Contrastive learning is a paradigm for learning representations from unlabelled data that has been highly successful for image and text data. Several recent works have examined contrastive losses to claim that contrastive models effectively learn spectral embeddings, while few works show relations between (wide) contrastive models and kernel principal component analysis (PCA). However, it is not known if trained contrastive models indeed correspond to kernel methods or PCA. In this work, we analyze the training dynamics of two-layer contrastive models, with non-linear activation, and answer when these models are close to PCA or kernel methods. It is well known in the supervised setting that neural networks are equivalent to neural tangent kernel (NTK) machines, and that the NTK of infinitely wide networks remains constant during training. We provide the first convergence results of NTK for contrastive losses, and present a nuanced picture: NTK of wide networks remains almost constant for cosine similarity based contrastive losses, but not for losses based on dot product similarity. We further study the training dynamics of contrastive models with orthogonality constraints on output layer, which is implicitly assumed in works relating contrastive learning to spectral embedding. Our deviation bounds suggest that representations learned by contrastive models are close to the principal components of a certain matrix computed from random features. We empirically show that our theoretical results possibly hold beyond two-layer networks.

Inference for Variational Autoencoders (VAEs) consists of learning two models: (1) a generative model, which transforms a simple distribution over a latent space into the distribution over observed data, and (2) an inference model, which approximates the posterior of the latent codes given data. The two components are learned jointly via a lower bound to the generative model's log marginal likelihood. In early phases of joint training, the inference model poorly approximates the latent code posteriors. Recent work showed that this leads optimization to get stuck in local optima, negatively impacting the learned generative model. As such, recent work suggests ensuring a high-quality inference model via iterative training: maximizing the objective function relative to the inference model before every update to the generative model. Unfortunately, iterative training is inefficient, requiring heuristic criteria for reverting from iterative to joint training for speed. Here, we suggest an inference method that trains the generative and inference models independently. It approximates the posterior of the true model a priori; fixing this posterior approximation, we then maximize the lower bound relative to only the generative model. By conventional wisdom, this approach should rely on the true prior and likelihood of the true model to approximate its posterior (which are unknown). However, we show that we can compute a deterministic, model-agnostic posterior approximation (MAPA) of the true model's posterior. We then use MAPA to develop a proof-of-concept inference method. We present preliminary results on low-dimensional synthetic data that (1) MAPA captures the trend of the true posterior, and (2) our MAPA-based inference performs better density estimation with less computation than baselines. Lastly, we present a roadmap for scaling the MAPA-based inference method to high-dimensional data.

Stochastic optimization methods in modern machine learning often require carefully tuning sensitive algorithmic parameters at significant cost in time, computation, and expertise. This reality has led to sustained interest in developing adaptive (or parameter-free) algorithms that require minimal or no tuning [6, 8, 12, 21, 22, 24, 26, 29, 35-39, 43, 45-47]. However, a basic theoretical question remains open: Are existing methods "as adaptive as possible," or is there substantial room for improvement? Put differently, is there a fundamental price to be paid (in terms of rate of convergence) for not knowing the problem parameters in advance? To address these questions, we must formally define what it means for an adaptive algorithm to be efficient. The standard notion of minimax optimality [1] does not suffice, since it does not constrain the algorithm to be agnostic to the parameters defining the function class; stochastic gradient descent (SGD) is in many cases minimax optimal, but its step size requires problemspecific tuning. To motivate our solution, we observe that guarantees for adaptive algorithms admit the following interpretation: assuming that the input problem satisfies certain assumptions (e.g., Lipschitz continuity, smoothness, etc.) the adaptive algorithm attains performance close to the best performance that is possible to guarantee given only these assumptions.

Deep Neural Networks (DNNs) do not inherently compute or exhibit empirically-justified task confidence. In mission critical applications, it is important to both understand associated DNN reasoning and its supporting evidence. In this paper, we propose a novel Bayesian approach to extract explanations, justifications, and uncertainty estimates from DNNs. Our approach is efficient both in terms of memory and computation, and can be applied to any black box DNN without any retraining, including applications to anomaly detection and out-of-distribution detection tasks. We validate our approach on the CIFAR-10 dataset, and show that it can significantly improve the interpretability and reliability of DNNs.

