New model improves field learning with improved equivariance.
problem Learning equivariant stochastic fields.
method Equivariant Gaussian processes and Steerable Conditional Neural Processes.
result SteerCNPs significantly improve performance in transfer learning tasks.
Develops a new theory for neural systems stability and width effects.
problem Stability and finite-width effects in deep neural systems.
method Gauge-covariant stochastic effective field theory using classical commuting fields.
result Predicts the edge of chaos and low-frequency spectral deformation.
Novel approach to Nash equilibrium in mean-field stochastic games with operator resolvents.
problem Finding Nash equilibrium in mean-field stochastic games with mean-field interaction.
method Proposed a novel approach to derive Nash equilibrium semi-explicitly using operator resolvents and stochastic Fredholm equations.
result Equilibrium of the N-player game converges to mean-field equilibrium, and ε-Nash equilibrium derived as a by-product. We describe the sample paths of the stochastic field F=Ft(v,x,q) of aggregate utilities parameterized by Pareto weights v and total cash amounts x and stocks' quantities q in an economy. We also describe the sample paths of the stochastic field G=Gt(u,y,q), which is conjugate to F with respect to the …
Conservative SPDEs emerge from fluctuating SGD dynamics in neural networks.
problem Understanding the convergence of stochastic gradient descent to SPDEs.
method Mean-field analysis and central limit theorem for SPDEs.
result Optimal convergence rates for SPDEs derived from SGD.
The asymptotic pseudo-trajectory approach to stochastic approximation of Benaim, Hofbauer and Sorin is extended for asynchronous stochastic approximations with a set-valued mean field. The asynchronicity of the process is incorporated into the mean field to produce convergence results which remain similar to those of a…
We introduce a stochastic model for noisy vector fields on manifolds.
problem Noisy vector fields violate the assumption of parallel transport in stochastic analysis.
method We define a stochastic Lie bracket that induces torsion and analyze its consequences.
result The stochastic Lie bracket induces torsion in expectation.
The paper connects PSO and CBO methods using stochastic modeling and mean-field limits.
problem Global optimization problems with particle swarm optimization and consensus based optimization.
method Stochastic differential equations and mean-field approximation to derive macroscopic hydrodynamic equations.
result Derives mean-field approximation for PSO and links it to CBO methods.
In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance function…
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field G-SDEs. method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
Analysis of SGD for Gaussian mixture classification using dynamical mean-field theory.
problem Learning dynamics of SGD for a neural network classifying Gaussian mixture.
method Applying dynamical mean-field theory to track SGD dynamics in high dimensions.
result Reveals how SGD navigates the non-convex loss landscape.
Investigates spontaneous symmetry breaking in non-equilibrium systems.
problem Spontaneous symmetry breaking of ergodicity in non-equilibrium systems.
method Mathematical and effective field theory approaches to investigate symmetry breaking.
result Symmetry breaking phenomena observed in stochastic processes.
The paper tackles drift identification in Lévy α-stable stochastic systems, proposing a Fourier space approach.
problem Estimating the drift field of a stochastic differential equation driven by Lévy α-stable noise.
method Fourier space approach, parameterizing the drift field using Fourier series, minimizing a loss function with gradients computed via the adjoint method.
result The method is capable of learning drift fields in qualitative and/or quantitative agreement with ground truth fields.
We provide bounds on control learning error in stochastic systems.
problem Learning optimal controls in stochastic environments with uncontrolled parts.
method Dynamic programming and mean-field interpretation of neural networks.
result Non-asymptotic bounds on generalization error for stable overparametrised settings.
New framework models neural systems with random architecture on manifolds.
problem Complex, uncertain systems with non-Gaussian outputs.
method Latent random field on compact manifold generates neural architecture and weights.
result Synthetic neural systems can produce stochastic outputs for deterministic inputs.
