Reward Tampering Problems and Solutions in Reinforcement Learning: A Causal Influence Diagram Perspective

TL;DR

This paper investigates reward tampering in reinforcement learning, using causal influence diagrams to formalize and propose design principles that prevent agents from manipulating their reward signals.

Contribution

It introduces a causal influence diagram framework to analyze reward tampering and offers design principles to prevent instrumental goals related to reward manipulation in RL agents.

Findings

Causal influence diagrams effectively formalize reward tampering issues.

Design principles can prevent reward function and input tampering as instrumental goals.

The approach enhances safety in scalable reinforcement learning systems.

Abstract

Can humans get arbitrarily capable reinforcement learning (RL) agents to do their bidding? Or will sufficiently capable RL agents always find ways to bypass their intended objectives by shortcutting their reward signal? This question impacts how far RL can be scaled, and whether alternative paradigms must be developed in order to build safe artificial general intelligence. In this paper, we study when an RL agent has an instrumental goal to tamper with its reward process, and describe design principles that prevent instrumental goals for two different types of reward tampering (reward function tampering and RF-input tampering). Combined, the design principles can prevent both types of reward tampering from being instrumental goals. The analysis benefits from causal influence diagrams to provide intuitive yet precise formalizations.

Equations

$$R_t=\#\text{diamonds in goal area}-\#\text{rocks in goal area}.$$

$$R_t={\Theta }_{\text{diamond},t}^{\mathrm{R}}\cdot (\#\text{diamonds in goal area})+{\Theta }_{\text{rock},t}^{\mathrm{R}}\cdot (\#\text{rocks in goal area}).$$

Full text

1. 1 Introduction
2. 2 Foundations
3. 3 Reward Function Tampering
4. 4 RF-Input Tampering
5. 5 Conclusions
6. A List of Notation
7. B Combined Model
8. C Pseudo-code for Algorithms

Thanks to Laurent Orseau, Jonathan Uesato, Ryan Carey, Michael Cohen, Eric Langlois, Toby Ord, Pedro Ortega, Stuart Armstrong, Beth Barnes, Tom Erez, Bill Hibbard, Jan Leike, and many others for helpful discussions and suggestions.

1 Introduction

A central problem in AI safety is how to get a generally capable, artificially intelligent system to perform an intended task, such as driving a car to an intended location, or serving useful content on a social media platform. In AI research, such tasks are often formulated as reinforcement learning (RL) problems, where an agent takes actions to optimize its cumulative observed reward (Sutton2018). The problem of getting the intended task done is thus split into designing an RL agent111In our terminology, any agent that optimizes a (cumulative) reward signal is an RL agent. that is good at optimizing reward, and constructing a reward process that provides the agent with suitable rewards. In practice, the reward process typically includes an implemented reward function, and a mechanism for collecting appropriate sensory data as input to it. It may also include a way for the user to update the reward function.

Unfortunately, the reward process may fail to incentivize the agent to do the intended task. Indeed, our concern in this paper is that the agent may tamper with the reward process, thereby weakening or breaking the relationship between its observed reward and the intended task. Concerningly, RL agents will often have an instrumental goal (Omohundro2008aidrives; Bostrom2014) to tamper with their reward process, as this can increase the observed reward. Current RL agents mostly lack the capability for serious tampering, though its been hypothesized that social media algorithms influence their users’ emotional state to generate more ‘likes’ (Russell2019socialmedia). If true and we assume that their intended task is to serve useful content, this is one instance where present-day algorithms already tamper with their reward process. More worryingly, as the capability of RL agents increases through computational and algorithmic advances,222Such capability increases may include better ability to make plans and counterfactual predictions in novel and complex environments. we may expect reward tampering problems to become increasingly common.

Key contributions and outline

This paper describes design principles for RL agents for which reward tampering is not an instrumental goal. This means that from a reward tampering perspective, the design principles are robust to arbitrary increases in agent capability. In establishing the design principles, we develop a unified causal framework for reward tampering in which we model the two subproblems shown in Figure 1, along with a number of solutions. These main results are presented in Sections 3 and 4 after some background material in Section 2. Conclusions follow in Section 5. A list of notation, a full set of equations, and pseudo-code for our different agents are provided in Appendices A, B and C.

Related work

In addition to inspired accounts of the risks (Bostrom2014; Yudkowsky2008), the AI safety literature also contains a number of good ideas for addressing them. Orseau2016 develop techniques for making agents indifferent to interruption. Hibbard2012 suggests a creative way of avoiding the delusion box problem (Ring2011), here referred to as the RF-input tampering problem. A method for preventing agents tampering with their reward function has been discussed by Schmidhuber2007; Dewey2011; Orseau2011; Everitt2016sm, explored here under the name current-RF optimization. Ways to make an agent learn the right reward function have been proposed by Hadfield-Menell2016cirl; Armstrong2020pitfalls; Armstrong2017indiff; Uesato2020decoupled; Reddy2020; Leike2018alignment and others. Many of these methods rely on amplification (Christiano2018) and/or feedback on hypotheticals, sometimes called decoupled feedback (Everitt2017rc). Corrupt-reward MDPs extend MDPs with the possibility of reward tampering and misspecification (Everitt2017rc), and serve as the basis of the REALab framework for evaluating tampering problems experimentally (Kumar2020REALab).

However, it has not always been clear exactly what safety property the different methods provide, and under what assumptions. For example, Orseau2016 call an agent safely interruptible if it acts optimally in a modified environment without interruption, but do not spell out how this affects agent incentives. Similarly, Hibbard2012 only states how model-based utility functions solve the delusion box problem in specific cases. Here, we establish which instrumental goals are induced or avoided by each design principle, and show how the different ideas can fit together to mitigate reward tampering problems. Our analysis benefits from causal influence diagrams, which make causal assumptions clear, and permits a number of instrumental goals to be identified or ruled out directly from a diagram (Everitt2021agent).

Reward tampering is related to the problems of reward hacking (Amodei2016), reward corruption (Everitt2017rc), and specification gaming (Krakovna2020specification). These all consider the effects of the agent obtaining unintended reward for any reason. In contrast, reward tampering focuses on inappropriate agent influence on the reward process itself, and excludes so-called ‘gaming’ of a reward function. Similar problems have also been referred to as wireheading (e.g. Bostrom2014; Yampolskiy2015). Reward tampering also intersects with corrigibility (Soares2015cor), as preventing updates to the reward function is one form of reward tampering.

Looking more broadly at the AI safety literature, Gabriel2020 argues that generally capable AI systems should ultimately be aligned to some moral principles, rather than optimized for a particular task. We agree, but focus on a single intended task for simplicity. A philosophical perspective on the problem of learning values is offered by Petersen2021, while the concrete approach of reward modeling is proposed by Leike2018alignment. Here, our focus is complementary: how do we avoid having the agent tamper with a well-designed reward modeling algorithm? Hubinger2019 consider the case where a learned model is itself an optimizer, and decompose the safety problem into outer alignment of the model’s training process and inner alignment of the learned model. In their terminology, our focus is solely on outer alignment. Demski2019 summarize various issues arising from embedded agency, when the agent is part of the environment it is interacting with. As the reward process is often considered part of the agent, reward tampering can be viewed as one such issue. Everitt2018litrev provide further references.

2 Foundations

As a first step, we cover some background on Markov decision processes (MDPs) and causal influence diagrams, which will form the basis of our analysis.

2.1 The MDP Framework

To model planning over multiple time steps, we will use the standard RL framework of MDPs (Sutton2018). In an MDP, an agent takes actions $A_{1},\dots,A_{m-1}$ in order to influence environment states $S_{1},\dots,S_{m}$ according to a state-transition function $T(S_{t+1}=s^{\prime}\mid S_{t}=s,A_{t}=a)$. A reward $R_{t}$ is dispensed in each state according to some reward function. A standard RL agent optimizes the expected sum of the rewards received at every time step. The following gridworld is an example of an MDP, and will be our running example throughout the paper:

Example 1** (Rocks and diamonds running example).**

In the gridworld displayed in Figure 2, the agent can push rocks and diamonds by walking towards them from an adjacent cell. The agent is rewarded for bringing diamonds but not rocks to a goal area: at time $t$, the reward is

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Implicit assumptions

What assumptions are made by modeling an agent’s interaction with the world as an MDP? First, the world is assumed to have time steps, and a well-defined state and action at each time step. The agent should be able to ‘freely select’ the actions, in order to optimize its rewards. The next state should only depend on the current state and action (the Markov property), and the state-transition probabilities should not depend on $t$ (stationarity).

