Nicolas Christianson

Nicolas Christianson, Tinashe Handina, Adam Wierman

We consider the problem of convex function chasing with black-box advice, where an online decision-maker aims to minimize the total cost of making and switching between decisions in a normed vector space, aided by black-box advice such as the decisions of a machine-learned algorithm. The decision-maker seeks cost comparable to the advice when it performs well, known as $\textit{consistency}$, while also ensuring worst-case $\textit{robustness}$ even when the advice is adversarial. We first consider the common paradigm of algorithms that switch between the decisions of the advice and a competitive algorithm, showing that no algorithm in this class can improve upon 3-consistency while staying robust. We then propose two novel algorithms that bypass this limitation by exploiting the problem's convexity. The first, INTERP, achieves $(\sqrt{2}+ε)$-consistency and $\mathcal{O}(\frac{C}{ε^2})$-robustness for any $ε> 0$, where $C$ is the competitive ratio of an algorithm for convex function chasing or a subclass thereof. The second, BDINTERP, achieves $(1+ε)$-consistency and $\mathcal{O}(\frac{CD}ε)$-robustness when the problem has bounded diameter $D$. Further, we show that BDINTERP achieves near-optimal consistency-robustness trade-off for the special case where cost functions are $α$-polyhedral.

Adam Lechowicz, Nicolas Christianson, Bo Sun et al.

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.

Bo Sun, Jerry Huang, Nicolas Christianson et al.

The burgeoning field of algorithms with predictions studies the problem of using possibly imperfect machine learning predictions to improve online algorithm performance. While nearly all existing algorithms in this framework make no assumptions on prediction quality, a number of methods providing uncertainty quantification (UQ) on machine learning models have been developed in recent years, which could enable additional information about prediction quality at decision time. In this work, we investigate the problem of optimally utilizing uncertainty-quantified predictions in the design of online algorithms. In particular, we study two classic online problems, ski rental and online search, where the decision-maker is provided predictions augmented with UQ describing the likelihood of the ground truth falling within a particular range of values. We demonstrate that non-trivial modifications to algorithm design are needed to fully leverage the UQ predictions. Moreover, we consider how to utilize more general forms of UQ, proposing an online learning framework that learns to exploit UQ to make decisions in multi-instance settings.

Christopher Yeh, Nicolas Christianson, Alan Wu et al.

Machine learning can significantly improve performance for decision-making under uncertainty across a wide range of domains. However, ensuring robustness guarantees requires well-calibrated uncertainty estimates, which can be difficult to achieve with neural networks. Moreover, in high-dimensional settings, there may be many valid uncertainty estimates, each with its own performance profile - i.e., not all uncertainty is equally valuable for downstream decision-making. To address this problem, this paper develops an end-to-end framework to learn uncertainty sets for conditional robust optimization in a way that is informed by the downstream decision-making loss, with robustness and calibration guarantees provided by conformal prediction. In addition, we propose to represent general convex uncertainty sets with partially input-convex neural networks, which are learned as part of our framework. Our approach consistently improves upon two-stage estimate-then-optimize baselines on concrete applications in energy storage arbitrage and portfolio optimization.

Adam Lechowicz, Nicolas Christianson, Bo Sun et al.

We introduce and study spatiotemporal online allocation with deadline constraints ($\mathsf{SOAD}$), a new online problem motivated by emerging challenges in sustainability and energy. In $\mathsf{SOAD}$, an online player completes a workload by allocating and scheduling it on the points of a metric space $(X, d)$ while subject to a deadline $T$. At each time step, a service cost function is revealed that represents the cost of servicing the workload at each point, and the player must irrevocably decide the current allocation of work to points. Whenever the player moves this allocation, they incur a movement cost defined by the distance metric $d(\cdot, \ \cdot)$ that captures, e.g., an overhead cost. $\mathsf{SOAD}$ formalizes the open problem of combining general metrics and deadline constraints in the online algorithms literature, unifying problems such as metrical task systems and online search. We propose a competitive algorithm for $\mathsf{SOAD}$ along with a matching lower bound establishing its optimality. Our main algorithm, \textsc{ST-CLIP}, is a learning-augmented algorithm that takes advantage of predictions (e.g., forecasts of relevant costs) and achieves an optimal consistency-robustness trade-off. We evaluate our proposed algorithms in a simulated case study of carbon-aware spatiotemporal workload management, an application in sustainable computing that schedules a delay-tolerant batch compute job on a distributed network of data centers. In these experiments, we show that \textsc{ST-CLIP} substantially improves on heuristic baseline methods.

Adam Lechowicz, Nicolas Christianson, Bo Sun et al.

