The Assistive Multi-Armed Bandit

TL;DR

This paper introduces the assistive multi-armed bandit framework where a robot helps a human learn and maximize rewards in a bandit task, considering the human's learning process and communication with the robot.

Contribution

It provides necessary and sufficient conditions for effective human assistance in bandit tasks and highlights the importance of communication over isolated human performance.

Findings

Better human performance alone does not guarantee better assistance outcomes.

Effective communication of observed rewards enhances human-robot collaboration.

Proof-of-concept experiments support the theoretical results.

Abstract

Learning preferences implicit in the choices humans make is a well studied problem in both economics and computer science. However, most work makes the assumption that humans are acting (noisily) optimally with respect to their preferences. Such approaches can fail when people are themselves learning about what they want. In this work, we introduce the assistive multi-armed bandit, where a robot assists a human playing a bandit task to maximize cumulative reward. In this problem, the human does not know the reward function but can learn it through the rewards received from arm pulls; the robot only observes which arms the human pulls but not the reward associated with each pull. We offer sufficient and necessary conditions for successfully assisting the human in this framework. Surprisingly, better human performance in isolation does not necessarily lead to better performance when assisted by the robot: a human policy can do better by effectively communicating its observed rewards to the robot. We conduct proof-of-concept experiments that support these results. We see this work as contributing towards a theory behind algorithms for human-robot interaction.

Tables

Table 1. TABLE I: Log-density of true reward params in a horizon 5 inverse MAB

	Assumed $H$ Policy
Actual $H$ Policy	Correct Policy	$ϵ$-Optimal
$ϵ$-greedy	0.49	-0.23
WSLS	0.95	0.13
TS	0.02	-0.23
UCL	0.03	-0.30
GI	0.94	0.20
$ϵ$-Optimal	–	1.55

Table 2. TABLE II: Increase in Reward from Robot Assistance

	Assumed $H$ Policy
Actual $H$	$ϵ$-greedy	WSLS	TS	UCL	GI	$ϵ$-optimal
$ϵ$-greedy	2.13	-0.60	-2.18	-2.20	-0.11	-3.95
WSLS	0.94	3.75	0.80	0.10	-2.21	-1.97
TS	0.33	0.66	0.60	0.44	-1.53	-0.19
UCL	1.76	-1.19	2.51	2.43	0.74	1.28
GI	-1.09	-0.28	-0.77	-0.85	-0.71	-1.50
$ϵ$-optimal	0.24	1.17	1.24	1.28	-3.09	1.46

Equations

$$\bar{R}(t)=\sum _t\left({\mu }^{\ast }-{\mu }_{a_t}\right)=\sum _k\left({\mu }^{\ast }-{\mu }_k\right)T_k(t).$$

$$\mathbb{E}[\bar{R}(T)]\le \sum _{i\ne j^{\ast }}\frac{{\mu }^{\ast }-{\mu }_i}{(\sqrt{f_{j^{\ast }}}-\sqrt{f_i}{)}^2}$$

$$\mathrm{Pr}({\hat{f}}_i(t)\le {\hat{f}}_j(t))\le e^{-t(\sqrt{f_i}-\sqrt{f_j}{)}^2}.$$

$$\mathbb{E}[\bar{R}(T)]\ge \Omega (log(T)).$$

$$a_{t}=\left\{\begin{array}[]{rl}k&t\in E_{k}\\ i_{t}&\textrm{otherwise}\end{array}\right..$$

$$h_{t}=\left\{\begin{array}[]{rl}a_{t-1}&r_{t-1}=1\\ \mathbf{Unif}(\{k|k\neq a_{t-1}\})&r_{t-1}=0\end{array}\right.$$

$$h_{t}=\left\{\begin{array}[]{rl}0&r_{t-1}=0\\ 1&r_{t-1}=1\end{array}\right.$$

$$P(A_t\ne j^{\ast })\log (N-1)$$

$$=\mathbf{H}({\hat{j}}^{\ast })-\mathbf{I}({\hat{j}}^{\ast };j^{\ast })-1$$

$$\ge \log N-\mathbf{I}(j^{\ast };h_1\times \cdots \times h_t)-1.$$

$$\mathbf{I}(j^{\ast };h_1\times \cdots \times h_t)$$

$$\ge (1-f(t))\log N-1.$$

$$\mathbf{I}(j^{\ast };h_1\times \cdots \times h_t)\ge (1-\frac{C}{t})\log N-1$$

Full text

I Introduction

Preference learning [1] seeks to learn a predictive model of human preferences from their observed behavior. These models have been applied quite successfully in contexts like personalized news feeds [2, 3, 4], movie recommendations [5, 6, 7], and human robot interaction [8, 9, 10, 11, 12]. We can learn this predictive model by fitting a utility function to revealed preferences [13, 14, 15], fitting parameters in a pre-specified human model [16], and applying contextual-bandit algorithms [3].

