Intractable posterior distributions of parameters with intractable normalizing constants depending upon the parameters are known as doubly intractable posterior distributions. The terminology itself indicates that obtaining Bayesian inference from such posteriors is doubly difficult compared to traditional intractable posteriors where the normalizing constants are tractable and admit traditional Markov Chain Monte Carlo (MCMC) solutions. As can be anticipated, a plethora of MCMC-based methods have originated in the literature to deal with doubly intractable distributions. Yet, it remains very much unclear if any of the methods can satisfactorily sample from such posteriors, particularly in high-dimensional setups. In this article, we consider efficient Monte Carlo and importance sampling approximations of the intractable normalizing constant for a few values of the parameters, and Gaussian process interpolations for the remaining values of the parameters, using the approximations. We then incorporate this strategy within the exact iid sampling framework developed in Bhattacharya (2021a) and Bhattacharya (2021b), and illustrate the methodology with simulation experiments comprising a two-dimensional normal-gamma posterior, a two-dimensional Ising model posterior, a two-dimensional Strauss process posterior and a 100-dimensional autologistic model posterior. In each case we demonstrate great accuracy of our methodology, which is also computationally extremely efficient, often taking only a few minutes for generating 10, 000 iid realizations on 80 processors.
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We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter $p$. We present several complexity-theoretic hardness results that explain the difficulty in approximating MAP inference and the normalizing constant for E-DPPs. We first prove that unconstrained MAP inference for an $n \times n$ matrix is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta n}$, where $\beta = 10^{-10^{13}} $. This result improves upon the best-known inapproximability factor of $(\frac{9}{8}-\epsilon)$, and rules out the existence of any polynomial-factor approximation algorithm assuming $\textsf{P} \neq \textsf{NP}$. We then show that log-determinant maximization is $\textsf{NP}$-hard to approximate within a factor of $\frac{5}{4}$ for the unconstrained case and within a factor of $1+10^{-10^{13}}$ for the size-constrained monotone case. In particular, log-determinant maximization does not admit a polynomial-time approximation scheme unless $\textsf{P} = \textsf{NP}$. As a corollary of the first result, we demonstrate that the normalizing constant for E-DPPs of any (fixed) constant exponent $p \geq \beta^{-1} = 10^{10^{13}}$ is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta pn}$, which is in contrast to the case of $p \leq 1$ admitting a fully polynomial-time randomized approximation scheme.
Federated learning (FedL) has emerged as a popular technique for distributing model training over a set of wireless devices, via iterative local updates (at devices) and global aggregations (at the server). In this paper, we develop parallel successive learning (PSL), which expands the FedL architecture along three dimensions: (i) Network, allowing decentralized cooperation among the devices via device-to-device (D2D) communications. (ii) Heterogeneity, interpreted at three levels: (ii-a) Learning: PSL considers heterogeneous number of stochastic gradient descent iterations with different mini-batch sizes at the devices; (ii-b) Data: PSL presumes a dynamic environment with data arrival and departure, where the distributions of local datasets evolve over time, captured via a new metric for model/concept drift. (ii-c) Device: PSL considers devices with different computation and communication capabilities. (iii) Proximity, where devices have different distances to each other and the access point. PSL considers the realistic scenario where global aggregations are conducted with idle times in-between them for resource efficiency improvements, and incorporates data dispersion and model dispersion with local model condensation into FedL. Our analysis sheds light on the notion of cold vs. warmed up models, and model inertia in distributed machine learning. We then propose network-aware dynamic model tracking to optimize the model learning vs. resource efficiency tradeoff, which we show is an NP-hard signomial programming problem. We finally solve this problem through proposing a general optimization solver. Our numerical results reveal new findings on the interdependencies between the idle times in-between the global aggregations, model/concept drift, and D2D cooperation configuration.
The construction of generalizable and transferable models is a fundamental goal of statistical learning. Learning with the multi-source data helps improve model generalizability and is integral to many important statistical problems, including group distributionally robust optimization, minimax group fairness, and maximin projection. This paper considers multiple high-dimensional regression models for the multi-source data. We introduce the covariate shift maximin effect as a group distributionally robust model. This robust model helps transfer the information from the multi-source data to the unlabelled target population. Statistical inference for the covariate shift maximin effect is challenging since its point estimator may have a non-standard limiting distribution. We devise a novel {\it DenseNet} sampling method to construct valid confidence intervals for the high-dimensional maximin effect. We show that our proposed confidence interval achieves the desired coverage level and attains a parametric length. Our proposed DenseNet sampling method and the related theoretical analysis are of independent interest in addressing other non-regular or non-standard inference problems. We demonstrate the proposed method over a large-scale simulation and genetic data on yeast colony growth under multiple environments.
Skills or low-level policies in reinforcement learning are temporally extended actions that can speed up learning and enable complex behaviours. Recent work in offline reinforcement learning and imitation learning has proposed several techniques for skill discovery from a set of expert trajectories. While these methods are promising, the number K of skills to discover is always a fixed hyperparameter, which requires either prior knowledge about the environment or an additional parameter search to tune it. We first propose a method for offline learning of options (a particular skill framework) exploiting advances in variational inference and continuous relaxations. We then highlight an unexplored connection between Bayesian nonparametrics and offline skill discovery, and show how to obtain a nonparametric version of our model. This version is tractable thanks to a carefully structured approximate posterior with a dynamically-changing number of options, removing the need to specify K. We also show how our nonparametric extension can be applied in other skill frameworks, and empirically demonstrate that our method can outperform state-of-the-art offline skill learning algorithms across a variety of environments. Our code is available at //github.com/layer6ai-labs/BNPO .
