DDEQs extend DEQs to discrete measure inputs using Wasserstein gradient flows.
problem Applying DEQs to discrete measure inputs like sets or point clouds.
method Wasserstein gradient flows for finding fixed points of discrete measures under permutation-invariance.
result DDEQs can compete with state-of-the-art models in tasks like point cloud classification and completion.
Deep equilibrium models estimate latent variables from data.
problem Estimating latent variables from data.
method Generalized exponential family models, deep equilibrium networks.
result Deep equilibrium models solve MAP estimates for latent and transformation parameters.
New model outperforms Neural ODEs while being more efficient.
problem Stable convergence and existence guarantees for implicit-depth models.
method Developed Monotone Operator Equilibrium Network (monDEQ) based on monotone operator theory.
result MonDEQ models outperform Neural ODEs and are more computationally efficient.
Deep learning models replicate Kyle's model's market equilibrium.
problem Understanding market equilibria in asymmetric information settings.
method Using deep neural networks to model agents in Kyle's single period model.
result Trained networks' behavior converges to Kyle's predicted equilibrium.
Discovery of atomistic systems with desirable properties is a major challenge in chemistry and material science. Here we introduce a novel, autoregressive, convolutional deep neural network architecture that generates molecular equilibrium structures by sequentially placing atoms in three-dimensional space. The model e…
Wide neural networks converge to Gaussian processes, improving generalization.
problem Understanding the generalization of wide neural networks, especially deep equilibrium models.
method Investigation of deep equilibrium models (DEQs) with infinite-depth layers, focusing on their convergence to Gaussian processes as width and depth approach infinity.
result Wide DEQs converge to Gaussian processes, maintaining generalization performance.
We present a new approach to modeling sequential data: the deep equilibrium model (DEQ). Motivated by an observation that the hidden layers of many existing deep sequence models converge towards some fixed point, we propose the DEQ approach that directly finds these equilibrium points via root-finding. Such a method is…
New method stabilizes DEQ models by regularizing Jacobian of fixed-point equations.
problem Stability and performance of DEQ models.
method Jacobian regularization to stabilize DEQ models.
result Significant stabilization of fixed-point convergence in DEQ models.
Computing equilibrium states in condensed-matter many-body systems, such as solvated proteins, is a long-standing challenge. Lacking methods for generating statistically independent equilibrium samples in "one shot", vast computational effort is invested for simulating these system in small steps, e.g., using Molecular…
DEQs and explicit networks are nearly equivalent for Gaussian mixtures.
problem Understanding the equivalence between DEQs and explicit neural networks.
method Random matrix theory and analysis of kernel matrices.
result A shallow explicit network can mimic the kernel of a DEQ.
A new method uses deep learning to predict rare events in complex systems.
problem Predicting rare and extreme events in non-equilibrium systems.
method A deep learning approach that minimizes the geometrical action.
result The method accurately predicts rare events in various complex systems.
Machine learning detects tipping points in complex systems.
problem Detecting abrupt shifts in complex dynamical systems.
method Equilibrium-informed neural networks (EINNs) trained on candidate equilibrium states.
result EINNs can identify critical thresholds in nonlinear systems.
Gradient descent dynamics studied for DEQs in linear and single-index models.
problem Understanding gradient descent dynamics for DEQs.
method Rigorously studied gradient descent dynamics for DEQs in linear and single-index models.
result Gradient descent converges to a global minimizer for linear DEQs and single-index models.
Proposes a deep learning method for solving complex financial games with delays.
problem Financial modeling with multi-agent interactions and delayed effects.
method Parameterizes controls using recurrent neural networks and trains them with modified fictitious play.
result Demonstrates effectiveness on finance problems with known solutions and new problems with derived Nash equilibria.
MDEQ models learn multi-resolution features efficiently.
problem Large-scale, hierarchical pattern recognition.
method Implicit differentiation, multiscale deep equilibrium model.
result MDEQs achieve performance on par with recent models.
New approaches improve adversarial robustness of DEQs.
problem Adversarial vulnerability of DEQs.
method Developed approaches to estimate intermediate gradients and integrate them into attacking pipelines.
result Demonstrated adversarial robustness of DEQs competitive with deep networks.
