Hamiltonian Monte Carlo on ReLU networks is inefficient due to large local error.
problem Inefficiency of Hamiltonian Monte Carlo on ReLU neural networks.
method Analysis of Hamiltonian Monte Carlo with leapfrog integrator for Bayesian neural network inference.
result Leapfrog HMC for ReLU networks has a large local error rate of Ω ( ε ) Ω(ε) Ω ( ε ) , leading to inefficiency. RQMC improves kernel-based learning by reducing deterministic error and offering computational advantages.
problem Improving kernel-based learning methods to reduce deterministic error and computational complexity.
method Randomized quasi-Monte Carlo (RQMC) methods applied to random feature approximations.
result RQMC methods improve deterministic approximation error bound from O P ( 1 / M ) O_P(1/\sqrt{M}) O P ( 1/ M ) to O ( 1 / M ) O(1/M) O ( 1/ M ) , matching QMC methods. This study compares MC and QMC methods for derivative pricing, showing QMC's superior convergence rates.
problem Improving derivative pricing accuracy and efficiency in high-dimensional settings.
method Compared Monte Carlo and quasi-Monte Carlo techniques, focusing on convergence rates and low-discrepancy sequences.
result Quasi-Monte Carlo methods achieve superior convergence rates and reduce root mean square error in derivative pricing.
Estimates for neural network risk nearly match Monte Carlo error rates.
problem Understanding the performance of two-layer neural networks.
method Established a priori estimates for the population risk of two-layer neural networks.
result The new estimates are nearly optimal and depend only on function norms, not model parameters.
RQMC improves QMC by providing practical error bounds for financial applications.
problem Lack of practical error estimates in QMC methods.
method Combines Sobol LDS with randomized scrambling methods.
result RQMC outperforms standard QMC in convergence rates and provides error bounds.
The paper provides mean-square error bounds for stochastic approximation algorithms.
problem Error bounds for recursive equations with Markovian disturbances.
method Analysis of mean-square error for stochastic approximation algorithms.
result Mean-square error achieves the optimal rate of O ( 1 / n ) O(1/n) O ( 1/ n ) under certain conditions. Combines normalizing flows and quasi-Monte Carlo for improved numerical integration.
problem Improving the efficiency of numerical integration methods.
method Uses normalizing flows to approximate distributions and quasi-Monte Carlo for sampling.
result Demonstrates an estimator with significantly lower variance.
New algorithm speeds up MCMC for complex distributions.
problem Efficient sampling from complex, high-dimensional distributions.
method Numerical Generalized Randomized Hamiltonian Monte Carlo with state-dependent event rates.
result Approximates Hamiltonian trajectories for robust sampling.
We apply multilevel Monte Carlo for option pricing problems using exponential Lévy models with a uniform timestep discretisation to monitor the running maximum required for lookback and barrier options. The numerical results demonstrate the computational efficiency of this approach. We derive estimates of the convergen…
Improves QMC for complex distributions using transport maps.
problem Challenges in applying QMC to general target distributions.
method Train a transport map to approximate target distributions, ensuring RQMC achieves superior error rates.
result Transport QMC achieves faster convergence rates than standard Monte Carlo under mild conditions.
PEMC uses ML to enhance Monte Carlo simulations, reducing variance and runtime.
problem Computational inefficiency in Monte Carlo simulations for complex tasks.
method Prediction-Enhanced Monte Carlo (PEMC) framework that uses ML surrogates as predictors.
result PEMC provides unbiased evaluations with reduced variance and runtime compared to standard Monte Carlo.
New method reduces Monte Carlo error in option pricing and Greeks estimation.
problem Reducing Monte Carlo error in option pricing and Greeks estimation.
method Denoised Monte Carlo technique for LSV models.
result Reduces Monte Carlo error by an order of magnitude.
The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio s / d s/d s / d , where s s s and d d d encode the smoothness and dimension of the integrand. However, an empirical investigation re…
The paper improves Monte Carlo methods for optimization problems.
problem Efficiently solving optimization problems with biased Monte Carlo estimators.
method Introduces Multilevel Monte Carlo (MLMC) within Sample Average Approximation (SAA).
result Establishes uniform convergence and sample complexity for MLMC in SAA.
New algorithms improve Langevin Monte Carlo efficiency.
problem High computational cost of classical Langevin Monte Carlo.
method Integrates ensemble feature into LMC, constraining gradient approximations.
result Constrained Ensemble Langevin Monte Carlo reduces gradient computation.
