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Tactical Optimism and Pessimism for Deep Reinforcement Learning

Ted Moskovitz
Gatsby Unit, UCL
[email protected]
&Jack Parker-Holder
University of Oxford
[email protected]
Aldo Pacchiano
Microsoft Research
[email protected]
&Michael Arbel
Université Grenoble Alpes, Inria, CNRS
[email protected]
&Michael I. Jordan
University of California, Berkeley
[email protected]
Grenoble INP, LJK,38000 Grenoble. Work mostly completed at the Gatsby Unit.

Abstract

In recent years, deep off-policy actor-critic algorithms have become a dominant approach to reinforcement learning for continuous control. One of the primary drivers of this improved performance is the use of pessimistic value updates to address function approximation errors, which previously led to disappointing performance. However, a direct consequence of pessimism is reduced exploration, running counter to theoretical support for the efficacy of optimism in the face of uncertainty. So which approach is best? In this work, we show that the most effective degree of optimism can vary both across tasks and over the course of learning. Inspired by this insight, we introduce a novel deep actor-critic framework, Tactical Optimistic and Pessimistic (TOP) estimation, which switches between optimistic and pessimistic value learning online. This is achieved by formulating the selection as a multi-arm bandit problem. We show in a series of continuous control tasks that TOP outperforms existing methods which rely on a fixed degree of optimism, setting a new state of the art in challenging pixel-based environments. Since our changes are simple to implement, we believe these insights can easily be incorporated into a multitude of off-policy algorithms.

1 Introduction

Reinforcement learning (RL) has begun to show significant empirical success in recent years, with value function approximation via deep neural networks playing a fundamental role in this success Mnih et al. (2015); Silver et al. (2016); Badia et al. (2020). However, this success has been achieved in a relatively narrow set of problem domains, and an emerging set of challenges arises when one considers placing RL systems in larger systems. In particular, the use of function approximators can lead to a positive bias in value computation Thrun and Schwartz (1993), and therefore systems that surround the learner do not receive an honest assessment of that value. One can attempt to turn this vice into a virtue, by appealing to a general form of the optimism-under-uncertainty principle—overestimation of the expected reward can trigger exploration of states and actions that would otherwise not be explored. Such exploration can be dangerous, however, if there is not a clear understanding of the nature of the overestimation.

This tension has not been resolved in the recent literature on RL approaches to continuous-control problems. On the one hand, some authors seek to correct the overestimation, for example by using the minimum of two value estimates as a form of approximate lower bound Fujimoto et al. (2018). This approach can be seen as a form of pessimism with respect to the current value function. On the other hand, Ciosek et al. (2019) have argued that the inherent optimism of approximate value estimates is actually useful for encouraging exploration of the environment and/or action space. Interestingly, both sides have used their respective positions to derive state-of-the-art algorithms. How can this be, if their views are seemingly opposed? Our key hypothesis is the following:

The degree of estimation bias, and subsequent efficacy of an optimistic strategy, varies as a function of the environment, the stage of optimization, and the overall context in which a learner is embedded.

This hypothesis motivates us to view optimism/pessimism as a spectrum and to investigate procedures that actively move along that spectrum during the learning process. We operationalize this idea by measuring two forms of uncertainty that arise during learning: aleatoric uncertainty and epistemic uncertainty. These notions of uncertainty, and their measurement, are discussed in detail in Section 5.1. We then further aim to control the effects of these two kinds of uncertainty, making the following learning-theoretic assertion:

When the level of bias is unknown, an adaptive strategy can be highly effective.

In this work, we investigate these hypotheses via the development of a new framework for value estimation in deep RL that we refer to as Tactical Optimism and Pessimism (TOP). This approach acknowledges the inherent uncertainty in the level of estimation bias present, and rather than adopt a blanket optimistic or pessimistic strategy, it estimates the optimal approach on the fly, by formulating the optimism/pessimism dilemma as a multi-armed bandit problem. Furthermore, TOP explicitly isolates the aleatoric and epistemic uncertainty by representing the environmental return using a distributional critic and model uncertainty with an ensemble. The overall concept is summarized in Figure 1.

We show in a series of experiments that not only does the efficacy of optimism indeed vary as we suggest, but TOP is able to capture the best of both worlds, achieving a new state of the art for challenging continuous control problems.

Our main contributions are as follows:

•

Our work shows that the efficacy of optimism for a fixed function approximator varies across environments and during training for reinforcement learning with function approximation.
•

We propose a novel framework for value estimation, Tactical Optimism and Pessimism (TOP), which learns to balance optimistic and pessimistic value estimation online. TOP frames the choice of the degree of optimism or pessimism as a multi-armed bandit problem.
•

Our experiments demonstrate that these insights, which require only simple changes to popular algorithms, lead to state-of-the-art results on both state- and pixel-based control.

Refer to caption — Figure 1: Visualization of the TOP framework. Blue arrows denote stochastic variables.

2 Related Work

Much of the recent success of off-policy actor-critic algorithms build on DDPG Lillicrap et al. (2016), which extended the deterministic policy gradient Silver et al. (2016) approach to off-policy learning with deep networks, using insights from DQN Mnih et al. (2015). Like D4PG (Barth-Maron et al., 2018), we combine DPG with distributional value estimation. However, unlike D4PG, we use two critics, a quantile representation rather than a categorical distribution Bellemare et al. (2017), and, critically, we actively manage the tradeoff between optimism and pessimism. We also note several other success stories in the actor-critic vein, including TD3, SAC, DrQ, and PI-SAC Fujimoto et al. (2018); Haarnoja et al. (2018); Yarats et al. (2021); Lee et al. (2020); these represent the state-of-the-art for continuous control and will serve as a baseline for our experiments.

