In Table 1, we compare our agnostic learning results. Our results in this setting come from Theorem 3.3. We note that the sample complexity for Diakonikolas et al. To prove Lemma 3.5, we use the following result of Y ehudai and Shamir [35]. We first consider the case when σ satisfies Assumption 3.1.

Althoughintuitive,sucha nativelabelassignment strategycannot revealtheunderlying semantic similarity between aquery anditspositivesandnegatives,andimpairs performance, since some negatives are semantically similar to the query or even share the same semantic class as the query.

Engineering design problems frequently require solving systems of partial differential equations with boundary conditions specified on object geometries in the form of a triangular mesh. These boundary geometries are provided by a designer and are problem dependent. The efficiency of the design process greatly benefits from fast turnaround times when repeatedly solving PDEs on various geometries. However, most current work that uses machine learning to speed up the solution process relies heavily on a fixed parameterization of the geometry, which cannot be changed after training. This severely limits the possibility of reusing a trained model across a variety of design problems. In this work, we propose a novel neural operator architecture which accepts boundary geometry, in the form of triangular meshes, as input and produces an approximate solution to a given PDE as output. Once trained, the model can be used to rapidly estimate the PDE solution over a new geometry, without the need for retraining or representation of the geometry with a pre-specified parameterization.

This view of the data generating process is also adopted and promoted by popular existing works.

Indeed, recent studies have shown that gradient-based training methods effectively regularize the solution by implicitly minimizing a certain complexity measure of the model [V ardi, 2022].

Leveraging distributed computing resources and decentralized data iscrucial, ifnot necessary,for large-scale machine learning applications.

Algorithm 1Federated Accelerated Stochastic Gradient Descent ( FEDAC) 1: procedureFEDAC( , , , ).See Eqs. Gregory Francis Coppola.Iterative Parameter Mixingfor Distributed Large-Margin Trainingof Structured Predictorsfor Natural Language Processing.