The study on generalization performance of ranking algorithms is one of the fundamental issues in ranking learning theory. Although several generalization bounds have been proposed based on different measures, the convergence rates of the existing bounds are usually at most O (√1/ n ), where n is the size of data set. In this paper, we derive novel generalization bounds for the regularized ranking in reproducing kernel Hilbert space via integral operator of kernel function. We prove that the rates of our bounds are much faster than (√1/ n ). Specifically, we first introduce a notion of local Rademacher complexity for ranking, called local ranking  Rademacher complexity, which is used to measure the complexity of the space of loss functions of the ranking. Then, we use the local ranking Rademacher complexity to obtain a basic generalization bound. Finally, we establish the relationship between the local Rademacher complexity and the eigenvalues of integral operator, and further derive sharp generalization bounds of faster convergence rate.

We consider the problem of identifying the causal direction between two discrete random variables using observational data. Unlike previous work, we keep the most general functional model but make an assumption on the unobserved exogenous variable: Inspired by Occam's razor, we assume that the exogenous variable is simple in the true causal direction. We quantify simplicity using Renyi entropy. Our main result is that, under natural assumptions, if the exogenous variable has low H0 entropy (cardinality) in the true direction, it must have high H0 entropy in the wrong direction. We establish several algorithmic hardness results about estimating the minimum entropy exogenous variable. We show that the problem of finding the exogenous variable with minimum H1 entropy (Shannon Entropy) is equivalent to the problem of finding minimum joint entropy given n marginal distributions, also known as minimum entropy coupling problem. We propose an efficient greedy algorithm for the minimum entropy coupling problem, that for n=2 provably finds a local optimum. This gives a greedy algorithm for finding the exogenous variable with minimum Shannon entropy. Our greedy entropy-based causal inference algorithm has similar performance to the state of the art additive noise models in real datasets. One advantage of our approach is that we make no use of the values of random variables but only their distributions. Our method can therefore be used for causal inference for both ordinal and also categorical data, unlike additive noise models.

We propose a new variant of the group activity selection problem (GASP), where the agents are placed on a social network and activities can only be assigned to connected subgroups. We show that if multiple groupscan simultaneously engage in the same activity, finding a stable outcome is easy as long as the networkis acyclic. In contrast, if each activity can be assigned to a single group only, finding stable outcomes becomes computationally intractable, even if the underlying network is very simple: the problem of determining whether a given instance of a GASP admits a Nash stable outcome turns out to be NP-hard when the social network is a path, a star, or if the size of each connected component is bounded by a constant.On the other hand, we obtain fixed-parameter tractability results for this problem with respectto the number of activities.

Machine learning is the subfield of computer science that gives computers the ability to learn without being explicitly programmed (Arthur Samuel, 1959). Evolved from the study of pattern recognition and computational learning theory in artificial intelligence, machine learning explores the study and construction of algorithms that can learn from and make predictions on data – such algorithms overcome following strictly static program instructions by making data driven predictions or decisions, through building a model from sample inputs. Machine learning is employed in a range of computing tasks where designing and programming explicit algorithms is infeasible; example applications include spam filtering, detection of network intruders or malicious insiders working towards a data breach, optical character recognition (OCR), search engines and computer vision. Machine learning is closely related to (and often overlaps with) computational statistics, which also focuses in prediction-making through the use of computers. It has strong ties to mathematical optimization, which delivers methods, theory and application domains to the field. Machine learning is sometimes conflated with data mining, where the latter subfield focuses more on exploratory data analysis and is known as unsupervised learning.

The supply of able ML designers has yet to catch up to this demand. A major reason for this is that ML is just plain tricky. This tutorial introduces the basics of Machine Learning theory, laying down the common themes and concepts, making it easy to follow the logic and get comfortable with the topic. So what exactly is "machine learning" anyway? ML is actually a lot of things. The field is quite vast and is expanding rapidly, being continually partitioned and sub-partitioned ad nauseam into different sub-specialties and types of machine learning.

Machine learning is the subfield of computer science that gives computers the ability to learn without being explicitly programmed (Arthur Samuel, 1959).[1] Evolved from the study of pattern recognition and computational learning theory in artificial intelligence,[2] machine learning explores the study and construction of algorithms that can learn from and make predictions on data[3] – such algorithms overcome following strictly static program instructions by making data driven predictions or decisions,[4]:2 through building a model from sample inputs. Machine learning is employed in a range of computing tasks where designing and programming explicit algorithms is infeasible; example applications include spam filtering, detection of network intruders or malicious insiders working towards a data breach,[5] optical character recognition (OCR),[6] search engines and computer vision. Machine learning is closely related to (and often overlaps with) computational statistics, which also focuses in prediction-making through the use of computers. It has strong ties to mathematical optimization, which delivers methods, theory and application domains to the field. Machine learning is sometimes conflated with data mining,[7] where the latter subfield focuses more on exploratory data analysis and is known as unsupervised learning.[4]:vii[8]

Lego robot used to reverse engineer the e.dentifier2 smartcard reader (picture courtesy of Chalupar13).

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In first part we explored the statistical model underlying the machine learning problem, and used it to formalize the problem in terms of obtaining the minimum generalization error. By noting that we cannot directly evaluate the generalization error of an ML model, we continued in the second part by establishing a theory that relates this elusive generalization error to another error metric that we can actually evaluate, which is the empirical error. That is: the generalization error (or the risk) $R(h)$ is bounded by the empirical risk (or the training error) plus a term that is proportionate to the complexity (or the richness) of the hypothesis space $ \mathcal{H} $, the dataset size $N$, and the degree of certainty $1 - \delta$ about the bound. Starting from this part, and based on this simplified theoretical result, we'll begin to draw some practical concepts for the process of solving the ML problem. We'll start by trying to get more intuition about why a more complex hypothesis space is bad.