The principle of optimism in the face of uncertainty (Audibert et al., 2007; Kocsis and Szepesvári, 2006; Zhang and Yao, 2019) provides a design tool for algorithms that trade off exploitation (maximization of the reward) against the need to explore state-action pairs with high epistemic uncertainty. The theoretical tool for evaluating the success of such designs is the notion of regret, which captures the loss incurred by failing to explore. Regret bounds have long been used in research on multi-armed bandits, and they have begun to become more prominent in RL as well, both in the tabular setting Jaksch et al. (2010); Filippi et al. (2010); Fruit et al. (2018); Azar et al. (2017); Bartlett and Tewari (2012); Tossou et al. (2019), and in the setting of function approximation Jin et al. (2020); Yang and Wang (2020). However, optimistic approaches have had limited empirical success when combined with deep neural networks in RL Ciosek et al. (2019). To be successful, these approaches need to be optimistic enough to upper bound the true value function while maintaining low estimation error Pacchiano et al. (2020a). This becomes challenging when using function approximation, and the result is often an uncontrolled, undesirable overestimation bias.

Recently, there has been increasing evidence in support of the efficacy of adaptive algorithms Ball et al. (2020); Schaul et al. (2019); Penedones et al. (2019); Parker-Holder et al. (2020). An example is Agent57 Badia et al. (2020), the first agent to outperform the human baseline for all 57 games in the Arcade Learning Environment (Bellemare et al., 2012). Agent57 adaptively switches among different exploration strategies. Our approach differs in that it aims to achieve a similar goal by actively varying the level of optimism in its value estimates.

Finally, our work is also related to automated RL (AutoRL), as we can consider TOP to be an example of an on-the-fly learning procedure Co-Reyes et al. (2021); Oh et al. (2020); Kirsch et al. (2020). An exciting area of future work will be to consider the interplay between the degree of optimism and model hyperparameters such as architecture and learning rate, and whether they can be adapted simultaneously.

3 Preliminaries

Reinforcement learning considers the problem of training an agent to interact with its environment so as to maximize its cumulative reward. Typically, a task and environment are cast as a Markov decision process (MDP), formally defined as a tuple $(\mathcal{S},\mathcal{A},p,r,\gamma)$ , where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the space of possible actions, $p:\mathcal{S}\times\mathcal{A}\to\mathcal{P}(\mathcal{S})$ is a transition kernel, $r:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ is the reward function, and $\gamma\in[0,1)$ is a discounting factor. For a given policy $\pi$ , the return $Z^{\pi}=\sum_{t}\gamma^{t}r_{t}$ , is a random variable representing the sum of discounted rewards observed along one trajectory of states obtained from following $\pi$ until some time horizon $T$ , potentially infinite. Given a parameterization of the set of policies, $\{\pi_{\theta}:\theta\in\Theta\}$ , the goal is to update $\theta$ so as to maximize the expected return, or discounted cumulative reward, $J(\theta)=\mathbb{E}_{\pi}\left[\sum_{t}\gamma^{t}r_{t}\right]=\operatorname{\mathbb{E}}[Z^{\pi}]$ .

Actor-critic algorithms are a framework for solving this problem in which the policy $\pi$ , here known as the actor, is trained to maximize expected return, while making use of a critic that evaluates the actions of the policy. Typically, the critic takes the form of a value function which predicts the expected return under the current policy, $Q^{\pi}(s,a)\coloneqq\operatorname{\mathbb{E}}_{\pi}[Z_{t}|s_{t}=s,a_{t}=a]$ . When the state space is large, $Q^{\pi}$ may be parameterized by a model with parameters $\phi$ . The deterministic policy gradient (DPG) theorem Silver et al. (2014) shows that gradient ascent on $J$ can be performed via

\displaystyle\nabla_{\theta}J(\theta)=\operatorname{\mathbb{E}}_{\pi}[\nabla_{a}Q^{\pi}(s,a)|_{a=\pi(s)}\nabla_{\theta}\pi_{\theta}(s)].

(1)

The critic is updated separately, usually via SARSA Sutton and Barto (2018), which, given a transition $s_{t},a_{t}\to r_{t+1},s_{t+1}$ , forms a learning signal via semi-gradient descent on the squared temporal difference (TD) error, $\delta_{t}^{2}$ , where

\displaystyle\begin{split}\delta_{t}\coloneqq y_{t}-Q^{\pi}(s_{t},a_{t})=r_{t+1}+\gamma Q^{\pi}(s_{t+1},\pi(s_{t+1}))-Q^{\pi}(s_{t},a_{t}),\end{split}

(2)

and where $y_{t}$ is the Bellman target. Rather than simply predicting the mean of the return $Z^{\pi}$ under the current policy, it can be advantageous to learn a full distribution of $Z^{\pi}$ given the current state and action, $\mathcal{Z}^{\pi}(s_{t},a_{t})$ Bellemare et al. (2017); Dabney et al. (2018b, a); Rowland et al. (2019). In this framework, the return distribution is typically parameterized via a set of $K$ functionals of the distribution (e.g., quantiles or expectiles) which are learned via minimization of an appropriate loss function. For example, the $k$ th quantile of the distribution at state $s$ and associated with action $a$ , $q_{k}(s,a)$ , can be learned via gradient descent on the Huber loss (Huber, 1964) of the distributional Bellman error, $\delta_{k}=\hat{Z}-q_{k}(s,a)$ , for $\hat{Z}\sim\mathcal{Z}^{\pi}(\cdot|s,a)$ . While $\hat{Z}$ is formally defined as a sample from the return distribution, $\delta_{k}$ is typically computed in practice as $K^{-1}\sum_{j=1}^{K}r+\gamma q_{j}(s,a)-q_{k}(s,a)$ (Dabney et al., 2018b).

4 Optimism versus Pessimism

Reducing overestimation bias with pessimism

It was observed by Thrun and Schwartz (1993) that Q-learning (Watkins and Dayan, 1992) with function approximation is biased towards overestimation. Noting that this overestimation bias can introduce instability in training, Fujimoto et al. (2018) introduced the Twin Delayed Deep Deterministic (TD3) policy gradient algorithm to correct for the bias. TD3 can be viewed as a pessimistic heuristic in which values are estimated via a SARSA-like variant of double Q-learning (Hasselt, 2010) and the Bellman target is constructed by taking the minimum of two critics:

\displaystyle y_{t}=r_{t+1}+\gamma\min_{i\in\{1,2\}}Q_{\phi_{i}}^{\pi}(s,\pi_{\theta}(s)+\epsilon).

