The aim of this paper is fourfold. Firstly, we introduce and study the f-ultra-harmonic maps. Secondly, we recall the geometric dynamics generated by a first order normal PDE system and we give original results regarding the geometric dynamics generated by other first order PDE systems. Thirdly, we determine the Gauss …
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48 results for “First-order dynamics”
CEFOL uses deep learning for dynamic programming with recursive utility.
problem Challenges in solving dynamic programming problems with recursive utility.
method Introduces a separate neural network for certainty equivalent, uses first-order optimality conditions to learn value and policy functions.
result CEFOL achieves high accuracy in learning value and policy functions, matching VFI benchmarks.
Study stabilizes second-order systems to first-order dynamics.
problem Stabilizing second-order systems to first-order dynamics.
method Feedback control of second-order systems on manifolds.
result Second-order systems can globally exponentially stabilize first-order dynamics for fully actuated systems.
First order discretizations of Langevin diffusion can achieve better generalization error with additional smoothness assumptions.
problem Analyzing generalization error for first order discretizations of Langevin diffusion.
method Providing a sufficient smoothness condition to show that first order methods can achieve arbitrarily runtime complexity for a given expected generalization error.
result First order methods can achieve arbitrarily runtime complexity with additional smoothness assumptions.
We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical s…
The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control pro…
Complexity of normalized stochastic first-order methods with momentum under heavy-tailed noisemath.OC
New methods solve optimization problems with heavy-tailed noise, improving upon existing complexity bounds.
problem Optimization problems with heavy-tailed noise and weakly average smoothness.
method Normalized stochastic first-order methods with Polyak, multi-extrapolated, and recursive momentum.
result First-order oracle complexity results for finding approximate stochastic stationary points under heavy-tailed noise.
Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they…
SONODEs and ANODEs improve learning of second order dynamics.
problem Learning dynamics governed by second order laws.
method Extended adjoint sensitivity method and theoretical analysis of ANODEs.
result SONODEs and ANODEs can learn higher order dynamics efficiently.
A new first-order sampler improves diffusion probabilistic model sampling quality.
problem The belief that first-order methods are inherently slower for diffusion probabilistic model sampling.
method A novel training-free, first-order sampler that approximates the forward-value evaluation via a one-step lookahead predictor.
result The proposed sampler provably approximates the ideal forward-value trajectory while retaining first-order convergence and can improve sample quality under the same NFE budget.
Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysismath.ST
New inequalities help optimize first-order algorithms for statistical risk analysis.
problem Optimizing first-order iterative algorithms for statistical risk analysis.
method Introducing basic inequalities that connect implicit and explicit regularization.
result The basic inequalities translate the number of iterations into an effective regularization coefficient.
Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
problem Accelerating convex optimization
method Hamiltonian dynamics
result Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
DEO uses gradient information to escape saddle points in neural networks.
problem Training deep neural networks struggles with flat regions and saddle points.
method Dimer-Enhanced Optimization (DEO) uses gradient information to estimate curvature and escape saddle points.
result DEO improves training efficiency and performance compared to standard first-order methods.
Deep learning solves dynamic programming with recursive utility.
problem Challenges in solving high-dimensional discrete-time dynamic programming problems with recursive utility.
method Certainty Equivalent Learning (CEL) algorithm that learns certainty-equivalent value directly with neural networks.
result Accurate value and policy approximations in high-dimensional problems, comparable to VFI in some cases.
Study stability of trading strategy under market perturbations.
problem Dynamic stability of trading strategy under market changes.
method Established reverse conjugacy characterizations, proved continuity and convergence of indirect utility process.
result Continuity and first-order convergence of indirect utility process under market perturbations.
Study dynamic risk measures with distributional uncertainty using optimal transport.
problem Risk robustification under distributional uncertainty in Markovian models.
method Characterize risk measures via convex monotone semigroups and optimal transport costs.
result Identify generator and correction terms for dynamic risk measures under different scaling regimes.
