Alessandro Abate

Lewis Hammond, James Fox, Tom Everitt et al. · deepmind

Causal reasoning and game-theoretic reasoning are fundamental topics in artificial intelligence, among many other disciplines: this paper is concerned with their intersection. Despite their importance, a formal framework that supports both these forms of reasoning has, until now, been lacking. We offer a solution in the form of (structural) causal games, which can be seen as extending Pearl's causal hierarchy to the game-theoretic domain, or as extending Koller and Milch's multi-agent influence diagrams to the causal domain. We then consider three key questions: i) How can the (causal) dependencies in games - either between variables, or between strategies - be modelled in a uniform, principled manner? ii) How may causal queries be computed in causal games, and what assumptions does this require? iii) How do causal games compare to existing formalisms? To address question i), we introduce mechanised games, which encode dependencies between agents' decision rules and the distributions governing the game. In response to question ii), we present definitions of predictions, interventions, and counterfactuals, and discuss the assumptions required for each. Regarding question iii), we describe correspondences between causal games and other formalisms, and explain how causal games can be used to answer queries that other causal or game-theoretic models do not support. Finally, we highlight possible applications of causal games, aided by an extensive open-source Python library.

Edwin Hamel-De le Court, Thom Badings, Alessandro Abate et al.

Shielding is an effective approach to formally guarantee the safety of reinforcement learning agents in Markov decision processes (MDPs). However, existing shielding techniques typically assume knowledge of the safety-relevant transition dynamics - a requirement that is seldom met in practice. To address this limitation, we introduce a novel shielding framework for robust MDPs (RMDPs), i.e., MDPs with sets of transition probabilities. We define safety as the satisfaction of a linear temporal logic (LTL) formula with a certain threshold probability under the worst-case transition probabilities of the RMDP. We prove that our shielding framework is both sound and optimal for the RMDP: every policy admissible by the shield is safe, and conversely, every safe RMDP policy is admissible by the shield. We combine our approach with existing sampling methods for learning transition probabilities of MDPs with probably approximately correct (PAC) guarantees. This combination enables the construction of shields for MDPs that, with high confidence, guarantee safety while remaining minimally restrictive. Our experiments show that our shields for learned RMDPs guarantee safety in unknown MDPs while recovering strong expected return as the number of samples increases.

Hosein Hasanbeig, Daniel Kroening, Alessandro Abate

LCRL is a software tool that implements model-free Reinforcement Learning (RL) algorithms over unknown Markov Decision Processes (MDPs), synthesising policies that satisfy a given linear temporal specification with maximal probability. LCRL leverages partially deterministic finite-state machines known as Limit Deterministic Buchi Automata (LDBA) to express a given linear temporal specification. A reward function for the RL algorithm is shaped on-the-fly, based on the structure of the LDBA. Theoretical guarantees under proper assumptions ensure the convergence of the RL algorithm to an optimal policy that maximises the satisfaction probability. We present case studies to demonstrate the applicability, ease of use, scalability, and performance of LCRL. Owing to the LDBA-guided exploration and LCRL model-free architecture, we observe robust performance, which also scales well when compared to standard RL approaches (whenever applicable to LTL specifications). Full instructions on how to execute all the case studies in this paper are provided on a GitHub page that accompanies the LCRL distribution www.github.com/grockious/lcrl.

Scott R. Jeen, Alessandro Abate, Jonathan M. Cullen

Heating and cooling systems in buildings account for 31\% of global energy use, much of which are regulated by Rule Based Controllers (RBCs) that neither maximise energy efficiency nor minimise emissions by interacting optimally with the grid. Control via Reinforcement Learning (RL) has been shown to significantly improve building energy efficiency, but existing solutions require access to building-specific simulators or data that cannot be expected for every building in the world. In response, we show it is possible to obtain emission-reducing policies without such knowledge a priori--a paradigm we call zero-shot building control. We combine ideas from system identification and model-based RL to create PEARL (Probabilistic Emission-Abating Reinforcement Learning) and show that a short period of active exploration is all that is required to build a performant model. In experiments across three varied building energy simulations, we show PEARL outperforms an existing RBC once, and popular RL baselines in all cases, reducing building emissions by as much as 31\% whilst maintaining thermal comfort. Our source code is available online via https://enjeeneer.io/projects/pearl .

Scott R. Jeen, Alessandro Abate, Jonathan M. Cullen

Heating and cooling systems in buildings account for 31% of global energy use, much of which are regulated by Rule Based Controllers (RBCs) that neither maximise energy efficiency nor minimise emissions by interacting optimally with the grid. Control via Reinforcement Learning (RL) has been shown to significantly improve building energy efficiency, but existing solutions require access to building-specific simulators or data that cannot be expected for every building in the world. In response, we show it is possible to obtain emission-reducing policies without such knowledge a priori--a paradigm we call zero-shot building control. We combine ideas from system identification and model-based RL to create PEARL (Probabilistic Emission-Abating Reinforcement Learning) and show that a short period of active exploration is all that is required to build a performant model. In experiments across three varied building energy simulations, we show PEARL outperforms an existing RBC once, and popular RL baselines in all cases, reducing building emissions by as much as 31% whilst maintaining thermal comfort. Our source code is available online via https://enjeeneer.io/projects/pearl/

Majid Zamani, Peyman Mohajerin Esfahani, Rupak Majumdar et al.