Machine learning (ML) methods, which fit to data the parameters of a given parameterized model class, have garnered significant interest as potential methods for learning surrogate models for complex engineering systems for which traditional simulation is expensive. However, in many scientific and engineering settings, generating high-fidelity data on which to train ML models is expensive, and the available budget for generating training data is limited. ML models trained on the resulting scarce high-fidelity data have high variance and are sensitive to vagaries of the training data set. We propose a new multifidelity training approach for scientific machine learning that exploits the scientific context where data of varying fidelities and costs are available; for example high-fidelity data may be generated by an expensive fully resolved physics simulation whereas lower-fidelity data may arise from a cheaper model based on simplifying assumptions. We use the multifidelity data to define new multifidelity Monte Carlo estimators for the unknown parameters of linear regression models, and provide theoretical analyses that guarantee the approach's accuracy and improved robustness to small training budgets. Numerical results verify the theoretical analysis and demonstrate that multifidelity learned models trained on scarce high-fidelity data and additional low-fidelity data achieve order-of-magnitude lower model variance than standard models trained on only high-fidelity data of comparable cost. This illustrates that in the scarce data regime, our multifidelity training strategy yields models with lower expected error than standard training approaches.

Label corruption, where training samples have incorrect labels, can significantly degrade the performance of machine learning models. This corruption often arises from non-expert labeling or adversarial attacks. Acquiring large, perfectly labeled datasets is costly, and retraining large models from scratch when a clean dataset becomes available is computationally expensive. To address this challenge, we propose Post-Training Correction, a new paradigm that adjusts model parameters after initial training to mitigate label noise, eliminating the need for retraining. We introduce Verifix, a novel Singular Value Decomposition (SVD) based algorithm that leverages a small, verified dataset to correct the model weights using a single update. Verifix uses SVD to estimate a Clean Activation Space and then projects the model's weights onto this space to suppress activations corresponding to corrupted data. We demonstrate Verifix's effectiveness on both synthetic and real-world label noise. Experiments on the CIFAR dataset with 25% synthetic corruption show 7.36% generalization improvements on average. Additionally, we observe generalization improvements of up to 2.63% on naturally corrupted datasets like WebVision1.0 and Clothing1M.

Microcanonical gradient descent is a sampling procedure for energy-based models allowing for efficient sampling of distributions in high dimension. It works by transporting samples from a high-entropy distribution, such as Gaussian white noise, to a low-energy region using gradient descent. We put this model in the framework of normalizing flows, showing how it can often overfit by losing an unnecessary amount of entropy in the descent. As a remedy, we propose a mean-field microcanonical gradient descent that samples several weakly coupled data points simultaneously, allowing for better control of the entropy loss while paying little in terms of likelihood fit. We study these models in the context of financial time series, illustrating the improvements on both synthetic and real data.

In many learning problems, ranging from clustering to ranking through metric learning, empirical estimates of the risk functional consist of an average over tuples (e.g., pairs or triplets) of observations, rather than over individual observations. In this paper, we focus on how to best implement a stochastic approximation approach to solve such risk minimization problems. We argue that in the largescale setting, gradient estimates should be obtained by sampling tuples of data points with replacement (incomplete U-statistics) instead of sampling data points without replacement (complete U-statistics based on subsamples). We develop a theoretical framework accounting for the substantial impact of this strategy on the generalization ability of the prediction model returned by the Stochastic Gradient Descent (SGD) algorithm. It reveals that the method we promote achieves a much better trade-off between statistical accuracy and computational cost. Beyond the rate bound analysis, experiments on AUC maximization and metric learning provide strong empirical evidence of the superiority of the proposed approach.

We analyze in this paper a random feature map based on a theory of invariance (I-theory) introduced in [1]. More specifically, a group invariant signal signature is obtained through cumulative distributions of group-transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as we show that this feature map defines an expected Haar-integration kernel that is invariant to the specified group action. We show how this non-linear random feature map approximates this group invariant kernel uniformly on a set of N points. Moreover, we show that it defines a function space that is dense in the equivalent Invariant Reproducing Kernel Hilbert Space. Finally, we quantify error rates of the convergence of the empirical risk minimization, as well as the reduction in the sample complexity of a learning algorithm using such an invariant representation for signal classification, in a classical supervised learning setting.

One of the central questions in statistical learning theory is to determine the conditions under which agents can learn from experience. This includes the necessary and sufficient conditions for generalization from a given finite training set to new observations. In this paper, we prove that algorithmic stability in the inference process is equivalent to uniform generalization across all parametric loss functions. We provide various interpretations of this result. For instance, a relationship is proved between stability and data processing, which reveals that algorithmic stability can be improved by post-processing the inferred hypothesis or by augmenting training examples with artificial noise prior to learning. In addition, we establish a relationship between algorithmic stability and the size of the observation space, which provides a formal justification for dimensionality reduction methods. Finally, we connect algorithmic stability to the size of the hypothesis space, which recovers the classical PAC result that the size (complexity) of the hypothesis space should be controlled in order to improve algorithmic stability and improve generalization.