Real-world clinical problems are often characterized by multimodal data, usually associated with incomplete views and limited sample sizes in their cohorts, posing significant limitations for machine learning algorithms. In this work, we propose a Bayesian approach designed to efficiently handle these challenges while providing interpretable solutions. Our approach integrates (1) a generative formulation to capture cross-view relationships with a semi-supervised strategy, and (2) a discriminative task-oriented formulation to identify relevant information for specific downstream objectives. This dual generative-discriminative formulation offers both general understanding and task-specific insights; thus, it provides an automatic imputation of the missing views while enabling robust inference across different data sources. The potential of this approach becomes evident when applied to the multimodal clinical data, where our algorithm is able to capture and disentangle the complex interactions among biological, psychological, and sociodemographic modalities.

We present BALDUR, a novel Bayesian algorithm designed to deal with multi-modal datasets and small sample sizes in high-dimensional settings while providing explainable solutions. To do so, the proposed model combines within a common latent space the different data views to extract the relevant information to solve the classification task and prune out the irrelevant/redundant features/data views. Furthermore, to provide generalizable solutions in small sample size scenarios, BALDUR efficiently integrates dual kernels over the views with a small sample-to-feature ratio. Finally, its linear nature ensures the explainability of the model outcomes, allowing its use for biomarker identification. This model was tested over two different neurodegeneration datasets, outperforming the state-of-the-art models and detecting features aligned with markers already described in the scientific literature.

We propose a mean-field approximation that dramatically reduces the computational complexity of solving stochastic dynamic games. We pro- vide conditions that guarantee our method approximates an equilibrium as the number of agents grow. We then derive a performance bound to assess how well the approximation performs for any given number of agents. We apply our method to an important class of problems in ap- plied microeconomics. We show with numerical experiments that we are able to greatly expand the set of economic problems that can be analyzed computationally.

Now from the above figure we can say the error is very small as the gaussian distribution is conveniently satisfying the job. Another word'field' is basically coming from electromagnetism theory of physics, as this approximation methodology incorporates the impact of nearby neighbors in making the decision, thus incorporating the fields impacts of all neighbors. Step 3: now the expression of KL divergence is used to differentiate and minimize the distance between the posterior and approximation. By taking q(z k) as common and treating the remain multiplier as a constant and for the second object for P(z*), it becomes as Expected value for the random variable q(z 1)*q(z 2) … q(z i) where i! Let us take a working example to understand the concept of mean field approximation in more practical manner.

Have you ever asked a question, why do we need to calculate the exact Posterior distribution? To understand the answer of the above question, I would like you to re-visit our basic Baye's rule. So, what if we try and approximate our posterior! Will it impact our results? The computation of the exact posterior of the above distribution is very difficult.

Mean field theory provides an effective way of scaling multiagent reinforcement learning algorithms to environments with many agents that can be abstracted by a virtual mean agent. In this paper, we extend mean field multiagent algorithms to multiple types. The types enable the relaxation of a core assumption in mean field games, which is that all agents in the environment are playing almost similar strategies and have the same goal. We conduct experiments on three different testbeds for the field of many agent reinforcement learning, based on the standard MAgents framework. We consider two different kinds of mean field games: a) Games where agents belong to predefined types that are known a priori and b) Games where the type of each agent is unknown and therefore must be learned based on observations. We introduce new algorithms for each type of game and demonstrate their superior performance over state of the art algorithms that assume that all agents belong to the same type and other baseline algorithms in the MAgent framework.

Submodularity is one of the most well-studied properties of problem classes in combinatorial optimization and many applications of machine learning and data mining, with strong implications for guaranteed optimization. In this thesis, we investigate the role of submodularity in provable non-convex optimization and validation of algorithms. A profound understanding which classes of functions can be tractably optimized remains a central challenge for non-convex optimization. By advancing the notion of submodularity to continuous domains (termed "continuous submodularity"), we characterize a class of generally non-convex and non-concave functions -- continuous submodular functions, and derive algorithms for approximately maximizing them with strong approximation guarantees. Meanwhile, continuous submodularity captures a wide spectrum of applications, ranging from revenue maximization with general marketing strategies, MAP inference for DPPs to mean field inference for probabilistic log-submodular models, which renders it as a valuable domain knowledge in optimizing this class of objectives. Validation of algorithms is an information-theoretic framework to investigate the robustness of algorithms to fluctuations in the input/observations and their generalization ability. We investigate various algorithms for one of the paradigmatic unconstrained submodular maximization problem: MaxCut. Due to submodularity of the MaxCut objective, we are able to present efficient approaches to calculate the algorithmic information content of MaxCut algorithms. The results provide insights into the robustness of different algorithmic techniques for MaxCut.

Finding the optimal signal timing strategy is a difficult task for the problem of large-scale traffic signal control (TSC). Multi-Agent Reinforcement Learning (MARL) is a promising method to solve this problem. However, there is still room for improvement in extending to large-scale problems and modeling the behaviors of other agents for each individual agent. In this paper, a new MARL, called Cooperative double Q-learning (Co-DQL), is proposed, which has several prominent features. It uses a highly scalable independent double Q-learning method based on double estimators and the UCB policy, which can eliminate the over-estimation problem existing in traditional independent Q-learning while ensuring exploration. It uses mean field approximation to model the interaction among agents, thereby making agents learn a better cooperative strategy. In order to improve the stability and robustness of the learning process, we introduce a new reward allocation mechanism and a local state sharing method. In addition, we analyze the convergence properties of the proposed algorithm. Co-DQL is applied on TSC and tested on a multi-traffic signal simulator. According to the results obtained on several traffic scenarios, Co- DQL outperforms several state-of-the-art decentralized MARL algorithms. It can effectively shorten the average waiting time of the vehicles in the whole road system.

The mean field variational Bayes method is becoming increasingly popular in statistics and machine learning. Its iterative Coordinate Ascent Variational Inference algorithm has been widely applied to large scale Bayesian inference. See Blei et al. (2017) for a recent comprehensive review. Despite the popularity of the mean field method there exist remarkably little fundamental theoretical justifications. To the best of our knowledge, the iterative algorithm has never been investigated for any high dimensional and complex model. In this paper, we study the mean field method for community detection under the Stochastic Block Model. For an iterative Batch Coordinate Ascent Variational Inference algorithm, we show that it has a linear convergence rate and converges to the minimax rate within $\log n$ iterations. This complements the results of Bickel et al. (2013) which studied the global minimum of the mean field variational Bayes and obtained asymptotic normal estimation of global model parameters. In addition, we obtain similar optimality results for Gibbs sampling and an iterative procedure to calculate maximum likelihood estimation, which can be of independent interest.