Symbolic approaches to the control design over complex systems employ the construction of finite-state models that are related to the original control systems, then use techniques from finite-state synthesis to compute controllers satisfying specifications given in a temporal logic, and finally translate the synthesized schemes back as controllers for the concrete complex systems. Such approaches have been successfully developed and implemented for the synthesis of controllers over non-probabilistic control systems. In this paper, we extend the technique to probabilistic control systems modeled by controlled stochastic differential equations. We show that for every stochastic control system satisfying a probabilistic variant of incremental input-to-state stability, and for every given precision $\varepsilon>0$, a finite-state transition system can be constructed, which is $\varepsilon$-approximately bisimilar (in the sense of moments) to the original stochastic control system. Moreover, we provide results relating stochastic control systems to their corresponding finite-state transition systems in terms of probabilistic bisimulation relations known in the literature. We demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic. The discussed technique enables a new, automated, correct-by-construction controller synthesis approach for stochastic control systems, which are common mathematical models employed in many safety critical systems subject to structured uncertainty and are thus relevant for cyber-physical applications.

Joar Skalse, Matthew Farrugia-Roberts, Stuart Russell et al.

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.

Joar Skalse, Lucy Farnik, Sumeet Ramesh Motwani et al. · berkeley

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.

Thom Badings, Licio Romao, Alessandro Abate et al.

Controllers for dynamical systems that operate in safety-critical settings must account for stochastic disturbances. Such disturbances are often modeled as process noise in a dynamical system, and common assumptions are that the underlying distributions are known and/or Gaussian. In practice, however, these assumptions may be unrealistic and can lead to poor approximations of the true noise distribution. We present a novel controller synthesis method that does not rely on any explicit representation of the noise distributions. In particular, we address the problem of computing a controller that provides probabilistic guarantees on safely reaching a target, while also avoiding unsafe regions of the state space. First, we abstract the continuous control system into a finite-state model that captures noise by probabilistic transitions between discrete states. As a key contribution, we adapt tools from the scenario approach to compute probably approximately correct (PAC) bounds on these transition probabilities, based on a finite number of samples of the noise. We capture these bounds in the transition probability intervals of a so-called interval Markov decision process (iMDP). This iMDP is, with a user-specified confidence probability, robust against uncertainty in the transition probabilities, and the tightness of the probability intervals can be controlled through the number of samples. We use state-of-the-art verification techniques to provide guarantees on the iMDP and compute a controller for which these guarantees carry over to the original control system. In addition, we develop a tailored computational scheme that reduces the complexity of the synthesis of these guarantees on the iMDP. Benchmarks on realistic control systems show the practical applicability of our method, even when the iMDP has hundreds of millions of transitions.

Zifan Wang, Yulong Gao, Siyi Wang et al.

Distributional reinforcement learning (DRL) enhances the understanding of the effects of the randomness in the environment by letting agents learn the distribution of a random return, rather than its expected value as in standard RL. At the same time, a main challenge in DRL is that policy evaluation in DRL typically relies on the representation of the return distribution, which needs to be carefully designed. In this paper, we address this challenge for a special class of DRL problems that rely on linear quadratic regulator (LQR) for control, advocating for a new distributional approach to LQR, which we call \emph{distributional LQR}. Specifically, we provide a closed-form expression of the distribution of the random return which, remarkably, is applicable to all exogenous disturbances on the dynamics, as long as they are independent and identically distributed (i.i.d.). While the proposed exact return distribution consists of infinitely many random variables, we show that this distribution can be approximated by a finite number of random variables, and the associated approximation error can be analytically bounded under mild assumptions. Using the approximate return distribution, we propose a zeroth-order policy gradient algorithm for risk-averse LQR using the Conditional Value at Risk (CVaR) as a measure of risk. Numerical experiments are provided to illustrate our theoretical results.

Ilya Tkachev, Alessandro Abate

This work is devoted to the formal verification of specifications over general discrete-time Markov processes, with an emphasis on infinite-horizon properties. These properties, formulated in a modal logic known as PCTL, can be expressed through value functions defined over the state space of the process. The main goal is to understand how structural features of the model (primarily the presence of absorbing sets) influence the uniqueness of the solutions of corresponding Bellman equations. Furthermore, this contribution shows that the investigation of these structural features leads to new computational techniques to calculate the specifications of interest: the emphasis is to derive approximation techniques with associated explicit convergence rates and formal error bounds.

Thom Badings, Licio Romao, Alessandro Abate et al.

