The field of Machine Learning has changed significantly since the 1970s. However, its most basic principle, Empirical Risk Minimization (ERM), remains unchanged. We propose Functional Risk Minimization~(FRM), a general framework where losses compare functions rather than outputs. This results in better performance in supervised, unsupervised, and RL experiments. In the FRM paradigm, for each data point $(x_i,y_i)$ there is function $f_{\theta_i}$ that fits it: $y_i = f_{\theta_i}(x_i)$. This allows FRM to subsume ERM for many common loss functions and to capture more realistic noise processes. We also show that FRM provides an avenue towards understanding generalization in the modern over-parameterized regime, as its objective can be framed as finding the simplest model that fits the training data.

Probabilistic weather forecasting is critical for decision-making in high-impact domains such as flood forecasting, energy system planning or transportation routing, where quantifying the uncertainty of a forecast -- including probabilities of extreme events -- is essential to guide important cost-benefit trade-offs and mitigation measures. Traditional probabilistic approaches rely on producing ensembles from physics-based models, which sample from a joint distribution over spatio-temporally coherent weather trajectories, but are expensive to run. An efficient alternative is to use a machine learning (ML) forecast model to generate the ensemble, however state-of-the-art ML forecast models for medium-range weather are largely trained to produce deterministic forecasts which minimise mean-squared-error. Despite improving skills scores, they lack physical consistency, a limitation that grows at longer lead times and impacts their ability to characterize the joint distribution. We introduce GenCast, a ML-based generative model for ensemble weather forecasting, trained from reanalysis data. It forecasts ensembles of trajectories for 84 weather variables, for up to 15 days at 1 degree resolution globally, taking around a minute per ensemble member on a single Cloud TPU v4 device. We show that GenCast is more skillful than ENS, a top operational ensemble forecast, for more than 96\% of all 1320 verification targets on CRPS and Ensemble-Mean RMSE, while maintaining good reliability and physically consistent power spectra. Together our results demonstrate that ML-based probabilistic weather forecasting can now outperform traditional ensemble systems at 1 degree, opening new doors to skillful, fast weather forecasts that are useful in key applications.

\textit{Graph neural networks} (GNNs) are effective models for many dynamical systems consisting of entities and relations. Although most GNN applications assume a single type of entity and relation, many situations involve multiple types of interactions. \textit{Relational inference} is the problem of inferring these interactions and learning the dynamics from observational data. We frame relational inference as a \textit{modular meta-learning} problem, where neural modules are trained to be composed in different ways to solve many tasks. This meta-learning framework allows us to implicitly encode time invariance and infer relations in context of one another rather than independently, which increases inference capacity. Framing inference as the inner-loop optimization of meta-learning leads to a model-based approach that is more data-efficient and capable of estimating the state of entities that we do not observe directly, but whose existence can be inferred from their effect on observed entities. To address the large search space of graph neural network compositions, we meta-learn a \textit{proposal function} that speeds up the inner-loop simulated annealing search within the modular meta-learning algorithm, providing two orders of magnitude increase in the size of problems that can be addressed.

Global medium-range weather forecasting is critical to decision-making across many social and economic domains. Traditional numerical weather prediction uses increased compute resources to improve forecast accuracy, but cannot directly use historical weather data to improve the underlying model. We introduce a machine learning-based method called "GraphCast", which can be trained directly from reanalysis data. It predicts hundreds of weather variables, over 10 days at 0.25 degree resolution globally, in under one minute. We show that GraphCast significantly outperforms the most accurate operational deterministic systems on 90% of 1380 verification targets, and its forecasts support better severe event prediction, including tropical cyclones, atmospheric rivers, and extreme temperatures. GraphCast is a key advance in accurate and efficient weather forecasting, and helps realize the promise of machine learning for modeling complex dynamical systems.

From CNNs to attention mechanisms, encoding inductive biases into neural networks has been a fruitful source of improvement in machine learning. Auxiliary losses are a general way of encoding biases in order to help networks learn better representations by adding extra terms to the loss function. However, since they are minimized on the training data, they suffer from the same generalization gap as regular task losses. Moreover, by changing the loss function, the network is optimizing a different objective than the one we care about. In this work we solve both problems: first, we take inspiration from transductive learning and note that, after receiving an input but before making a prediction, we can fine-tune our models on any unsupervised objective. We call this process tailoring, because we customize the model to each input. Second, we formulate a nested optimization (similar to those in meta-learning) and train our models to perform well on the task loss after adapting to the tailoring loss. The advantages of tailoring and meta-tailoring are discussed theoretically and demonstrated empirically on several diverse examples: encoding inductive conservation laws from physics to improve predictions, improving local smoothness to increase robustness to adversarial examples, and using contrastive losses on the query image to improve generalization. The key to successful generalization in machine learning is the encoding of useful inductive biases. A variety of mechanisms, from parameter tying to data augmentation, have proven useful but there is no systematic strategy for designing and implementing these biases. Auxiliary losses are a paradigm for encoding a wide variety of biases, constraints and objectives, helping networks learn better representations and generalize more broadly.

We explore the use of graph neural networks (GNNs) to model spatial processes in which there is a priori graphical structure. Similar to finite element analysis, we assign nodes of a GNN to spatial locations and use a computational process defined on the graph to model the relationship between an initial function defined over a space and a resulting function in the same space. We use GNNs as a computational substrate, and show that the locations of the nodes in space as well as their connectivity can be optimized to focus on the most complex parts of the space. Moreover, this representational strategy allows the learned input-output relationship to generalize over the size of the underlying space and run the same model at different levels of precision, trading computation for accuracy. We demonstrate this method on a traditional PDE problem, a physical prediction problem from robotics, and a problem of learning to predict scene images from novel viewpoints.

Modular meta-learning is a new framework that generalizes to unseen datasets by combining a small set of neural modules in different ways. In this work we propose abstract graph networks: using graphs as abstractions of a system's subparts without a fixed assignment of nodes to system subparts, for which we would need supervision. We combine this idea with modular meta-learning to get a flexible framework with combinatorial generalization to new tasks built in. We then use it to model the pushing of arbitrarily shaped objects from little or no training data.

Many prediction problems, such as those that arise in the context of robotics, have a simplifying underlying structure that could accelerate learning. In this paper, we present a strategy for learning a set of neural network modules that can be combined in different ways. We train different modular structures on a set of related tasks and generalize to new tasks by composing the learned modules in new ways. We show this improves performance in two robotics-related problems.

In many applications that involve processing high-dimensional data, it is important to identify a small set of entities that account for a significant fraction of detections. Rather than formalize this as a clustering problem, in which all detections must be grouped into hard or soft categories, we formalize it as an instance of the frequent items or heavy hitters problem, which finds groups of tightly clustered objects that have a high density in the feature space. We show that the heavy hitters formulation generates solutions that are more accurate and effective than the clustering formulation. In addition, we present a novel online algorithm for heavy hitters, called HAC, which addresses problems in continuous space, and demonstrate its effectiveness on real video and household domains.