We study algorithmic applications of a natural discretization for the hard-sphere model and the Widom-Rowlinson model in a region $\mathbb{V}\subset\mathbb{R}^d$. These models are used in statistical physics to describe mixtures of one or multiple particle types subjected to hard-core interactions. For each type, particles follow a Poisson point process with a type specific activity parameter (fugacity). The Gibbs distribution is characterized by the mixture of these point processes conditioned that no two particles are closer than a type-dependent distance threshold. A key part in better understanding the Gibbs distribution is its normalizing constant, called partition function. We give sufficient conditions that the partition function of a discrete hard-core model on a geometric graph based on a point set $X \subset \mathbb{V}$ closely approximates those of such continuous models. Previously, this was only shown for the hard-sphere model on cubic regions $\mathbb{V}=[0, \ell)^d$ when $X$ is exponential in the volume of the region $\nu(\mathbb{V})$, limiting algorithmic applications. In the same setting, our refined analysis only requires a quadratic number of points, which we argue to be tight. We use our improved discretization results to approximate the partition functions of the hard-sphere model and the Widom-Rowlinson efficiently in $\nu(\mathbb{V})$. For the hard-sphere model, we obtain the first quasi-polynomial deterministic approximation algorithm for the entire fugacity regime for which, so far, only randomized approximations are known. Furthermore, we simplify a recently introduced fully polynomial randomized approximation algorithm. Similarly, we obtain the best known deterministic and randomized approximation bounds for the Widom-Rowlinson model. Moreover, we obtain approximate sampling algorithms for the respective spin systems within the same fugacity regimes.
A challenging aspect of the bandit problem is that a stochastic reward is observed only for the chosen arm and the rewards of other arms remain missing. The dependence of the arm choice on the past context and reward pairs compounds the complexity of regret analysis. We propose a novel multi-armed contextual bandit algorithm called Doubly Robust (DR) Thompson Sampling employing the doubly-robust estimator used in missing data literature to Thompson Sampling with contexts (\texttt{LinTS}). Different from previous works relying on missing data techniques (\citet{dimakopoulou2019balanced}, \citet{kim2019doubly}), the proposed algorithm is designed to allow a novel additive regret decomposition leading to an improved regret bound with the order of $\tilde{O}(\phi^{-2}\sqrt{T})$, where $\phi^2$ is the minimum eigenvalue of the covariance matrix of contexts. This is the first regret bound of \texttt{LinTS} using $\phi^2$ without the dimension of the context, $d$. Applying the relationship between $\phi^2$ and $d$, the regret bound of the proposed algorithm is $\tilde{O}(d\sqrt{T})$ in many practical scenarios, improving the bound of \texttt{LinTS} by a factor of $\sqrt{d}$. A benefit of the proposed method is that it utilizes all the context data, chosen or not chosen, thus allowing to circumvent the technical definition of unsaturated arms used in theoretical analysis of \texttt{LinTS}. Empirical studies show the advantage of the proposed algorithm over \texttt{LinTS}.
We propose a general and scalable approximate sampling strategy for probabilistic models with discrete variables. Our approach uses gradients of the likelihood function with respect to its discrete inputs to propose updates in a Metropolis-Hastings sampler. We show empirically that this approach outperforms generic samplers in a number of difficult settings including Ising models, Potts models, restricted Boltzmann machines, and factorial hidden Markov models. We also demonstrate the use of our improved sampler for training deep energy-based models on high dimensional discrete data. This approach outperforms variational auto-encoders and existing energy-based models. Finally, we give bounds showing that our approach is near-optimal in the class of samplers which propose local updates.
Approaches based on deep neural networks have achieved striking performance when testing data and training data share similar distribution, but can significantly fail otherwise. Therefore, eliminating the impact of distribution shifts between training and testing data is crucial for building performance-promising deep models. Conventional methods assume either the known heterogeneity of training data (e.g. domain labels) or the approximately equal capacities of different domains. In this paper, we consider a more challenging case where neither of the above assumptions holds. We propose to address this problem by removing the dependencies between features via learning weights for training samples, which helps deep models get rid of spurious correlations and, in turn, concentrate more on the true connection between discriminative features and labels. Extensive experiments clearly demonstrate the effectiveness of our method on multiple distribution generalization benchmarks compared with state-of-the-art counterparts. Through extensive experiments on distribution generalization benchmarks including PACS, VLCS, MNIST-M, and NICO, we show the effectiveness of our method compared with state-of-the-art counterparts.
Adversarial training is among the most effective techniques to improve the robustness of models against adversarial perturbations. However, the full effect of this approach on models is not well understood. For example, while adversarial training can reduce the adversarial risk (prediction error against an adversary), it sometimes increase standard risk (generalization error when there is no adversary). Even more, such behavior is impacted by various elements of the learning problem, including the size and quality of training data, specific forms of adversarial perturbations in the input, model overparameterization, and adversary's power, among others. In this paper, we focus on \emph{distribution perturbing} adversary framework wherein the adversary can change the test distribution within a neighborhood of the training data distribution. The neighborhood is defined via Wasserstein distance between distributions and the radius of the neighborhood is a measure of adversary's manipulative power. We study the tradeoff between standard risk and adversarial risk and derive the Pareto-optimal tradeoff, achievable over specific classes of models, in the infinite data limit with features dimension kept fixed. We consider three learning settings: 1) Regression with the class of linear models; 2) Binary classification under the Gaussian mixtures data model, with the class of linear classifiers; 3) Regression with the class of random features model (which can be equivalently represented as two-layer neural network with random first-layer weights). We show that a tradeoff between standard and adversarial risk is manifested in all three settings. We further characterize the Pareto-optimal tradeoff curves and discuss how a variety of factors, such as features correlation, adversary's power or the width of two-layer neural network would affect this tradeoff.
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.
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