Deep CapsNet improves sign language recognition from wearable IMUs.
problem Continuous recognition of sign language from wearable devices.
method Custom CapsNet architecture using deep capsule networks and game theory.
result Improved accuracy of 94% and 92.50% for 3 and 5 routings respectively, compared to 87.99% for CNN.
RevDEQs improve performance on tasks with exact gradients and fewer function evaluations.
problem Inexact gradient calculation in DEQs leads to unstable training and requires regularisation or many function evaluations.
method Introduce Reversible Deep Equilibrium Models (RevDEQs) that allow for exact gradient calculation, no regularisation, and far fewer function evaluations.
result RevDEQs significantly improve performance on language modelling and image classification tasks.
DEQs converge to optimal solutions with mild over-parameterization.
problem Training over-parameterized deep equilibrium models.
method Solves equilibrium point directly, uses gradient descent, and analyzes convergence via linear rate.
result Gradient descent converges to a globally optimal solution at a linear rate for quadratic loss.
The paper challenges the belief that more inner iterations at test time improve performance in implicit deep learning.
problem The performance improvement of implicit deep learning models with increased inner iterations at test time.
method Theoretical analysis of a simple setting, validation on implicit deep learning problems.
result Overparametrization plays a key role; increasing the number of iterations at test time does not improve performance for overparametrized networks.
We propose a deep neural network-based algorithm to identify the Markovian Nash equilibrium of general large N-player stochastic differential games. Following the idea of fictitious play, we recast the N-player game into N decoupled decision problems (one for each player) and solve them iteratively. The individua…
Why and how that deep learning works well on different tasks remains a mystery from a theoretical perspective. In this paper we draw a geometric picture of the deep learning system by finding its analogies with two existing geometric structures, the geometry of quantum computations and the geometry of the diffeomorphic…
In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric N-player non-zero-sum stochastic differential games, for which we refer as \emph{deep fictitious play}, a multi-stage learning pro…
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
DREAM learns optimal strategies in imperfect games without needing a simulator.
problem Learning optimal strategies in imperfect-information games with multiple agents.
method DREAM is a deep reinforcement learning algorithm that converges to Nash Equilibria and coarse correlated equilibria.
result DREAM achieves state-of-the-art performance in benchmark games and is competitive with simulator-based algorithms.
Deep fictitious play converges to Nash equilibrium in stochastic differential games.
problem Finding Nash equilibrium in large stochastic differential games.
method Decouples the game into sub-optimization problems and solves each player's optimal strategy with deep BSDE method.
result Deep fictitious play converges to the true Nash equilibrium.
This paper connects deep neural networks to game theory, revealing their congestion game properties.
problem Understanding the behavior of deep neural networks using game theory.
method Analyzing deep neural networks as congestion games and applying game theory results.
result Deep neural networks exhibit congestion game properties, linking them to game theory.
Deep equilibrium models converge globally without explicit computation.
problem Global convergence of deep learning models with implicit layers.
method Analysis of gradient dynamics and proof of convergence rate.
result Deep equilibrium models converge to global optimum at a linear rate.
Deep neural networks show some layers better align with data than others.
problem Understanding why some layers in deep neural networks better align with data.
method Introducing the Equilibrium Hypothesis to connect alignment pattern to signal propagation.
result The Equilibrium Hypothesis explains the ascent-descent pattern of alignment in deep neural networks.
This paper extends the convergence rate of DEQs with ReLU to any general activation.
problem Proving global convergence rate for DEQs with general activations.
method Developed a novel population Gram matrix and new form of dual activation with Hermite polynomial expansion.
result Gradient descent converges to a globally optimal solution at a linear rate for DEQs with general activations.
Deep RL solves complex macroeconomic models.
problem Solving dynamic stochastic general equilibrium models with bounded rationality.
method Using deep reinforcement learning to model agents as neural networks.
result Artificially intelligent agents can solve models in all policy regimes.
A new method relaxes molecules without needing non-equilibrium data.
problem Molecular relaxation requires understanding non-equilibrium structures.
method MoreRed: molecular relaxation by reverse diffusion with time step prediction.
result MoreRed learns a simpler pseudo potential energy surface.
