arXiv subjects

Difan Zou

Publications and source records attributed to Difan Zou.

At least 91 records · Page 5Linked to original sources

arXiv2024

The Implicit Bias of Adam on Separable Data

Adam has become one of the most favored optimizers in deep learning problems. Despite its success in practice, numerous mysteries persist regarding its theoretical understanding. In this paper, we study the implicit bias of Adam in linear logistic regression. Specifically, we show that when the training data are linearly separable, Adam converges towards a linear classifier that achieves the maximum $\ell_\infty$-margin. Notably, for a general class of diminishing learning rates, this convergence occurs within polynomial time. Our result shed light on the difference between Adam and (stochastic) gradient descent from a theoretical perspective.

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arXiv2024

Improving Group Robustness on Spurious Correlation Requires Preciser Group Inference

Standard empirical risk minimization (ERM) models may prioritize learning spurious correlations between spurious features and true labels, leading to poor accuracy on groups where these correlations do not hold. Mitigating this issue often requires expensive spurious attribute (group) labels or relies on trained ERM models to infer group labels when group information is unavailable. However, the significant performance gap in worst-group accuracy between using pseudo group labels and using oracle group labels inspires us to consider further improving group robustness through preciser group inference. Therefore, we propose GIC, a novel method that accurately infers group labels, resulting in improved worst-group performance. GIC trains a spurious attribute classifier based on two key properties of spurious correlations: (1) high correlation between spurious attributes and true labels, and (2) variability in this correlation between datasets with different group distributions. Empirical studies on multiple datasets demonstrate the effectiveness of GIC in inferring group labels, and combining GIC with various downstream invariant learning methods improves worst-group accuracy, showcasing its powerful flexibility. Additionally, through analyzing the misclassifications in GIC, we identify an interesting phenomenon called semantic consistency, which may contribute to better decoupling the association between spurious attributes and labels, thereby mitigating spurious correlation. The code for GIC is available at https://github.com/yujinhanml/GIC.

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arXiv2024

A Human-Like Reasoning Framework for Multi-Phases Planning Task with Large Language Models

Recent studies have highlighted their proficiency in some simple tasks like writing and coding through various reasoning strategies. However, LLM agents still struggle with tasks that require comprehensive planning, a process that challenges current models and remains a critical research issue. In this study, we concentrate on travel planning, a Multi-Phases planning problem, that involves multiple interconnected stages, such as outlining, information gathering, and planning, often characterized by the need to manage various constraints and uncertainties. Existing reasoning approaches have struggled to effectively address this complex task. Our research aims to address this challenge by developing a human-like planning framework for LLM agents, i.e., guiding the LLM agent to simulate various steps that humans take when solving Multi-Phases problems. Specifically, we implement several strategies to enable LLM agents to generate a coherent outline for each travel query, mirroring human planning patterns. Additionally, we integrate Strategy Block and Knowledge Block into our framework: Strategy Block facilitates information collection, while Knowledge Block provides essential information for detailed planning. Through our extensive experiments, we demonstrate that our framework significantly improves the planning capabilities of LLM agents, enabling them to tackle the travel planning task with improved efficiency and effectiveness. Our experimental results showcase the exceptional performance of the proposed framework; when combined with GPT-4-Turbo, it attains $10\times$ the performance gains in comparison to the baseline framework deployed on GPT-4-Turbo.

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arXiv2024

Faster Sampling via Stochastic Gradient Proximal Sampler

Stochastic gradients have been widely integrated into Langevin-based methods to improve their scalability and efficiency in solving large-scale sampling problems. However, the proximal sampler, which exhibits much faster convergence than Langevin-based algorithms in the deterministic setting Lee et al. (2021), has yet to be explored in its stochastic variants. In this paper, we study the Stochastic Proximal Samplers (SPS) for sampling from non-log-concave distributions. We first establish a general framework for implementing stochastic proximal samplers and establish the convergence theory accordingly. We show that the convergence to the target distribution can be guaranteed as long as the second moment of the algorithm trajectory is bounded and restricted Gaussian oracles can be well approximated. We then provide two implementable variants based on Stochastic gradient Langevin dynamics (SGLD) and Metropolis-adjusted Langevin algorithm (MALA), giving rise to SPS-SGLD and SPS-MALA. We further show that SPS-SGLD and SPS-MALA can achieve $ε$-sampling error in total variation (TV) distance within $\tilde{\mathcal{O}}(dε^{-2})$ and $\tilde{\mathcal{O}}(d^{1/2}ε^{-2})$ gradient complexities, which outperform the best-known result by at least an $\tilde{\mathcal{O}}(d^{1/3})$ factor. This enhancement in performance is corroborated by our empirical studies on synthetic data with various dimensions, demonstrating the efficiency of our proposed algorithm.