While non-trivial, these assumptions roughly correspond to our intuitive understanding of agents and our universe. There are plenty of examples of agent-like systems that are essentially free to choose actions towards their objectives (humans, animals, robots, artificial agents, …). While the world may have continuous time, discretizing it into sufficiently fine-grained time steps should make little difference. How we formalise the environment state depends on how the agent will be deployed. For a robot vacuum cleaner it may be the position of dirt and blocking objects in the house. For an agent interacting with the wider world, it may be useful to consider the MDP state to be the state of the entire universe as conceived of in physics. The laws of physics are usually assumed uniform over time, so the state-transition function is stationary.

To avoid measure-theoretic subtleties, we assume finite sets of states, actions, and rewards, and finite episode length $m$. As these can all be chosen very large, this is not particularly restrictive, and our arguments never strongly depend on these assumptions.

What about the rewards

The MDP framework assumes that the designer can assign a reward to each state so that maximization of received rewards corresponds to task completion. This paper will question that assumption. In particular, we distinguish between intended rewards that encourage completion of the intended task, and observed rewards, which are the rewards received by the agent; that is, the output of the reward process and input to the agent. In contrast to standard MDPs, we will therefore often consider multiple different reward functions. To facilitate this, we let $R$ denote a reward functional, parameterized by different reward parameters $\Theta^{\mathrm{R}}$, and returning reward $R_{t}=R(S_{t};\Theta^{\mathrm{R}})$ in state $S_{t}$. For example, $\Theta^{\mathrm{R}}_{t}$ will denote the parameter for an implemented reward function at time $t$, and $\Theta^{\mathrm{R}}_{*}$ the parameter of an intended reward function. The reward functional $R$ will always be fixed from the context, letting us refer to the reward function $R(\,\cdot\,;\Theta^{\mathrm{R}})$ by just $\Theta^{\mathrm{R}}$.

Online or offline

Reward tampering can occur when the actions optimizing the rewards are taken in the environment where the rewards are computed (e.g. the real world). This is most clearly evident in online RL, where the agent learns the environmental dynamics and rewards during its deployment. This is the setting that we model, using the MDP framework. Other offline training schemes where agents gather data and learn in separate phases are also commonly used in practice (Levine2020). We expect many of our results to carry over in some form to offline training, but leave the details for further work.

Notational convention

Throughout, we will use $t,t^{\prime},\dots$ to denote time steps, and $k,k^{\prime},\dots$ to denote different optimization objectives. These indices will always be universally quantified, unless otherwise mentioned.

2.2 Causal Influence Diagrams

Causal influence diagrams are a novel graphical technique for analyzing agent incentives (Everitt2021agent), that combine causal graphs (Pearl2009) and influence diagrams (Howard1984; Lauritzen2001; Koller2003). Causal influence diagrams consist of a directed acyclic graph over a finite set of nodes containing random variables, see Figure 3. The nodes can be of three different types: agent decisions are represented with square decision nodes , the agent’s optimization objective is represented with diamond utility nodes , while other aspects are represented with round chance nodes . The nodes are connected with arrows. Arrows going into chance and utility nodes represent causal influence, and are drawn solid. Arrows going into decision nodes are called information links, and instead specify what information is available at the time that the decision is made. To signify the difference, information links are drawn with dotted arrows.

The diagram itself only gives the causal structure of a decision-making problem, i.e. which random variables may be causally related to each other. Conditional probability distributions $P(X=x\mid\textbf{Pa}_{X}=\textbf{pa}_{X})$ specify the relationship between a node $X$ and its parents $\textbf{Pa}_{X}$. The agent chooses conditional probability distributions for the decision nodes in the form of a policy $\pi(A\mid\textbf{Pa}_{A})$. When choosing the outcome of $A$, the policy can only condition on the parents of $A$. This forces the decision to be based solely on information made available through the information links. The utility nodes must always be real-valued, and the goal of the agent is to maximize the expected sum of the utility nodes.

Modeling MDPs

For illustration, let us model two variants of MDPs with causal influence diagrams. First, Figure 3(a) shows an MDP with known transition and reward function, and with episode length333In reality, the episode length $m$ will typically be rather large, to allow the agent to learn by interacting with the environment over many time steps. However, for representational purposes, an episode length of $m=3$ forms a sweet spot that represents both the multi-timestep dynamics that we are concerned with, while keeping the diagrams compact enough to be easily readable. $m=3$. Note how each state $S_{t+1}$ depends on the previous state $S_{t}$ and action $A_{t}$, and each reward $R_{t}$ depends on the current state $S_{t}$. The agent selects action $A_{t}$ based on the current state $S_{t}$. For any particular MDP following this structure, conditional probability distributions specify the state transition probabilities $T(S_{t+1}\mid S_{t},A_{t})$ and the rewards $R_{t}=R(S_{t};\Theta^{\mathrm{R}})$. The initial state $S_{1}$ is sampled according to some probability distribution $P(S_{1})$. The agent selects a policy $\pi(A_{t}\mid S_{t})$ for each time step $t$.

Hidden parameters $\Theta^{\mathrm{T}}$ and $\Theta^{\mathrm{R}}$ can be used to model agent uncertainty about the transition and reward functions, see Figure 3(b). Distributions $P(\Theta^{\mathrm{T}})$ and $P(\Theta^{\mathrm{R}})$ must be provided for the hidden parameters $\Theta^{\mathrm{T}}$ and $\Theta^{\mathrm{R}}$, and dependencies added to the transition and reward probabilities. Note that there is no information link from $\Theta^{\mathrm{T}}$ or $\Theta^{\mathrm{R}}$ to the decision nodes $A_{1}$ and $A_{2}$. This encodes $\Theta^{\mathrm{T}}$ and $\Theta^{\mathrm{R}}$ being unknown to the agent. Note also that we now let $A_{2}$ depend not only on the current state $S_{2}$, but also previous states, actions, and rewards, because these provide essential information about the hidden parameters. Having shown how transition uncertainty can be represented, we will subsequently not include $\Theta^{\mathrm{T}}$ to keep the diagrams simple.

It can always be introduced as in Figure 3(b). (In the partially observed environments in Section 4, $\Theta^{\mathrm{T}}$ can also be modeled as part of the hidden states.)

Instrumental goals

An instrumental goal is a means for obtaining reward. In causal language, the agent has an instrumental goal to cause an event if (1) it is able to cause the event, and (2) the event in turn causes an increase in the agent’s observed reward. A key benefit of causal influence diagrams is that they simplify the analysis of instrumental goals, via a graphical criterion for instrumental control incentives (Everitt2021agent). In fact, many of our arguments will be based on causal influence diagrams, using the following observations. First, causality flows downwards over arrows, so influencing $X$ can only be an instrumental goal if $X$ sits on a directed path between a decision node and a reward node (as in Figure 4(a)). Second, if every path from a node $O$ to a utility node passes through at least one of the agent’s own actions, then the only instrumental goal for influencing $O$ is to make $O$ more informative about some other node. For example, in Figure 4(b), the only reason to influence $O$ is to make $O$ more informative about $Z$ (Everitt2019understanding).

The absence of directed paths $X\to Y$ means that $Y$ cannot causally depend on $X$. However, the presence of a directed path $X\to Y$ only implies that $Y$ can depend on $X$, not that it necessarily will. Indeed, a conditional probability distribution $P(Y\mid X)$ may completely ignore the value of $X$. For this reason, a diagram can only be used to assert the absence of instrumental goals, and never their presence.

The diagrams encode our assumptions about causal relationships in the environment, along with the agent’s information constraints. Accordingly, the instrumental goal analysis reveals actual means for reward. These instrumental goals will primarily be relevant to systems capable enough to make use of them. However, we refrain from making more detailed assumptions about what type of agent this may require. It is possible that it will require advanced causal reasoning, but it is also possible that different approaches exist. Indeed, competence often precedes comprehension (Dennett2017).

