91影视破解版

Abstract:

Many methods for global optimization, from Langevin dynamics and simulated annealing to consensus-based particle methods, use dynamical systems to explore an objective landscape and approach its minima. This talk asks how optimal control can guide the design of such dynamics. We reformulate the global optimization problem as an infinite-horizon optimal control problem, whose Hamilton鈥揓acobi鈥揃ellman (HJB) equation yields a value function that encodes the global structure of the objective. The gradient of this value function gives a feedback control that drives trajectories toward the global minimizers of general non-convex functions. We establish explicit convergence rates for both the variational behaviour of the value function and the pathwise convergence of the resulting trajectories.

Abstract:

TBA

Abstract:

TBA

Abstract:

TBA

Abstract:

TBA

Abstract:

TBA

Abstract:

We advocate for a new statistical principle that combines the most desirable aspects of both parameter inference and density estimation. This leads us to the predictively oriented posterior, which expresses uncertainty as a consequence of predictive ability. We show that these posteriors converge to the predictively optimal model average and predictively dominate both classical and generalised Bayes posterior predictive distributions. We further put forward an approach for computation based on an approximation whose density can be evaluated point-wise allowing for standard Markov chain Monte Carlo methods, and whose approximation error decays rapidly.

Abstract:

Filtering, the problem of estimating the probability distribution of a system's states given partial and noisy observations, is essential for weather forecasting, along with other areas of science and engineering. Obtaining the filtering distribution is generally intractable for high-dimensional, nonlinear systems. The ensemble Kalman filter (EnKF) approximates the filtering distribution with an ensemble of interacting particles, employing a Gaussian ansatz for the joint distribution of the state and observation at each observation time. The EnKF is robust, but the Gaussian ansatz limits accuracy. We address this shortcoming by using machine learning to map the forecast distribution and observation to the filtering distribution. We propose cost functions that are minimised uniquely at the true filtering distribution. By time-averaging over trajectories in ergodic dynamical systems, the map can be learned and subsequently used for future filtering; this is a form of amortised Bayesian inference. We focus on learning ensemble-based filters within a mean field framework. We demonstrate the approach using a set transformer neural architecture, which is invariant to ensemble permutations. The learned filtering algorithms outperform state-of-the-art methods for filtering chaotic systems. They also perform well in challenging highly non-Gaussian and multimodal problems where the EnKF fails. Once learned at a given ensemble size, the learned map can be applied to other ensemble sizes with minimal fine-tuning.

Abstract:

Hesterberg (1988) introduced a proposal for self-normalized importance sampling (SNIS) which minimizes the asymptotic variance under i.i.d. sampling. The Hesterberg proposal can provide substantial variance reduction even w.r.t to i.i.d. Monte Carlo: the variance of the former can be finite when the latter is infinite.

We develop and study the convergence properties of an adaptive Monte Carlo framework whose asymptotic variance can be close to that of the Hesterberg proposal.

In particular, we prove a central limit theorem (CLT) which includes unbounded test functions that are not square-integrable. Therefore, our schemes can achieve a finite asymptotic variance when the i.i.d. Monte Carlo estimator has infinite variance. We show connections with existing notions of asymptotic optimality for adaptive importance sampling, and with waste-free sequential Monte Carlo samplers.

We demonstrate that our method performs better than the alternatives and, often, close to the Hesterberg proposal, in Bayesian statistics applications where unbounded test functions naturally arise. 

Abstract:

In this talk, I will discuss a particular inverse problem arising from optimal transport (OT). In many modern pipelines, however, the transport cost is not known a priori and must be inferred from data. This leads to inverse optimal transport (iOT): recover the ground cost (or metric parameters) from an observed optimal coupling. On the other hand, modern computational pipelines typically exploit an entropic regularization variant (eOT) of OT. I will discuss iOT in a regime that is both mathematically delicate and practically unavoidable: the entropic regularization level is small (approaching the unregularized OT model), while the coupling is observed through a finite number of samples.