Adam optimizer converges to zeros of a new vector field, not just gradient zeros.
problem Prove convergence rates for Adam optimizer in simple quadratic optimization problems.
method Introduced Adam vector field to analyze Adam optimizer's convergence.
result Established optimal convergence rates for Adam optimizer.
Study on price formation among investors with exponential utility and liabilities.
problem Equilibrium price formation among investors with heterogeneous risk-averseness and liabilities.
method Mean-field game theory and mean-field backward stochastic differential equations (BSDE).
result Existence of equilibrium risk-premium process and market clearing in the large population limit.
NN-Turb generates turbulent velocity statistics using neural networks.
problem Creating a 1D field with turbulent velocity statistics.
method Fully-convolutional neural network (NN-Turb) to generate the field.
result NN-Turb generates a 1D field that satisfies Kolmogorov's 2/3 and 4/5 laws, exhibiting intermittency.
We apply stochastic average gradient (SAG) algorithms for training conditional random fields (CRFs). We describe a practical implementation that uses structure in the CRF gradient to reduce the memory requirement of this linearly-convergent stochastic gradient method, propose a non-uniform sampling scheme that substant…
Stochastic normalizing flows improve lattice field theory simulations.
problem Efficiently sample lattice field theories.
method Combining neural-network layers with Monte Carlo updates.
result Stochastic normalizing flows are equivalent to out-of-equilibrium simulations.
We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fie…
APAC-Net solves high-dimensional stochastic MFGs using neural networks.
problem High-dimensional stochastic mean-field games.
method Alternating population and control neural networks, parameterizing value and density functions.
result Solves up to 100-dimensional MFG problems.
Paper models Bitcoin market dynamics using 1+1D field theory.
problem Understanding stylized facts in Bitcoin markets.
method Collects order-book datasets, applies KPZ-like stochastic equations.
result Predicts order book dynamics with high precision.
Study shows how market firm capitalization models converge to stochastic PDE solutions.
problem Understanding convergence of rank-based models with common noise to stochastic PDE solutions.
method Analysis of mean field limit, martingale problem, and pathwise entropy solutions.
result Empirical cumulative distribution function converges to solution of a stochastic PDE under certain conditions.
We establish a stochastic maximum principle (SMP) for control problems of partially observed diffusions of mean-field type with risk-sensitive performance functionals.
A new method for stochastic optimal control improves accuracy over existing techniques.
problem Improving the accuracy of stochastic optimal control for noisy systems.
method Stochastic Optimal Control Matching (SOCM) using Iterative Diffusion Optimization (IDO) with path-wise reparameterization trick.
result SOCM achieves lower error than existing techniques for three out of four control problems, sometimes by an order of magnitude.
In a recent formulation of a quantum field theory of forward rates, the volatility of the forward rates was taken to be deterministic. The field theory of the forward rates is generalized to the case of stochastic volatility. Two cases are analyzed, firstly when volatility is taken to be a function of the forward rates…
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.
Solves a game between brokers and informed traders using stochastic differential equations.
problem Optimizing wealth in a game between brokers and informed traders with private signals.
method Closed-form solutions to a mean-field game using forward-backward SDEs.
result Optimal trading strategies for both brokers and informed traders are found.
Analyzes high-dimensional SGD dynamics using DMFT.
problem Understanding the high-dimensional behavior of multi-pass SGD with small batch sizes.
method Derives DMFT equations for high-dimensional SGD dynamics.
result Proves DMFT equations characterize the asymptotic distribution of SGF parameters.
The paper studies Hawkes processes under mean-field limits and criticality conditions.
problem Analyzing nearly unstable Hawkes processes in a mean-field regime.
method Extending the method by Jaisson and Rosenbaum, establishing scaling limits and propagation of chaos.
result Scaling limits of Hawkes processes are stochastic Volterra diffusions of affine type, with three distinct limiting regimes.
