Transformers owe much of their empirical success in natural language processing to the self-attention blocks. Recent perspectives interpret attention blocks as interacting particle systems, whose mean-field limits correspond to gradient flows of interaction energy functionals on probability density spaces equipped with Wasserstein-$2$-type metrics. We extend this viewpoint by introducing accelerated attention blocks derived from inertial Nesterov-type dynamics on density spaces. In our proposed architecture, tokens carry both spatial (feature) and velocity variables. The time discretization and the approximation of accelerated density dynamics yield Hamiltonian momentum attention blocks, which constitute the proposed accelerated attention architectures. In particular, for linear self-attention, we show that the attention blocks approximate a Stein variational gradient flow, using a bilinear kernel, of a potential energy. In this setting, we prove that elliptically contoured probability distributions are preserved by the accelerated attention blocks. We present implementable particle-based algorithms and demonstrate that the proposed accelerated attention blocks converge faster than the classical attention blocks while preserving the number of oracle calls.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The bottom line of this paper is an efficient algorithm for finding maximum likelihood estimators for elliptically contoured distributions, a class of densities that includes the Gaussian and various generalizations of it. For the Gaussian itself, that optimization is straightforward, it's the generalizations where the new algorithm provides real advantages. One could argue that this focus on a relatively arcane family of distributions (Kotz-type) limits the utility of this paper. But I think it's actually the other way round: The paper may spark new interest at NIPS in these models.

This paper develops (conic) geometric optimisation on the cone of hpd matrices, which allows us to globally optimise a large class of nonconvex functions of hpd matrices. Specifically, we first use the Riemannian manifold structure of the hpd cone for studying functions that are nonconvex in the Euclidean sense but are geodesically convex (g-convex), hence globally optimisable. We then go beyond g-convexity, and exploit the conic geometry of hpd matrices to identify another class of functions that remain amenable to global optimisation without requiring g-convexity. We present key results that help recognise g-convexity and also the additional structure alluded to above. We illustrate our ideas by applying them to likelihood maximisation for a broad family of elliptically contoured distributions: for this maximisation, we derive novel, parameter free fixed-point algorithms. To our knowledge, ours are the most general results on geometric optimisation of hpd matrices known so far. Experiments show that advantages of using our fixed-point algorithms.

In this paper we develop \emph{geometric optimisation} for globally optimising certain nonconvex loss functions arising in the modelling of data via elliptically contoured distributions (ECDs). We exploit the remarkable structure of the convex cone of positive definite matrices which allows one to uncover hidden geodesic convexity of objective functions that are nonconvex in the ordinary Euclidean sense. Going even beyond manifold convexity we show how further metric properties of HPD matrices can be exploited to globally optimise several ECD log-likelihoods that are not even geodesic convex. We present key results that help recognise this geometric structure, as well as obtain efficient fixed-point algorithms to optimise the corresponding objective functions. To our knowledge, ours are the most general results on geometric optimisation of HPD matrices known so far.

This paper develops (conic) geometric optimisation on the cone of hpd matrices, which allows us to globally optimise a large class of nonconvex functions of hpd matrices. Specifically, we first use the Riemannian manifold structure of the hpd cone for studying functions that are nonconvex in the Euclidean sense but are geodesically convex (g-convex), hence globally optimisable. We then go beyond g-convexity, and exploit the conic geometry of hpd matrices to identify another class of functions that remain amenable to global optimisation without requiring g-convexity. We present key results that help recognise g-convexity and also the additional structure alluded to above. We illustrate our ideas by applying them to likelihood maximisation for a broad family of elliptically contoured distributions: for this maximisation, we derive novel, parameter free fixed-point algorithms. To our knowledge, ours are the most general results on geometric optimisation of hpd matrices known so far. Experiments show that advantages of using our fixed-point algorithms.