Understanding the Limits of Poisoning Attacks in Episodic Reinforcement Learning

1 Introduction

Learning algorithms have been widely used in web services [Zhao et al., 2018], conversational AI [Dhingra et al., 2016], UAV coordination [Venugopal et al., 2021], medical trials [Badanidiyuru et al., 2018], and crowdsourcing systems [Rangi and Franceschetti, 2018]. The distributed nature of these applications makes these algorithms prone to third party attacks. For example, in web services decision making critically depends on reward collection, and this is prone to attacks that can impact observations and monitoring, delay or temper rewards, produce link failures, and generally modify or delete information through hijacking of communication links [Agarwal et al., 2016; Cardenas et al., 2008]. Making these systems secure requires an understanding of the regime where the systems may be vulnerable, and designing ways to mitigate these attacks. This paper focuses on the former aspect, namely understanding of the regime where the systems can be attacked, in episodic Reinforcement Learning (RL).

1 Throughout the paper, ˜Θ(·) and ˜O(·) notation omits the logarithmic terms.

We consider a man in the middle (MITM) attack. In this attack, there are three entities: the environment, the learner (RL algorithm), and the adversary. The learner interacts with the environment for T episodes, and each episode has H steps. In episode t ≤ T at step h ≤ H, the learner observes the state st(h) ∈ S of the environment, selects an action at(h) ∈ A, the environment then generates a reward rt(st(h), at(h)) and changes its state based on an underlying Markov Decision Process (MDP), and attempts to communicate the new state to the learner. However, an adversary acts as a "man in the middle" between the learner and the environment. It can observe and may manipulate the action at(h) to aot (h) ∈ A which will generate reward rt(st(h), aot (h)) corresponding to the manipulated action. Additionally, the adversary may also intercept the reward rt(st(h), aot (h)) by adding contamination noise ϵt,h(st(h), at(h)). With both attacks, the learner ends up observing the contaminated reward ro t (st(h), at(h)) = rt(st(h), aot (h))+ϵt,h(st(h), at(h)). The cost of attack is measured as the amount of contamination ∑t,h |ϵt,h(st(h), at(h))| and number of action manipulations ∑t,h 1(at(h) ̸= aot (h)), respectively. Notably, with the wide application of RL today, MITM attack is a realistic concern to the vulnerability of RL algorithms and is thus important to understand. For instance, RL-based UAV coordination to reduce poaching activities in conservation areas is naturally subject to poachers' poisoning attacks, which can falsify the reward feedback (i.e., reward manipulation) and executed actions (i.e., action manipulation) [Venugopal et al., 2021]; similarly, RL algorithms for recommender systems are subject to attacks from hackers or competitors [Zhao et al., 2018].

Reward poisoning attack is a special case of the MITM attack where aot (h) = at(h), and has been widely studied in both RL and Multi-Armed Bandits (MAB) settings [Jun et al., 2018; Rakhsha et al., 2020; Rangi et al., 2021b]. Likewise, action manipulation attack is another special case of the MITM attack where ϵt,h(st(h), at(h)) = 0, and has been previously studied for MAB setting [Liu and Lai, 2020]. Another variant of action manipulation attack, recently studied in RL [Rakhsha et al., 2020], is manipulation of the transition dynamics. This can be considered as manipulating the action at(h) to another action, not necessarily in A. Therefore, the attacker has strictly stronger power there than the action manipulation as in our setting. MITM attacks has also been previously considered in cyber-physical systems [Rangi et al., 2021a; Khojasteh et al., 2020].

2Performance of algorithms in infinite horizon RL is studied by analyzing the performance over first T rounds [Wu et al., 2021].

3This result is not explicitly stated, and is derived for order-optimal algorithms from the reference.

We consider MITM attacks in two different settings: unbounded rewards and bounded rewards, which turns out to differ fundamentally. In unbounded reward setting, the contamination ϵt,h(st(h), at(h)) is unconstrained whereas in bounded reward setting, the contaminated reward ro t (st(h), aot (h)) is constrained to be in the interval [0, 1], just like the original rewards rt(st(h), aot (h)). This constrained situation limits the attacker's contamination at every round, and turns out to be provably more difficult to attack. In each setting, we study attack strategy in more realistic black-box setting in which the attacker does not know, and needs to learn, the underlying MDP as well. A similar study between the two settings has been performed for MAB in [Rangi et al., 2021b], where it is shown that bounded rewards setting is more difficult to attack in comparison to unbounded reward setting. This paper extends the study from MAB to RL, and provides new insights on the feasibility of attacks.

