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Abstract:We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components: we only require that the covariance matrices have bounded condition number and that the means and covariances lie in a ball of bounded radius. We give an algorithm that draws $d^{\mathrm{poly}(k/\varepsilon)}$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler whose output distribution is $\varepsilon$-far from the unknown mixture in total variation. Prior works for this problem either (i) required exponential runtime in the dimension $d$, (ii) placed strong assumptions on the instance (e.g., spherical covariances or clusterability), or (iii) had doubly exponential dependence on the number of components $k$. Our approach departs from commonly used techniques for this problem like the method of moments. Instead, we leverage a recently developed reduction, based on diffusion models, from distribution learning to a supervised learning task called score matching. We give an algorithm for the latter by proving a structural result showing that the score function of a Gaussian mixture can be approximated by a piecewise-polynomial function, and there is an efficient algorithm for finding it. To our knowledge, this is the first example of diffusion models achieving a state-of-the-art theoretical guarantee for an unsupervised learning task.

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Abstract:Deep networks typically learn concepts via classifiers, which involves setting up a model and training it via gradient descent to fit the concept-labeled data. We will argue instead that learning a concept could be done by looking at its moment statistics matrix to generate a concrete representation or signature of that concept. These signatures can be used to discover structure across the set of concepts and could recursively produce higher-level concepts by learning this structure from those signatures. When the concepts are `intersected', signatures of the concepts can be used to find a common theme across a number of related `intersected' concepts. This process could be used to keep a dictionary of concepts so that inputs could correctly identify and be routed to the set of concepts involved in the (latent) generation of the input.

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Abstract:Recent works have shown that diffusion models can learn essentially any distribution provided one can perform score estimation. Yet it remains poorly understood under what settings score estimation is possible, let alone when practical gradient-based algorithms for this task can provably succeed. In this work, we give the first provably efficient results along these lines for one of the most fundamental distribution families, Gaussian mixture models. We prove that gradient descent on the denoising diffusion probabilistic model (DDPM) objective can efficiently recover the ground truth parameters of the mixture model in the following two settings: 1) We show gradient descent with random initialization learns mixtures of two spherical Gaussians in $d$ dimensions with $1/\text{poly}(d)$-separated centers. 2) We show gradient descent with a warm start learns mixtures of $K$ spherical Gaussians with $\Omega(\sqrt{\log(\min(K,d))})$-separated centers. A key ingredient in our proofs is a new connection between score-based methods and two other approaches to distribution learning, the EM algorithm and spectral methods.

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Authors:Giannis Daras, Kulin Shah, Yuval Dagan, Aravind Gollakota, Alexandros G. Dimakis, Adam Klivans

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Abstract:We present the first diffusion-based framework that can learn an unknown distribution using only highly-corrupted samples. This problem arises in scientific applications where access to uncorrupted samples is impossible or expensive to acquire. Another benefit of our approach is the ability to train generative models that are less likely to memorize individual training samples since they never observe clean training data. Our main idea is to introduce additional measurement distortion during the diffusion process and require the model to predict the original corrupted image from the further corrupted image. We prove that our method leads to models that learn the conditional expectation of the full uncorrupted image given this additional measurement corruption. This holds for any corruption process that satisfies some technical conditions (and in particular includes inpainting and compressed sensing). We train models on standard benchmarks (CelebA, CIFAR-10 and AFHQ) and show that we can learn the distribution even when all the training samples have $90\%$ of their pixels missing. We also show that we can finetune foundation models on small corrupted datasets (e.g. MRI scans with block corruptions) and learn the clean distribution without memorizing the training set.

