Parameterized Exploration

TL;DR

Parameterized Exploration (PE) is a new method for tuning exploration schedules in decision problems, considering time horizon and knowledge state, leading to improved performance across various bandit and MDP tasks.

Contribution

Introduces Parameterized Exploration, a simple family of methods that adapt exploration based on problem horizon and knowledge, outperforming un-tuned methods in multiple settings.

Findings

PE outperforms un-tuned exploration methods in bandit and MDP tasks.

Performance of PE depends on the accuracy of the dynamics model.

PE adapts exploration to the decision problem's characteristics.

Abstract

We introduce Parameterized Exploration (PE), a simple family of methods for model-based tuning of the exploration schedule in sequential decision problems. Unlike common heuristics for exploration, our method accounts for the time horizon of the decision problem as well as the agent's current state of knowledge of the dynamics of the decision problem. We show our method as applied to several common exploration techniques has superior performance relative to un-tuned counterparts in Bernoulli and Gaussian multi-armed bandits, contextual bandits, and a Markov decision process based on a mobile health (mHealth) study. We also examine the effects of the accuracy of the estimated dynamics model on the performance of PE.

Tables

Table 1. Table 1: Bernoulli MAB: Mean cumulative regrets and standard errors (in parentheses) of different methods for different number of arms over 192 replicates. t = 1 , 2 , … , 50 . 𝑡 1 2 … 50 t=1,2,\dots,50.

	Mean Cumulative Regrets (SE)
Methods	2 arms	5 arms	10 arms
Tuned $ϵ$-greedy	1.64(0.08)	3.08(0.25)	4.60(0.35)
$ϵ$-greedy ($ϵ=0.05$)	2.78(0.32)	2.67(0.20)	4.58(0.29)
$ϵ$-greedy ($ϵ=0.1$)	2.54(0.24)	2.76(0.19)	4.96(0.27)
${ϵ}_t=0.5/t$	2.40(0.30)	2.87(0.22)	4.43(0.29)
Tuned TS	1.02(0.15)	2.88(0.20)	4.33(0.27)
TS	2.39(0.20)	3.41(0.18)	6.45(0.25)
${\eta }^t=1/t$	2.14(0.30)	2.57(0.21)	4.50(0.33)
Tuned UCB	1.08(0.13)	3.11(0.21)	4.21(0.28)
UCB ($\alpha =0.05$)	1.89(0.15)	3.28(0.18)	5.92(0.25)
${\alpha }^t=0.5-0.45/t$	2.44(0.34)	3.15(0.24)	4.62(0.29)
Gittins index	1.50(0.20)	2.72(0.18)	4.37(0.24)

Table 2. Table 2: Gaussian MAB: Mean cumulative regrets and standard errors (in parentheses) of different methods for different σ 𝜎 \sigma generative models over 192 replicates ( σ = 0.1 𝜎 0.1 \sigma=0.1 ), 384 replicates ( σ = 1 𝜎 1 \sigma=1 ). t = 1 , 2 , ⋯ , 50 . 𝑡 1 2 ⋯ 50 t=1,2,\cdots,50. Peformance of tuned methods in blue. Performance of method with best mean performance in red where different from the tuned variant.

