# Machine Learning and Data Science

## Results

### Regularization of the Kernel Matrix via Covariance Matrix Shrinkage Estimation

The kernel trick concept, formulated as an inner product in a feature space, facilitates powerful extensions to many well-known algorithms. While the kernel matrix involves inner products in the feature space, the sample covariance matrix of the data requires outer products. Therefore, their spectral properties are tightly connected. This allows us to examine the kernel matrix through the sample covariance matrix in the feature space and vice versa. The use of kernels often involves a large number of features, compared to the number of observations. In this scenario, the sample covariance matrix is not well-conditioned nor is it necessarily invertible, mandating a solution to the problem of estimating high-dimensional covariance matrices under small sample size conditions. We tackle this problem through the use of a shrinkage estimator that offers a compromise between the sample covariance matrix and a well-conditioned matrix (also known as the "target") with the aim of minimizing the mean-squared error (MSE). We propose a distribution-free kernel matrix regularization approach that is tuned directly from the kernel matrix, avoiding the need to address the feature space explicitly. Numerical simulations demonstrate that the proposed regularization is effective in classification tasks.

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### Sequential Inverse Approximation of a Regularized Sample Covariance Matrix

One of the goals in scaling sequential machine learning methods pertains to dealing with high-dimensional data spaces. A key related challenge is that many methods heavily depend on obtaining the inverse covariance matrix of the data. It is well known that covariance matrix estimation is problematic when the number of observations is relatively small compared to the number of variables. A common way to tackle this problem is through the use of a shrinkage estimator that offers a compromise between the sample covariance matrix and a well-conditioned matrix, with the aim of minimizing the mean-squared error. We derived sequential update rules to approximate the inverse shrinkage estimator of the covariance matrix. The approach paves the way for improved large-scale machine learning methods that involve sequential updates.

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### Drawing Sound Conclusions from Noisy Judgments

The quality of a search engine is typically evaluated using hand-labeled data sets, where the labels indicate the relevance of documents to queries. Often the number of labels needed is too large to be created by the best annotators, and so less accurate labels (e.g. from crowdsourcing) must be used. This introduces errors in the labels, and thus errors in standard precision metrics (such as P@k and DCG); the lower the quality of the judge, the more errorful the labels, consequently the more inaccurate the metric. We introduce equations and algorithms that can adjust the metrics to the values they would have had if there were no annotation errors.

This is especially important when two search engines are compared by comparing their metrics. We give examples where one engine appeared to be statistically significantly better than the other, but the effect disappeared after the metrics were corrected for annotation error. In other words the evidence supporting a statistical difference was illusory, and caused by a failure to account for annotation error.

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### Astro: A Predictive Model for Anomaly Detection and Feedback-based Scheduling on Hadoop

The sheer growth in data volume and Hadoop cluster size make it a significant challenge to diagnose and locate problems in a production-level cluster environment efficiently and within a short period of time. Often times, the distributed monitoring systems are not capable of detecting a problem well in advance when a large-scale Hadoop cluster starts to deteriorate i n performance or becomes unavailable. Thus, inc o m i n g workloads, scheduled between the time when cluster starts to deteriorate and the time when the problem is identified, suffer from longer execution times. As a result, both reliability and throughput of the cluster reduce significantly. In this paper, we address this problem by proposing a system called Astro, which consists of a predictive model and an extension to the Hadoop scheduler. The predictive model in Astro takes into account a rich set of cluster behavioral information that are collected by monitoring processes and model them using machine learning algorithms to predict future behavior of the cluster. The Astro predictive model detects anomalies in the cluster and also identifies a ranked set of metrics that have contributed the most towards the problem. The Astro scheduler uses the prediction outcome and the list of metrics to decide whether it needs to move and reduce workloads from the problematic cluster nodes or to prevent additional workload allocations to them, in order to improve both throughput and reliability of the cluster. The results demonstrate that the Astro scheduler improves usage of cluster compute resources significantly by 64.23% compared to traditional Hadoop. Furthermore, the runtime of the benchmark application reduced by 26.68% during the time of anomaly, thus improving the cluster throughput.

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