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arXiv2024

Improving Implicit Regularization of SGD with Preconditioning for Least Square Problems

Stochastic gradient descent (SGD) exhibits strong algorithmic regularization effects in practice and plays an important role in the generalization of modern machine learning. However, prior research has revealed instances where the generalization performance of SGD is worse than ridge regression due to uneven optimization along different dimensions. Preconditioning offers a natural solution to this issue by rebalancing optimization across different directions. Yet, the extent to which preconditioning can enhance the generalization performance of SGD and whether it can bridge the existing gap with ridge regression remains uncertain. In this paper, we study the generalization performance of SGD with preconditioning for the least squared problem. We make a comprehensive comparison between preconditioned SGD and (standard \& preconditioned) ridge regression. Our study makes several key contributions toward understanding and improving SGD with preconditioning. First, we establish excess risk bounds (generalization performance) for preconditioned SGD and ridge regression under an arbitrary preconditions matrix. Second, leveraging the excessive risk characterization of preconditioned SGD and ridge regression, we show that (through construction) there exists a simple preconditioned matrix that can make SGD comparable to (standard \& preconditioned) ridge regression. Finally, we show that our proposed preconditioning matrix is straightforward enough to allow robust estimation from finite samples while maintaining a theoretical improvement. Our empirical results align with our theoretical findings, collectively showcasing the enhanced regularization effect of preconditioned SGD.

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arXiv2024

Reverse Transition Kernel: A Flexible Framework to Accelerate Diffusion Inference

To generate data from trained diffusion models, most inference algorithms, such as DDPM, DDIM, and other variants, rely on discretizing the reverse SDEs or their equivalent ODEs. In this paper, we view such approaches as decomposing the entire denoising diffusion process into several segments, each corresponding to a reverse transition kernel (RTK) sampling subproblem. Specifically, DDPM uses a Gaussian approximation for the RTK, resulting in low per-subproblem complexity but requiring a large number of segments (i.e., subproblems), which is conjectured to be inefficient. To address this, we develop a general RTK framework that enables a more balanced subproblem decomposition, resulting in $\tilde O(1)$ subproblems, each with strongly log-concave targets. We then propose leveraging two fast sampling algorithms, the Metropolis-Adjusted Langevin Algorithm (MALA) and Underdamped Langevin Dynamics (ULD), for solving these strongly log-concave subproblems. This gives rise to the RTK-MALA and RTK-ULD algorithms for diffusion inference. In theory, we further develop the convergence guarantees for RTK-MALA and RTK-ULD in total variation (TV) distance: RTK-ULD can achieve $ε$ target error within $\tilde{\mathcal O}(d^{1/2}ε^{-1})$ under mild conditions, and RTK-MALA enjoys a $\mathcal{O}(d^{2}\log(d/ε))$ convergence rate under slightly stricter conditions. These theoretical results surpass the state-of-the-art convergence rates for diffusion inference and are well supported by numerical experiments.

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arXiv2024

What Can Transformer Learn with Varying Depth? Case Studies on Sequence Learning Tasks

We study the capabilities of the transformer architecture with varying depth. Specifically, we designed a novel set of sequence learning tasks to systematically evaluate and comprehend how the depth of transformer affects its ability to perform memorization, reasoning, generalization, and contextual generalization. We show a transformer with only one attention layer can excel in memorization but falls short in other tasks. Then, we show that exhibiting reasoning and generalization ability requires the transformer to have at least two attention layers, while context generalization ability may necessitate three attention layers. Additionally, we identify a class of simple operations that a single attention layer can execute, and show that the complex tasks can be approached as the combinations of these simple operations and thus can be resolved by stacking multiple attention layers. This sheds light on studying more practical and complex tasks beyond our design. Numerical experiments corroborate our theoretical findings.

