A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We …
A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First- and second-order models exhibit stability properties that make them a…
Second-order guarantees for federated learning algorithms.
problem Non-convex optimization in federated learning with saddle-points as bottlenecks.
method Drawing on recent results on second-order optimality in centralized and decentralized settings, establish second-order guarantees for federated learning algorithms.
result Established second-order guarantees for federated learning algorithms.
Two new algorithms improve federated optimization under second-order similarity.
problem Federated learning under communication constraints and second-order similarity.
method SVRP and Catalyzed SVRP algorithms combining proximal point evaluations, client sampling, and variance reduction.
result Achieves superior performance and uniformly improves upon existing algorithms for federated optimization under second-order similarity and strong convexity.
Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and Sølna (2011, CUP) analyzes models in which the volatility of the underlying i…
We study the Hamiltonian formalisms of the second order degenerate Clèment and Sarıoğlu-Tekin Lagrangians. The Dirac-Bergmann constraint algorithm is employed while arriving at the total Hamiltonian functions and the Hamilton's equations on the associated momemtum phase spaces whereas the Gotay-Nester-Hinds algorithm i…
For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in…
In this paper we put together some tools from differential topology and analysis in order to study second order semi-linear partial differential equations on a Riemannian manifold M. We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [J DIFFER GEO…
Stochastic gradient Langevin dynamics (SGLD) is a fundamental algorithm in stochastic optimization. Recent work by Zhang et al. [2017] presents an analysis for the hitting time of SGLD for the first and second order stationary points. The proof in Zhang et al. [2017] is a two-stage procedure through bounding the Cheege…
Modern proximal and stochastic gradient descent (SGD) methods are believed to efficiently minimize large composite objective functions, but such methods have two algorithmic challenges: (1) a lack of fast or justified stop conditions, and (2) sensitivity to the objective function's conditioning. In response to the firs…
In this paper, we report a "new" continuity path which links the constant scalar curvature equation to a second order elliptic equation. This is largely an expository article where we describes various aspects of geometry and analysis associated with path.
We study the sensitivity of the expected utility maximization problem in a continuous semi-martingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value …
This work analyzes how preconditioning affects generalization in machine learning models.
problem The impact of preconditioning on the generalization of machine learning models.
method An asymptotic bias-variance decomposition of the generalization error for ridgeless regression under various preconditioners.
result The optimal preconditioner depends on label noise, model specification, and signal alignment, with NGD potentially better under certain conditions.
We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and interpolate between holomorphic gauge and the Wu-Mandelstam-Liebran…
We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points. We show that a simple perturbed version of stochastic recursiv…
Virtually any model we use in machine learning to make predictions does not perfectly represent reality. So, most of the learning happens under model misspecification. In this work, we present a novel analysis of the generalization performance of Bayesian model averaging under model misspecification and i.i.d. data usi…
Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M=M_1 x M_2 where each factor is uniquely determined as follows: M_2 is a Ri…
Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce …
Exact second-order optimization for deep learning reduces computational cost and improves performance.
problem Inadequate use of second-order optimization methods in deep learning due to high computational cost and non-convexity.
method Developed an exact stochastic second-order Newton method that addresses the non-convexity issue and provides an expression for the stochastic Hessian.
result Exact second-order Newton direction formula and its application in deep learning datasets.
A dynamical system on the total space of the fibre bundle of second order accelerations, T2M, is defined as a third order vector field S on T2M, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that …
Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.
problem Exploring properties of second order symmetric parallel tensors in generalized f.pk-space forms.
method Analyzes the properties of second order symmetric parallel tensors and deduces the existence or non-existence of certain tensors and hypersurfaces.
result There does not exist second order skew-symmetric parallel tensor in f.pk-space form. There is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurface.
problem Inaccurate separation of sources with temporal autocorrelations and mixed spectra.
method Estimates spectral density functions and line spectra using cubic splines and indicator functions, then maximizes the Whittle likelihood function.
result Outperforms existing ICA methods in simulations and EEG data applications.
Paper improves particle variational inference by optimizing generalization error bound.
problem Improving the diversity of models in particle variational inference to enhance generalization.
method Develops a new second-order Jensen inequality with a repulsion term based on the loss function, leading to a tighter generalization error bound.
result The proposed PVI optimizes the generalization error bound directly, improving performance compared to existing methods.
Current training methods for deep neural networks boil down to very high dimensional and non-convex optimization problems which are usually solved by a wide range of stochastic gradient descent methods. While these approaches tend to work in practice, there are still many gaps in the theoretical understanding of key as…