Kinjal Basu

Low discrepancy constructions in the triangle

Most quasi-Monte Carlo research focuses on sampling from the unit cube. Many problems, especially in computer graphics, are defined via quadrature over the unit triangle. Quasi-Monte Carlo methods for the triangle have been developed by Pillards and Cools [J. Comput. Appl. Math., 174 (2005), pp. 29--42] and by Brandolini et al. [``A Koksma--Hlawka inequality for simplices,” in Trends in Harmonic Analysis, Springer, 2013, pp. 33--46]. This paper presents two quasi-Monte Carlo constructions in the triangle with a vanishing discrepancy. The first is a version of the van der Corput sequence customized to the unit triangle. It is an extensible digital construction that attains a discrepancy below

Almost empty monochromatic triangles in planar point sets

12/{\sqrt{N}}

Online Parameter Selection for Web-based Ranking Problems

. The second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number. It attains a discrepancy of

Quasi-Monte Carlo for an Integrand with a Singularity Along a Diagonal in the Square

O(\log(N)/N),

Scrambled Geometric Net Integration Over General Product Spaces

which is the best possible rate. Previous work strongly indicated that such a discrepancy was possible, but no constructions were available. S...

Transformations and Hardy--Krause Variation

For positive integers c , s ź 1 , let M 3 ( c , s ) be the least integer such that any set of at least M 3 ( c , s ) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s = 0 , which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M 3 ( 1 , 0 ) = 3 , M 3 ( 2 , 0 ) = 9 and M 3 ( c , 0 ) = ∞ , for c ź 3 . In this paper we extend these results when c ź 2 and s ź 1 . We prove that the least integer λ 3 ( c ) such that M 3 ( c , λ 3 ( c ) ) < ∞ satisfies: ź c - 1 2 ź ź λ 3 ( c ) ź c - 2 , where c ź 2 . Moreover, the exact values of M 3 ( c , s ) are determined for small values of c and s . We also conjecture that λ 3 ( 4 ) = 1 , and verify it for sufficiently large Horton sets.

Quasi-Monte Carlo tractability of high dimensional integration over products of simplices

Web-based ranking problems involve ordering different kinds of items in a list or grid to be displayed in mediums like a website or a mobile app. In most cases, there are multiple objectives or metrics like clicks, viral actions, job applications, advertising revenue and others that we want to balance. Constructing a serving algorithm that achieves the desired tradeoff among multiple objectives is challenging, especially for more than two objectives. In addition, it is often not possible to estimate such a serving scheme using offline data alone for non-stationary systems with frequent online interventions. We consider a large-scale online application where metrics for multiple objectives are continuously available and can be controlled in a desired fashion by changing certain control parameters in the ranking model. We assume that the desired balance of metrics is known from business considerations. Our approach models the balance criteria as a composite utility function via a Gaussian process over the space of control parameters. We show that obtaining a solution can be equated to finding the maximum of the Gaussian process, practically obtainable via Bayesian optimization. However, implementing such a scheme for large-scale applications is challenging. We provide a novel framework to do so and illustrate its efficacy in the context of LinkedIn Feed. In particular, we show the effectiveness of our method by using both offline simulations as well as promising online A/B testing results. At the time of writing this paper, the method described was fully deployed on the LinkedIn Feed.

Constrained Multi-Slot Optimization for Ranking Recommendations

Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube [0, 1]d or at isolated possibly unknown points within [0, 1]d. Here we consider functions on the square [0, 1]2 that may become singular as the point approaches the diagonal line x1 = x2, and we study three quadrature methods. The first method splits the square into two triangles separated by a region around the line of singularity, and applies recently developed triangle QMC rules to the two triangular parts. For functions with a singularity ‘no worse than |x1 − x2|−A is’ for 0 < A < 1 that method yields an error of \(O( (\log (n)/n)^{(1-A)/2})\). We also consider methods extending the integrand into a region containing the singularity and show that method will not improve upon using two triangles. Finally, we consider transforming the integrand to have a more QMC-friendly singularity along the boundary of the square. This then leads to error rates of O(n−1+𝜖+A) when combined with some corner-avoiding Halton points or with randomized QMC but it requires some stronger assumptions on the original singular integrand.