Understanding Self-Distillation and Partial Label Learning in Multi-Class Classification with Label Noise

Self-distillation (SD) is the process of training a student model using the outputs of a teacher model, with both models sharing the same architecture. Our study theoretically examines SD in multi-class classification with cross-entropy loss, exploring both multi-round SD and SD with refined teacher outputs, inspired by partial label learning (PLL). By deriving a closed-form solution for the student model's outputs, we discover that SD essentially functions as label averaging among instances with high feature correlations. Initially beneficial, this averaging helps the model focus on feature clusters correlated with a given instance for predicting the label. However, it leads to diminishing performance with increasing distillation rounds. Additionally, we demonstrate SD's effectiveness in label noise scenarios and identify the label corruption condition and minimum number of distillation rounds needed to achieve 100% classification accuracy. Our study also reveals that one-step distillation with refined teacher outputs surpasses the efficacy of multi-step SD using the teacher's direct output in high noise rate regimes.

Efficient Algorithms for Exact Graph Matching on Correlated Stochastic Block Models with Constant Correlation

We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

Detection problems in the spiked matrix models

We study the statistical decision process of detecting the low-rank signal from various signal-plus-noise type data matrices, known as the spiked random matrix models. We first show that the principal component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non-Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked random matrices, which generalize the Baik-Ben Arous-P\'{e}ch\'{e} (BBP) transition. We also prove the central limit theorem for the linear spectral statistics for the spiked random matrices and propose a hypothesis test based on it, which does not depend on the distribution of the signal or the noise. When the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix with additive noise. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.

Data Valuation Without Training of a Model

Many recent works on understanding deep learning try to quantify how much individual data instances influence the optimization and generalization of a model, either by analyzing the behavior of the model during training or by measuring the performance gap of the model when the instance is removed from the dataset. Such approaches reveal characteristics and importance of individual instances, which may provide useful information in diagnosing and improving deep learning. However, most of the existing works on data valuation require actual training of a model, which often demands high-computational cost. In this paper, we provide a training-free data valuation score, called complexity-gap score, which is a data-centric score to quantify the influence of individual instances in generalization of two-layer overparameterized neural networks. The proposed score can quantify irregularity of the instances and measure how much each data instance contributes in the total movement of the network parameters during training. We theoretically analyze and empirically demonstrate the effectiveness of the complexity-gap score in finding 'irregular or mislabeled' data instances, and also provide applications of the score in analyzing datasets and diagnosing training dynamics.

Recovering Top-Two Answers and Confusion Probability in Multi-Choice Crowdsourcing

Crowdsourcing has emerged as an effective platform to label a large volume of data in a cost- and time-efficient manner. Most previous works have focused on designing an efficient algorithm to recover only the ground-truth labels of the data. In this paper, we consider multi-choice crowdsourced labeling with the goal of recovering not only the ground truth but also the most confusing answer and the confusion probability. The most confusing answer provides useful information about the task by revealing the most plausible answer other than the ground truth and how plausible it is. To theoretically analyze such scenarios, we propose a model where there are top-two plausible answers for each task, distinguished from the rest of choices. Task difficulty is quantified by the confusion probability between the top two, and worker reliability is quantified by the probability of giving an answer among the top two. Under this model, we propose a two-stage inference algorithm to infer the top-two answers as well as the confusion probability. We show that our algorithm achieves the minimax optimal convergence rate. We conduct both synthetic and real-data experiments and demonstrate that our algorithm outperforms other recent algorithms. We also show the applicability of our algorithms in inferring the difficulty of tasks and training neural networks with the soft labels composed of the top-two most plausible classes.

Rank-1 Matrix Completion with Gradient Descent and Small Random Initialization

The nonconvex formulation of matrix completion problem has received significant attention in recent years due to its affordable complexity compared to the convex formulation. Gradient descent (GD) is the simplest yet efficient baseline algorithm for solving nonconvex optimization problems. The success of GD has been witnessed in many different problems in both theory and practice when it is combined with random initialization. However, previous works on matrix completion require either careful initialization or regularizers to prove the convergence of GD. In this work, we study the rank-1 symmetric matrix completion and prove that GD converges to the ground truth when small random initialization is used. We show that in logarithmic amount of iterations, the trajectory enters the region where local convergence occurs. We provide an upper bound on the initialization size that is sufficient to guarantee the convergence and show that a larger initialization can be used as more samples are available. We observe that implicit regularization effect of GD plays a critical role in the analysis, and for the entire trajectory, it prevents each entry from becoming much larger than the others.

Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices

We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. Assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. From the mean and the variance of the limiting Gaussian, we also compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test.

A Worker-Task Specialization Model for Crowdsourcing: Efficient Inference and Fundamental Limits

Crowdsourcing system has emerged as an effective platform to label data with relatively low cost by using non-expert workers. However, inferring correct labels from multiple noisy answers on data has been a challenging problem, since the quality of answers varies widely across tasks and workers. Many previous works have assumed a simple model where the order of workers in terms of their reliabilities is fixed across tasks, and focused on estimating the worker reliabilities to aggregate answers with different weights. We propose a highly general $d$-type worker-task specialization model in which the reliability of each worker can change depending on the type of a given task, where the number $d$ of types can scale in the number of tasks. In this model, we characterize the optimal sample complexity to correctly infer labels with any given recovery accuracy, and propose an inference algorithm achieving the order-wise optimal bound. We conduct experiments both on synthetic and real-world datasets, and show that our algorithm outperforms the existing algorithms developed based on strict model assumptions.

Detection of Signal in the Spiked Rectangular Models

We consider the problem of detecting signals in the rank-one signal-plus-noise data matrix models that generalize the spiked Wishart matrices. We show that the principal component analysis can be improved by pre-transforming the matrix entries if the noise is non-Gaussian. As an intermediate step, we prove a sharp phase transition of the largest eigenvalues of spiked rectangular matrices, which extends the Baik-Ben Arous-P\'ech\'e (BBP) transition. We also propose a hypothesis test to detect the presence of signal with low computational complexity, based on the linear spectral statistics, which minimizes the sum of the Type-I and Type-II errors when the noise is Gaussian.

Self-Diagnosing GAN: Diagnosing Underrepresented Samples in Generative Adversarial Networks

Despite remarkable performance in producing realistic samples, Generative Adversarial Networks (GANs) often produce low-quality samples near low-density regions of the data manifold. Recently, many techniques have been developed to improve the quality of generated samples, either by rejecting low-quality samples after training or by pre-processing the empirical data distribution before training, but at the cost of reduced diversity. To guarantee both the quality and the diversity, we propose a simple yet effective method to diagnose and emphasize underrepresented samples during training of a GAN. The main idea is to use the statistics of the discrepancy between the data distribution and the model distribution at each data instance. Based on the observation that the underrepresented samples have a high average discrepancy or high variability in discrepancy, we propose a method to emphasize those samples during training of a GAN. Our experimental results demonstrate that the proposed method improves GAN performance on various datasets, and it is especially effective in improving the quality of generated samples with minor features.