(3)

Here $\epsilon\sim\text{clip}(\mathcal{N}(0,s^{2}),-c,c)$ is drawn from a clipped Gaussian distribution ( $c$ is a constant). This added noise is used for smoothing in order to prevent the actor from overfitting to narrow peaks in the value function. Secondly, TD3 delays policy updates, updating value estimates several times between each policy gradient step. By taking the minimum of two separate critics and increasing the number of critic updates for each policy update, this approach takes a pessimistic view on the policy’s value in order to reduce overestimation bias. These ideas have become ubiquitous in state-of-the-art continuous control algorithms (Ball and Roberts, 2021), such as SAC, RAD, (PI)-SAC Haarnoja et al. (2018); Laskin et al. (2020a); Lee et al. (2020).

Optimism in the face of uncertainty

While it is valuable to attempt to correct for overestimation of the value function, it is also important to recall that overestimation can be viewed as a form of optimism, and as such can provide a guide for exploration, a necessary ingredient in theoretical treatments of RL in terms of regret

(Jin et al., 2018; Jaksch et al., 2010; Azar et al., 2017). In essence, the effect of optimistic value estimation is to induce the agent to explore regions of the state space with high epistemic uncertainty, encouraging further data collection in unexplored regions. Moreover, Ciosek et al. (2019) found that reducing value estimates, as done in pessimistic algorithms, can lead to pessimistic underexploration, in which actions that could lead to experience that gives the agent a better long-term reward. To address this problem, Ciosek et al. (2019) introduced the Optimistic Actor-Critic (OAC) algorithm, which trains an exploration policy using an optimistic upper bound on the value function while constructing targets for learning using the lower bound of Fujimoto et al. (2018). OAC demonstrated improved performance compared to SAC, hinting at a complex interplay between optimism and pessimism in deep RL algorithms.

Trading off optimism and pessimism

As we have discussed, there are arguments for both optimism and pessimism in RL. Optimism can aid exploration, but if there is significant estimation error, then a more pessimistic approach may be needed to stabilize learning. Moreover, both approaches have led to algorithms that are supported by strong empirical evidence. We aim to reconcile these seemingly contradictory perspectives by hypothesizing that the relative contributions of these two ingredients can vary depending on the nature of the task, with relatively simple settings revealing predominantly one aspect. As an illustrative example, we trained “Optimistic” and “Pessimistic” versions of the same deep actor-critic algorithm (details in Figure 5) for two different tasks and compared their performance in Figure 2. As we can see, in the HalfCheetah task, the Optimistic agent outperforms the Pessimistic agent, while in the Hopper task, the opposite is true. This result suggests that the overall phenomenon is multi-faceted and active management of the overall optimism-pessimism trade-off is necessary. Accordingly, in the current paper we propose the use of an adaptive approach in which the degree of optimism or pessimism is adjusted dynamically during training. As a consequence of this approach, the optimal degree of optimism can vary across tasks and over the course of a single training run as the model improves. Not only does this approach reconcile the seemingly contradictory perspectives in the literature, but it also can outperform each individual framework in a wider range of tasks.

5 Tactical Optimistic and Pessimistic Value Estimation

TOP is based on the idea of adaptive optimism in the face of uncertainty. We begin by discussing how TOP represents uncertainty and then turn to a description of the mechanism by which TOP dynamically adapts during learning.

5.1 Representing uncertainty in TOP estimation

TOP distinguishes between two types of uncertainty—aleatoric uncertainty and epistemic uncertainty—and represents them using two separate mechanisms.

Aleatoric uncertainty reflects the noise that is inherent to the environment regardless of the agent’s understanding of the task. Following Bellemare et al. (2017); Dabney et al. (2018b, a); Rowland et al. (2019), TOP represents this uncertainty by learning the full return distribution, $\mathcal{Z}^{\pi}(s,a)$ , for a given policy $\pi$ and state-action pair $(s,a)$ rather than only the expected return, $Q^{\pi}(s,a)=\mathbb{E}[Z^{\pi}(s,a)]$ , $Z^{\pi}(s,a)\sim\mathcal{Z}^{\pi}(\cdot|s,a)$ . Note here we are using $Z^{\pi}(s,a)$ to denote the return random variable and $\mathcal{Z}^{\pi}(s,a)$ to denote the return distribution of policy $\pi$ for state-action pair $(s,a)$ . Depending on the stochasticity of the environment, the distribution $\mathcal{Z}^{\pi}(s,a)$ is more or less spread out, thereby acting as a measure of aleatoric uncertainty.

Epistemic uncertainty reflects lack of knowledge about the environment and is expected to decrease as the agent gains experience. TOP uses this uncertainty to quantify how much an optimistic belief about the return differs from a pessimistic one. Following Ciosek et al. (2019), we model epistemic uncertainty via a Gaussian distribution with mean $\bar{Z}(s,a)$ and standard deviation $\sigma(s,a)$ of the quantile estimates across multiple critics as follows:

\displaystyle Z^{\pi}(s,a)\stackrel{{\scriptstyle d}}{{=}}\bar{Z}(s,a)+\epsilon\sigma(s,a),

(4)

where $\stackrel{{\scriptstyle d}}{{=}}$ indicates equality in distribution. However, unlike in Ciosek et al. (2019), where the parameters of the Gaussian are deterministic, we treat both $\bar{Z}(s,a)$ and $\sigma(s,a)$ as random variables underlying a Bayesian representation of aleatoric uncertainty. As we describe next, only $\mathcal{Z}^{\pi}(s,a)$ is modeled (via a quantile representation), hence $\bar{Z}(s,a)$ and $\sigma(s,a)$ are unknown. Proposition 1 shows how to recover them from $Z^{\pi}(s,a)$ and is proven in Appendix E.

Proposition 1.