Paper analyzes Gibbs and Langevin Monte Carlo for interpolation regime, showing generalization from low errors.
problem Analyzing Gibbs and Langevin Monte Carlo in overparameterized interpolation regime.
method Data-dependent bounds and stability under approximation with Langevin Monte Carlo.
result Generalization is signaled by small training errors in noisy regime, with bounds stable under approximation.
This study compares MC and QMC methods for likelihood functions.
problem Approximating the normalizing constant of posterior distributions and marginal likelihoods.
method Characterizes the integration error of MC and QMC methods for likelihood functions.
result QMC outperforms MC under certain conditions, especially in high dimensions.
A new tamed stochastic gradient Hamiltonian Monte Carlo algorithm for superlinearly growing stochastic gradients.
problem Sampling and stochastic optimization problems with superlinearly growing stochastic gradients.
method Tamed Stochastic Gradient Hamiltonian Monte Carlo (tSGHMC) algorithm.
result Established a non-asymptotic error bound in Wasserstein-2 distance with a convergence rate of 1 / 4 1/4 1/4 . New algorithms improve MCMC efficiency for complex distributions.
problem High variance and low effective sample size in MCMC samplers.
method Antithetic Riemannian Manifold and Quantum-Inspired Hamiltonian Monte Carlo.
result Improved effective sample size and variance reduction.
Bayes Error Rate estimators are evaluated for accuracy and sample requirements.
problem Evaluating the accuracy and sample requirements of Bayes Error Rate estimators.
method Monte Carlo simulations with synthetic data and real-world scenarios.
result k-Nearest Neighbor (kNN) is the most accurate non-parametric estimator.
New optimised adaptive importance samplers converge faster than standard methods.
problem Improving Monte Carlo estimators for target distributions.
method Optimised adaptive importance samplers using convex optimisation of χ 2 χ^2 χ 2 -divergence. result Convergence rate of O ( 1 / N ) \mathcal{O}(1/\sqrt{N}) O ( 1/ N ) for optimised samplers, with explicit iteration and sample dependence. The paper improves probabilistic herding methods using Gibbs distributions.
problem Improving integration accuracy over Monte Carlo quadrature in infinite-dimensional RKHS.
method Developed a Gibbs distribution over quadrature nodes to minimize MMD.
result The Gibbs distribution outperforms i.i.d. Monte Carlo in integration accuracy.
New algorithm MTMC reduces MCMC evaluation costs.
problem High-dimensional sampling with intractable posterior evaluations.
method Iteratively updated approximation of posterior distribution for acceptance rate.
result Approximation converges to true posterior as iterations increase.
Importance weighting is a general way to adjust Monte Carlo integration to account for draws from the wrong distribution, but the resulting estimate can be highly variable when the importance ratios have a heavy right tail. This routinely occurs when there are aspects of the target distribution that are not well captur…
Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal targ…
A fast Monte Carlo method for additive processes and option pricing.
problem Efficiently pricing path-dependent options with additive processes.
method Developed a fast Monte Carlo scheme for additive processes, analyzing and reducing numerical error sources.
result Shows significant reduction in error (1 bp or below) for pricing path-dependent options.
SLMC improves sampling efficiency for high-dimensional distributions.
problem Sampling from high-dimensional distributions is computationally challenging.
method SLMC projects Langevin updates onto subsampled eigenblocks of a time-varying preconditioner.
result SLMC offers superior adaptability and computational efficiency compared to traditional methods.
Paper uses Monte Carlo simulations to predict retirement portfolios.
problem Retirement financial planning uncertainty.
method Monte Carlo simulations incorporating inflation, interest rates, etc.
result Probabilistic prediction of IRA and 401(k) values.
This paper analyzes error bounds for biased SMC samplers in conditional sampling.
problem Analyzing error bounds for biased SMC samplers in conditional sampling.
method Develops a non-asymptotic error analysis for SMC samplers with biased mutation kernels.
result Derives the first non-asymptotic error bound for conditional sampling with score-based diffusion models.
The paper explores how control variates can reduce variance in Monte Carlo simulations, especially for Sobolev functions.
problem Efficiency of control variates in reducing variance for Monte Carlo simulations.
method Study of a specific quadrature rule using nonparametric regression-adjusted control variates.
result A specific quadrature rule can improve the Monte Carlo rate and achieve the minimax optimal rate under sufficient smoothness assumptions.
New algorithm improves sampling efficiency for complex systems.
problem Sampling with Markov chain Monte Carlo methods.
method Discretizing smooth Itô diffusions with stochastic Runge-Kutta.
result Sampling iterates converge to target distribution faster.
