The problem of selecting an algorithm that appears most suitable for a specific instance of an algorithmic problem class, such as the Boolean satisfiability problem, is called instance-specific algorithm selection. Over the past decade, the problem has received considerable attention, resulting in a number of different methods for algorithm selection. Although most of these methods are based on machine learning, surprisingly little work has been done on meta learning, that is, on taking advantage of the complementarity of existing algorithm selection methods in order to combine them into a single superior algorithm selector. In this paper, we introduce the problem of meta algorithm selection, which essentially asks for the best way to combine a given set of algorithm selectors. We present a general methodological framework for meta algorithm selection as well as several concrete learning methods as instantiations of this framework, essentially combining ideas of meta learning and ensemble learning. In an extensive experimental evaluation, we demonstrate that ensembles of algorithm selectors can significantly outperform single algorithm selectors and have the potential to form the new state of the art in algorithm selection.

We extend the theory of PAC learning in a way which allows to model a rich variety of learning tasks where the data satisfy special properties that ease the learning process. For example, tasks where the distance of the data from the decision boundary is bounded away from zero. The basic and simple idea is to consider partial concepts: these are functions that can be undefined on certain parts of the space. When learning a partial concept, we assume that the source distribution is supported only on points where the partial concept is defined. This way, one can naturally express assumptions on the data such as lying on a lower dimensional surface or margin conditions. In contrast, it is not at all clear that such assumptions can be expressed by the traditional PAC theory. In fact we exhibit easy-to-learn partial concept classes which provably cannot be captured by the traditional PAC theory. This also resolves a question posed by Attias, Kontorovich, and Mansour 2019. We characterize PAC learnability of partial concept classes and reveal an algorithmic landscape which is fundamentally different than the classical one. For example, in the classical PAC model, learning boils down to Empirical Risk Minimization (ERM). In stark contrast, we show that the ERM principle fails in explaining learnability of partial concept classes. In fact, we demonstrate classes that are incredibly easy to learn, but such that any algorithm that learns them must use an hypothesis space with unbounded VC dimension. We also find that the sample compression conjecture fails in this setting. Thus, this theory features problems that cannot be represented nor solved in the traditional way. We view this as evidence that it might provide insights on the nature of learnability in realistic scenarios which the classical theory fails to explain.

Machine learning is a branch of computer science that has the potential to transform epidemiologic sciences. Amid a growing focus on "Big Data," it offers epidemiologists new tools to tackle problems for which classical methods are not well-suited. In order to critically evaluate the value of integrating machine learning algorithms and existing methods, however, it is essential to address language and technical barriers between the two fields that can make it difficult for epidemiologists to read and assess machine learning studies. Here, we provide an overview of the concepts and terminology used in machine learning literature, which encompasses a diverse set of tools with goals ranging from prediction to classification to clustering. We provide a brief introduction to 5 common machine learning algorithms and 4 ensemble-based approaches. We then summarize epidemiologic applications of machine learning techniques in the published literature. We recommend approaches to incorporate machine learning in epidemiologic research and discuss opportunities and challenges for integrating machine learning and existing epidemiologic research methods. Machine learning is a branch of computer science that broadly aims to enable computers to "learn" without being directly programmed (1). It has origins in the artificial intelligence movement of the 1950s and emphasizes practical objectives and applications, particularly prediction and optimization. Computers "learn" in machine learning by improving their performance at tasks through "experience" (2, p. xv). In practice, "experience" usually means fitting to data; hence, there is not a clear boundary between machine learning and statistical approaches. Indeed, whether a given methodology is considered "machine learning" or "statistical" often reflects its history as much as genuine differences, and many algorithms (e.g., least absolute shrinkage and selection operator (LASSO), stepwise regression) may or may not be considered machine learning depending on who you ask. Still, despite methodological similarities, machine learning is philosophically and practically distinguishable. At the liberty of (considerable) oversimplification, machine learning generally emphasizes predictive accuracy over hypothesis-driven inference, usually focusing on large, high-dimensional (i.e., having many covariates) data sets (3, 4). Regardless of the precise distinction between approaches, in practice, machine learning offers epidemiologists important tools. In particular, a growing focus on "Big Data" emphasizes problems and data sets for which machine learning algorithms excel while more commonly used statistical approaches struggle. This primer provides a basic introduction to machine learning with the aim of providing readers a foundation for critically reading studies based on these methods and a jumping-off point for those interested in using machine learning techniques in epidemiologic research.

"Tell me who your friends are and I will tell you who you are" As the saying goes -- "A person is known by the company he keeps" and it sounds quite intuitive because people in the same company share similar interests. Data points that are closer to a particular data point will portray similarity in properties and there is a high possibility that they belong to the same class. Here these close data points are called "neighbors" for that particular data point. So, today we gonna discuss one famous Machine Learning algorithm called "K-Nearest Neighbours" or KNN simply. "KNN is a supervised, non-parametric and lazy learning algorithm." Non-parametric means there is no assumption for underlying data distribution. In other words, the model structure is determined from the dataset.

