We present a new non-negative matrix factorization model for $(0,1)$ bounded-support data based on the doubly non-central beta (DNCB) distribution, a generalization of the beta distribution. The expressiveness of the DNCB distribution is particularly useful for modeling DNA methylation datasets, which are typically highly dispersed and multi-modal; however, the model structure is sufficiently general that it can be adapted to many other domains where latent representations of $(0,1)$ bounded-support data are of interest. Although the DNCB distribution lacks a closed-form conjugate prior, several augmentations let us derive an efficient posterior inference algorithm composed entirely of analytic updates. Our model improves out-of-sample predictive performance on both real and synthetic DNA methylation datasets over state-of-the-art methods in bioinformatics. In addition, our model yields meaningful latent representations that accord with existing biological knowledge.

We study a stochastic program where the probability distribution of the uncertain problem parameters is unknown and only indirectly observed via finitely many correlated samples generated by an unknown Markov chain with $d$ states. We propose a data-driven distributionally robust optimization model to estimate the problem's objective function and optimal solution. By leveraging results from large deviations theory, we derive statistical guarantees on the quality of these estimators. The underlying worst-case expectation problem is nonconvex and involves $\mathcal O(d^2)$ decision variables. Thus, it cannot be solved efficiently for large $d$. By exploiting the structure of this problem, we devise a customized Frank-Wolfe algorithm with convex direction-finding subproblems of size $\mathcal O(d)$. We prove that this algorithm finds a stationary point efficiently under mild conditions. The efficiency of the method is predicated on a dimensionality reduction enabled by a dual reformulation. Numerical experiments indicate that our approach has better computational and statistical properties than the state-of-the-art methods.

State-of-the-art methods for scalable Gaussian processes use iterative algorithms, requiring fast matrix vector multiplies (MVMs) with the covariance kernel. The Structured Kernel Interpolation (SKI) framework accelerates these MVMs by performing efficient MVMs on a grid and interpolating back to the original space. In this work, we develop a connection between SKI and the permutohedral lattice used for high-dimensional fast bilateral filtering. Using a sparse simplicial grid instead of a dense rectangular one, we can perform GP inference exponentially faster in the dimension than SKI. Our approach, Simplex-GP, enables scaling SKI to high dimensions, while maintaining strong predictive performance. We additionally provide a CUDA implementation of Simplex-GP, which enables significant GPU acceleration of MVM based inference.

Clustering is a technique of dividing the population or data points, grouping them into different clusters on the basis of similarity and dissimilarity between them. It's helps in determining the intrinsic group among the unlabeled data points. In this project we will be using Taxi dataset ( can be downloaded from Kaggle) and perform clustering Geolocation Data using K-Means and demostrate how to use DBSCAN Density-Based Spatial Clustering of Applications with Noise (DBSCAN) which discovers clusters of different shapes and sizes from data containing noise and outliers and HDBSCAN -- Hierarchical Density-Based Spatial Clustering of Applications with Noise which performs DBSCAN over varying epsilon values and integrates the result to find a clustering that gives the best stability over epsilon. Folium makes it easy to visualize data that's been manipulated in Python on an interactive leaflet map. It enables both the binding of data to a map for choropleth visualizations as well as passing rich vector/raster/HTML visualizations as markers on the map. The library has a number of built-in tilesets from OpenStreetMap, Mapbox, and Stamen, and supports custom tilesets with Mapbox or Cloudmade API keys.

The "cold posterior effect" (CPE) in Bayesian deep learning describes the uncomforting observation that the predictive performance of Bayesian neural networks can be significantly improved if the Bayes posterior is artificially sharpened using a temperature parameter T<1. The CPE is problematic in theory and practice and since the effect was identified many researchers have proposed hypotheses to explain the phenomenon. However, despite this intensive research effort the effect remains poorly understood. In this work we provide novel and nuanced evidence relevant to existing explanations for the cold posterior effect, disentangling three hypotheses: 1. The dataset curation hypothesis of Aitchison (2020): we show empirically that the CPE does not arise in a real curated data set but can be produced in a controlled experiment with varying curation strength. 2. The data augmentation hypothesis of Izmailov et al. (2021) and Fortuin et al. (2021): we show empirically that data augmentation is sufficient but not necessary for the CPE to be present. 3. The bad prior hypothesis of Wenzel et al. (2020): we use a simple experiment evaluating the relative importance of the prior and the likelihood, strongly linking the CPE to the prior. Our results demonstrate how the CPE can arise in isolation from synthetic curation, data augmentation, and bad priors. Cold posteriors observed "in the wild" are therefore unlikely to arise from a single simple cause; as a result, we do not expect a simple "fix" for cold posteriors.