New algorithms optimize constrained problems faster, avoiding full set optimization.
problem Optimizing constrained problems efficiently and quickly.
method Designing accelerated first-order algorithms that avoid full set optimization.
result Proved convergence to stationary points in nonconvex settings and accelerated rates in convex settings.
This work proposes a meta-learning approach for better adaptation of source code models.
problem Adapting source code models to unseen local contexts.
method Formulated as a meta-learning problem, selecting targeted information for adaptation.
result Improved performance in code auto-completion tasks, especially for identifiers and literals.
We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We …
In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g.~the volume standing at the top of the book). We study the fluctuations of the …
Paper analyzes adversarial dynamics in neural networks.
problem Vulnerability of neural networks to adversarial perturbations.
method Analyzed the dynamics of maximization step in adversarial training.
result Projected gradient ascent finds a local maximum in polynomial iterations.
A new method optimizes diffusion models for fine-tuning tasks efficiently.
problem Optimizing diffusion models for downstream tasks using nested bilevel structures.
method Formalizes the challenge as a generative bilevel optimization problem and introduces a first-order bilevel framework.
result Our method outperforms existing fine-tuning and hyperparameter search baselines.
The exploration-exploitation trade-off is among the central challenges of reinforcement learning. The optimal Bayesian solution is intractable in general. This paper studies to what extent analytic statements about optimal learning are possible if all beliefs are Gaussian processes. A first order approximation of learn…
Recent computational strategies based on screening tests have been proposed to accelerate algorithms addressing penalized sparse regression problems such as the Lasso. Such approaches build upon the idea that it is worth dedicating some small computational effort to locate inactive atoms and remove them from the dictio…
Using agent-based modelling, empirical evidence and physical ideas, such as the energy function and the fact that the phase space must have twice the dimension of the configuration space, we argue that the stochastic differential equations which describe the motion of financial prices with respect to real world probabi…
Adaptive Stochastic Gradient Langevin Dynamics: Taming Convergence and Saddle Point Escape Timecs.LG
In this paper, we propose a new adaptive stochastic gradient Langevin dynamics (ASGLD) algorithmic framework and its two specialized versions, namely adaptive stochastic gradient (ASG) and adaptive gradient Langevin dynamics(AGLD), for non-convex optimization problems. All proposed algorithms can escape from saddle poi…
We analyze training dynamics in Gaussian mixture models using a comparison theorem.
problem Analyzing training algorithms with Gaussian mixture data.
method Applying a Gaussian comparison theorem to a specific family of training algorithms.
result Validated dynamic mean-field expressions and provided iterative refinement schemes.
In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures natura…
New model predicts video sequences with latent dynamics.
problem Predicting video sequences with inherent uncertainty.
method Introduces a novel stochastic temporal model with latent dynamics.
result Latent model outperforms prior state-of-the-art methods.
Machine learning predicts synchronization transitions in unknown systems.
problem Predicting synchronization transitions in systems with unknown equations.
method Developed a 'parameter-aware' machine learning scheme using reservoir computing or echo state networks.
result Machine learning accurately predicts synchronization transitions, including hysteresis loops.
The paper explores the geometry of holomorphic flows and orbits.
problem Understanding the local geometry of holomorphic flows and their equilibria.
method Analyzing the local geometry of first-order equilibria and higher-order equilibria under holomorphic conditions.
result Holomorphic Poincaré-Bendixson theorem: bounded non-periodic orbits are homoclinic or heteroclinic.
Analyzes Poisson structures on solution spaces of Hamiltonian field theories.
problem Defining Poisson bracket structures on solution spaces of first order Hamiltonian field theories.
method Examines mechanical point systems and field theories without gauge symmetries, introduces symplectic structures; for gauge theory, free electrodynamics, a pre-symplectic tensor is used to induce a Poisson structure.
result Existence of Poisson structures on solution spaces of Hamiltonian field theories, including free electrodynamics.