The quantile function $q_{\bar{Z}(s,a)}$ for the distribution of $\bar{Z}$ is given by:

\displaystyle q_{\bar{Z}(s,a)}=\mathbb{E}_{\epsilon}\left[q_{Z^{\pi}(s,a)}\right],

(5)

where $q_{Z^{\pi}(s,a)}$ is the quantile function of $\mathcal{Z}^{\pi}(s,a)$ knowing $\epsilon$ and $\sigma(s,a)$ and $\mathbb{E}_{\epsilon}$ denotes the expectation w.r.t. $\epsilon\sim\mathcal{N}(0,1)$ . Moreover, $\sigma^{2}(s,a)$ satisfies:

\displaystyle\sigma^{2}(s,a)

\displaystyle=\mathbb{E}_{\epsilon}[\|\bar{Z}(s,a)-Z^{\pi}\|^{2}].

(6)

Quantile approximation

Following Dabney et al. (2018b), TOP represents the return distribution $Z^{\pi}(s,a)$ using a quantile approximation, meaning that it forms $K$ statistics, $q^{(k)}(s,a)$ , to serve as an approximation of the quantiles of $\mathcal{Z}^{\pi}(s,a)$ . The quantiles $q^{(k)}(s,a)$ can be learned as the outputs of a parametric function—in our case, a deep neural network—with parameter vector $\phi$ . To measure epistemic uncertainty, TOP stores two estimates, $\mathcal{Z}^{\pi}_{1}(s,a)$ and $\mathcal{Z}^{\pi}_{2}(s,a)$ , with respective quantile functions $q_{1}^{(k)}(s,a)$ and $q_{2}^{(k)}(s,a)$ and parameters $\phi_{1}$ and $\phi_{2}$ . This representation allows for straightforward estimation of the mean $\bar{Z}(s,a)$ and variance $\sigma(s,a)$ in Equation 4 using Proposition 1. Indeed, applying Equations 5 and 6 and treating $Z^{\pi}_{1}(s,a)$ and $Z^{\pi}_{2}(s,a)$ as exchangeable draws from Equation 4, we approximate the quantiles $q_{\bar{Z}(s,a)}$ and $q_{\sigma(s,a)}$ of the distribution of $\bar{Z}(s,a)$ and $\sigma(s,a)$ as follows:

\displaystyle\begin{aligned} \bar{q}^{(k)}(s,a)=\frac{1}{2}\left(q_{1}^{(k)}(s,a)+q_{2}^{(k)}(s,a)\right),\qquad\sigma^{(k)}(s,a)=\sqrt{\sum_{i=1}^{2}\left(q_{i}^{(k)}(s,a)-\bar{q}^{(k)}(s,a)\right)^{2}}.\end{aligned}

(7)

Next, we will show these approximations can be used to define an exploration strategy for the agent.

5.2 An uncertainty-based strategy for exploration

We use the quantile estimates defined in Equation 7 to construct a belief distribution $\tilde{\mathcal{Z}}^{\pi}(s,a)$ over the expected return whose quantiles are defined by

\displaystyle q_{\tilde{Z}^{\pi}(s,a)}=q_{\bar{Z}(s,a)}+\beta q_{\sigma(s,a)}.

(8)

This belief distribution $\tilde{\mathcal{Z}}^{\pi}(s,a)$ is said be optimistic when $\beta\geq 0$ and pessimistic when $\beta<0$ . The amplitude of optimism or pessimism is measured by $\sigma(s,a)$ , which quantifies epistemic uncertainty. The degree of optimism depends on $\beta$ and is adjusted dynamically during training, as we will see in Section 5.3. Note that $\beta$ replaces $\epsilon\sim\mathcal{N}(0,1)$ , making the belief distribution non-Gaussian.

Learning the critics. TOP uses the belief distribution in Equation 8 to form a target for both estimates of the distribution, $\mathcal{Z}^{\pi}_{1}(s,a)$ and $\mathcal{Z}^{\pi}_{2}(s,a)$ . To achieve this, TOP computes an approximation of $\tilde{\mathcal{Z}}^{\pi}(s,a)$ using $K$ quantiles $\tilde{q}^{(k)}=\bar{q}^{k}+\beta\sigma^{(k)}$ . The temporal difference error for each $\mathcal{Z}^{\pi}_{i}(s,a)$ is given by $\delta_{i}^{(j,k)}:=r+\gamma\tilde{q}^{(j)}-q_{i}^{(k)}$ with $i\in\{1,2\}$ and where $(j,k)$ ranges over all possible combinations of quantiles. Finally, following the quantile regression approach in Dabney et al. (2018b), we minimize the Huber loss $\mathcal{L}_{\mathrm{Huber}}$ evaluated at each distributional error $\delta_{i}^{(j,k)}$ , which provides a gradient signal to learn the distributional critics as given by Equation 9:

\displaystyle\Delta\phi_{i}\propto\sum_{1\leq k,j\leq K}\nabla_{\phi_{i}}\mathcal{L}_{\mathrm{Huber}}(\delta_{i}^{(j,k)}).

(9)

The overall process is summarized in Algorithm 2.

Learning the actor.

The actor is trained to maximize the expected value $\tilde{Q}(s,a)$ under the belief distribution $\tilde{\mathcal{Z}}^{\pi}(s,a)$ . Using the quantile approximation, $\tilde{Q}(s,a)$ is simply given as an average over $\tilde{q}^{(k)}$ : $\tilde{Q}(s,a)=\frac{1}{K}\sum_{k=1}^{K}\tilde{q}^{(k)}(s,a)$ . The update of the actor follows via the DPG gradient:

\displaystyle\Delta\theta\propto\nabla_{a}\tilde{Q}(s,a)|_{a=\pi_{\theta}(s)}\nabla_{\theta}\pi_{\theta}(s).

(10)

This process is summarized in Algorithm 3. To reduce variance and leverage past experience, the critic and actor updates in Equations 10 and 9 are both averaged over $N$ transitions, $(s,a,r,s^{\prime})_{n=1}^{N}$ , sampled from a replay buffer $\mathcal{B}$ (Lin, 1992).

