To understand how deep learning works, it is crucial to understand the training dynamics of neural networks. Several interesting hypotheses about these dynamics have been made based on empirically observed phenomena, but there exists a limited theoretical understanding of when and why such phenomena occur. In this paper, we consider the training dynamics of gradient flow on kernel least-squares objectives, which is a limiting dynamics of SGD trained neural networks. Using precise high-dimensional asymptotics, we characterize the dynamics of the fitted model in two "worlds": in the Oracle World the model is trained on the population distribution and in the Empirical World the model is trained on a sampled dataset. We show that under mild conditions on the kernel and $L^2$ target regression function the training dynamics undergo three stages characterized by the behaviors of the models in the two worlds. Our theoretical results also mathematically formalize some interesting deep learning phenomena. Specifically, in our setting we show that SGD progressively learns more complex functions and that there is a "deep bootstrap" phenomenon: during the second stage, the test error of both worlds remain close despite the empirical training error being much smaller. Finally, we give a concrete example comparing the dynamics of two different kernels which shows that faster training is not necessary for better generalization.

Curriculum learning (CL) is a commonly used machine learning training strategy. However, we still lack a clear theoretical understanding of CL's benefits. In this paper, we study the benefits of CL in the multitask linear regression problem under both structured and unstructured settings. For both settings, we derive the minimax rates for CL with the oracle that provides the optimal curriculum and without the oracle, where the agent has to adaptively learn a good curriculum. Our results reveal that adaptive learning can be fundamentally harder than the oracle learning in the unstructured setting, but it merely introduces a small extra term in the structured setting. To connect theory with practice, we provide justification for a popular empirical method that selects tasks with highest local prediction gain by comparing its guarantees with the minimax rates mentioned above.

We study the supervised clustering problem under the two-component anisotropic Gaussian mixture model in high dimensions and in the non-asymptotic setting. We first derive a lower and a matching upper bound for the minimax risk of clustering in this framework. We also show that in the high-dimensional regime, the linear discriminant analysis (LDA) classifier turns out to be sub-optimal in the minimax sense. Next, we characterize precisely the risk of $\ell_2$-regularized supervised least squares classifiers. We deduce the fact that the interpolating solution may outperform the regularized classifier, under mild assumptions on the covariance structure of the noise. Our analysis also shows that interpolation can be robust to corruption in the covariance of the noise when the signal is aligned with the "clean" part of the covariance, for the properly defined notion of alignment. To the best of our knowledge, this peculiar phenomenon has not yet been investigated in the rapidly growing literature related to interpolation. We conclude that interpolation is not only benign but can also be optimal, and in some cases robust.

The recent introduction of thermodynamic integration techniques has provided a new framework for understanding and improving variational inference (VI). In this work, we present a careful analysis of the thermodynamic variational objective (TVO), bridging the gap between existing variational objectives and shedding new insights to advance the field. In particular, we elucidate how the TVO naturally connects the three key variational schemes, namely the importance-weighted VI, Renyi-VI, and MCMC-VI, which subsumes most VI objectives employed in practice. To explain the performance gap between theory and practice, we reveal how the pathological geometry of thermodynamic curves negatively affects TVO. By generalizing the integration path from the geometric mean to the weighted Holder mean, we extend the theory of TVO and identify new opportunities for improving VI. This motivates our new VI objectives, named the Holder bounds, which flatten the thermodynamic curves and promise to achieve a one-step approximation of the exact marginal log-likelihood. A comprehensive discussion on the choices of numerical estimators is provided. We present strong empirical evidence on both synthetic and real-world datasets to support our claims.

Teaching plays a fundamental role in human learning. Typically, a human teaching strategy would involve assessing a student's knowledge progress for tailoring the teaching materials in a way that enhances the learning progress. A human teacher would achieve this by tracing a student's knowledge over important learning concepts in a task. Albeit, such teaching strategy is not well exploited yet in machine learning as current machine teaching methods tend to directly assess the progress on individual training samples without paying attention to the underlying learning concepts in a learning task. In this paper, we propose a novel method, called Knowledge Augmented Data Teaching (KADT), which can optimize a data teaching strategy for a student model by tracing its knowledge progress over multiple learning concepts in a learning task. Specifically, the KADT method incorporates a knowledge tracing model to dynamically capture the knowledge progress of a student model in terms of latent learning concepts. Then we develop an attention pooling mechanism to distill knowledge representations of a student model with respect to class labels, which enables to develop a data teaching strategy on critical training samples. We have evaluated the performance of the KADT method on four different machine learning tasks including knowledge tracing, sentiment analysis, movie recommendation, and image classification. The results comparing to the state-of-the-art methods empirically validate that KADT consistently outperforms others on all tasks.