	Mean cumulative regrets (SE)
	2 arms		5 arms		10 arms
Methods	$\sigma =1$	$\sigma =0.1$	$\sigma =1$	$\sigma =0.1$	$\sigma =1$	$\sigma =0.1$
Tuned $ϵ$-greedy	4.76(0.30)	0.14(0.08)	5.92(0.31)	0.89(0.11)	8.43(0.27)	0.74(0.06)
$ϵ$-greedy (${ϵ}^t=0.05$)	4.62(0.30)	0.39(0.04)	5.64(0.26)	0.83(0.09)	8.65(0.27)	1.28(0.06)
$ϵ$-greedy (${ϵ}^t=0.1$)	4.78(0.28)	0.74(0.04)	5.37(0.24)	0.98(0.09)	8.80(0.25)	1.98(0.07)
$ϵ$-greedy (${ϵ}^t=0.5/t$)	4.98(0.31)	0.32(0.02)	5.46(0.27)	0.77(0.10)	8.35(0.28)	1.07(0.07)
Tuned TS	4.76(0.32)	0.06(0.01)	6.00(0.29)	0.72(0.09)	8.66(0.27)	0.71(0.05)
TS $\left({\tau }^t=1\right)$	5.09(0.29)	0.25(0.01)	7.14(0.23)	1.28(0.04)	10.30(0.21)	2.18(0.05)
TS (${\tau }^t=1/t$)	5.29(0.35)	0.05(0.01)	6.16(0.32)	0.90(0.11)	8.78(0.29)	0.74(0.05)
Tuned UCB	5.41(0.35)	0.02(0.01)	5.52(0.31)	0.44(0.09)	8.33(0.28)	0.44(0.06)
UCB (${\alpha }^t=0.05$)	5.37(0.34)	0.08(0.01)	6.80(0.32)	0.70(0.04)	9.52(0.25)	1.27(0.04)
UCB (${\alpha }^t=0.5-0.45/t$)	5.02(0.34)	0.09(0.08)	5.86(0.29)	0.85(0.11)	8.61(0.29)	0.64(0.05)

Table 3. Table 3: Contextual bandit performance of ϵ italic-ϵ \epsilon -greedy variants; 96 replicates of T = 50 𝑇 50 T=50 time steps.

Methods	Mean cumulative regret (SE)
Tuned $ϵ$-greedy	1.31(0.07)
$ϵ=0.05$	1.47(0.10)
${ϵ}_t=t^{-1}$	2.12(0.08)
${ϵ}_t=0.5t^{-1}$	1.95(0.08)
${ϵ}_t={0.8}^t$	2.13(0.08)

Table 4. Table 4: Comparison of ϵ italic-ϵ \epsilon -greedy variants in the glucose problem in terms of mean cumulative reward (MCRew). 96 replicates for T=25, 192 replicates for T=50 (as these were considerably higher-variance). Conditional glucose distribution estimated using i) (correctly specified) AR(2) linear model, ii) (incorrectly specified) AR(1) linear model, and iii) (correctly specified) AR(2) nonparametric conditional density estimator.

Time horizon	Method	MCRew (SE)
Tuned (AR(2) linear)	-13.08(0.55)
	Tuned (AR(1) linear)	-13.39(0.84)
	Tuned (AR(2) NP)	-30.40(1.97)
T=25	${ϵ}^t=0.05$	-17.57(0.53)
	${ϵ}^t=t^{-1}$	-20.02(1.25)
	${ϵ}^t=0.5t^{-1}$	-17.80(0.90)
	${ϵ}^t={0.8}^t$	-22.93(1.79)
Tuned (AR(2) linear)	-16.92(0.42)
	Tuned (AR(1) linear)	-25.10(3.45)
	Tuned (AR(2) NP)	-52.80(4.45)
T=50	${ϵ}^t=0.05$	-39.56(2.55)
	${ϵ}^t=t^{-1}$	-57.23(4.24)
	${ϵ}^t=0.5t^{-1}$	-33.83(3.41)
	${ϵ}^t={0.8}^t$	-52.93(4.45)

Equations

$${\Gamma }_{{ϵ}^t}({\mathbf{H}}^t)=\{\begin{array}{l}\arg \underset{i=1,\dots ,k}{\max }{\bar{U}}_i^t,\text{with probability }1-\frac{{ϵ}^t}{k},\\ j,\text{with probability }\frac{{ϵ}^t}{k},\text{ for }j=1,\dots ,k,\end{array}$$

