The aim of Inverse Reinforcement Learning (IRL) is to infer a reward function R from a policy π . This problem is difficult, for several reasons. First of all, there are typically multiple reward functions which are compatible with a given policy; this means that the reward function is only partially identifiable , and that IRL contains a certain fundamental degree of ambiguity. Secondly, in order to infer R from π , an IRL algorithm must have a behavioural model of how π relates to R . However, the true relationship between human preferences and human behaviour is very complex, and practically impossible to fully capture with a simple model. This means that the behavioural model in practice will be misspecified , which raises the worry that it might lead to unsound inferences if applied to real-world data. In this paper, we provide a comprehensive mathematical analysis of partial identifiability and misspecification in IRL. Specifically, we fully characterise and quantify the ambiguity of the reward function for all of the behavioural models that are most common in the current IRL literature. We also provide necessary and sufficient conditions that describe precisely how the observed demonstrator policy may differ from each of the standard behavioural models before that model leads to faulty inferences about the reward function R . In addition to this, we introduce a cohesive framework for reasoning about partial identifiability and misspecification in IRL, together with several formal tools that can be used to easily derive the partial identifiability and misspecification robustness of new IRL models, or analyse other kinds of reward learning algorithms.

The aim of inverse reinforcement learning (IRL) is to infer an agent's preferences from observing their behaviour. Usually, preferences are modelled as a reward function, R, and behaviour is modelled as a policy, pi. One of the central difficulties in IRL is that multiple preferences may lead to the same observed behaviour. That is, R is typically underdetermined by pi, which means that R is only partially identifiable. Recent work has characterised the extent of this partial identifiability for different types of agents, including optimal and Boltzmann-rational agents. However, work so far has only considered agents that discount future reward exponentially: this is a serious limitation, especially given that extensive work in the behavioural sciences suggests that humans are better modelled as discounting hyperbolically. In this work, we newly characterise partial identifiability in IRL for agents with non-exponential discounting: our results are in particular relevant for hyperbolical discounting, but they also more generally apply to agents that use other types of (non-exponential) discounting. We significantly show that generally IRL is unable to infer enough information about R to identify the correct optimal policy, which entails that IRL alone can be insufficient to adequately characterise the preferences of such agents.

In reinforcement learning, specifying reward functions that capture the intended task can be very challenging. Reward learning aims to address this issue by *learning* the reward function. However, a learned reward model may have a low error on the data distribution, and yet subsequently produce a policy with large regret. We say that such a reward model has an *error-regret mismatch*. The main source of an error-regret mismatch is the distributional shift that commonly occurs during policy optimization. In this paper, we mathematically show that a sufficiently low expected test error of the reward model guarantees low worst-case regret, but that for any *fixed* expected test error, there exist realistic data distributions that allow for error-regret mismatch to occur. We then show that similar problems persist even when using policy regularization techniques, commonly employed in methods such as RLHF. We hope our results stimulate the theoretical and empirical study of improved methods to learn reward models, and better ways to measure their quality reliably.

Implementing a reward function that perfectly captures a complex task in the real world is impractical. As a result, it is often appropriate to think of the reward function as a *proxy* for the true objective rather than as its definition. We study this phenomenon through the lens of *Goodhart’s law*, which predicts that increasing optimisation of an imperfect proxy beyond some critical point decreases performance on the true objective. First, we propose a way to *quantify* the magnitude of this effect and *show empirically* that optimising an imperfect proxy reward often leads to the behaviour predicted by Goodhart’s law for a wide range of environments and reward functions. We then provide a *geometric explanation* for why Goodhart's law occurs in Markov decision processes. We use these theoretical insights to propose an *optimal early stopping method* that provably avoids the aforementioned pitfall and derive theoretical *regret bounds* for this method. Moreover, we derive a training method that maximises worst-case reward, for the setting where there is uncertainty about the true reward function. Finally, we evaluate our early stopping method experimentally. Our results support a foundation for a theoretically-principled study of reinforcement learning under reward misspecification.