The paper explores how symmetries and noise in SGD influence parameter dynamics.
problem Understanding the dynamics of parameter updates in SGD with symmetries.
method Proved the existence of noise equilibria and showed their role in balancing gradient noise.
result Gradient noise creates a systematic motion of parameters to a unique fixed point, called noise equilibria.
Improves adversarial robustness of DEQ models by regulating neural dynamics.
problem Limited adversarial robustness of DEQ models.
method Interprets DEQs as neural dynamics, uses entropy reduction and random intermediate states.
result Significantly increases adversarial robustness of DEQ models.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
We propose a novel approach to addressing the vanishing (or exploding) gradient problem in deep neural networks. We construct a new architecture for deep neural networks where all layers (except the output layer) of the network are a combination of rotation, permutation, diagonal, and activation sublayers which are all…
A multi-layer deep Gaussian process (DGP) model is a hierarchical composition of GP models with a greater expressive power. Exact DGP inference is intractable, which has motivated the recent development of deterministic and stochastic approximation methods. Unfortunately, the deterministic approximation methods yield a…
HomoODE connects DEQs and Neural ODEs via homotopy continuation, improving accuracy and memory efficiency.
problem Connecting DEQs and Neural ODEs for better model performance and efficiency.
method Established a connection between DEQs and Neural ODEs using homotopy continuation, proposing HomoODE.
result HomoODE outperforms existing implicit models in accuracy and memory consumption.
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.
Deep-MacroFin uses neural networks to solve complex economic models efficiently.
problem Solving high-dimensional partial differential equations in continuous time economics.
method Leverages deep learning, specifically Multi-Layer Perceptrons and Kolmogorov-Arnold Networks, optimized with HJB equations.
result Offers a more efficient solution (5imes less memory, 40imes fewer FLOPs) for 50D economic models. Deep learning solves and estimates complex financial models.
problem Estimating and solving continuous-time financial models.
method Uses deep learning to solve and estimate models simultaneously.
result Demonstrates advantages like generality and large state space handling.
We address the challenge of designing optimal adversarial noise algorithms for settings where a learner has access to multiple classifiers. We demonstrate how this problem can be framed as finding strategies at equilibrium in a two-player, zero-sum game between a learner and an adversary. In doing so, we illustrate the…
Analyzes how financial network dependencies can lead to multiple equilibrium outcomes and optimal bailout strategies.
problem Multiple equilibrium outcomes in financial networks due to dependency cycles.
method Characterized necessary and sufficient conditions for bank solvency, and provided upper bounds on optimal bailout payments.
result Minimum bailout payments needed to ensure systemic solvency and prevent cascading defaults.
New learning algorithm mimics biological neural networks.
problem Biologically implausible backpropagation for directed neural networks.
method Introduces new neuronal dynamics and learning rule for arbitrary architectures, sparsity-inducing pruning method, and dynamical-systems characterization.
result Prunes irrelevant connections and improves learning efficiency.
Neural network models colloidal particle dynamics in non-equilibrium systems.
problem Analyzing non-equilibrium dynamics of many-body colloidal systems.
method Combining power functional theory and machine learning, training a neural network to predict internal force fields.
result The neural network accurately predicts dynamics in non-equilibrium systems, in good agreement with simulations.
This paper tackles task offloading in edge computing systems with dynamic interactions.
problem Dynamic interactions among mobile users in an edge computing system.
method Modelled as a non-cooperative stochastic game, transformed into a Markov decision process, and solved using online deep reinforcement learning.
result The proposed deep RL scheme balances age of information and energy consumption.
FNO-DEQ solves steady-state PDEs as fixed points, outperforming traditional FNOs.
problem Lack of understanding in designing neural network architectures for PDEs.
method Proposes FNO-DEQ, a deep equilibrium architecture that solves steady-state PDEs as fixed points.
result FNO-DEQ outperforms FNO-based architectures in predicting solutions to steady-state PDEs.
Develops a method for non-equilibrium importance sampling to estimate expectations and constants.
problem Estimating expectations and normalization constants for complex high-dimensional distributions.
method Generates samples from a base distribution, transports them using a velocity field, and averages along flowlines.
result The method can achieve zero-variance estimation and significantly reduces variance compared to vanilla estimators.