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Learning-Based Algorithms for Graph Searching Problems

Adela Frances DePavia, Erasmo Tani, Ali Vakilian

TL;DR

The first formal guarantees on unknown weighted graphs on unknown weighted graphs are established and lower bounds showing that the algorithms proposed have optimal or nearly-optimal dependence on the prediction error are provided.

Abstract

We consider the problem of graph searching with prediction recently introduced by Banerjee et al. (2022). In this problem, an agent, starting at some vertex $r$ has to traverse a (potentially unknown) graph $G$ to find a hidden goal node $g$ while minimizing the total distance travelled. We study a setting in which at any node $v$, the agent receives a noisy estimate of the distance from $v$ to $g$. We design algorithms for this search task on unknown graphs. We establish the first formal guarantees on unknown weighted graphs and provide lower bounds showing that the algorithms we propose have optimal or nearly-optimal dependence on the prediction error. Further, we perform numerical experiments demonstrating that in addition to being robust to adversarial error, our algorithms perform well in typical instances in which the error is stochastic. Finally, we provide alternative simpler performance bounds on the algorithms of Banerjee et al. (2022) for the case of searching on a known graph, and establish new lower bounds for this setting.

Learning-Based Algorithms for Graph Searching Problems

TL;DR

The first formal guarantees on unknown weighted graphs on unknown weighted graphs are established and lower bounds showing that the algorithms proposed have optimal or nearly-optimal dependence on the prediction error are provided.

Abstract

We consider the problem of graph searching with prediction recently introduced by Banerjee et al. (2022). In this problem, an agent, starting at some vertex has to traverse a (potentially unknown) graph to find a hidden goal node while minimizing the total distance travelled. We study a setting in which at any node , the agent receives a noisy estimate of the distance from to . We design algorithms for this search task on unknown graphs. We establish the first formal guarantees on unknown weighted graphs and provide lower bounds showing that the algorithms we propose have optimal or nearly-optimal dependence on the prediction error. Further, we perform numerical experiments demonstrating that in addition to being robust to adversarial error, our algorithms perform well in typical instances in which the error is stochastic. Finally, we provide alternative simpler performance bounds on the algorithms of Banerjee et al. (2022) for the case of searching on a known graph, and establish new lower bounds for this setting.
Paper Structure (39 sections, 24 theorems, 111 equations, 11 figures, 2 tables, 4 algorithms)

This paper contains 39 sections, 24 theorems, 111 equations, 11 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Organization
  4. Summary Of Results
  5. Exploration Under Absolute Error
  6. Relative Errors
  7. Planning Problems
  8. Technical Preliminaries and Notation
  9. Notation For Exploration Algorithms
  10. Metric Embeddings
  11. Exploration Under Absolute Error
  12. Exploration Under Relative Error
  13. Planning With Relative Error
  14. Planning
  15. Planning Bounds Via Metric Embeddings
  16. ...and 24 more sections

Key Result

Theorem 1

There is an algorithm for searching arbitrary (potentially directed) graphs which finds the goal $g$ by traveling a distance of at most $\operatorname{OPT} + \,\mathcal{E}_1^- + n\cdot \mathcal{E}_\infty^+$, where $\mathcal{E}_1^- \stackrel{\mathrm{ def}}{=} \sum_{v \in V} \max \left\{0, d(v,g)-f(v

Figures (11)

  • Figure 1: The lower bound construction for the proof of Lemmas \ref{['lemma:lowerbound-E1-planning']} and \ref{['lemma:lowerbound-E1-planning-trees']}.
  • Figure 2: Performance of Algorithms \ref{['alg:greedy-l1-search']} and \ref{['alg:weighted-vareps-unknown-search']} against random errors. Experimental procedures are detailed in Section \ref{['sec:further_experiments']}. The number of nodes is fixed over all graph topologies and error settings. LEFT: Average and standard deviation of $\mathrm{ALG}-\mathrm{OPT}$ incurred by Algorithm \ref{['alg:greedy-l1-search']} over 2000 independent random trials for varying values of $\mathcal{E}_1$. RIGHT: Average and standard deviation of $\mathrm{ALG}/\mathrm{OPT}$ incurred by Algorithm \ref{['alg:weighted-vareps-unknown-search']} over 2000 independent random trials for varying values of $\varepsilon$.
  • Figure 3: A comparison of the performance of \ref{['alg:greedy-l1-search']} with the Smallest Prediction heuristic. We plot the average and the standard deviation of the performance of \ref{['alg:greedy-l1-search']} and that of the Smallest Prediction heuristic against the magnitude of the error vector $\mathcal{E}_1$. Experiments in this figure were conducted on random trees; for analogous results on other graph topologies, see Figure \ref{['fig:baseline_comparison_all']} in \ref{['sec:further_experiments']}.
  • Figure 4: Comparing Algorithm \ref{['alg:greedy-l1-search']} versus A$^*$ search on a random tree with randomly generated errors. The same set of predictions is provided to both algorithms. While the set of nodes visited by the two algorithms is comparable, the computational model in A$^*$ places no penalty on traversal distance so that algorithm has a tendency to double-back on itself, leading to a more expensive tour. Nodes are colored by prediction value and labeled with the order in which they are first visited by the relevant algorithm. Tour cost is taken to be $\sum d(v_i, v_{i+1})$ for indices $i$ ordered according to when a node is first visited: the tour cost for A$^*$ omits costs incurred by re-expanding a node within the execution of the algorithm.
  • Figure 5: The construction in the proof of Theorem \ref{['thm:optimal_E1_lowerbounds']}.
  • ...and 6 more figures

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1: Distortion
  • Corollary 5
  • Theorem 6
  • Proposition 7
  • Theorem 8
  • Proposition 9
  • ...and 37 more