arXiv research

Improved root-finding method for smooth functions.

Root-finding methods improve efficiency of conformal prediction sets.

Optimizes AMM markets with a new framework reducing complex optimization to simpler root finding.

Develops new algorithms for solving root-finding problems in large-scale settings.

Develops efficient method for nonconvex problems using Regula Falsi.

Improved bounds for Black-Scholes volatility lead to faster root-finding.

New method solves root-finding problems with faster convergence.

Method estimates noise variance in Gaussian process regression.

Algorithm removes leaves to find root in uniform trees.

We consider numerical schemes for root finding of noisy responses through generalizing the Probabilistic Bisection Algorithm (PBA) to the more practical context where the sampling distribution is unknown and location-dependent. As in standard PBA, we rely on a knowledge state for the approximate posterior of the root l…

MSLs use parallelizable root-finding for efficient ODE and PDE solutions.

A new line search rule improves support recovery in high-dimensional data.

Develops a method for learning sparse generalized linear models in high-dimensional data.

Probabilistic Bisection Algorithm performs root finding based on knowledge acquired from noisy oracle responses. We consider the generalized PBA setting (G-PBA) where the statistical distribution of the oracle is unknown and location-dependent, so that model inference and Bayesian knowledge updating must be performed s…

The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.

Newton's method solves variational problems on manifolds.

Efficient neural networks compute various differential operators cheaply.

Recently, the decentralized optimization problem is attracting growing attention. Most existing methods are deterministic with high per-iteration cost and have a convergence rate quadratically depending on the problem condition number. Besides, the dense communication is necessary to ensure the convergence even if the …

Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. In data-rich environments, the SOEC objective and constraints contain expectations defined with respect to large datasets. Therefore, efficient algorithms for solvi…

Smooth flows for physical systems with smooth energies and forces.

Modified BA algorithm computes RD and DR functions efficiently.

Estimates and optimizes UBSR risk in recursive settings.

Improved MLMC method for robust and efficient probability and density estimation.

The probabilistic bisection algorithm (PBA) solves a class of stochastic root-finding problems in one dimension by successively updating a prior belief on the location of the root based on noisy responses to queries at chosen points. The responses indicate the direction of the root from the queried point, and are incor…

We develop a conditional sampling scheme for pricing knock-out barrier options under the Linear Transformations (LT) algorithm from Imai and Tan (2006). We compare our new method to an existing conditional Monte Carlo scheme from Glasserman and Staum (2001), and show that a substantial variance reduction is achieved. W…

We simplify SVI volatility smile constraints for three sub-SVIs without numerical methods.

The paper analyzes the performance of constant step-size stochastic approximation algorithms.

We create precise formulas for VIX option implied volatility.

We propose a quasi-Monte Carlo algorithm for pricing knock-out and knock-in barrier options under the Heston (1993) stochastic volatility model. This is done by modifying the LT method from Imai and Tan (2006) for the Heston model such that the first uniform variable does not influence the stochastic volatility path an…

Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time…

New algorithm reduces regret in both adversarial and stochastic contexts.

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data s…

We investigate the problem of computing a nested expectation of the form $\mathbb{P}[\mathbb{E}[X|Y] \!\geq\!0]\!=\!\mathbb{E}[\textrm{H}(\mathbb{E}[X|Y])]$ where $\textrm{H}$ is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfo…

With the purpose of examining biased updates in variance-reduced stochastic gradient methods, we introduce SVAG, a SAG/SAGA-like method with adjustable bias. SVAG is analyzed in a cocoercive root-finding setting, a setting which yields the same results as in the usual smooth convex optimization setting for the ordinary…

Two accelerated extragradient methods converge at $O(1/k)$ rate for co-hypomonotone inclusions.

The paper finds formulas for word lengths and conjugacy classes in surface groups.

Deep equilibrium models converge globally without explicit computation.

MLE and CVE are equivalent under exponential families, leading to faster and more stable EM algorithms.

New method smooths integrands for efficient option pricing.

DEQs converge to optimal solutions with mild over-parameterization.

Efficient algorithm reduces communication costs in sparse regression.

We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the uni…

We present a new approach to modeling sequential data: the deep equilibrium model (DEQ). Motivated by an observation that the hidden layers of many existing deep sequence models converge towards some fixed point, we propose the DEQ approach that directly finds these equilibrium points via root-finding. Such a method is…

This paper compares VaR estimation methods under tail misspecification, finding importance sampling underestimates VaR.

Spiking neuronal networks are usually simulated with three main simulation schemes: the classical time-driven and event-driven schemes, and the more recent hybrid scheme. All three schemes evolve the state of a neuron through a series of checkpoints: equally spaced in the first scheme and determined neuron-wise by spik…

Optimal posterior distributions improve SVM classifiers and parameter selection.

A new algorithm reduces the time and space complexity for multinomial logistic bandits.

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…