Average-case Complexity of Teaching Convex Polytopes via Halfspace Queries

We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a $\phi$-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is $\Theta(d)$, which is in sharp contrast to the worst-case teaching complexity of $\Theta(n)$. If instead, we consider the average-case learning complexity, the bounds have a dependency on $n$ as $\Theta(n)$ for i.i.d. queries and $\Theta(d \log(n))$ for actively chosen queries by the learner. Our proof techniques are based on novel insights from computational geometry, which allow us to count the number of convex polytopes and faces in a Euclidean space depending on the arrangement of halfspaces. Our insights allow us to establish a tight bound on the average-case complexity for $\phi$-separable dichotomies, which generalizes the known $\mathcal{O}(d)$ bound on the average number of "extreme patterns" in the classical computational geometry literature (Cover, 1965).

Environment Shaping in Reinforcement Learning using State Abstraction

One of the central challenges faced by a reinforcement learning (RL) agent is to effectively learn a (near-)optimal policy in environments with large state spaces having sparse and noisy feedback signals. In real-world applications, an expert with additional domain knowledge can help in speeding up the learning process via \emph{shaping the environment}, i.e., making the environment more learner-friendly. A popular paradigm in literature is \emph{potential-based reward shaping}, where the environment's reward function is augmented with additional local rewards using a potential function. However, the applicability of potential-based reward shaping is limited in settings where (i) the state space is very large, and it is challenging to compute an appropriate potential function, (ii) the feedback signals are noisy, and even with shaped rewards the agent could be trapped in local optima, and (iii) changing the rewards alone is not sufficient, and effective shaping requires changing the dynamics. We address these limitations of potential-based shaping methods and propose a novel framework of \emph{environment shaping using state abstraction}. Our key idea is to compress the environment's large state space with noisy signals to an abstracted space, and to use this abstraction in creating smoother and more effective feedback signals for the agent. We study the theoretical underpinnings of our abstraction-based environment shaping, and show that the agent's policy learnt in the shaped environment preserves near-optimal behavior in the original environment.

Task-agnostic Exploration in Reinforcement Learning

Efficient exploration is one of the main challenges in reinforcement learning (RL). Most existing sample-efficient algorithms assume the existence of a single reward function during exploration. In many practical scenarios, however, there is not a single underlying reward function to guide the exploration, for instance, when an agent needs to learn many skills simultaneously, or multiple conflicting objectives need to be balanced. To address these challenges, we propose the \textit{task-agnostic RL} framework: In the exploration phase, the agent first collects trajectories by exploring the MDP without the guidance of a reward function. After exploration, it aims at finding near-optimal policies for $N$ tasks, given the collected trajectories augmented with \textit{sampled rewards} for each task. We present an efficient task-agnostic RL algorithm, \textsc{UCBZero}, that finds $\epsilon$-optimal policies for $N$ arbitrary tasks after at most $\tilde O(\log(N)H^5SA/\epsilon^2)$ exploration episodes. We also provide an $\Omega(\log (N)H^2SA/\epsilon^2)$ lower bound, showing that the $\log$ dependency on $N$ is unavoidable. Furthermore, we provide an $N$-independent sample complexity bound of \textsc{UCBZero} in the statistically easier setting when the ground truth reward functions are known.

The Teaching Dimension of Q-learning

In this paper, we initiate the study of sample complexity of teaching, termed as "teaching dimension" (TDim) in the literature, for Q-learning. While the teaching dimension of supervised learning has been studied extensively, these results do not extend to reinforcement learning due to the temporal constraints posed by the underlying Markov Decision Process environment. We characterize the TDim of Q-learning under different teachers with varying control over the environment, and present matching optimal teaching algorithms. Our TDim results provide the minimum number of samples needed for reinforcement learning, thus complementing standard PAC-style RL sample complexity analysis. Our teaching algorithms have the potential to speed up RL agent learning in applications where a helpful teacher is available.

Policy Teaching via Environment Poisoning: Training-time Adversarial Attacks against Reinforcement Learning

We study a security threat to reinforcement learning where an attacker poisons the learning environment to force the agent into executing a target policy chosen by the attacker. As a victim, we consider RL agents whose objective is to find a policy that maximizes average reward in undiscounted infinite-horizon problem settings. The attacker can manipulate the rewards or the transition dynamics in the learning environment at training-time and is interested in doing so in a stealthy manner. We propose an optimization framework for finding an \emph{optimal stealthy attack} for different measures of attack cost. We provide sufficient technical conditions under which the attack is feasible and provide lower/upper bounds on the attack cost. We instantiate our attacks in two settings: (i) an \emph{offline} setting where the agent is doing planning in the poisoned environment, and (ii) an \emph{online} setting where the agent is learning a policy using a regret-minimization framework with poisoned feedback. Our results show that the attacker can easily succeed in teaching any target policy to the victim under mild conditions and highlight a significant security threat to reinforcement learning agents in practice.

