The matrix normal model, i.e., the family of Gaussian matrix-variate distributions whose covariance matrices are the Kronecker product of two lower dimensional factors, is frequently used to model matrix-variate data. The tensor normal model generalizes this family to Kronecker products of three or more factors. We study the estimation of the Kronecker factors of the covariance matrix in the matrix and tensor normal models. For the above models, we show that the maximum likelihood estimator (MLE) achieves nearly optimal nonasymptotic sample complexity and nearly tight error rates in the Fisher-Rao and Thompson metrics. In contrast to prior work, our results do not rely on the factors being well-conditioned or sparse, nor do we need to assume an accurate enough initial guess. For the matrix normal model, all our bounds are minimax optimal up to logarithmic factors, and for the tensor normal model our bounds for the largest factor and for overall covariance matrix are minimax optimal up to constant factors provided there are enough samples for any estimator to obtain constant Frobenius error. In the same regimes as our sample complexity bounds, we show that the flip-flop algorithm, a practical and widely used iterative procedure to compute the MLE, converges linearly with high probability. Our main technical insight is that, given enough samples, the negative log-likelihood function is strongly geodesically convex in the geometry on positive-definite matrices induced by the Fisher information metric. This strong convexity is determined by the expansion of certain random quantum channels.

AI models have garnered significant research attention towards predictive task automation. However, a stationary training environment is an underlying assumption for most models and such models simply do not work on non-stationary data since a stationary relationship is learned. The existing solutions propose making data stationary prior to model training and evaluation. This leads to loss of trend and seasonal patterns which are vital components for learning temporal dependencies of the system under study. This research aims to address this limitation by proposing a method for enforcing stationary behaviour within the latent space while preserving trend and seasonal information. The method deploys techniques including Differencing, Time-series decomposition, and Latent Space Arithmetic (LSA), to learn information vital for efficient approximation of trend and seasonal information which is then stored as embeddings within the latent space of a Variational Autoencoder (VAE). The approach's ability to preserve trend and seasonal information was evaluated on two time-series non-stationary datasets. For predictive performance evaluation, four deep learning models were trained on the latent vector representations of the datasets after application of the proposed method and all models produced competitive results in comparison with state-of-the-art techniques using RMSE as the performance metric.

Modern large-scale recommender systems are built upon computation-intensive infrastructure and usually suffer from a huge difference in traffic between peak and off-peak periods. In peak periods, it is challenging to perform real-time computation for each request due to the limited budget of computational resources. The recommendation with a cache is a solution to this problem, where a user-wise result cache is used to provide recommendations when the recommender system cannot afford a real-time computation. However, the cached recommendations are usually suboptimal compared to real-time computation, and it is challenging to determine the items in the cache for each user. In this paper, we provide a cache-aware reinforcement learning (CARL) method to jointly optimize the recommendation by real-time computation and by the cache. We formulate the problem as a Markov decision process with user states and a cache state, where the cache state represents whether the recommender system performs recommendations by real-time computation or by the cache. The computational load of the recommender system determines the cache state. We perform reinforcement learning based on such a model to improve user engagement over multiple requests. Moreover, we show that the cache will introduce a challenge called critic dependency, which deteriorates the performance of reinforcement learning. To tackle this challenge, we propose an eigenfunction learning (EL) method to learn independent critics for CARL. Experiments show that CARL can significantly improve the users' engagement when considering the result cache. CARL has been fully launched in Kwai app, serving over 100 million users.

The problem of estimating a matrix based on a set of its observed entries is commonly referred to as the matrix completion problem. In this work, we specifically address the scenario of binary observations, often termed as 1-bit matrix completion. While numerous studies have explored Bayesian and frequentist methods for real-value matrix completion, there has been a lack of theoretical exploration regarding Bayesian approaches in 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our contributions include obtaining concentration results for the fractional posterior and demonstrating its effectiveness in recovering the underlying parameter matrix. We accomplish this using two distinct types of prior distributions: low-rank factorization priors and a spectral scaled Student prior, with the latter requiring fewer assumptions. Importantly, our results exhibit an adaptive nature by not mandating prior knowledge of the rank of the parameter matrix. Our findings are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.

