We study the problem of online convex optimization (OCO) under unknown linear constraints that are either static, or stochastically time-varying. For this problem, we introduce an algorithm that we term Optimistically Safe OCO (OSOCO) and show that it enjoys $\tilde{O}(\sqrt{T})$ regret and no constraint violation. In the case of static linear constraints, this improves on the previous best known $\tilde{O}(T^{2/3})$ regret under the same assumptions. In the case of stochastic time-varying constraints, our work supplements existing results that show $O(\sqrt{T})$ regret and $O(\sqrt{T})$ cumulative violation under more general convex constraints and a different set of assumptions. In addition to our theoretical guarantees, we also give numerical results that further validate the effectiveness of our approach.

In this work, we study online convex optimization with a fixed constraint function $g : \mathbb{R}^d \rightarrow \mathbb{R}$. Prior work on this problem has shown $O(\sqrt{T})$ regret and cumulative constraint satisfaction $\sum_{t=1}^{T} g(x_t) \leq 0$, while only accessing the constraint value and subgradient at the played actions $g(x_t), \partial g(x_t)$. Using the same constraint information, we show a stronger guarantee of anytime constraint satisfaction $g(x_t) \leq 0 \forall t \in [T]$, and matching $O(\sqrt{T})$ regret guarantees. These contributions are thanks to our approach of using Polyak feasibility steps to ensure constraint satisfaction, without sacrificing regret. Specifically, after each step of online gradient descent, our algorithm applies a subgradient descent step on the constraint function where the step-size is chosen according to the celebrated Polyak step-size. We further validate this approach with numerical experiments.

We consider a collection of linear stochastic bandit problems, each modeling the random response of different agents to proposed interventions, coupled together by a global safety constraint. We assume a central coordinator must choose actions to play on each bandit with the objective of regret minimization, while also ensuring that the expected response of all agents satisfies the global safety constraints at each round, in spite of uncertainty about the bandits' parameters. The agents consider their observed responses to be private and in order to protect their sensitive information, the data sharing with the central coordinator is performed under local differential privacy (LDP). However, providing higher level of privacy to different agents would have consequences in terms of safety and regret. We formalize these tradeoffs by building on the notion of the sharpness of the safety set - a measure of how the geometric properties of the safe set affects the growth of regret - and propose a unilaterally unimprovable vector of privacy levels for different agents given a maximum regret budget.

The safe linear bandit problem is a version of the classical stochastic linear bandit problem where the learner’s actions must satisfy an uncertain constraint at all rounds. Due its applicability to many real-world settings, this problem has received considerable attention in recent years. By leveraging a novel approach that we call directional optimism, we find that it is possible to achieve improved regret guarantees for both well-separated problem instances and action sets that are finite star convex sets. Furthermore, we propose a novel algorithm for this setting that improves on existing algorithms in terms of empirical performance, while enjoying matching regret guarantees. Lastly, we introduce a generalization of the safe linear bandit setting where the constraints are convex and adapt our algorithms and analyses to this setting by leveraging a novel convex-analysis based approach.

We study two model selection settings in stochastic linear bandits (LB). In the first setting, which we refer to as feature selection, the expected reward of the LB problem is in the linear span of at least one of $M$ feature maps (models). In the second setting, the reward parameter of the LB problem is arbitrarily selected from $M$ models represented as (possibly) overlapping balls in $\mathbb R^d$. However, the agent only has access to misspecified models, i.e., estimates of the centers and radii of the balls. We refer to this setting as parameter selection. For each setting, we develop and analyze a computationally efficient algorithm that is based on a reduction from bandits to full-information problems. This allows us to obtain regret bounds that are not worse (up to a $\sqrt{\log M}$ factor) than the case where the true model is known. This is the best reported dependence on the number of models $M$ in these settings. Finally, we empirically show the effectiveness of our algorithms using synthetic and real-world experiments.

In this work we investigate meta-learning (or learning-to-learn) approaches in multi-task linear stochastic bandit problems that can originate from multiple environments. Inspired by the work of [1] on meta-learning in a sequence of linear bandit problems whose parameters are sampled from a single distribution (i.e., a single environment), here we consider the feasibility of meta-learning when task parameters are drawn from a mixture distribution instead. For this problem, we propose a regularized version of the OFUL algorithm that, when trained on tasks with labeled environments, achieves low regret on a new task without requiring knowledge of the environment from which the new task originates. Specifically, our regret bound for the new algorithm captures the effect of environment misclassification and highlights the benefits over learning each task separately or meta-learning without recognition of the distinct mixture components.

Many applications require a learner to make sequential decisions given uncertainty regarding both the system's payoff function and safety constraints. In safety-critical systems, it is paramount that the learner's actions do not violate the safety constraints at any stage of the learning process. In this paper, we study a stochastic bandit optimization problem where the unknown payoff and constraint functions are sampled from Gaussian Processes (GPs) first considered in [1]. We develop a safe variant of GP-UCB called SGP-UCB, with necessary modifications to respect safety constraints at every round. The algorithm has two distinct phases. The first phase seeks to estimate the set of safe actions in the decision set, while the second phase follows the GP-UCB decision rule. Our main contribution is to derive the first sub-linear regret bounds for this problem. We numerically compare SGP-UCB against existing safe Bayesian GP optimization algorithms.

