A key challenge in automated formal reasoning is the intractable search space, which grows exponentially with the depth of the proof. This branching is caused by the large number of candidate proof tactics which can be applied to a given goal. Nonetheless, many of these tactics are semantically similar or lead to an execution error, wasting valuable resources in both cases. We address the problem of effectively pruning this search, using only synthetic data generated from previous proof attempts. We first demonstrate that it is possible to generate semantically aware tactic representations which capture the effect on the proving environment, likelihood of success and execution time. We then propose a novel filtering mechanism which leverages these representations to select semantically diverse and high quality tactics, using Determinantal Point Processes. Our approach, 3D-Prover, is designed to be general, and to augment any underlying tactic generator. We demonstrate the effectiveness of 3D-Prover on the miniF2F-valid and miniF2F-test benchmarks by augmenting the ReProver LLM. We show that our approach leads to an increase in the overall proof rate, as well as a significant improvement in the tactic success rate, execution time and diversity.

Approximate k-Nearest Neighbour (ANN) methods are often used for mining information and aiding machine learning on large scale high-dimensional datasets. ANN methods typically differ in the index structure used for accelerating searches, resulting in various recall/runtime trade-off points. For applications with static datasets, runtime constraints and dataset properties can be used to empirically select an ANN method with suitable operating characteristics. However, for applications with dynamic datasets, which are subject to frequent online changes (like addition of new samples), there is currently no consensus as to which ANN methods are most suitable. Traditional evaluation approaches do not consider the computational costs of updating the index structure, as well as the rate and size of index updates. To address this, we empirically evaluate 5 popular ANN methods on two main applications (online data collection and online feature learning) while taking into account these considerations. Two dynamic datasets are used, derived from the SIFT1M dataset with 1 million samples and the DEEP1B dataset with 1 billion samples. The results indicate that the often used k-d trees method is not suitable on dynamic datasets as it is slower than a straightforward baseline exhaustive search method. For online data collection, the Hierarchical Navigable Small World Graphs method achieves a consistent speedup over baseline across a wide range of recall rates. For online feature learning, the Scalable Nearest Neighbours method is faster than baseline for recall rates below 75%.

Artificial Intelligence for Theorem Proving has given rise to a plethora of benchmarks and methodologies, particularly in Interactive Theorem Proving (ITP). Research in the area is fragmented, with a diverse set of approaches being spread across several ITP systems. This presents a significant challenge to the comparison of methods, which are often complex and difficult to replicate. Addressing this, we present BAIT, a framework for fair and streamlined comparison of learning approaches in ITP. We demonstrate BAIT's capabilities with an in-depth comparison, across several ITP benchmarks, of state-of-the-art architectures applied to the problem of formula embedding. We find that Structure Aware Transformers perform particularly well, improving on techniques associated with the original problem sets. BAIT also allows us to assess the end-to-end proving performance of systems built on interactive environments. This unified perspective reveals a novel end-to-end system that improves on prior work. We also provide a qualitative analysis, illustrating that improved performance is associated with more semantically-aware embeddings. By streamlining the implementation and comparison of Machine Learning algorithms in the ITP context, we anticipate BAIT will be a springboard for future research.

We study the problem of imputing missing values in a dataset, which has important applications in many domains. The key to missing value imputation is to capture the data distribution with incomplete samples and impute the missing values accordingly. In this paper, by leveraging the fact that any two batches of data with missing values come from the same data distribution, we propose to impute the missing values of two batches of samples by transforming them into a latent space through deep invertible functions and matching them distributionally. To learn the transformations and impute the missing values simultaneously, a simple and well-motivated algorithm is proposed. Our algorithm has fewer hyperparameters to fine-tune and generates high-quality imputations regardless of how missing values are generated. Extensive experiments over a large number of datasets and competing benchmark algorithms show that our method achieves state-of-the-art performance.

Reinforcement learning can effectively learn amortised design policies for designing sequences of experiments. However, current methods rely on contrastive estimators of expected information gain, which require an exponential number of contrastive samples to achieve an unbiased estimation. We propose an alternative lower bound estimator, based on the cross-entropy of the joint model distribution and a flexible proposal distribution. This proposal distribution approximates the true posterior of the model parameters given the experimental history and the design policy. Our estimator requires no contrastive samples, can achieve more accurate estimates of high information gains, allows learning of superior design policies, and is compatible with implicit probabilistic models. We assess our algorithm's performance in various tasks, including continuous and discrete designs and explicit and implicit likelihoods.

