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For a widely-studied data model and general loss and sample-hardening functions we prove that the Supervised Contrastive Learning (SCL), Hard-SCL (HSCL), and Unsupervised Contrastive Learning (UCL) risks are minimized by representations that exhibit Neural Collapse (NC), i.e., the class means form an Equianglular Tight Frame (ETF) and data from the same class are mapped to the same representation. We also prove that for any representation mapping, the HSCL and Hard-UCL (HUCL) risks are lower bounded by the corresponding SCL and UCL risks. Although the optimality of ETF is known for SCL, albeit only for InfoNCE loss, its optimality for HSCL and UCL under general loss and hardening functions is novel. Moreover, our proofs are much simpler, compact, and transparent. We empirically demonstrate, for the first time, that ADAM optimization of HSCL and HUCL risks with random initialization and suitable hardness levels can indeed converge to the NC geometry if we incorporate unit-ball or unit-sphere feature normalization. Without incorporating hard negatives or feature normalization, however, the representations learned via ADAM suffer from dimensional collapse (DC) and fail to attain the NC geometry.

In typical medical image classification problems, labeled data is scarce while unlabeled data is more available. Semi-supervised learning and self-supervised learning are two different research directions that can improve accuracy by learning from extra unlabeled data. Recent methods from both directions have reported significant gains on traditional benchmarks. Yet past benchmarks do not focus on medical tasks and rarely compare self- and semi- methods together on an equal footing. Furthermore, past benchmarks often handle hyperparameter tuning suboptimally. First, they may not tune hyperparameters at all, leading to underfitting. Second, when tuning does occur, it often unrealistically uses a labeled validation set that is much larger than the training set. Therefore currently published rankings might not always corroborate with their practical utility This study contributes a systematic evaluation of self- and semi- methods with a unified experimental protocol intended to guide a practitioner with scarce overall labeled data and a limited compute budget. We answer two key questions: Can hyperparameter tuning be effective with realistic-sized validation sets? If so, when all methods are tuned well, which self- or semi-supervised methods achieve the best accuracy? Our study compares 13 representative semi- and self-supervised methods to strong labeled-set-only baselines on 4 medical datasets. From 20000+ GPU hours of computation, we provide valuable best practices to resource-constrained practitioners: hyperparameter tuning is effective, and the semi-supervised method known as MixMatch delivers the most reliable gains across 4 datasets., Comment: CVPR 2024

Through minimization of an appropriate loss function such as the InfoNCE loss, contrastive learning (CL) learns a useful representation function by pulling positive samples close to each other while pushing negative samples far apart in the embedding space. The positive samples are typically created using "label-preserving" augmentations, i.e., domain-specific transformations of a given datum or anchor. In absence of class information, in unsupervised CL (UCL), the negative samples are typically chosen randomly and independently of the anchor from a preset negative sampling distribution over the entire dataset. This leads to class-collisions in UCL. Supervised CL (SCL), avoids this class collision by conditioning the negative sampling distribution to samples having labels different from that of the anchor. In hard-UCL (H-UCL), which has been shown to be an effective method to further enhance UCL, the negative sampling distribution is conditionally tilted, by means of a hardening function, towards samples that are closer to the anchor. Motivated by this, in this paper we propose hard-SCL (H-SCL) {wherein} the class conditional negative sampling distribution {is tilted} via a hardening function. Our simulation results confirm the utility of H-SCL over SCL with significant performance gains {in downstream classification tasks.} Analytically, we show that {in the} limit of infinite negative samples per anchor and a suitable assumption, the {H-SCL loss} is upper bounded by the {H-UCL loss}, thereby justifying the utility of H-UCL {for controlling} the H-SCL loss in the absence of label information. Through experiments on several datasets, we verify the assumption as well as the claimed inequality between H-UCL and H-SCL losses. We also provide a plausible scenario where H-SCL loss is lower bounded by UCL loss, indicating the limited utility of UCL in controlling the H-SCL loss.

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycentric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing., Comment: ICML 2022

We study the problem of designing hard negative sampling distributions for unsupervised contrastive representation learning. We propose and analyze a novel min-max framework that seeks a representation which minimizes the maximum (worst-case) generalized contrastive learning loss over all couplings (joint distributions between positive and negative samples subject to marginal constraints) and prove that the resulting min-max optimum representation will be degenerate. This provides the first theoretical justification for incorporating additional regularization constraints on the couplings. We re-interpret the min-max problem through the lens of Optimal Transport (OT) theory and utilize regularized transport couplings to control the degree of hardness of negative examples. Through experiments we demonstrate that the negative samples generated from our designed negative distribution are more similar to the anchor than those generated from the baseline negative distribution. We also demonstrate that entropic regularization yields negative sampling distributions with parametric form similar to that in a recent state-of-the-art negative sampling design and has similar performance in multiple datasets. Utilizing the uncovered connection with OT, we propose a new ground cost for designing the negative distribution and show improved performance of the learned representation on downstream tasks compared to the representation learned when using squared Euclidean cost., Comment: Minor changes - fixing typos, minor rephrasing in abstract and introduction

This paper shows that a popular approach to the supervised embedding of documents for classification, namely, contrastive Word Mover's Embedding, can be significantly enhanced by adding interpretability. This interpretability is achieved by incorporating a clustering promoting mechanism into the contrastive loss. On several public datasets, we show that our method improves significantly upon existing baselines while providing interpretation to the clusters via identifying a set of keywords that are the most representative of a particular class. Our approach was motivated in part by the need to develop Natural Language Processing (NLP) methods for the \textit{novel problem of assessing student work for scientific writing and thinking} - a problem that is central to the area of (educational) Learning Sciences (LS). In this context, we show that our approach leads to a meaningful assessment of the student work related to lab reports from a biology class and can help LS researchers gain insights into student understanding and assess evidence of scientific thought processes., Comment: 8 pages, 4 figures

Qualitative analysis of verbal data is of central importance in the learning sciences. It is labor-intensive and time-consuming, however, which limits the amount of data researchers can include in studies. This work is a step towards building a statistical machine learning (ML) method for achieving an automated support for qualitative analyses of students' writing, here specifically in score laboratory reports in introductory biology for sophistication of argumentation and reasoning. We start with a set of lab reports from an undergraduate biology course, scored by a four-level scheme that considers the complexity of argument structure, the scope of evidence, and the care and nuance of conclusions. Using this set of labeled data, we show that a popular natural language modeling processing pipeline, namely vector representation of words, a.k.a word embeddings, followed by Long Short Term Memory (LSTM) model for capturing language generation as a state-space model, is able to quantitatively capture the scoring, with a high Quadratic Weighted Kappa (QWK) prediction score, when trained in via a novel contrastive learning set-up. We show that the ML algorithm approached the inter-rater reliability of human analysis. Ultimately, we conclude, that machine learning (ML) for natural language processing (NLP) holds promise for assisting learning sciences researchers in conducting qualitative studies at much larger scales than is currently possible.