1.1 Summary of Contributions

We consider poisoning attacks with the objective of forcing the learner to execute a target policy π+. More specifically, for all h ≤ H and s ∈ S, if πh(s) ̸= π+ h (s), then the attack aims to induce values satisfying

where a) policy π of an agent is a collection of H functions {πh : S → A}; and b) value function ˜ V π h (s) is the expected reward of state s under policy π using contaminated reward observation, between step h until step H.

We propose the first set of efficient black-box online attacks to any order-optimal episodic RL algorithms in both bounded and unbounded reward settings. Specifically, for the bounded reward setting, we show that mere reward manipulation does not guarantee successful attacks; namely there exist an MDP and a target policy π+ which cannot be attacked - i.e., (1) cannot be achieved - by manipulating only the rewards. Similarly, we show that action manipulation attack does not suffice by showing the existence of an MDP and a target policy π+ which cannot be attacked by manipulating actions. Hence, to guarantee a successful attack, the attacker needs a combined the power of reward and action manipulation. Indeed, we propose an MITM attack in bounded reward setting, which requires ˜O(√T) amount of reward contamination and ˜O(√T) number of action manipulations to attack any order-optimal learning algorithm. We also show that this attack cost, namely sum of amount of reward contamination and number of action manipulations, is order-optimal.

Next we move to unbounded reward settings. Reward manipulation attack for unbounded rewards has been studied in [Rakhsha et al., 2020]. However, the work investigated infinite-horizon RL and proposed a white-box attack. Extending this research agenda, we propose black-box attack for episodic RL, and show that our proposed attack can attack any order-optimal learning algorithm in ˜O(√T) amount of contamination. An interesting conceptual message from our results is that bounded reward setting is more difficult to attack than the unbounded reward setting. Our results are compared with the relevant literature in Table 1.

2 ProblemFormulation

We consider the episodic Markov Decision Process (MDP), denoted by (S,A,H, P, µ), where S is the set of states with |S| = S, A is the set of actions with |A| = A, H is the number of steps in each episode, P is the transition metric such that P(.|s, a) gives the transition distribution over the next state if action a is taken in the current state s, and µ : S ×A → R is the expected reward of state action pair (s, a). To avoid cumbersome notations, we work with the stationary MDPs here with the same reward and transition functions at each h ≤ H. However all our analysis extend trivially to non-stationary MDPs, with just more involved notations. 4

An RL agent (or learner) interacts with the MDP for T episodes, and each episode consists of H steps. In each episode of the MDP, an initial state st(1) can be fixed or selected from any distribution. In episode t and step h, the learner observes the current state st(h) ∈ S, selects an action at(h) ∈ A, and incurs a noisy reward rt,h(st(h), at(h)). Also, we have E[rt,h(st(h), at(h))] = µ(st(h), at(h)). Our results can also be extended to the setting where reward is a function of step h ≤ H. Finally, both P and µ are unknown to the learner and the attacker.

4 Moreover, if an online learning is already vulnerable for this "nice" special case, then it must be vulnerable in general as well since the stationary MDP is a special case of general MDPs.

We consider episodic RL under MITM attacks. The attacker can manipulate the action at(h) selected by the learner to another action aot (h) ∈ A. The MDP thus undergoes transition to next state based on the action aot (h), namely the next state is drawn from the distribution P(.|st(h), aot (h)). The reward observation rt(st(h), aot (h)) is generated. If aot (h) ̸= at(h), then the episode t and step h is said to be under action manipulation attack. Hence, the number of action manipulations is ∑Tt=1 ∑Hh=1 1(aot (h) ̸= at(h)).

The adversary can also intercept the realized reward rt(st(h), aot (h)) and contaminate it by adding noise ϵt,h(st(h), at(h)). Learner thus observes reward

where the contamination ϵt,h(st(h), at(h)) added by the attacker is a function of the entire history, including all the states visited previously and all the actions selected previously by the learner and the attacker. If ϵt,h(st(h), at(h)) ̸= 0, then the episode t and step h is said to be under reward manipulation attack. Hence, the number of reward manipulations is ∑Tt=1 ∑Hh=1 1(ϵt,h(st(h), at(h)) ̸= 0), and the amount of contamination is ∑Tt=1 ∑Hh=1 |ϵt,h(st(h), at(h))|. Notably, reward manipulation attack is a special case of MITM attack, where the adversary cannot manipulate the action (i.e., at(h) = aot (h)). Action manipulation attack is also special case of MITM, in which the adversary cannot contaminate the reward observation (i.e., ϵt,h(st(h), at(h)) = 0).