In the special case of $\beta=-1/\sqrt{2}$ , the average of Equation 8 reduces to $\min_{i}Z_{i}^{\pi}(s,a)$ and Equation 10 recovers a distributional version of TD3, a pessimistic algorithm. On the other hand, when $\beta\geq 0$ , the learning target is optimistic with respect to the current value estimates, recovering a procedure that can be viewed as a distributional version of the optimistic algorithm of Ciosek et al. (2019). However, in our case, when $\beta\geq 0$ the learning target is also optimistic. Hence, Equations 9 and 10 can be seen as a generalization of the existing literature to a distributional framework that can recover both optimistic and pessimistic value estimation depending on the sign of $\beta$ . In the next section we propose a principled way to adapt $\beta$ during training to benefit from both the pessimistic and optimistic facets of our approach.

5.3 Optimism and pessimism as a multi-arm bandit problem

As we have seen (see Figure 2), the optimal degree of optimism or pessimism for a given algorithm may vary across environments. As we shall see, it can also be beneficial to be more or less optimistic over the course of a single training run. It is therefore sensible for an agent to adapt its degree of optimism dynamically in response to feedback from the environment. In our framework, the problem can be cast in terms of the choice of $\beta$ . Note that the evaluation of the effect of $\beta$ is a form of bandit feedback, where learning episodes tell us about the absolute level of performance associated with a particular value of $\beta$ , but do not tell us about relative levels. We accordingly frame the problem as a multi-armed bandit problem, using the Exponentially Weighted Average Forecasting algorithm (Cesa-Bianchi and Lugosi, 2006). In our setting, each bandit arm represents a particular value of $\beta$ , and we consider $D$ experts making recommendations from a discrete set of values $\{\beta_{d}\}_{d=1}^{D}$ . After sampling a decision $d_{m}\in\{1,\dots,D\}$ at episode $m$ , we form a distribution $\mathbf{p}_{m}\in\Delta_{D}$ of the form $\mathbf{p}_{m}(d)\propto\exp\left(w_{m}(d)\right)$ . The learner receives a feedback signal, $f_{m}\in\mathbb{R}$ , based on this choice. The parameter $w_{m}$ is updated as follows:

w_{m+1}(d)=\begin{cases}w_{m}(d)+\eta\frac{f_{m}}{\mathbf{p}_{m}(d)}&\text{if }d=d_{m}\\ w_{m}(d)&\text{otherwise},\end{cases}

(11)

for a step size parameter $\eta>0$ . Intuitively, if the feedback signal obtained is high and the current probability of selecting a given arm is low, the likelihood of selecting that arm again will increase. For the feedback signal $f_{m}$ , we use improvement in performance. Concretely, we set $f_{m}=R_{m}-R_{m-1}$ , where $R_{m}$ is the cumulative reward obtained in episode $m$ . Henceforth, we denote by $\mathbf{p}^{\beta}_{m}$ the exponential weights distribution over $\beta$ values at episode $m$ .

Our approach can be thought of as implementing a form of model selection similar to that of Pacchiano et al. (2020d), where instead of maintaining distinct critics for each optimism choice, we simply update the same pair of critics using the choice of $\beta$ proposed by the bandit algorithm. For a more thorough discussion of TOP’s connection to model selection, see Appendix D.

Algorithm 1 TOP-TD3

1: Initialize critic networks

Q_{\phi_{1}}

Q_{\phi_{2}}

and actor

\pi_{\theta}

Initialize target networks

\phi_{1}^{\prime}\leftarrow\phi_{1}

\phi_{2}^{\prime}\leftarrow\phi_{2}

\theta^{\prime}\leftarrow\theta

Initialize replay buffer and bandit probabilities

\mathcal{B}\leftarrow\emptyset,

\mathbf{p}_{1}^{\beta}\leftarrow\mathcal{U}([0,1]^{D})

2: for episode in

m=1,2,\dots

3: Initialize episode reward

R_{m}\leftarrow 0

4: Sample optimism

\beta_{m}\sim\mathbf{p}_{m}^{\beta}

5: for time step

t=1,2,\dots,T

6: Select noisy action

a_{t}=\pi_{\theta}(s_{t})+\epsilon

\epsilon\sim\mathcal{N}(0,s^{2})

, obtain

r_{t+1},s_{t+1}

7: Add to total reward

R_{m}\leftarrow R_{m}+r_{t+1}

8: Store transition

\mathcal{B}\leftarrow\mathcal{B}\cup\{(s_{t},a_{t},r_{t+1},s_{t+1})\}

9: Sample

N

transitions

\mathcal{T}=\left(s,a,r,s^{\prime}\right)_{n=1}^{N}\sim\mathcal{B}

10: UpdateCritics

\left(\mathcal{T},\beta_{m},\theta^{\prime},\phi_{1}^{\prime},\phi_{2}^{\prime}\right)

11: if

t\mod b

then

12: UpdateActor

(\mathcal{T},\beta_{m},\theta,\phi_{1},\phi_{2})

13: Update

\phi_{i}^{\prime}

\phi_{i}^{\prime}\leftarrow\tau\phi_{i}+(1-\tau)\phi_{i}^{\prime}

i\in\{1,2\}

14: Update

\theta^{\prime}

\theta^{\prime}\leftarrow\tau\theta+(1-\tau)\theta^{\prime}

15: end for

16: Update bandit

\mathbf{p}^{\beta}

weights using Equation 11

17: end for

5.4 The TOP framework

The general TOP framework can be applied to any off-policy actor-critic architecture. As an example, an integration of the procedure with TD3 (TOP-TD3) is shown in Algorithm 1, with key differences from TD3 highlighted in purple. Like TD3, we apply target networks, which use slow-varying averages of the current parameters, $\theta,\phi_{1},\phi_{2}$ , to provide stable updates for the critic functions. The target parameters $\theta^{\prime},\phi_{1}^{\prime},\phi_{2}^{\prime}$ are updated every $b$ time steps along with the policy. We use two critics, which has been shown to be sufficient for capturing epistemic uncertainty (Ciosek et al., 2019). However, it is likely that the ensemble would be more effective with more value estimates, as demonstrated in Osband et al. (2016).

