A new training method uses multilevel minimization for machine learning.
problem Training machine learning models with high variance and low efficiency.
method Constructs a multilevel hierarchy by reducing sample size and internally trains surrogate models with fewer samples.
result The multilevel method enhances model training efficiency compared to subsampled Newton's and variance reduction methods.
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of refining time-grids to reduce statistical approximation errors in an adaptive and…
A new clustering method using transportation distance for multilevel data.
problem Multilevel clustering problems, especially with large datasets.
method Probabilistic approach based on composite transportation distance, solving a joint optimization problem over probability measures.
result Efficient and scalable solution for multilevel datasets, demonstrated on synthetic and real data.
A multilevel optimization method for constrained problems.
problem Regularized constrained linear inverse problems with box constraints.
method Geometric multilevel optimization with varying discretization levels.
result Preserves feasibility of updates while speeding up computations.
Faster SVMs trained with multilevel approach.
problem Training time inefficiency for SVMs on large datasets.
method Label propagation algorithm to construct a hierarchy of smaller SVM problems.
result Up to orders of magnitude faster than previous fastest algorithm.
Accelerates MCMC sampling for large-scale problems using machine learning.
problem Efficiently sampling large-scale Bayesian inference problems with high computational cost.
method Integrates low-fidelity machine learning models into a multilevel MCMC framework.
result Significantly accelerates multilevel sampling by a factor of two with similar accuracy.
A new method for multilevel clustering using Wasserstein distances.
problem Simultaneously partitioning data in groups and discovering group patterns.
method Joint optimization over spaces of discrete probability measures with Wasserstein distances.
result Consistency properties for estimates of local and global clusters established.
With the advent of massive data sets much of the computational science and engineering community has moved toward data-intensive approaches in regression and classification. However, these present significant challenges due to increasing size, complexity and dimensionality of the problems. In particular, covariance mat…
The paper discusses building ETF risk models using a multilevel classification taxonomy.
problem Building accurate risk models for ETFs.
method First, build a multilevel classification taxonomy for ETFs. Then, use this taxonomy to define risk factors and build risk models.
result The approach can accurately classify and model ETF risks.
Improved Bayesian regression for large datasets using multilevel Gibbs sampling.
problem Efficiently handling large-scale Bayesian regression with complex posterior distributions.
method Developed a multilevel Gibbs sampler for linear mixed models, incorporating data clustering and correlated samples for variance reduction.
result Significant speed-up achieved for Bayesian regression without sacrificing predictive performance.
Flexible framework assesses multilevel data group heterogeneity.
problem Multilevel data structure complicates model selection.
method Flexible framework for assessing differences between levels of grouping variables.
result Framework reliably identifies relevant multilevel components.
MiME learns EHR data structure for predictive healthcare tasks.
problem Data insufficiency in EHR for predictive healthcare tasks.
method Leverages multilevel structure of EHR data and learns multilevel embedding.
result MiME outperforms baseline methods in diverse evaluation settings.
Enhances SBI accuracy with multilevel Monte Carlo for expensive simulators.
problem Limited accuracy in SBI due to expensive simulators.
method Multilevel Monte Carlo techniques for cost-effective SBI.
result Significant enhancement in SBI accuracy with fixed computational budget.
Efficiently computes tree-Wasserstein barycenter for large-scale multilevel clustering and scalable Bayes.
problem Large-scale multilevel clustering and scalable Bayes problems.
method Proposes an efficient algorithm for tree-Wasserstein barycenter and variants.
result Significantly improves efficiency in computation and memory usage for large-scale applications.
A method learns to solve multilevel combinatorial problems with two players.
problem Multilevel combinatorial optimization problems with multiple players.
method Value-based multi-agent reinforcement learning in a graph neural network framework.
result Close to optimal solutions on graphs up to 100 nodes, with a significant speedup.
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…
Efficiently price VIX options using multilevel Monte Carlo in rough Bergomi model.
problem Pricing VIX options in a rough Bergomi model with high computational complexity.
method Combining rectangle discretization, Cholesky sampling, and multilevel Monte Carlo.
result Reduced computational complexity to O(ε−2log2(ε)) and asymptotically optimal O(ε−2). Monte Carlo is a simple and flexible tool that is widely used in computational finance. In this context, it is common for the quantity of interest to be the expected value of a random variable defined via a stochastic differential equation. In 2008, Giles proposed a remarkable improvement to the approach of discretizin…
Detects anomalies in vector fields without distributional assumptions.
problem Detecting anomalies in high-dimensional, non-stationary vector fields.
method Optimal Karhunen-Loeve expansion, multilevel orthogonal subspaces, hypothesis tests.
result Reliable anomaly detection without distributional assumptions.
Estimates log marginal likelihood using multilevel Monte Carlo.
problem Estimating log marginal likelihood accurately.
method Unbiased multilevel Monte Carlo estimator.
result Validates application in variational Bayes.
A new multilevel framework speeds up ResNet training.
problem Training deep residual networks (ResNets) is time-consuming.
method Formulates ResNets as dynamical systems and uses time-dependent optimal control problems.
result Enhanced training of ResNets with multilevel auxiliary networks achieves significant speedup.
Improved multilevel scheme for value-at-risk computation.
problem Discontinuity in Heaviside function affects value-at-risk computation.
method Adaptive multilevel stochastic approximation to mitigate discontinuity.
result Best complexity improved to O(ε−2∣lnε∣25). New training method for neural nets using multilevel entropic regularization.
problem Training efficiency and generalization bounds for neural nets.
method Multilevel relative entropy, chaining mutual information, Gibbs posterior distribution.
result Proves the Gibbs posterior achieves the unique minimum of the empirical risk minimization problem.
Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the key features in achieving a high rate of multilevel variance convergence, and su…
Develops a fast algorithm for fitting multilevel factor models.
problem Fitting multilevel factor models with covariance structure.
method Novel expectation-maximization algorithm tailored for multilevel factor models.
result Shows efficient computation of inverse of positive definite MLR matrix.
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
Option valuation problems are often solved using standard Monte Carlo (MC) methods. These techniques can often be enhanced using several strategies especially when one discretizes the dynamics of the underlying asset, of which we assume follows a diffusion process. We consider the combination of two methodologies in th…
We investigate the extension of the multilevel Monte Carlo path simulation method to jump-diffusion SDEs. We consider models with finite rate activity, using a jump-adapted discretisation in which the jump times are computed and added to the standard uniform dis- cretisation times. The key component in multilevel analy…
In this paper a novel modification of the multilevel Monte Carlo approach, allowing for further significant complexity reduction, is proposed. The idea of the modification is to use the method of control variates to reduce variance at level zero. We show that, under a proper choice of control variates, one can reduce t…
MLMC boosts Bayesian optimization's look-ahead efficiency.
problem Efficiently computing nested expectations in Bayesian optimization.
method Multilevel Monte Carlo (MLMC) for nested operations.
result MLMC achieves MC convergence rate for nested operations, improving BO performance.
We propose a novel approach to the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our method involves a joint optimization formulation over several spaces of discrete…
Paper proposes a new algorithm to reduce derivative pricing computation time.
problem Derivative pricing computational inefficiency.
method Combines multilevel Richardson-Romberg and importance sampling.
result Reduces computational time while maintaining accuracy.
In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multil…
New neural network models speed up Bayesian multilevel modeling.
problem Complex computational challenges in Bayesian multilevel modeling.
method Probabilistic neural network architectures that leverage multilevel model factorization.
result Efficient posterior inference on unseen datasets with near-instant results.
The paper introduces a multilevel initialization method for deep neural networks.
problem Training very deep neural networks with layer-parallel methods.
method Continuous interpretation of training as optimal control, using time-dependent ODEs for neural network discretization, and a refinement strategy across the time domain.
result The method creates deep networks with good initializations from coarser networks, reducing training time and providing regularization.
Proposes a method to reduce parallel complexity of MLMC in SGD.
problem Poor scalability of MLMC in SGD on parallel platforms.
method Proposes a delayed MLMC gradient estimator to reduce parallel complexity.
result Proves reduction in average parallel complexity per iteration at the cost of slightly worse convergence rate.
We describe general multilevel Monte Carlo methods that estimate the price of an Asian option monitored at m fixed dates. Our approach yields unbiased estimators with standard deviation O(ε) in O(m+(1/ε)2) expected time for a variety of processes including the Black-Scholes model, Merton's jump-diffusion mod…
The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milst…
A new weighted MLMC method improves efficiency in Monte Carlo simulations.
problem Improving efficiency in Monte Carlo simulations with correlated coarse level approximations.
method Generalization of MLMC to any number of levels with control variates and weights.
result Significant efficiency improvements possible, especially when coarse level approximations are poorly correlated.
This work is motivated by the needs of predictive analytics on healthcare data as represented by Electronic Medical Records. Such data is invariably problematic: noisy, with missing entries, with imbalance in classes of interests, leading to serious bias in predictive modeling. Since standard data mining methods often …
New method estimates nested expectations with biased and antithetic sampling.
problem Estimating nested expectations with biased and antithetic sampling.
method Nested multilevel Monte Carlo with biased and antithetic sampling.
result Estimator achieves order ε^(-2) asymptotic cost.
Bayesian inference for deep neural networks using trace-class priors and MLMC.
problem Efficient Bayesian inference for deep neural networks.
method Trace-class neural network priors and Multilevel Monte Carlo method.
result Optimal computational complexity for Bayesian inference of TNN models.
Proposes a method for multilevel explanations of black-box models.
problem Need for explanations at intermediate or group levels, especially for GDPR compliance.
method Meta-method that builds a multilevel explanation tree using local explainability methods.
result Effective multilevel explanations for groups of data points, including novel test points.
Model trains passing events on a bridge using multilevel Gaussian process.
problem Represent aggregate train-passing events from a bridge monitoring system.
method Formulate a combined model with low-rank approximation hierarchical Gaussian process, incorporating domain expertise as constraints.
result Allow for simulation of previously unobserved train types.
We study the use of the multilevel Monte Carlo technique in the context of the calculation of Greeks. The pathwise sensitivity analysis differentiates the path evolution and reduces the payoff's smoothness. This leads to new challenges: the inapplicability of pathwise sensitivities to non-Lipschitz payoffs often makes …
In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O(Δt) with MLMC we can reduce the computational complexity to estimate expected value…
In medical domain, data features often contain missing values. This can create serious bias in the predictive modeling. Typical standard data mining methods often produce poor performance measures. In this paper, we propose a new method to simultaneously classify large datasets and reduce the effects of missing values.…
New algorithms reduce complexity for learning in MDPs with entropy regularization.
problem Efficient learning for MDPs with large or continuous state and action spaces.
method Multilevel Monte Carlo (MLMC) algorithms integrating fixed-point iteration and stochastic approximation of the Bellman operator.
result MLMC with unbiased approximation of the Bellman operator achieves polynomial sample complexity.