The classical multi-armed bandit is a class of sequential decision making problems where selecting actions incurs costs that are sampled independently from an unknown underlying distribution. Bandit algorithms have many applications in safety critical systems, where several constraints must be respected during the run of the algorithm in spite of uncertainty about problem parameters. This paper formulates a generalized linear stochastic multi-armed bandit problem with generalized linear safety constraints that depend on an unknown parameter vector. In this setting, we propose a Safe UCB-GLM algorithm for which we provide general and problem-dependent regret bounds.

We study how adding unknown linear safety constraints affects the performance of Thompson Sampling in the linear stochastic bandit problem. The additional constraints must be met at each round in spite of uncertainty about the environment requiring that the learner acts conservatively in choosing her actions. In this setting, we propose Safe-LTS, the first safe Thompson Sampling based algorithm, and we prove that it achieves no-regret learning. We obtain regrets that have the same dependence on the total number of rounds (modulo logarithmic factors) as Safe-UCB, a recently proposed safe algorithm that uses the upper confidence bound principle. Finally, we provide numerical simulations that demonstrate the efficacy of our algorithm.

We study stage-wise conservative linear stochastic bandits: an instance of bandit optimization, which accounts for (unknown) safety constraints that appear in applications such as online advertising and medical trials. At each stage, the learner must choose actions that not only maximize cumulative reward across the entire time horizon, but further satisfy a linear baseline constraint that takes the form of a lower bound on the instantaneous reward. For this problem, we present two novel algorithms, stage-wise conservative linear Thompson Sampling (SCLTS) and stage-wise conservative linear UCB (SCLUCB), that respect the baseline constraints and enjoy probabilistic regret bounds of order $\mathcal{O}(\sqrt{T} \log^{3/2}T)$ and $\mathcal{O}(\sqrt{T} \log T)$, respectively. Notably, the proposed algorithms can be adjusted with only minor modifications to tackle different problem variations, such as, constraints with bandit-feedback, or an unknown sequence of baseline rewards. We discuss these and other improvements over the state-of-the art. For instance, compared to existing solutions, we show that SCLTS plays the (non-optimal) baseline action at most $\mathcal{O}(\log{T})$ times (compared to $\mathcal{O}(\sqrt{T})$). Finally, we make connections to another studied form of safety-constraints that takes the form of an upper bound on the instantaneous reward. While this incurs additional complexity to the learning process as the optimal action is not guaranteed to belong to the safe-set at each round, we show that SCLUCB can properly adjust in this setting via a simple modification.

Bandit algorithms have various application in safety-critical systems, where it is important to respect the system constraints that rely on the bandit's unknown parameters at every round. In this paper, we formulate a linear stochastic multi-armed bandit problem with safety constraints that depend (linearly) on an unknown parameter vector. As such, the learner is unable to identify all safe actions and must act conservatively in ensuring that her actions satisfy the safety constraint at all rounds (at least with high probability). For these bandits, we propose a new UCB-based algorithm called Safe-LUCB, which includes necessary modifications to respect safety constraints. The algorithm has two phases. During the pure exploration phase the learner chooses her actions at random from a restricted set of safe actions with the goal of learning a good approximation of the entire unknown safe set. Once this goal is achieved, the algorithm begins a safe exploration-exploitation phase where the learner gradually expands their estimate of the set of safe actions while controlling the growth of regret. We provide a general regret bound for the algorithm, as well as a problem dependent bound that is connected to the location of the optimal action within the safe set. We then propose a modified heuristic that exploits our problem dependent analysis to improve the regret.

At present, the power grid has tight control over its dispatchable generation capacity but a very coarse control on the demand. Energy consumers are shielded from making price-aware decisions, which degrades the efficiency of the market. This state of affairs tends to favor fossil fuel generation over renewable sources. Because of the technological difficulties of storing electric energy, the quest for mechanisms that would make the demand for electricity controllable on a day-to-day basis is gaining prominence. The goal of this paper is to provide one such mechanisms, which we call Digital Direct Load Scheduling (DDLS). DDLS is a direct load control mechanism in which we unbundle individual requests for energy and digitize them so that they can be automatically scheduled in a cellular architecture. Specifically, rather than storing energy or interrupting the job of appliances, we choose to hold requests for energy in queues and optimize the service time of individual appliances belonging to a broad class which we refer to as "deferrable loads". The function of each neighborhood scheduler is to optimize the time at which these appliances start to function. This process is intended to shape the aggregate load profile of the neighborhood so as to optimize an objective function which incorporates the spot price of energy, and also allows distributed energy resources to supply part of the generation dynamically.

Electrical Vehicles are gaining increasing attention, due to the opportunities and challenges they present for the energy market. On the one hand, they will allow to drastically reduce the need for oil; on the other hand they may require a significant shift in the day to day management of the electricity generation. This paper is concerned with finding appropriate models for residential load in light of a widespread penetration of electric vehicles. The analysis is aimed at finding a SmartGrid solution that would enable us to optimize the generation dispatch in real time and allow to plug cars in any SmartGrid enabled plug. The key idea is to discriminate between regular load and the load due to the EVs, gathering in real time aggregate information about the sensed EV arrivals and their associated charging times in a demand matrix, that can be readily used to optimize the dispatch, while updating without real time constraints the billing record for the EV.