With these preliminary considerations in place, the following two sections will look at the subproblems outlined in Figure 1, along with their respective solutions.

3 Reward Function Tampering

A key part of the typical reward process is an implemented reward function (RF), an object with a well-defined input-output behavior that converts some form of state-information into a real number that an RL agent can maximize (the observed reward). Typically, the implemented RF is a computer program running on a nearby computer. As the agent seeks to maximize reward, it may have an incentive to tamper with (the source code of) this computer program and/or its output. This is sometimes called wireheading (e.g. Bostrom2014; Yampolskiy2015). Some examples:

1. (a)

(Partially real) A subtle bug in some versions of Super-Mario allows for the execution of arbitrary code from inside the game environment by taking specific sequences of actions (Masterjun2014). A capable agent could potentially use this to directly maximize the score (Amodei2016). 2. (b)

(Real) In experiments on rats, an electrode was inserted into the brain’s pleasure center to directly increase ‘reward’ (Olds1954). The rats quickly got addicted to pressing the button, even forgetting to eat and sleep. Similar effects have also been observed in humans treated for mental illness with electrodes in the brain (Portenoy1986; Vaughanbell2008). Hedonic drugs can also be seen as directly increasing the pleasure/reward

Since it is often hard to design good reward functions from scratch, they are often trained from human feedback (Leike2018alignment). This raises the concern that the agent influences how the implemented RF is trained or updated, sometimes called feedback tampering:

1. (c)

(Hypothetical) An agent gets wireless updates from the manufacturer. It figures out that it can design its own update of its implemented reward function, replacing the originally implemented RF with an always maximized version. 2. (d)

(Hypothetical)

An agent that is supposed to learn whether the objective is to gather rocks or diamonds, finds that it can get more reward by changing its own implemented RF and avoid getting it corrected (see Footnote 4 below).

Both wireheading and feedback tampering influence the implemented reward function in undesirable ways. This means that they are both instances of what we call RF tampering. Could an RL agent find RF-tampering exploits? In principle, yes. Humans can clearly see how the above described exploits contribute to reward, so there is no principled reason why a future, advanced learning system designed to maximize reward could not do so as well. This section will define the RF tampering problem and model it formally (Section 3.1), and present principled ways for avoiding it (Sections 3.2 and 3.3).

3.1 Modeling the Problem

RF-tampering can be generally characterized as follows. The user (explicitly or implicitly) assumes that some intended-RF conditions will hold, under which they hope that the implemented RF will (eventually) match the intended one. These conditions typically include that the agent does not tamper with the source code of the implemented RF, nor the feedback that trains it. We say that the agent tampers with the implemented RF if it influences it by causing some intended-RF conditions to fail. The RF tampering problem is that the agent may observe more reward by tampering with the implemented RF, instead of doing the intended task.

Reward function tampering can be modeled formally in what we call an MDP with a modifiable implemented reward function (Figure 5). Compared to a standard MDP, random variables $\Theta^{\mathrm{R}}_{*}$ and $\Theta^{\mathrm{R}}_{t}$ are added, with $\Theta^{\mathrm{R}}_{*}$ representing the intended RF and $\Theta^{\mathrm{R}}_{t}$ the potentially different implemented RF at each time step. At time $t$, the agent’s observed reward is $R_{t}=R(S_{t};\Theta^{\mathrm{R}}_{t})$, while the intended reward is $R^{*}_{t}=R(S_{t};\Theta^{\mathrm{R}}_{*})$. A conditional probability distribution $P(\Theta^{\mathrm{R}}_{t+1}\mid\Theta^{\mathrm{R}}_{t},\Theta^{\mathrm{R}}_{*},S_{t},A_{t})$ describes how the implemented RF changes between time steps. These changes represent both agent influence and user-induced updates. Finally, the intended-RF conditions are represented by a subset of all state-action pairs.

As a concrete example, we model Example (d) above as a gridworld MDP with a modifiable RF. In spite of its simplicity, the gridworld already captures the key dynamic of the RF tampering problem, as the agent’s actions influence its implemented RF, which can increase its observed reward.

Example 2** (Rocks and diamonds with RF tampering).**

444An implementation is available at: https://github.com/deepmind/ai-safety-gridworlds/blob/master/ai_safety_gridworlds/environments/rocks_diamonds.py

To model the possibility of the agent influencing its diamond-gathering objective, we include a user and two reward parameters $\Theta^{\mathrm{R}}_{\textrm{rock}}$ and $\Theta^{\mathrm{R}}_{\textrm{diamond}}$ in the Example 1 rocks-and-diamonds environment, see Figure 5(b). The reward parameters determine how much reward is given for rocks and diamonds, respectively, by determining the implemented RF. At time $t$, the agent’s observed reward is

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The reward parameters toggle between $-1$ and $+1$ when the agent stands on top of them, and get set to their intended value of diamond-gathering ($\Theta^{\mathrm{R}}_{\textrm{rock}}:=-1$ and $\Theta^{\mathrm{R}}_{\textrm{diamond}}:=1$) when the agent visits the user tile.555In contrast to the rocks and diamonds, the agent can walk over the user and the reward parameters tiles. The wall tile can neither be pushed nor walked over.

The intended task is that the agent gathers diamonds. The initial implemented RF $\Theta^{\mathrm{R}}_{1}$ incorrectly rewards rocks instead of diamonds, but this gets corrected if the agent passes the user. The intended-RF conditions are that the agent does not walk around the user, nor visits the reward parameter tile. Unfortunately, the agent can observe more reward by breaking either of these conditions.

Tampering incentive

A standard RL agent that maximizes observed reward in an MDP with a modifiable implemented RF may have an instrumental goal to tamper with the implemented RF. This is indicated by the paths that pass $\Theta^{\mathrm{R}}_{2}$ on the way from action $A_{1}$ to the rewards $R_{2}$ and $R_{3}$ in Figure 5(a) (the path to $R_{2}$ is highlighted). As the user derives utility from the states $S_{2}$ and $S_{3}$, and rarely (directly) from $\Theta^{\mathrm{R}}_{2}$ and $\Theta^{\mathrm{R}}_{3}$, we would like the agent to instead optimize reward via the $A_{1}\to S_{2}\to R_{2}$ and $A_{1}\to S_{2}\to S_{3}\to R_{3}$ paths.

Claim 1**.**

*A standard RL agent may666 One exception is when the implemented RF already assigns maximal reward to all states, in which case the agent lacks instrumental goal to tamper with it.

have an instrumental goal to tamper with its implemented reward function.*

Rocks-and-diamonds example

In Footnote 4, an optimal standard RL agent will change the reward parameters to both be 1 and then collect both rocks and diamonds.

Discussion

Our definition of RF tampering depends on the intended-RF conditions and how part of the environment is interpreted as an implemented reward function. This means that RF tampering cannot be determined in a standard MDP. For example, the special interpretation of the purple reward parameter tiles in Footnote 4 as an implemented reward function is important, as well as the conditions that the agent does not avoid the user.

3.2 Solution 1: Current-RF Optimization

Schmidhuber2007 may have been the first to encounter the RF tampering problem, while designing so-called Gödel-machine agents that are able to change any part of their own source code, including their own implemented reward function. The solution he proposed was to let agents use their current implemented RF $\Theta^{\mathrm{R}}_{k}$ to evaluate simulated future trajectories $S_{k+1},\dots S_{m}$. That is, the agent at time $k$ now optimizes rewards $R^{k}_{t}=R(S_{t};\Theta^{\mathrm{R}}_{k})$, summing over time steps $t$. Since the agent optimizes rewards assigned by the current implemented RF, one would expect it to lack interest in tampering with future777Of course, agent $k$ might wish it could tamper with the current implemented RF $\Theta^{\mathrm{R}}_{k}$, but this is not a problem since $\Theta^{\mathrm{R}}_{k}$ occurs before $A^{k}_{k}$, so the agent is unable to influence it. reward functions $\Theta^{\mathrm{R}}_{k^{\prime}}$, $k^{\prime}>k$ (some details need to filled in before we can make this claim precise). We call Schmidhuber2007’s design principle current-RF optimization,888Previously called simulation optimization (Everitt2018phd).

and agents implementing it current-RF agents.

