This dissertation studies a fundamental open challenge in deep learning theory: why do deep networks generalize well even while being overparameterized, unregularized and fitting the training data to zero error? In the first part of the thesis, we will empirically study how training deep networks via stochastic gradient descent implicitly controls the networks' capacity. Subsequently, to show how this leads to better generalization, we will derive {\em data-dependent} {\em uniform-convergence-based} generalization bounds with improved dependencies on the parameter count. Uniform convergence has in fact been the most widely used tool in deep learning literature, thanks to its simplicity and generality. Given its popularity, in this thesis, we will also take a step back to identify the fundamental limits of uniform convergence as a tool to explain generalization. In particular, we will show that in some example overparameterized settings, {\em any} uniform convergence bound will provide only a vacuous generalization bound. With this realization in mind, in the last part of the thesis, we will change course and introduce an {\em empirical} technique to estimate generalization using unlabeled data. Our technique does not rely on any notion of uniform-convergece-based complexity and is remarkably precise. We will theoretically show why our technique enjoys such precision. We will conclude by discussing how future work could explore novel ways to incorporate distributional assumptions in generalization bounds (such as in the form of unlabeled data) and explore other tools to derive bounds, perhaps by modifying uniform convergence or by developing completely new tools altogether.

Bootstrap is a principled and powerful frequentist statistical tool for uncertainty quantification. Unfortunately, standard bootstrap methods are computationally intensive due to the need of drawing a large i.i.d. bootstrap sample to approximate the ideal bootstrap distribution; this largely hinders their application in large-scale machine learning, especially deep learning problems. In this work, we propose an efficient method to explicitly \emph{optimize} a small set of high quality "centroid" points to better approximate the ideal bootstrap distribution. We achieve this by minimizing a simple objective function that is asymptotically equivalent to the Wasserstein distance to the ideal bootstrap distribution. This allows us to provide an accurate estimation of uncertainty with a small number of bootstrap centroids, outperforming the naive i.i.d. sampling approach. Empirically, we show that our method can boost the performance of bootstrap in a variety of applications.

On top of recommending the excellent autobiography of Stanislaw Ulam, this post is about using the software Stan, but not directly to perform inference, instead to obtain R functions to evaluate a target's probability density function and its gradient. With which, one can implement custom methods, while still benefiting from the great work of the Stan team on the "modeling language" side. As a proof of concept I have implemented a plain Hamiltonian Monte Carlo sampler for a random effect logistic regression model (taken from a course on Multilevel Models by Germán Rodríguez), a coupling of that HMC algorithm (as in "Unbiased Hamiltonian Monte Carlo with couplings", see also this very recent article on the topic of coupling HMC), and then upper bounds on the total variation distance between the chain and its limiting distribution, as in "Estimating Convergence of Markov chains with L-Lag Couplings".

Many deep learning models pick up objectives using the gradient-descent method. Gradient-descent optimization needs a big number of training samples for a model to converge. That creates it out of shape for few-shot learning. We train our models to learn to achieve a sure objective in generic deep learning models. However, humans train to learn any objective. There are different optimization methods that emphasize learn-to-learn mechanisms.

This article will discuss how to graph, organize, and set-up data using sklearn, pandas, and NumPy in reference to the Kaggle project. I am going to be using Jupyter Labs, and the code will be based on that. Sklearn: Sklearn is a machine learning software in Python's library. The main features are used for statistical modeling for topics such as regression. The sklearn API can be referenced here.

The binomial distribution is a probability distribution that applies to binomial experiments. The binomial distribution may be imagined as the probability distribution of a number of heads that appear on a coin flip in a specific experiment comprising of a fixed number of coin flips. In this blog post, we will learn binomial distribution with the help of examples. If you are an aspiring data scientist looking forward to learning/understand the binomial distribution in a better manner, this post might be very helpful. The binomial distribution is a discrete probability distribution that represents the probabilities of binomial random variables in a binomial experiment.

