Optimistic Safety for Linearly-Constrained Online Convex Optimization

The setting of online convex optimization (OCO) under unknown constraints has garnered significant attention in recent years. In this work, we consider a version of this problem with static linear constraints that the player receives noisy feedback of and must always satisfy. By leveraging our novel design paradigm of optimistic safety, we give an algorithm for this problem that enjoys $\tilde{\mathcal{O}}(\sqrt{T})$ regret. This improves on the previous best regret bound of $\tilde{\mathcal{O}}(T^{2/3})$ while using only slightly stronger assumptions of independent noise and an oblivious adversary. Then, by recasting this problem as OCO under time-varying stochastic linear constraints, we show that our algorithm enjoys the same regret guarantees in such a setting and never violates the constraints in expectation. This contributes to the literature on OCO under time-varying stochastic constraints, where the state-of-the-art algorithms enjoy $\tilde{\mathcal{O}}(\sqrt{T})$ regret and $\tilde{\mathcal{O}}(\sqrt{T})$ violation when the constraints are convex and the player receives full feedback. Additionally, we provide a version of our algorithm that is more computationally efficient and give numerical experiments comparing it with benchmark algorithms.

Exploiting Problem Geometry in Safe Linear Bandits

The safe linear bandit problem is a version of the classic linear bandit problem where the learner's actions must satisfy an uncertain linear constraint at all rounds. Due its applicability to many real-world settings, this problem has received considerable attention in recent years. We find that by exploiting the geometry of the specific problem setting, we can achieve improved regret guarantees for both well-separated problem instances and action sets that are finite star convex sets. Additionally, we propose a novel algorithm for this setting that chooses problem parameters adaptively and enjoys at least as good regret guarantees as existing algorithms. 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. Simulation results show improved performance over existing algorithms for a variety of randomly sampled settings.

The Impact of the Geometric Properties of the Constraint Set in Safe Optimization with Bandit Feedback

We consider a safe optimization problem with bandit feedback in which an agent sequentially chooses actions and observes responses from the environment, with the goal of maximizing an arbitrary function of the response while respecting stage-wise constraints. We propose an algorithm for this problem, and study how the geometric properties of the constraint set impact the regret of the algorithm. In order to do so, we introduce the notion of the sharpness of a particular constraint set, which characterizes the difficulty of performing learning within the constraint set in an uncertain setting. This concept of sharpness allows us to identify the class of constraint sets for which the proposed algorithm is guaranteed to enjoy sublinear regret. Simulation results for this algorithm support the sublinear regret bound and provide empirical evidence that the sharpness of the constraint set impacts the performance of the algorithm.

Predicting Parameters for Modeling Traffic Participants

Accurately modeling the behavior of traffic participants is essential for safely and efficiently navigating an autonomous vehicle through heavy traffic. We propose a method, based on the intelligent driver model, that allows us to accurately model individual driver behaviors from only a small number of frames using easily observable features. On average, this method makes prediction errors that have less than 1 meter difference from an oracle with full-information when analyzed over a 10-second horizon of highway driving. We then validate the efficiency of our method through extensive analysis against a competitive data-driven method such as Reinforcement Learning that may be of independent interest.

Collaborative Multi-agent Stochastic Linear Bandits

We study a collaborative multi-agent stochastic linear bandit setting, where $N$ agents that form a network communicate locally to minimize their overall regret. In this setting, each agent has its own linear bandit problem (its own reward parameter) and the goal is to select the best global action w.r.t. the average of their reward parameters. At each round, each agent proposes an action, and one action is randomly selected and played as the network action. All the agents observe the corresponding rewards of the played actions and use an accelerated consensus procedure to compute an estimate of the average of the rewards obtained by all the agents. We propose a distributed upper confidence bound (UCB) algorithm and prove a high probability bound on its $T$-round regret in which we include a linear growth of regret associated with each communication round. Our regret bound is of order $\mathcal{O}\Big(\sqrt{\frac{T}{N \log(1/|\lambda_2|)}}\cdot (\log T)^2\Big)$, where $\lambda_2$ is the second largest (in absolute value) eigenvalue of the communication matrix.

Multi-Environment Meta-Learning in Stochastic Linear Bandits

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.

Robust Distributed Optimization With Randomly Corrupted Gradients

In this paper, we propose a first-order distributed optimization algorithm that is provably robust to Byzantine failures-arbitrary and potentially adversarial behavior, where all the participating agents are prone to failure. We model each agent's state over time as a two-state Markov chain that indicates Byzantine or trustworthy behaviors at different time instants. We set no restrictions on the maximum number of Byzantine agents at any given time. We design our method based on three layers of defense: 1) Temporal gradient averaging, 2) robust aggregation, and 3) gradient normalization. We study two settings for stochastic optimization, namely Sample Average Approximation and Stochastic Approximation, and prove that for strongly convex and smooth non-convex cost functions, our algorithm achieves order-optimal statistical error and convergence rates.

Parameter and Feature Selection in Stochastic Linear Bandits

We study two model selection settings in stochastic linear bandits (LB). In the first 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. In the second 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). For each setting, we develop and analyze an 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. Our parameter selection algorithm is OFUL-style and the one for feature selection is based on the SquareCB algorithm. We also show that the regret of our parameter selection algorithm scales logarithmically with model misspecification.

Stage-wise Conservative Linear Bandits

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 O(\sqrt{T} \log^{3/2}T) and 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 actions. 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 O(\log{T}) times (compared to 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.

Regret Bounds for Safe Gaussian Process Bandit Optimization

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 [Srinivas et al., 2010]. 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.