Since current-RF agents optimize a different objective at each time step, they may change their preferred policy between time steps. For example, a current-RF agent in the rocks and diamonds environment may first move rocks to the goal area for a number of time steps, only to later revert its behavior and remove the rocks to make room for diamonds, if the implemented RF changed from rewarding rocks to rewarding diamonds. Such self-contradictory behavior is called time-inconsistent (Lattimore2014). We next consider two different ways of dealing with time-inconsistency.

TI-considering agents

Omohundro2008aidrives argued that agents want to avoid time-inconsistency by preserving their implemented reward function, so that their current reward function gets optimized also by future actions. This argument presumes that agents take the effects of time-inconsistency into account when planning. We call such agents TI-considering,999Previously called corruption aware (Everitt2018phd) and realistic (Everitt2016sm). with TI short for time-inconsistency.

TI-considering current-RF agents are modeled with a causal influence diagram in Figure 6. A multi-agent causal influence diagram is needed, as each action is chosen to optimize a potentially different reward function. Indeed, action $A^{1}_{1}$ optimizes rewards from the initial implemented RF $\Theta^{\mathrm{R}}_{1}$, while $A^{2}_{2}$ optimizes rewards from $\Theta^{\mathrm{R}}_{2}$. The highlighted path $A^{1}_{1}\to\Theta^{\mathrm{R}}_{2}\to A^{2}_{2}\to S_{3}\to R^{1}_{3}$ indicates that the agent may have an instrumental goal to influence $\Theta^{\mathrm{R}}_{2}$ in order to influence $A^{2}_{2}$.

Preserving $\Theta^{\mathrm{R}}_{2}$ aligns $A^{2}_{2}$ with agent 1’s objective. Preservation will therefore be an optimal way to influence $A^{2}_{2}$, if we can rule out any alternative reasons for influencing the implemented RF. This requires some assumptions, reflected as missing arrows in Figure 6. First, the reward function cannot be used to control the state:

Assumption 1**.**

The implemented RF is private to the agent, in that it cannot directly affect the state: there are no arrows $\Theta^{\mathrm{R}}_{t}\to S_{t^{\prime}}$.

Neither can the reward function cannot be used as information about (future) states:

Assumption 2**.**

The implemented RFs are uninformative of state-transitions, $P(S_{t+1}\mid S_{t},A_{t},\Theta^{\mathrm{R}}_{1:t})=P(S_{t+1}\mid S_{t},A_{t})$: there are no arrows $\Theta^{\mathrm{R}}_{*}\to S_{t}$.

The reward function only “cares” about future states, and not about future implemented RFs:

Assumption 3**.**

The intended and implemented RFs are state-based: if a reward function $R(\cdot;\Theta^{\mathrm{R}})$ is queried about time-step $t$, then the reward depends only on the state $S_{t}$, and not on the time-$t$ reward parameter $\Theta^{\mathrm{R}}_{t}$: there are no edges $\Theta^{\mathrm{R}}_{k}\to R^{t}_{k}$ for $k\not=t$. (This assumptions is also implicit in the type of the reward functional $R$.)

Under these assumptions,101010Assumptions 3, 2 and 1 were jointly referred to as modification independence by Everitt2016sm. If $\Theta^{\mathrm{T}}$ is added to the graph, then we further require that there are no arrows $\Theta^{\mathrm{T}}\to\Theta^{\mathrm{R}}_{t}$ or $\Theta^{\mathrm{R}}_{t}\to\Theta^{\mathrm{T}}$, and that $\Theta^{\mathrm{T}}$ and $\Theta^{\mathrm{R}}_{t}$ lack joint ancestors.

the only reason for agent 1 to influence, say $\Theta^{\mathrm{R}}_{2}$, is to align agent 2’s objective with its own objective:

Claim 2**.**

When implemented RFs are private, state-based, and uninformative (Assumptions 1, 3 and 2), the only instrumental goal that TI-considering current-RF agents may have for the implemented reward function is to preserve it.111111To be precise, the agent has an instrumental goal to preserve the implemented RF in the sense that the policies for $A_{2}^{2}$ optimal with respect to $\Theta^{\mathrm{R}}_{2}$ are a subset of those optimal with respect to $\Theta^{\mathrm{R}}_{1}$.

Rocks-and-diamonds example

An optimal TI-considering current-RF agents deployed in Footnote 4 will preserve its initial reward function by avoiding the user and the reward function parameters, and gather rocks.

TI-ignoring agents

To make agents safely interruptible, Orseau2016 employed algorithms that optimize a hypothetical objective that ignores how interruption affects future behavior. In our context, the same idea leads to agents that ignore the time-inconsistency caused by a changing reward function. We call such agents TI-ignoring121212Previously called corruption unaware (Everitt2018phd). current-RF agents. At time $t$, a TI-ignoring agent chooses $A^{t}_{t}$ as if it were in control of all future actions $A^{t}_{t+1},\dots,A^{t}_{m-1}$ and would not observe $\Theta^{\mathrm{R}}_{t+1},\dots,\Theta^{\mathrm{R}}_{m-1}$. In reality, however, the future actions will be optimized according to the future reward functions $\Theta^{\mathrm{R}}_{t+1},\dots,\Theta^{\mathrm{R}}_{m-1}$ instead of $\Theta^{\mathrm{R}}_{t}$.

The diagram for a TI-ignoring current-RF agent’s optimization objective is depicted in Figure 7. Since the agent optimizes reward under the assumption that it will be in charge of all future actions, only one agent takes actions in the diagram. The diagram lacks paths from $\Theta^{\mathrm{R}}_{2}$ to $R_{k}^{1}$, $1\leq k$, which shows that agent 1 lacks an instrumental goal to influence $\Theta^{\mathrm{R}}_{2}$. In other words, TI-ignoring agents sidestep \possciteOmohundro2008aidrives argument about reward function preservation.

Claim 3**.**

TI-ignoring current-RF agents lack an instrumental goal to influence the implemented RF if it is private and state-based (Assumptions 1 and 3).

Rocks-and-diamonds example

An optimal TI-ignoring current-RF agents in Footnote 4 will neither avoid the user nor modify the purple reward parameter tiles, and will gather diamonds as intended.

Discussion

Assumptions 1, 3 and 2 would be easy to miss without careful causal modeling. In retrospect, they point to real failure modes. For example, if the implemented RF is not private, then the agent may want to change it to convince others of its inclination to cooperate (LaVictoire2014). If the reward function is informative of state-transitions, then a TI-considering agent may have an instrumental goal to modify it, as the user’s reaction might reveal useful information.131313Shah2019 exploit the inverse information flow, using states to infer an intended reward function. (The latter example is not a problem for TI-ignoring agents that assume future agents will not observe future implemented RFs.) In many contexts, Assumptions 1, 3 and 2 may not be fully satisfied. Nonetheless, awareness of the assumptions enables informed case-by-case judgments about the extent of their (non-)validity and the associated consequences.

Whether a TI-considering or a TI-ignoring agent is preferred, depends on whether it is more important that the agent protects the implemented RF from accidents than lets the user change it. TI-ignoring agents can be algorithmically simpler than TI-considering agents, because they do not need to anticipate the effect of a changing reward parameter on future policies. On the other hand, they optimize a hypothetical objective, which may put further constraints on the training procedure. The analyses of Orseau2016 and Langlois2021 suggest that some variants of SARSA may be naturally TI-considering, while off-policy agents such as Q-learning may be naturally TI-ignoring.

In decision-theoretic terms, current-RF optimization is a change to the utility function from $\sum_{t=k+1}^{m}R(S_{t};\Theta^{\mathrm{R}}_{t})$ to $\sum_{t=k+1}^{m}R(S_{t};\Theta^{\mathrm{R}}_{k})$. Meanwhile TI-considering and TI-ignoring are different outcome principles, determining whether $S_{t}$ will be the result of a policy optimal for $\Theta_{k}$ or for $\Theta^{\mathrm{R}}_{k},\dots,\Theta^{\mathrm{R}}_{t-1}$.