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arXiv2024

On the Benefits of Over-parameterization for Out-of-Distribution Generalization

In recent years, machine learning models have achieved success based on the independently and identically distributed assumption. However, this assumption can be easily violated in real-world applications, leading to the Out-of-Distribution (OOD) problem. Understanding how modern over-parameterized DNNs behave under non-trivial natural distributional shifts is essential, as current theoretical understanding is insufficient. Existing theoretical works often provide meaningless results for over-parameterized models in OOD scenarios or even contradict empirical findings. To this end, we are investigating the performance of the over-parameterized model in terms of OOD generalization under the general benign overfitting conditions. Our analysis focuses on a random feature model and examines non-trivial natural distributional shifts, where the benign overfitting estimators demonstrate a constant excess OOD loss, despite achieving zero excess in-distribution (ID) loss. We demonstrate that in this scenario, further increasing the model's parameterization can significantly reduce the OOD loss. Intuitively, the variance term of ID loss remains low due to orthogonality of long-tail features, meaning overfitting noise during training generally doesn't raise testing loss. However, in OOD cases, distributional shift increases the variance term. Thankfully, the inherent shift is unrelated to individual x, maintaining the orthogonality of long-tail features. Expanding the hidden dimension can additionally improve this orthogonality by mapping the features into higher-dimensional spaces, thereby reducing the variance term. We further show that model ensembles also improve OOD loss, akin to increasing model capacity. These insights explain the empirical phenomenon of enhanced OOD generalization through model ensembles, supported by consistent simulations with theoretical results.

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arXiv2024

How Many Pretraining Tasks Are Needed for In-Context Learning of Linear Regression?

Transformers pretrained on diverse tasks exhibit remarkable in-context learning (ICL) capabilities, enabling them to solve unseen tasks solely based on input contexts without adjusting model parameters. In this paper, we study ICL in one of its simplest setups: pretraining a linearly parameterized single-layer linear attention model for linear regression with a Gaussian prior. We establish a statistical task complexity bound for the attention model pretraining, showing that effective pretraining only requires a small number of independent tasks. Furthermore, we prove that the pretrained model closely matches the Bayes optimal algorithm, i.e., optimally tuned ridge regression, by achieving nearly Bayes optimal risk on unseen tasks under a fixed context length. These theoretical findings complement prior experimental research and shed light on the statistical foundations of ICL.

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arXiv2024

An Improved Analysis of Langevin Algorithms with Prior Diffusion for Non-Log-Concave Sampling

Understanding the dimension dependency of computational complexity in high-dimensional sampling problem is a fundamental problem, both from a practical and theoretical perspective. Compared with samplers with unbiased stationary distribution, e.g., Metropolis-adjusted Langevin algorithm (MALA), biased samplers, e.g., Underdamped Langevin Dynamics (ULD), perform better in low-accuracy cases just because a lower dimension dependency in their complexities. Along this line, Freund et al. (2022) suggest that the modified Langevin algorithm with prior diffusion is able to converge dimension independently for strongly log-concave target distributions. Nonetheless, it remains open whether such property establishes for more general cases. In this paper, we investigate the prior diffusion technique for the target distributions satisfying log-Sobolev inequality (LSI), which covers a much broader class of distributions compared to the strongly log-concave ones. In particular, we prove that the modified Langevin algorithm can also obtain the dimension-independent convergence of KL divergence with different step size schedules. The core of our proof technique is a novel construction of an interpolating SDE, which significantly helps to conduct a more accurate characterization of the discrete updates of the overdamped Langevin dynamics. Our theoretical analysis demonstrates the benefits of prior diffusion for a broader class of target distributions and provides new insights into developing faster sampling algorithms.

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arXiv2024

PRES: Toward Scalable Memory-Based Dynamic Graph Neural Networks

Memory-based Dynamic Graph Neural Networks (MDGNNs) are a family of dynamic graph neural networks that leverage a memory module to extract, distill, and memorize long-term temporal dependencies, leading to superior performance compared to memory-less counterparts. However, training MDGNNs faces the challenge of handling entangled temporal and structural dependencies, requiring sequential and chronological processing of data sequences to capture accurate temporal patterns. During the batch training, the temporal data points within the same batch will be processed in parallel, while their temporal dependencies are neglected. This issue is referred to as temporal discontinuity and restricts the effective temporal batch size, limiting data parallelism and reducing MDGNNs' flexibility in industrial applications. This paper studies the efficient training of MDGNNs at scale, focusing on the temporal discontinuity in training MDGNNs with large temporal batch sizes. We first conduct a theoretical study on the impact of temporal batch size on the convergence of MDGNN training. Based on the analysis, we propose PRES, an iterative prediction-correction scheme combined with a memory coherence learning objective to mitigate the effect of temporal discontinuity, enabling MDGNNs to be trained with significantly larger temporal batches without sacrificing generalization performance. Experimental results demonstrate that our approach enables up to a 4x larger temporal batch (3.4x speed-up) during MDGNN training.