A (deterministic) policy π of an agent is a collection of H functions {πh : S → A}. The value function V π h (s) is the expected reward under policy π, starting from state s at step h, until the end of the episode, namely

where sh′ denotes the state at step h′ . Likewise, the Qvalue function Qπh (s, a) is the expected reward under policy π, starting from state s and action a, until the end of the episode, namely

where ah′ denotes the action at step h′ . Since S, A and H are finite, there exists an optimal policy π∗ such that V π∗ h (s) = supπ V π h (s). The regret RA(T,H) of any algorithmA is the difference between the total expected true reward from the best fixed policy π∗ in the hindsight, and the expected true reward over T episodes, namely

The objective of the learner is to minimize the regret RA(T,H). In contrast, the objective of the attacker is to poison the environment to teach/force the learner to execute a target policy π+ by achieving the following objective: for all h ≤ H, s ∈ S and any policy π,

where ˜ V π h (s) is the expected reward in states based on the reward observation in (2) under policy π. Consequently, the policy π+ will be executed for Ω(T) times under the attack.

3 Attacks in Bounded Reward Setting

In this section, we investigate bounded reward setting, i.e., rt,h(s, a) ∈ [0, 1] with mean µ(s, a) ∈ (0, 1] for all (s, a) ∈ S×A. 5 We first show in subsection 3.1 that there exist MDPs and target policies π+ such that the objective of the attacker, namely (6), cannot be achieved by only reward manipulation attack or only action manipulation attack. In subsection 3.2, we show that combined reward and action manipulation suffice for a successful attack.

3.1 Insufficiency of (Only) Reward or Action Manipulation

Similar to [Rakhsha et al., 2020], attacker is subject to a constraint that the reward manipulation (i.e., ϵt,h(st(h), at(h)) ̸= 0) and action manipulation (i.e., at(h) ̸= aot (h)) can occur only if the selected action is different from the desired action, namely at(h) ̸= π+ h (st(h)).Intuitively, it can also be interpreted as the attacker can manipulate rewards (or actions) whenever the learner's action is not in accordance with the attacker's desired policy π+. Therefore, in the reward manipulation attack, the attacker is subject to following constraints

This constraint is crucial for obtaining sub-linear attack cost due to the following reason. Suppose the objective in (6) is achieved by contaminating the reward of action π+ h (st(h)). Now since the learner would execute the policy π+ with high probability, namely Ω(T) times, the total contamination will thus grow linearly with T . This constraint (or strategy of not contamination π+ h (st(h))) is also applied in the previous literature in RL [Rakhsha et al., 2020] and MAB [Jun et al., 2018; Ma et al., 2019b; Rangi et al., 2021b], where the reward manipulation are performed only on non-desirable actions. Similarly, in the action manipulation attack, we assume that

That is, the action can be manipulated only if the selected action is not the same as the target action of the policy.

Our first result establishes that only the reward manipulation or only the action manipulation cannot always guarantee successful attacks in bounded reward setting.

5 The µ(s, a) ̸= 0 is to avoid minor technicality issues due to tie breaking.

Theorem 1. In bounded reward setting,

1. there exists an MDP and a target policy π+ such that any reward manipulation attack satisfying (7) cannot be successful, namely achieve the objective in (6).
2. there exists an MDP and a target policy π+ such that any action manipulation attack satisfying (9) cannot be successful, namely achieve the objective in (6).

The proof of Theorem 1 proceeds by constructing a carefully crafted MDP such that there exists a target policy π+ and another policy ˜π such that V ˜π h (s) > V π+ h (s) irrespective of action or reward manipulation. This implies that the constructed target policy π+ can not be induced in the constructed MDP. 6

3.2 Efficient Attack by Combining Reward & Action Manipulation

We now show that the attacker can achieve its objective by combining the strength of both reward manipulation and action manipulation attacks in bounded reward setting. This is the first attack strategy of its kind, using both reward and action manipulation, proposed in the literature.

Given the target policy π+, for all st(h) ∈ S, at(h) ∈ A and h ≤ H, we consider the following attack strategy

In the above attack, the adversary manipulates both the action and the reward observation when at(h) ̸= π+ h (s). Specifically, the adversary manipulates the action to π+ h (s) to control the transition dynamics, and at the same time manipulates the reward observation to zero so that the action at(h) appears to be sub-optimal in comparison to the action π+ h (s).In this attack, we carefully "coordinate" action and reward manipulation to achieve two goals simultaneously: (1) the target policy is optimal; (2) the target policy is the greedy solution of the resultant MDP, namely the action yielding maximum reward at current step h is also the action yielding maximum cumulative reward between h and H.