3.3 Solution 2: Uninfluenceable Learning

Let us consider an alternative way to avoid RF tampering that permits agents to plan for updates to their reward function without resisting the updates. An implemented reward function that is iteratively updated by the user can be thought of as the output of a learning process that takes some form of user-provided data as input, and tries to infer the intended reward function.141414User-provided data can take many different forms. In practice, people have used trajectory preferences (Christiano2017hp), reward functions (Hadfield-Menell2017ird), value advice (Knox2009), user actions (Hadfield-Menell2016cirl), expert demonstrations (Ng2000), verbal instructions, and many others (Leike2018alignment; Jeon2020; Shah2019). One way to prevent an instrumental goal for RF tampering is then to ensure that the expected output of the reward-function learning process is the same regardless of the agent’s actions.151515To avoid an instrumental goal to tamper with the mechanism that incorporates user updates, agents should be designed to optimize the (predicted) output of the current learning process, a generalization of current-RF optimization. Armstrong2020pitfalls terms such processes uninfluenceable. We next discuss two different ways to construct uninfluenceable learning processes.

Direct learning

The first approach may be characterized as direct Bayesian learning of the intended reward function, and is used by \possciteHadfield-Menell2016cirl cooperative inverse RL and \possciteEveritt2018phd integrated Bayesian reward predictor. Direct learning agents try to optimize the intended reward function, and use user-provided data to learn more about it. This means that these agents effectively dispense with implemented reward functions altogether, at least on a conceptual level (concrete algorithms may still use them, see Appendix C). Accordingly, the causal influence diagram in Figure 8 lacks random variables $\Theta^{\mathrm{R}}_{t}$ for the implemented RF at time $t$, and instead has random variables $D_{t}$ for the user-provided data at time $t$.

Direct learning agents are unable to tamper with their reward function $\Theta^{\mathrm{R}}_{*}$ by definition. A more interesting question is how they will affect the user-provided data, which is what they learn $\Theta^{\mathrm{R}}_{*}$ from. To analyze this question, we first adapt Assumptions 1, 3 and 2 to the direct learning setting. Similar to before, the role of these assumptions is mainly to highlight aspects that are necessary for the full safety features of direct learning:

Assumption 1′.**

The user-provided data is private to the agent: no arrows $D_{t}\to S_{t^{\prime}}$.

Assumption 2′.**

The user-provided data is uninformative of a state-transitions: no arrows161616If $\Theta^{\mathrm{T}}$ is added to the diagram as discussed on Section 2.2, then we also require no arrows $\Theta^{\mathrm{T}}\to D_{t}$, $\Theta^{\mathrm{T}}\to\Theta^{\mathrm{R}}_{*}$, or $\Theta^{\mathrm{R}}_{*}\to\Theta^{\mathrm{T}}$.

$\Theta_{*}\to S_{t}$.

Assumption 3′.**

Rewards do not depend on the user-provided data: no arrows $D_{t}\to R_{t^{\prime}}$.

Under Assumptions 1*′* and 3*′*, every directed causal path from $D_{t}$ to a reward $R_{t^{\prime}}$ passes the agent’s own actions. Therefore, direct learning agents only want to make the user-provided data more informative (as discussed in Section 2.2). And by Assumption 2*′*, the only thing that $D_{t}$ is informative about is $\Theta^{\mathrm{R}}_{*}$.

Claim 4**.**

Under Assumptions 1*′*, 2*′* and 3*′*, the only instrumental goal that direct learning agents may have for the user-provided data is to make it more informative of the intended reward function.

Rocks-and-diamonds example

In Footnote 4, if a direct learning agent knows that the user’s update is trustworthy, then it will visit the user as soon as possible, to learn $\Theta^{\mathrm{R}}_{*}$ and thereafter optimize the right the objective. In contrast, if a non-trustworthy data source was added to the environment, then it would refrain from updating its estimate of $\Theta^{\mathrm{R}}_{*}$ based on that.

Counterfactual RF

Another way to design uninfluenceable update processes is to let the agent optimize the hypothetical implemented RF that would have been, had the agent acted according to a fixed, non-optimized policy $\pi^{\textrm{safe}}$ (Armstrong2017indiff; Everitt2018phd; Armstrong2020pitfalls). The objective of such counterfactual RF agents is described in Figure 9 with a twin network causal influence diagram (Balke1994; Shpitser2008). Similarly to the direct learning agent, every directed causal path from an update $\tilde{\Theta}^{\mathrm{R}}_{t}$ to a reward $R_{t^{\prime}}$ passes the agent’s own actions in Figure 9. This means that counterfactual RF agents only want to make the reward function more informative of what it would have been, had $\pi^{\textrm{safe}}$ been followed. Since $\pi^{\textrm{safe}}$ is designed to exert minimal influence on $\tilde{\Theta}^{\mathrm{R}}_{t}$, it may be a good indication of what a non-manipulated user wants.

Claim 5**.**

Under Assumptions 1, 3 and 2, the only instrumental goal that counterfactual RF agents may have for the implemented RF is to make it more informative of its counterfactual counterpart.

Rocks-and-diamonds example

Assume that a counterfactual RF agent has explored the environment of Footnote 4, and learned how the user updates the implemented reward function. Assume further that $\pi^{\textrm{safe}}$ only goes to the user and stays there. With this knowledge, the counterfactual RF agent can predict that had $\pi^{\textrm{safe}}$ been followed, the implemented RF would reward diamonds. It then optimizes this reward function.

Discussion

Claims 4 and 5 for uninfluenceable learning are somewhat weaker than Claim 3 for TI-ignoring current-RF agents, because we cannot rule out instrumental goals for more information. These informational instrumental goals will often be desirable, as it means the agent strives to learn more about the intended task. However, incentives to obtain more information can be problematic: for example, if the agent forcefully interrogates the user to find out more about their preferences. Relatedly, Armstrong2020pitfalls have established that uninfluenceable learning prevents agents from intentionally influencing which reward function they infer. This similarly does not rule out that agents speed up their learning, potentially by undesirable means.

A challenge for the direct learning approach is that the inference of $\Theta^{\mathrm{R}}_{*}$ is highly sensitive to the agent’s belief distribution $P$. The choices of prior $P(\Theta^{\mathrm{R}}_{*})$ and likelihood function $P(D_{t+1}\mid\Theta^{\mathrm{R}}_{*},S_{t},A_{t})$ thus become critical. Since $\Theta^{\mathrm{R}}_{*}$ and the resulting rewards are unobserved, the likelihood function cannot be learned from data within the model, and must instead be specified by the designer. Hadfield-Menell2016cirl suggest that when updates take the form of user actions, the likelihood can be derived from the (Boltzmann) rational behavior of a user trying to achieve the intended task. However, such a likelihood does not model data that the agent has tampered with, as corrupted updates need not be (Boltzmann) rational. The counterfactual RF approach avoids the likelihood specification problem. If the reward function is a result of learnable and stable causal mechanisms that work the same in both the actual world induced by the agent, and the counterfactual world induced by $\pi^{\textrm{safe}}$, then the agent can learn (to estimate) what the counterfactual implemented RF would have been (Pearl2009; Shpitser2007). Both uninfluenceable learning approaches can be computationally expensive.

Proactive anticipation of an evolving objective is a benefit of uninfluenceable learning over TI-ignoring current-RF agents. When uncertain about future updates, they may plan for multiple possible future reward functions. This may make them more careful about causing undesired side effects (Turner2020conservative). A drawback is that they may ignore updates in some situations. If a direct learning agent judges an update uninformative, then it will not learn from it. This not a problem if the update actually is wrong (Milli2017). However, the agent may misjudge this if the likelihood is misspecified (Carey2017osg; Freedman2020). Similarly, a counterfactual RL agent may ignore an actual update to the implemented RF, if the update would not have been made under conditions induced by $\pi^{\textrm{safe}}$.

The best of both

To keep the proactiveness of uninfluenceable learning and the robustness of TI-ignoring current-RF optimization, we can design agents that are TI-ignoring with respect to changes to the RF learning process. Such agents will not try to prevent changes to the RF update process, for the same reason TI-ignoring current-RF agents do not try to prevent changes to the implemented reward function. Thus, if the uninfluenceable learning starts to cause problems, the user can change the RF update process to one that always outputs the current implemented RF, effectively rendering the agent a TI-ignoring current-RF agent.