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arXiv2024

Towards Robust Graph Incremental Learning on Evolving Graphs

Incremental learning is a machine learning approach that involves training a model on a sequence of tasks, rather than all tasks at once. This ability to learn incrementally from a stream of tasks is crucial for many real-world applications. However, incremental learning is a challenging problem on graph-structured data, as many graph-related problems involve prediction tasks for each individual node, known as Node-wise Graph Incremental Learning (NGIL). This introduces non-independent and non-identically distributed characteristics in the sample data generation process, making it difficult to maintain the performance of the model as new tasks are added. In this paper, we focus on the inductive NGIL problem, which accounts for the evolution of graph structure (structural shift) induced by emerging tasks. We provide a formal formulation and analysis of the problem, and propose a novel regularization-based technique called Structural-Shift-Risk-Mitigation (SSRM) to mitigate the impact of the structural shift on catastrophic forgetting of the inductive NGIL problem. We show that the structural shift can lead to a shift in the input distribution for the existing tasks, and further lead to an increased risk of catastrophic forgetting. Through comprehensive empirical studies with several benchmark datasets, we demonstrate that our proposed method, Structural-Shift-Risk-Mitigation (SSRM), is flexible and easy to adapt to improve the performance of state-of-the-art GNN incremental learning frameworks in the inductive setting.

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arXiv2024

Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo

To sample from a general target distribution $p_*\propto e^{-f_*}$ beyond the isoperimetric condition, Huang et al. (2023) proposed to perform sampling through reverse diffusion, giving rise to Diffusion-based Monte Carlo (DMC). Specifically, DMC follows the reverse SDE of a diffusion process that transforms the target distribution to the standard Gaussian, utilizing a non-parametric score estimation. However, the original DMC algorithm encountered high gradient complexity, resulting in an exponential dependency on the error tolerance $ε$ of the obtained samples. In this paper, we demonstrate that the high complexity of DMC originates from its redundant design of score estimation, and proposed a more efficient algorithm, called RS-DMC, based on a novel recursive score estimation method. In particular, we first divide the entire diffusion process into multiple segments and then formulate the score estimation step (at any time step) as a series of interconnected mean estimation and sampling subproblems accordingly, which are correlated in a recursive manner. Importantly, we show that with a proper design of the segment decomposition, all sampling subproblems will only need to tackle a strongly log-concave distribution, which can be very efficient to solve using the Langevin-based samplers with a provably rapid convergence rate. As a result, we prove that the gradient complexity of RS-DMC only has a quasi-polynomial dependency on $ε$, which significantly improves exponential gradient complexity in Huang et al. (2023). Furthermore, under commonly used dissipative conditions, our algorithm is provably much faster than the popular Langevin-based algorithms. Our algorithm design and theoretical framework illuminate a novel direction for addressing sampling problems, which could be of broader applicability in the community.

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arXiv2023

Benign Oscillation of Stochastic Gradient Descent with Large Learning Rates

In this work, we theoretically investigate the generalization properties of neural networks (NN) trained by stochastic gradient descent (SGD) algorithm with large learning rates. Under such a training regime, our finding is that, the oscillation of the NN weights caused by the large learning rate SGD training turns out to be beneficial to the generalization of the NN, which potentially improves over the same NN trained by SGD with small learning rates that converges more smoothly. In view of this finding, we call such a phenomenon "benign oscillation". Our theory towards demystifying such a phenomenon builds upon the feature learning perspective of deep learning. Specifically, we consider a feature-noise data generation model that consists of (i) weak features which have a small $\ell_2$-norm and appear in each data point; (ii) strong features which have a larger $\ell_2$-norm but only appear in a certain fraction of all data points; and (iii) noise. We prove that NNs trained by oscillating SGD with a large learning rate can effectively learn the weak features in the presence of those strong features. In contrast, NNs trained by SGD with a small learning rate can only learn the strong features but makes little progress in learning the weak features. Consequently, when it comes to the new testing data which consist of only weak features, the NN trained by oscillating SGD with a large learning rate could still make correct predictions consistently, while the NN trained by small learning rate SGD fails. Our theory sheds light on how large learning rate training benefits the generalization of NNs. Experimental results demonstrate our finding on "benign oscillation".