Capturing uncertainty in models of complex dynamical systems is crucial to designing safe controllers. Stochastic noise causes aleatoric uncertainty, whereas imprecise knowledge of model parameters leads to epistemic uncertainty. Several approaches use formal abstractions to synthesize policies that satisfy temporal specifications related to safety and reachability. However, the underlying models exclusively capture aleatoric but not epistemic uncertainty, and thus require that model parameters are known precisely. Our contribution to overcoming this restriction is a novel abstraction-based controller synthesis method for continuous-state models with stochastic noise and uncertain parameters. By sampling techniques and robust analysis, we capture both aleatoric and epistemic uncertainty, with a user-specified confidence level, in the transition probability intervals of a so-called interval Markov decision process (iMDP). We synthesize an optimal policy on this iMDP, which translates (with the specified confidence level) to a feedback controller for the continuous model with the same performance guarantees. Our experimental benchmarks confirm that accounting for epistemic uncertainty leads to controllers that are more robust against variations in parameter values.

Joar Skalse, Alessandro Abate

The aim of Inverse Reinforcement Learning (IRL) is to infer a reward function $R$ from a policy $π$. To do this, we need a model of how $π$ 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.

Joar Skalse, Lewis Hammond, Charlie Griffin et al.

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, demonstrating their practical applicability. 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.

Alessandro Abate, Daniel Contro, Mirco Giacobbe et al.

Certification methods for stochastic systems provide sufficient proof rules, based on real-valued supermartingale certificates, to determine the almost-sure satisfaction of $ω$-regular properties (and therefore of linear temporal logic) over general state spaces, encompassing both countably infinite and continuous state spaces. Conversely, reinforcement learning (RL) methods for $ω$-regular tasks have received considerable attention, but they typically lack formal guarantees that the learned policy satisfies the specification, except possibly for finite state and action spaces. We bridge these two lines of research by establishing a novel theoretical connection: under an appropriate reward, the value function associated to a policy that almost surely satisfies an $ω$-regular property encodes a Streett supermartingale certificate for that specification. Our results, validated experimentally on finite Markov decision processes, hold for finite, countably infinite, and continuous state spaces, suggesting a principled route to certificate synthesis via RL.

Alessandro Abate, Alec Edwards, Mirco Giacobbe

We present a novel method for the safety verification of nonlinear dynamical models that uses neural networks to represent abstractions of their dynamics. Neural networks have extensively been used before as approximators; in this work, we make a step further and use them for the first time as abstractions. For a given dynamical model, our method synthesises a neural network that overapproximates its dynamics by ensuring an arbitrarily tight, formally certified bound on the approximation error. For this purpose, we employ a counterexample-guided inductive synthesis procedure. We show that this produces a neural ODE with non-deterministic disturbances that constitutes a formal abstraction of the concrete model under analysis. This guarantees a fundamental property: if the abstract model is safe, i.e., free from any initialised trajectory that reaches an undesirable state, then the concrete model is also safe. By using neural ODEs with ReLU activation functions as abstractions, we cast the safety verification problem for nonlinear dynamical models into that of hybrid automata with affine dynamics, which we verify using SpaceEx. We demonstrate that our approach performs comparably to the mature tool Flow* on existing benchmark nonlinear models. We additionally demonstrate and that it is effective on models that do not exhibit local Lipschitz continuity, which are out of reach to the existing technologies.

Erik Y. Wang, Sumeet Motwani, James V. Roggeveen et al.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of over 100 predominantly unsolved problems spanning 8 domains in computational and applied mathematics, paired with an open-source evaluation framework for automated verification. Our benchmark targets a class of problems where discovery is hard, requiring meaningful mathematical insight, but verification is computationally efficient and simple. Because these solutions are unknown, HorizonMath is immune to data contamination, and most state-of-the-art models score near 0%. Existing research-level benchmarks instead rely on formal proof verification or manual review, both of which are expensive to scale. Using this platform, we find two problems for which GPT 5.4 Pro proposes solutions that improve on the best-known published results, representing potential novel contributions (pending expert review). We release HorizonMath as an open challenge and a growing community resource, where correct solutions to problems in the unsolved problem classes could constitute novel results in the mathematical literature.

Matthew Wicker, Luca Laurenti, Andrea Patane et al.

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.

Adrien Banse, Licio Romao, Alessandro Abate et al.

We introduce an adaptive refinement procedure for smart, and scalable abstraction of dynamical systems. Our technique relies on partitioning the state space depending on the observation of future outputs. However, this knowledge is dynamically constructed in an adaptive, asymmetric way. In order to learn the optimal structure, we define a Kantorovich-inspired metric between Markov chains, and we use it as a loss function. Our technique is prone to data-driven frameworks, but not restricted to. We also study properties of the above mentioned metric between Markov chains, which we believe could be of application for wider purpose. We propose an algorithm to approximate it, and we show that our method yields a much better computational complexity than using classical linear programming techniques.

Alessandro Abate, Alec Edwards, Mirco Giacobbe et al.

We present a data-driven approach to the quantitative verification of probabilistic programs and stochastic dynamical models. Our approach leverages neural networks to compute tight and sound bounds for the probability that a stochastic process hits a target condition within finite time. This problem subsumes a variety of quantitative verification questions, from the reachability and safety analysis of discrete-time stochastic dynamical models, to the study of assertion-violation and termination analysis of probabilistic programs. We rely on neural networks to represent supermartingale certificates that yield such probability bounds, which we compute using a counterexample-guided inductive synthesis loop: we train the neural certificate while tightening the probability bound over samples of the state space using stochastic optimisation, and then we formally check the certificate's validity over every possible state using satisfiability modulo theories; if we receive a counterexample, we add it to our set of samples and repeat the loop until validity is confirmed. We demonstrate on a diverse set of benchmarks that, thanks to the expressive power of neural networks, our method yields smaller or comparable probability bounds than existing symbolic methods in all cases, and that our approach succeeds on models that are entirely beyond the reach of such alternative techniques.