The set of local modes and the ridge lines estimated from a dataset are important summary characteristics of the data-generating distribution. In this work, we consider estimating the local modes and ridges from point cloud data in a product space with two or more Euclidean/directional metric spaces. Specifically, we generalize the well-known (subspace constrained) mean shift algorithm to the product space setting and illuminate some pitfalls in such generalization. We derive the algorithmic convergence of the proposed method, provide practical guidelines on the implementation, and demonstrate its effectiveness on both simulated and real datasets.

Multi-head attention is a driving force behind state-of-the-art transformers which achieve remarkable performance across a variety of natural language processing (NLP) and computer vision tasks. It has been observed that for many applications, those attention heads learn redundant embedding, and most of them can be removed without degrading the performance of the model. Inspired by this observation, we propose Transformer with a Mixture of Gaussian Keys (Transformer-MGK), a novel transformer architecture that replaces redundant heads in transformers with a mixture of keys at each head. These mixtures of keys follow a Gaussian mixture model and allow each attention head to focus on different parts of the input sequence efficiently. Compared to its conventional transformer counterpart, Transformer-MGK accelerates training and inference, has fewer parameters, and requires less FLOPs to compute while achieving comparable or better accuracy across tasks. Transformer-MGK can also be easily extended to use with linear attentions. We empirically demonstrate the advantage of Transformer-MGK in a range of practical applications including language modeling and tasks that involve very long sequences. On the Wikitext-103 and Long Range Arena benchmark, Transformer-MGKs with 4 heads attain comparable or better performance to the baseline transformers with 8 heads.

Empirical risk minimization (ERM) is a principle in statistical learning. Through ERM, we can measure the performance of a family of learning algorithms based on a set of observed training data empirically without knowing the true distribution of the data and derive theoretical bounds on the performance. ERM is routinely applied in a wide range of learning problems such as regression, classification, and clustering. In recent years, with the increasing popularity in privacy-preserving machine learning that satisfies formal privacy guarantees such as differential privacy (DP) [10], the topic of privacy-preserving ERM has also been investigated. Generally speaking, differentially private empirical risk minimization (DP-ERM) can be realized by perturbing the output (estimation or prediction), the objective function (input), or iteratively during the algorithmic optimization, given an ERM problem. For output perturbation, randomization mechanisms need to be applied every time a new output is released; for iterative algorithmic perturbation, each iteration incurs a privacy loss, careful planning and implementation of privacy accounting methods to minimize the overall privacy loss is critical. In this paper, we focus on differentially private perturbation of objective functions. Once an objective function is perturbed, the subsequent optimization does not incur additional privacy loss and all outputs generated from the optimization are also differentially private.

We propose an approach for learning robust acoustic models in adverse environments, characterized by a significant mismatch between training and test conditions. This problem is of paramount importance for the deployment of speech recognition systems that need to perform well in unseen environments. Our approach is an instance of vicinal risk minimization, which aims to improve risk estimates during training by replacing the delta functions that define the empirical density over the input space with an approximation of the marginal population density in the vicinity of the training samples. More specifically, we assume that local neighborhoods centered at training samples can be approximated using a mixture of Gaussians, and demonstrate theoretically that this can incorporate robust inductive bias into the learning process. We characterize the individual mixture components implicitly via data augmentation schemes, designed to address common sources of spurious correlations in acoustic models. To avoid potential confounding effects on robustness due to information loss, which has been associated with standard feature extraction techniques (e.g., FBANK and MFCC features), we focus our evaluation on the waveform-based setting. Our empirical results show that the proposed approach can generalize to unseen noise conditions, with 150% relative improvement in out-of-distribution generalization compared to training using the standard risk minimization principle. Moreover, the results demonstrate competitive performance relative to models learned using a training sample designed to match the acoustic conditions characteristic of test utterances (i.e., optimal vicinal densities).