Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to be connected to the mirror descent framework. Using a kernel density estimate to build the sampling policy, the uniform convergence is established under mild conditions. Finally, several practical ways to choose the regularization parameter are discussed and the benefits of the proposed approach are illustrated empirically.
相關內容
A trivializing map is a field transformation whose Jacobian determinant exactly cancels the interaction terms in the action, providing a representation of the theory in terms of a deterministic transformation of a distribution from which sampling is trivial. Recently, a proof-of-principle study by Albergo, Kanwar and Shanahan [arXiv:1904.12072] demonstrated that approximations of trivializing maps can be `machine-learned' by a class of invertible, differentiable neural models called \textit{normalizing flows}. By ensuring that the Jacobian determinant can be computed efficiently, asymptotically exact sampling from the theory of interest can be performed by drawing samples from a simple distribution and passing them through the network. From a theoretical perspective, this approach has the potential to become more efficient than traditional Markov Chain Monte Carlo sampling techniques, where autocorrelations severely diminish the sampling efficiency as one approaches the continuum limit. A major caveat is that it is not yet understood how the size of models and the cost of training them is expected to scale. As a first step, we have conducted an exploratory scaling study using two-dimensional $\phi^4$ with up to $20^2$ lattice sites. Although the scope of our study is limited to a particular model architecture and training algorithm, initial results paint an interesting picture in which training costs grow very quickly indeed. We describe a candidate explanation for the poor scaling, and outline our intentions to clarify the situation in future work.
In recent years, gradient based Meta-RL (GMRL) methods have achieved remarkable successes in either discovering effective online hyperparameter for one single task (Xu et al., 2018) or learning good initialisation for multi-task transfer learning (Finn et al., 2017). Despite the empirical successes, it is often neglected that computing meta gradients via vanilla backpropagation is ill-defined. In this paper, we argue that the stochastic meta-gradient estimation adopted by many existing MGRL methods are in fact biased; the bias comes from two sources: 1) the compositional bias that is inborn in the structure of compositional optimisation problems and 2) the bias of multi-step Hessian estimation caused by direct automatic differentiation. To better understand the meta gradient biases, we perform the first of its kind study to quantify the amount for each of them. We start by providing a unifying derivation for existing GMRL algorithms, and then theoretically analyse both the bias and the variance of existing gradient estimation methods. On understanding the underlying principles of bias, we propose two mitigation solutions based on off-policy correction and multi-step Hessian estimation techniques. Comprehensive ablation studies have been conducted and results reveals: (1) The existence of these two biases and how they influence the meta-gradient estimation when combined with different estimator/sample size/step and learning rate. (2) The effectiveness of these mitigation approaches for meta-gradient estimation and thereby the final return on two practical Meta-RL algorithms: LOLA-DiCE and Meta-gradient Reinforcement Learning.
This study concerns probability distribution estimation of sample maximum. The traditional approach is the parametric fitting to the limiting distribution - the generalized extreme value distribution; however, the model in finite cases is misspecified to a certain extent. We propose a plug-in type of the kernel distribution estimator which does not need model specification. It is proved that both asymptotic convergence rates depend on the tail index and the second order parameter. As the tail gets light, the degree of misspecification of the parametric fitting becomes large, that means the convergence rate becomes slow. In the Weibull cases, which can be seen as the limit of tail-lightness, only the nonparametric distribution estimator keeps its consistency. Finally, we report results of numerical experiments and two real case studies.
In federated learning (FL) problems, client sampling plays a key role in the convergence speed of training algorithm. However, while being an important problem in FL, client sampling is lack of study. In this paper, we propose an online learning with bandit feedback framework to understand the client sampling problem in FL. By adapting an Online Stochastic Mirror Descent algorithm to minimize the variance of gradient estimation, we propose a new adaptive client sampling algorithm. Besides, we use online ensemble method and doubling trick to automatically choose the tuning parameters in the algorithm. Theoretically, we show dynamic regret bound with comparator as the theoretically optimal sampling sequence; we also include the total variation of this sequence in our upper bound, which is a natural measure of the intrinsic difficulty of the problem. To the best of our knowledge, these theoretical contributions are novel to existing literature. Moreover, by implementing both synthetic and real data experiments, we show empirical evidence of the advantages of our proposed algorithms over widely-used uniform sampling and also other online learning based sampling strategies in previous studies. We also examine its robustness to the choice of tuning parameters. Finally, we discuss its possible extension to sampling without replacement and personalized FL objective. While the original goal is to solve client sampling problem, this work has more general applications on stochastic gradient descent and stochastic coordinate descent methods.
Algorithms based on normalizing flows are emerging as promising machine learning approaches to sampling complicated probability distributions in a way that can be made asymptotically exact. In the context of lattice field theory, proof-of-principle studies have demonstrated the effectiveness of this approach for scalar theories, gauge theories, and statistical systems. This work develops approaches that enable flow-based sampling of theories with dynamical fermions, which is necessary for the technique to be applied to lattice field theory studies of the Standard Model of particle physics and many condensed matter systems. As a practical demonstration, these methods are applied to the sampling of field configurations for a two-dimensional theory of massless staggered fermions coupled to a scalar field via a Yukawa interaction.
The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.
We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.
Importance sampling is one of the most widely used variance reduction strategies in Monte Carlo rendering. In this paper, we propose a novel importance sampling technique that uses a neural network to learn how to sample from a desired density represented by a set of samples. Our approach considers an existing Monte Carlo rendering algorithm as a black box. During a scene-dependent training phase, we learn to generate samples with a desired density in the primary sample space of the rendering algorithm using maximum likelihood estimation. We leverage a recent neural network architecture that was designed to represent real-valued non-volume preserving ('Real NVP') transformations in high dimensional spaces. We use Real NVP to non-linearly warp primary sample space and obtain desired densities. In addition, Real NVP efficiently computes the determinant of the Jacobian of the warp, which is required to implement the change of integration variables implied by the warp. A main advantage of our approach is that it is agnostic of underlying light transport effects, and can be combined with many existing rendering techniques by treating them as a black box. We show that our approach leads to effective variance reduction in several practical scenarios.
In two-phase image segmentation, convex relaxation has allowed global minimisers to be computed for a variety of data fitting terms. Many efficient approaches exist to compute a solution quickly. However, we consider whether the nature of the data fitting in this formulation allows for reasonable assumptions to be made about the solution that can improve the computational performance further. In particular, we employ a well known dual formulation of this problem and solve the corresponding equations in a restricted domain. We present experimental results that explore the dependence of the solution on this restriction and quantify imrovements in the computational performance. This approach can be extended to analogous methods simply and could provide an efficient alternative for problems of this type.
Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191