Alessandro Abate, Yousif Almulla, James Fox et al.

Training reinforcement learning (RL) agents using scalar reward signals is often infeasible when an environment has sparse and non-Markovian rewards. Moreover, handcrafting these reward functions before training is prone to misspecification, especially when the environment's dynamics are only partially known. This paper proposes a novel pipeline for learning non-Markovian task specifications as succinct finite-state `task automata' from episodes of agent experience within unknown environments. We leverage two key algorithmic insights. First, we learn a product MDP, a model composed of the specification's automaton and the environment's MDP (both initially unknown), by treating the product MDP as a partially observable MDP and using the well-known Baum-Welch algorithm for learning hidden Markov models. Second, we propose a novel method for distilling the task automaton (assumed to be a deterministic finite automaton) from the learnt product MDP. Our learnt task automaton enables the decomposition of a task into its constituent sub-tasks, which improves the rate at which an RL agent can later synthesise an optimal policy. It also provides an interpretable encoding of high-level environmental and task features, so a human can readily verify that the agent has learnt coherent tasks with no misspecifications. In addition, we take steps towards ensuring that the learnt automaton is environment-agnostic, making it well-suited for use in transfer learning. Finally, we provide experimental results compared with two baselines to illustrate our algorithm's performance in different environments and tasks.

Luke Rickard, Thom Badings, Licio Romao et al.

Automated synthesis of provably correct controllers for cyber-physical systems is crucial for deployment in safety-critical scenarios. However, hybrid features and stochastic or unknown behaviours make this problem challenging. We propose a method for synthesising controllers for Markov jump linear systems (MJLSs), a class of discrete-time models for cyber-physical systems, so that they certifiably satisfy probabilistic computation tree logic (PCTL) formulae. An MJLS consists of a finite set of stochastic linear dynamics and discrete jumps between these dynamics that are governed by a Markov decision process (MDP). We consider the cases where the transition probabilities of this MDP are either known up to an interval or completely unknown. Our approach is based on a finite-state abstraction that captures both the discrete (mode-jumping) and continuous (stochastic linear) behaviour of the MJLS. We formalise this abstraction as an interval MDP (iMDP) for which we compute intervals of transition probabilities using sampling techniques from the so-called 'scenario approach', resulting in a probabilistically sound approximation. We apply our method to multiple realistic benchmark problems, in particular, a temperature control and an aerial vehicle delivery problem.

James Fox, Matt MacDermott, Lewis Hammond et al.

Multi-agent influence diagrams (MAIDs) are a popular game-theoretic model based on Bayesian networks. In some settings, MAIDs offer significant advantages over extensive-form game representations. Previous work on MAIDs has assumed that agents employ behavioural policies, which set independent conditional probability distributions over actions for each of their decisions. In settings with imperfect recall, however, a Nash equilibrium in behavioural policies may not exist. We overcome this by showing how to solve MAIDs with forgetful and absent-minded agents using mixed policies and two types of correlated equilibrium. We also analyse the computational complexity of key decision problems in MAIDs, and explore tractable cases. Finally, we describe applications of MAIDs to Markov games and team situations, where imperfect recall is often unavoidable.

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.

Patrick Benjamin, Alessandro Abate

Methods like multi-agent reinforcement learning struggle to scale with growing population size. Mean-field games (MFGs) are a game-theoretic approach that can circumvent this by finding a solution for an abstract infinite population, which can then be used as an approximate solution for the $N$-agent problem. However, classical mean-field algorithms usually only work under restrictive conditions. We take steps to address this by introducing networked communication to MFGs, in particular to settings that use a single, non-episodic run of $N$ decentralised agents to simulate the infinite population, as is likely to be most reasonable in real-world deployments. We prove that our architecture's sample guarantees lie between those of earlier theoretical algorithms for the centralised- and independent-learning architectures, varying dependent on network structure and the number of communication rounds. However, the sample guarantees of the three theoretical algorithms do not actually result in practical convergence times. We thus contribute practical enhancements to all three algorithms allowing us to present their first empirical demonstrations. We then show that in practical settings where the theoretical hyperparameters are not observed, giving fewer loops but poorer estimation of the Q-function, our communication scheme still respects the earlier theoretical analysis: it considerably accelerates learning over the independent case, which hardly seems to learn at all, and often performs similarly to the centralised case, while removing the restrictive assumption of the latter. We provide ablations and additional studies showing that our networked approach also has advantages over both alternatives in terms of robustness to update failures and to changes in population size.

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.

Irmak Sağlam, Mahdi Nazeri, Alessandro Abate et al.

