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Informative cluster size (ICS) arises in situations with clustered data where a latent relationship exists between the number of participants in a cluster and the outcome measures. Although this phenomenon has been sporadically reported in statistical literature for nearly two decades now, further exploration is needed in certain statistical methodologies to avoid potentially misleading inferences. For inference about population quantities without covariates, inverse cluster size reweightings are often employed to adjust for ICS. Further, to study the effect of covariates on disease progression described by a multistate model, the pseudo-value regression technique has gained popularity in time-to-event data analysis. We seek to answer the question: "How to apply pseudo-value regression to clustered time-to-event data when cluster size is informative?" ICS adjustment by the reweighting method can be performed in two steps; estimation of marginal functions of the multistate model and fitting the estimating equations based on pseudo-value responses, leading to four possible strategies. We present theoretical arguments and thorough simulation experiments to ascertain the correct strategy for adjusting for ICS. A further extension of our methodology is implemented to include informativeness induced by the intra-cluster group size. We demonstrate the methods in two real-world applications: (i) to determine predictors of tooth survival in a periodontal study, and (ii) to identify indicators of ambulatory recovery in spinal cord injury patients who participated in locomotor-training rehabilitation.

We consider the problem of finding the matching map between two sets of $d$-dimensional noisy feature-vectors. The distinctive feature of our setting is that we do not assume that all the vectors of the first set have their corresponding vector in the second set. If $n$ and $m$ are the sizes of these two sets, we assume that the matching map that should be recovered is defined on a subset of unknown cardinality $k^*\le \min(n,m)$. We show that, in the high-dimensional setting, if the signal-to-noise ratio is larger than $5(d\log(4nm/\alpha))^{1/4}$, then the true matching map can be recovered with probability $1-\alpha$. Interestingly, this threshold does not depend on $k^*$ and is the same as the one obtained in prior work in the case of $k = \min(n,m)$. The procedure for which the aforementioned property is proved is obtained by a data-driven selection among candidate mappings $\{\hat\pi_k:k\in[\min(n,m)]\}$. Each $\hat\pi_k$ minimizes the sum of squares of distances between two sets of size $k$. The resulting optimization problem can be formulated as a minimum-cost flow problem, and thus solved efficiently. Finally, we report the results of numerical experiments on both synthetic and real-world data that illustrate our theoretical results and provide further insight into the properties of the algorithms studied in this work.

Markov decision processes (MDPs) are formal models commonly used in sequential decision-making. MDPs capture the stochasticity that may arise, for instance, from imprecise actuators via probabilities in the transition function. However, in data-driven applications, deriving precise probabilities from (limited) data introduces statistical errors that may lead to unexpected or undesirable outcomes. Uncertain MDPs (uMDPs) do not require precise probabilities but instead use so-called uncertainty sets in the transitions, accounting for such limited data. Tools from the formal verification community efficiently compute robust policies that provably adhere to formal specifications, like safety constraints, under the worst-case instance in the uncertainty set. We continuously learn the transition probabilities of an MDP in a robust anytime-learning approach that combines a dedicated Bayesian inference scheme with the computation of robust policies. In particular, our method (1) approximates probabilities as intervals, (2) adapts to new data that may be inconsistent with an intermediate model, and (3) may be stopped at any time to compute a robust policy on the uMDP that faithfully captures the data so far. Furthermore, our method is capable of adapting to changes in the environment. We show the effectiveness of our approach and compare it to robust policies computed on uMDPs learned by the UCRL2 reinforcement learning algorithm in an experimental evaluation on several benchmarks.

Stochastic versions of proximal methods have gained much attention in statistics and machine learning. These algorithms tend to admit simple, scalable forms, and enjoy numerical stability via implicit updates. In this work, we propose and analyze a stochastic version of the recently proposed proximal distance algorithm, a class of iterative optimization methods that recover a desired constrained estimation problem as a penalty parameter $\rho \rightarrow \infty$. By uncovering connections to related stochastic proximal methods and interpreting the penalty parameter as the learning rate, we justify heuristics used in practical manifestations of the proximal distance method, establishing their convergence guarantees for the first time. Moreover, we extend recent theoretical devices to establish finite error bounds and a complete characterization of convergence rates regimes. We validate our analysis via a thorough empirical study, also showing that unsurprisingly, the proposed method outpaces batch versions on popular learning tasks.

We consider the problem of performing Bayesian inference in probabilistic models where observations are accompanied by uncertainty, referred to as `uncertain evidence'. In many real-world scenarios, such uncertainty stems from measurement errors associated with observable quantities in probabilistic models. We explore how to interpret uncertain evidence, and by extension the importance of proper interpretation as it pertains to inference about latent variables. We consider a recently-proposed method `stochastic evidence' as well as revisit two older methods: Jeffrey's rule and virtual evidence. We devise concrete guidelines on how to account for uncertain evidence and we provide new insights, particularly regarding consistency. To showcase the impact of different interpretations of the same uncertain evidence, we carry out experiments in which we compare inference results associated with each interpretation.