Dynamic model improves static economics by incorporating time effects.
problem Static economics overlooks time-dependent phenomena, limiting model accuracy.
method Signals-based approach to reinterpret microeconomic theory, using utility function.
result Dynamic models provide better comparisons with empirical observations.
New method for efficient probabilistic deep state-space models.
problem Efficient inference for probabilistic deep state-space models.
method Deterministic inference algorithm for ProDSSM with neural network weights.
result Superior balance between predictive performance and computational budget.
Hamiltonian Descent Methodsmath.OC
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at th…
We formulate and study a general family of (continuous-time) stochastic dynamics for accelerated first-order minimization of smooth convex functions. Building on an averaging formulation of accelerated mirror descent, we propose a stochastic variant in which the gradient is contaminated by noise, and study the resultin…
Stochastic Gradient Langevin Dynamics infuses isotropic gradient noise to SGD to help navigate pathological curvature in the loss landscape for deep networks. Isotropic nature of the noise leads to poor scaling, and adaptive methods based on higher order curvature information such as Fisher Scoring have been proposed t…
Improved sampling efficiency with Random Reshuffling for Langevin dynamics.
problem Sampling efficiency in stochastic gradient algorithms.
method Random Reshuffling for Stochastic Gradient Langevin Dynamics (SGLD).
result Random Reshuffling leads to improved performance in sampling.
This work extends identifiability analysis to sequential latent variable models, focusing on Switching Dynamical Systems.
problem Identifying latent variables in sequential data models.
method Proved identifiability of Markov Switching Models and established conditions for Switching Dynamical Systems.
result Identifiability of latent variables and non-linear mappings in Switching Dynamical Systems up to affine transformations.
New conditional risk measures called conditional generalized quantiles defined and characterized.
problem Developing new risk measures for dynamic risk assessment.
method Propose and characterize conditional generalized quantiles using expected utility model and equivalent conditions.
result Characterized conditional generalized quantiles as well-defined and equivalent to a conditional first order condition.
fSGLD optimizes deep learning by favoring flat regions in the loss landscape.
problem Understanding and improving the behavior and generalization of deep learning algorithms.
method Flatness-Aware Stochastic Gradient Langevin Dynamics (fSGLD) that biases learning towards flat basins.
result fSGLD targets a flatness-biased Gibbs distribution with explicit excess risk guarantees.
Transforms game optimization dynamics into frequency domain for precise hyperparameter analysis.
problem Analyzing convergence of hyperparameters in game optimization.
method Frequency-domain framework using High-Resolution Differential Equations (HRDEs) and Laplace transforms.
result Derives precise convergence criteria for the Lookahead algorithm.
An algorithm is presented for momentum gradient descent optimization based on the first-order differential equation of the Newtonian dynamics. The fictitious mass is introduced to the dynamics of momentum for regularizing the adaptive stepsize of each individual parameter. The dynamic relaxation is adapted for stochast…
Dynamic Boltzmann Machine (DyBM) has been shown highly efficient to predict time-series data. Gaussian DyBM is a DyBM that assumes the predicted data is generated by a Gaussian distribution whose first-order moment (mean) dynamically changes over time but its second-order moment (variance) is fixed. However, in many fi…
Proposes a new variational principle for Einstein gravity.
problem Formulating Einstein gravity as a gauge theory for the conformal group.
method First order formulation of conformal tractor geometry, variational principle based on abstract principal bundle.
result Provides first order field equations without requiring supplementary constraints.
In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian) an explicit second-order expansion formula for the power investor's value function - seen as a function of the underlying market price of risk process - is provided. T…
Examines how first-order differential operators can be equivalently transformed.
problem Equivalence of first-order linear differential operators.
method Discussion of equivalency transformations.
result Explains how first-order differential operators can be transformed equivalently.
New vector fields integrate first-order ODEs.
problem Integrating first-order ODEs.
method Relation between Riemannian manifolds and ODEs integration.
result Integration procedure for first-order ODEs.