This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in topic models. The data consists of observed multinomial counts of $p$ words across $n$ independent documents. In topic models, the $p\times n$ expected word frequency matrix is assumed to be factorized as a $p\times K$ word-topic matrix $A$ and a $K\times n$ topic-document matrix $T$. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns of $A$ are viewed as $p$-dimensional mixture components that are common to all documents while columns of $T$ are viewed as the $K$-dimensional mixture weights that are document specific and are allowed to be sparse. The main interest is to provide sharp, finite sample, $\ell_1$-norm convergence rates for estimators of the mixture weights $T$ when $A$ is either known or unknown. For known $A$, we suggest MLE estimation of $T$. Our non-standard analysis of the MLE not only establishes its $\ell_1$ convergence rate, but reveals a remarkable property: the MLE, with no extra regularization, can be exactly sparse and contain the true zero pattern of $T$. We further show that the MLE is both minimax optimal and adaptive to the unknown sparsity in a large class of sparse topic distributions. When $A$ is unknown, we estimate $T$ by optimizing the likelihood function corresponding to a plug in, generic, estimator $\hat{A}$ of $A$. For any estimator $\hat{A}$ that satisfies carefully detailed conditions for proximity to $A$, the resulting estimator of $T$ is shown to retain the properties established for the MLE. The ambient dimensions $K$ and $p$ are allowed to grow with the sample sizes. Our application is to the estimation of 1-Wasserstein distances between document generating distributions. We propose, estimate and analyze new 1-Wasserstein distances between two probabilistic document representations.

Learning data representations that are useful for various downstream tasks is a cornerstone of artificial intelligence. While existing methods are typically evaluated on downstream tasks such as classification or generative image quality, we propose to assess representations through their usefulness in downstream control tasks, such as reaching or pushing objects. By training over 10,000 reinforcement learning policies, we extensively evaluate to what extent different representation properties affect out-of-distribution (OOD) generalization. Finally, we demonstrate zero-shot transfer of these policies from simulation to the real world, without any domain randomization or fine-tuning. This paper aims to establish the first systematic characterization of the usefulness of learned representations for real-world OOD downstream tasks.

Parameter reconstruction is a common problem in optical nano metrology. It generally involves a set of measurements, to which one attempts to fit a numerical model of the measurement process. The model evaluation typically involves to solve Maxwell's equations and is thus time consuming. This makes the reconstruction computationally demanding. Several methods exist for fitting the model to the measurements. On the one hand, Bayesian optimization methods for expensive black-box optimization enable an efficient reconstruction by training a machine learning model of the squared sum of deviations. On the other hand, curve fitting algorithms, such as the Levenberg-Marquardt method, take the deviations between all model outputs and corresponding measurement values into account which enables a fast local convergence. In this paper we present a Bayesian Target Vector Optimization scheme which combines these two approaches. We compare the performance of the presented method against a standard Levenberg-Marquardt-like algorithm, a conventional Bayesian optimization scheme, and the L-BFGS-B and Nelder-Mead simplex algorithms. As a stand-in for problems from nano metrology, we employ a non-linear least-square problem from the NIST Standard Reference Database. We find that the presented method generally uses fewer calls of the model function than any of the competing schemes to achieve similar reconstruction performance.

Bayesian models based on the Dirichlet process and other stick-breaking priors have been proposed as core ingredients for clustering, topic modeling, and other unsupervised learning tasks. Prior specification is, however, relatively difficult for such models, given that their flexibility implies that the consequences of prior choices are often relatively opaque. Moreover, these choices can have a substantial effect on posterior inferences. Thus, considerations of robustness need to go hand in hand with nonparametric modeling. In the current paper, we tackle this challenge by exploiting the fact that variational Bayesian methods, in addition to having computational advantages in fitting complex nonparametric models, also yield sensitivities with respect to parametric and nonparametric aspects of Bayesian models. In particular, we demonstrate how to assess the sensitivity of conclusions to the choice of concentration parameter and stick-breaking distribution for inferences under Dirichlet process mixtures and related mixture models. We provide both theoretical and empirical support for our variational approach to Bayesian sensitivity analysis.

Power grid operation subject to an extreme event requires decision-making by human operators under stressful condition with high cognitive load. Decision support under adverse dynamic events, specially if forecasted, can be supplemented by intelligent proactive control. Power system operation during wildfires require resiliency-driven proactive control for load shedding, line switching and resource allocation considering the dynamics of the wildfire and failure propagation. However, possible number of line- and load-switching in a large system during an event make traditional prediction-driven and stochastic approaches computationally intractable, leading operators to often use greedy algorithms. We model and solve the proactive control problem as a Markov decision process and introduce an integrated testbed for spatio-temporal wildfire propagation and proactive power-system operation. We transform the enormous wildfire-propagation observation space and utilize it as part of a heuristic for proactive de-energization of transmission assets. We integrate this heuristic with a reinforcement-learning based proactive policy for controlling the generating assets. Our approach allows this controller to provide setpoints for a part of the generation fleet, while a myopic operator can determine the setpoints for the remaining set, which results in a symbiotic action. We evaluate our approach utilizing the IEEE 24-node system mapped on a hypothetical terrain. Our results show that the proposed approach can help the operator to reduce load loss during an extreme event, reduce power flow through lines that are to be de-energized, and reduce the likelihood of infeasible power-flow solutions, which would indicate violation of short-term thermal limits of transmission lines.

