasoning by LLMs. However, integrating structured knowledge from KGs into LLMs remains challenging, as most current KG-enhanced LLM methods directly convert the KG into linearized text triples, which is not as expressive as the original structured data. To address this, we introduce GraphVis, which conserves the intricate graph structure through the visual modality to enhance the comprehension of KGs with the aid of Large Vision Language Models (LVLMs). Our approach incorporates a unique curriculum fine-tuning scheme which first instructs LVLMs to recognize basic graphical features from the images, and subsequently incorporates reasoning on QA tasks with the visual graphs. This cross-modal methodology not only markedly enhances performance on standard textual QA but also shows improved zero-shot VQA performance by utilizing synthetic graph images to augment the data for VQA tasks. We present comprehensive evaluations across commonsense reasoning QA benchmarks, where GraphVis provides an average improvement of 11.1% over its base model and outperforms existing KG-enhanced LLM approaches. Across VQA benchmarks such as ScienceQA that share similar scientific diagram images, GraphVis provides a notable gain of 4.32%.

The $k$-sparse parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-sparse parity problem with sign stochastic gradient descent, a variant of stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that this approach can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\le O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{\Theta(k)}$ neurons, matching the established $\Omega(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with sign SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. To the best of our knowledge, this is the first result that matches the SQ lower bound for solving $k$-sparse parity problem using gradient-based methods.

Modern deep learning models with great expressive power can be trained to overfit the training data but still generalize well. This phenomenon is referred to as benign overfitting. Recently, a few studies have attempted to theoretically understand benign overfitting in neural networks. However, these works are either limited to neural networks with smooth activation functions or to the neural tangent kernel regime. How and when benign overfitting can occur in ReLU neural networks remains an open problem. In this work, we seek to answer this question by establishing algorithm-dependent risk bounds for learning two-layer ReLU convolutional neural networks with label-flipping noise. We show that, under mild conditions, the neural network trained by gradient descent can achieve near-zero training loss and Bayes optimal test risk. Our result also reveals a sharp transition between benign and harmful overfitting under different conditions on data distribution in terms of test risk. Experiments on synthetic data back up our theory.

The challenge of overfitting, in which the model memorizes the training data and fails to generalize to test data, has become increasingly significant in the training of large neural networks. To tackle this challenge, Sharpness-Aware Minimization (SAM) has emerged as a promising training method, which can improve the generalization of neural networks even in the presence of label noise. However, a deep understanding of how SAM works, especially in the setting of nonlinear neural networks and classification tasks, remains largely missing. This paper fills this gap by demonstrating why SAM generalizes better than Stochastic Gradient Descent (SGD) for a certain data model and two-layer convolutional ReLU networks. The loss landscape of our studied problem is nonsmooth, thus current explanations for the success of SAM based on the Hessian information are insufficient. Our result explains the benefits of SAM, particularly its ability to prevent noise learning in the early stages, thereby facilitating more effective learning of features. Experiments on both synthetic and real data corroborate our theory.

The implicit bias towards solutions with favorable properties is believed to be a key reason why neural networks trained by gradient-based optimization can generalize well. While the implicit bias of gradient flow has been widely studied for homogeneous neural networks (including ReLU and leaky ReLU networks), the implicit bias of gradient descent is currently only understood for smooth neural networks. Therefore, implicit bias in non-smooth neural networks trained by gradient descent remains an open question. In this paper, we aim to answer this question by studying the implicit bias of gradient descent for training two-layer fully connected (leaky) ReLU neural networks. We showed that when the training data are nearly-orthogonal, for leaky ReLU activation function, gradient descent will find a network with a stable rank that converges to $1$, whereas for ReLU activation function, gradient descent will find a neural network with a stable rank that is upper bounded by a constant. Additionally, we show that gradient descent will find a neural network such that all the training data points have the same normalized margin asymptotically. Experiments on both synthetic and real data backup our theoretical findings.

Deep neural networks (DNNs) have revealed severe vulnerability to adversarial perturbations, beside empirical adversarial training for robustness, the design of provably robust classifiers attracts more and more attention. Randomized smoothing methods provide the certified robustness with agnostic architecture, which is further extended to a provable robustness framework using f-divergence. While these methods cannot be applied to smoothing measures with bounded support set such as uniform probability measure due to the use of likelihood ratio in their certification methods. In this paper, we generalize the $f$-divergence-based framework to a Wasserstein-distance-based and total-variation-distance-based framework that is first able to analyze robustness properties of bounded support set smoothing measures both theoretically and experimentally. By applying our methodology to uniform probability measures with support set $l_p (p=1,2,\infty\text{ and general})$ ball, we prove negative certified robustness properties with respect to $l_q (q=1, 2, \infty)$ perturbations and present experimental results on CIFAR-10 dataset with ResNet to validate our theory. And it is also worth mentioning that our certification procedure only costs constant computation time.