Bayesian approaches developed to solve the optimal design of sequential experiments are mathematically elegant but computationally challenging. Recently, techniques using amortization have been proposed to make these Bayesian approaches practical, by training a parameterized policy that proposes designs efficiently at deployment time. However, these methods may not sufficiently explore the design space, require access to a differentiable probabilistic model and can only optimize over continuous design spaces. Here, we address these limitations by showing that the problem of optimizing policies can be reduced to solving a Markov decision process (MDP). We solve the equivalent MDP with modern deep reinforcement learning techniques. Our experiments show that our approach is also computationally efficient at deployment time and exhibits state-of-the-art performance on both continuous and discrete design spaces, even when the probabilistic model is a black box.

We propose a novel approach to interactive theorem-proving (ITP) using deep reinforcement learning. Unlike previous work, our framework is able to prove theorems both end-to-end and from scratch (i.e., without relying on example proofs from human experts). We formulate the process of ITP as a Markov decision process (MDP) in which each state represents a set of potential derivation paths. The agent learns to select promising derivations as well as appropriate tactics within each derivation using deep policy gradients. This structure allows us to introduce a novel backtracking mechanism which enables the agent to efficiently discard (predicted) dead-end derivations and restart the derivation from promising alternatives. Experimental results show that the framework provides comparable performance to that of the approaches that use human experts, and that it is also capable of proving theorems that it has never seen during training. We further elaborate the role of each component of the framework using ablation studies.

Neuroscience studies of human decision-making abilities commonly involve subjects completing a decision-making task while BOLD signals are recorded using fMRI. Hypotheses are tested about which brain regions mediate the effect of past experience, such as rewards, on future actions. One standard approach to this is model-based fMRI data analysis, in which a model is fitted to the behavioral data, i.e., a subject's choices, and then the neural data are parsed to find brain regions whose BOLD signals are related to the model's internal signals. However, the internal mechanics of such purely behavioral models are not constrained by the neural data, and therefore might miss or mischaracterize aspects of the brain. To address this limitation, we introduce a new method using recurrent neural network models that are flexible enough to be jointly fitted to the behavioral and neural data. We trained a model so that its internal states were suitably related to neural activity during the task, while at the same time its output predicted the next action a subject would execute. We then used the fitted model to create a novel visualization of the relationship between the activity in brain regions at different times following a reward and the choices the subject subsequently made. Finally, we validated our method using a previously published dataset. We found that the model was able to recover the underlying neural substrates that were discovered by explicit model engineering in the previous work, and also derived new results regarding the temporal pattern of brain activity.

Neuroscience studies of human decision-making abilities commonly involve subjects completing a decision-making task while BOLD signals are recorded using fMRI. Hypotheses are tested about which brain regions mediate the effect of past experience, such as rewards, on future actions. One standard approach to this is model-based fMRI data analysis, in which a model is fitted to the behavioral data, i.e., a subject's choices, and then the neural data are parsed to find brain regions whose BOLD signals are related to the model's internal signals. However, the internal mechanics of such purely behavioral models are not constrained by the neural data, and therefore might miss or mischaracterize aspects of the brain. To address this limitation, we introduce a new method using recurrent neural network models that are flexible enough to be jointly fitted to the behavioral and neural data. We trained a model so that its internal states were suitably related to neural activity during the task, while at the same time its output predicted the next action a subject would execute. We then used the fitted model to create a novel visualization of the relationship between the activity in brain regions at different times following a reward and the choices the subject subsequently made. Finally, we validated our method using a previously published dataset. We found that the model was able to recover the underlying neural substrates that were discovered by explicit model engineering in the previous work, and also derived new results regarding the temporal pattern of brain activity.

We propose a network structure discovery model for continuous observations that generalizes linear causal models by incorporating a Gaussian process (GP) prior on a network-independent component, and random sparsity and weight matrices as the network-dependent parameters. This approach provides flexible modeling of network-independent trends in the observations as well as uncertainty quantification around the discovered network structure. We establish a connection between our model and multi-task GPs and develop an efficient stochastic variational inference algorithm for it. Furthermore, we formally show that our approach is numerically stable and in fact numerically easy to carry out almost everywhere on the support of the random variables involved. Finally, we evaluate our model on three applications, showing that it outperforms previous approaches. We provide a qualitative and quantitative analysis of the structures discovered for domains such as the study of the full genome regulation of the yeast Saccharomyces cerevisiae.