Exploring the Power of Rescaling

The goal of our research is a comprehensive exploration of the power of rescaling to improve the efficiency of various algorithms for linear optimization and related problems. Linear optimization and linear feasibility problemsarguably yield the fundamental problems of optimization. Advances in solvingthese problems impact the core of optimization theory, and consequently itspractical applications. The development and analysis of solution methods for linear optimization is one of the major topics in optimization research. Although the polynomial time ellipsoid method has excellent theoretical properties,however it turned out to be inefficient in practice.Still today, in spite of the dominance of interior point methods, various algorithms, such as perceptron algorithms, rescaling perceptron algorithms,von Neumann algorithms, Chubanov\u27s method, and linear optimization related problems,such as the colorful feasibility problem -- whose complexity status is still undecided --are studied.Motivated by the successful application of a rescaling principle on the perceptron algorithm,our research aims to explore the power of rescaling on other algorithms too,and improve their computational complexity. We focus on algorithms forsolving linear feasibility and related problems, whose complexity depend on a quantity $\rho$, which is a condition number for measuring the distance to the feasibility or infeasibility of the problem.These algorithms include the von Neumann algorithm and the perceptron algorithm. First, we discuss the close duality relationship between the perceptron and the von Neumann algorithms. This observation allows us to transit one algorithm as a variant of the other, as well as we can transit their complexity results. The discovery of this duality not only provides a profound insight into both of the algorithms, but also results in new variants of the algorithms.Based on this duality relationship, we propose a deterministic rescaling von Neumann algorithm. It computationally outperforms the original von Neumann algorithm. Though its complexity has not been proved yet, we construct a von Neumann example which shows that the rescaling steps cannot keep the quantity $\rho$ increasing monotonically. Showing a monotonic increase of $\rho$ is a common technique used to prove the complexity of rescaling algorithms. Therefore, this von Neumann example actually shows that another proof method needs to be discovered in order to obtain the complexity of this deterministic rescaling von Neumann algorithm. Furthermore, this von Neumann example serves as the foundation of a perceptron example, which verifies that $\rho$ is not always increasing after one rescaling step in the polynomial time deterministic rescaling perceptron algorithm either.After that, we adapt the idea of Chubanov\u27s method to our rescaling frame and develop a polynomial-time column-wise rescaling von Neumann algorithm. Chubanov recently proposed a simple polynomial-time algorithm for solving homogeneous linear systems with positive variables. The Basic Procedure of Chubanov\u27s method can either find a feasible solution, or identify an upper bound for at least one coordinate of any feasible solution. The column-wise rescaling von Neumann algorithm combines the Basic Procedure with column-wise rescaling to identify zero coordinates in all feasible solutions and remove the corresponding columns from the coefficient matrix. This is the first variant of the von Neumann algorithm with polynomial-time complexity. Furthermore, compared with the original von Neumann algorithm which returns an approximate solution, this rescaling variant guarantees an exact solution for feasible problems.Finally, we develop the methodology of higher order rescaling and propose a higher-order perceptron algorithm.We implement the perceptron improvement phase by utilizing parallel processors.Therefore, in a multi-core environment we may obtain several rescaling vectors without extra wall-clock time.Once we use these rescaling vectors in a single higher-order rescaling step, better rescaling ratesmay be expected and thus computational efficiency is improved

Machine Learning Methods for Attack Detection in the Smart Grid

Attack detection problems in the smart grid are posed as statistical learning problems for different attack scenarios in which the measurements are observed in batch or online settings. In this approach, machine learning algorithms are used to classify measurements as being either secure or attacked. An attack detection framework is provided to exploit any available prior knowledge about the system and surmount constraints arising from the sparse structure of the problem in the proposed approach. Well-known batch and online learning algorithms (supervised and semi-supervised) are employed with decision and feature level fusion to model the attack detection problem. The relationships between statistical and geometric properties of attack vectors employed in the attack scenarios and learning algorithms are analyzed to detect unobservable attacks using statistical learning methods. The proposed algorithms are examined on various IEEE test systems. Experimental analyses show that machine learning algorithms can detect attacks with performances higher than the attack detection algorithms which employ state vector estimation methods in the proposed attack detection framework.Comment: 14 pages, 11 Figure

Constant Rate Approximate Maximum Margin Algorithms

We present a new class of perceptron-like algorithms with margin in which the “effective” learning rate, defined as the ratio of the learning rate to the length of the weight vector, remains constant. We prove that the new algorithms converge in a finite number of steps and show that there exists a limit of the parameters involved in which convergence leads to classification with maximum margin

Selective Sampling with Drift

Recently there has been much work on selective sampling, an online active learning setting, in which algorithms work in rounds. On each round an algorithm receives an input and makes a prediction. Then, it can decide whether to query a label, and if so to update its model, otherwise the input is discarded. Most of this work is focused on the stationary case, where it is assumed that there is a fixed target model, and the performance of the algorithm is compared to a fixed model. However, in many real-world applications, such as spam prediction, the best target function may drift over time, or have shifts from time to time. We develop a novel selective sampling algorithm for the drifting setting, analyze it under no assumptions on the mechanism generating the sequence of instances, and derive new mistake bounds that depend on the amount of drift in the problem. Simulations on synthetic and real-world datasets demonstrate the superiority of our algorithms as a selective sampling algorithm in the drifting setting

Optimal coloured perceptrons

Ashkin-Teller type perceptron models are introduced. Their maximal capacity per number of couplings is calculated within a first-step replica-symmetry-breaking Gardner approach. The results are compared with extensive numerical simulations using several algorithms.Comment: 8 pages in Latex with 2 eps figures, RSB1 calculations has been adde

Identifying Real Estate Opportunities using Machine Learning

The real estate market is exposed to many fluctuations in prices because of existing correlations with many variables, some of which cannot be controlled or might even be unknown. Housing prices can increase rapidly (or in some cases, also drop very fast), yet the numerous listings available online where houses are sold or rented are not likely to be updated that often. In some cases, individuals interested in selling a house (or apartment) might include it in some online listing, and forget about updating the price. In other cases, some individuals might be interested in deliberately setting a price below the market price in order to sell the home faster, for various reasons. In this paper, we aim at developing a machine learning application that identifies opportunities in the real estate market in real time, i.e., houses that are listed with a price substantially below the market price. This program can be useful for investors interested in the housing market. We have focused in a use case considering real estate assets located in the Salamanca district in Madrid (Spain) and listed in the most relevant Spanish online site for home sales and rentals. The application is formally implemented as a regression problem that tries to estimate the market price of a house given features retrieved from public online listings. For building this application, we have performed a feature engineering stage in order to discover relevant features that allows for attaining a high predictive performance. Several machine learning algorithms have been tested, including regression trees, k-nearest neighbors, support vector machines and neural networks, identifying advantages and handicaps of each of them.Comment: 24 pages, 13 figures, 5 table