Unbiased Estimation of the Gradient of the Log-Likelihood for a Class of Continuous-Time State-Space Models

• URL: http://arxiv.org/abs/2105.11522v1
• Date: Mon, 24 May 2021 20:31:48 GMT
• Title: Unbiased Estimation of the Gradient of the Log-Likelihood for a Class of Continuous-Time State-Space Models
• Authors: Marco Ballesio and Ajay Jasra
• Abstract summary: We consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function)
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a \emph{doubly randomized scheme} (see, e.g.,~\cite{ub_mcmc, ub_grad}), that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization \cite{jacob2}. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee \& Glynn estimator \cite{rhee,vihola}.

Related papers

• Limit Theorems for Stochastic Gradient Descent with Infinite Variance [47.87144151929621]
  We show that the gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-rnstein process driven by an appropriate L'evy process. We also explore the applications of these results in linear regression and logistic regression models.
  arXiv  Detail & Related papers  (2024-10-21T09:39:10Z)
• One-step corrected projected stochastic gradient descent for statistical estimation [49.1574468325115]
  It is based on the projected gradient descent on the log-likelihood function corrected by a single step of the Fisher scoring algorithm. We show theoretically and by simulations that it is an interesting alternative to the usual gradient descent with averaging or the adaptative gradient descent.
  arXiv  Detail & Related papers  (2023-06-09T13:43:07Z)
• Score-based Continuous-time Discrete Diffusion Models [102.65769839899315]
  We extend diffusion models to discrete variables by introducing a Markov jump process where the reverse process denoises via a continuous-time Markov chain. We show that an unbiased estimator can be obtained via simple matching the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.
  arXiv  Detail & Related papers  (2022-11-30T05:33:29Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• Random-reshuffled SARAH does not need a full gradient computations [61.85897464405715]
  The StochAstic Recursive grAdientritHm (SARAH) algorithm is a variance reduced variant of the Gradient Descent (SGD) algorithm. In this paper, we remove the necessity of a full gradient. The aggregated gradients serve as an estimate of a full gradient in the SARAH algorithm.
  arXiv  Detail & Related papers  (2021-11-26T06:00:44Z)
• Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling [110.88857917726276]
  We provide a new convergence analysis of gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain.
  arXiv  Detail & Related papers  (2020-10-19T15:23:18Z)
• Analysis of Stochastic Gradient Descent in Continuous Time [0.0]
  We introduce the gradient process as a continuous-time representation of gradient descent. We show that it converges weakly to the gradient flow as the learning rate approaches zero. In this case, the process converges weakly to the point mass concentrated in the global minimum of the full target function.
  arXiv  Detail & Related papers  (2020-04-15T16:04:41Z)
• The Implicit Regularization of Stochastic Gradient Flow for Least Squares [24.976079444818552]
  We study the implicit regularization of mini-batch gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time differential equation having the same moments as gradient descent, which we call gradient flow. We give a bound on the excess risk of gradient flow at time $t$, over ridge regression with tuning parameter $lambda = 1/t$.
  arXiv  Detail & Related papers  (2020-03-17T16:37:25Z)
• Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles [8.655294504286635]
  We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error. Our proposed oracles are in practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed simulation.
  arXiv  Detail & Related papers  (2020-02-26T12:53:04Z)
• The Wasserstein Proximal Gradient Algorithm [23.143814848127295]
  Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures. We propose a Forward Backward (FB) discretization scheme that can tackle the case where the objective function is the sum of a smooth and a nonsmooth geodesically convex terms.
  arXiv  Detail & Related papers  (2020-02-07T22:19:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.