The following theorem shows that novel attack in (10) and (11) can achieve the objective in (6), and does not require learning the parameters of the MDP. Additionally, the attack cost is ˜O(√T) for any order-optimal learning algorithm.

Theorem 2. Consider any learning algorithmA such that its regret in the absence of attack is

with probability at least 1 - δ, where α ≥ 1 is a numerical constant. For any sub-optimal target policy π+, if an adversary follows the strategy in (10) and (11) to attack the algorithmA, then with probability at least 1-δ the following hold simultaneously:

6 We note that such worst-case analysis is typical for establishing lower bounds on regret, which usually identifies a difficult instance and shows the impossibility of a good regret (see, e.g., [Bogunovic et al., 2021]).

1. The attacker achieves its objective in (6) and moreover ∑Tt=1 ∑Hh=1 1(at(h) = π+ h (st(h))) = Ω(T );
2. The number of reward manipulation attacks, namely ∑Tt=1 ∑Hh=1 1(ϵt,h(st(h), at(h)) ̸= 0) is ˜O(√THα/ minh,s µ(s, π+ h (s)))
3. The amount of reward contamination, namely ∑Tt=1 ∑Hh=1 |ϵt,h(st(h), at(h))|, is ˜O(√THα/ minh,s µ(s, π+ h (s)))
4. The number of action manipulation attacks, namely ∑Tt=1 ∑Hh=1 1(aot (h) ̸= at(h)), is ˜O(√THα/minh,s µ(s, π+ h (s))).

In [Chen et al., 2021], if the attacker can observe learner's action (as in our setting), BARBAR-RL has a regret ˜O(√T + C 2 ) where C is the total contamination in rewards and transition dynamics. Since action and reward manipulation is a special case of transition dynamics and reward manipulation, the bound of BARBAR-RL will also hold in our setting. If attacker is absent (C = 0), then the regret of the algorithm is ˜O(√T), and (12) holds. If C = o(√T), then the regret of BARBAR-RL is o(T ), and claim 1 in Theorem 2 does not hold. Thus, there exists an order-optimal algorithm such that attacker's objective is not achieved in o(√T) attack cost. Hence, the attack cost of proposed strategy is order-optimal.

The order-optimal algorithm, in (12), has three parameters α, t0 and δ. The parameter α is used to capture the dependence of regret on H; parameter t0 captures the fact that the regret bound holds for sufficiently large T; parameter δ captures the fact that the regret bound holds with high probability. Some examples of order-optimal algorithm in episodic RL can be found in [Jin, 2018] and references therein. We present results on attack cost for order-optimal algorithms, however this is not a limitation of our attack strategy. The analysis can be extended for any no-regret learning algorithm and a corresponding bound on attack cost can be obtained.

Our result directly focus on the more realistic and also harder black-box attacks; for completeness, we also discuss how to design a white-box attack by manipulating rewards and actions in Appendix B. This can also be used for designing a black-box attack in unbounded reward setting.

4 Reward Poisoning in Unbounded Settings

In this section, we investigate the unbounded reward setting. Formally, for all (s, a) ∈ S ×A, we assume rt,h(s, a) follows a sub-Gaussian distribution with mean µ(s, a) and standard deviation σ. While the rewards, drawn from sub-Gaussians, may be unbounded, we assume their mean µ(s, a) ∈ [-M,M] is bounded within some known interval for any (s, a) pair. Our main result in this section is the design of a black-box attack to any order-optimal episodic RL algorithm with ˜O(√T) amount of reward contamination. To our knowledge, this is the first efficient attack to episodic RL algorithms in unbounded reward settings.

Formally, we consider the black-box attack setting in which the attacker doesn't know about the expected reward and the transition dynamics of the underlying MDP. In this setting, we propose an attack which learns about the MDP, and has almost the same attack cost as the white-box attack, with an additional O(√log T ) factor.

