The study analyzes implicit biases in neural networks using backward error analysis.
problem Analyzing implicit biases in multitask and continual learning settings.
method Backward error analysis to compute implicit training biases, deriving modified losses with three terms.
result The conflict term, measuring gradient alignment, is a new quantity in continual learning.
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
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…
FBSJNN solves PIDEs and FBSDEJs with deep learning, offering theoretical and numerical efficiency.
problem Solving Partial Integro-Differential Equations and Forward-Backward Stochastic Differential Equations with Jumps.
method FBSJNN framework using a single neural network for both solution approximation and non-local integral.
result FBSJNN achieves numerical solutions with a relative error of 10−3, demonstrating efficiency. The paper analyzes the probabilistic structure of DDPMs and bounds their sampling error.
problem Understanding and controlling errors in discrete-time DDPMs.
method Structural analysis of score functions, Schrödinger's problem, and FBSDEs.
result Explicit upper bound for total variation distance between sampling and target distributions.
In this paper we propose a new kind of high order numerical scheme for backward stochastic differential equations(BSDEs). Unlike the traditional θ-scheme, we reduce truncation errors by taking θ carefully for every subinterval according to the characteristics of integrands. We give error estimates of this nonlinear…
The detrending moving average (DMA) algorithm is a widely used technique to quantify the long-term correlations of non-stationary time series and the long-range correlations of fractal surfaces, which contains a parameter θ determining the position of the detrending window. We develop multifractal detrending moving a…
HA-SME models SGD dynamics with Hessian info for better escaping behaviors.
problem Capturing the escaping behaviors of SGD from stationary points.
method HA-SME, a novel SDE with Hessian info in drift and diffusion.
result HA-SME achieves best approximation error and recovers SGD dynamics for quadratics.
The study examines backward compatibility issues in ML systems, especially with noisy data.
problem Backward compatibility challenges in ML systems, especially with noisy data.
method Empirical analysis of ML systems across different architectures and datasets, focusing on data shifts and noise.
result Backward compatibility issues arise even without data shift due to optimization stochasticity and training on large-scale noisy datasets can significantly decrease compatibility.
Gradient-based methods for games suffer from discrete update steps that cause drift, affecting performance.
problem Gradient-based methods for two-player games suffer from drift due to discrete update steps.
method Derived modified continuous dynamical systems to closely follow the discrete dynamics of games.
result Identified distinct components of discretization drift that can alter or destabilize game performance.
New method preserves convergence rates in gradient-based optimization.
problem How to discretize gradient-based optimization systems while preserving stability and convergence rates.
method Geometric framework for dissipative symplectic integration.
result Dissipative symplectic integrators preserve rates of convergence up to a controlled error.
Improved numerical solution for BSDEs with reduced boundary errors.
problem Boundary errors in numerical solution of BSDEs.
method Modified damping and shifting schemes to transform target function into a bounded periodic function, applying Fourier transforms.
result Significant reduction in boundary errors with improved accuracy and convergence.
Recurrent neural networks' hidden state can be reconstructed from its past, providing a theoretical framework for stability and tracking.
problem Hidden-state stability in RNNs
method Backward coherence analysis
result Almost-sure convergence, rates under mixing, interpretable limiting representation, finite pathwise stopping times, and theoretical framework for time-uniform confidence sequences.
Novel approach finds implicit regularisation in two-player games using BEA.
problem Understanding implicit regularisation in two-player games.
method Using backward error analysis to construct continuous-time flows with gradient-eligible vector fields.
result Identifies new implicit regularisation effects in two-player games.
We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility,…
Deep signature algorithm for pricing path-dependent options.
problem Pricing path-dependent options with complex payoff functions.
method Extended backward scheme for state-dependent FBSDEs with reflections, incorporating signature layer for path-dependent FBSDEs.
result Convergence analysis of the algorithm with explicit dependence on truncation order and neural network approximation errors.
Backpropagation algorithm is indispensable for the training of feedforward neural networks. It requires propagating error gradients sequentially from the output layer all the way back to the input layer. The backward locking in backpropagation algorithm constrains us from updating network layers in parallel and fully l…
Study BSΔE on lattices for asset price analysis.
problem Optimal investment and market equilibrium analysis in asset price models.
method Backward stochastic difference equations on lattices.
result Applications to optimal investment and market equilibrium analysis.
The paper analyzes error propagation in dynamic programming for stochastic control and option pricing.
problem Error propagation in dynamic programming for stochastic control and option pricing.
method Formulated a general dynamic programming framework, used RKHSs for nonparametric regression, and Monte Carlo subsampling for estimating continuation value.
result Proposed a rigorous error decomposition and control mechanism for error propagation in dynamic programming.
Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential equations into the theoretical framework of diffusion approximation, extending the …
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.
SGD converges with perturbed forward-backward passes, explained by geometric amplification.
problem Analyzing convergence of SGD with perturbed forward-backward passes in composite optimization.
method Characterized propagation and amplification of perturbations, derived convergence guarantees for non-convex and PL objectives.
result Perturbations cascade through the computational graph, affecting convergence order under specific conditions.
We study the convergence rate of stochastic optimization of exact (NP-hard) objectives, for which only biased estimates of the gradient are available. We motivate this problem in the context of learning the structure and parameters of Ising models. We first provide a convergence-rate analysis of deterministic errors fo…
Deep model improves option pricing for CSI 300 index with sentiment and volatility features.
problem Challenges in real market option pricing, especially with constant volatility assumption.
method Deep Forward-Backward Stochastic Differential Equation (FBSDE) framework with dual-network architecture.
result Significant reduction in MAE and MAPE compared to BSM model.