Inverse reinforcement learning (IRL) aims to infer an agent's *preferences* (represented as a reward function $R$) from their *behaviour* (represented as a policy $\pi$). To do this, we need a *behavioural model* of how $\pi$ relates to $R$. In the current literature, the most common behavioural models are *optimality*, *Boltzmann-rationality*, and *causal entropy maximisation*. However, the true relationship between a human's preferences and their behaviour is much more complex than any of these behavioural models. This means that the behavioural models are *misspecified*, which raises the concern that they may lead to systematic errors if applied to real data. In this paper, we analyse how sensitive the IRL problem is to misspecification of the behavioural model. Specifically, we provide necessary and sufficient conditions that completely characterise how the observed data may differ from the assumed behavioural model without incurring an error above a given threshold. In addition to this, we also characterise the conditions under which a behavioural model is robust to small perturbations of the observed policy, and we analyse how robust many behavioural models are to misspecification of their parameter values (such as e.g. the discount rate). Our analysis suggests that the IRL problem is highly sensitive to misspecification, in the sense that very mild misspecification can lead to very large errors in the inferred reward function.

In order to solve a task using reinforcement learning, it is necessary to first formalise the goal of that task as a *reward function*. However, for many real-world tasks, it is very difficult to manually specify a reward function that never incentivises undesirable behaviour. As a result, it is increasingly popular to use *reward learning algorithms*, which attempt to *learn* a reward function from data. However, the theoretical foundations of reward learning are not yet well-developed. In particular, it is typically not known when a given reward learning algorithm with high probability will learn a reward function that is safe to optimise. This means that reward learning algorithms generally must be evaluated empirically, which is expensive, and that their failure modes are difficult to anticipate in advance. One of the roadblocks to deriving better theoretical guarantees is the lack of good methods for *quantifying* the difference between reward functions. In this paper we provide a solution to this problem, in the form of a class of pseudometrics on the space of all reward functions that we call STARC (STAndardised Reward Comparison) metrics. We show that STARC metrics induce both an upper and a lower bound on worst-case regret, which implies that our metrics are tight, and that any metric with the same properties must be bilipschitz equivalent to ours. Moreover, we also identify a number of issues with reward metrics proposed by earlier works. Finally, we evaluate our metrics empirically, to demonstrate their practical efficacy. STARC metrics can be used to make both theoretical and empirical analysis of reward learning algorithms both easier and more principled.

Most algorithms in reinforcement learning (RL) require that the objective is formalised with a Markovian reward function. However, it is well-known that certain tasks cannot be expressed by means of an objective in the Markov rewards formalism, motivating the study of alternative objective-specification formalisms in RL such as Linear Temporal Logic and Multi-Objective Reinforcement Learning. To date, there has not yet been any thorough analysis of how these formalisms relate to each other in terms of their expressivity. We fill this gap in the existing literature by providing a comprehensive comparison of 17 salient objective-specification formalisms. We place these formalisms in a preorder based on their expressive power, and present this preorder as a Hasse diagram. We find a variety of limitations for the different formalisms, and argue that no formalism is both dominantly expressive and straightforward to optimise with current techniques. For example, we prove that each of Regularised RL, (Outer) Nonlinear Markov Rewards, Reward Machines, Linear Temporal Logic, and Limit Average Rewards can express a task that the others cannot. The significance of our results is twofold. First, we identify important expressivity limitations to consider when specifying objectives for policy optimization. Second, our results highlight the need for future research which adapts reward learning to work with a greater variety of formalisms, since many existing reward learning methods assume that the desired objective takes a Markovian form. Our work contributes towards a more cohesive understanding of the costs and benefits of different RL objective-specification formalisms.

The aim of Inverse Reinforcement Learning (IRL) is to infer a reward function R from a policy pi. To do this, we need a model of how pi relates to R. In the current literature, the most common models are optimality, Boltzmann rationality, and causal entropy maximisation. One of the primary motivations behind IRL is to infer human preferences from human behaviour. However, the true relationship between human preferences and human behaviour is much more complex than any of the models currently used in IRL. This means that they are misspecified, which raises the worry that they might lead to unsound inferences if applied to real-world data. In this paper, we provide a mathematical analysis of how robust different IRL models are to misspecification, and answer precisely how the demonstrator policy may differ from each of the standard models before that model leads to faulty inferences about the reward function R. We also introduce a framework for reasoning about misspecification in IRL, together with formal tools that can be used to easily derive the misspecification robustness of new IRL models.

It is often very challenging to manually design reward functions for complex, real-world tasks. To solve this, one can instead use reward learning to infer a reward function from data. However, there are often multiple reward functions that fit the data equally well, even in the infinite-data limit. This means that the reward function is only partially identifiable. In this work, we formally characterise the partial identifiability of the reward function given several popular reward learning data sources, including expert demonstrations and trajectory comparisons. We also analyse the impact of this partial identifiability for several downstream tasks, such as policy optimisation. We unify our results in a framework for comparing data sources and downstream tasks by their invariances, with implications for the design and selection of data sources for reward learning.