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions $\mathbf{x}_t$ in a metric space $(X,d)$ to simultaneously minimize their hitting cost $f_t(\mathbf{x}_t)$ and switching cost as determined by the metric. Over the time horizon $T$, the player must satisfy a long-term demand constraint $\sum_{t} c(\mathbf{x}_t) \geq 1$, where $c(\mathbf{x}_t)$ denotes the fraction of demand satisfied at time $t$. Such problems can find a wide array of applications to online resource allocation in sustainable energy/computing systems. We devise optimal competitive and learning-augmented algorithms for the case of bounded hitting cost gradients and weighted $\ell_1$ metrics, and further show that our proposed algorithms perform well in numerical experiments.

Christopher Yeh, Nicolas Christianson, Adam Wierman et al.

While deep learning models often achieve high predictive accuracy, their predictions typically do not come with any provable guarantees on risk or reliability, which are critical for deployment in high-stakes applications. The framework of conformal risk control (CRC) provides a distribution-free, finite-sample method for controlling the expected value of any bounded monotone loss function and can be conveniently applied post-hoc to any pre-trained deep learning model. However, many real-world applications are sensitive to tail risks, as opposed to just expected loss. In this work, we develop a method for controlling the general class of Optimized Certainty-Equivalent (OCE) risks, a broad class of risk measures which includes as special cases the expected loss (generalizing the original CRC method) and common tail risks like the conditional value-at-risk (CVaR). Furthermore, standard post-hoc CRC can degrade average-case performance due to its lack of feedback to the model. To address this, we introduce "conformal risk training," an end-to-end approach that differentiates through conformal OCE risk control during model training or fine-tuning. Our method achieves provable risk guarantees while demonstrating significantly improved average-case performance over post-hoc approaches on applications to controlling classifiers' false negative rate and controlling financial risk in battery storage operation.

Adam Lechowicz, Nicolas Christianson, Mohammad Hajiesmaili et al.

We introduce and study a class of online problems called online smoothed demand management $(\texttt{OSDM})$, motivated by paradigm shifts in grid integration and energy storage for large energy consumers such as data centers. In $\texttt{OSDM}$, an operator makes two decisions at each time step: an amount of energy to be purchased, and an amount of energy to be delivered (i.e., used for computation). The difference between these decisions charges (or discharges) the operator's energy storage (e.g., a battery). Two types of demand arrive online: base demand, which must be covered at the current time, and flexible demand, which can be satisfied at any time before a demand-specific deadline $Δ_t$. The operator's goal is to minimize a cost (subject to above constraints) that combines a cost of purchasing energy, a cost for delivering energy (if applicable), and smoothness penalties on the purchasing and delivery rates to discourage fluctuations and encourage ``grid healthy'' decisions. $\texttt{OSDM}$ generalizes several problems in the online algorithms literature while being the first to fully model applications of interest. We propose a competitive algorithm for $\texttt{OSDM}$ called $\texttt{PAAD}$ (partitioned accounting & aggregated decisions) and show it achieves the optimal competitive ratio. To overcome the pessimism typical of worst-case analysis, we also propose a novel learning framework that provides guarantees on the worst-case competitive ratio (i.e., to provide robustness against nonstationarity) while allowing end-to-end differentiable learning of the best algorithm on historical instances of the problem. We evaluate our algorithms in a case study of a grid-integrated data center with battery storage, showing that $\texttt{PAAD}$ effectively solves the problem and end-to-end learning achieves substantial performance improvements compared to $\texttt{PAAD}$.

Sizhe Li, Nicolas Christianson, Tongxin Li

Algorithms with predictions} has emerged as a powerful framework to combine the robustness of traditional online algorithms with the data-driven performance benefits of machine-learned (ML) predictions. However, most existing approaches in this paradigm are overly conservative, {as they do not leverage problem structure to optimize performance in a prediction-specific manner}. In this paper, we show that such prediction-specific performance criteria can enable significant performance improvements over the coarser notions of consistency and robustness considered in prior work. Specifically, we propose a notion of \emph{strongly-optimal} algorithms with predictions, which obtain Pareto optimality not just in the worst-case tradeoff between robustness and consistency, but also in the prediction-specific tradeoff between these metrics. We develop a general bi-level optimization framework that enables systematically designing strongly-optimal algorithms in a wide variety of problem settings, and we propose explicit strongly-optimal algorithms for several classic online problems: deterministic and randomized ski rental, and one-max search. Our analysis reveals new structural insights into how predictions can be optimally integrated into online algorithms by leveraging a prediction-specific design. To validate the benefits of our proposed framework, we empirically evaluate our algorithms in case studies on problems including dynamic power management and volatility-based index trading. Our results demonstrate that prediction-specific, strongly-optimal algorithms can significantly improve performance across a variety of online decision-making settings.