We study the robustness properties of ℓ1 norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the ℓ1 estimator for measurements errors consisting of outliers coupled with noise. We…
A new algorithm solves high-dimensional nonlinear BSDEs using deep learning.
problem Solving high-dimensional nonlinear backward stochastic differential equations (BSDEs).
method Backward differential deep learning, reformulating BSDEs as differential deep learning problems, using Malliavin calculus, discretizing integrals with Euler-Maruyama method, approximating processes with DNNs, backwardly optimizing DNN parameters.
result The proposed algorithm efficiently approximates solutions and their derivatives for high-dimensional BSDEs.
We propose a numerical algorithm for backward stochastic differential equations based on time discretization and trigonometric wavelets. This method combines the effectiveness of Fourier-based methods and the simplicity of a wavelet-based formula, resulting in an algorithm that is both accurate and easy to implement. F…
Paper learns dictionaries for sparse signal recovery using automatic differentiation.
problem Learning dictionaries for sparse signal recovery from noisy data.
method Approximates reconstructions using FB algorithm and learns dictionaries with projected gradient descent.
result Successfully learns 1D TV dictionary from piecewise constant signals.
Statistical analysis of regularization in continual learning tasks.
problem Understanding how regularization affects model performance in sequential learning.
method Derivation of convergence rates, iterative update formula, and optimal hyperparameters for generalized ℓ2-regularization.
result Optimal hyperparameters balance forward and backward knowledge transfer, improving model performance.
We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward a…
Solves wealth maximization problem using variational analysis.
problem Maximizing expected utility of terminal wealth.
method Variational analysis, forward-backward stochastic differential equation (FBSDE).
result Characterization and solutions for various utility functions.
Algorithm for hedging American options with transaction costs.
problem Hedging American options considering transaction costs.
method Backward Hedging algorithm minimizing loss function.
result Optimal hedging strategy determined by minimizing loss function.
Deep-learning method solves BSVIEs and coupled systems.
problem High-dimensional, time-inconsistent stochastic control problems.
method Trains a neural network to approximate solution fields directly.
result Non-asymptotic error bound and scalable performance.
We establish existence, uniqueness and regularity of solution results for a class of backward stochastic partial differential equations with singular terminal condition. The equation describes the value function of non-Markovian stochastic optimal control problem in which the terminal state of the controlled process is…
New method reveals insights about stochastic optimization methods using modified equations.
problem Understanding the qualitative behavior of stochastic optimization algorithms.
method Developed a class of stochastic differential equations to approximate the dynamics of stochastic optimization methods.
result Mean-square stability of the modified equation provides qualitative insights about stochastic coordinate descent.
Kernel learning FBSDE filter improves nonlinear filtering efficiency.
problem Nonlinear filtering problem in high-dimensional systems.
method Iterative and adaptive meshfree approach using forward backward SDE and KDE.
result Rigorous convergence analysis provided, supporting empirical results.
Study Nash equilibrium between broker and informed trader in dealer and lit markets.
problem Nash equilibrium between broker and informed trader in dealer and lit markets with partial information.
method Convex analysis, FBSDEs, polynomial approximation.
result Existence and uniqueness of Nash equilibrium for short time horizons.
New algorithm solves complex equations using deep learning.
problem High-dimensional nonlinear PDEs and BSDEs.
method Iterated time discretization, deep neural networks, stochastic gradient descent.
result Increased accuracy and reduced complexity compared to existing methods.
Second-order estimator improves continuous-time policy evaluation.
problem Estimating value surfaces from discrete data with time-inhomogeneous dynamics.
method Moment-matching coefficients for high-order generator regression.
result Second-order estimator consistently outperforms Bellman baseline.
A new method detects changes in data sequences by comparing backward and forward confidence sequences.
problem Detecting changes in data sequences over time.
method Constructing and comparing backward and forward confidence sequences.
result The method successfully detects changes with strong nonasymptotic guarantees.
Unified framework for inference in complex nonlinear processes.
problem Challenges in inferring nonlinear continuous stochastic processes with sparse observations and complex topologies.
method Neural Backward Filtering Forward Guiding (NBFFG) framework that constructs a variational posterior using a proxy linear-Gaussian process.
result Empirical results show NBFFG outperforms baselines on synthetic benchmarks and high-dimensional phylogenetic analysis tasks.
The paper relaxes constraints on predictive coding models, making them more biologically plausible.
problem Neurophysiological models of predictive coding are not fully biologically plausible.
method The paper relaxes constraints on standard predictive coding algorithms by removing neurally implausible features.
result The removal of neurally implausible features does not significantly affect learning performance.
Adam's hyperparameters implicitly regularize solutions, penalizing or impeding loss gradients' norms.
problem Implicit regularization in Adam's hyperparameters and training stage.
method Backward error analysis and ODE approximations to study Adam's behavior.
result Adam's implicit regularization depends on hyperparameters and training stage, involving different norms.
Improved state estimation in nonlinear models using amortized backward variational inference.
problem State estimation in general state-space models.
method Amortized backward variational inference with neural network parameters.
result Linear growth of variational approximation error in number of observations.
The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs), introduced in [21], uses a uniform space grid. In this paper we utilize a tree-like spatial discretization that approximates the BSDE on the tree, so that no spatial interpolation procedure is necessary. In…
Deep learning is also known as hierarchical learning, where the learner _learns_ to represent a complicated target function by decomposing it into a sequence of simpler functions to reduce sample and time complexity. This paper formally analyzes how multi-layer neural networks can perform such hierarchical learning _ef…
Deep learning solves non-Markovian FBSDEs for utility maximization.
problem Solving utility maximization problems under rough volatility.
method Deep learning-based numerical methods for non-Markovian fully coupled FBSDEs.
result Error estimates and convergence provided for the deep learning approach.