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arXiv2023

Benign Overfitting in Two-Layer ReLU Convolutional Neural Networks for XOR Data

Modern deep learning models are usually highly over-parameterized so that they can overfit the training data. Surprisingly, such overfitting neural networks can usually still achieve high prediction accuracy. To study this "benign overfitting" phenomenon, a line of recent works has theoretically studied the learning of linear models and two-layer neural networks. However, most of these analyses are still limited to the very simple learning problems where the Bayes-optimal classifier is linear. In this work, we investigate a class of XOR-type classification tasks with label-flipping noises. We show that, under a certain condition on the sample complexity and signal-to-noise ratio, an over-parameterized ReLU CNN trained by gradient descent can achieve near Bayes-optimal accuracy. Moreover, we also establish a matching lower bound result showing that when the previous condition is not satisfied, the prediction accuracy of the obtained CNN is an absolute constant away from the Bayes-optimal rate. Our result demonstrates that CNNs have a remarkable capacity to efficiently learn XOR problems, even in the presence of highly correlated features.

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arXiv2023

The Implicit Bias of Batch Normalization in Linear Models and Two-layer Linear Convolutional Neural Networks

We study the implicit bias of batch normalization trained by gradient descent. We show that when learning a linear model with batch normalization for binary classification, gradient descent converges to a uniform margin classifier on the training data with an $\exp(-Ω(\log^2 t))$ convergence rate. This distinguishes linear models with batch normalization from those without batch normalization in terms of both the type of implicit bias and the convergence rate. We further extend our result to a class of two-layer, single-filter linear convolutional neural networks, and show that batch normalization has an implicit bias towards a patch-wise uniform margin. Based on two examples, we demonstrate that patch-wise uniform margin classifiers can outperform the maximum margin classifiers in certain learning problems. Our results contribute to a better theoretical understanding of batch normalization.

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arXiv2023

Finite-Sample Analysis of Learning High-Dimensional Single ReLU Neuron

This paper considers the problem of learning a single ReLU neuron with squared loss (a.k.a., ReLU regression) in the overparameterized regime, where the input dimension can exceed the number of samples. We analyze a Perceptron-type algorithm called GLM-tron (Kakade et al., 2011) and provide its dimension-free risk upper bounds for high-dimensional ReLU regression in both well-specified and misspecified settings. Our risk bounds recover several existing results as special cases. Moreover, in the well-specified setting, we provide an instance-wise matching risk lower bound for GLM-tron. Our upper and lower risk bounds provide a sharp characterization of the high-dimensional ReLU regression problems that can be learned via GLM-tron. On the other hand, we provide some negative results for stochastic gradient descent (SGD) for ReLU regression with symmetric Bernoulli data: if the model is well-specified, the excess risk of SGD is provably no better than that of GLM-tron ignoring constant factors, for each problem instance; and in the noiseless case, GLM-tron can achieve a small risk while SGD unavoidably suffers from a constant risk in expectation. These results together suggest that GLM-tron might be preferable to SGD for high-dimensional ReLU regression.

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arXiv2023

Per-Example Gradient Regularization Improves Learning Signals from Noisy Data

Gradient regularization, as described in \citet{barrett2021implicit}, is a highly effective technique for promoting flat minima during gradient descent. Empirical evidence suggests that this regularization technique can significantly enhance the robustness of deep learning models against noisy perturbations, while also reducing test error. In this paper, we explore the per-example gradient regularization (PEGR) and present a theoretical analysis that demonstrates its effectiveness in improving both test error and robustness against noise perturbations. Specifically, we adopt a signal-noise data model from \citet{cao2022benign} and show that PEGR can learn signals effectively while suppressing noise. In contrast, standard gradient descent struggles to distinguish the signal from the noise, leading to suboptimal generalization performance. Our analysis reveals that PEGR penalizes the variance of pattern learning, thus effectively suppressing the memorization of noises from the training data. These findings underscore the importance of variance control in deep learning training and offer useful insights for developing more effective training approaches.

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