**Convex Function**

In mathematics, a real-valued function f(x) defined on an interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. Well-known examples of convex functions are the quadratic function f(x)=x^2 and the exponential function f(x)=e^x for any real number x. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a (strictly) convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and, as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always less than or equal to the expected value of the convex function of the random variable. This result, known as Jensen’s inequality, underlies many important inequalities (including, for instance, the arithmeticâ€“geometric mean inequality and Hölder’s inequality). Exponential growth is a special case of convexity. Exponential growth narrowly means “increasing at a rate proportional to the current value”, while convex growth generally means “increasing at an increasing rate (but not necessarily proportionally to current value)”. … **Data-Driven Threshold Machine (DTM)**

We present a novel distribution-free approach, the data-driven threshold machine (DTM), for a fundamental problem at the core of many learning tasks: choose a threshold for a given pre-specified level that bounds the tail probability of the maximum of a (possibly dependent but stationary) random sequence. We do not assume data distribution, but rather relying on the asymptotic distribution of extremal values, and reduce the problem to estimate three parameters of the extreme value distributions and the extremal index. We specially take care of data dependence via estimating extremal index since in many settings, such as scan statistics, change-point detection, and extreme bandits, where dependence in the sequence of statistics can be significant. Key features of our DTM also include robustness and the computational efficiency, and it only requires one sample path to form a reliable estimate of the threshold, in contrast to the Monte Carlo sampling approach which requires drawing a large number of sample paths. We demonstrate the good performance of DTM via numerical examples in various dependent settings. … **Feature Evolvable Streaming Learning**

Learning with streaming data has attracted much attention during the past few years. Though most studies consider data stream with fixed features, in real practice the features may be evolvable. For example, features of data gathered by limited-lifespan sensors will change when these sensors are substituted by new ones. In this paper, we propose a novel learning paradigm: Feature Evolvable Streaming Learning where old features would vanish and new features will occur. Rather than relying on only the current features, we attempt to recover the vanished features and exploit it to improve performance. Specifically, we learn two models from the recovered features and the current features, respectively. To benefit from the recovered features, we develop two ensemble methods. In the first method, we combine the predictions from two models and theoretically show that with assistance of old features, the performance on new features can be improved. In the second approach, we dynamically select the best single prediction and establish a better performance guarantee when the best model switches. Experiments on both synthetic and real data validate the effectiveness of our proposal. …

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**03**
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Dec 2017

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