Taking the Fourier integral theorem as our starting point, in this paper we focus on natural Monte Carlo and fully nonparametric estimators of multivariate distributions and conditional distribution functions. We do this without the need for any estimated covariance matrix or dependence structure between variables. These aspects arise immediately from the integral theorem. Being able to model multivariate data sets using conditional distribution functions we can study a number of problems, such as prediction for Markov processes, estimation of mixing distribution functions which depend on covariates, and general multivariate data. Estimators are explicit Monte Carlo based and require no recursive or iterative algorithms.

Unsupervised representation learning has recently received lots of interest due to its powerful generalizability through effectively leveraging large-scale unlabeled data. There are two prevalent approaches for this, contrastive learning and generative pre-training, where the former learns representations from instance-wise discrimination tasks and the latter learns them from estimating the likelihood. These seemingly orthogonal approaches have their own strengths and weaknesses. Contrastive learning tends to extract semantic information and discards details irrelevant for classifying objects, making the representations effective for discriminative tasks while degrading robustness to out-of-distribution data. On the other hand, the generative pre-training directly estimates the data distribution, so the representations tend to be robust but not optimal for discriminative tasks. In this paper, we show that we could achieve the best of both worlds by a hybrid training scheme. Specifically, we demonstrated that a transformer-based encoder-decoder architecture trained with both contrastive and generative losses can learn highly discriminative and robust representations without hurting the generative performance. We extensively validate our approach on various tasks.

While deep and large pre-trained models are the state-of-the-art for various natural language processing tasks, their huge size poses significant challenges for practical uses in resource constrained settings. Recent works in knowledge distillation propose task-agnostic as well as task-specific methods to compress these models, with task-specific ones often yielding higher compression rate. In this work, we develop a new task-agnostic distillation framework XtremeDistilTransformers that leverages the advantage of task-specific methods for learning a small universal model that can be applied to arbitrary tasks and languages. To this end, we study the transferability of several source tasks, augmentation resources and model architecture for distillation. We evaluate our model performance on multiple tasks, including the General Language Understanding Evaluation (GLUE) benchmark, SQuAD question answering dataset and a massive multi-lingual NER dataset with 41 languages. We release three distilled task-agnostic checkpoints with 13MM, 22MM and 33MM parameters obtaining SOTA performance in several tasks.

With the popularity of Machine Learning (ML) solutions, algorithms and data have been released faster than the capacity of processing them. In this context, the problem of Algorithm Recommendation (AR) is receiving a significant deal of attention recently. This problem has been addressed in the literature as a learning task, often as a Meta-Learning problem where the aim is to recommend the best alternative for a specific dataset. For such, datasets encoded by meta-features are explored by ML algorithms that try to learn the mapping between meta-representations and the best technique to be used. One of the challenges for the successful use of ML is to define which features are the most valuable for a specific dataset since several meta-features can be used, which increases the meta-feature dimension. This paper presents an empirical analysis of Feature Selection and Feature Extraction in the meta-level for the AR problem. The present study was focused on three criteria: predictive performance, dimensionality reduction, and pipeline runtime. As we verified, applying Dimensionality Reduction (DR) methods did not improve predictive performances in general. However, DR solutions reduced about 80% of the meta-features, obtaining pretty much the same performance as the original setup but with lower runtimes. The only exception was PCA, which presented about the same runtime as the original meta-features. Experimental results also showed that various datasets have many non-informative meta-features and that it is possible to obtain high predictive performance using around 20% of the original meta-features. Therefore, due to their natural trend for high dimensionality, DR methods should be used for Meta-Feature Selection and Meta-Feature Extraction.

The invariance principle from causality is at the heart of notable approaches such as invariant risk minimization (IRM) that seek to address out-of-distribution (OOD) generalization failures. Despite the promising theory, invariance principle-based approaches fail in common classification tasks, where invariant (causal) features capture all the information about the label. Are these failures due to the methods failing to capture the invariance? Or is the invariance principle itself insufficient? To answer these questions, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. In contrast to the linear regression tasks, we show that for linear classification tasks we need much stronger restrictions on the distribution shifts, or otherwise OOD generalization is impossible. Furthermore, even with appropriate restrictions on distribution shifts in place, we show that the invariance principle alone is insufficient. We prove that a form of the information bottleneck constraint along with invariance helps address key failures when invariant features capture all the information about the label and also retains the existing success when they do not. We propose an approach that incorporates both of these principles and demonstrate its effectiveness in several experiments.