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Abstract:In this paper, we propose deep architectures for learning instance specific abstain (reject option) binary classifiers. The proposed approach uses double sigmoid loss function as described by Kulin Shah and Naresh Manwani in ("Online Active Learning of Reject Option Classifiers", AAAI, 2020), as a performance measure. We show that the double sigmoid loss is classification calibrated. We also show that the excess risk of 0-d-1 loss is upper bounded by the excess risk of double sigmoid loss. We derive the generalization error bounds for the proposed architecture for reject option classifiers. To show the effectiveness of the proposed approach, we experiment with several real world datasets. We observe that the proposed approach not only performs comparable to the state-of-the-art approaches, it is also robust against label noise. We also provide visualizations to observe the important features learned by the network corresponding to the abstaining decision.

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Abstract:In supervised learning, it is known that overparameterized neural networks with one hidden layer provably and efficiently learn and generalize, when trained using stochastic gradient descent with sufficiently small learning rate and suitable initialization. In contrast, the benefit of overparameterization in unsupervised learning is not well understood. Normalizing flows (NFs) constitute an important class of models in unsupervised learning for sampling and density estimation. In this paper, we theoretically and empirically analyze these models when the underlying neural network is one-hidden-layer overparameterized network. Our main contributions are two-fold: (1) On the one hand, we provide theoretical and empirical evidence that for a class of NFs containing most of the existing NF models, overparametrization hurts training. (2) On the other hand, we prove that unconstrained NFs, a recently introduced model, can efficiently learn any reasonable data distribution under minimal assumptions when the underlying network is overparametrized.

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Abstract:Group-fairness in classification aims for equality of a predictive utility across different sensitive sub-populations, e.g., race or gender. Equality or near-equality constraints in group-fairness often worsen not only the aggregate utility but also the utility for the least advantaged sub-population. In this paper, we apply the principles of Pareto-efficiency and least-difference to the utility being accuracy, as an illustrative example, and arrive at the Rawls classifier that minimizes the error rate on the worst-off sensitive sub-population. Our mathematical characterization shows that the Rawls classifier uniformly applies a threshold to an ideal score of features, in the spirit of fair equality of opportunity. In practice, such a score or a feature representation is often computed by a black-box model that has been useful but unfair. Our second contribution is practical Rawlsian fair adaptation of any given black-box deep learning model, without changing the score or feature representation it computes. Given any score function or feature representation and only its second-order statistics on the sensitive sub-populations, we seek a threshold classifier on the given score or a linear threshold classifier on the given feature representation that achieves the Rawls error rate restricted to this hypothesis class. Our technical contribution is to formulate the above problems using ambiguous chance constraints, and to provide efficient algorithms for Rawlsian fair adaptation, along with provable upper bounds on the Rawls error rate. Our empirical results show significant improvement over state-of-the-art group-fair algorithms, even without retraining for fairness.

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Abstract:Active learning is an important technique to reduce the number of labeled examples in supervised learning. Active learning for binary classification has been well addressed. However, active learning of reject option classifier is still an unsolved problem. In this paper, we propose novel algorithms for active learning of reject option classifiers. We develop an active learning algorithm using double ramp loss function. We provide mistake bounds for this algorithm. We also propose a new loss function called double sigmoid loss function for reject option and corresponding active learning algorithm. We provide extensive experimental results to show the effectiveness of the proposed algorithms. The proposed algorithms efficiently reduce the number of label examples required.

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Abstract:In this paper, we propose a novel mixture of expert architecture for learning polyhedral classifiers. We learn the parameters of the classifierusing an expectation maximization algorithm. Wederive the generalization bounds of the proposedapproach. Through an extensive simulation study, we show that the proposed method performs comparably to other state-of-the-art approaches.

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Abstract:In this paper, we propose a new sparse and robust reject option classifier based on minimization of $l_1$ regularized risk under double ramp loss $L_{dr,\rho}$. We use DC programming to find the risk minimizer. The algorithm solves a sequence of linear programs to learn the reject option classifier. Moreover, we show that the risk under $L_{dr,\rho}$ is minimized by generalized Bayes classifier in the reject option setting. We also provide the excess risk bound for $L_{dr,\rho}$. We show the effectiveness of the proposed approach by experimenting it on several real world datasets.

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