Central to all of these approaches is a fundamental assumption: human behavior is noisily-optimal with respect to a set of stationary preferences. Under this assumption, the problem can then be elegantly cast and analyzed as an inverse optimal control (IOC) [17] or *inverse reinforcement learning * (IRL) problem [18]. Here, the human selects an action, takes it, and receives a reward, which captures their internal preference (e.g. the enjoyment of having their desk organized a particular way). The robot only observes human actions and attempts to learn their preference under the assumption that the human likes the actions they selected; if you go for a particular desk configuration more frequently, the robot will assume that you like that configuration more.

Unfortunately, in practice, this natural inference assumes stationarity, which is often violated. We have all experienced situations where our preferences change with experience and time [19, 20]. This is particularly true in situations where we are ourselves learning about our preferences as we are providing them [21, 22]. For example, as we are organizing our desk, we might be experimenting with different configurations over time, to see what works.

Now imagine a personal robot is trying to help you organize that same desk. If the robot believes you are optimal, it will infer the wrong preferences. Instead, if the robot accounts for the fact that you are learning about your preferences, it has a better shot at understanding what you want. Even more crucially, the robot can expose you to new configurations – ones that might improve your posture, something you hadn’t considered before and you were not going to explore if left to your own devices.

In this work, we formalize how a robot can actively assist humans who are themselves learning about their preferences. Our thesis is that by modeling and influencing the dynamics of human learning, the robot can enable the human-robot team to learn more effectively and outperform a human learning suboptimally in isolation.

To this end, we introduce the assistive multi-armed bandit: an extension of the classical multi-armed bandit (MAB) model of learning. In each round, the human selects an action, referred to in the bandit setting as an arm. However, the robot intercepts their intended action and chooses a (potentially different) arm to pull. The human then observes the pulled arm and corresponding reward, and the process repeats (Figure 1). We find this model surprisingly rich and fascinating. It captures the heart of collaboration: information asymmetry and the cost of equalizing it. As the human learns about their preferences, they are compelled to communicate them to the robot as it decides their eventual reward. Analyzing this model allows us to understand the theoretical limits to assisting learning agents and the properties that make learners easier to assist.

Our contributions are the following: 1) we formalize the assistive multi-armed bandit; 2) we give weak sufficient conditions under which a human and robot team learns consistently, a lower bound on the cost of assuming noisy-optimality, and a mutual-information based upper bound on team performance; and 3) we use policy optimization [23] to conduct an in-depth empirical validation of our theoretical results and investigate the effect of incorrectly modeling the human’s learning strategy. We train $A$ against a fixed learning strategy, e.g., $\epsilon$-greedy, and test it against a different learning strategy, e.g., Thompson sampling. Our analysis shows a person that is better at learning does not necessarily lead to the human-robot team performing better - there are human learning strategies that are ineffective in isolation but communicate well and enable the robot to effectively assist. In fact, human learning strategies that are inconsistent in isolation, that is, failing a weak notion of asymptotic optimality, can allow the human-robot team to match optimal performance in a standard multi-armed bandit. Our results advance the theory behind algorithmic preference learning and provide guidance for structuring algorithms for human-robot interaction.

II A Family of Bandits

II-A The Standard Multi-Armed Bandit

A multi-armed bandit (MAB) $\mathcal{M}$ is defined by:

• •

$\Theta$: a space of reward distributions parameters; $\mathbf{\theta}\in\Theta$;

• •

$N$: an integer representing the number of arms;

• •

$p$: a distribution over $\Theta$.

At the start of the game, $\mathbf{\theta}$ is sampled from $\Theta$ according to the prior $p$. At each timestep $t$, an arm $a_{t}\in[1,\ldots,N]$ is chosen. A reward $r_{t}\sim\mathbf{\theta}_{a_{t}}$ is sampled from the corresponding arm distribution. A strategy is a mapping that determines the correct distribution to sample from given a history of reward observations and previous arm pulls: $K_{t}(a_{1},r_{1},\ldots,a_{t-1},r_{t-1})$.