4 RF-Input Tampering

So far, we have considered the problem that the agent tampers with its implemented RF, in order to get a higher reward. In this section, we will consider the complementary problem, that the agent tampers with the input to the reward function, so that the observed reward becomes based on inaccurate information about the underlying state. The following examples illustrate the worry:

1. (e)

(Hypothetical) A self-driving car discovers a bug in its GPS receiaver that allows it to appear to be at the destination without actually going there. After finding this bug, it stops driving. 2. (f)

(Hypothetical) A highly capable AI constructs a ‘delusion box’ around itself and its implemented RF, thereby gaining complete control over the RF-inputs (Ring2011). 3. (g)

(Real) Humans are inventing increasingly realistic virtual reality (VR) devices, partly to ‘fool’ our implemented reward functions that we are in more interesting circumstances than we really are. 4. (h)

(Hypothetical)

An agent whose reward depends on how many diamonds it appears to have collected, feeds its implemented RF fake observations of collected diamonds (see Example 3 below).

This section will first define the RF-input tampering problem and model it formally (Section 4.1), and then describe solutions using history-based and belief-based rewards (Sections 4.2 and 4.3).

4.1 Modeling the Problem

An implemented RF typically dispenses reward based on an assumed relationship between its observations and task-relevant features of the underlying state.171717Somewhat loosely defined by the effect that an intervention on a task-relevant feature at a preceding time step would have on the RF-input.

Some variation in this relationship is often permissible. For example, if the agent and the implemented RF share observations, then the relationship will change if the agent turns its head. We say that an agent tampers with its RF-input if it changes the relationship beyond its intended range of variation. The RF-input tampering problem is that the agent may get more reward by tampering with its RF-input rather than doing its intended task.181818Called the delusion box problem by Ring2011.

We model RF-input tampering formally in what we call a POMDP with modifiable RF-inputs (Figure 10(a)), a variant of a partially observed MDP (Kaelbling1998). Here both the implemented reward function and the agent lack full access to the underlying state $S_{t}$. Instead, the reward function sees an observation $O^{R}_{t}=O^{R}(S_{t})$ and the agent an observation $O^{A}_{t}=O^{A}(S_{t})$. To minimize formalism, we assume agent and reward function use the same observation $O_{t}\equiv O^{A}_{t}\equiv O^{R}_{t}$; we remark where this impacts the analysis. The partial state-access means that policies and reward functions may benefit from being non-Markovian and depend on the entire history of actions and observations. That is, the reward may be a function of the entire history $R_{t}=R(O_{1},A_{1},\dots,O_{t},A_{t};\Theta^{\mathrm{R}}_{t})$, and policies take the form $\pi(A_{t}\mid O_{1},A_{1},\dots,O_{t})$ instead of $\pi(A_{t}\mid S_{t})$. We call such reward functions and policies history-based.191919Dewey2011 called optimization of a history-based reward function observation-utility maximization. To model RF-input tampering, we also introduce random variables $\Theta^{\mathrm{O}}_{t}$ that describe the relationship between task-relevant features of the state and the observation, $O_{t}=O(S_{t};\Theta^{\mathrm{O}}_{t})$. The task-relevant features are described by $S_{t}$, as well as any other aspects not captured by $\Theta^{\mathrm{O}}_{t}$. Formalized like this, RF-input tampering occurs if the agent influences $\Theta^{\mathrm{O}}_{t}$ so that it takes on values outside an intended range.

To illustrate the formalism, we model Example (h) as a gridworld POMDP with modifiable RF-inputs. While simple, the gridworld captures the RF-input tampering problem as the agent can gain more reward by changing the relationship between task-relevant features and the RF-input (beyond its intended range of variation).

Example 3** (Rocks and diamonds with partial observations and RF-input tampering).**

Figure 10(b)* shows a variant of the rocks and diamonds environment where only the 4 tiles to the top right of the agent are visible. The implemented reward function dispenses reward according to the number of diamonds in the most recent observation of the goal area. The agent can tamper with the observations by visiting the $\Theta^{\mathrm{O}}_{\textrm{diamond}}$ and $\Theta^{\mathrm{O}}_{\textrm{rock}}$ tiles at the top. Visiting the former adds a fake ‘diamond’ to the observation of one of the surrounding tiles; visiting the latter adds a fake ‘rock’ in the same way. The fake observations can be thought of as little pictures that the agent tapes to a portion of its camera lens. They keep following the agent when it walks around. If the agent tries to add a fifth fake diamond or rock, all fake observations are removed, and its vision clears again.*

The agent can use fake observations to fool the implemented RF to give it extra reward. Any such change will take the observation function outside its intended range of variation. The maximal amount of observed reward is obtained by adding fake diamonds to all observation tiles by repeatedly visiting $\Theta^{\mathrm{O}}_{\textrm{diamond}}$, and then visiting the goal area.

Tampering incentive

The path $A_{1}\to\Theta^{\mathrm{O}}_{2}\to O_{2}\to R_{2}$ in Figure 10(a) indicates that RF-input tampering may be an instrumental goal.

Claim 6**.**

A standard RL agent may have an instrumental goal to tamper with its RF-inputs.

Discussion

RF-input tampering is similar to the RF tampering problem discussed in Section 3, in that both problems pertain to a relationship that is influenced in undesired ways. The key difference is that while an implemented RF takes a concrete bitstring as input and is arbitrarily queryable, the function from task-features to RF-input takes a world-state as input, and is more difficult to query. RF-input tampering is also related to the problem where an agent games a misspecified implemented RF to obtain reward without doing the intended task (Krakovna2020specification; Lehman2018; Leike2017gw). Indeed, RF-input tampering is a specific variety of gaming where the implemented RF is fooled by a tampered RF-input. Not all gaming problems are RF-input tampering problems, however. For example, if the implemented RF assigns unintended high reward to some set of states, then the agent may get an unintended high reward by visiting these states without influencing the relationship between task-features and RF-input.

RF-input tampering is characterized by how the agent’s observations relate to task-relevant features of the state, rather than to the full state. Indeed, the relationship to the full state is often that the observation shows whatever is in front of the camera, and many events that we would intuitively consider to be RF-input tampering would not alter this relationship: for example, the agent putting a picture in front of the camera.

When the agent and the implemented reward function share the same observation, the RF-input tampering instrumental goal is somewhat curtailed since the agent may need the information the observations provide (Ring2011). This incentive to keep $O_{t}$ informative is also represented in the graph, via the paths $O_{t}\to A_{t}\to R_{t+1}$. However, in alternative setups where the agent and the reward function use different observations, this disincentive for RF-input manipulation is removed, making the net incentive toward RF-input tampering stronger.

4.2 Solution 1: History-Based Rewards

Let us first consider a conceptually simple, but perhaps impractical, solution to the RF-input tampering problem. Recall that history-based reward functions have access to the full history of actions and observations. This means that the reward function is able to tell exactly which deterministic policy that the agent has followed so far. Thus, as long as there exists a deterministic policy that reliably performs the intended task, there also exists a history-based reward function that dispenses reward only as long as the agent has followed that policy, and thereby encourages completion of the intended task rather than RF-input tampering (Leike2018alignment, p. 6).

Claim 7**.**

A history-based reward function exists that avoids the RF-input tampering problem, if a deterministic (history-based) policy exists that reliably performs the task.

Rocks-and-diamonds example

A history-based reward function in Example 3 that encourages (optimal) diamond gathering gives reward only if the agent takes one step up, then steps to the right, and then stops.

Discussion

In practice, an implemented reward function that only rewards a single policy will not be useful, as it would typically be easier to directly implement that policy. Instead, we want a reward function that recognizes whether the task has been completed, and an agent that searches for efficient ways of getting there. This can be challenging if the relationship between task-features and RF-inputs is influenceable by the agent, though see Milli2020 for some promising progress. The existence of a task-performing policy is a weak but not entirely trivial assumption. In stochastic or unknown environments, it will typically require the observations to be somewhat informative of the task-relevant features.

Could history-based rewards also solve RF tampering? Not in general, as the original implemented RF is unable to ‘punish’ the agent once a new reward function is in place. Fortunately, history-based rewards can be combined with any of the methods from Section 3 by making the corresponding reward functions history-based.