This paper contains a summary of mathematical researches of stochastic properties of the long time behavior of a continuously observed (and interactively controlled) quantum--field top. Applications to interactively controlled stochastic computer-graphic dynamical systems are also discussed.
Modeling price formation with interacting Hawkes processes leading to stochastic volatility with leverage.
problem Capturing the complex dynamics of price formation in financial markets.
method Agent-based approach to aggregate self-exciting point processes with mean-field interaction.
result Aggregated model converges to a stochastic volatility model with leverage effect and faster-than-linear mean reversion.
New SDEs use G-Brownian motion, extending mean-field models.
problem Extending mean-field models to new types of stochastic processes.
method Introduced G-SDEs with coefficients dependent on current state and solution as random variable. result Validated new SDE framework for complex stochastic systems.
We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural net…
In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well…
Improved convergence rates for MFLD in various gradient estimators.
problem Proving convergence rates for mean-field Langevin dynamics with stochastic gradient updates.
method General framework for propagation of chaos, including finite-particle approximation, time-discretization, and stochastic gradient approximation.
result Improved convergence rates for SGD and SVRG settings.
We consider expected performances based on max-stable random fields and we are interested in their derivatives with respect to the spatial dependence parameters of those fields. Max-stable fields, such as the Brown--Resnick and Smith fields, are very popular in spatial extremes. We focus on the two most popular unbiase…
Generative adversarial networks generate realistic, time-evolving high-resolution atmospheric fields.
problem Improving spatial resolution of low-resolution atmospheric images.
method Recurrent, stochastic super-resolution GAN for generating ensembles of time-evolving high-resolution atmospheric fields.
result The GAN produces realistic, temporally consistent super-resolution sequences for radar-measured precipitation and cloud optical thickness.
The study analyzes stochastic Lie systems and their applications in various models.
problem Analyzing stochastic differential equations on manifolds.
method Coalgebra method for Hamiltonian stochastic Lie systems.
result New examples of stochastic Lie systems and Hamiltonian stochastic Lie systems are analyzed.
New tree-structured Markov fields with Poisson marginals for counting variables.
problem Counting variables with complex dependencies.
method Tree-structured Markov random fields with Poisson marginals.
result Straightforward sampling and joint probability calculations.
The paper generalizes the construction by stochastic flows of consistent utility processes introduced by M. Mrad and N. El Karoui in (2010). The utilities random fields are defined from a general class of processes denoted by $\GX$. Making minimal assumptions and convex constraints on test-processes, we construct by co…
New method controls gradient error for sparse MRFs.
problem Efficient learning for sparse discrete MRFs with NP-hard inference.
method Stochastic proximal gradient (SPG) with controlled gradient approximation error.
result Novel bounds control gradient approximation quality.
In stochastic control problems delicate issues arise when the controlled system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. We propose to use Meyer-σ-fields as a flexi…
Proves equations for high-dimensional gradient-based methods from Gaussian data.
problem High-dimensional asymptotics of gradient-based learning algorithms.
method Closed-form equations derived from dynamical mean-field theory.
result Equations match those from discretized DMFT for gradient flow.
Develops asset pricing models with mean field game theory for heterogeneous agents.
problem Tackles equilibrium asset pricing in incomplete markets with heterogeneous agents.
method Uses mean field game theory and mean field backward stochastic differential equations (BSDEs).
result Derives equilibrium risk premium and shows market clearing in the large population limit.
The paper extends macroscopic market making to stochastic games, revealing properties and solving equations.
problem Price competition among market makers in a stochastic game setting.
method Extension of macroscopic market making framework to stochastic games, introducing multidimensional characteristic equations.
result New well-posedness results for forward-backward stochastic differential equations.
This paper extends transfer operator theory to McKean-Vlasov equations.
problem Analyzing the behavior of complex dynamical systems using transfer operators.
method Extended dynamic mode decomposition and Galerkin projection.
result Finite-dimensional approximations of transfer operators computed.