We address the synthesis of control policies for unknown discrete-time stochastic dynamical systems to satisfy temporal logic objectives. We present a data-driven, abstraction-based control framework that integrates online learning with novel incremental game-solving. Under appropriate continuity assumptions, our method abstracts the system dynamics into a finite stochastic (2.5-player) game graph derived from data. Given a requirement over time on this graph, we compute the winning region -- i.e., the set of initial states from which the objective is satisfiable -- in the resulting game, together with a corresponding control policy. Our main contribution is the construction of abstractions, winning regions and control policies \emph{incrementally}, as data about the system dynamics accumulates. Concretely, our algorithm refines under- and over-approximations of reachable sets for each state-action pair as new data samples arrive. These refinements induce structural modifications in the game graph abstraction -- such as the addition or removal of nodes and edges -- which in turn modify the winning region. Crucially, we show that these updates are inherently monotonic: under-approximations only grow, over-approximations only shrink, and the winning region only expands. We exploit this monotonicity by defining an objective-induced ranking function on the nodes of the abstract game that increases monotonically as new data samples are incorporated. These ranks underpin our novel incremental game-solving algorithm, which employs customized gadgets (DAG-like subgames) within a rank-lifting algorithm to efficiently update the winning region. Numerical case studies demonstrate significant computational savings compared to the baseline approach, which re-solves the entire game from scratch whenever new data samples arrive.

Charlie Griffin, Louis Thomson, Buck Shlegeris et al.

To evaluate the safety and usefulness of deployment protocols for untrusted AIs, AI Control uses a red-teaming exercise played between a protocol designer and an adversary. This paper introduces AI-Control Games, a formal decision-making model of the red-teaming exercise as a multi-objective, partially observable, stochastic game. We also introduce methods for finding optimal protocols in AI-Control Games, by reducing them to a set of zero-sum partially observable stochastic games. We apply our formalism to model, evaluate and synthesise protocols for deploying untrusted language models as programming assistants, focusing on Trusted Monitoring protocols, which use weaker language models and limited human assistance. Finally, we demonstrate the utility of our formalism by showcasing improvements over empirical studies in existing settings, evaluating protocols in new settings, and analysing how modelling assumptions affect the safety and usefulness of protocols.

Daniel Jarne Ornia, Licio Romao, Lewis Hammond et al.

Policy robustness in Reinforcement Learning may not be desirable at any cost: the alterations caused by robustness requirements from otherwise optimal policies should be explainable, quantifiable and formally verifiable. In this work we study how policies can be maximally robust to arbitrary observational noise by analysing how they are altered by this noise through a stochastic linear operator interpretation of the disturbances, and establish connections between robustness and properties of the noise kernel and of the underlying MDPs. Then, we construct sufficient conditions for policy robustness, and propose a robustness-inducing scheme, applicable to any policy gradient algorithm, that formally trades off expected policy utility for robustness through lexicographic optimisation, while preserving convergence and sub-optimality in the policy synthesis.

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.

Alessandro Abate, Mirco Giacobbe, Sergey Ichtchenko et al.

We introduce a general methodology for the construction of sound and complete proof rules for the almost-sure and quantitative acceptance of reactivity properties on time-homogeneous Markov chains with general state spaces. Reactivity captures $ω$-regular properties and subsumes linear temporal logic. Our core technical result establishes that every reactivity property admits decomposition into multiple obligations of almost-sure termination into absorbing regions, and that appropriate absorbing regions always exist on general state spaces. This enables the extension of every complete proof rule for almost-sure termination into a proof rule for reactivity that is complete in the almost-sure case, and complete up to an arbitrarily small $\varepsilon$-approximation in the quantitative case. We apply our new methodology to recent results on sound and complete supermartingale certificates for almost-sure termination in the special case of countably infinite state spaces, alongside standard results on quantitative safety. As a result, we obtain the first sound and complete supermartingale certificates for almost-sure $ω$-regular properties and the first sound and $\varepsilon$-complete supermartingale certificates for quantitative $ω$-regular properties on time-homogeneous Markov chains with countably infinite state spaces.

Alessandro Abate, Giuseppe De Giacomo, Mathias Jackermeier et al.

We study multi-task reinforcement learning (RL), a setting in which an agent learns a single, universal policy capable of generalising to arbitrary, possibly unseen tasks. We consider tasks specified as linear temporal logic (LTL) formulae, which are commonly used in formal methods to specify properties of systems, and have recently been successfully adopted in RL. In this setting, we present a novel task embedding technique leveraging a new generation of semantic LTL-to-automata translations, originally developed for temporal synthesis. The resulting semantically labelled automata contain rich, structured information in each state that allow us to (i) compute the automaton efficiently on-the-fly, (ii) extract expressive task embeddings used to condition the policy, and (iii) naturally support full LTL. Experimental results in a variety of domains demonstrate that our approach achieves state-of-the-art performance and is able to scale to complex specifications where existing methods fail.

Alessandro Riccardi, Thom Badings, Luca Laurenti et al.

