Category: JMLR

How can we effectively exploit the collected samples when solving a continuous control task with Reinforcement Learning? Recent results have empirically demonstrated that multiple policy optimization steps can be performed with the same batch by using off-distribution techniques based on importance sampling. However, when dealing with off-distribution optimization, it is essential to take into account the uncertainty introduced by the importance sampling process. In this paper, we propose and analyze a class of model-free, policy search algorithms that extend the recent Policy Optimization via Importance Sampling (Metelli et al., 2018) by incorporating two advanced variance reduction techniques: per-decision and multiple importance sampling. For both of them, we derive a high-probability bound, of independent interest, and then we show how to employ it to define a suitable surrogate objective function that can be used for both action-based and parameter-based settings. The resulting algorithms are finally evaluated on a set of continuous control tasks, using both linear and deep policies, and compared with modern policy optimization methods.

Structure learning of Gaussian graphical models typically involves careful tuning of penalty parameters, which balance the tradeoff between data fidelity and graph sparsity. Unfortunately, this tuning is often a “black art” requiring expert experience or brute-force search. It is therefore tempting to develop tuning-free algorithms that can determine the sparsity of the graph adaptively from the observed data in an automatic fashion. In this paper, we propose a novel approach, named BISN (Bayesian inference of Sparse Networks), for automatic Gaussian graphical model selection. Specifically, we regard the off-diagonal entries in the precision matrix as random variables and impose sparse-promoting horseshoe priors on them, resulting in automatic sparsity determination. With the help of stochastic gradients, an efficient variational Bayes algorithm is derived to learn the model. We further propose a decaying recursive stochastic gradient (DRSG) method to reduce the variance of the stochastic gradients and to accelerate the convergence. Our theoretical analysis shows that the time complexity of BISN scales only quadratically with the dimension, whereas the theoretical time complexity of the state-of-the-art methods for automatic graphical model selection is typically a third-order function of the dimension. Furthermore, numerical results show that BISN can achieve comparable or better performance than the state-of-the-art methods in terms of structure recovery, and yet its computational time is several orders of magnitude shorter, especially for large dimensions.

In many real-world settings, a team of agents must coordinate its behaviour while acting in a decentralised fashion. At the same time, it is often possible to train the agents in a centralised fashion where global state information is available and communication constraints are lifted. Learning joint action-values conditioned on extra state information is an attractive way to exploit centralised learning, but the best strategy for then extracting decentralised policies is unclear. Our solution is QMIX, a novel value-based method that can train decentralised policies in a centralised end-to-end fashion. QMIX employs a mixing network that estimates joint action-values as a monotonic combination of per-agent values. We structurally enforce that the joint-action value is monotonic in the per-agent values, through the use of non-negative weights in the mixing network, which guarantees consistency between the centralised and decentralised policies. To evaluate the performance of QMIX, we propose the StarCraft Multi-Agent Challenge (SMAC) as a new benchmark for deep multi-agent reinforcement learning. We evaluate QMIX on a challenging set of SMAC scenarios and show that it significantly outperforms existing multi-agent reinforcement learning methods.

Many methods for machine learning rely on approximate inference from intractable probability distributions. Variational inference approximates such distributions by tractable models that can be subsequently used for approximate inference. Learning sufficiently accurate approximations requires a rich model family and careful exploration of the relevant modes of the target distribution. We propose a method for learning accurate GMM approximations of intractable probability distributions based on insights from policy search by using information-geometric trust regions for principled exploration. For efficient improvement of the GMM approximation, we derive a lower bound on the corresponding optimization objective enabling us to update the components independently. Our use of the lower bound ensures convergence to a stationary point of the original objective. The number of components is adapted online by adding new components in promising regions and by deleting components with negligible weight. We demonstrate on several domains that we can learn approximations of complex, multimodal distributions with a quality that is unmet by previous variational inference methods, and that the GMM approximation can be used for drawing samples that are on par with samples created by state-of-the-art MCMC samplers while requiring up to three orders of magnitude less computational resources.

