arXiv research

Unified view of federated learning and distributed RL using local stochastic approximation.

New method improves Euler approximation for local stochastic volatility models.

Neural networks can approximate complex stochastic equations well.

Paper approximates rough stochastic local volatility models for efficient computation.

Stochastic gradient descent optimizes Nyström samples for kernel matrix approximation.

Paper proposes faster method to find local minima in nonconvex optimization.

Paper develops Gaussian approximations and bootstrap for federated LSA with trade-off bounds.

We design a stochastic algorithm to train any smooth neural network to $\varepsilon$-approximate local minima, using $O(\varepsilon^{-3.25})$ backpropagations. The best result was essentially $O(\varepsilon^{-4})$ by SGD. More broadly, it finds $\varepsilon$-approximate local minima of any smooth nonconvex function in …

New method for CMS derivatives pricing using Watanabe's expansions.

Stochastic approximation proves asymptotic normality for non-smooth problems.

Optimizers find approximate global minima in non-convex problems.

We empirically evaluate a stochastic annealing strategy for Bayesian posterior optimization with variational inference. Variational inference is a deterministic approach to approximate posterior inference in Bayesian models in which a typically non-convex objective function is locally optimized over the parameters of t…

We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.

We derive asymptotic expansions for the prices of a variety of European and barrier-style claims in a general local-stochastic volatility setting. Our method combines Taylor series expansions of the diffusion coefficients with an expansion in the correlation parameter between the underlying asset and volatility process…

We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for the both the value function and the optimal investment strategy. We also analyze the `implied Sharpe ratio' and derive a series a…

This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(ε^{-3.5})$ stochastic gradien…

Existence of calibrated local stochastic volatility models proven for non-regular coefficients.

DP-SEP privatizes EP by refining a single factor per data point.

New method for sampling from complex distributions using stochastic localization.

Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local approximations that are iteratively refined for each datapoint. EP can offer analytic and comp…

New deep learning method solves complex BSDEs efficiently.

We study the Heston-Cox-Ingersoll-Ross++ stochastic-local volatility model in the context of foreign exchange markets and propose a Monte Carlo simulation scheme which combines the full truncation Euler scheme for the stochastic volatility component and the stochastic domestic and foreign short interest rates with the …

HA-SME models SGD dynamics with Hessian info for better escaping behaviors.

Stochastic algo learns from evolving data, achieving optimal performance.

Heavy-tailed distributions emerge in SGD's parameter evolution.

The paper explores local-correlation models for pricing complex financial contracts.

A new measure $k$-variance captures local distributional shape.

New bounds show diffusion models converge nearly linearly in data dimension.

Improves posterior approximation speed for Dirichlet process mixture models.

We find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar Lévy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness a…

A new method for creating simpler models from complex ones.

In this work, we provide theoretical guarantees for reward decomposition in deterministic MDPs. Reward decomposition is a special case of Hierarchical Reinforcement Learning, that allows one to learn many policies in parallel and combine them into a composite solution. Our approach builds on mapping this problem into a…

Most models for barrier pricing are designed to let a market maker tune the model-implied covariance between moves in the asset spot price and moves in the implied volatility skew. This is often implemented with a local volatility/stochastic volatility mixture model, where the mixture parameter tunes that covariance. T…

This article suggests that deterministic Gradient Descent, which does not use any stochastic gradient approximation, can still exhibit stochastic behaviors. In particular, it shows that if the objective function exhibit multiscale behaviors, then in a large learning rate regime which only resolves the macroscopic but n…

Local network community detection aims to find a single community in a large network, while inspecting only a small part of that network around a given seed node. This is much cheaper than finding all communities in a network. Most methods for local community detection are formulated as ad-hoc optimization problems. In…

DSVNP uses global and local latent variables for improved neural process predictions.

Study variance-reduced method for estimating fixed points in Banach spaces.

Recent work has argued that stochastic gradient descent can approximate the Bayesian uncertainty in model parameters near local minima. In this work we develop a similar correspondence for minibatch natural gradient descent (NGD). We prove that for sufficiently small learning rates, if the model predictions on the trai…

Improved stochastic optimization outperforms standard methods.

This paper deals with the exact calibration of semidiscretized stochastic local volatility (SLV) models to their underlying semidiscretized local volatility (LV) models. Under an SLV model, it is common to approximate the fair value of European-style options by semidiscretizing the backward Kolmogorov equation using fi…

New algorithm solves complex equations using deep learning.

In this paper we discuss the basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expan…

We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order $\mathcal{O}(1/{k^{2/3}})$ as long as those differential estimations are sufficientl…

Gaussian Processes (GPs) are powerful non-parametric Bayesian regression models that allow exact posterior inference, but exhibit high computational and memory costs. In order to improve scalability of GPs, approximate posterior inference is frequently employed, where a prominent class of approximation techniques is ba…

Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local-stochastic volatility setting. Our price approximations require only a normal CDF and our implied volatility approximations are fully explicit (ie, they require no spec…

We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small ste…

Derivative-free method solves stochastic optimization problems with noisy objectives and constraints.

Corrects local error estimates for UBU integrator in SDEs, improving complexity guarantees.