Our ignorance of the winnability percentage of the solitaire card game ‘Klondike’ has been described as “one of the embarrassments of applied mathematics” by Yan, Diaconis, Rusmevichientong, and Roy. Klondike, the game in the Windows Solitaire program, is just one of many single-player card games, generically called ‘patience’ or ‘solitaire’ games, for which players have long wanted to know how likely a particular game is to be winnable. A number of different games have been studied empirically in the academic literature and by non-academic enthusiasts. Here we show that a single general purpose Artificial Intelligence program named ‘Solvitaire’ can be used to determine the winnability percentage of 73 variants of 35 different single-player card games with a 95% confidence interval of ±0.1% or better. For example, we report the winnability of Klondike as 81.945% ± 0.084% (in the ‘thoughtful’ variant where the player knows the rank and suit of all cards), a 30-fold reduction in confidence interval over the best previous result. The vast majority of our results are either entirely new or represent significant improvements on previous knowledge. Solvitaire uses depth-first search and exploits a number of AI techniques including transposition tables, symmetry breaking, dominances, and streamliners. We give the first correctness proofs of two key dominances for patience games.

The computational difficulties of large language model (LLM) inference remain a significant obstacle to their widespread deployment. The need for many applications to support long input sequences and process them in large batches typically causes token-generation to be bottlenecked by data transfer. For this reason, we introduce **SparQ Attention**, a technique for increasing the inference throughput of LLMs by utilising memory bandwidth more efficiently within the attention layers, through selective fetching of the cached history. Our proposed technique can be applied directly to off-the-shelf LLMs during inference, without requiring any modification to the pre-training setup or additional fine-tuning. We show that SparQ Attention brings up to 8x savings in attention data transfers without substantial drops in accuracy, by evaluating Llama 2 and 3, Mistral, Gemma and Pythia models on a wide range of downstream tasks.

We present unit scaling, a paradigm for designing deep learning models that simplifies the use of low-precision number formats. Training in FP16 or the recently proposed FP8 formats offers substantial efficiency gains, but can lack sufficient range for out-of-the-box training. Unit scaling addresses this by introducing a principled approach to model numerics: seeking unit variance of all weights, activations and gradients at initialisation. Unlike alternative methods, this approach neither requires multiple training runs to find a suitable scale nor has significant computational overhead. We demonstrate the efficacy of unit scaling across a range of models and optimisers. We further show that existing models can be adapted to be unit-scaled, training BERT-Large in FP16 and then FP8 with no degradation in accuracy.

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.