Periodicity is often studied in timeseries modelling with autoregressive methods but is less popular in the kernel literature, particularly for higher dimensional problems such as in textures, crystallography, and quantum mechanics. Large datasets often make modelling periodicity untenable for otherwise powerful non-parametric methods like Gaussian Processes (GPs) which typically incur an $\mathcal{O}(N^3)$ computational burden and, consequently, are unable to scale to larger datasets. To this end we introduce a method termed \emph{Index Set Fourier Series Features} to tractably exploit multivariate Fourier series and efficiently decompose periodic kernels on higher-dimensional data into a series of basis functions. We show that our approximation produces significantly less predictive error than alternative approaches such as those based on random Fourier features and achieves better generalisation on regression problems with periodic data.

We examine the squared error loss landscape of shallow linear neural networks. By utilizing a regularizer on the training samples, we show---with significantly milder assumptions than previous works---that the corresponding optimization problems have benign geometric properties: there are no spurious local minima and the Hessian at every saddle point has at least one negative eigenvalue. This means that at every saddle point there is a directional negative curvature which algorithms can utilize to further decrease the objective value. These geometric properties imply that many local search algorithms---including gradient descent, which is widely utilized for training neural networks---can provably solve the training problem with global convergence. The additional regularizer has no effect on the global minimum value; rather, it plays a useful role in shrinking the set of critical points. Experiments show that this additional regularizer also speeds the convergence of iterative algorithms for solving the training optimization problem in certain cases.

An algorithm is presented for momentum gradient descent optimization based on the first-order differential equation of the Newtonian dynamics. The fictitious mass is introduced to the dynamics of momentum for regularizing the adaptive stepsize of each individual parameter. The dynamic relaxation is adapted for stochastic optimization of nonlinear objective functions through an explicit time integration with varying damping ratio. The adaptive stepsize is optimized for each individual neural network layer based on the number of inputs. The adaptive stepsize for every parameter over the entire neural network is uniformly optimized with one upper bound, independent of sparsity, for better overall convergence rate. The numerical implementation of the algorithm is similar to the Adam Optimizer, possessing computational efficiency, similar memory requirements, etc. There are three hyper-parameters in the algorithm with clear physical interpretation. Preliminary trials show promise in performance and convergence.

Linear Regression – Simple Linear Regression- there is only independent variable. Multiple Linear Regression- refers to defining a relationship between independent and dependent variables Logistic Regression – A super simple form of regression analysis in which the outcome variable is binary or dichotomous. Helps to estimate adjusted prevalence rates, adjusted for potential confounders (sociodemographic or clinical characteristics) Linear Discriminant Analysis – A generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events. Classification and Regression Trees- Decision trees are are an important type of algorithm for predictive modeling machine learning. A greedy algorithm based on divide and conquer rule.

This course will take you through the process of predictive analytics/predictive modeling. A statistical technique or machine learning algorithm is borrowed to help predict an outcome. The goal of this course is to start you on your journey to becoming a top data scientist. To do that, you need to understand the methodology or methods at your disposal in solving these problems. By using a famous example (the titanic disaster), we will show you how to understand the problem in-front of you, how to explore your data, pre-process your data, how to create your first model, how to improve model accuracy, and look at some evaluation metrics.

When food delivery services talk breathlessly about machine learning, feel free to roll your eyes: it's baked salmon they're dropping off, not Bayesian statistics. Insurance is another kettle of fish altogether. The birth of statistics is usually dated to 1662, when John Graunt calculated the probabilities of Londoners surviving to a given age. Lloyds of London started shortly thereafter, and advances in statistics and insurance have been inseparable ever since. Insurers, of course, have machines too, but the machine's'secret power' is its ability to extract prophetic insights from inhuman quantities of data.

In the era of "big data", it is becoming more of a challenge to not only build state-of-the-art predictive models, but also gain an understanding of what's really going on in the data. For example, it is often of interest to know which, if any, of the predictors in a fitted model are relatively influential on the predicted outcome. Some modern algorithms---like random forests and gradient boosted decision trees---have a natural way of quantifying the importance or relative influence of each feature. Other algorithms---like naive Bayes classifiers and support vector machines---are not capable of doing so and model-free approaches are generally used to measure each predictor's importance. In this paper, we propose a standardized, model-based approach to measuring predictor importance across the growing spectrum of supervised learning algorithms. Our proposed method is illustrated through both simulated and real data examples. The R code to reproduce all of the figures in this paper is available in the supplementary materials.

Knowledge distillation (KD) consists of transferring knowledge from one machine learning model (the teacher}) to another (the student). Commonly, the teacher is a high-capacity model with formidable performance, while the student is more compact. By transferring knowledge, one hopes to benefit from the student's compactness. %we desire a compact model with performance close to the teacher's. We study KD from a new perspective: rather than compressing models, we train students parameterized identically to their teachers. Surprisingly, these {Born-Again Networks (BANs), outperform their teachers significantly, both on computer vision and language modeling tasks. Our experiments with BANs based on DenseNets demonstrate state-of-the-art performance on the CIFAR-10 (3.5%) and CIFAR-100 (15.5%) datasets, by validation error. Additional experiments explore two distillation objectives: (i) Confidence-Weighted by Teacher Max (CWTM) and (ii) Dark Knowledge with Permuted Predictions (DKPP). Both methods elucidate the essential components of KD, demonstrating a role of the teacher outputs on both predicted and non-predicted classes. We present experiments with students of various capacities, focusing on the under-explored case where students overpower teachers. Our experiments show significant advantages from transferring knowledge between DenseNets and ResNets in either direction.

Single-trial classification of event-related potentials in electroencephalogram (EEG) signals is a very important paradigm of brain-computer interface (BCI). Because of individual differences, usually some subject-specific calibration data are required to tailor the classifier for each subject. Transfer learning has been extensively used to reduce such calibration data requirement, by making use of auxiliary data from similar/relevant subjects/tasks. However, all previous research assumes that all auxiliary data have been labeled. This paper considers a more general scenario, in which part of the auxiliary data could be unlabeled. We propose active semi-supervised transfer learning (ASTL) for offline BCI calibration, which integrates active learning, semi-supervised learning, and transfer learning. Using a visual evoked potential oddball task and three different EEG headsets, we demonstrate that ASTL can achieve consistently good performance across subjects and headsets, and it outperforms some state-of-the-art approaches in the literature.

Users in various web and mobile applications are vulnerable to attribute inference attacks, in which an attacker leverages a machine learning classifier to infer a target user's private attributes (e.g., location, sexual orientation, political view) from its public data (e.g., rating scores, page likes). Existing defenses leverage game theory or heuristics based on correlations between the public data and attributes. These defenses are not practical. Specifically, game-theoretic defenses require solving intractable optimization problems, while correlation-based defenses incur large utility loss of users' public data. In this paper, we present AttriGuard, a practical defense against attribute inference attacks. AttriGuard is computationally tractable and has small utility loss. Our AttriGuard works in two phases. Suppose we aim to protect a user's private attribute. In Phase I, for each value of the attribute, we find a minimum noise such that if we add the noise to the user's public data, then the attacker's classifier is very likely to infer the attribute value for the user. We find the minimum noise via adapting existing evasion attacks in adversarial machine learning. In Phase II, we sample one attribute value according to a certain probability distribution and add the corresponding noise found in Phase I to the user's public data. We formulate finding the probability distribution as solving a constrained convex optimization problem. We extensively evaluate AttriGuard and compare it with existing methods using a real-world dataset. Our results show that AttriGuard substantially outperforms existing methods. Our work is the first one that shows evasion attacks can be used as defensive techniques for privacy protection.