Assessing the validity of a real-world system with respect to given quality criteria is a common yet costly task in industrial applications due to the vast number of required real-world tests. Validating such systems by means of simulation offers a promising and less expensive alternative, but requires an assessment of the simulation accuracy and therefore end-to-end measurements. Additionally, covariate shifts between simulations and actual usage can cause difficulties for estimating the reliability of such systems. In this work, we present a validation method that propagates bounds on distributional discrepancy measures through a composite system, thereby allowing us to derive an upper bound on the failure probability of the real system from potentially inaccurate simulations. Each propagation step entails an optimization problem, where -- for measures such as maximum mean discrepancy (MMD) -- we develop tight convex relaxations based on semidefinite programs. We demonstrate that our propagation method yields valid and useful bounds for composite systems exhibiting a variety of realistic effects. In particular, we show that the proposed method can successfully account for data shifts within the experimental design as well as model inaccuracies within the used simulation.

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. Recently a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ was proved \cite{Betal}. This characterization was then used to give new combinatorial lower bounds on the MLT of any graph. We continue this line of research by exploiting combinatorial rigidity results to compute the MLT precisely for several families of graphs. These include graphs with at most $9$ vertices, graphs with at most 24 edges, every graph sufficiently close to a complete graph and graphs with bounded degrees.

The graph retrieval problem is to search in a large corpus of graphs for ones that are most similar to a query graph. A common consideration for scoring similarity is the maximum common subgraph (MCS) between the query and corpus graphs, usually counting the number of common edges (i.e., MCES). In some applications, it is also desirable that the common subgraph be connected, i.e., the maximum common connected subgraph (MCCS). Finding exact MCES and MCCS is intractable, but may be unnecessary if ranking corpus graphs by relevance is the goal. We design fast and trainable neural functions that approximate MCES and MCCS well. Late interaction methods compute dense representations for the query and corpus graph separately, and compare these representations using simple similarity functions at the last stage, leading to highly scalable systems. Early interaction methods combine information from both graphs right from the input stages, are usually considerably more accurate, but slower. We propose both late and early interaction neural MCES and MCCS formulations. They are both based on a continuous relaxation of a node alignment matrix between query and corpus nodes. For MCCS, we propose a novel differentiable network for estimating the size of the largest connected common subgraph. Extensive experiments with seven data sets show that our proposals are superior among late interaction models in terms of both accuracy and speed. Our early interaction models provide accuracy competitive with the state of the art, at substantially greater speeds.

AI is undergoing a paradigm shift with the rise of models (e.g., BERT, DALL-E, GPT-3) that are trained on broad data at scale and are adaptable to a wide range of downstream tasks. We call these models foundation models to underscore their critically central yet incomplete character. This report provides a thorough account of the opportunities and risks of foundation models, ranging from their capabilities (e.g., language, vision, robotics, reasoning, human interaction) and technical principles(e.g., model architectures, training procedures, data, systems, security, evaluation, theory) to their applications (e.g., law, healthcare, education) and societal impact (e.g., inequity, misuse, economic and environmental impact, legal and ethical considerations). Though foundation models are based on standard deep learning and transfer learning, their scale results in new emergent capabilities,and their effectiveness across so many tasks incentivizes homogenization. Homogenization provides powerful leverage but demands caution, as the defects of the foundation model are inherited by all the adapted models downstream. Despite the impending widespread deployment of foundation models, we currently lack a clear understanding of how they work, when they fail, and what they are even capable of due to their emergent properties. To tackle these questions, we believe much of the critical research on foundation models will require deep interdisciplinary collaboration commensurate with their fundamentally sociotechnical nature.

To address the sparsity and cold start problem of collaborative filtering, researchers usually make use of side information, such as social networks or item attributes, to improve recommendation performance. This paper considers the knowledge graph as the source of side information. To address the limitations of existing embedding-based and path-based methods for knowledge-graph-aware recommendation, we propose Ripple Network, an end-to-end framework that naturally incorporates the knowledge graph into recommender systems. Similar to actual ripples propagating on the surface of water, Ripple Network stimulates the propagation of user preferences over the set of knowledge entities by automatically and iteratively extending a user's potential interests along links in the knowledge graph. The multiple "ripples" activated by a user's historically clicked items are thus superposed to form the preference distribution of the user with respect to a candidate item, which could be used for predicting the final clicking probability. Through extensive experiments on real-world datasets, we demonstrate that Ripple Network achieves substantial gains in a variety of scenarios, including movie, book and news recommendation, over several state-of-the-art baselines.