We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads, causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria may fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a $K$-approximate equilibrium can always be guaranteed, where $K \approx 1.1974$ is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a $K$-approximate equilibrium. The factor $K$ is tight, meaning that there is an instance that does not admit an $\alpha$-approximate equilibrium for any $\alpha

We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.

You've found the right Statistics and Probability with Excel course! This course will teach you the skill to apply statistics and data analysis tools to various business applications. How this course will help you? A Verifiable Certificate of Completion is presented to all students who undertake this course on Probability and Statistics in Excel. If you are a business manager, or business analyst or an executive, or a student who wants to learn Probability and Statistics concepts and apply these techniques to real-world problems of the business function, this course will give you a solid base for Probability and Statistics by teaching you the most important concepts of Probability and Statistics and how to implement them in MS Excel.

VIP Cheat Sheets - Deep Learning by Stanford's CS 229 Students Download whole PDF of Supervised Learning Cheatsheet: From Here VIP Cheat Sheets - Machine Learning Tips by Stanford's CS 229 Students Download whole PDF of Supervised Learning Cheatsheet: From Here VIP Refresher: Probabilities and Statistics Cheatsheet Download whole PDF of Probability and Statistics Cheatsheets: From Here VIP Refresher: Linear Algebra and Calculus Cheat Sheets Download whole PDF of Linear Algebra and Calculus Cheatsheets: From Here You may like this: 100 Free Machine Learning Books Super VIP Cheat Sheet: Machine Learning Download whole PDF of Super VIP Machine Learning Cheat Sheet: From Here

If you've started looking into behind the scenes of popular machine learning algorithms, you might have come across the term "linear algebra". The term seems scary, but it isn't really so. Many of the machine learning algorithms rely on linear algebra because it provides the ability to "vectorize" them, making them computationally fast and efficient. Linear algebra is a vast branch of Mathematics, and not all of its knowledge is required in understanding and building machine learning algorithms, so our focus will be on the basic topics related to machine learning. NumPy implementations for each of the operations are also included at the end of each topic.

Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grow to infinity. We show that the Euclidean gradient flow of a suitable function of the edge-weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our set-up, and the examples have been worked out in detail.

We solve university level probability and statistics questions by program synthesis using OpenAI's Codex, a Transformer trained on text and fine-tuned on code. We transform course problems from MIT's 18.05 Introduction to Probability and Statistics and Harvard's STAT110 Probability into programming tasks. We then execute the generated code to get a solution. Since these course questions are grounded in probability, we often aim to have Codex generate probabilistic programs that simulate a large number of probabilistic dependencies to compute its solution. Our approach requires prompt engineering to transform the question from its original form to an explicit, tractable form that results in a correct program and solution. To estimate the amount of work needed to translate an original question into its tractable form, we measure the similarity between original and transformed questions. Our work is the first to introduce a new dataset of university-level probability and statistics problems and solve these problems in a scalable fashion using the program synthesis capabilities of large language models.

We solve MIT's Linear Algebra 18.06 course and Columbia University's Computational Linear Algebra COMS3251 courses with perfect accuracy by interactive program synthesis. This surprisingly strong result is achieved by turning the course questions into programming tasks and then running the programs to produce the correct answers. We use OpenAI Codex with zero-shot learning, without providing any examples in the prompts, to synthesize code from questions. We quantify the difference between the original question text and the transformed question text that yields a correct answer. Since all COMS3251 questions are not available online the model is not overfitting. We go beyond just generating code for questions with numerical answers by interactively generating code that also results visually pleasing plots as output. Finally, we automatically generate new questions given a few sample questions which may be used as new course content. This work is a significant step forward in solving quantitative math problems and opens the door for solving many university level STEM courses by machine.

Optimization of parameterized quantum circuits is indispensable for applications of near-term quantum devices to computational tasks with variational quantum algorithms (VQAs). However, the existing optimization algorithms for VQAs require an excessive number of quantum-measurement shots in estimating expectation values of observables or iterating updates of circuit parameters, whose cost has been a crucial obstacle for practical use. To address this problem, we develop an efficient framework, \textit{stochastic gradient line Bayesian optimization} (SGLBO), for the circuit optimization with fewer measurement shots. The SGLBO reduces the cost of measurement shots by estimating an appropriate direction of updating the parameters based on stochastic gradient descent (SGD) and further by utilizing Bayesian optimization (BO) to estimate the optimal step size in each iteration of the SGD. We formulate an adaptive measurement-shot strategy to achieve the optimization feasibly without relying on precise expectation-value estimation and many iterations; moreover, we show that a technique of suffix averaging can significantly reduce the effect of statistical and hardware noise in the optimization for the VQAs. Our numerical simulation demonstrates that the SGLBO augmented with these techniques can drastically reduce the required number of measurement shots, improve the accuracy in the optimization, and enhance the robustness against the noise compared to other state-of-art optimizers in representative tasks for the VQAs. These results establish a framework of quantum-circuit optimizers integrating two different optimization approaches, SGD and BO, to reduce the cost of measurement shots significantly.

During the past two decades, methods for identifying groups with different trends in longitudinal data have become of increasing interest across many areas of research. To support researchers, we summarize the guidance from the literature regarding longitudinal clustering. Moreover, we present a selection of methods for longitudinal clustering, including group-based trajectory modeling (GBTM), growth mixture modeling (GMM), and longitudinal k-means (KML). The methods are introduced at a basic level, and strengths, limitations, and model extensions are listed. Following the recent developments in data collection, attention is given to the applicability of these methods to intensive longitudinal data (ILD). We demonstrate the application of the methods on a synthetic dataset using packages available in R.