In this paper, we study the expressivity of scalar, Markovian reward functions in Reinforcement Learning (RL), and identify several limitations to what they can express. Specifically, we look at three classes of RL tasks; multi-objective RL, risk-sensitive RL, and modal RL. For each class, we derive necessary and sufficient conditions that describe when a problem in this class can be expressed using a scalar, Markovian reward. Moreover, we find that scalar, Markovian rewards are unable to express most of the instances in each of these three classes. We thereby contribute to a more complete understanding of what standard reward functions can and cannot express. In addition to this, we also call attention to modal problems as a new class of problems, since they have so far not been given any systematic treatment in the RL literature. We also briefly outline some approaches for solving some of the problems we discuss, by means of bespoke RL algorithms.

In this work we introduce reinforcement learning techniques for solving lexicographic multi-objective problems. These are problems that involve multiple reward signals, and where the goal is to learn a policy that maximises the first reward signal, and subject to this constraint also maximises the second reward signal, and so on. We present a family of both action-value and policy gradient algorithms that can be used to solve such problems, and prove that they converge to policies that are lexicographically optimal. We evaluate the scalability and performance of these algorithms empirically, and demonstrate their applicability in practical settings. As a more specific application, we show how our algorithms can be used to impose safety constraints on the behaviour of an agent, and compare their performance in this context with that of other constrained reinforcement learning algorithms.

We provide the first formal definition of \textbf{reward hacking}, a phenomenon where optimizing an imperfect proxy reward function, $\mathcal{\tilde{R}}$, leads to poor performance according to the true reward function, $\mathcal{R}$. We say that a proxy is \textbf{unhackable} if increasing the expected proxy return can never decrease the expected true return.Intuitively, it might be possible to create an unhackable proxy by leaving some terms out of the reward function (making it ``narrower'') or overlooking fine-grained distinctions between roughly equivalent outcomes, but we show this is usually not the case.A key insight is that the linearity of reward (in state-action visit counts) makes unhackability a very strong condition. In particular, for the set of all stochastic policies, two reward functions can only be unhackable if one of them is constant.We thus turn our attention to deterministic policies and finite sets of stochastic policies, where non-trivial unhackable pairs always exist, and establish necessary and sufficient conditions for the existence of simplifications, an important special case of unhackability.Our results reveal a tension between using reward functions to specify narrow tasks and aligning AI systems with human values.

Newcomblike decision problems have been studied extensively in the decision theory literature, but they have so far been largely absent in the reinforcement learning literature. In this paper we study value-based reinforcement learning algorithms in the Newcomblike setting, and answer some of the fundamental theoretical questions about the behaviour of such algorithms in these environments. We show that a value-based reinforcement learning agent cannot converge to a policy that is not \emph{ratifiable}, i.e., does not only choose actions that are optimal given that policy. This gives us a powerful tool for reasoning about the limit behaviour of agents -- for example, it lets us show that there are Newcomblike environments in which a reinforcement learning agent cannot converge to any optimal policy. We show that a ratifiable policy always exists in our setting, but that there are cases in which a reinforcement learning agent normally cannot converge to it (and hence cannot converge at all). We also prove several results about the possible limit behaviours of agents in cases where they do not converge to any policy.

Deep neural networks (DNNs) generalise remarkably well in the overparameterised regime, suggesting a strong inductive bias towards functions with low generalisation error. We empirically investigate this bias by calculating, for a range of architectures and datasets, the probability $P_{SGD}(f\mid S)$ that an overparameterised DNN, trained with stochastic gradient descent (SGD) or one of its variants, converges on a function $f$ consistent with a training set $S$. We also use Gaussian processes to estimate the Bayesian posterior probability $P_{B}(f\mid S)$ that the DNN expresses $f$ upon random sampling of its parameters, conditioned on $S$. Our main findings are that $P_{SGD}(f\mid S)$ correlates remarkably well with $P_{B}(f\mid S)$ and that $P_{B}(f\mid S)$ is strongly biased towards low-error and low complexity functions. These results imply that strong inductive bias in the parameter-function map (which determines $P_{B}(f\mid S)$), rather than a special property of SGD, is the primary explanation for why DNNs generalise so well in the overparameterised regime. While our results suggest that the Bayesian posterior $P_{B}(f\mid S)$ is the first order determinant of $P_{SGD}(f\mid S)$, there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on $P_{SGD}(f\mid S)$ and/or $P_{B}(f\mid S)$, can shed light on the way that variations in architecture or hyperparameter settings such as batch size, learning rate, and optimiser choice, affect DNN performance.