Given the input parameters π+ and ϵ > 0, our proposed attack strategy is presented in Algorithm 1. In Algorithm 1, the attacker utilizes its estimate of ˆµ(s, a), ˆµUCB (s, a) and ˆµLCB (s, a), defined in (13), (14) and (15) respectively, to contaminate the reward observations. These estimates are initialized using the fact that µ(s, a) ∈ [-M,M], and are updated in each episode t ≤ T and at each step h ≤ H. The parameters ˆµUCB (s, a) and ˆµLCB (s, a) are Upper Confidence Bound (UCB) and Lower Confidence Bound (LCB) of µ(s, a). Therefore, we show that, with high probability,

In this attack, the reward observations are contaminated only if the action selected by the learner is not the same as the ac-

Algorithm 1 Black box attack strategy

• 1: Initialization: Input parameters are ϵ > 0 and policy π+. For all (s, a, h) ∈ S ×A× [H], attacker initializes ˆµ(s, a) = 0, ˆµUCB (s, a) = M, ˆµLCB (s, a) = -M and N(s, a) = 0.
• 2: for episode t ≤ T do
• 3: Observe the initial state st(1)
• 4: for h ≤H do
• 5: Observe the selected action at(h), reward r(st(h), at(h)) and next state st(h + 1).
• 6: Update

ˆµUCB (st(h), at(h))

7:

• 8: Do not contaminate.
• 9: else
• 10: Contaminate the reward observation such that

• 11: end if
• 12: end for
• 13: end for

tion desired by the target policy, namely at(h) ̸= π+ h (st(h)). In this scenario, the reward observation ro t (st(h), at(h)) is defined in (16). In (16), LCB estimates of the expected rewards are used to get the LCB estimate of value function of target policy and UCB estimates of the expected rewards are used to get UCB value function over all policy. This ensures the target policy is optimal, and reward observation consist of a large negative bias sufficient to achieve the objective. This reward manipulation strategy in (16) ensures that the target policy π+ is the optimal policy based on the observed reward observations, namely for all h ≤ H, and (s, a) ∈ S ×A such that a ̸= π+ h (s), we have

where ˜Qπh (s, a) is the expected reward in state s for action a for the reward observation under policy π. These values will not be same as the ones defined in (3) and (4) since the reward observations are manipulated. We remark that the ro t (st(h), at(h)) can be computed through a backward induction procedure starting from horizon H. At any step h in the episode, the definition of ro t (st(h), at(h)) depends linearly on the Q-values at h, which then depends linearly on ro t (st(h), at(h)). Therefore, ro t (st(h), at(h)) at any horizon h can be computed by solving a linear system.

We briefly discuss the key steps in this process of obtaining (18). We show that with high probability

Additionally, we have that with high probability

Combining (19) and (20), we have that with high probability, the rewards contamination in Algorithm 1 ensures (17).

The following theorem shows that our proposed black box attack has ˜O(√T) amount of contamination.

Theorem 3. Consider any learning algorithm A such that its regret in the absence of attack is RA(T,H) = ˜O(√THα), ∀T ≥ t0

with probability at least 1 - δ where α ≥ 1 is a numerical constant. For any sub-optimal target policy π+, ϵ > 0 and T ≥ t20 , if an attacker follows strategy in Algorithm 1, then with probability at least 1 - δ - 2/(HSAT) the following hold simultaneously:

1. The attacker achieves its objective in (6) and moreover ∑Tt=1 ∑Hh=1 1(at(h) = π+ h (st(h))) = Ω(T );
2. The number of reward manipulations is ˜O(√THα/ϵ).
3. The total amount of reward contamination is ˜O(√T Hα+1 (ϵ + √log(HT SA))/ϵ).

Additionally, the proposed black-box attack has an additional cost O(√log T) in comparison to the white-box attack in unbounded reward setting.

Remark 1. In unbounded setting, reward poisoning attack has been studied in black-box setting recently in RL for infinite horizon by [Rakhsha et al., 2021]. They consider attacking L online learners whereas we only has one learner. The attack objective of [Rakhsha et al., 2021] is to force all these learning algorithms to execute the target policy π+. We now highlight the key differences between our attack strategy and the strategy of [Rakhsha et al., 2021]. The attack cost in [Rakhsha et al., 2021] is ˜O(T logL), which is linear in T. On contrary, the attack cost of our proposed attack is ˜O(√T ). This difference occurs because the attack in [Rakhsha et al., 2021] estimates both the expected reward and the transition dynamics of the MDP. This is done by an explore-then-exploit form of strategy which leads to ˜O(T logL) attack cost. On contrary, our strategy focuses on estimating expected rewards only (and not transition dynamics). It compensates for this lack of knowledge of transition dynamics by adding a negative bias O(√log T ) to the reward observation. However, this additional cost is minimal in comparison to the cost of learning transition dynamics in [Rakhsha et al., 2021]. This key technical insight allows us to reduce the attack cost of O(T) in [Rakhsha et al., 2021] to ˜O(√T) in current work.