Full text

Machine Learning, ICML

1 Introduction

While significant attention has been paid in the reinforcement learning literature to domains such as game-playing, in which large amounts of data may be generated from a reliable simulator, quantitiative researchers are also interested in decision problems in which data are scarce, noisy, and expensive. Examples from the statistical literature include the estimation of optimal treatment regimes from observational or randomized studies (Robins, 1986; Murphy, 2003; Robins, 2004; Chakraborty et al., 2014); just-in-time adaptive interventions for mobile health applications (Ertefaie, 2014; Luckett et al., 2018); and the control of disease outbreaks (Laber et al., 2018). In such cases, computational efficiency is of secondary importance while data-efficiency is paramount. Moreover, in such decision problems, time horizons may be short and the best-performing exploration strategies must account for the number of remaining decisions.

With a view to such problems, we introduce a simple class of algorithms under the heading Parameterized Exploration (PE). Given a class of exploration strategies, e.g., $\epsilon$-greedy (Watkins, 1989; Sutton and Barto, 1998) or upper-confidence bound exploration (Lai and Robbins, 1985) the PE algorithm simply tunes the exploration schedule using an estimated model of the underlying system dynamics. This approach leverages the current state of knowledge about the generative model as well as the time horizon.

Perhaps most similar to our algorithm in the recent literature is the ”Noisy Net,” introduced by (Fortunato et al., 2017), in which exploration in the deep reinforcement learning setting is induced by adding parameterized noise to the weights of the neural network, and tuning the parameters governing the exploration noise via gradient descent. Our method is similar in spirit to the “meta-gradient” approach of Xu et al. (2018) for tuning reinforcement learning hyperparameters. However, each of these approaches is model-free, while our method leverages an estimator of the environment dynamics to tune the rate of exploration. Finally, the literature on Thompson sampling (Thompson, 1933) contains several examples of modifying the posterior used to induce exploration in order to vary the degree of exploration. Chapelle and Li (2011) present empirical results which suggest that a modified version of Thompson sampling in which the posterior variance is shrunk may outperform standard Thompson sampling, and in the context of clinical trials, Thall and Wathen (2007) present a variant of Thompson sampling in which randomization probabilities are adjusted according to the remaining number of patients in the trial.

2 Setup and notation

For simplicity, we will consider multi-armed bandits (MABs) in presenting the setup and method; the extension of the proposed methods to more general sequential decision settings is straightforward.

Let $T<\infty$ be the time horizon, $k$ be the number of possible actions, and $D_{1},\dots,D_{k}$ be the unknown reward distributions of each decision with means $\mu_{1},\dots,\mu_{k}$; define $\mu^{*}=\max_{i}\mu_{i}$. At each time $t=1,\dots,T$, let $A^{t}\in\{1,\dots,k\}$ denote the selected action and $U^{t}\sim D_{A^{t}}$ the observed reward. Write the observed sequences of actions and rewards up to time $t$ as {\mbox{\boldmathA}}^{t}=(A^{1},\dots,A^{t}) and {\mbox{\boldmathU}}^{t}=(U^{1},\dots,U^{t}), and define the history until time $t$ to be {\mathbf{H}}^{t}=({\mbox{\boldmathA}}^{t},{\mbox{\boldmathU}}^{t}). The goal is to take a sequence of actions such that the expected cumulative regret \mathbb{E}\Big{[}\sum_{t=1}^{T}\Big{(}\mu^{*}-U^{t}\Big{)}\Big{]} is minimized.

Define a learning algorithm as a sequence {\mbox{\boldmath\Gamma}}=(\Gamma^{1},\dots,\Gamma^{T}), where for each $t$, $\Gamma^{t}:\mathcal{H}^{t}\longrightarrow\mathcal{S}_{\mathcal{A}}$, where $\mathcal{S}_{\mathcal{A}}$ is the set of probability distributions over the set $\mathcal{A}$ of actions, and $\mathcal{H}^{t}$ is the space in which ${\mathbf{H}}^{t}$ takes values.