6 Experiments

The key question we seek to address with our experiments is whether augmenting state-of-the-art off-policy actor-critic methods with TOP can increase their performance on challenging continuous-control benchmarks. We also test our assumption that the relative performance of optimistic and pessimistic strategies should vary across environments and across training regimes. We perform ablations to ascertain the relative contributions of different components of the framework to performance. Our code is available at https://github.com/tedmoskovitz/TOP.

State-based control To address our first question, we augmented TD3 (Fujimoto et al., 2018) with TOP (TOP-TD3) and evaluated its performance on seven state-based continuous-control tasks from the MuJoCo framework (Todorov et al., 2012) via OpenAI Gym (Brockman et al., 2016). As baselines, we also trained standard TD3 (Fujimoto et al., 2018), SAC (Haarnoja et al., 2018), OAC (Ciosek et al., 2019), as well as two ablations of TOP. The first, QR-TD3, is simply TD3 with distributional critics, and the second, non-distributional (ND) TOP-TD3, is our bandit framework applied to TD3 without distributional value estimation. TD3, SAC, and OAC use their default hyperparameter settings, with TOP and its ablations using the same settings as TD3. For tactical optimism, we set the possible $\beta$ values to be $\{-1,0\}$ , such that $\beta=-1$ corresponds to a pessimistic lower bound, and $\beta=0$ corresponds to simply using the average of the critic. It’s important to note that $\beta=0$ is an optimistic setting, as the mean is biased towards optimism. We also tested the effects of different settings for $\beta$ (Appendix, Figure 6). Hyperparameters were kept constant across all environments. Further details can be found in Appendix B. We trained all algorithms for one million time steps and repeated each experiment with ten random seeds. To determine statistical significance, we used a two-sided t-test.

Table 1: Average reward over ten trials on Mujoco tasks, trained for 1M time steps.

\pm

values denote one standard deviation across trials. Values within one standard deviation of the highest performance are listed in bold.

\star

indicates that gains over base TD3 are statistically significant (

p<0.05

Task TOP-TD3 ND TOP-TD3 QR-TD3 TD3 OAC SAC Humanoid $\mathbf{5899{\pm 142}}^{\star}$ 5445 5003 5386 5349 5315 HalfCheetah $\mathbf{13144\pm 701}^{\star}$ 12477 11170 9566 11723 10815 Hopper $\mathbf{3688\pm 33}^{\star}$ 3458 3392 3390 2896 2237 Walker2d $\mathbf{5111\pm 220}^{\star}$ 4832 4560 4412 4786 4984 Ant $\mathbf{6336\pm 181}^{\star}$ 6096 5642 4242 4761 3421 InvDoublePend $\mathbf{9337\pm 20}^{\star}$ $\mathbf{9330}$ 9299 8582 $\mathbf{9356}$ $\mathbf{9348}$ Reacher $\mathbf{-3.85\pm 0.96}$ $\mathbf{-3.91}$ $\mathbf{-3.95}$ $\mathbf{-4.22}$ $\mathbf{-4.15}$ $\mathbf{-4.14}$

Our results, displayed in Figure 3 and Table 1, demonstrate that TOP-TD3 is able to outperform or match baselines across all environments, with state-of-the-art performance in the 1M time step regime for the challenging Humanoid task. In addition, we see that TOP-TD3 matches the best optimistic and pessimistic performance for HalfCheetah and Hopper in Fig. 2. Without access to raw scores for all environments we cannot make strong claims of statistical significance. However, it is worth noting that the mean minus one standard deviation of TOP-RAD outperforms the mean performance all baselines in five out of the seven environments considered.

Table 2: Final average reward over ten trials on DMControl tasks for 100k and 500k time steps.

\pm

values denote one unit of std error across trials. Values within one standard deviation of the highest performance are listed in bold.

\star

indicates that gains over base RAD are statistically significant (

p<0.05

Task (100k) TOP-RAD ND TOP-RAD QR-RAD RAD DrQ PI-SAC CURL PlaNet Dreamer Cheetah, Run $\mathbf{512\pm 14}^{\star}$ 382 406 419 344 460 299 307 235 Finger, Spin $832\pm 93$ 769 682 729 901 $\mathbf{957}$ 767 560 341 Walker, Walk $541\pm 44^{\star}$ 413 436 391 612 514 403 221 277 Cartpole, Swing $734\pm 24^{\star}$ 540 610 632 759 816 582 563 326 Reacher, Easy $530\pm 50^{\star}$ 372 410 385 601 758 538 82 314 Cup, Catch $\mathbf{919\pm 17}^{\star}$ 850 777 488 913 933 769 710 246 Task (500k) TOP-RAD ND TOP-RAD QR-RAD RAD DrQ PI-SAC CURL PlaNet Dreamer Cheetah, Run $\mathbf{803\pm 11}^{\star}$ 564 650 548 660 801 518 568 570 Finger, Spin $\mathbf{917\pm 97}$ 883 758 $\mathbf{902}$ $\mathbf{938}$ $\mathbf{957^{*}}$ $\mathbf{926}$ 718 796 Walker, Walk $\mathbf{966\pm 13}^{\star}$ 822 831 771 921 946 902 478 897 Cartpole, Swing $\mathbf{886\pm 5}^{\star}$ 863 850 842 868 816 845 787 762 Reacher, Easy $\mathbf{957\pm 24}^{\star}$ 810 930 797 942 950 929 588 793 Cup, Catch $\mathbf{994\pm 0.6}^{\star}$ 994 991 963 963 933 959 939 879

Pixel-based control We next consider a suite of challenging pixel-based environments, to test the scalability of TOP to high-dimensional regimes. We introduce TOP-RAD, a new algorithm that dynamically switches between optimism and pessimism while using SAC with data augmentation (as in Laskin et al. (2020a)). We evaluate TOP-RAD on both the 100k and 500k benchmarks on six tasks from the DeepMind (DM) Control Suite (Tassa et al., 2018). In addition to the original RAD, we also report performance from DrQ (Yarats et al., 2021), PI-SAC (Lee et al., 2020), CURL (Laskin et al., 2020b), PlaNet (Hafner et al., 2019) and Dreamer (Hafner et al., 2020), representing state-of-the-art methods. All algorithms use their standard hyperparameter settings, with TOP using the same settings as in the state-based tasks, with no further tuning. We report results for both settings averaged over ten seeds (Table 2). We see that TOP-RAD sets a new state of the art in every task except one (Finger, Spin), and in that case there is still significant improvement compared to standard RAD. Note that this is a very simple method, requiring only the a few lines of change versus vanilla RAD—and yet the gains over the baseline method are sizeable.