Study on interest rate model with jumps, proving strong convergence in simulations.
problem Analytical solutions for complex interest rate models with jumps are difficult.
method Employed truncated Euler-Maruyama techniques to prove strong convergence.
result Justified strong convergence for Monte Carlo calibration and valuation.
DALMC provides non-asymptotic error bounds for generative models.
problem Efficiently generating samples from complex data distributions.
method Analysis of diffusion paths and Langevin Monte Carlo.
result Theoretical guarantees for a class of generative models.
We introduce cylindrical projections to simulate infinite-dimensional occupation flows of diffusions.
problem Computational intractability of infinite-dimensional occupation flows of diffusions.
method Introduce cylindrical projections to approximate the occupation flow via a finite-dimensional system.
result Strong convergence of cylindrical projections to the initial process with derived rates.
Algorithm reduces historical expected shortfall computation by focusing on worst-case scenarios.
problem Computing the historical expected shortfall efficiently and accurately.
method Multi-step algorithm using Monte Carlo simulations to identify and reduce the number of worst-case scenarios.
result Non-asymptotic bounds for the L p-error of the expected shortfall estimator are derived.
Improved Least-Squares Monte Carlo with finite-difference ansatz.
problem Improving accuracy and stability in option pricing.
method Constructing an ansatz using finite-difference solution for conditional expected continuation payoffs.
result Reduces mean squared error and final pricing error.
New wavelets use Monte Carlo for efficient discretization.
problem Efficiently discretizing continuous wavelets on general domains.
method Defined continuous wavelets via spectral calculus and proposed a Monte Carlo discretization.
result Convergence of Monte Carlo wavelets under natural regularity assumptions.
Adaptive SAA solves large-scale stochastic linear programs efficiently.
problem Solving large-scale two-stage stochastic linear programs.
method Iterative algorithm with adaptive sample size and warm starts.
result The algorithm converges to the true solution set with a probabilistic guarantee.
Many machine learning problems involve Monte Carlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performance of MCVI crucially depends on the variance of its stochastic gradients. We propose variance reduction by means of Quasi-Monte Carlo (QMC) sam…
We introduce interacting particle Markov chain Monte Carlo (iPMCMC), a PMCMC method based on an interacting pool of standard and conditional sequential Monte Carlo samplers. Like related methods, iPMCMC is a Markov chain Monte Carlo sampler on an extended space. We present empirical results that show significant improv…
This study improves audit sampling by using sequential procedures with statistical guarantees.
problem Improving audit efficiency and reliability with statistical methods.
method Formulated as a sequential testing problem, defining null and alternative hypotheses, stopping and decision rules, and exact boundary conditions.
result Exact design yields ex ante control of decision error probabilities, and simulation-based implementation approximates this design.
HAVER improves error bounds for estimating the largest mean in machine learning tasks.
problem Estimating the largest mean among multiple distributions.
method Proposes HAVER, a novel algorithm for maximum mean estimation.
result HAVER achieves better error bounds than the oracle in many cases.
Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistica…
This paper introduces a set of algorithms for Monte-Carlo Bayesian reinforcement learning. Firstly, Monte-Carlo estimation of upper bounds on the Bayes-optimal value function is employed to construct an optimistic policy. Secondly, gradient-based algorithms for approximate upper and lower bounds are introduced. Finally…
New method prices interest rate derivatives without Monte Carlo, achieving high accuracy and speed.
problem Arbitrage-free pricing of path-dependent interest rate derivatives using infinite-dimensional models.
method Casting the stochastic pricing problem as a deterministic PDE solved by FINNs, which minimize violations of the PDE and boundary conditions.
result FINNs achieve pricing accuracy within 0.04 to 0.07 cents per dollar of contract value compared to Monte Carlo benchmarks.
Proposes a new SPVM model for RVM with more flexible priors.
problem Improper priors on multiple penalty parameters in RVM lead to improper posteriors.
method Introduces a single penalty approach (SPRVM) and a semi-Bayesian fitting method.
result SPRVM allows for more flexible priors and has proven conditions for posterior propriety.
Paper develops Monte-Carlo estimators for CoVaR, a key risk measure.
problem Estimating CoVaR, a critical risk measure in finance.
method Developed Monte-Carlo and importance-sampling estimators for CoVaR.
result Optimal rates of convergence for both estimators: n − 1 / 3 n^{-1/3} n − 1/3 and n − 1 / 2 n^{-1/2} n − 1/2 . New algorithm trains deep neural networks without global optimization.
problem Training deep neural networks efficiently and without global optimization.
method Uses random complex exponential activation functions and Markov Chain Monte Carlo sampling.
result Consistently attains theoretical approximation rate for residual networks.