Adaptive Reward-Poisoning Attacks against Reinforcement Learning

In reward-poisoning attacks against reinforcement learning (RL), an attacker can perturb the environment reward $r_t$ into $r_t+\delta_t$ at each step, with the goal of forcing the RL agent to learn a nefarious policy. We categorize such attacks by the infinity-norm constraint on $\delta_t$: We provide a lower threshold below which reward-poisoning attack is infeasible and RL is certified to be safe; we provide a corresponding upper threshold above which the attack is feasible. Feasible attacks can be further categorized as non-adaptive where $\delta_t$ depends only on $(s_t,a_t, s_{t+1})$, or adaptive where $\delta_t$ depends further on the RL agent's learning process at time $t$. Non-adaptive attacks have been the focus of prior works. However, we show that under mild conditions, adaptive attacks can achieve the nefarious policy in steps polynomial in state-space size $|S|$, whereas non-adaptive attacks require exponential steps. We provide a constructive proof that a Fast Adaptive Attack strategy achieves the polynomial rate. Finally, we show that empirically an attacker can find effective reward-poisoning attacks using state-of-the-art deep RL techniques.

Understanding the Power and Limitations of Teaching with Imperfect Knowledge

Machine teaching studies the interaction between a teacher and a student/learner where the teacher selects training examples for the learner to learn a specific task. The typical assumption is that the teacher has perfect knowledge of the task---this knowledge comprises knowing the desired learning target, having the exact task representation used by the learner, and knowing the parameters capturing the learning dynamics of the learner. Inspired by real-world applications of machine teaching in education, we consider the setting where teacher's knowledge is limited and noisy, and the key research question we study is the following: When does a teacher succeed or fail in effectively teaching a learner using its imperfect knowledge? We answer this question by showing connections to how imperfect knowledge affects the teacher's solution of the corresponding machine teaching problem when constructing optimal teaching sets. Our results have important implications for designing robust teaching algorithms for real-world applications.

Learning to Switch Between Machines and Humans

Reinforcement learning algorithms have been mostly developed and evaluated under the assumption that they will operate in a fully autonomous manner---they will take all actions. However, in safety critical applications, full autonomy faces a variety of technical, societal and legal challenges, which have precluded the use of reinforcement learning policies in real-world systems. In this work, our goal is to develop algorithms that, by learning to switch control between machines and humans, allow existing reinforcement learning policies to operate under different automation levels. More specifically, we first formally define the learning to switch problem using finite horizon Markov decision processes. Then, we show that, if the human policy is known, we can find the optimal switching policy directly by solving a set of recursive equations using backwards induction. However, in practice, the human policy is often unknown. To overcome this, we develop an algorithm that uses upper confidence bounds on the human policy to find a sequence of switching policies whose total regret with respect to the optimal switching policy is sublinear. Simulation experiments on two important tasks in autonomous driving---lane keeping and obstacle avoidance---demonstrate the effectiveness of the proposed algorithms and illustrate our theoretical findings.

Preference-Based Batch and Sequential Teaching: Towards a Unified View of Models

Algorithmic machine teaching studies the interaction between a teacher and a learner where the teacher selects labeled examples aiming at teaching a target hypothesis. In a quest to lower teaching complexity and to achieve more natural teacher-learner interactions, several teaching models and complexity measures have been proposed for both the batch settings (e.g., worst-case, recursive, preference-based, and non-clashing models) as well as the sequential settings (e.g., local preference-based model). To better understand the connections between these different batch and sequential models, we develop a novel framework which captures the teaching process via preference functions $\Sigma$. In our framework, each function $\sigma \in \Sigma$ induces a teacher-learner pair with teaching complexity as $\TD(\sigma)$. We show that the above-mentioned teaching models are equivalent to specific types/families of preference functions in our framework. This equivalence, in turn, allows us to study the differences between two important teaching models, namely $\sigma$ functions inducing the strongest batch (i.e., non-clashing) model and $\sigma$ functions inducing a weak sequential (i.e., local preference-based) model. Finally, we identify preference functions inducing a novel family of sequential models with teaching complexity linear in the VC dimension of the hypothesis class: this is in contrast to the best known complexity result for the batch models which is quadratic in the VC dimension.

Towards Deployment of Robust AI Agents for Human-Machine Partnerships

We study the problem of designing AI agents that can robustly cooperate with people in human-machine partnerships. Our work is inspired by real-life scenarios in which an AI agent, e.g., a virtual assistant, has to cooperate with new users after its deployment. We model this problem via a parametric MDP framework where the parameters correspond to a user's type and characterize her behavior. In the test phase, the AI agent has to interact with a user of unknown type. Our approach to designing a robust AI agent relies on observing the user's actions to make inferences about the user's type and adapting its policy to facilitate efficient cooperation. We show that without being adaptive, an AI agent can end up performing arbitrarily bad in the test phase. We develop two algorithms for computing policies that automatically adapt to the user in the test phase. We demonstrate the effectiveness of our approach in solving a two-agent collaborative task.