Does the efficacy of optimism vary across environments? To provide insight into how TOP’s degree of optimism changes across tasks and over the course of learning, we plotted the average arm choice made by the bandit algorithm over time for each environment in Figure 4. Optimistic choices were given a value of 1 and and pessimistic selections were assigned 0. A mean of 0.5 indicates that $\beta=0$ (optimism) and $\beta=-1$ (pessimism) were equally likely.

From the plot, we can see that in some environments (e.g., Humanoid and Walker, Walk), TOP learned to be more optimistic over time, while in others (e.g., Hopper and Finger, Spin), the agent became more pessimistic. Importantly, these changes were not always monotonic. On Ant, for example, TOP becomes steadily more pessimistic until around halfway through training, at which point it switches and grows more optimistic over time. The key question, then, is whether this flexibility contributes to improved performance.

To investigate this, we compared TOP to two baselines, a “Pessimistic" version in which $\beta=-1$ for every episode, and an “Optimistic" version in which $\beta$ is fixed to 0. If TOP is able to accurately gauge the degree of optimism that’s effective for a given task, then it should match the best performing baseline in each task even if these vary. We tested this hypothesis in the HalfCheetah and Hopper environments, and obtained the results shown in Figure 5. We see TOP matches the Optimistic performance for HalfCheetah and the Pessimistic performance in Hopper. This aligns with Figure 4, where we see that TOP does indeed favor a more Optimistic strategy for HalfCheetah, with a more Pessimistic one for Hopper. This result can be seen as connected to the bandit regret guarantees referenced in Section 5.3, in which an adaptive algorithm is able to perform at least as well as the best fixed optimism choice in hindsight.

7 Conclusion

We demonstrated empirically that differing levels of optimism are useful across tasks and over the course of learning. As previous deep actor-critic algorithms rely on a fixed degree of optimism, we introduce TOP, which is able to dynamically adapt its value estimation strategy, accounting for both aleatoric and epistemic uncertainty to optimize performance. We then demonstrate that TOP is able to outperform state-of-the-art approaches on challenging continuous control tasks while appropriately modulating its degree of optimism.

One limitation of TOP is that the available settings for $\beta$ are pre-specified. It would be interesting to learn $\beta$ , either through a meta-learning or Bayesian framework. Nevertheless, we believe that the bandit framework provides a useful, simple-to-implement template for adaptive optimism that could be easily be applied to other settings in RL. Other future avenues could involve adapting other parameters online, such as regularization (Pacchiano et al., 2020c), using natural gradient methods (Moskovitz et al., 2021; Arbel et al., 2019), constructing the belief distribution from more than two critics, and learning a weighting over quantiles rather than simply taking the mean. This would induce a form of optimism and/or pessimism specifically with respect to aleatoric uncertainty and has connections to risk-sensitive RL, as described by Dabney et al. (2018a); Ma et al. (2019).

Acknowledgements

We’d like to thank Rishabh Agarwal and Maneesh Sahani for their useful comments and suggestions on earlier drafts. We’d also like to thank Tao Huang for pointing out an error in our training and evaluation setup for the DMC tasks.

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Appendix A Additional Experimental Results

The results for different settings of $\beta$ for TOP-TD3 on Hopper and HalfCheetah are presented in Figure 6.

Reward curves for TOP-RAD and RAD on pixel-based tasks from the DM Control Suite are shown in Figure 7.

Appendix B Further Experimental Details

All experiments were run on an internal cluster containing a mixture of GeForce GTX 1080, GeForce 2080, and Quadro P5000 GPUs. Each individual run was performed on a single GPU and lasted between 3 and 18 hours, depending on the task and GPU model. The Mujoco OpenAI Gym tasks licensing information is given at https://github.com/openai/gym/blob/master/LICENSE.md, and the DM control tasks are licensed under Apache License 2.0.

Our baseline implementations for TD3 and SAC are the same as those from Ball and Roberts [2021]. They can be found at https://github.com/fiorenza2/TD3_PyTorch and https://github.com/fiorenza2/SAC_PyTorch. We use the same base hyperparameters across all experiments, displayed in Table 3.

Hyperparameter	TOP	TD3	SAC
Collection Steps	1000	1000	1000
Random Action Steps	10000	10000	10000
Network Hidden Layers	256:256	256:256	256:256
Learning Rate	$3\times 10^{-4}$	$3\times 10^{-4}$	$3\times 10^{-4}$
Optimizer	Adam	Adam	Adam
Replay Buffer Size	$1\times 10^{6}$	$1\times 10^{6}$	$1\times 10^{6}$
Action Limit	$[-1,1]$	$[-1,1]$	$[-1,1]$
Exponential Moving Avg. Parameters	$5\times 10^{-3}$	$5\times 10^{-3}$	$5\times 10^{-3}$
(Critic Update:Environment Step) Ratio	1	1	1
(Policy Update:Environment Step) Ratio	2	2	1
Has Target Policy?	Yes	Yes	No
Expected Entropy Target	N/A	N/A	$-\mathrm{dim}(\mathcal{A})$
Policy Log-Variance Limits	N/A	N/A	$[-20,2]$
Target Policy $\sigma$	0.2	0.2	N/A
Target Policy Clip Range	$[-0.5,0.5]$	$[-0.5,0.5]$	N/A
Rollout Policy $\sigma$	$0.1$	$0.1$	N/A
Number of Quantiles	50	N/A	N/A
Huber parameter $\kappa$	1.0	N/A	N/A
Bandit Learning Rate	0.1	N/A	N/A
$\beta$ Options	$\{-1,0\}$	N/A	N/A

Table 3: Mujoco hyperparameters, used for all experiments.