In the previous tutorial, we wrote an optimization function that will output the learned w and b parameters. Now we can use w and b to predict the labels for our dataset X. So, in this tutorial, we will implement the predict() function. So we will implement a prediction function, but first, let's see what are the inputs and outputs to it: If we'll run our new function on previous values "predict(w, b, X)" we should receive the following results: From our results, we could say that we predicted two cats and one dog. But because input was not real images but just simple random test numbers, our predictions also don't mean anything.

Magnetic resonance imaging (MRI) is a medical imaging technique used to capture volumetric image sequences of internal soft tissues, such as cardiac muscles. In comparison to X-Ray imaging (XR) and Computer Tomography (CT), MRI provides images with improved structural details via finer spatial resolutions. Cardiac MRI (CMR) focuses on the heart, allowing trained cardiologists to measure heart parameters, for example the mass of the cardiac muscle (myocardium mass), the volumes of blood cavities (atrial and ventricular volumes) and the amount of blood pumped per heartbeat (ejection fraction) [Peng et al., 2016]. Those parameters are used to assess how healthy is the heart, by recognizing early conditions and signs before the onset of infarcts and other complications. Due to the size and complexity of CMR sequences, complex techniques are required to produce detailed analyses; one of these techniques is deep learning (DL). Many of the tasks and goals related to the cardiac functional analysis - for example segmentation of structures [Bernard et al., 2018], estimation of heart parameters [Xue et al., 2018], and detection of diseases [Khened et al., 2017] - have benefited from DL methods. For even better results, research in DL has pointed out that models based on convolutional neural networks (CNN) have had a higher efficacy when provided with regions-of-interest (ROI) either explicitly or implicitly [Xue et al., 2018]. The detection of ROIs, usually named ROI proposal, is a preprocessing step whose goal is to identify the most prominent regions of an image (frame) for discovering clinically-relevant artifacts. The explicit ROI proposal approaches usually follow a combination of methods, for example: (a) pipelining a segmentation and a regression network; (b) preprocessing the input with a region proposal algorithm [He et al., 2015] or with a CNN [Wu et al., 2020]; or (c) by using manual cropping [Xue et al., 2017].

Reward-Weighted Regression (RWR) belongs to a family of widely known iterative Reinforcement Learning algorithms based on the Expectation-Maximization framework. In this family, learning at each iteration consists of sampling a batch of trajectories using the current policy and fitting a new policy to maximize a return-weighted log-likelihood of actions. Although RWR is known to yield monotonic improvement of the policy under certain circumstances, whether and under which conditions RWR converges to the optimal policy have remained open questions. In this paper, we provide for the first time a proof that RWR converges to a global optimum when no function approximation is used.

Models like LASSO and ridge regression are extensively used in practice due to their interpretability, ease of use, and strong theoretical guarantees. Cross-validation (CV) is widely used for hyperparameter tuning in these models, but do practical optimization methods minimize the true out-of-sample loss? A recent line of research promises to show that the optimum of the CV loss matches the optimum of the out-of-sample loss (possibly after simple corrections). It remains to show how tractable it is to minimize the CV loss. In the present paper, we show that, in the case of ridge regression, the CV loss may fail to be quasiconvex and thus may have multiple local optima. We can guarantee that the CV loss is quasiconvex in at least one case: when the spectrum of the covariate matrix is nearly flat and the noise in the observed responses is not too high. More generally, we show that quasiconvexity status is independent of many properties of the observed data (response norm, covariate-matrix right singular vectors and singular-value scaling) and has a complex dependence on the few that remain. We empirically confirm our theory using simulated experiments.

We show that the Riemannian gradient descent algorithm on the low-rank matrix manifold almost surely escapes some spurious critical points on the boundary of the manifold. Given that the low-rank matrix manifold is an incomplete set, this result is the first to overcome this difficulty and partially justify the global use of the Riemannian gradient descent on the manifold. The spurious critical points are some rank-deficient matrices that capture only part of the SVD components of the ground truth. They exhibit very singular behavior and evade the classical analysis of strict saddle points. We show that using the dynamical low-rank approximation and a rescaled gradient flow, some of the spurious critical points can be converted to classical strict saddle points, which leads to the desired result. Numerical experiments are provided to support our theoretical findings.

In this work we explore the limiting dynamics of deep neural networks trained with stochastic gradient descent (SGD). We find empirically that long after performance has converged, networks continue to move through parameter space by a process of anomalous diffusion in which distance travelled grows as a power law in the number of gradient updates with a nontrivial exponent. We reveal an intricate interaction between the hyperparameters of optimization, the structure in the gradient noise, and the Hessian matrix at the end of training that explains this anomalous diffusion. To build this understanding, we first derive a continuous-time model for SGD with finite learning rates and batch sizes as an underdamped Langevin equation. We study this equation in the setting of linear regression, where we can derive exact, analytic expressions for the phase space dynamics of the parameters and their instantaneous velocities from initialization to stationarity. Using the Fokker-Planck equation, we show that the key ingredient driving these dynamics is not the original training loss, but rather the combination of a modified loss, which implicitly regularizes the velocity, and probability currents, which cause oscillations in phase space. We identify qualitative and quantitative predictions of this theory in the dynamics of a ResNet-18 model trained on ImageNet. Through the lens of statistical physics, we uncover a mechanistic origin for the anomalous limiting dynamics of deep neural networks trained with SGD.