The focus of the present work is the (theoretical) approximation of a solution of the p(x)-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a truncation of the nonlinearity from below and from above by using a pair of positive cut-off parameters. We will then verify that, for any such pair, a damped Ka\v{c}anov scheme generates a sequence converging to a solution of the relaxed equation. Subsequently, it will be shown that the solutions of the relaxed problems converge to the solution of the original problem in the discrete setting. Finally, the discrete solutions of the unrelaxed problem converge to the continuous solution. Our work will finally be rounded up with some numerical experiments that underline the analytical findings.
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Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.
Existing results for the estimation of the L\'evy measure are mostly limited to the onedimensional setting. We apply the spectral method to multidimensional L\'evy processes in order to construct a nonparametric estimator for the multivariate jump distribution. We prove convergence rates for the uniform estimation error under both a low- and a high-frequency observation regime. The method is robust to various dependence structures. Along the way, we present a uniform risk bound for the multivariate empirical characteristic function and its partial derivatives. The method is illustrated with simulation examples.
Stein Variational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm. Despite its wide usage, understanding the theoretical properties of SVGD has remained a challenging problem. For sampling from a Gaussian target, the SVGD dynamics with a bilinear kernel will remain Gaussian as long as the initializer is Gaussian. Inspired by this fact, we undertake a detailed theoretical study of the Gaussian-SVGD, i.e., SVGD projected to the family of Gaussian distributions via the bilinear kernel, or equivalently Gaussian variational inference (GVI) with SVGD. We present a complete picture by considering both the mean-field PDE and discrete particle systems. When the target is strongly log-concave, the mean-field Gaussian-SVGD dynamics is proven to converge linearly to the Gaussian distribution closest to the target in KL divergence. In the finite-particle setting, there is both uniform in time convergence to the mean-field limit and linear convergence in time to the equilibrium if the target is Gaussian. In the general case, we propose a density-based and a particle-based implementation of the Gaussian-SVGD, and show that several recent algorithms for GVI, proposed from different perspectives, emerge as special cases of our unified framework. Interestingly, one of the new particle-based instance from this framework empirically outperforms existing approaches. Our results make concrete contributions towards obtaining a deeper understanding of both SVGD and GVI.
Stein thinning is a promising algorithm proposed by (Riabiz et al., 2022) for post-processing outputs of Markov chain Monte Carlo (MCMC). The main principle is to greedily minimize the kernelized Stein discrepancy (KSD), which only requires the gradient of the log-target distribution, and is thus well-suited for Bayesian inference. The main advantages of Stein thinning are the automatic remove of the burn-in period, the correction of the bias introduced by recent MCMC algorithms, and the asymptotic properties of convergence towards the target distribution. Nevertheless, Stein thinning suffers from several empirical pathologies, which may result in poor approximations, as observed in the literature. In this article, we conduct a theoretical analysis of these pathologies, to clearly identify the mechanisms at stake, and suggest improved strategies. Then, we introduce the regularized Stein thinning algorithm to alleviate the identified pathologies. Finally, theoretical guarantees and extensive experiments show the high efficiency of the proposed algorithm.
We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our theta-scheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order $\mathcal{O}(h^r)$ for the theta-scheme, where $h$ is the step length of the space variable and $r \in (0,1)$ is related to the H\"older continuity of the solution of the continuous problem and some of its derivatives.
The success of many healthcare programs depends on participants' adherence. We consider the problem of scheduling interventions in low resource settings (e.g., placing timely support calls from health workers) to increase adherence and/or engagement. Past works have successfully developed several classes of Restless Multi-armed Bandit (RMAB) based solutions for this problem. Nevertheless, all past RMAB approaches assume that the participants' behaviour follows the Markov property. We demonstrate significant deviations from the Markov assumption on real-world data on a maternal health awareness program from our partner NGO, ARMMAN. Moreover, we extend RMABs to continuous state spaces, a previously understudied area. To tackle the generalised non-Markovian RMAB setting we (i) model each participant's trajectory as a time-series, (ii) leverage the power of time-series forecasting models to learn complex patterns and dynamics to predict future states, and (iii) propose the Time-series Arm Ranking Index (TARI) policy, a novel algorithm that selects the RMAB arms that will benefit the most from an intervention, given our future state predictions. We evaluate our approach on both synthetic data, and a secondary analysis on real data from ARMMAN, and demonstrate significant increase in engagement compared to the SOTA, deployed Whittle index solution. This translates to 16.3 hours of additional content listened, 90.8% more engagement drops prevented, and reaching more than twice as many high dropout-risk beneficiaries.
We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and provide a theoretical estimate of the error. We conclude with some numerical experiments to further support the theoretical results.
Markov Switching models have had increasing success in time series analysis due to their ability to capture the existence of unobserved discrete states in the dynamics of the variables under study. This result is generally obtained thanks to the inference on states derived from the so--called Hamilton filter. One of the open problems in this framework is the identification of the number of states, generally fixed a priori; it is in fact impossible to apply classical tests due to the problem of the nuisance parameters present only under the alternative hypothesis. In this work we show, by Monte Carlo simulations, that fuzzy clustering is able to reproduce the parametric state inference derived from the Hamilton filter and that the typical indices used in clustering to determine the number of groups can be used to identify the number of states in this framework. The procedure is very simple to apply, considering that it is performed (in a nonparametric way) independently of the data generation process and that the indicators we use are present in most statistical packages. A final application on real data completes the analysis.
This article proposes a meta-learning method for estimating the conditional average treatment effect (CATE) from a few observational data. The proposed method learns how to estimate CATEs from multiple tasks and uses the knowledge for unseen tasks. In the proposed method, based on the meta-learner framework, we decompose the CATE estimation problem into sub-problems. For each sub-problem, we formulate our estimation models using neural networks with task-shared and task-specific parameters. With our formulation, we can obtain optimal task-specific parameters in a closed form that are differentiable with respect to task-shared parameters, making it possible to perform effective meta-learning. The task-shared parameters are trained such that the expected CATE estimation performance in few-shot settings is improved by minimizing the difference between a CATE estimated with a large amount of data and one estimated with just a few data. Our experimental results demonstrate that our method outperforms the existing meta-learning approaches and CATE estimation methods.
In this paper, we study the statistical efficiency of Reinforcement Learning in Mean-Field Control (MFC) and Mean-Field Game (MFG) with general function approximation. We introduce a new concept called Mean-Field Model-Based Eluder Dimension (MBED), which subsumes a rich family of Mean-Field RL problems. Additionally, we propose algorithms based on Optimistic Maximal Likelihood Estimation, which can return an $\epsilon$-optimal policy for MFC or an $\epsilon$-Nash Equilibrium policy for MFG, with sample complexity polynomial w.r.t. relevant parameters and independent of the number of states, actions and the number of agents. Notably, our results only require a mild assumption of Lipschitz continuity on transition dynamics and avoid strong structural assumptions in previous work. Finally, in the tabular setting, given the access to a generative model, we establish an exponential lower bound for MFC setting, while providing a novel sample-efficient model elimination algorithm to approximate equilibrium in MFG setting. Our results reveal a fundamental separation between RL for single-agent, MFC, and MFG from the sample efficiency perspective.
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