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Joar Skalse, Alessandro Abate

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.

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Alexandros E. Tzikas, Licio Romao, Mert Pilanci, Alessandro Abate, Mykel J. Kochenderfer

Many machine learning applications require operating on a spatially distributed dataset. Despite technological advances, privacy considerations and communication constraints may prevent gathering the entire dataset in a central unit. In this paper, we propose a distributed sampling scheme based on the alternating direction method of multipliers, which is commonly used in the optimization literature due to its fast convergence. In contrast to distributed optimization, distributed sampling allows for uncertainty quantification in Bayesian inference tasks. We provide both theoretical guarantees of our algorithm's convergence and experimental evidence of its superiority to the state-of-the-art. For our theoretical results, we use convex optimization tools to establish a fundamental inequality on the generated local sample iterates. This inequality enables us to show convergence of the distribution associated with these iterates to the underlying target distribution in Wasserstein distance. In simulation, we deploy our algorithm on linear and logistic regression tasks and illustrate its fast convergence compared to existing gradient-based methods.

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Joar Skalse, Alessandro Abate

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.

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Rohan Mitta, Hosein Hasanbeig, Jun Wang, Daniel Kroening, Yiannis Kantaros, Alessandro Abate

This paper addresses the problem of maintaining safety during training in Reinforcement Learning (RL), such that the safety constraint violations are bounded at any point during learning. In a variety of RL applications the safety of the agent is particularly important, e.g. autonomous platforms or robots that work in proximity of humans. As enforcing safety during training might severely limit the agent's exploration, we propose here a new architecture that handles the trade-off between efficient progress and safety during exploration. As the exploration progresses, we update via Bayesian inference Dirichlet-Categorical models of the transition probabilities of the Markov decision process that describes the environment dynamics. This paper proposes a way to approximate moments of belief about the risk associated to the action selection policy. We construct those approximations, and prove the convergence results. We propose a novel method for leveraging the expectation approximations to derive an approximate bound on the confidence that the risk is below a certain level. This approach can be easily interleaved with RL and we present experimental results to showcase the performance of the overall architecture.

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Alec Edwards, Andrea Peruffo, Alessandro Abate

This paper presents Fossil 2.0, a new major release of a software tool for the synthesis of certificates (e.g., Lyapunov and barrier functions) for dynamical systems modelled as ordinary differential and difference equations. Fossil 2.0 is much improved from its original release, including new interfaces, a significantly expanded certificate portfolio, controller synthesis and enhanced extensibility. We present these new features as part of this tool paper. Fossil implements a counterexample-guided inductive synthesis (CEGIS) loop ensuring the soundness of the method. Our tool uses neural networks as templates to generate candidate functions, which are then formally proven by an SMT solver acting as an assertion verifier. Improvements with respect to the first release include a wider range of certificates, synthesis of control laws, and support for discrete-time models.

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Thom Badings, Nils Jansen, Licio Romao, Alessandro Abate

Automated synthesis of correct-by-construction controllers for autonomous systems is crucial for their deployment in safety-critical scenarios. Such autonomous systems are naturally modeled as stochastic dynamical models. The general problem is to compute a controller that provably satisfies a given task, represented as a probabilistic temporal logic specification. However, factors such as stochastic uncertainty, imprecisely known parameters, and hybrid features make this problem challenging. We have developed an abstraction framework that can be used to solve this problem under various modeling assumptions. Our approach is based on a robust finite-state abstraction of the stochastic dynamical model in the form of a Markov decision process with intervals of probabilities (iMDP). We use state-of-the-art verification techniques to compute an optimal policy on the iMDP with guarantees for satisfying the given specification. We then show that, by construction, we can refine this policy into a feedback controller for which these guarantees carry over to the dynamical model. In this short paper, we survey our recent research in this area and highlight two challenges (related to scalability and dealing with nonlinear dynamics) that we aim to address with our ongoing research.

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Matthew Wicker, Luca Laurenti, Andrea Patane, Nicola Paoletti, Alessandro Abate, Marta Kwiatkowska

Model-based reinforcement learning seeks to simultaneously learn the dynamics of an unknown stochastic environment and synthesise an optimal policy for acting in it. Ensuring the safety and robustness of sequential decisions made through a policy in such an environment is a key challenge for policies intended for safety-critical scenarios. In this work, we investigate two complementary problems: first, computing reach-avoid probabilities for iterative predictions made with dynamical models, with dynamics described by Bayesian neural network (BNN); second, synthesising control policies that are optimal with respect to a given reach-avoid specification (reaching a "target" state, while avoiding a set of "unsafe" states) and a learned BNN model. Our solution leverages interval propagation and backward recursion techniques to compute lower bounds for the probability that a policy's sequence of actions leads to satisfying the reach-avoid specification. Such computed lower bounds provide safety certification for the given policy and BNN model. We then introduce control synthesis algorithms to derive policies maximizing said lower bounds on the safety probability. We demonstrate the effectiveness of our method on a series of control benchmarks characterized by learned BNN dynamics models. On our most challenging benchmark, compared to purely data-driven policies the optimal synthesis algorithm is able to provide more than a four-fold increase in the number of certifiable states and more than a three-fold increase in the average guaranteed reach-avoid probability.

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Joar Skalse, Lucy Farnik, Sumeet Ramesh Motwani, Erik Jenner, Adam Gleave, Alessandro Abate

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 predict 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.

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Alec Edwards, Andrea Peruffo, Alessandro Abate

An emerging branch of control theory specialises in certificate learning, concerning the specification of a desired (possibly complex) system behaviour for an autonomous or control model, which is then analytically verified by means of a function-based proof. However, the synthesis of controllers abiding by these complex requirements is in general a non-trivial task and may elude the most expert control engineers. This results in a need for automatic techniques that are able to design controllers and to analyse a wide range of elaborate specifications. In this paper, we provide a general framework to encode system specifications and define corresponding certificates, and we present an automated approach to formally synthesise controllers and certificates. Our approach contributes to the broad field of safe learning for control, exploiting the flexibility of neural networks to provide candidate control and certificate functions, whilst using SMT-solvers to offer a formal guarantee of correctness. We test our framework by developing a prototype software tool, and assess its efficacy at verification via control and certificate synthesis over a large and varied suite of benchmarks.

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Alec Edwards, Mirco Giacobbe, Alessandro Abate

Neural abstractions have been recently introduced as formal approximations of complex, nonlinear dynamical models. They comprise a neural ODE and a certified upper bound on the error between the abstract neural network and the concrete dynamical model. So far neural abstractions have exclusively been obtained as neural networks consisting entirely of $ReLU$ activation functions, resulting in neural ODE models that have piecewise affine dynamics, and which can be equivalently interpreted as linear hybrid automata. In this work, we observe that the utility of an abstraction depends on its use: some scenarios might require coarse abstractions that are easier to analyse, whereas others might require more complex, refined abstractions. We therefore consider neural abstractions of alternative shapes, namely either piecewise constant or nonlinear non-polynomial (specifically, obtained via sigmoidal activations). We employ formal inductive synthesis procedures to generate neural abstractions that result in dynamical models with these semantics. Empirically, we demonstrate the trade-off that these different neural abstraction templates have vis-a-vis their precision and synthesis time, as well as the time required for their safety verification (done via reachability computation). We improve existing synthesis techniques to enable abstraction of higher-dimensional models, and additionally discuss the abstraction of complex neural ODEs to improve the efficiency of reachability analysis for these models.

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