Abstract We present —a Dafny-style verifier for imperative programs embedded in the Lean proof assistant. Like Dafny, supports reasoning about effectful programs featuring mutable state, loops, and non-determinism. Unlike Dafny, seamlessly combines automated SMT-based proofs with the interactive proof mode of the Lean proof assistant, in which it is embedded, thus enabling multi-modal proofs. Implemented as a Lean library, enjoys interaction with the rest of the Lean ecosystem, and in particular, with its automation tactics and rich library of mathematical theories. In this paper, we give a tour of ’s features, outline the techniques underlying its implementation, and evaluate its performance and expressivity in comparison with Dafny.

Multi-modal program verification is a process of validating code against its specification using both dynamic and symbolic techniques, and proving its correctness by a combination of automated and interactive machine-assisted tools. In order to be trustworthy, such verification tools must themselves come with formal soundness proofs, establishing that any program verified in them against a certain specification does not violate the specification’s statement when executed. Verification tools that are proven sound in a general-purpose proof assistant with a small trusted core are commonly referred to as foundational . We present a framework that facilitates and streamlines construction of program verifiers that are both foundational and multi-modal. Our approach adopts the well-known idea of monadic shallow embedding of an executable program semantics into the programming language of a theorem prover based on higher-order logic, in our case, the Lean proof assistant. We provide a library of monad transformers for such semantics, encoding a variety of computational effects, including state, divergence, exceptions, and non-determinism. The key theoretical innovation of our work are monad transformer algebras that enable automated derivation of the respective sound verification condition generators. We show that proofs of the resulting verification conditions enjoy automation using off-the-shelf SMT solvers and allow for an interactive proof mode when automation fails. To demonstrate versatility of our framework, we instantiated it to embed two foundational multi-modal verifiers into Lean for reasoning about (1) distributed protocol safety and (2) Dafny-style specifications of imperative programs, and used them to mechanically verify a number of non-trivial case studies.

Recent multi-task learning research argues against unitary scalarization, where training simply minimizes the sum of the task losses. Several ad-hoc multi-task optimization algorithms have instead been proposed, inspired by various hypotheses about what makes multi-task settings difficult. The majority of these optimizers require per-task gradients, and introduce significant memory, runtime, and implementation overhead. We show that unitary scalarization, coupled with standard regularization and stabilization techniques from single-task learning, matches or improves upon the performance of complex multi-task optimizers in popular supervised and reinforcement learning settings. We then present an analysis suggesting that many specialized multi-task optimizers can be partly interpreted as forms of regularization, potentially explaining our surprising results. We believe our results call for a critical reevaluation of recent research in the area.

Recent research has shown that graph neural networks (GNNs) can learn policies for locomotion control that are as effective as a typical multi-layer perceptron (MLP), with superior transfer and multi-task performance. However, results have so far been limited to training on small agents, with the performance of GNNs deteriorating rapidly as the number of sensors and actuators grows. A key motivation for the use of GNNs in the supervised learning setting is their applicability to large graphs, but this benefit has not yet been realised for locomotion control. We show that poor scaling in GNNs is a result of increasingly unstable policy updates, caused by overfitting in parts of the network during training. To combat this, we introduce Snowflake, a GNN training method for high-dimensional continuous control that freezes parameters in selected parts of the network. Snowflake significantly boosts the performance of GNNs for locomotion control on large agents, now matching the performance of MLPs while offering superior transfer properties.

This paper introduces the deep coordination graph (DCG) for collaborative multi-agent reinforcement learning. DCG strikes a flexible trade-off between representational capacity and generalization by factoring the joint value function of all agents according to a coordination graph into payoffs between pairs of agents. The value can be maximized by local message passing along the graph, which allows training of the value function end-to-end with Q-learning. Payoff functions are approximated with deep neural networks that employ parameter sharing and low-rank approximations to significantly improve sample efficiency. We show that DCG can solve predator-prey tasks that highlight the relative overgeneralization pathology, as well as challenging StarCraft II micromanagement tasks.

We present Graph-Q-SAT, a branching heuristic for a Boolean SAT solver trained with value-based reinforcement learning (RL) using Graph Neural Networks for function approximation. Solvers using Graph-Q-SAT are complete SAT solvers that either provide a satisfying assignment or proof of unsatisfiability, which is required for many SAT applications. The branching heuristics commonly used in SAT solvers make poor decisions during their warm-up period, whereas Graph-Q-SAT is trained to examine the structure of the particular problem instance to make better decisions early in the search. Training Graph-Q-SAT is data efficient and does not require elaborate dataset preparation or feature engineering. We train Graph-Q-SAT using RL interfacing with MiniSat solver and show that Graph-Q-SAT can reduce the number of iterations required to solve SAT problems by 2-3X. Furthermore, it generalizes to unsatisfiable SAT instances, as well as to problems with 5X more variables than it was trained on. We show that for larger problems, reductions in the number of iterations lead to wall clock time reductions, the ultimate goal when designing heuristics. We also show positive zero-shot transfer behavior when testing Graph-Q-SAT on a task family different from that used for training. While more work is needed to apply Graph-Q-SAT to reduce wall clock time in modern SAT solving settings, it is a compelling proof-of-concept showing that RL equipped with Graph Neural Networks can learn a generalizable branching heuristic for SAT search.

We propose CAVIA for meta-learning, a simple extension to MAML that is less prone to meta-overfitting, easier to parallelise, and more interpretable. CAVIA partitions the model parameters into two parts: context parameters that serve as additional input to the model and are adapted on individual tasks, and shared parameters that are meta-trained and shared across tasks. At test time, only the context parameters are updated, leading to a low-dimensional task representation. We show empirically that CAVIA outperforms MAML for regression, classification, and reinforcement learning. Our experiments also highlight weaknesses in current benchmarks, in that the amount of adaptation needed in some cases is small.

Learning from demonstration (LfD) is useful in settings where hand-coding behaviour or a reward function is impractical. It has succeeded in a wide range of problems but typically relies on manually generated demonstrations or specially deployed sensors and has not generally been able to leverage the copious demonstrations available in the wild: those that capture behaviours that were occurring anyway using sensors that were already deployed for another purpose, e.g., traffic camera footage capturing demonstrations of natural behaviour of vehicles, cyclists, and pedestrians. We propose video to behaviour (ViBe), a new approach to learn models of behaviour from unlabelled raw video data of a traffic scene collected from a single, monocular, initially uncalibrated camera with ordinary resolution. Our approach calibrates the camera, detects relevant objects, tracks them through time, and uses the resulting trajectories to perform LfD, yielding models of naturalistic behaviour. We apply ViBe to raw videos of a traffic intersection and show that it can learn purely from videos, without additional expert knowledge.

The performance of policy gradient methods is sensitive to hyperparameter settings that must be tuned for any new application. Widely used grid search methods for tuning hyperparameters are sample inefficient and computationally expensive. More advanced methods like Population Based Training that learn optimal schedules for hyperparameters instead of fixed settings can yield better results, but are also sample inefficient and computationally expensive. In this paper, we propose Hyperparameter Optimisation on the Fly (HOOF), a gradient-free algorithm that requires no more than one training run to automatically adapt the hyperparameter that affect the policy update directly through the gradient. The main idea is to use existing trajectories sampled by the policy gradient method to optimise a one-step improvement objective, yielding a sample and computationally efficient algorithm that is easy to implement. Our experimental results across multiple domains and algorithms show that using HOOF to learn these hyperparameter schedules leads to faster learning with improved performance.