The deployment of autonomous systems in safety-critical environments requires control policies that guarantee satisfaction of complex control specifications. These systems are commonly modeled as nonlinear discrete-time stochastic systems. A~popular approach to computing a policy that provably satisfies a complex control specification is to construct a finite-state abstraction, often represented as a Markov decision process (MDP) with intervals of transition probabilities, i.e., an interval MDP (IMDP). However, existing abstraction techniques compute a \emph{single policy}, thus leaving no room for online cost or performance optimization, e.g., of energy consumption. To overcome this limitation, we propose a novel IMDP abstraction technique that yields a \emph{set of policies}, each of which satisfies the control specification with a certain minimum probability. We can thus use any online control algorithm to search through this set of verified policies while retaining the guaranteed satisfaction probability of the entire policy set. In particular, we employ model predictive control (MPC) to minimize a desired cost function that is independent of the control specification considered in the abstraction. Our experiments demonstrate that our approach yields better control performance than state-of-the-art single-policy abstraction techniques, with a small degradation of the guarantees.

Yannik Schnitzer, Alessandro Abate, David Parker

Learning-based approaches to verifying unknown Markov decision processes (MDPs) often employ uncertain MDPs. These models use, for example, confidence intervals to capture transition uncertainty and allow synthesis of policies that are robust to this uncertainty. However, this approach typically quantifies uncertainty independently for individual transition probabilities, ignoring dependencies due to shared latent quantities. We propose to learn such models using parametric MDPs (pMDPs), where transition probabilities are expressions over a set of parameters. We project statistical uncertainty from empirical transition frequencies onto the pMDP's parameter space, yielding a probably approximately correct (PAC) uncertainty model for the underlying MDP that respects the algebraic dependencies between transitions. The resulting models are algorithmically challenging to solve, so we propose a hierarchy of sound polytopic outer approximations of the induced confidence set. We implement and evaluate our approach, demonstrating substantially tighter uncertainty estimates than classical interval-based uncertain MDP learning techniques.

Thom Badings, Nils Jansen, Licio Romao et al.

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.

Alex Mallen, Charlie Griffin, Misha Wagner et al.

An AI control protocol is a plan for usefully deploying AI systems that aims to prevent an AI from intentionally causing some unacceptable outcome. This paper investigates how well AI systems can generate and act on their own strategies for subverting control protocols whilst operating statelessly (without shared memory between contexts). To do this, an AI system may need to reliably generate optimal plans in each context, take actions with well-calibrated probabilities, and coordinate plans with other instances of itself without communicating. We develop Subversion Strategy Eval, a suite of eight environments, covering a range of protocols and strategic capabilities, and six sets of affordances that help isolate individual capabilities. We implement the evaluation in Inspect-AI and release it open-source. We evaluate Claude 3.5 models, including helpful-only versions, as well as OpenAI reasoning models. None of the models demonstrate substantial capability in strategizing to subvert control protocols statelessly. However, providing models with additional affordances, such as the ability to share a plan between contexts, can substantially improve performance. We hope our evaluations can act as a leading indicator for when models are capable of subverting control protocols and also relax the worst-case assumption of perfect strategic ability in AI control evaluations.

Ondrej Bajgar, Alessandro Abate, Konstantinos Gatsis et al.

The goal of Bayesian inverse reinforcement learning (IRL) is recovering a posterior distribution over reward functions using a set of demonstrations from an expert optimizing for a reward unknown to the learner. The resulting posterior over rewards can then be used to synthesize an apprentice policy that performs well on the same or a similar task. A key challenge in Bayesian IRL is bridging the computational gap between the hypothesis space of possible rewards and the likelihood, often defined in terms of Q values: vanilla Bayesian IRL needs to solve the costly forward planning problem - going from rewards to the Q values - at every step of the algorithm, which may need to be done thousands of times. We propose to solve this by a simple change: instead of focusing on primarily sampling in the space of rewards, we can focus on primarily working in the space of Q-values, since the computation required to go from Q-values to reward is radically cheaper. Furthermore, this reversion of the computation makes it easy to compute the gradient allowing efficient sampling using Hamiltonian Monte Carlo. We propose ValueWalk - a new Markov chain Monte Carlo method based on this insight - and illustrate its advantages on several tasks.

Nikolaus Vertovec, Frederik Baymler Mathiesen, Thom Badings et al.

Control barrier functions (CBFs) are a popular tool for safety certification of nonlinear dynamical control systems. Recently, CBFs represented as neural networks have shown great promise due to their expressiveness and applicability to a broad class of dynamics and safety constraints. However, verifying that a trained neural network is indeed a valid CBF is a computational bottleneck that limits the size of the networks that can be used. To overcome this limitation, we present a novel framework for verifying neural CBFs based on piecewise linear upper and lower bounds on the conditions required for a neural network to be a CBF. Our approach is rooted in linear bound propagation (LBP) for neural networks, which we extend to compute bounds on the gradients of the network. Combined with McCormick relaxation, we derive linear upper and lower bounds on the CBF conditions, thereby eliminating the need for computationally expensive verification procedures. Our approach applies to arbitrary control-affine systems and a broad range of nonlinear activation functions. To reduce conservatism, we develop a parallelizable refinement strategy that adaptively refines the regions over which these bounds are computed. Our approach scales to larger neural networks than state-of-the-art verification procedures for CBFs, as demonstrated by our numerical experiments.