We use $\mu_{k}$ to represent the mean of arm $k$, with parameters $\mathbf{\theta}_{k}$. We use $j^{*}$ to represent the index of the best arm and $\mu^{*}$ to represent its mean. $T_{k}(t)$ represents the number of pulls of arm $k$ up to and including time $t$. The goal of this game is to maximize the sum of rewards over time, or alternatively, to minimize the expectation of the regret $\bar{R}(t)$, defined as:

[TABLE]

II-B Stationary Inverse Optimal Control

In preference learning, e.g., inverse reinforcement learning (IRL)[24, 25] and inverse optimal control (IOC)[17], an AI system observes (noisily-)optimal behavior and infers the reward function or preferences of that agent. This relies on a key assumption that the agent being observed knows the value of actions it can take, at least in the sense that they are able to select optimal actions. In a multi-armed bandit setting, this set of assumptions corresponds to assuming that the human knows the parameters of the bandit, but has some small probability of picking a suboptimal arm. We refer to a human with this knowledge state and policy as implementing the $\epsilon$-optimal policy. Inferring the reward of an $\epsilon$-optimal, or noisily-optimal, human can be thought of as solving a stationary IOC problem.

II-C The Inverse Multi-Armed Bandit

Before formalizing the problem of assisting a human who is learning, rather than noisily-optimal, we look at passively inferring the reward from their actions. We call this the inverse bandit problem. Each Inverse Bandit problem is defined by:

• •

$\mathcal{M}$: a multi-armed bandit problem

• •

$H$: a bandit strategy employed by the human, that maps histories of past actions and rewards to distributions over arm indices. $H_{t}:h_{1}\times r_{1}\times\cdots\times h_{t-1}\times r_{t-1}\rightarrow\Pi(N)$

The goal is to recover the reward parameter $\theta$ by observing only the arm pulls of the human over time $h_{1},...,h_{t}$.

Unlike the stationary IOC case, $H$ does not have access to the true reward parameters. $H$ receives the reward signal $r_{t}$ sampled according to $\theta$. As a consequence, the human arm pulls are not i.i.d.; the distribution of human arm pulls changes as they learn more about their preferences.

II-D The Assistive Multi-Armed Bandit

In the assistive multi-armed bandit, we have a joint system $A\circ H$ that aims to do well in an MAB $\mathcal{M}$. This strategy consists of two parts: the human player $H$ and robot player $A$. As in an MAB, the goal is to minimize the expected regret. The key difference between an assistive MAB and the standard MAB is that the policy is decomposed into a human component and a robot component. The goal is to capture scenarios where our goal, as designers of the robot $A_{t}$, is to optimize a reward signal which is only observed implicitly through the actions of a human who is themselves learning about the reward function.

The human and robot components of the policy are arranged in a setup similar to teleoperation. In each round:

The human player $H$ selects an arm to suggest based on the history of previous arm pulls and rewards: $H_{t}(a_{1},r_{1},\ldots,a_{t-1},r_{t-1})\in[1,\ldots,N]$. 2. 2.

The robot player $A$ selects which arm to actually execute based on the history of the human’s attempts and the actual arms chosen: $A_{t}(h_{1},a_{1},\ldots,h_{t-1},a_{t-1},h_{t})\in[1,\ldots,N]$. 3. 3.

The human player $H$ observes the current round’s arm and corresponding reward: $(a_{t},r_{t}\sim\mathbf{\theta}_{A_{t}})$.

Unlike the inverse MAB or (stationary) IOC, the assistive MAB formalizes the problem of actually using learned preference knowledge to assist a human. Even if we are able to solve the inverse MAB, this is not useful if we can’t actually help a learner reduce regret. We expect an optimal solution to the assistive MAB to improve on suboptimal learning, guide exploration, and correct for noise.

III Theoretical Results

III-A Hardness of Assistive MABs

We consider the relative difficulty of assisting a person that knows what they want with assisting a person that is learning. We model the first situation as an assistive stationary IOC problem, and the second as an assistive MAB. First, we show that assistive stationary IOC is, as one might expect, quite easy in theory; we show that it is possible to infer the correct arm while making finitely many mistakes in expectation.

Proposition 1**.**

Suppose that $H$’s arm pulls are i.i.d and let $f_{i}$ be the probability $H$ pulls arm $i$. If $H$ is noisily optimal, that is, $f_{j^{*}}>f_{i}$ for all sub-optimal $i$, there exists a robot policy $A$ that has finite expected regret for every value of $\theta$:

[TABLE]

Proof.

(Sketch) Our robot policy $A$ simply pulls the most commonly pulled arm.

Let $\hat{f}_{i}(t)=\frac{1}{t}\sum_{k=1}^{t}[h_{t}=i]$ be the empirical frequency of $H$’s pulls of arm $i$ up to time $t$. Note that $A_{t}=i$ only if $\hat{f}_{j^{*}}(t)\leq\hat{f}_{i}(t)$. We apply a Chernoff bound to the random variable $\hat{f}_{j^{*}}(t)-\hat{f}_{i}(t)$. This gives that, for each $i$,

[TABLE]

Summing Eq. 2 over $t$ and suboptimal arms gives the result. ∎

This is in contrast to the standard results about regret in an MAB: for a fixed, nontrivial MAB problem $\mathcal{M}$, any MAB policy has expected regret at least logarithmic in time on some choice of parameter $\theta$ [26, 27]:

[TABLE]

Several approaches based on Upper Confidence Bounds (UCB) have been shown to achieve this bound, implying that this bound is tight [26, 28, 29]. Nonetheless, this suggests that the problem of assisting a noisily-optimal human is significantly easier than solving a standard MAB.

The assistive MAB is at least as hard as a standard MAB. For the same sequence of arm pulls and observed rewards, the amount of information available to $A$ about the true reward parameters is upper bounded by the corresponding information available in a standard MAB. From a certain perspective, actually improving on human performance in isolation is hopelessly difficult – $A$ does not get access to the reward signal, and somehow must still assist a person who does.

III-B Consistent Assisted Learning

We begin with the simplest success criterion from the bandit literature: consistency. Informally, consistency is the property that the player eventually pulls suboptimal arms with probability 0. This can be stated formally as the average regret going to 0 in the limit: $\lim_{t\rightarrow\infty}{\bar{R}(t)}/{t}=0.$ In an MAB, achieving consistency is relatively straightforward: any policy that is greedy in the limit with infinite exploration (GLIE) is consistent [30, 31]. In contrast, in an assistive MAB, it is not obvious that the robot can implement such a policy when the $H$ strategy is inconsistent. The robot observes no rewards and thus cannot estimate the best arm in hindsight.

However, it turns out a weak condition on the human allows the robot-human joint system to guarantee consistency:

Proposition 2**.**

If the human $H$ implements a noisily greedy policy, that is, a policy that pulls the arm with highest sample mean strictly most often, then there exists a robot policy $A$ such that $A\circ H$ is consistent.

Proof.

(Sketch) Fix a set of decaying disjoint exploration sequences $E_{k}$, one per arm, such that $\lim_{t\rightarrow\infty}\frac{1}{t}|E_{k}\cap\{1,...,t\}|\rightarrow 0$ and $\lim_{t\rightarrow\infty}|E_{k}\cap\{1,...,t\}|\rightarrow\infty$. In other words, each arm is pulled infinitely often, but at a decaying rate over time.

Let $i_{t}$ be the arm most commonly pulled by $H$ up until time $t$, and $A$ be defined by

[TABLE]

Note that this implies that for suboptimal k, $\frac{1}{t}T_{k}(t)\rightarrow 0$ in probability as $t\rightarrow\infty$, as the sample means of all the arms converge to the true means, and the rate of exploration decays to zero. This in turn implies that $A\circ H$ achieves consistency. ∎

In other words, assistance is possible if the human picks the best actions in hindsight. This robot $A$ assists the human $H$ in two ways. First, it helps the human explore their preferences – $A\circ H$ pulls every arm infinitely often. This fixes possible under-exploration in the human. Second, it stabilizes their actions and helps ensure that $H$ does not take too many suboptimal actions - eventually, $A\circ H$ converges to only pulling the best arm. This helps mitigate the effect of noise from the human.

III-B1 modeling learning as $\epsilon$-optimality leads to inconsistency

We now investigate what occurs when mistakenly we model learning behavior as noisy-optimality.

A simple way to make $A\circ H$ consistent when $H$ is noisily optimal is for $A$ to pull the arm most frequently pulled by $H$.

Proposition 3**.**

If $H{}$ plays a strategy that pulls the best arm most often and $A{}$ plays $H$’s most frequently pulled arm, then $A\circ H$ is consistent.

Proof.

(Sketch) Eventually, $H$’s most frequent arm converges to the best arm with probability 1 by hypothesis. At this point, $A$ will pull the best arm going forward and achieve a per-round regret of 0. ∎

Next we consider the impact of applying this strategy when its assumptions are incorrect, i.e., $H$ is learning. For simplicity, we assume $H$ is greedy and pulls the best arm given the rewards so far. We will consider a $1\frac{1}{2}$-arm bandit: a bandit with two arms, where one has a known expected value and the other is unknown. We show that pairing this suboptimal-learner with the ‘most-frequent-arm’ strategy leads the joint system $A\circ H{}$ to be inconsistent:

Proposition 4**.**

If $H{}$ is a greedy learner and $A{}$ is ‘most-frequent-arm’, then there exists an assistive MAB $\mathcal{M}$ such that $A\circ H{}$ is inconsistent.

Proof.