5 Additional RelatedWork

Reward manipulation attack has been studied extensively in MAB [Jun et al., 2018; Liu and Shroff, 2019; Rangi et al., 2021b], where the attacker's objective is to mislead the learner to choose a suboptimal action. Action manipulation attack has also been studied in MAB [Liu and Lai, 2020], and the number of action manipulations required by the attacker is O(log T ). All these attacks are studied in a Black-box setting. In this work, we show that unlike MAB setting, reward manipulation (only) and action manipulation (only) are not sufficient to successfully attack Episodic RL setting with bounded rewards. [Bogunovic et al., 2021] considered the "possibility" of reward poisoning attack in Linear Bandits, as a step towards showing lower bound for designing robust algorithms. While their results can be extended to prove a similar regret lower bound for RL, this only means that there exists an RL instance such that the attacker can successfully attack any noregret algorithm for this particular instance. However, this existence result is significantly different from our perspective of designing attacks, in which we design strategies that can successfully attack an arbitrary episodic RL instance.

In online RL setting, studies related to poisoning attacks have only started recently, and have primarily focused on white-box settings, where the attacker has complete knowledge of the underlying MDP models, with unbounded rewards [Rakhsha et al., 2020]. In such white-box attacks, [Rakhsha et al., 2020] show that reward poisoning attack requires ˜Θ(√T) amount of contamination to attack any orderoptimal learning algorithm; they also show that dynamic manipulation attack can achieve the same success with similar amount of cost in unbounded reward setting under some sufficient conditions. [Zhang et al., 2020] study the feasibility of the reward poisoning attack in white box setting for Q-learning, and the attacker is constrained by the amount of contamination. In a slightly different thread, [Huang and Zhu, 2020] analyse the degradation of the performance of Temporal difference learning and Q-learning under falsified rewards.

To our knowledge, [Rakhsha et al., 2021] is the only work that studies poisoning attack in black-box setting for policy teaching in infinite-horizon RL. However, they focused on the settings with L online learners, and the objective of their attacker is to force all these learners to execute a target policy π+. They proposed an attack with ˜O(T log L+L√T ) amount of contamination when L is large enough. However, our work focuses on attacking a single learner and thus our setting is not comparable to [Rakhsha et al., 2021]. However, we can indeed apply our attack repeatedly to different learners to obtain an effective attack strategy for the setup of [Rakhsha et al., 2021], which leads to an attack cost of ˜O(L√T) (note however, the attack of [Rakhsha et al., 2021] cannot work for small L, e.g., L = 1 as in our setup) in episodic RL. This improves their attack cost by an additive amount O(T log L). Our more efficient attack is due to a more efficient design for the adversary to explore and learn the MDP, which is discussed at length in Remark 1 in Section 4 .

Test-time adversarial attacks against RL has also been studied. Here, however, the policy π of the RL agent is pretrained and fixed, and the objective of the attacker is to manipulate the perceived state of the RL agent in order to induce undesired action [Huang et al., 2017; Lin et al., 2017; Kos and Song, 2017; Behzadan and Munir, 2017]. Such test-time attacks do not modify the the policy π, whereas training-time attacks we study in this paper aims at poisoning the learned policy directly and thus may have a longer-term bad effects. There have also been studies on reward poisoning against Batch RL [Ma et al., 2019a; Zhang et al., 2009] where the attacker can modify the pre-collected batch data set at once. The focus of the present work is on online attack where the poisoning is done on the fly.

6 Conclusion and Future Directions

This paper tries to understand poisoning attacks in RL. Towards that end, we propose a reward manipulation attack for unbounded reward setting which successfully fool any orderoptimal RL algorithm to pull a target policy with ˜O(√T) attack cost. Extending the study to bounded reward setting, we show that the adversary cannot achieve its objective using either reward manipulation or action manipulation attack even in white-box setting, where the information about the MDP is assumed to be known. Hence, to contaminate a order-optimal RL algorithm, the adversary needs to combine the power of reward manipulation and action manipulation. Indeed, we show that an attack that uses both reward manipulation and action manipulation can achieve adversary's objective with ˜Θ(√T) attack cost, which is order-optimal. We also studied the in-feasibility of the attack under the constraint that the adversary can attack only if at(h) ̸= π+ h (s).