We study generalization properties of distributed algorithms in the setting of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We first investigate distributed stochastic gradient methods (SGM), with mini-batches and multi-passes over the data. We show that optimal generalization error bounds (up to logarithmic factor) can be retained for distributed SGM provided that the partition level is not too large. We then extend our results to spectral algorithms (SA), including kernel ridge regression (KRR), kernel principal component regression, and gradient methods. Our results show that distributed SGM has a smaller theoretical computational complexity, compared with distributed KRR and classic SGM. Moreover, even for a general non-distributed SA, they provide optimal, capacity-dependent convergence rates, for the case that the regression function may not be in the RKHS in the well-conditioned regimes.

Decision trees are flexible models that are well suited for many statistical regression problems. In the Bayesian framework for regression trees, Markov Chain Monte Carlo (MCMC) search algorithms are required to generate samples of tree models according to their posterior probabilities. The critical component of such MCMC algorithms is to construct "good" Metropolis-Hastings steps to update the tree topology. Such algorithms frequently suffer from poor mixing and local mode stickiness; therefore, the algorithms are slow to converge. Hitherto, authors have primarily used discrete-time birth/death mechanisms for Bayesian (sums of) regression tree models to explore the tree-model space. These algorithms are efficient, in terms of computation and convergence, only if the rejection rate is low which is not always the case. We overcome this issue by developing a novel search algorithm which is based on a continuous-time birth-death Markov process. The search algorithm explores the tree-model space by jumping between parameter spaces corresponding to different tree structures. The jumps occur in continuous time corresponding to the birth-death events which are modeled as independent Poisson processes. In the proposed algorithm, the moves between models are always accepted which can dramatically improve the convergence and mixing properties of the search algorithm. We provide theoretical support of the algorithm for Bayesian regression tree models and demonstrate its performance in a simulated example.

We study the theoretical properties of image denoising via total variation penalized least-squares. We define the total vatiation in terms of the two-dimensional total discrete derivative of the image and show that it gives rise to denoised images that are piecewise constant on rectangular sets. We prove that, if the true image is piecewise constant on just a few rectangular sets, the denoised image converges to the true image at a parametric rate, up to a log factor. More generally, we show that the denoised image enjoys oracle properties, that is, it is almost as good as if some aspects of the true image were known. In other words, image denoising with total variation regularization leads to an adaptive reconstruction of the true image.

Standard high-dimensional regression methods assume that the underlying coefficient vector is sparse. This might not be true in some cases, in particular in presence of hidden, confounding variables. Such hidden confounding can be represented as a high-dimensional linear model where the sparse coefficient vector is perturbed. For this model, we develop and investigate a class of methods that are based on running the Lasso on preprocessed data. The preprocessing step consists of applying certain spectral transformations that change the singular values of the design matrix. We show that, under some assumptions, one can achieve the usual Lasso $\ell_1$-error rate for estimating the underlying sparse coefficient vector, despite the presence of confounding. Our theory also covers the Lava estimator (Chernozhukov et al., 2017) for a special model class. The performance of the methodology is illustrated on simulated data and a genomic dataset.

Iterative methods for fitting a Gaussian Random Field (GRF) model via maximum likelihood (ML) estimation requires solving a nonconvex optimization problem. The problem is aggravated for anisotropic GRFs where the number of covariance function parameters increases with the dimension. Even evaluation of the likelihood function requires $O(n^3)$ floating point operations, where $n$ denotes the number of data locations. In this paper, we propose a new two-stage procedure to estimate the parameters of second-order stationary GRFs. First, a convex likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse precision (inverse covariance) matrix to the observed data. Second, the parameters of the covariance function are estimated by solving a least squares problem. Theoretical error bounds for the solutions of stage I and II problems are provided, and their tightness are investigated.

We study derivative-free methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linear-quadratic systems, and study various settings of driving noise and reward feedback. Our main theoretical result provides an explicit bound on the sample or evaluation complexity: we show that these methods are guaranteed to converge to within any pre-specified tolerance of the optimal policy with a number of zero-order evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive driving noise and random initialization, as well as the settings of one-point and two-point reward feedback. Our theory is corroborated by simulations of derivative-free methods in application to these systems. Along the way, we derive convergence rates for stochastic zero-order optimization algorithms when applied to a certain class of non-convex problems.