Yannik Schnitzer, Alessandro Abate, David Parker

We present a data-driven approach for producing policies that are provably robust across unknown stochastic environments. Existing approaches can learn models of a single environment as an interval Markov decision processes (IMDP) and produce a robust policy with a probably approximately correct (PAC) guarantee on its performance. However these are unable to reason about the impact of environmental parameters underlying the uncertainty. We propose a framework based on parametric Markov decision processes (MDPs) with unknown distributions over parameters. We learn and analyse IMDPs for a set of unknown sample environments induced by parameters. The key challenge is then to produce meaningful performance guarantees that combine the two layers of uncertainty: (1) multiple environments induced by parameters with an unknown distribution; (2) unknown induced environments which are approximated by IMDPs. We present a novel approach based on scenario optimisation that yields a single PAC guarantee quantifying the risk level for which a specified performance level can be assured in unseen environments, plus a means to trade-off risk and performance. We implement and evaluate our framework using multiple robust policy generation methods on a range of benchmarks. We show that our approach produces tight bounds on a policy's performance with high confidence.

Jacques Cloete, Mathias Jackermeier, Ioannis Havoutis et al.

A central challenge in multi-task reinforcement learning (RL) is to train generalist policies capable of performing tasks not seen during training. To facilitate such generalization, linear temporal logic (LTL) has recently emerged as a powerful formalism for specifying structured, temporally extended tasks to RL agents. While existing approaches to LTL-guided multi-task RL demonstrate successful generalization across LTL specifications, they are unable to generalize to unseen vocabularies of propositions (or "symbols"), which describe high-level events in LTL. We present PlatoLTL, a novel approach that enables policies to zero-shot generalize not only compositionally across LTL formula structures, but also parametrically across propositions. We achieve this by treating propositions as instances of parameterized predicates rather than discrete symbols, allowing policies to learn shared structure across related propositions. We propose a novel architecture that embeds and composes predicates to represent LTL specifications, and demonstrate successful zero-shot generalization to novel propositions and tasks across challenging environments.

Nelson Gardner-Challis, Jonathan Bostock, Georgiy Kozhevnikov et al.

AIs are increasingly being deployed with greater autonomy and capabilities, which increases the risk that a misaligned AI may be able to cause catastrophic harm. Untrusted monitoring -- using one untrusted model to oversee another -- is one approach to reducing risk. Justifying the safety of an untrusted monitoring deployment is challenging because developers cannot safely deploy a misaligned model to test their protocol directly. In this paper, we develop upon existing methods for rigorously demonstrating safety based on pre-deployment testing. We relax assumptions that previous AI control research made about the collusion strategies a misaligned AI might use to subvert untrusted monitoring. We develop a taxonomy covering passive self-recognition, causal collusion (hiding pre-shared signals), acausal collusion (hiding signals via Schelling points), and combined strategies. We create a safety case sketch to clearly present our argument, explicitly state our assumptions, and highlight unsolved challenges. We identify conditions under which passive self-recognition could be a more effective collusion strategy than those studied previously. Our work builds towards more robust evaluations of untrusted monitoring.

Lewis Hammond, James Fox, Tom Everitt et al.

Multi-agent influence diagrams (MAIDs) are a popular form of graphical model that, for certain classes of games, have been shown to offer key complexity and explainability advantages over traditional extensive form game (EFG) representations. In this paper, we extend previous work on MAIDs by introducing the concept of a MAID subgame, as well as subgame perfect and trembling hand perfect equilibrium refinements. We then prove several equivalence results between MAIDs and EFGs. Finally, we describe an open source implementation for reasoning about MAIDs and computing their equilibria.

Francesco Cosentino, Harald Oberhauser, Alessandro Abate

Given a discrete probability measure supported on $N$ atoms and a set of $n$ real-valued functions, there exists a probability measure that is supported on a subset of $n+1$ of the original $N$ atoms and has the same mean when integrated against each of the $n$ functions. If $ N \gg n$ this results in a huge reduction of complexity. We give a simple geometric characterization of barycenters via negative cones and derive a randomized algorithm that computes this new measure by "greedy geometric sampling". We then study its properties, and benchmark it on synthetic and real-world data to show that it can be very beneficial in the $N\gg n$ regime. A Python implementation is available at \url{https://github.com/FraCose/Recombination_Random_Algos}.

David "davidad" Dalrymple, Joar Skalse, Yoshua Bengio et al. · mit

Ensuring that AI systems reliably and robustly avoid harmful or dangerous behaviours is a crucial challenge, especially for AI systems with a high degree of autonomy and general intelligence, or systems used in safety-critical contexts. In this paper, we will introduce and define a family of approaches to AI safety, which we will refer to as guaranteed safe (GS) AI. The core feature of these approaches is that they aim to produce AI systems which are equipped with high-assurance quantitative safety guarantees. This is achieved by the interplay of three core components: a world model (which provides a mathematical description of how the AI system affects the outside world), a safety specification (which is a mathematical description of what effects are acceptable), and a verifier (which provides an auditable proof certificate that the AI satisfies the safety specification relative to the world model). We outline a number of approaches for creating each of these three core components, describe the main technical challenges, and suggest a number of potential solutions to them. We also argue for the necessity of this approach to AI safety, and for the inadequacy of the main alternative approaches.

