Gradual Domain Adaptation via Manifold-Constrained Distributionally Robust Optimization

The aim of this paper is to address the challenge of gradual domain adaptation within a class of manifold-constrained data distributions. In particular, we consider a sequence of $T\ge2$ data distributions $P_1,\ldots,P_T$ undergoing a gradual shift, where each pair of consecutive measures $P_i,P_{i+1}$ are close to each other in Wasserstein distance. We have a supervised dataset of size $n$ sampled from $P_0$, while for the subsequent distributions in the sequence, only unlabeled i.i.d. samples are available. Moreover, we assume that all distributions exhibit a known favorable attribute, such as (but not limited to) having intra-class soft/hard margins. In this context, we propose a methodology rooted in Distributionally Robust Optimization (DRO) with an adaptive Wasserstein radius. We theoretically show that this method guarantees the classification error across all $P_i$s can be suitably bounded. Our bounds rely on a newly introduced {\it {compatibility}} measure, which fully characterizes the error propagation dynamics along the sequence. Specifically, for inadequately constrained distributions, the error can exponentially escalate as we progress through the gradual shifts. Conversely, for appropriately constrained distributions, the error can be demonstrated to be linear or even entirely eradicated. We have substantiated our theoretical findings through several experimental results.

Unlabeled Out-Of-Domain Data Improves Generalization

We propose a novel framework for incorporating unlabeled data into semi-supervised classification problems, where scenarios involving the minimization of either i) adversarially robust or ii) non-robust loss functions have been considered. Notably, we allow the unlabeled samples to deviate slightly (in total variation sense) from the in-domain distribution. The core idea behind our framework is to combine Distributionally Robust Optimization (DRO) with self-supervised training. As a result, we also leverage efficient polynomial-time algorithms for the training stage. From a theoretical standpoint, we apply our framework on the classification problem of a mixture of two Gaussians in $\mathbb{R}^d$, where in addition to the $m$ independent and labeled samples from the true distribution, a set of $n$ (usually with $n\gg m$) out of domain and unlabeled samples are gievn as well. Using only the labeled data, it is known that the generalization error can be bounded by $\propto\left(d/m\right)^{1/2}$. However, using our method on both isotropic and non-isotropic Gaussian mixture models, one can derive a new set of analytically explicit and non-asymptotic bounds which show substantial improvement on the generalization error compared ERM. Our results underscore two significant insights: 1) out-of-domain samples, even when unlabeled, can be harnessed to narrow the generalization gap, provided that the true data distribution adheres to a form of the "cluster assumption", and 2) the semi-supervised learning paradigm can be regarded as a special case of our framework when there are no distributional shifts. We validate our claims through experiments conducted on a variety of synthetic and real-world datasets.

On Statistical Learning of Simplices: Unmixing Problem Revisited

Learning of high-dimensional simplices from uniformly-sampled observations, generally known as the "unmixing problem", is a long-studied task in computer science. More recently, a significant interest is focused on this problem from other areas, such as computational biology and remote sensing. In this paper, we have studied the Probably Approximately Correct (PAC)-learnability of simplices with a focus on sample complexity. Our analysis shows that a sufficient sample size for PAC-learning of $K$-simplices is only $O\left(K^2\log K\right)$, yielding a huge improvement over the existing results, i.e. $O\left(K^{22}\right)$. Moreover, a novel continuously-relaxed optimization scheme is proposed which is guaranteed to achieve a PAC-approximation of the simplex, followed by a corresponding scalable algorithm whose performance is extensively tested on synthetic and real-world datasets. Experimental results show that not only being comparable to other existing strategies on noiseless samples, our method is superior to the state-of-the-art in noisy cases. The overall proposed framework is backed with solid theoretical guarantees and provides a rigorous framework for future research in this area.

Reliable Learning of Bernoulli Mixture Models

In this paper, we have derived a set of sufficient conditions for reliable clustering of data produced by Bernoulli Mixture Models (BMM), when the number of clusters is unknown. A BMM refers to a random binary vector whose components are independent Bernoulli trials with cluster-specific frequencies. The problem of clustering BMM data arises in many real-world applications, most notably in population genetics where researchers aim at inferring the population structure from multilocus genotype data. Our findings stipulate a minimum dataset size and a minimum number of Bernoulli trials (or genotyped loci) per sample, such that the existence of a clustering algorithm with a sufficient accuracy is guaranteed. Moreover, the mathematical intuitions and tools behind our work can help researchers in designing more effective and theoretically-plausible heuristic methods for similar problems.