Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning-Augmented Algorithms for $k$-median via Online Learning

Anish Hebbar, Rong Ge, Amit Kumar, Debmalya Panigrahi

Abstract

The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm's solution and that of an optimal solution for that instance. We apply this framework to the classic $k$-median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed $k$-median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence.

Learning-Augmented Algorithms for $k$-median via Online Learning

Abstract

The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm's solution and that of an optimal solution for that instance. We apply this framework to the classic -median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed -median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence.
Paper Structure (23 sections, 19 theorems, 108 equations, 10 figures, 2 algorithms)

This paper contains 23 sections, 19 theorems, 108 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Reduction from Learn-Median to Learn-Bounded-Median
  5. Fractional Algorithm via Online Mirror Descent with Hyperbolic Entropy Regularizer
  6. Integral Solutions via Online Rounding
  7. Experimental Results
  8. Closing Remarks
  9. Reduction from Learn-Median to Learn-Bounded-Median
  10. Bounded Instances of Learn-Median
  11. Reduction
  12. Detailed Proofs for Fractional Algorithm
  13. Basic facts about the 1-median problem
  14. Regularizer and its properties
  15. Algorithm
  16. ...and 8 more sections

Key Result

Theorem 2.1

Given an instance for the Learn-Median problem, there exists an algorithm $\mathcal{A}$ that maps each sub-instance $V_t$ to a sub-instance $R_t = \mathcal{A}(V_t)$, resulting in a Learn-Bounded-Median instance. If solutions $Y_1,...,Y_T$ for the new instance of Learn-Bounded-Median satisfy where $\alpha \ge 1$ and $R = R_0 \cup R_1 \ldots \cup R_T$, then we have

Figures (10)

  • Figure 1: For i.i.d. instances (Uniform Square and Multiple Clusters), the optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions (top figure) and approximation ratios - avg. and std. dev. over 10 random instances (bottom figure).
  • Figure 2: Experimental Results for Dynamic Instances
  • Figure 3: (Uniform Square): The optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios – avg. and std. dev. over 10 random instances (right) for $k=2,3,6$.
  • Figure 4: (Uniform Rectangle): The optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios – avg. and std. dev. over 10 random instances (right) for $k=2,3,6$.
  • Figure 5: (Multiple Clusters): the optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios - avg. and std. dev. over 10 random instances (right) for $k=4,8,12,16$.
  • ...and 5 more figures

Theorems & Definitions (51)

  • Theorem 2.1
  • Definition 3.1: $\beta$-hyperbolic entropy ghai2020exponentiated
  • Lemma 3.2
  • Lemma 3.3
  • Claim A.1
  • proof
  • proof : Proof of Theorem \ref{['thm:reduction']}
  • proof
  • proof
  • Corollary B.3
  • ...and 41 more