Online Conversion with Switching Costs: Robust and Learning-Augmented Algorithms

TL;DR

This work introduces competitive (robust) threshold-based algorithms for both the minimization and maximization variants of this problem, and shows they are optimal among deterministic online algorithms.

Abstract

We introduce and study online conversion with switching costs, a family of online problems that capture emerging problems at the intersection of energy and sustainability. In this problem, an online player attempts to purchase (alternatively, sell) fractional shares of an asset during a fixed time horizon with length $T$. At each time step, a cost function (alternatively, price function) is revealed, and the player must irrevocably decide an amount of asset to convert. The player also incurs a switching cost whenever their decision changes in consecutive time steps, i.e., when they increase or decrease their purchasing amount. We introduce competitive (robust) threshold-based algorithms for both the minimization and maximization variants of this problem, and show they are optimal among deterministic online algorithms. We then propose learning-augmented algorithms that take advantage of untrusted black-box advice (such as predictions from a machine learning model) to achieve significantly better average-case performance without sacrificing worst-case competitive guarantees. Finally, we empirically evaluate our proposed algorithms using a carbon-aware EV charging case study, showing that our algorithms substantially improve on baseline methods for this problem.

Figures (15)

• Figure 1: The decisions made by different algorithms and the optimal solution during a 9-hour EV charging session with 10kW of local solar capacity and $\beta = 30$. The total energy requested is $\sim$12.9 kWh.
• Figure 2: Overall performance improvement of RORO-min and RO-Advice-min ($\epsilon\sim 1.86$) over baseline threshold $\sqrt{UL}$ algorithm and one-way trading algorithm. The empirical competitive ratio measures performance, and we report the average and the 95$^\text{th}$ percentile (i.e., worst-case) improvement for all experiments.
• Figure 3: Cumulative distribution functions (CDFs) of empirical competitive ratios for all tested algorithms, in experiments testing the impact of differently-sized local solar generation. Each simulated solar system is described using the DC rating. Switching cost is fixed $\beta = 20$, and RO-Advice uses CarbonCast predictions.
• Figure 4: Heat map of empirical competitive ratios for algorithms in a set of experiments testing the impact of different switching costs. Local solar generation is fixed to $0$, and RO-Advice uses CarbonCast predictions with $\epsilon \thicksim 1.86$. Note that the simple threshold and one-way trading algorithms are switching-oblivious, while our proposed RORO and RO-Advice algorithms are aware of the switching cost.
• Figure 5: Heat map of empirical competitive ratios for algorithms in a set of experiments testing the consistency and robustness of RO-Advice-min by simulating advice from perfect ($\zeta = 0$) to adversarial ($\zeta = 1$). No local solar, and four version of RO-Advice-min, with $\epsilon \in [{\thicksim} 2.97, {\thicksim} 2.23, {\thicksim} 1.48, {\thicksim} 0.74]$. The black-box advice algorithm naïvely plays the simulated advice at each time step.

Theorems & Definitions (22)

• Claim 3.1
• definition 1: Threshold function $\phi$ for OCS-min
• definition 2: RORO instantiation for OCS-min (RORO-min)
• Theorem 3.2
• Theorem 3.3
• Theorem 4.1
• Theorem 4.2
• definition 3: Black-box advice model for OCS
• definition 4: RO-Advice instantiation for OCS-min (RO-Advice-min)
• Theorem 4.3