Patrick Benjamin, Alessandro Abate

Recent algorithms allow decentralised agents, possibly connected via a communication network, to learn equilibria in mean-field games from a non-episodic run of the empirical system. However, these algorithms are for tabular settings: this computationally limits the size of agents' observation space, meaning the algorithms cannot handle anything but small state spaces, nor generalise beyond policies depending only on the agent's local state to so-called 'population-dependent' policies. We address this limitation by introducing function approximation to the existing setting, drawing on the Munchausen Online Mirror Descent method that has previously been employed only in finite-horizon, episodic, centralised settings. While this permits us to include the mean field in the observation for players' policies, it is unrealistic to assume decentralised agents have access to this global information: we therefore also provide new algorithms allowing agents to locally estimate the global empirical distribution, and to improve this estimate via inter-agent communication. We prove theoretically that exchanging policy information helps networked agents outperform both independent and even centralised agents in function-approximation settings. Our experiments demonstrate this happening empirically, and show that the communication network allows decentralised agents to estimate the mean field for population-dependent policies.

Christoph Weinhuber, Yannik Schnitzer, Alessandro Abate et al.

We study LTLf synthesis with multiple properties, where satisfying all properties may be impossible. Instead of enumerating subsets of properties, we compute in one fixed-point computation the relation between product-game states and the goal sets that are realizable from them, and we synthesize strategies achieving maximal realizable sets. We develop a fully symbolic algorithm that introduces Boolean goal variables and exploits monotonicity to represent exponentially many goal combinations compactly. Our approach substantially outperforms enumeration-based baselines, with speedups of up to two orders of magnitude.

Yannik Schnitzer, Mathias Jackermeier, Alessandro Abate et al.

Multi-task reinforcement learning trains generalist policies that can execute multiple tasks. While recent years have seen significant progress, existing approaches rarely provide formal performance guarantees, which are indispensable when deploying policies in safety-critical settings. We present an approach for computing high-confidence guarantees on the performance of a multi-task policy on tasks not seen during training. Concretely, we introduce a new generalisation bound that composes (i) per-task lower confidence bounds from finitely many rollouts with (ii) task-level generalisation from finitely many sampled tasks, yielding a high-confidence guarantee for new tasks drawn from the same arbitrary and unknown distribution. Across state-of-the-art multi-task RL methods, we show that the guarantees are theoretically sound and informative at realistic sample sizes.

Mattia Giuri, Mathias Jackermeier, Alessandro Abate

Linear temporal logic (LTL) is a compelling framework for specifying complex, structured tasks for reinforcement learning (RL) agents. Recent work has shown that interpreting LTL instructions as finite automata, which can be seen as high-level programs monitoring task progress, enables learning a single generalist policy capable of executing arbitrary instructions at test time. However, existing approaches fall short in environments where multiple high-level events (i.e., atomic propositions) can be true at the same time and potentially interact in complicated ways. In this work, we propose a novel approach to learning a multi-task policy for following arbitrary LTL instructions that addresses this shortcoming. Our method conditions the policy on sequences of simple Boolean formulae, which directly align with transitions in the automaton, and are encoded via a graph neural network (GNN) to yield structured task representations. Experiments in a complex chess-based environment demonstrate the advantages of our approach.

Alessandro Abate, Mirco Giacobbe, Yannik Schnitzer

We introduce a data-driven approach to computing finite bisimulations for state transition systems with very large, possibly infinite state space. Our novel technique computes stutter-insensitive bisimulations of deterministic systems, which we characterize as the problem of learning a state classifier together with a ranking function for each class. Our procedure learns a candidate state classifier and candidate ranking functions from a finite dataset of sample states; then, it checks whether these generalise to the entire state space using satisfiability modulo theory solving. Upon the affirmative answer, the procedure concludes that the classifier constitutes a valid stutter-insensitive bisimulation of the system. Upon a negative answer, the solver produces a counterexample state for which the classifier violates the claim, adds it to the dataset, and repeats learning and checking in a counterexample-guided inductive synthesis loop until a valid bisimulation is found. We demonstrate on a range of benchmarks from reactive verification and software model checking that our method yields faster verification results than alternative state-of-the-art tools in practice. Our method produces succinct abstractions that enable an effective verification of linear temporal logic without next operator, and are interpretable for system diagnostics.

Alessandro Abate, Sergiy Bogomolov, Alec Edwards et al.

We present a novel technique for online safety verification of autonomous systems, which performs reachability analysis efficiently for both bounded and unbounded horizons by employing neural barrier certificates. Our approach uses barrier certificates given by parameterized neural networks that depend on a given initial set, unsafe sets, and time horizon. Such networks are trained efficiently offline using system simulations sampled from regions of the state space. We then employ a meta-neural network to generalize the barrier certificates to state space regions that are outside the training set. These certificates are generated and validated online as sound over-approximations of the reachable states, thus either ensuring system safety or activating appropriate alternative actions in unsafe scenarios. We demonstrate our technique on case studies from linear models to nonlinear control-dependent models for online autonomous driving scenarios.