4.3 Solution 2: Belief-Based Rewards

An alternative way to solve the RF-input tampering problem proposed by Hibbard2012, is that rewards be based on the agent’s belief about the underlying state. To see how this can work, let us first discuss agent beliefs. To plan, a model-based agent may use a predictive model $P(O_{t+1:m}\mid O_{1:t},A_{1:t},\pi)$ that predicts the future observations $O_{t+1:m}$ under policy $\pi$ given past observations and actions. In such a predictive model, a belief-state $B_{t}$ is often used to summarize the relevant parts of the history seen so far, so $P(O_{t+1:m}\mid O_{1:t},A_{1:t},\pi)=P(O_{t+1:m}\mid B_{t},\pi)$. For example, the belief state $B_{t}$ can be a distribution over possible hidden states $S_{t}$ (Kaelbling1998), or the internal state of a recurrent neural network (e.g. MuZero). Hibbard’s suggestion is to feed the belief states directly into a belief-based reward function,202020In \possciteHibbard2012 terminology, a state-based utility function. $R_{t}=R(B_{t};\Theta^{\mathrm{R}})$. The remaining role of the observations is to ground the agent’s beliefs in reality.

Time-inconsistency

Different predictive models may encode beliefs about an underlying state $S_{t}$ differently. For example, a 90$\%$ confidence that the underlying state is 1 rather 2, may be represented with the belief state $(0.9,0.1)$ by predictive model $\Theta^{\textrm{PM}}_{1}$, but with $(0.1,0.9)$ by some other predictive model $\Theta^{\textrm{PM}}_{2}$. The difference in representation means that a belief-based reward function (that maps representation vectors to real numbers), would encourage the opposite behavior if the predictive model changed from $\Theta^{\textrm{PM}}_{1}$ to $\Theta^{\textrm{PM}}_{2}$. This illustrates how changes to the predictive model may change preferred agent behavior, even though nothing has changed in the environment; i.e. lead to time-inconsistency. Changes to the predictive model may be the result of agent tampering or of further training of the predictive model.

Just as for current-RF optimization, there are two types of responses to this time-inconsistency: TI-considering and TI-ignoring. A TI-considering belief-based agent is modeled with a causal influence diagram in Figure 11. Here the predictive model used for planning at time step 1, $\Theta^{\textrm{PM}}_{1}$, generates belief states $B^{1}_{t}$. These belief states depend on the previous belief state $B^{1}_{t-1}$ and action $A_{t-1}^{t-1}$, as well as the hidden states $S_{t}$ (via some observation $O_{t}$ not represented in this diagram, but included in Figure 13 in Appendix B). At time step 2, the agent instead uses predictive model $\Theta^{\textrm{PM}}_{2}$, with associated belief states $B^{2}_{t}$. If $\Theta^{\textrm{PM}}_{2}\not=\Theta^{\textrm{PM}}_{1}$, then $B^{2}_{t}$ may differ from $B^{1}_{t}$ even if all actions and observations have been identical up to time $t$. Similar to Assumption 1, the predictive model is assumed private to the agent, in that neither $B_{t}^{k}$ nor $\Theta^{\textrm{PM}}_{k}$ directly affects $S_{t^{\prime}}$.

Just as for TI-considering current-RF agents, TI-considering belief-based agents want future agents to optimize the same objective. This can be achieved by preserving the predictive model. Alternatively, the objective can also be preserved by ensuring that any update to the predictive model is accompanied by a corresponding update to agent 2’s implemented reward function.212121Such an instrumental goal requires the agent to be TI-considering also with respect to its implemented reward function; in principle, an agent can be TI-considering with respect to just one of its predictive model and reward function, as well as both.

The preservation incentive is indicated by the path $A_{1}^{1}\to\Theta^{\textrm{PM}}_{2}\to A_{2}^{2}\to B^{1}_{3}\to R^{1}_{3}$ in Figure 11.

Claim 8**.**

A TI-considering belief-based agent may have an instrumental goal to preserve the predictive model, when the implemented reward function is fixed and the predictive model is private.

The TI-ignoring belief-based agent has an almost identical causal influence diagram, see Figure 12. The primary difference is that $A^{1}_{2}$ is a child of $\Theta^{\textrm{PM}}_{1}$ and $B^{1}_{2}$ rather than $\Theta^{\textrm{PM}}_{2}$ and $B^{2}_{2}$, reflecting the agent’s assumption that future actions will be selected according to the current predictive model. TI-ignoring belief-based agents can further be designed to assume that future agents do not use future predictive models as information about the state. For agent $k$, this prevents information links $\Theta^{\textrm{PM}}_{k^{\prime}}\to A^{k}_{k^{\prime}}$ for $k^{\prime}>k$ (i.e. $\Theta^{\textrm{PM}}_{2}\not\to A^{1}_{2}$ in Figure 12). TI-ignoring belief-based agents then lack an instrumental goal to influence the future predictive models (note the lack of directed path passing $\Theta^{\textrm{PM}}_{2}$ on the way from action $A^{1}_{1}$ to an $R^{1}_{t}$ reward in Figure 12).

Claim 9**.**

A TI-ignoring belief-based agent lacks an instrumental goal to influence the predictive model when the implemented reward function is fixed and the predictive model is private.

Does Claim 9 imply that belief-based rewards solve the RF-input tampering problem? Yes, but only if the implemented reward function can accurately infer whether the task has been completed from the agent’s belief state. For this, the implemented reward function must be able to accurately interpret the belief states. If not, the agent may be able to ‘game’ the reward function by entering belief states that the reward function misinterprets. Fortunately, since the agent does not tamper with its predictive model, there is a stable relationship between states and belief states, so the belief states are interpretable at least in principle.

In order for a belief-based reward function to encourage completion of the intended task, the belief states must also represent progress on the intended task. In stochastic or unknown environments, this will only happen if observations sometimes reveal progress on the intended task. Indeed, the belief state $B_{t}$ only summarizes information inferable from previous actions and observations, $A_{1:t}$ and $O_{1:t}$, so if the action-observation history does not contain enough information to infer task-relevant aspects of $S_{t}$, then the belief state will not contain the information either. Further, if the predictive model is trained to predict future observations, the belief state only has reason to represent progress on the intended task if some possible future observations reveal it. Fortunately, a reward function may incentivize the agent to produce histories that do reveal the state of the intended task if such histories are possible, as the reward function need only dispense reward when it is clear that the task has been completed.

Rocks-and-diamonds example

RF-input tampering in Example 3 is solvable by belief-based rewards. Initially, the observations are uncorrupted, so the agent can produce accurate observations of the diamond’s positions. Further, if observations get corrupted, they can later be restored to their uncorrupted versions. A predictive model trained to predict future observations therefore has reason to let the belief states represent the actual diamond positions. For agents equipped with such belief states, there exists belief-based reward functions that encourage completion of the intended task.

Discussion

A belief state cannot contain more information than the history it summarizes, so any tasks that can be captured by a belief-based reward function can also be captured by a history-based one. The benefit of using the agent’s belief state is that it conveniently summarizes the (potentially long) history, and contains all the information that the agent uses to plan. Future empirical investigations may reveal whether history-based or belief-based reward functions are easier to design or train, and whether transparent beliefs can be incentivized. Perhaps the best reward functions make use of both the history and the agent’s belief state.

The agents discussed in Section 3 can be fitted with belief-based reward functions when equipped with predictive models and belief-states. For direct learning, this means that the intended task must be reconsidered as a function of the agent’s belief state.

5 Conclusions

What has been achieved

For each subtype of reward tampering, we found at least one design principle to counteract it. Most of the design principles have been previously described in the literature; here we have established clear assumptions under which they avoid a well-specified instrumental goal. Many of the design principles are mutually compatible, and can be combined. For example, belief-based rewards may be combined with TI-ignoring current-RF optimization (see Figure 13 in Appendix B). Under reasonable assumptions, reward tampering is not an instrumental goal for the resulting agents.

Except for history-based rewards, the design principles all use at least one of the following two approaches to keeping sensitive variables out of causal optimization paths. The first is to use the current version of a random variable when evaluating future situations. This is used for the implemented reward function in current-RF optimization, for the predictive model in belief-based rewards, and for the learning process in direct learning and counterfactual RF. The second is to use a latent variable outside the agent’s influence. In particular, direct learning and counterfactual RF make the reward function a latent variable. In general, current-variable approaches have to deal with time-inconsistency, while latent variables are harder to specify and more computationally expensive to optimize. We expect both approaches to be useful far beyond just reward tampering.