State-of-the-art artificial neural networks (ANNs) require labelled data or feedback between layers, are often biologically implausible, and are vulnerable to adversarial attacks that humans are not susceptible to. On the other hand, Hebbian learning in winner-take-all (WTA) networks, is unsupervised, feed-forward, and biologically plausible. However, an objective optimization theory for WTA networks has been missing, except under very limiting assumptions. Here we derive formally such a theory, based on biologically plausible but generic ANN elements. Through Hebbian learning, network parameters maintain a Bayesian generative model of the data. There is no supervisory loss function, but the network does minimize cross-entropy between its activations and the input distribution. The key is a "soft" WTA where there is no absolute "hard" winner neuron, and a specific type of Hebbian-like plasticity of weights and biases. We confirm our theory in practice, where, in handwritten digit (MNIST) recognition, our Hebbian algorithm, SoftHebb, minimizes cross-entropy without having access to it, and outperforms the more frequently used, hard-WTA-based method. Strikingly, it even outperforms supervised end-to-end backpropagation, under certain conditions. Specifically, in a two-layered network, SoftHebb outperforms backpropagation when the training dataset is only presented once, when the testing data is noisy, and under gradient-based adversarial attacks. Adversarial attacks that confuse SoftHebb are also confusing to the human eye. Finally, the model can generate interpolations of objects from its input distribution.

Under the Bayesian brain hypothesis, behavioural variations can be attributed to different priors over generative model parameters. This provides a formal explanation for why individuals exhibit inconsistent behavioural preferences when confronted with similar choices. For example, greedy preferences are a consequence of confident (or precise) beliefs over certain outcomes. Here, we offer an alternative account of behavioural variability using R\'enyi divergences and their associated variational bounds. R\'enyi bounds are analogous to the variational free energy (or evidence lower bound) and can be derived under the same assumptions. Importantly, these bounds provide a formal way to establish behavioural differences through an $\alpha$ parameter, given fixed priors. This rests on changes in $\alpha$ that alter the bound (on a continuous scale), inducing different posterior estimates and consequent variations in behaviour. Thus, it looks as if individuals have different priors, and have reached different conclusions. More specifically, $\alpha \to 0^{+}$ optimisation leads to mass-covering variational estimates and increased variability in choice behaviour. Furthermore, $\alpha \to + \infty$ optimisation leads to mass-seeking variational posteriors and greedy preferences. We exemplify this formulation through simulations of the multi-armed bandit task. We note that these $\alpha$ parameterisations may be especially relevant, i.e., shape preferences, when the true posterior is not in the same family of distributions as the assumed (simpler) approximate density, which may be the case in many real-world scenarios. The ensuing departure from vanilla variational inference provides a potentially useful explanation for differences in behavioural preferences of biological (or artificial) agents under the assumption that the brain performs variational Bayesian inference.

Fully Bayesian approaches to sequential decision-making assume that problem parameters are generated from a known prior, while in practice, such information is often lacking, and needs to be estimated through learning. This problem is exacerbated in decision-making setups with partial information, where using a misspecified prior may lead to poor exploration and inferior performance. In this work we prove, in the context of stochastic linear bandits and Gaussian priors, that as long as the prior estimate is sufficiently close to the true prior, the performance of an algorithm that uses the misspecified prior is close to that of the algorithm that uses the true prior. Next, we address the task of learning the prior through metalearning, where a learner updates its estimate of the prior across multiple task instances in order to improve performance on future tasks. The estimated prior is then updated within each task based on incoming observations, while actions are selected in order to maximize expected reward. In this work we apply this scheme within a linear bandit setting, and provide algorithms and regret bounds, demonstrating its effectiveness, as compared to an algorithm that knows the correct prior. Our results hold for a broad class of algorithms, including, for example, Thompson Sampling and Information Directed Sampling.

Energy-based models (EBMs) are explicit generative models which work by considering Gibbs measures defined through an energy function f, with a probability density proportional to exp( βf(x)), where β is the inverse temperature. Such models originate in statistical physics [Gibbs, 2010, Ruelle, 1969], and have become a fundamental modeling tool in statistics and machine learning [Wainwright and Jordan, 2008, Ranzato et al., 2007, LeCun et al., 2006, Du and Mordatch, 2019, Song and Kingma, 2021]. Given data samples from a target distribution, the learning algorithms for EBMs attempt to estimate an energy function f to model the samples density. The resulting learned model can then be used to obtain new samples, typically through Markov Chain Monte Carlo (MCMC) techniques. The standard method to train EBMs is maximum likelihood estimation, i.e. the learned energy is the one maximizing the likelihood of the target samples, within a certain function class. One generic approach for this is to use gradient descent, where gradients may be approximated using MCMC samples from the trained model. However, this is computationally difficult for highly non-convex trained energies, which in recent years has motivated a myriad of alternative losses to learn EBM energies, such as the popular score matching; see [Song and Kingma, 2021] for a review. EBMs also have structural connections with maximum entropy (maxent) models, which have been studied for decades through Fenchel duality. Dai et al. [2019b] was the first work to leverage similar duality arguments

Causal discovery from observational data is an important tool in many branches of science. Under certain assumptions it allows scientists to explain phenomena, predict, and make decisions. In the large sample limit, sound and complete causal discovery algorithms have been previously introduced, where a directed acyclic graph (DAG), or its equivalence class, representing causal relations is searched. However, in real-world cases, only finite training data is available, which limits the power of statistical tests used by these algorithms, leading to errors in the inferred causal model. This is commonly addressed by devising a strategy for using as few as possible statistical tests. In this paper, we introduce such a strategy in the form of a recursive wrapper for existing constraint-based causal discovery algorithms, which preserves soundness and completeness. It recursively clusters the observed variables using the normalized min-cut criterion from the outset, and uses a baseline causal discovery algorithm during backtracking for learning local sub-graphs. It then combines them and ensures completeness. By an ablation study, using synthetic data, and by common real-world benchmarks, we demonstrate that our approach requires significantly fewer statistical tests, learns more accurate graphs, and requires shorter run-times than the baseline algorithm.