We study the dynamics of two local optimization algorithms, online stochastic gradient descent (SGD) and gradient flow, within the framework of the multi-spiked tensor model in the high-dimensional regime. This multi-index model arises from the tensor principal component analysis (PCA) problem, which aims to infer $r$ unknown, orthogonal signal vectors within the $N$-dimensional unit sphere through maximum likelihood estimation from noisy observations of an order-$p$ tensor. We determine the number of samples and the conditions on the signal-to-noise ratios (SNRs) required to efficiently recover the unknown spikes from natural initializations. Specifically, we distinguish between three types of recovery: exact recovery of each spike, recovery of a permutation of all spikes, and recovery of the correct subspace spanned by the signal vectors. We show that with online SGD, it is possible to recover all spikes provided a number of sample scaling as $N^{p-2}$, aligning with the computational threshold identified in the rank-one tensor PCA problem [Ben Arous, Gheissari, Jagannath 2020, 2021]. For gradient flow, we show that the algorithmic threshold to efficiently recover the first spike is also of order $N^{p-2}$. However, recovering the subsequent directions requires the number of samples to scale as $N^{p-1}$. Our results are obtained through a detailed analysis of a low-dimensional system that describes the evolution of the correlations between the estimators and the spikes. In particular, the hidden vectors are recovered one by one according to a sequential elimination phenomenon: as one correlation exceeds a critical threshold, all correlations sharing a row or column index decrease and become negligible, allowing the subsequent correlation to grow and become macroscopic. The sequence in which correlations become macroscopic depends on their initial values and on the associated SNRs.

Expected free energy (EFE) is a central quantity in active inference which has recently gained popularity due to its intuitive decomposition of the expected value of control into a pragmatic and an epistemic component. While numerous conjectures have been made to justify EFE as a decision making objective function, the most widely accepted is still its intuitiveness and resemblance to variational free energy in approximate Bayesian inference. In this work, we take a bottom up approach and ask: taking EFE as given, what's the resulting agent's optimality gap compared with a reward-driven reinforcement learning (RL) agent, which is well understood? By casting EFE under a particular class of belief MDP and using analysis tools from RL theory, we show that EFE approximates the Bayes optimal RL policy via information value. We discuss the implications for objective specification of active inference agents.

We demonstrate that a wide array of machine learning algorithms are specific instances of one single paradigm: reciprocal learning. These instances range from active learning over multi-armed bandits to self-training. We show that all these algorithms do not only learn parameters from data but also vice versa: They iteratively alter training data in a way that depends on the current model fit. We introduce reciprocal learning as a generalization of these algorithms using the language of decision theory. This allows us to study under what conditions they converge. The key is to guarantee that reciprocal learning contracts such that the Banach fixed-point theorem applies. In this way, we find that reciprocal learning algorithms converge at linear rates to an approximately optimal model under relatively mild assumptions on the loss function, if their predictions are probabilistic and the sample adaption is both non-greedy and either randomized or regularized. We interpret these findings and provide corollaries that relate them to specific active learning, self-training, and bandit algorithms.

Schr\"{o}dinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr\"{o}dinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schr\"{o}dinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schr\"{o}dinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.

The Variational Bayesian method (VB) is used to solve the probability distributions of latent variables with the minimum free energy criterion. This criterion is not easy to understand, and the computation is complex. For these reasons, this paper proposes the Semantic Variational Bayes' method (SVB). The Semantic Information Theory the author previously proposed extends the rate-distortion function R(D) to the rate-fidelity function R(G), where R is the minimum mutual information for given semantic mutual information G. SVB came from the parameter solution of R(G), where the variational and iterative methods originated from Shannon et al.'s research on the rate-distortion function. The constraint functions SVB uses include likelihood, truth, membership, similarity, and distortion functions. SVB uses the maximum information efficiency (G/R) criterion, including the maximum semantic information criterion for optimizing model parameters and the minimum mutual information criterion for optimizing the Shannon channel. For the same tasks, SVB is computationally simpler than VB. The computational experiments in the paper include 1) using a mixture model as an example to show that the mixture model converges as G/R increases; 2) demonstrating the application of SVB in data compression with a group of error ranges as the constraint; 3) illustrating how the semantic information measure and SVB can be used for maximum entropy control and reinforcement learning in control tasks with given range constraints, providing numerical evidence for balancing control's purposiveness and efficiency. Further research is needed to apply SVB to neural networks and deep learning.