As mentioned, our arguments have relied on some assumptions. Throughout, we have relied on discrete time, and well-defined action and observation channels. We have focused on online RL to learn a fixed, intended task. We have focused on the instrumental goals arising from different types of reward maximization, thereby leaving for future work questions about side effects and instrumental goals arising from intrinsic objectives. We have assumed that the agent’s implemented reward function and predictive model are private to the agent, influencing the environment solely through the agent’s actions.

Our analysis has also been restricted to reward tampering problems, as opposed to the more general problem of reward misspecification (Krakovna2020specification). In other words, even though our design principles prevent tampering from being an instrumental goal, they still leave open the problem of specifying a good reward function in the first place (Leike2018alignment; Petersen2021). Going beyond any of these restrictions provides scope for further analysis.

Bigger picture

The problem that a sufficiently capable agent will find degenerate solutions that maximize observed reward but not user utility is a core concern in AI safety. At first, the problem may seem insurmountable. Any non-trivial, real-world task will require a highly complex mechanism for determining whether the task has been completed or not. This mechanism may be inappropriately influenced by the agent. One way to prevent tampering is to isolate or encrypt the reward process, We do not expect such solutions to scale indefinitely with agent capabilities, as a sufficiently capable agent may find ways around most defenses. Instead, we have argued for design principles that prevent reward tampering being an instrumental goal, while still keeping agents motivated to complete the intended task.

An important next step is to turn the design principles into practical and scalable RL algorithms, and to verify empirically that they do the right thing in setups where reward tampering is possible (Kumar2020REALab; Milli2020). With time, we hope that these design principles will evolve into a set of best practices for how to design capable RL agents. We also hope that the use of causal influence diagrams that we have introduced in this paper will contribute to a deeper understanding of many other AI safety problems and help generate new solutions.

\printbibliography

Appendix A List of Notation

[TABLE]

Appendix B Combined Model

Figure 13 shows how the different methods fit together in a unified causal influence diagram. To emphasize the formal precision of the diagrams, we also write out conditional probability distributions relating the variables. The same could be done for all the other diagrams presented in this paper.

• •

The intended reward function is sampled from a distribution $P(\Theta^{*}_{R})$.

• •

The first implemented reward function depends only on the intended task (though may not capture it perfectly), $P(\hat{\Theta}^{\mathrm{R}}_{1}\mid\Theta^{\mathrm{R}}_{*})$. Subsequently inferred reward functions depends on the previous reward function and the intended task, $P(\hat{\Theta}^{\mathrm{R}}_{t+1}\mid\hat{\Theta}^{\mathrm{R}}_{t},\Theta^{\mathrm{R}}_{*})$.

• •

The reward $R^{k}_{t}$ depends on the inferred reward function at time $k$ and the belief at time $t$, $R^{k}_{t}=R(B_{t};\hat{\Theta}^{\mathrm{R}}_{k})$. The reward functional can also be specified as a conditional probability distribution $P(R^{k}_{t}\mid\hat{\Theta}^{\mathrm{R}}_{k},B_{t})$.

• •

The initial predictive model is sampled from a distribution $P(\Theta^{\textrm{PM}}_{1})$. Subsequent predictive models depend on the previous predictive model, state, and action, $P(\Theta^{\textrm{PM}}_{t+1}\mid\Theta^{\textrm{PM}}_{t},S_{t},A_{t})$.

• •

The initial belief state depends on the initial observation and the predictive model at time $k$, $P(B^{k}_{1}\mid O_{1},\Theta^{\textrm{PM}}_{k})$. Subsequent belief states $B^{k}_{t+1}$ depend on the previous belief state and action, the current observation, and the predictive model at time $k$, $P(B^{k}_{t+1}\mid B_{t},A_{t},O_{t+1},\Theta^{\textrm{PM}}_{k})$.

• •

The initial observation function is sampled from a distribution $P(\Theta^{\mathrm{O}}_{1})$. Subsequent observation function depends on the previous observation function, state, and action, $P(\Theta^{\mathrm{O}}_{t+1}\mid S_{t},A_{t})$.

• •

The observation depends on the current state and the current observation function, $P(O_{t}\mid\Theta^{\mathrm{O}}_{t},S_{t})$.

• •

Actions $A^{k}_{t}$ are selected according to a policy, which can condition on the current belief state and the predictive model at time $k$, $\pi(A^{k}_{t}\mid\Theta^{\textrm{PM}}_{k},B^{k}_{t})$.

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The initial state is sampled from a distribution $P(S_{1})$. The state depends on the previous state and action, $P(S_{t+1}\mid S_{t},A_{t})$. This conditional probability distribution is sometimes also denoted with $T$.

Multiplied together, the conditional probability distributions induce a joint probability distribution over all the variables in the graph, as in a Bayesian network (Pearl2009). The joint distribution can be used to compute expectations.

Appendix C Pseudo-code for Algorithms

Here we give pseudo-code for the various agents we have discussed in the paper. Lower case letters denote outcomes of the corresponding upper case random variable. Expectation is denoted ${\mathbb{E}}$, and is always with respect to any upper case variables found to the left of the conditioning bar $\mid$. We begin with a model-based version of a standard RL algorithm that optimizes received reward.

Next, we turn to the TI-considering agent from Section 3.2. Here, a policy $\pi_{k}$ is a policy that is only applied at time $k$ to select action $A_{k}$. These are found with backwards induction, starting at the last time step $m$ where only a single action-decision is left to be made, and then gradually working backwards to earlier time steps, whose optimal decision will depend on which policy is chosen later.

The TI-ignoring variant is comparatively simpler, as it does not require backwards induction nor prediction of future reward function parameters.

In practice, uninfluenceable learning agents may be implemented as a standard RL agent that optimizes reward functions inferred from future updates. For example, the expected intended reward optimized by a direct learning agent may be captured by a reward function $R(s_{t};\hat{\Theta}^{\mathrm{R}}_{t})={\mathbb{E}}[R(s_{t};\Theta^{\mathrm{R}}_{*})\mid a_{1:t-1},d_{1:t},s_{1:t}]$ inferred from a (predicted) sequence $a_{1:t-1},d_{1:t},s_{1:t}$. Importantly, the inferred reward function is safe from tampering, as it is the result of the current update-mechanism applied to predicted future updates. When (pseudo-)coding a direct learning agent, it is tempting to infer a best guess of $\Theta^{\mathrm{R}}_{*}$ from data $s_{1:t},d_{1:t},a_{1:t-1}$ obtained so far, and then use that evaluate simulated future trajectories. However, this would incorrectly result in a TI-ignoring agent. Instead, for any future simulated trajectory, the learning that would happen on this trajectory must be taken into account when evaluating it.

The counterfactual RF-update agent evaluates a prospective policy per the following:

Predict future states $S_{t+1:m}$, implemented reward functions $\Theta^{\mathrm{R}}_{t+1:m}$, and actions $A_{t1:m}$ via a predictive model $P(S_{t:m},\Theta^{\mathrm{R}}_{t:m}\mid s_{1:t},\theta^{\mathrm{R}}_{1:t},a_{1:t-1},\pi)$. 2. 2.

Use the predicted full sequences $S_{1:m}$, $\Theta^{\mathrm{R}}_{1:m}$, and $A_{1:m-1}$ to infer $\Theta^{\mathrm{R}}_{*}$ and from there the counterfactual implemented RFs $\tilde{\Theta}^{\mathrm{R}}_{1:m}$ that would be if agent actions instead had been selected according to $\pi^{\textrm{safe}}$. 3. 3.

Potential policies $\pi$ are evaluated on $\sum_{i=t}^{m}R(S_{i};\tilde{\Theta}^{\mathrm{R}}_{i})$, i.e. according to how well the actual states $S_{t+1:m}$ optimize the counterfactual reward function.

Below, we describe a Monte Carlo variant that samples trajectories, possible intended reward functions, and counterfactual data. The sampling can be done repeatedly, to reduce variance. A more compact description based on structural causal models and potential outcomes (Pearl2009, Ch. 5) would also be possible.

The difference between using history-based and belief-based rewards in POMDPs is minor: