Study Markov chain gradient descent in Hilbert spaces for quadratic loss.
problem Approximating optimal solutions for quadratic loss functions.
method Developed a Markov chain-based stochastic gradient algorithm in Hilbert spaces.
result Established probabilistic upper bounds on convergence.
Deviation inequalities for stochastic approximation methods.
problem Establishing bounds on the deviation of stochastic approximation methods.
method Martingale approximation method for separately Lipschitz functions.
result Established various deviation inequalities for stochastic approximation by averaging and minimization.
Study shows TD(0) with linear approx. converges for reversible Markov chains.
problem TD(0) divergence with off-policy and function approximation.
method Analyzes standard TD(0) with reversible Markov chains, adapting stochastic approximation framework.
result Establishes convergence with probability one for projected Bellman error = 0.
Estimates Markov chain variance efficiently without storing samples.
problem Estimating the asymptotic variance of Markov chain functions.
method Linear stochastic approximation of Poisson equation solution.
result Optimal MSE convergence rate with finite sample guarantees.
We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however i…
Uniform TD(0) bound derived for function approximation with Markov noise.
problem Uniform concentration bound for TD(0) with function approximation.
method Contractive stochastic approximation, martingale and Markov noises, Poisson equation, relaxed concentration inequalities.
result Uniform all-time concentration bound for TD(0) with linear function approximation.
Probabilistic models are conceptually powerful tools for finding structure in data, but their practical effectiveness is often limited by our ability to perform inference in them. Exact inference is frequently intractable, so approximate inference is often performed using Markov chain Monte Carlo (MCMC). To achieve the…
The paper estimates key metrics for linear models with Markov or hidden Markov sources.
problem Estimating free energy, mutual information, and MMSE for linear models with specific signal priors.
method Replica analysis in statistical physics, focusing on Markov and hidden Markov sources.
result The linear model with Markov or hidden Markov sources can be simplified into decoupled AWGN channels.
Markov chains and diffusion processes are indispensable tools in machine learning and statistics that are used for inference, sampling, and modeling. With the growth of large-scale datasets, the computational cost associated with simulating these stochastic processes can be considerable, and many algorithms have been p…
Investor selects portfolios based on news attention in a hidden Markov model.
problem Mean-variance portfolio selection in a dynamic attention context.
method Closed-loop equilibrium strategies via extended HJB equation and Markov chain approximation.
result Equilibrium strategies found through iterative algorithm and numerical examples.
We study the problem of identity testing of markov chains. In this setting, we are given access to a single trajectory from a markov chain with unknown transition matrix Q and the goal is to determine whether Q=P for some known matrix P or Dist(P,Q)≥ε where Dist is suitably defined. In r…
Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great interest to describe and understand the space of such classes. However, with curren…
The paper studies how quickly samples from Langevin dynamics become independent.
problem Understanding the dependence between samples along Langevin dynamics and related algorithms.
method Measures dependence via Φ-mutual information and proves strong data processing inequalities. result The Φ-mutual information between samples decreases exponentially to zero. Study approximates financial market with discrete-time models.
problem Approximating continuous-time financial market models with discrete-time.
method Constructs discrete-time market models with Markov switching and proves convergence.
result Discrete-time models converge to continuous-time Black-Scholes model with Markov switching.
New bounds for SMC show its advantage over MCMC in multimodal distributions.
problem Estimating expectations under multimodal distributions with slow global mixing.
method Proves finite sample complexities for SMC with local mixing times, addressing bias through sequential resampling.
result SMC provides fully polynomial time approximation for multimodal problems.
Proposes MIVI for efficient posterior estimation and design of MCMC transitions.
problem Efficiently estimating posterior distributions in constrained time.
method Combines variational inference and MCMC with a variational distribution and optimized Markov chain.
result Optimized Markov chain improves variational distribution and vice versa, leading to more accurate posteriors.
Paper introduces a diagnostic for approximate inference methods.
problem Estimating errors in probabilistic inference algorithms, especially for approximate methods.
method Repeatedly simulate datasets from the prior and perform inference on each, estimating a symmetric KL-divergence.
result A diagnostic for approximate inference methods can be estimated using symmetric KL-divergence.
New method simulates sticky boundaries in multidimensional diffusions.
problem Simulating sticky boundaries in multidimensional diffusions.
method Approximate sticky diffusion by a Markov chain, using either finite difference or matching local moments.
result Validates both construction methods for first-order simulation schemes.
We analyze a new Markov chain model for better sampling and optimization.
problem Developing a new Markov chain model for improved sampling and optimization.
method We introduce a new class of Ito chains with arbitrary noise and inexact drift/diffusion coefficients, proving a bound in W2-distance. result Our analysis provides improved or first results for various applications like SGLD, sampling, and boosting.
Novel bounds improve TD learning consistency in RL.
problem Analyzing Temporal Difference learning's performance.
method High-dimensional concentration inequalities and Berry-Esseen bounds for Markov chain induced martingales.
result Sharp high-probability consistency guarantee for TD learning, matching asymptotic variance up to logarithmic factors.
The paper analyzes stability of random matrix products with Markovian noise.
problem Analyzing stability of random matrix products with Markovian noise.
method Using a super-Lyapunov drift condition and controlled growth of matrix-valued functions, the paper provides an exponential stability result for the p-th moment of random matrix product.
result Finite-time p-th moment bounds for linear stochastic approximation and TD learning algorithms.
The paper tackles learning from non-irreducible Markov chains, proving learnability and generalization bounds.
problem Learning from temporal dependent data with non-irreducible Markov chains.
method Uniform convergence and generalization bounds for sample error under uniform ergodicity.
result Learnability and generalization bounds for approximate sample error minimization algorithm.
