Active-Learning a Convex Body in Low Dimensions

Consider a set $$P\subseteq \mathbb {R}^d$$ of n points, and a convex body $$C$$ provided via a separation oracle. The task at hand is to decide for each point of $$P$$ if it is in $$C$$ using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using queries, where is the size of the largest subset of points of $$P$$ in convex position. In 2D, we provide an algorithm that efficiently generates these adaptive queries. Furthermore, we show that in two dimensions one can solve this problem using oracle queries, where is a lower bound on the minimum number of queries that any algorithm for this specific instance requires. Finally, we consider other variations on the problem, such as using the fewest number of queries to decide if $$C$$ contains all points of $$P$$ . As an application of the above, we show that the discrete geometric median of a point set P in $$\mathbb {R}^2$$ can be computed in expected time.

[1]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[2]  D. Angluin.  Queries and Concept Learning , 1988 .

[3]  Natan Rubin,et al.  An Improved Bound for Weak Epsilon-Nets in the Plane , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[4]  Dana Angluin,et al.  Queries and concept learning , 1988, Machine Learning.

[5]  Shachar Lovett,et al.  Active Classification with Comparison Queries , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[6]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1989, Discret. Comput. Geom..

[7]  Imre Bárány,et al.  Longest convex chains , 2009, Random Struct. Algorithms.

[8]  Andrey Kupavskii.  The VC-dimension of k-vertex d-polytopes , 2020, ArXiv.

[9]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[10]  David M. Mount,et al.  Space Exploration via Proximity Search , 2016, Discret. Comput. Geom..

[11]  Kenneth Y. Goldberg,et al.  An efficient proximity probing algorithm for metrology , 2013, 2013 IEEE International Conference on Automation Science and Engineering (CASE).

[12]  Timothy M. Chan.  An optimal randomized algorithm for maximum Tukey depth , 2004, SODA '04.

[13]  R. Dudley.  Metric Entropy of Some Classes of Sets with Differentiable Boundaries , 1974 .

[14]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[15]  Jivri Matouaek.  New constructions of weak epsilon-nets , 2003, SoCG 2003.

[16]  David A. Cohn,et al.  Improving generalization with active learning , 1994, Machine Learning.

[17]  Imre Bárány,et al.  A generalization of carathéodory's theorem , 1982, Discret. Math..

[18]  Jakub W. Pachocki,et al.  Geometric median in nearly linear time , 2016, STOC.

[19]  Kurt Mehlhorn,et al.  Dynamic fractional cascading , 1990, Algorithmica.

[20]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[21]  David A. Cohn,et al.  Improving generalization with active learning , 1994, Machine-mediated learning.

[22]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[23]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[24]  Imre Bárány,et al.  Longest convex chains, {\it Random Structures and Alg.} {\bf 35} (2009), 137--162, (with G. Ambrus) , 2009 .

[25]  Juan Ferrera,et al.  An Introduction to Nonsmooth Analysis , 2013 .

[26]  Burr Settles,et al.  Active Learning Literature Survey , 2009 .

[27]  Micha Sharir,et al.  A Nearly Quadratic Bound for Point-Location in Hyperplane Arrangements, in the Linear Decision Tree Model , 2018, Discrete & Computational Geometry.

[28]  Sariel Har-Peled,et al.  Active-Learning a Convex Body in Low Dimensions , 2019, Algorithmica.

[29]  Russell Greiner,et al.  Optimistic Active-Learning Using Mutual Information , 2007, IJCAI.