Alex Davies

Johannes Bausch, Andrew W Senior, Francisco J H Heras et al.

Quantum error-correction is a prerequisite for reliable quantum computation. Towards this goal, we present a recurrent, transformer-based neural network which learns to decode the surface code, the leading quantum error-correction code. Our decoder outperforms state-of-the-art algorithmic decoders on real-world data from Google's Sycamore quantum processor for distance 3 and 5 surface codes. On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk, leakage, and analog readout signals, and sustains its accuracy far beyond the 25 cycles it was trained on. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.

Alex Davies, Nirav Ajmeri

Social network analysis faces profound difficulties in sharing data between researchers due to privacy and security concerns. A potential remedy to this issue are synthetic networks, that closely resemble their real counterparts, but can be freely distributed. generating synthetic networks requires the creation of network topologies that, in application, function as realistically as possible. Widely applied models are currently rule-based and can struggle to reproduce structural dynamics. Lead by recent developments in Graph Neural Network (GNN) models for network generation we evaluate the potential of GNNs for synthetic social networks. Our GNN use is specifically within a reasonable use-case and includes empirical evaluation using Maximum Mean Discrepancy (MMD). We include social network specific measurements which allow evaluation of how realistically synthetic networks behave in typical social network analysis applications. We find that the Gated Recurrent Attention Network (GRAN) extends well to social networks, and in comparison to a benchmark popular rule-based generation Recursive-MATrix (R-MAT) method, is better able to replicate realistic structural dynamics. We find that GRAN is more computationally costly than R-MAT, but is not excessively costly to employ, so would be effective for researchers seeking to create datasets of synthetic social networks.

Alex O. Davies, Nirav S. Ajmeri, Telmo M. Silva Filho

Large graphs are present in a variety of domains, including social networks, civil infrastructure, and the physical sciences to name a few. Graph generation is similarly widespread, with applications in drug discovery, network analysis and synthetic datasets among others. While GNN (Graph Neural Network) models have been applied in these domains their high in-memory costs restrict them to small graphs. Conversely less costly rule-based methods struggle to reproduce complex structures. We propose HIGGS (Hierarchical Generation of Graphs) as a model-agnostic framework of producing large graphs with realistic local structures. HIGGS uses GNN models with conditional generation capabilities to sample graphs in hierarchies of resolution. As a result HIGGS has the capacity to extend the scale of generated graphs from a given GNN model by quadratic order. As a demonstration we implement HIGGS using DiGress, a recent graph-diffusion model, including a novel edge-predictive-diffusion variant edge-DiGress. We use this implementation to generate categorically attributed graphs with tens of thousands of nodes. These HIGGS generated graphs are far larger than any previously produced using GNNs. Despite this jump in scale we demonstrate that the graphs produced by HIGGS are, on the local scale, more realistic than those from the rule-based model BTER.

Taylor Applebaum, Sam Blackwell, Alex Davies et al.

We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots $K$ and $K'$ that satisfy some mild assumptions, there is a diagram of their connected sum and $u(K) + u(K')$ unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.

Alex O. Davies, Riku W. Green, Nirav S. Ajmeri et al.

The principal benefit of unsupervised representation learning is that a pre-trained model can be fine-tuned where data or labels are scarce. Existing approaches for graph representation learning are domain specific, maintaining consistent node and edge features across the pre-training and target datasets. This has precluded transfer to multiple domains. We present Topology Only Pre-Training (ToP), a graph pre-training method based on node and edge feature exclusion. We show positive transfer on evaluation datasets from multiple domains, including domains not present in pre-training data, running directly contrary to assumptions made in contemporary works. On 75% of experiments, ToP models perform significantly $p \leq 0.01$ better than a supervised baseline. Performance is significantly positive on 85.7% of tasks when node and edge features are used in fine-tuning. We further show that out-of-domain topologies can produce more useful pre-training than in-domain. Under ToP we show better transfer from non-molecule pre-training, compared to molecule pre-training, on 79% of molecular benchmarks. Against the limited set of other generalist graph models ToP performs strongly, including against models with many orders of magnitude larger. These findings show that ToP opens broad areas of research in both transfer learning on scarcely populated graph domains and in graph foundation models.

Alex O. Davies, Telmo de Menezes e Silva Filho, Nirav Ajmeri

Tabular foundation models are pre-trained on one of three classes of corpus: curated datasets drawn from benchmark repositories, tables harvested at scale from the web, or synthetic tables sampled from a parametric generative prior. Despite the centrality of pre-training data to model performance, little is known about how these corpora relate to one another in distribution, and the impact this has on downstream performance. In this work we take three canonical, archetypal datasets used to train tabular foundation models; the T4 dataset represents web-scraped corpora, the TabFM dataset curated tables from Kaggle, and the TabICL dataset as the only well-used synthetic prior with publicly available parameters. We characterise each corpus using aggregate features over whole tables, columns and correlations, and compare them using discriminator AUCs and k-NN coverage metrics. We find that the TabICL synthetic prior occupies a narrow region of the space of real tables, that this mismatch cannot be closed by optimising prior hyper-parameters across more than 86 thousand configurations, and that curated and web-scraped corpora are broadly interchangeable on a distributional level in feature space. Surprisingly, the distributional gap between synthetic pre-training data and real tables has a clearly detectable effect on performance under neither feature-based proximity measures or TabICL's own internal representations, suggesting that coverage of the real-data distribution is not the primary driver of TabICL's generalisation.

Alex O. Davies, Nirav S. Ajmeri, Telmo de Menezes e Silva Filho

Graph learning on molecules makes use of information from both the molecular structure and the features attached to that structure. Much work has been conducted on biasing either towards structure or features, with the aim that bias bolsters performance. Identifying which information source a dataset favours, and therefore how to approach learning that dataset, is an open issue. Here we propose Noise-Noise Ratio Difference (NNRD), a quantitative metric for whether there is more useful information in structure or features. By employing iterative noising on features and structure independently, leaving the other intact, NNRD measures the degradation of information in each. We employ NNRD over a range of molecular tasks, and show that it corresponds well to a loss of information, with intuitive results that are more expressive than simple performance aggregates. Our future work will focus on expanding data domains, tasks and types, as well as refining our choice of baseline model.

Alex Davies, András Juhász, Marc Lackenby et al.

We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.

Alex Davies, Zoubin Ghahramani

We present Random Partition Kernels, a new class of kernels derived by demonstrating a natural connection between random partitions of objects and kernels between those objects. We show how the construction can be used to create kernels from methods that would not normally be viewed as random partitions, such as Random Forest. To demonstrate the potential of this method, we propose two new kernels, the Random Forest Kernel and the Fast Cluster Kernel, and show that these kernels consistently outperform standard kernels on problems involving real-world datasets. Finally, we show how the form of these kernels lend themselves to a natural approximation that is appropriate for certain big data problems, allowing $O(N)$ inference in methods such as Gaussian Processes, Support Vector Machines and Kernel PCA.