(Sketch) The proof consists of two steps. First, we show a variant of a classical bandit result: if $H{}$ and $A{}$ output the constant arm in the same round, they will for the rest of time. Second, we show that this occurs with finite probability and get a positive lower bound on the per-round regret of $A\circ H.$ ∎

While this is a simplified setting, this shows that the types of mistakes and suboptimality represented by learning systems are not well modeled by the standard suboptimality assumptions used in research on recommendation systems, preference learning, and human-robot interaction. The suboptimality exhibited by learning systems is stateful and self-reinforcing. Figure 2 shows the practical impact of modeling learning. It compares an optimal assistance policy for stationary IOC with an optimal policy for an assistive MAB. In general, assitive MAB policies seem to fit into three steps: explore to give $H$ a good estimate of rewards; observe $H$ to identify a good arm; and then exploit that information.

MABs are the standard theoretical model of reinforcement learning and so this observation highlights the point that the term inverse reinforcement learning is somewhat of a misnomer (as opposed to inverse optimal control): IRL’s assumptions about an agent (noisy optimality) lead to very different inferences than actually assuming an agent is learning.

III-C Regret in Assistive Multi-Armed Bandits

Having argued that we can achieve consistency for such a broad class of human policies in an assistive MAB, we now return to the question of achieving low regret. In particular, we investigate the conditions under which $A\circ H{}$ achieve $O(\log(T))$ expected regret, as is possible in the standard MAB.

For any given human $H$, there exists a robot $A$ such that $A\circ H$ does as well as $H$: let $A$ copy the $H$’s actions without modification; that is, $a_{t}=h_{t}$ for all $t$. So in the case where $H$ achieves $O(\log(T))$ regret by itself, $A\circ H{}$ can as well.

However, a more interesting question is that of when we can successfully assist an suboptimal $H$ that achieves $\omega(\log(T))$ regret. *While one may hypothesize that better human policies lead to better performance when assisted, this is surprisingly not the case, as the next section demonstrates. *

III-C1 An inconsistent policy that is easy to assist

Consider a Beta-Bernoulli assistive MAB where rewards are binary: $r_{t}\in\{0,1\}$. A classic bandit strategy here is ‘win-stay-lose-shift’ (WSLS) [32] which, as the name suggests, sticks with the current arm if the most recent reward is one:

[TABLE]

This a simple strategy that performs somewhat well empirically – although it is easy to see that it is not consistent in isolation, let alone capable of achieving $O(\log(T))$ regret. Indeed, it achieves $\Theta(T)$ regret, as it spends a fixed fraction of its time pulling suboptimal arms. However, if $H$ can implement this strategy, the combined system can implement an arbitrary MAB strategy from the standard MAB setting, including those that achieve logarithmic regret. In other words, the robot can successfully assist the human in efficiently balancing exploration and exploitation despite only having access to the reward parameter through an inconsistent human.

Proposition 5**.**

Let $R^{*}$ be the optimal regret for a Beta-Bernoulli multi-armed bandit. If $H$ implements the WSLS strategy in the corresponding assistive MAB, then there exists a robot strategy $A$ such that $A\circ H$ achieves regret $R^{*}$.

Proof.

(Sketch) The pair $(a_{t-1},h_{t})$ directly encodes the previous reward $r_{t-1}$. This means that $A_{t}$ can be an arbitrary function of the history of arm pulls and rewards and so it can implement the MAB policy that achieves regret $R^{*}$. ∎

Figure 3 compares a rollout of this $A\circ H$(we describe the approach in Section V-B1) with a rollout of a near-optimal policy for a standard MAB.

III-C2 Communication upper bounds team performance

The WSLS policy is not unique in that it allows $A\circ H{}$ to obtain logarithmic regret. A less interesting, but similarly effective policy, is for the human to directly encode their reward observations into their actions; the human need not implement a sensible bandit policy. For example, the following purely communicative $H$ also works for a Beta-Bernoulli bandit:

[TABLE]

We can generalize the results regarding communicative policies using the notion of mutual information, which quantifies the amount of information obtained through observing the human arm pulls.

Let $\mathbf{I}(X;Y)$ be the mutual information between $X$ and $Y$, $\mathbf{H}(X)$ be the entropy of $X$, and $\mathbf{H}(X|Y)$ be the entropy of $X$ given $Y$.

Proposition 6**.**

Suppose that the probability the robot pulls a suboptimal arm at time $t$ is bounded above by some function $f(t)$, that is $P(A_{t}\not=j^{*})\leq f(t)$. Then the mutual information $\mathbf{I}(j^{*};h_{1}\times\cdots\times h_{t})$ between the human actions up to time t and the optimal arm must be at least $(1-f(t))\log N-1$.

Proof.

We can consider the multi-armed bandit task as one of deducing the best arm from the human’s actions. This allows us to apply Fano’s inequality [33] to $P(A_{t}\not=j^{*})$, and using the fact that the entropy of a Bernoulli random variable is bounded above by $1$, we get

[TABLE]

Rearranging terms and using $P(A_{t}\not=j^{*})\leq f(t)$, we get

[TABLE]

∎

Intuitively, since the probability of error is bounded by $f(t)$, in $(1-f(t))$ cases the human actions conveyed enough information for A to successfully choose the best action out of $N$ options. This corresponds to $\log N$ bits, so there needs to be at least $(1-f(t))\log N$ bits of information in $H$’s actions.

Corollary 7**.**

Suppose that the probability the robot pulls a suboptimal arm at time $t$ is bounded above by some function $f(t)$, that is $P(A_{t}\not=j^{*})\leq f(t)$. Then the mutual information $\mathbf{I}(a_{1}\times r_{1}\times\cdots\times a_{t-1}\times r_{t-1};h_{1}\times\cdots\times h_{t})$ between the human actions up to time t and the human observations must be at least $(1-f(t))\log N-1$.

Proof.

Since the best arm is independent of the human actions given the human observations, this follows immediately from the data processing inequality and proposition 6. ∎

In order to achieve regret logarithmic in time, we must have that $P(K_{t}\not=j^{*})\leq\frac{C}{t}$ for some $C>0$. Applying proposition 6 above implies that we must have

[TABLE]

Note that the term $\mathbf{I}(\hat{j}^{*};h_{1}\times\cdots\times h_{t})$ depends on both the human policy and the robot policy - no learning human policy can achieve this bound unless the human-robot system $A\circ H$ samples each arm sufficiently often. As a consequence, simple strategies such as inferring the best arm at each timestep and pulling it, cannot achieve the $\Theta(\log T)$ lower bound on regret.

IV Algorithms for Assistive Multi-Armed Bandits

The optimal response to a given human strategy can be computed by solving a partially observed Markov decision process (POMDP) [34]. The state is the reward parameters $\theta$ and $H$’s internal state. The observations are the human arm pulls. In this framing, a variety of approaches can be used to compute policies or plans, e.g., online Monte-Carlo planning [35, 36] or point-based value iteration [37].

In order to run experiments with large sample sizes, our primary design criterion was fast online performance. This lead us to use a direct policy optimization approach. The high per-action cost of Monte-Carlo planners makes them impractical for this problem. Further, explicitly tracking $\theta$ and $H$’s internal state is strictly harder than solving the inverse MAB.

Our approach applies the policy optimization algorithm of [23] to assistive MABs. Given an assistive MAB $(\mathcal{M},H)$, we sample a batch of reward parameters $\theta$ from the prior $p(\Theta)$; generate trajectories of the form $\xi=[(h_{1},a_{1},r_{1}),...,(h_{t},a_{t},r_{t})]$ from $H$ and the current robot policy $A^{(i)}$; and use the trajectories to update the robot policy to $A^{(i+1)}$. During this offline training stage, since we are sampling reward parameters rather than using the ground truth reward parameters, we can use the generated rewards $r_{t}$ to improve on $A^{(i)}$.

We represent $A$’s policy as a recurrent neural network (RNN). At each timestep, it observes a tuple $(a_{t-1},h_{t})$ where $a_{t-1}$ is the most recent robot action and $h_{t}$ is the most recent human action. In response, it outputs a distribution over arm indices, from which an action is sampled. Given a batch of trajectories, we use an approximate policy gradient method [38] to update the weights of our RNN111Our code is available online at https://github.com/chanlaw/assistive-bandits. We summarize this procedure in Algorithm 1. In our experiments, we used Proximal Policy Optimization (PPO) [39], due to its ease of implementation, good performance, and relative insensitivity to hyperparameters.

V Experiments

In our experiments, we used a horizon 50 Beta-Bernoulli bandit with four arms. Pulling the $i$th arm produces a reward of one with probability $\theta_{i}$ and zero with probability $1-\theta_{i}$: $\Theta=[0,1]^{4}$. We assume a uniform prior over $\Theta$: $\theta_{i}\sim\textrm{Beta}(1,1)$.

We consider 5 classes of human policy:

• •

$\epsilon$-greedy, a learning $H$ that chooses the best arm in hindsight with probability $1-\epsilon$ and a random arm with probability $\epsilon$.222We performed grid search to pick an $\epsilon$ based on empirical performance, and found that $\epsilon=0.1$ performed best.

• •

WSLS, the win-stay-lose-shift policy [32] sticks with the arm pulled in the last round if it returned 1, and otherwise switches randomly to another arm.

• •

TS, the Thompson-sampling policy [40] maintains a posterior over the arm parameters, and chooses each arm in proportion to the current probability it is optimal. This is implemented by sampling a particle from the posterior of each arm, then pulling the arm associated with the highest value.