The idea of using quantum computers for machine learning has recently received attention both in academia and industry [1-13]. While proof of principle study have shown that some problems of mathematical interest quantum computers are useful [13], quantum advantage in machine learning algorithms for practical applications is still unclear [14]. On classical architectures, a first-principle theory of machine learning, especially the so-called deep learning that uses a large number of layers, is still in development. Early developments of the statistical learning theory provide rigorous guarantees on the learning capability in generic learning algorithms, but theoretical bounds obtained from information theory are sometimes weak in practical settings. The theory of neural tangent kernel (NTK) has been deemed an important tool to understand deep neural networks [15-21]. In the large-width limit, a generic neural network becomes nearly Gaussian when averaging over the initial weights and biases, and the learning capabilities become predictable. The NTK theory allows to derive analytical understanding of the neural networks dynamics, improving on statistical learning theory and shedding light on the underlying principle of deep learning [22-26]. In the quantum machine learning community, a similar first principle theory would help in understanding the training dynamics and selecting appropri-junyuliu@uchicago.edu

This tutorial will introduce Survival Analysis and its wide scope. Survival analysis is applicable to all fields that deal with time-dependent data: medicine, biology, engineering, marketing, finance, and many others. A data scientist in the role of a survival analyst is involved in questions concerning clinical trials, customer churn, time till violent death of Roman emperors, the failure rate of products or systems, and a practically infinite number of other situations that raise the question: "When will it happen, and what's its risk of failure or its chance of success?" After the introduction in chapter 1, we will explore a number of examples in chapter 2 -- and then complete the tutorial with bathtub-shaped lifetimes. To begin with: What is force of mortality? It is the conditional probability of death at a particular instant after having survived up to that instant. Also called the intensity of mortality or the instantaneous death rate in actuarial science. In fields that are less concerned about predicting the transition to the afterlife, it is better known as the hazard rate.

Let's say that you work with the sales and marketing team to reach your company's pre-set goals. While your company is doing well in terms of generating revenue and retaining customers, you can not help but think that it can do better. As things stand, the advertisements, promotions, and special offers are homogenous across all customers, which is a serious issue. Engaging with customers in a manner that they won't be receptive to is tantamount to wasting your advertising budget. After all, you don't want your company to spend its limited budget sending diaper coupons to college students or advertising gaming consoles to elderly women.

While significant progress has been made towards explaining black-box machine-learning (ML) models, there is still a distinct lack of diagnostic tools that elucidate the spatial behaviour of ML models in terms of predictive skill and variable importance. This contribution proposes spatial prediction error profiles (SPEPs) and spatial variable importance profiles (SVIPs) as novel model-agnostic assessment and interpretation tools for spatial prediction models with a focus on prediction distance. Their suitability is demonstrated in two case studies representing a regionalization task in an environmental-science context, and a classification task from remotely-sensed land cover classification. In these case studies, the SPEPs and SVIPs of geostatistical methods, linear models, random forest, and hybrid algorithms show striking differences but also relevant similarities. Limitations of related cross-validation techniques are outlined, and the case is made that modelers should focus their model assessment and interpretation on the intended spatial prediction horizon. The range of autocorrelation, in contrast, is not a suitable criterion for defining spatial cross-validation test sets. The novel diagnostic tools enrich the toolkit of spatial data science, and may improve ML model interpretation, selection, and design.

We consider the problem of upper bounding the expected log-likelihood sub-optimality of the maximum likelihood estimate (MLE), or a conjugate maximum a posteriori (MAP) for an exponential family, in a non-asymptotic way. Surprisingly, we found no general solution to this problem in the literature. In particular, current theories do not hold for a Gaussian or in the interesting few samples regime. After exhibiting various facets of the problem, we show we can interpret the MAP as running stochastic mirror descent (SMD) on the log-likelihood. However, modern convergence results do not apply for standard examples of the exponential family, highlighting holes in the convergence literature. We believe solving this very fundamental problem may bring progress to both the statistics and optimization communities.