[Paper Review] Online Computation with Untrusted Advice

Spyros Angelopoulos, Christoph Dürr+3 more|arXiv (Cornell University)|Jan 1, 2020

Optimization and Search Problems24 citations

TL;DR

This paper introduces a novel model for online computation with untrusted advice, where advice may be incorrect or adversarial. It proposes algorithms that balance performance under trusted vs. untrusted advice, proving Pareto-optimal competitive ratios for ski rental and online bidding, and establishing the first lower bounds on the trade-off between advice size and competitiveness in this setting.

ABSTRACT

The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner. To this end, we focus on well- studied online problems such as ski rental, online bidding, bin packing, and list update. For ski-rental and online bidding, we show how to obtain algorithms that are Pareto-optimal with respect to the competitive ratios achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in which Pareto-optimality is not necessarily guaranteed. For bin packing and list update, we give online algorithms with worst-case tradeoffs in their competitiveness, depending on whether the advice is trusted or not; this is motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging problem, but in which the competitiveness depends on the reliability of the advice. Furthermore, we demonstrate how to prove lower bounds, within this model, on the tradeoff between the number of advice bits and the competitiveness of any online algorithm. Last, we study the effect of randomization: here we show that for ski-rental there is a randomized algorithm that Pareto-dominates any deterministic algorithm with advice of any size. We also show that a single random bit is not always inferior to a single advice bit, as it happens in the standard model.

Motivation & Objective

• To address the impractical assumption in standard advice complexity that advice is always correct, by modeling advice as potentially untrusted or adversarial.
• To design online algorithms that are robust when advice is incorrect, yet efficient when advice is accurate, quantifying this trade-off.
• To formalize and analyze the performance of online algorithms using a dual competitive ratio: one for trusted advice (rA) and one for untrusted advice (wA).
• To establish the first lower bounds on the trade-off between advice size and competitiveness in the untrusted advice model.
• To compare the power of advice bits versus random bits in the presence of untrusted advice, showing that randomization can outperform determinism in untrusted settings.

Proposed method

• Propose a new model where online algorithms receive advice that may be incorrect, with performance measured by two competitive ratios: rA (trusted) and wA (untrusted).
• Define Pareto-optimality in terms of competitive ratios: an algorithm is Pareto-optimal if no other algorithm dominates it in both rA and wA.
• Use the ski rental and online bidding problems to construct randomized algorithms that achieve Pareto-optimal trade-offs between rA and wA.
• For bin packing and list update, design algorithms with worst-case competitiveness depending on advice trustworthiness, showing trade-offs between performance under trusted and untrusted advice.
• Prove lower bounds on the trade-off between advice size and competitiveness using reductions to known problems and adversarial input constructions.
• Compare advice and random bits by constructing randomized algorithms with 1 random bit and L advice bits that outperform deterministic algorithms with L+1 advice bits in untrusted settings.

Experimental results

Research questions

• RQ1 What is the optimal trade-off between competitive ratios under trusted and untrusted advice for online problems?
• RQ2 Can randomized algorithms outperform deterministic ones in the untrusted advice model, even with limited advice?
• RQ3 What are the fundamental limits (lower bounds) on the competitiveness achievable with a given number of advice bits when advice may be adversarial?
• RQ4 How does the power of advice bits compare to that of random bits in the untrusted advice setting?
• RQ5 Can robust online algorithms be designed that gracefully degrade when advice is incorrect, while still performing well when advice is correct?

Key findings

• For ski rental and online bidding, the paper constructs randomized algorithms that are Pareto-optimal, achieving competitive ratios (rA, wA) = (w ln w / (w-1), w) for any w > 1.
• It is proven that no deterministic algorithm with any advice size can achieve a better trade-off than (1+λ, 1+1/λ) for λ > 0, and that randomized algorithms Pareto-dominate all deterministic ones even with unbounded advice.
• For bounded online bidding, a randomized algorithm with 1 random bit and L advice bits achieves (1+ρ1²/(ρ1+ǫ), αw+ǫ)-competitiveness for all w > 4, where ρ1 = (w - √(w²-4w))/2 and α < 1.
• The untrusted competitive ratio αw is strictly less than w, showing that randomization improves robustness by a multiplicative factor α < 1 compared to deterministic algorithms.
• The trusted competitive ratio of the randomized algorithm degrades by a factor of (1+ρ1)/2 > 1 compared to the deterministic optimal, but this is offset by a significant improvement in untrusted performance.
• The paper establishes the first lower bounds on the trade-off between advice size and competitiveness in the untrusted advice model, showing inherent limits on performance improvement with advice.