Hyperparameter	Value
Augmentation	Crop - walker, walk; Translate - otherwise
Observation rendering	(100, 100)
Observation down/upsampling	(84, 84) (crop); (108, 108) (translate)
Replay buffer size	100000
Initial steps	1000
Stacked frames	3
Action repeat	2 finger, spin; walker, walk
	8 cartpole, swingup
	4 otherwise
Hidden units (MLP)	1024
Evaluation episodes	10
Optimizer	Adam
$(\beta_{1},\beta_{2})\to(f_{\theta},\pi_{\psi},Q_{\phi})$	$(0.9,0.999$
$(\beta_{1},\beta_{2})\to(\alpha)$	$(0.5,0.999$
Learning rate $(f_{\theta},\pi_{\psi},Q_{\phi})$	2e-4 cheetah, run
	1e-3 otherwise
Learning rate ( $\alpha$ )	1e-4
Batch size	128
$Q$ function EMA $\tau$	$0.01$
Critic target update freq	2
Convolutional layers	4
Number of filters	32
Nonlinearity	ReLu
Encoder EMA $\tau$	0.05
Latent dimension	50
Discount $\gamma$	0.99
Initial Temperature	0.1
Number of Quantiles	50
Huber parameter $\kappa$	1.0
Bandit Learning Rate	0.1
$\beta$ Options	$\{-1,0\}$

Table 4: DM Control hyperparameters for RAD and TOP-RAD; TOP-specific settings are in purple.

Appendix C Further Algorithm Details

The procedures for updating the critics and the actor for TOP-TD3 are described in detail in Algorithm 2 and Algorithm 3.

Algorithm 2 UpdateCritics

1: Input: Transitions

(s,a,r,s^{\prime})_{n=1}^{N}

, optimism parameter

\beta

, policy parameters

\theta

, critic parameters

\phi_{1}

and

\phi_{2}

2: Set smoothed target action (see Equation 3)

\tilde{a}=\pi_{\theta^{\prime}}(s^{\prime})+\epsilon,\quad\epsilon\sim\text{clip}(\mathcal{N}(0,s^{2}),-c,c)

3: Compute quantiles

\bar{q}^{(k)}(s^{\prime},\tilde{a})

and

\sigma^{(k)}(s^{\prime},\tilde{a})

using Equation 7.

4: Belief distribution:

\tilde{q}^{(k)}\leftarrow\bar{q}^{(k)}+\beta\sigma^{(k)}

5: Target

y^{(k)}\leftarrow r+\gamma\tilde{q}^{(k)}

6: Update critics using

\Delta\phi_{i}

from Equation 9.

Algorithm 3 UpdateActor

1: Input: Transitions

(s,a,r,s^{\prime})_{n=1}^{N}

, optimism parameter

\beta

, critic parameters

\phi_{1},\phi_{2}

, actor parameters

\theta

2: Compute quantiles

\bar{q}^{(k)}(s,a)

and

\sigma^{(k)}(s,a)

using Equation 7.

3: Belief distributions:

\tilde{q}^{(k)}\leftarrow\bar{q}^{(k)}+\beta\sigma^{(k)}

4: Compute values:

Q(s,a)\leftarrow K^{-1}\sum_{k=1}^{K}\tilde{q}^{(k)}

5: Update

\theta

\Delta\theta\propto N^{-1}\sum\nabla_{a}Q(s,a)\big{|}_{a=\pi_{\theta}(s)}\nabla_{\theta}\pi_{\theta}(s).

Appendix D Connection to Model Selection

In order to enable adaptation, we make use of an approach inspired by recent results in the model selection for contextual bandits literature. As opposed to the traditional setting of Multi-Armed Bandit problems, the ”arm” choices in the model selection setting are not stationary arms, but learning algorithms. The objective is to choose in an online manner, the best algorithm for the task at hand.The setting of model selection for contextual bandits is a much more challenging setting than selecting among rewards generated from a set of arms with fixed means. Algorithms such as CORRAL Agarwal et al. [2017], Pacchiano et al. [2020d] or regret balancing Pacchiano et al. [2020b] can be used to select among a collection of bandit algorithms designed to solve a particular bandit instance, while guaranteeing to incur a regret that scales with the best choice among them. Unfortunately, most of these techniques, perhaps as a result of their recent nature, have not been used in real deep learning systems and particularly not in deep RL.

While it may be impossible to show a precise theoretical result for our setting due to the function approximation regime we are working in, we do note that our approach is based on a framework that under the right settings can provide a meaningful regret bound. In figure 5 we show that our approach is able to adapt and compete against the best fixed optimistic choice in hindsight. These are precisely the types of guarantees that can be found in theoretical model selection works such as Agarwal et al. [2017], Pacchiano et al. [2020d, b]. What is more, beyond being able to compete against the best fixed choice, this flexibility may result in the algorithm outperforming any of these. In figure 5, Ant-v2 we show this to be the case.

Appendix E Proofs

Proof of Proposition 1.

Let $q_{Z^{\pi}}$ be the quantile function of $Z^{\pi}(s,a)$ knowing $\epsilon$ and $\sigma$ and $q_{\bar{Z}}$ be the quantile function of $\bar{Z}$ . Since $\epsilon$ and $\sigma$ are known, the quantile $q_{Z^{\pi}}$ is given by:

\displaystyle q_{Z^{\pi}}(u)=q_{\bar{Z}}(u)+\epsilon\sigma(s,a).

Therefore, recalling that $\epsilon$ has $0$ means and is independent from $\sigma$ , it follows that

\displaystyle q_{\bar{Z}}(u)=\mathbb{E}_{\epsilon}\left[q_{Z^{\pi}}(u)\right]

The second identity follows directly by definition of $Z^{\pi}(s,a)$ :

\displaystyle Z^{\pi}(s,a)=\bar{Z}(s,a)+\epsilon\sigma(s,a).

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