We consider variants of three of the most popular learning algorithms. The first is $\epsilon$-greedy: for a decision problem with $k$ actions available at time $t$, this algorithm takes the greedy (estimated-optimal) action with probability $1-\frac{\epsilon^{t}}{k}$, and selects actions uniformly at random otherwise. The second is upper confidence bound (UCB) exploration: this algorithm takes the action which has the greatest upper $(1-\alpha^{t})\times 100\%$ confidence bound on its mean reward. The third is Thompson sampling: in classical TS for multi-armed bandits, a sample is taken from the posterior distribution over the mean rewards, and the action which maximizes these posterior draws is taken (Thompson, 1933). However, we consider a more general class of TS algorithms. Let $C^{t}_{i}$ denote a confidence distribution (Schweder and Hjort, 2002) for the mean reward under action $i$ at time $t$; let $\omega_{i}^{t}$ denote the mean of $C_{i}^{t}$. At each time $t$ and action $i$ we draw a value $\tilde{\mu}_{i}^{t}$ from $C_{i}^{t}$ and select the action which maximizes $\omega_{i}^{t}+\tau^{t}(\tilde{\mu}_{i}^{t}-\omega_{i}^{t})$ for parameter $\tau^{t}$; alternatively, one could truncate $C_{i}^{t}$ to its $(1-\tau^{t})$ highest density region, sample means from this truncated distribution, and select the action that maximizes the sampled means. Thus, the amount of exploration in the three algorithms is dictated by the sequences $\{e^{t}\}_{t=1}^{T},\{\alpha^{t}\}_{t=1}^{T}$, and $\{\tau^{t}\}_{t=1}^{T}$, respectively.

We can write each of these learning algorithms in terms of their respective exploration parameters. For instance, the $\epsilon$-greedy algorithm can be written as

[TABLE]

where $\bar{U}^{t}_{i}$ is the sample mean of rewards observed from arm $i$ until time $t$. We write the subscript $\epsilon^{t}$ to emphasize the dependence on the exploration parameter $\epsilon^{t}$. More generally, we can write a decision rule with a generic exploration parameter $\eta^{t}$ as $\Gamma_{\eta^{t}}$.

In each case, it is clear that the optimal sequence $\{\eta^{t}\}_{t=1}^{T}$ is nonincreasing in $t$, as the value of exploring goes to 0 as we approach the time horizon. In the next section, we present a simple strategy for tuning the sequence $\{\eta^{t}\}_{t=1}^{T}$ for a given class of learning algorithms.

3 Parameterized exploration

In order to adaptively tune the rate of exploration, we propose to parameterize the sequence $\{\eta^{t}\}_{t=1}^{T}$ using a family of nonincreasing functions, such that for each $t$, $\eta^{t}=\eta(T,t,\theta)$ for some $\theta\in\Theta$. We consider the following class of functions: $\{\eta(T,t,\theta)=\frac{\theta_{0}}{1+\exp{[-\theta_{2}(T-t-\theta_{1})]}}:\theta=(\theta_{0},\theta_{1},\theta_{2})\in\Theta\}$, with $\Theta$ chosen such that $\eta(T,\cdot,\theta)$ is decreasing for each $\theta\in\Theta$. Then, if $\Gamma_{\eta}$ is a decision rule with exploration parameter $\eta$, each value of $\theta$ leads to a learning algorithm {\mbox{\boldmath\Gamma}}_{(\eta,T,\theta)}=(\Gamma_{\eta(T,1,\theta)},\dots,\Gamma_{\eta(T,T,\theta)}). We will refer to this algorithm as {\mbox{\boldmath\Gamma}}_{\theta}, suppressing the dependence on the time horizon $T$ and the class of functions $\eta$.

Let $\widehat{\mathcal{M}}^{t}$ be an estimator of the generative model $\mathcal{M}$ underlying the sequential decision problem at time $t$ — in the MAB setting, the generative model consists of the reward distributions at each arm, i.e., $\mathcal{M}=(D_{1},\dots,D_{k})$. Define R^{T}(\theta,\mathcal{M})=\mathbb{E}_{\mathcal{M},\theta}\Big{[}\sum_{t=1}^{T}\Big{(}\mu^{*}-U^{t}\Big{)}\Big{]} to be the cumulative regret until the horizon $T$ if actions are chosen according to {\mbox{\boldmath\Gamma}}_{\theta} and the true generative model is $\mathcal{M}$. Then, at each time step $t$ we can solve $\hat{\theta}^{t}=\arg\min_{\theta\in\Theta}R^{T}(\theta,\widehat{\mathcal{M}}^{t})$ and take the action $A^{t}=\Gamma_{\eta(T,t,\hat{\theta}^{t})}({\mathbf{H}}^{t})$.

However, in early episodes, point estimates $\widehat{\mathcal{M}}^{t}$ of $\mathcal{M}$ may be of low quality due to insufficient training data; for instance, if the variances of the reward distributions are simultaneously under-estimated and the ordering of estimated reward means is incorrect, this method may lead to under-exploration and therefore long sequences of suboptimal actions. (We observed such behavior in preliminary simulation experiments.) In order to account for uncertainty in $\widehat{\mathcal{M}}^{t}$, we can instead minimize the expected value of the above objective quantity against a confidence distribution $C^{t}$ for $\mathcal{M}$ at time $t$. That is, we can solve $\hat{\theta}^{t}=\arg\min_{\theta\in\Theta}\mathbb{E}_{\widetilde{\mathcal{M}}\sim C^{t}}R^{T}(\theta,\widetilde{\mathcal{M}})$. to get $\hat{\theta}^{t}$ and associated decision rule at each time. This variant of PE is presented in Algorithm 1.

4 Simulations

We present comparisons of tuned and un-tuned variants of the learning algorithms discussed above in multi-armed bandits (MABs), a normal-linear contextual bandit, and a continuous-state Markov decision process (MDP). In the MDP case, we modify the objective to maximize cumulative reward rather than minimize cumulative regret. In each case, we carry out the requisite optimizations using Gaussian process optimization, implemented in the Python package BayesianOptimization (Nogueira, 2018).

4.1 Bernoulli multi-armed bandit

We consider Bernoulli MABs with 2, 5,and 10 arms. Table 1 compares mean cumulative regrets of tuned and untuned variants of $\epsilon$-greedy, UCB, and Thompson sampling. We also report the performance of finite horizon Gittins index (Kaufmann et al., 2018) in the table; the finite horizon Gittins index policy is an important baseline given that it is approximately Bayes-optimal and therefore represents another approach to incorporating the full state of knowledge about the environment as well as the time horizon into the exploration. For each family of learning algorithms, the tuned variant is competitive with or outperforms the un-tuned variants. The tuned algorithms are also competitive with the Gittins index policy in most cases.

4.2 Gaussian multi-armed bandit

Following Kuleshov and Precup (2014), we test our algorithms on Gaussian MABs with reward means in $[0,1]$ and different choices of variances. We set the time horizon at $T=50$ given our focus on short-horizon decision problems. Table 2 displays the cumulative regrets for tuned and un-tuned variants of $\epsilon$-greedy, UCB, and Thompson sampling, with 2, 5, and 10 arms. The tuned variants outperform un-tuned counterparts across many settings, and when they do not have the lowest sample mean regret are within approximately one standard error of the methods which do perform best in sample mean.

4.3 Contextual bandits

In the contextual bandit setting, we considered the two-arm normal-linear contextual bandit of Lei et al. (2017), based on the Heartsteps applications for increasing physical activity Klasnja et al. (2015). In order to tune the exploration parameter, we fit a correctly specified multivariate normal model for the context distribution. The results are displayed in Table 3.

4.4 Controlling glucose: a continuous-state MDP

In this experiment, we consider an MDP in which the states are continuous. We simulate cohorts of patients with type 1 diabetes using a generative model based on the mobile health study of (Maahs et al., 2012). We only consider the action of whether to use insulin, so the action space is $\mathcal{A}=\{0,1\}$. The covariates observed for patient $i$ at time $t$ is average blood glucose level, total dietary intake, and total counts of physical activity, denoted by $(Gl_{i}^{t},Di_{i}^{t},Ex_{i}^{t})$ respectively. The action taken at time $t$ is denoted as $A^{t}$. Defining states $S^{t}=(Gl^{t},Di^{t},Ex^{t},Gl^{t-1},Di^{t-1},Ex^{t-1},A^{t-1})^{\intercal}$, glucose levels evolve according to the second-order autoregressive (AR(2)) process $Gl^{t}=\beta S^{t-1}+e^{t}$, where $e^{t}\sim N(0,5^{2})$ and $\beta=(10,0.9,0.1,-0.01,0.0,0.1,-0.01,-10,-4)$; and $Di^{t}\sim N(0,10^{2})$ with probability $0.6$, otherwise $Di^{t}=0$; similarly, $Ex^{t}\sim N(0,10^{2})$ with probability $0.6$, otherwise $Ex^{t}=0$. Thus the dynamics are Markovian with states $S^{t}$. The reward at each time step is given by $U^{t}\equiv\mathbbm{1}(Gl^{t}<70)[-0.005(Gl^{t})^{2}+0.95Gl^{t}-45]+\mathbbm{1}(Gl^{t}\geq 70)[-0.0002(Gl^{t})^{2}+0.022Gl^{t}-0.5]$, which decreases as $Gl^{t}$ departs from normal glucose levels. (This is a continuous variant of the discrete reward function used in Luckett et al. (2018).)

We simulate data for $n=15$ patients and time horizons of 25 and 50. In each case we use an $\epsilon$-greedy learning algorithm, where the greedy action is given by the argmax of the estimated conditional expected reward function, fit using random forest regression. We examined three approaches to estimating the glucose transition dynamics. First, we fit a correctly-specified AR(2) linear model using ordinary least squares. Second, we fit an (incorrectly-specified) AR(1) linear model in which glucose depends only the glucose, food intake, activity level, and action at the previous time. Finally, we model the conditional probability distribution of glucose at time $t$ given the glucose, food, activity, and treatments from times $t-1,t-2$ using a two-step procedure similar to that described in (Hansen, 2004): First, we estimate the conditional mean of glucose using a random forest estimator, and then the full conditional distribution of the residuals of the fitted models by the ratio of the kernel density estimators of the joint distribution of glucose and covariates to the marginal density of the covariates; the relevant bandwidths were selected using cross-validation. In each of these transition model estimators, activity and food are assumed to be i.i.d. over time points and patients, and their distributions are estimated by their empircal distributions.

The mean cumulative regrets incurred by each $\epsilon$-greedy variant for $T=25,50$ are displayed in Table 4. While the linear models, both correct and misspecified, do considerably better than each of the other methods, tuning with the nonparametric conditional density estimator performs worst or among the worst at both time horizons.

5 Future work

While we have only considered model-based tuning of exploration parameters, the method presented here could be used to tune other hyperparameters such as the discount factor used in the estimation of the optimal policy, or modify the objective function to improve (for instance) the operating characteristics of statistical tests of comparisons between patients receiving different treatments (see Williamson et al. (2017) for an alternative apprach to trading off expected in-sample rewards and statistical power in the context of clinical trials). Another direction for future work is constructing diagnostics to identify if the estimated system dynamics are of sufficiently high-quality to justify their use in tuning, as evidenced by the poor performance of the nonparametric conditional density estimator, which suffers from high variance. Finally, a theoretical regret analysis of our method is called for.