Geometrically Inspired Kernel Machines for Collaborative Learning Beyond Gradient Descent

This paper develops a novel mathematical framework for collaborative learning by means of geometrically inspired kernel machines which includes statements on the bounds of generalisation and approximation errors, and sample complexity. For classification problems, this approach allows us to learn bounded geometric structures around given data points and hence solve the global model learning problem in an efficient way by exploiting convexity properties of the related optimisation problem in a Reproducing Kernel Hilbert Space (RKHS). In this way, we can reduce classification problems to determining the closest bounded geometric structure from a given data point. Further advantages that come with our solution is that our approach does not require clients to perform multiple epochs of local optimisation using stochastic gradient descent, nor require rounds of communication between client/server for optimising the global model. We highlight that numerous experiments have shown that the proposed method is a competitive alternative to the state-of-the-art.

A novel framework for systematic propositional formula simplification based on existential graphs

This paper presents a novel simplification calculus for propositional logic derived from Peirce's existential graphs' rules of inference and implication graphs. Our rules can be applied to propositional logic formulae in nested form, are equivalence-preserving, guarantee a monotonically decreasing number of variables, clauses and literals, and maximise the preservation of structural problem information. Our techniques can also be seen as higher-level SAT preprocessing, and we show how one of our rules (TWSR) generalises and streamlines most of the known equivalence-preserving SAT preprocessing methods. In addition, we propose a simplification procedure based on the systematic application of two of our rules (EPR and TWSR) which is solver-agnostic and can be used to simplify large Boolean satisfiability problems and propositional formulae in arbitrary form, and we provide a formal analysis of its algorithmic complexity in terms of space and time. Finally, we show how our rules can be further extended with a novel n-ary implication graph to capture all known equivalence-preserving preprocessing procedures.