Regime-switching models, in particular Hidden Markov Models (HMMs) where the switching is driven by an unobservable Markov chain, are widely-used in financial applications, due to their tractability and good econometric properties. In this work we consider HMMs in continuous time with both constant and switching volati…
We develop continuous time Markov chain (CTMC) approximation of one-dimensional diffusions with a lower sticky boundary. Approximate solutions to the action of the Feynman-Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matri…
Recent developments in differentially private (DP) machine learning and DP Bayesian learning have enabled learning under strong privacy guarantees for the training data subjects. In this paper, we further extend the applicability of DP Bayesian learning by presenting the first general DP Markov chain Monte Carlo (MCMC)…
Paper approximates rough stochastic local volatility models for efficient computation.
problem No unified method for rough stochastic local volatility models.
method Semimartingale and continuous-time Markov chain approximation.
result Fast CTMC algorithm with weak convergence proved.
Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a…
Method calculates Parisian stopping times and option prices using Markov chains.
problem Computing distribution and pricing of Parisian stopping times under Markov processes.
method Continuous-time Markov chain approximation to solve for distribution and convergence analysis.
result Sharp convergence rate and efficient method for diffusion and jump models.
This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approxima…
Stochastic approximation algorithms show exponential progress bounds.
problem Analyzing the convergence of stochastic approximation algorithms.
method Developed geometric ergodicity proofs to establish exponential concentration bounds.
result Proved faster convergence rates for specific algorithms.
Optimizes MCMC chains with neural control variates.
problem Reducing variance in Markov Chain Monte Carlo (MCMC) simulations.
method Uses neural networks as control variates to minimize asymptotic variance.
result Derives optimal convergence rate under various ergodicity assumptions.
The paper proposes methods to estimate MCMC quality with couplings, bounding Wasserstein distance.
problem Improving MCMC efficiency without sacrificing asymptotic consistency.
method Estimators based on couplings of Markov chains to assess quality of asymptotically biased sampling methods.
result Empirical upper bounds of Wasserstein distance for assessing MCMC quality.
The paper develops fast Bayesian methods for estimating huge PVARs with competitive forecasts.
problem Computational and statistical issues in estimating PVARs with many parameters.
method Integrated rotated Gaussian approximations, exploiting domestic over international information, and fast approximations for international coefficients.
result Produces competitive forecasts quickly using a huge world economy model.
A new eigenvalue-based method speeds up Monte Carlo simulations.
problem Reducing the number of paths needed for accurate Monte Carlo simulations.
method Eigenvalue-based approximation of Markov Chain Monte Carlo.
result Significant variance reduction and comparable results to traditional Monte Carlo.
New method resolves time order in genetic mutation models.
problem Underspecification in modeling genetic mutation time evolution.
method Continuous-time Markov chains with additional independent items.
result Additional items help determine time order and resolve underspecification.
New method reduces computational cost for Bayesian inference.
problem High computational cost in Bayesian inference.
method Coreset MCMC, a Markov chain targeting coreset posterior.
result Coreset MCMC provides higher quality posterior approximations and reduced computational cost.
Efficiently infers coupled hidden Markov models with noisy discrete observations.
problem Intractable inference for coupled continuous-time Markov chains with discrete observations.
method Latent Interacting Particle Systems, look-ahead functions, twisted Sequential Monte Carlo sampling.
result Demonstrated effectiveness on latent SIRS model and wildfire spread dynamics.
The paper provides concentration inequalities for Markov chain variance estimators.
problem Estimating the variance of Markov chains with concentration properties.
method Martingale decomposition method for uniformly geometrically ergodic Markov chains.
result Explicit control of the p-th moment of the OBM estimator difference and dependence on p and mixing time.
A new method simulates a lazy version of a Markov chain for empirical inference.
problem Estimating and testing unknown Markov chains with limited data.
method Simulates an α-lazy version of an unknown Markov chain, making it ergodic.
result The pseudo spectral gap can be applied to non-ergodic Markov chains.
A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few works have provided "beyond worst-case" guarantees for such settings. For log-c…
Reduces identity testing of reversible Markov chains to simpler symmetric chain tests.
problem Testing identity of reversible Markov chains from a single trajectory.
method Using lumping-congruent Markov embeddings, the problem is simplified to testing symmetric chains over a larger state space.
result Achieves state-of-the-art sample complexity for identity testing.
Develops a new risk measure for Markov chains' asymptotic behavior.
problem Lack of risk measures for asymptotic regimes of Markov chains.
method Simulation-based approach using large deviations theory, density estimation, and stochastic approximation.
result Developed Asymptotic CVaR (ACVaR) for Markov chains.
In this paper we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, incl…
LGS-Net improves NCO performance on combinatorial optimization tasks.
problem NP-hard combinatorial optimization problems in logistics, manufacturing, and drug discovery.
method LGS-Net uses a latent space model that conditions on problem instances and introduces Latent Guided Sampling for efficient inference.
result Empirical results show state-of-the-art performance on benchmark routing tasks.
Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the t…
PDHAMS improves sampling for discrete distributions with quadratic potential functions.
problem Sampling discrete distributions efficiently and accurately.
method Integrates a second-order approximation of the potential function and uses Gaussian integral trick.
result PDHAMS yields superior performance compared to other methods.
New method improves Bayesian inference for large models.
problem Bayesian inference in large-scale models like neural networks.
method Non-parametric variational approximation with a modified Langevin algorithm.
result Improves convergence speed and accuracy compared to existing methods.
This paper analyzes the bias of inexact MCMC methods in high dimensions.
problem Understanding the bias of inexact MCMC methods in high-dimensional spaces.
method Establishing bounds on Wasserstein distances between inexact MCMC methods and target distributions.
result The asymptotic bias of ULA and uHMC depends on key quantities related to the target distribution or the stationary probability measure of the scheme.