Robustness and Consistency in Linear Quadratic Control with Untrusted Predictions

Abstract

We study the problem of learning-augmented predictive linear quadratic control. Our goal is to design a controller that balances consistency, which measures the competitive ratio when predictions are accurate, and robustness, which bounds the competitive ratio when predictions are inaccurate.

Online Allocation with Replenishable Budgets: Worst Case and Beyond

“…for 𝜋. In fact, considering a baseline algorithm for worst-case robustness is also a common practice in existing learning-augmented algorithms [19,39,52]. Thus, in the following, it suffices to consider (7) to achieve the best of both worlds: maximizing the average utility while bounding the worst-case utility (directly with respect to OACP or OACP+ and also indirectly with respect to 𝑂𝑃𝑇 ).…”

“…), and thus get a sufficient condition for the robust constriant(40) as +1 − 𝐵 • 𝑚,𝑡 +1 | + ≥ (𝜆 − 1) 𝑡 ∑︁ 𝑖=1 𝑓 𝑖 (𝑥 • 𝑖 ) − 𝑅. (41)By(39) and the monotonicity of ReLU operation, we have|𝐵 † 𝑚,𝑡 +1 − 𝐵 • 𝑚,𝑡 +1 | + ≤ |𝐵 † 𝑚,𝑡 +1 − 𝛾 B * 𝑚,𝑡 +1 − (1 − 𝛾)𝐵 † 𝑚,𝑡 +1 | + = 𝛾 |𝐵 † 𝑚,𝑡 +1 − B * 𝑚,𝑡 +1 | + . (42)Substituting the expressions of 𝑥 • 𝑡 and (42) into the inequality, the sufficient condition for the robust constraint(40) becomes−𝛾𝜆𝐿 𝑡 ∑︁ 𝑖=1 ∥ x * 𝑖 − 𝑥 † 𝑖 ∥ 1 − 𝛾𝜆𝐿 𝑀 ∑︁ 𝑚=1 |𝐵 † 𝑚,𝑡 +1 − B * 𝑚,𝑡 +1 | + ≥ (𝜆 − 1) 𝑡 ∑︁ 𝑖=1 𝑓 𝑖 (𝑥 • 𝑖 ) − 𝑅.…”

Contract Scheduling with Predictions

“…See also the survey (Mitzenmacher & Vassilvitskii, 2020). In particular, studies of Pareto-efficient algorithms with respect to consistency-robustness tradeoffs have become prominent recently in the context of online optimization problems with untrusted predictions (Angelopoulos et al, 2020;Sun, Lee, Hajiesmaili, Wierman, & Tsang, 2021;Wei & Zhang, 2020;Li, Yang, Qu, Shi, Yu, Wierman, & Low, 2022;Lee, Maghakian, Hajiesmaili, Li, Sitaraman, & Liu, 2021).…”

Competitive search in a network

“…The first objective is to find strategies of optimal, or near-optimal tradeoff between their consistency (namely, the competitive ratio assuming error-free prediction) and their robustness (namely, the competitive ratio assuming adversarially generated predictions). This is one of the standard methods of analyzing algorithms with predictions, since it establishes strong guarantees on worst-case (extreme) situations with respect to the quality of the prediction; see, e.g., [53,40,39,4,7] for applications to other online problems, and settings of incomplete information, more generally. Specifically, we are interested in showing both positive and negative results on the best-possible consistency that can be achieved by r-robust strategies, for any given r.…”

“…Last, we note that the Pareto-based approach has been applied in many works on online algorithms with ML predictions have been studied under this Pareto-based approach, e.g., (Sun et al, 2021, Wei and Zhang, 2020, Li et al, 2022, Lee et al, 2022, Christianson et al, 2023, Lykouris and Vassilvitskii, 2018.…”