Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
Universal approximation theorem for differentiable maps on infinite-dimensional manifolds
problem Approximation of differentiable maps on infinite-dimensional manifolds
method Weighted universal approximation theorem
result Universal approximation theorem for differentiable maps
Deep learning approximates geometric measures of planar curves.
problem Approximating differential invariants of planar curves.
method Utilizing deep neural networks to estimate geometric measures of planar curves.
result Deep neural networks can learn to overcome instabilities and sampling artifacts.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. Unified framework for Gaussian process methods in differential equations.
problem Fragmented approaches to Gaussian process methods in differential equations.
method Unified Bayesian perspective integrating differential equation constraints.
result Consolidation of existing methods and foundation for future research.
Efficient offline reinforcement learning with neural networks using differentiable function approximation.
problem Statistical efficiency of offline reinforcement learning with function approximators.
method Pessimistic fitted Q-learning (PFQL) and differentiable function approximation.
result Provably efficient offline reinforcement learning with differentiable function approximation.
Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.
problem Approximating differentiable maps on infinite-dimensional manifolds.
method Proves a weighted Nachbin theorem to establish universal approximation for differentiable maps, including derivatives.
result Linear functions of the signature can approximate path space functionals including their derivatives.
Framework purifies approximate differential privacy to pure differential privacy.
problem Achieving pure differential privacy from approximate differential privacy.
method Randomized post-processing with calibrated noise to eliminate δ parameter.
result First statistically and computationally efficient reduction from approximate DP to pure DP.
Stochastic differential equation approximation for linear TD(0) under Markovian noise
problem Temporal-difference learning with linear function approximation
method Stochastic differential equation approximation
result Explains the constant-stepsize error floor
Efficiently private clustering algorithms with tight approximation ratios.
problem Differentially private clustering of various types.
method Efficient algorithms achieving tight approximation ratios for clustering problems.
result Achieves approximation ratios similar to non-private algorithms with small additive errors.
Simple DP algorithms find approximate solutions for nonconvex ERM.
problem Finding approximate solutions to nonconvex ERM problems with privacy.
method Differential privacy, descent directions, line search, mini-batching, two-phase strategy.
result Effective algorithms for nonconvex ERM with privacy guarantees.
This paper introduces a spline-based method for nonparametric ADVI that handles complex posterior distributions.
problem Learning complex posterior distributions with skewness, multimodality, and bounded support.
method Develops a spline-based nonparametric approximation approach for ADVI.
result Establishes the asymptotic consistency of the derived lower bound for importance weighted autoencoder.
Global approximation for piecewise linear paths via signatures.
problem Global approximation theorems for piecewise linear paths.
method Using signatures of piecewise linear paths and their density in Lp-norms. result Linear functionals of signatures are dense in Lp-norms under an integrability condition. Differential privacy is a cryptographically-motivated definition of privacy which has gained significant attention over the past few years. Differentially private solutions enforce privacy by adding random noise to a function computed over the data, and the challenge in designing such algorithms is to control the added…
Develops a mathematical model for automatic differentiation in machine learning.
problem Current automatic differentiation lacks a simple mathematical model for machine learning.
method Articulates relationships between program differentiation and nonsmooth functions, provides a class of functions and nonsmooth calculus.
result Shows how nonsmooth calculus applies to stochastic approximation methods and evidence of artificial critical points.
Researchers develop methods to calibrate ABMs using Bayesian techniques.
problem Challenges in calibrating ABMs due to likelihood intractability and non-differentiability.
method Generalised variational inference for differentiable ABMs.
result Accurate Bayesian parameter inferences for differentiable ABMs demonstrated.
Efficiently infers latent SDEs with scalable memory and time costs.
problem Inference of latent SDEs with high time and memory complexity.
method Amortized reparametrization of expectations under linear SDEs, coupled with efficient gradient approximation.
result Achieves similar performance to adjoint sensitivities with fewer model evaluations.
NODEs can approximate a wide range of diffeomorphisms with strong guarantees.
problem The approximation power of NODEs under certain conditions.
method Leveraging a structure theorem of the diffeomorphism group.
result NODEs can approximate a large class of diffeomorphisms with a stronger guarantee.
Improved set prediction model using multiset-equivariant operations and approximate implicit differentiation.
problem Existing set prediction models struggle with multisets and cannot represent certain functions.
method Introduced multiset-equivariance, improved DSPN with approximate implicit differentiation, and applied to CLEVR object property prediction.
result Significantly improved object property prediction on CLEVR dataset.
Gaussian processes with differential privacy protect both inputs and outputs.
problem Previous DP methods only protected model outputs, not inputs.
method Sparse GP with private variational approximation, adjusting covariance for DP noise.
result Accurate models can be produced under strong privacy protection with sufficient data.
The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for eve…
We develop a scalable method for Bayesian neural networks with stochastic differential equations.
problem Uncertainty quantification in deep neural networks.
method Gradient-based stochastic variational inference in continuous-depth Bayesian neural networks.
result Gradient estimator with zero variance as the approximation improves.
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
The paper clarifies the approximation of SGD with Ito SDEs for finite learning rates.
problem Theoretical justification and experimental verification of the Ito SDE approximation for finite learning rates in SGD.
method An efficient simulation algorithm SVAG and a necessary condition test for the SDE approximation.
result The Ito SDE approximation can meaningfully capture training and generalization properties of deep nets with finite learning rates.
Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.
NeuroDiff improves neural network equivalence verification with fine-grained approximations.
problem Verifying the equivalence of compressed neural networks.
method Symbolic and fine-grained approximation technique for differential verification.
result NeuroDiff achieves up to 1000X speedup and 5X accuracy improvement.
Differential ML combines AAD with ML for fast, accurate financial derivatives pricing and risk management.
problem Computational bottlenecks in financial derivatives risk management.
method Novel algorithms using automatic adjoint differentiation (AAD) for training fast, accurate approximations in real-time.
result Convergence guarantees for fast, accurate pricing and risk approximations for arbitrary derivatives instruments.
Study approximates operator learning for PDEs using Fourier multipliers.
problem Approximating operator behavior for PDE simulations.
method Approximation of operator symbols in Fourier domain using semi-norms.
result Identifies conditions for achieving predefined approximation error.
Universal approximation for stochastic processes using Brownian motion.
problem Approximating stochastic processes with linear functionals.
method Establishing Lp-type universal approximation theorems for rough path spaces. result Linear functionals on the signature of time-extended Brownian motion can approximate any p-integrable stochastic process. Study efficient numerical methods for American basket options.
problem Valuation of American basket options.
method Partial differential complementarity problems (PDCPs) and efficient discretization.
result Approximations of American basket options are close and converge favourably.
INNs can approximate diverse functions despite layer restrictions.
problem Can INNs approximate sufficiently diverse functions?
method Developed a theoretical framework based on differential geometry to simplify the approximation problem of diffeomorphisms.
result INNs have the universal approximation property.
Improves privacy amplification by shuffling for differential privacy.
problem Enhancing privacy guarantees in systems with anonymous data contributions.
method Theoretical and numerical analysis of Rényi differential privacy parameters and privacy amplification by shuffling.
result First asymptotically optimal analysis of Rényi differential privacy parameters for shuffled outputs.
We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process r…
Study on Wasserstein distance for numerical approximations of stochastic differential equations.
problem Estimating the Wasserstein distance between stochastic differential equation distributions and their numerical approximations.
method Unified framework for analyzing different integrators and a novel splitting method for underdamped Langevin dynamics.
result A novel splitting method for underdamped Langevin dynamics with optimal complexity.
Kernel method approximates dynamical operators from data.
problem Estimating eigenfunctions of dynamical operators from data.
method Kernel-based approach in reproducing kernel Hilbert spaces.
result Eigenfunctions estimated via matrix eigenvalue problems.
A continuing challenge for machine learning is providing methods to perform computation on data while ensuring the data remains private. In this paper we build on the provable privacy guarantees of differential privacy which has been combined with Gaussian processes through the previously published \emph{cloaking metho…
Improved ADMM for convex distributed learning with differential privacy.
problem Privacy concerns in distributed learning with sensitive data.
method Approximate multi-step ADMM with calibrated noise.
result Higher utility and error bounds asymptotic to state-of-the-art.
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs numerically requires the approximation of nested conditional expectations, i.e., it…
NCDEs improve predictions for irregular time series data.
problem Theoretical understanding of NCDEs' performance and irregular time series effects.
method Combining CDE theory and neural net complexity measures.
result Generalization bound and detailed sampling and approximation bias analysis.
In this paper, we present a method for the accurate estimation of the derivative (aka.~sensitivity) of expectations of functions involving an indicator function by combining a stochastic algorithmic differentiation and a regression. The method is an improvement of the approach presented in [Risk Magazine April 2018]. T…
We compare the sample complexity of private learning [Kasiviswanathan et al. 2008] and sanitization~[Blum et al. 2008] under pure ε-differential privacy [Dwork et al. TCC 2006] and approximate (ε,δ)-differential privacy [Dwork et al. Eurocrypt 2006]. We show that the sample complexity of these tasks under approxima…
HDNNs can approximate any continuous function, proving their expressivity.
problem Lack of a comprehensive study on the expressivity of HDNNs.
method Discretization of Hamiltonian Neural Ordinary Differential Equations (HNN-ODEs).
result HDNNs can approximate any continuous function over a compact domain.
We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion o…
Study shows rate of convergence for particle approximation of PDEs in Wasserstein space.
problem Analyzing convergence rates for particle approximations of PDEs in Wasserstein space.
method Backward stochastic differential equations techniques.
result Proved a rate of convergence of order 1/N for pathwise error and 1/sqrt(N) for L2-error on the derivative.
Study compares methods for computing hypergradients in machine learning problems.
problem Computing exact hypergradients in machine learning is difficult.
method Investigates reverse mode iterative differentiation and approximate implicit differentiation methods.
result Unified analysis provides iteration complexity bounds and hierarchy of methods.
New DP optimization methods for sparse gradients, improving on existing algorithms.
problem Differentially private optimization with sparse gradients in high-dimensional settings.
method Improved bounds for mean estimation, pure- and approximate-DP algorithms for stochastic convex optimization.
result First nearly dimension-independent rates for DP optimization with sparse gradients.
New method tackles convergence issues in approximating FBSDEs.
problem Convergence issues in approximating coupled FBSDEs.
method Approximates initial condition for a family of FBSDEs, then uses it to approximate the original FBSDE.
result Method converges even when standard deep BSDE method fails.
Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this …