This paper proposes an adaptive behavioral decision-making method for autonomous vehicles (AVs) focusing on complex merging scenarios. Leveraging principles from non-cooperative game theory, we develop a vehicle interaction behavior model that defines key traffic elements and integrates a multifactorial reward function. Maximum entropy inverse reinforcement learning (IRL) is employed for behavior model parameter optimization. Optimal matching parameters can be obtained using the interaction behavior feature vector and the behavior probabilities output by the vehicle interaction model. Further, a behavioral decision-making method adapted to dynamic environments is proposed. By establishing a mapping model between multiple environmental variables and model parameters, it enables parameters online learning and recognition, and achieves to output interactive behavior probabilities of AVs. Quantitative analysis employing naturalistic driving datasets (highD and exiD) and real-vehicle test data validates the model's high consistency with human decision-making. In 188 tested interaction scenarios, the average human-like similarity rate is 81.73%, with a notable 83.12% in the highD dataset. Furthermore, in 145 dynamic interactions, the method matches human decisions at 77.12%, with 6913 consistence instances. Moreover, in real-vehicle tests, a 72.73% similarity with 0% safety violations are obtained. Results demonstrate the effectiveness of our proposed method in enabling AVs to make informed adaptive behavior decisions in interactive environments.

Bilevel programming has recently received attention in the literature, due to its wide range of applications, including reinforcement learning and hyper-parameter optimization. However, it is widely assumed that the underlying bilevel optimization problem is solved either by a single machine or in the case of multiple machines connected in a star-shaped network, i.e., federated learning setting. The latter approach suffers from a high communication cost on the central node (e.g., parameter server) and exhibits privacy vulnerabilities. Hence, it is of interest to develop methods that solve bilevel optimization problems in a communication-efficient decentralized manner. To that end, this paper introduces a penalty function based decentralized algorithm with theoretical guarantees for this class of optimization problems. Specifically, a distributed alternating gradient-type algorithm for solving consensus bilevel programming over a decentralized network is developed. A key feature of the proposed algorithm is to estimate the hyper-gradient of the penalty function via decentralized computation of matrix-vector products and few vector communications, which is then integrated within an alternating algorithm to obtain finite-time convergence analysis under different convexity assumptions. Our theoretical result highlights improvements in the iteration complexity of decentralized bilevel optimization, all while making efficient use of vector communication. Empirical results on both synthetic and real datasets demonstrate that the proposed method performs well in real-world settings.

High-dimensional linear regression under heavy-tailed noise or outlier corruption is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since the robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a projected sub-gradient descent algorithm for both the sparse linear regression and low-rank linear regression problems. The algorithm is not only computationally efficient with linear convergence but also statistically optimal, be the noise Gaussian or heavy-tailed with a finite 1 + epsilon moment. The convergence theory is established for a general framework and its specific applications to absolute loss, Huber loss and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of two-phase convergence. In phase one, the algorithm behaves as in typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator, which is already observed in the existing literature. Interestingly, during phase two, the algorithm converges linearly as if minimizing a smooth and strongly convex objective function, and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Numerical simulations confirm our theoretical discovery and showcase the superiority of our algorithm over prior methods.

Group synchronization refers to estimating a collection of group elements from the noisy pairwise measurements. Such a nonconvex problem has received much attention from numerous scientific fields including computer vision, robotics, and cryo-electron microscopy. In this paper, we focus on the orthogonal group synchronization problem with general additive noise models under incomplete measurements, which is much more general than the commonly considered setting of complete measurements. Characterizations of the orthogonal group synchronization problem are given from perspectives of optimality conditions as well as fixed points of the projected gradient ascent method which is also known as the generalized power method (GPM). It is well worth noting that these results still hold even without generative models. In the meantime, we derive the local error bound property for the orthogonal group synchronization problem which is useful for the convergence rate analysis of different algorithms and can be of independent interest. Finally, we prove the linear convergence result of the GPM to a global maximizer under a general additive noise model based on the established local error bound property. Our theoretical convergence result holds under several deterministic conditions which can cover certain cases with adversarial noise, and as an example we specialize it to the setting of the Erd\"os-R\'enyi measurement graph and Gaussian noise.