[Paper Review] Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization

Haotian Jiang, Janardhan Kulkarni+1 more|arXiv (Cornell University)|Oct 2, 2019

Mathematical Approximation and Integration28 references4 citations

TL;DR

This paper presents an efficient online algorithm for geometric discrepancy minimization in stochastic settings, where points arrive uniformly at random on a unit interval or square. It achieves sub-polynomial expected discrepancy—significantly improving upon the $~ O(\sqrt{n})$ of random coloring—by leveraging a novel potential function based on hyperbolic sine functions to balance intervals dynamically, and applies this to improve envy minimization in online allocation problems.

ABSTRACT

Consider a unit interval $[0,1]$ in which $n$ points arrive one-by-one independently and uniformly at random. On arrival of a point, the problem is to immediately and irrevocably color it in $\{+1,-1\}$ while ensuring that every interval $[a,b] \subseteq [0,1]$ is nearly-balanced. We define \emph{discrepancy} as the largest imbalance of any interval during the entire process. If all the arriving points were known upfront then we can color them alternately to achieve a discrepancy of $1$. What is the minimum possible expected discrepancy when we color the points online? We show that the discrepancy of the above problem is sub-polynomial in $n$ and that no algorithm can achieve a constant discrepancy. This is a substantial improvement over the trivial random coloring that only gets an $\widetilde{O}(\sqrt n)$ discrepancy. We then obtain similar results for a natural generalization of this problem to $2$-dimensions where the points arrive uniformly at random in a unit square. This generalization allows us to improve recent results of Benade et al.\cite{BenadeKPP-EC18} for the online envy minimization problem when the arrivals are stochastic.

Motivation & Objective

• To address the challenge of minimizing discrepancy in online geometric discrepancy problems where points arrive uniformly at random, rather than adversarially.
• To design an efficient online algorithm that achieves sub-polynomial expected discrepancy, beating the $\widetilde{O}(\sqrt{n})$ bound of random coloring.
• To apply the improved discrepancy bounds to online envy minimization, particularly in settings with stochastic arrivals.
• To demonstrate that no online algorithm can achieve constant discrepancy in the stochastic interval setting, despite offline solutions achieving discrepancy 1.
• To establish theoretical limits on online discrepancy by proving tightness of the separation lemma via tree-based constructions.

Proposed method

• The algorithm uses a potential function based on hyperbolic sine functions, $\sinh(\lambda x)$, to dynamically balance interval imbalances as points arrive.
• It defines a 'dangerous' region around each point to control cancellation effects in the potential function, ensuring that imbalances do not grow uncontrollably.
• The analysis introduces a separation lemma that bounds the sum of $\sinh(\lambda x)$ terms across pairs of points, showing that their combined imbalance is at least $\frac{8}{9}$ of the maximum individual imbalance.
• A recursive tree-based construction is used to prove the tightness of the separation lemma, demonstrating that the bound cannot be significantly improved.
• The method ensures that the expected discrepancy remains sub-polynomial in $n$ by controlling the growth of the potential function through careful balancing of positive and negative imbalances.
• The algorithm is online: it assigns colors $+1$ or $-1$ immediately and irrevocably upon point arrival, using only local information and the potential function.

Experimental results

Research questions

• RQ1 Can online algorithms achieve sub-polynomial expected discrepancy in geometric discrepancy problems with stochastic arrivals, beating the $\widetilde{O}(\sqrt{n})$ bound of random coloring?
• RQ2 Is it possible to achieve a discrepancy significantly smaller than $\widetilde{O}(\sqrt{n})$ in the online interval discrepancy problem with uniform random arrivals?
• RQ3 What is the fundamental limit of online discrepancy in the stochastic interval setting—can constant discrepancy be achieved?
• RQ4 How can improved discrepancy bounds be leveraged to enhance performance in online envy minimization problems?
• RQ5 Can the separation lemma for hyperbolic functions be shown to be tight, and what does this imply for the limits of discrepancy minimization?

Key findings

• The paper proves that no online algorithm can achieve constant discrepancy in the stochastic interval discrepancy problem, even with random arrivals.
• An efficient online algorithm is designed that achieves sub-polynomial expected discrepancy, significantly improving upon the $\widetilde{O}(\sqrt{n})$ bound of random coloring.
• The key technical contribution is a separation lemma showing that for any two points $x$ and $y$, the sum $|\sinh(\lambda x) + \sinh(\lambda y)|$ is at least $\frac{8}{9}$ of the maximum of $|\sinh(\lambda x)|$ and $|\sinh(\lambda y)|$, under certain conditions.
• The analysis demonstrates that the expected discrepancy grows slower than any polynomial in $n$, implying sub-polynomial growth.
• The method is extended to the 2D case of points in a unit square, yielding similar sub-polynomial discrepancy bounds.
• The results improve recent bounds in online envy minimization by Benade et al. [BKPP18], particularly in the stochastic arrival model.