Unexpected stimuli induce "error" or "surprise" signals in the brain. The theory of predictive coding promises to explain these observations in terms of Bayesian inference by suggesting that the cortex implements variational inference in a probabilistic graphical model. However, when applied to machine learning tasks, this family of algorithms has yet to perform on par with other variational approaches in high-dimensional, structured inference problems. To address this, we introduce a novel predictive coding algorithm for structured generative models, that we call divide-and-conquer predictive coding (DCPC). DCPC differs from other formulations of predictive coding, as it respects the correlation structure of the generative model and provably performs maximum-likelihood updates of model parameters, all without sacrificing biological plausibility. Empirically, DCPC achieves better numerical performance than competing algorithms and provides accurate inference in a number of problems not previously addressed with predictive coding. We provide an open implementation of DCPC in Pyro on Github.

In this paper, we propose a novel Risk-Aware Local Trajectory Planner (RALTPER) for autonomous vehicles in complex environments characterized by Gaussian uncertainty. The proposed method integrates risk awareness and trajectory planning by leveraging probabilistic models to evaluate the likelihood of collisions with dynamic and static obstacles. The RALTPER focuses on collision avoidance constraints for both the ego vehicle region and the Gaussian-obstacle risk region. Additionally, this work enhances the generalization of both vehicle and obstacle models, making the planner adaptable to a wider range of scenarios. Our approach formulates the planning problem as a nonlinear optimization, solved using the IPOPT solver within the CasADi environment. The planner is evaluated through simulations of various challenging scenarios, including complex, static, mixed environment and narrow single-lane avoidance of pedestrians. Results demonstrate that RALTPER achieves safer and more efficient trajectory planning particularly in navigating narrow areas where a more accurate vehicle profile representation is critical for avoiding collisions.

The E-learning environment offers greater flexibility compared to face-to-face interactions, allowing for adapting educational content to meet learners' individual needs and abilities through personalization and customization of e-content and the educational process. Despite the advantages of this approach, customizing the learning environment can reduce the costs of tutoring systems for similar learners by utilizing the same content and process for co-like learning groups. Various indicators for grouping learners exist, but many of them are conceptual, uncertain, and subject to change over time. In this article, we propose using the Felder-Silverman model, which is based on learning styles, to group similar learners. Additionally, we model the behaviors and actions of e-learners in a network environment using Fuzzy Set Theory (FST). After identifying the learning styles of the learners, co-like learning groups are formed, and each group receives adaptive content based on their preferences, needs, talents, and abilities. By comparing the results of the experimental and control groups, we determine the effectiveness of the proposed grouping method. In terms of "educational success," the weighted average score of the experimental group is 17.65 out of 20, while the control group achieves a score of 12.6 out of 20. Furthermore, the "educational satisfaction" of the experimental group is 67%, whereas the control group's satisfaction level is 37%.

In this paper we propose an extension of the classical Sobol' estimator for the estimation of variance based sensitivity indices. The approach assumes a linear correlation model between the input variables which is used to decompose the contribution of an input variable into a correlated and an uncorrelated part. This method provides sampling matrices following the original joint probability distribution which are used directly to compute the model output without any assumptions or approximations of the model response function.

The ability to detect out-of-distribution (OOD) inputs is critical to guarantee the reliability of classification models deployed in an open environment. A fundamental challenge in OOD detection is that a discriminative classifier is typically trained to estimate the posterior probability p(y|z) for class y given an input z, but lacks the explicit likelihood estimation of p(z) ideally needed for OOD detection. While numerous OOD scoring functions have been proposed for classification models, these estimate scores are often heuristic-driven and cannot be rigorously interpreted as likelihood. To bridge the gap, we propose Intrinsic Likelihood (INK), which offers rigorous likelihood interpretation to modern discriminative-based classifiers. Specifically, our proposed INK score operates on the constrained latent embeddings of a discriminative classifier, which are modeled as a mixture of hyperspherical embeddings with constant norm. We draw a novel connection between the hyperspherical distribution and the intrinsic likelihood, which can be effectively optimized in modern neural networks. Extensive experiments on the OpenOOD benchmark empirically demonstrate that INK establishes a new state-of-the-art in a variety of OOD detection setups, including both far-OOD and near-OOD. Code is available at https://github.com/deeplearning-wisc/ink.