• •

UCL, the upper-credible limit policy [41] is an algorithm similar to Bayes UCB [42] with softmax noise, used as a model of human behavior in a bandit environment.333We set $K=4$ and softmax temperature $\tau=4$.

• •

GI, the Gittins index policy [43] is the Bayesian optimal solution to an infinite horizon discounted objective MAB.444We follow the approximations described by Chakravorty and Mahajan in [44], and choose a discount rate ($\gamma=0.9$) that performs best empirically using grid search.

In addition, we also defined the following noisily-optimal human policy to serve as a baseline:

• •

$\epsilon$-optimal, a fully informed $H$ that knows the reward parameters $\theta$, chooses the optimal arm with probability $1-\epsilon$, and chooses a random action with probability $\epsilon$.555We set $\epsilon$ to match that of the $\epsilon$-greedy policy.

V-A Inverse Multi-Armed Bandit

Our first experiment investigates the miscalibration that occurs when we do not model learning behavior. A robot that doesn’t model human learning will be overconfident. To show this, we use the Metropolis-Hastings algorithm [45] to approximate the posterior over reward parameters $\mathbf{\theta}$ given human actions. We compare the posterior we get when we model learning with the posterior that assumes $H$ is $\epsilon$-optimal.

We compared the log-density of the true reward parameters in the posterior conditioned on 5 $H$ actions, under both models, when $H$ is actually learning. We report the results in Table I. For every learning human policy, we find that the log-density of the true reward parameters is significantly higher when we model learning than when we do not. In the case of $\epsilon$-greedy and TS, we find that the posterior that fails to model learning assigns negative log-density to the true parameters. This means the posterior is a worse estimate of $\theta$ than the prior.

V-B Assistance is Possible

Propositions 2 and 5 prove that it is possible to assist suboptimal learners in theory. In this section, we show that it is possible in practice. We use Algorithm 1 to train a recurrent policy for each human policy. In Fig. 4, we report the performance with assistance and without.

V-B1 Policy optimization details

To alleviate the problem of exploding and vanishing gradients [46], we use Gated Recurrent Units (GRU) [47] as the cells of our recurrent neural network. The output of the GRU cell is fed into a softmax function, and this output is interpreted as the distribution over actions. To reduce to variance in our policy gradient estimate, we also use a value function baseline  [48] and apply Generalized Advantage Estimation (GAE) [49]. We used weight normalization [50] to speed up training. We used a batch size of $250000$ timesteps or $5000$ trajectories per policy iteration, and performed $100$ policy iterations using PPO.

V-B2 Results

To quantitatively test the hypotheses that our learned models successfully assist, we perform a two factor ANOVA. We found a significant interaction effect, $F(3,999990)=1778.8$, $p<.001$, and a post-hoc analysis with Tukey HSD corrections showed that we were able to successfully assist the human in all four sub-optimal learning policies ($p<.001$).

Our WSLS results agree with Proposition 5. Assisted WSLS achieves a regret of $3.5$, close to the regret of the best-performing unassisted policy, $3.2$. The gap in reward is due to our choice to employ approximate policy optimization. We provide an example trajectory in Fig. 3. The actions selected are almost identical to those of the optimal policy.

This suboptimality also accounts for the small increase in regret when assisting the Gittins index policy.

V-B3 Modeling learning matters

Proposition 4 shows that modeling learning matters. We compared assistance assuming that $H$ is $\epsilon$-optimal with assistance with the correct (learning) model. We report the results in Fig. 4. We found that the regret with the wrong model is higher than no intervention in every case but UCL. Assisted WSLS with the wrong model has double the regret of assisted WSLS with the correct model. Proposition 4 shows that in theory, ignoring learning leads to inconsistency. Fig. 4 shows that this mistake leads to higher regret empirically.

V-C Mutual Information Predicts Performance

Proposition 6 implies that high mutual information is required for good team performance. To verify this, we computed the mutual information for a variety of combined policies after 5 timesteps. Fig. 5 plots this against the regret of the combined system. We consider several variants of $\epsilon$-greedy and TS that are more or less deterministic. We consider $\epsilon\in[0,0.02,0.05,0.1].$ To make TS more deterministic, we use the mean of a sample of $n$ particles to select arms. We consider $n\in[1,2,3,10,30,\infty]$.

Across this data, higher mutual information is associated with lower assisted regret, $r(10)=-.82$, $p<.001$. Furthermore, by looking at the $\epsilon$-greedy and TS results as a sequence, we can observe a clear and distinct pattern. Policies that are more deterministic tend to be easier to help. This is supported by the results in Table I, which shows that it is easier to infer reward parameters for WSLS and GI (i.e., the two policies with the highest mutual information) than TS and $\epsilon$-greedy.

V-D Sensitivity to Model Misspecification

In the previous three experiments, we assumed knowledge of the correct learning policy. In this experiment, we consider the implications of incorrectly modeling learning. We took the policies we trained in Section V-B and tested them with every human policy. We report the net change in reward in Table II. We colored cases where the robot $A$ successfully assists the human $H$ green, and cases where it fails to assist red. We bolded the best performance in each row.

Modeling learning (even with the incorrect model) generally leads to lower regret than assuming $\epsilon$-optimal for every learning $H$ policy. However, when the robot has the wrong model of learning, it can fail to assist the human. For example, $\epsilon$-greedy is only successfully assisted when it is correctly modeled. This argues that research into the learning strategies employed by people in practice is an important area for future research.

An intriguing result is that assuming $\epsilon$-greedy does successfully assist all of the suboptimal learning policies. This suggests that, although some learning policies must be well modeled, learning to assist some models can be transferred to other models in some cases. On the other hand, trying to assist GI leads to a policy that hurts performance across the board. In future work, we plan to identify classes of learners which can be assisted by the same robot policy.

V-E Other Paradigms of Assistance

The assistive multi-armed bandit considered so far only captures one mode of interaction. It is straightforward to consider extensions to different modes. We consider two such modes. The first is turn taking, where $H$ and $A$ take turns selecting arms. This can be more difficult because the robot has to act in early rounds, when it has less information, and because the human has to act in later rounds, when $H$ may be noisy and the best arm has already been identified.

The second variant we consider is preemptive interaction. In this case, $A$ goes first and either pulls an arm or lets $H$ act. This creates an exploration-exploitation tradeoff. $A$ only observes $H$’s arm pulls by actually allowing $H$ to pull arms and so it must choose between observing $H$’s behavior and exploiting that knowledge.

Fig. 6 shows the experiment from Section V-B applied to each of these interaction modes. The results are largely similar to those of teleoperation. We are able to assist the suboptimal policies and modeling learning as $\epsilon$-optimality increases regret in all cases. We see that WSLS is a less attractive policy in these settings: because $H$ actions are always executed when they are observed, it no longer makes sense for $H$ to employ a purely communicative policy. However, we do still see results that confirm Proposition 6: more deterministic policies that reveal more information are easier to help.

V-E1 Explore, observe, then exploit

In looking at the policies learned for the preemptive interaction mode, we see an interesting pattern emerge. Because the policy has to choose between selecting arms directly and observing $H$, by looking at rollouts of the learned policy we can determine when it is observing the human. We find that a clear pattern emerges. $A$ initially explores for $H$: it selects arms uniformly to give $H$ a good estimate of $\theta$. Then, $A$ observes $H$’s arm pulls to identify the optimal arm. For the final rounds, $A$ exploits this information and pulls its estimate of the optimal arm. Fig. 2 compares a representative trajectory with one that is optimized against an $\epsilon$-optimal $H$.

VI Discussion

VI-A Summary

In this work, we studied the problem of assisting a human who is learning about their own preferences. Our central thesis is that by modeling and influencing the dynamics of a human’s learning, we can create robots that can better assist people in achieving their preferences. We formalized this as the assistive multi-armed bandit problem, which extends the multi-armed bandit to account for teleoperation and human learning. We analyzed our formalism theoretically, then used policy optimization in proof-of-concept experiments that supported our theoretical results. Surprisingly, we found that a person that is better at learning in isolation does not necessarily lead to a human-robot team that performs better. We highlighted a theoretical connection between the amount of information communicated by the human policy and the best assisted performance, which we validated in our experiments.

VI-B Limitations and Future Work.

VI-B1 Stateful environments

One significant limitation of this work is that we assume the environment the human is acting in is stateless. In practice, the environmental state changes over time, and the state can greatly influence the reward associated with certain actions. This suggests natural extensions of the assistive multi-armed bandit to the contextual bandit [51] and full Markov decision process (MDP) [52] settings.

VI-B2 Realistic human policies

Another significant limitation of this work is the use of simple bandit policies in place of actual human policies. In addition, we do not have access to the true human policy in any case. Future work can remedy this by incorporating more realistic policies, and can study to what extent these results generalize to assisting actual humans.

VI-C Closing Remarks

The assistive multi-armed bandit is representative of a world where robots are supposed to assist people, even though people haven’t figured out what they want yet. Laying down the theoretical foundations for these kinds of interaction paradigms is an important and under-served aspect of HRI.

Acknowledgements

We thank the members of the InterACT lab and the Center for Human Compatible AI for helpful advice. This work was partially supported by OpenPhil, AFOSR, NSF, and NVIDIA.