Hierarchical Gaussian Filtering (HGF) networks allow for efficient updating of posterior distributions (beliefs) about hidden states of an agent's environment. HGF parent nodes can target the mean or variance of their children. New information entering at input nodes leads to a cascade of belief updates across the network according to one-step update equations for each node's mean and precision (inverse variance). However, the original form of the update equations for variance-targeting parents(volatility coupling) can in some regions of parameter space lead to negative posterior precision, a logical impossibility which causes the updating algorithm to terminate with an error. In this report, we introduce a modified quadratic approximation to the variational energy of volatility-coupled nodes that avoids negative posterior precision. The key idea is to interpolate between two quadratic expansions of the variational energy: one at the prior prediction and one at a second mode whose location is obtained in closed form via the Lambert W function. The resulting update equations are robust across the entire parameter space and faithfully track the variational posterior even for large prediction errors.

Tensor factorization is a powerful tool to analyse multi-way data. Recently proposed nonlinear factorization methods, although capable of capturing complex relationships, are computationally quite expensive and may suffer a severe learning bias in case of extreme data sparsity. Therefore, we propose a distributed, flexible nonlinear tensor factorization model, which avoids the expensive computations and structural restrictions of the Kronecker-product in the existing TGP formulations, allowing an arbitrary subset of tensorial entries to be selected for training. Meanwhile, we derive a tractable and tight variational evidence lower bound (ELBO) that enables highly decoupled, parallel computations and high-quality inference. Based on the new bound, we develop a distributed, key-value-free inference algorithm in the MAPREDUCE framework, which can fully exploit the memory cache mechanism in fast MAPREDUCE systems such as SPARK. Experiments demonstrate the advantages of our method over several state-of-the-art approaches, in terms of both predictive performance and computational efficiency.

In this section, we list frequently asked questions from researchers who help proofread this manuscript. These raised questions might also be relevant for others and help in better understanding the paper, so we include more detailed discussions here. This work considers the multi-input multi-output setting of multi-task learning under the episodic training mechanism. As shown in Table 1, we use "Heterogeneous tasks" to distinguish the different branches of multi-task learning: (1) single-input multi-output (SIMO) considers different tasks which have the same input and different supervision information. All tasks are related since they share the target space. This setting encourages deep models to deal with the insufficient data of each task by aggregating the training data from related tasks in the spirit of data augmentation. Meanwhile, "Episodic training" is used to describe the data-feeding strategy. Multi-task meta-learning also benefits from episodic training, but it follows the SIMO setting in every single episode and cannot sufficiently handle heterogeneous tasks.

In addition to the four appendix sections mentioned in our main paper, we would like to draw atten-1 tion to two additional experiments: one evaluating the practical training and coding time, and the2 other investigating the impact of the number of training samples. These two experiments, especially3 the later one, offer crucial insights and are detailed in Appendix E1 and Appendix E2, respectively.4 Algorithm 1 A* encoding Require: Proposal distribution pw and target distribution qw. In our experiments, we used global-bound depth-limited A*7 coding to achieve this [1]. We describe the encoding procedure in Algorithm 1 and the decoding8 procedure in Algorithm 2. For brevity, we refer to this particular variant of the algorithm as A*9 coding for the rest of the appendix.10

When considering a model selection or, more generally, an aggregation approach for adaptive statistical inference, it is often necessary to compute estimators over a wide range of model complexities including unnecessarily large models even when the true data-generating process is relatively simple, due to the lack of prior knowledge. This requirement can lead to substantial computational inefficiency. In this work, we propose a novel framework for efficient model aggregation called the early-stopped aggregation (ESA): instead of computing and aggregating estimators for all candidate models, we compute only a small number of simpler ones using an early-stopping criterion and aggregate only these for final inference. Our framework is versatile and applies to both Bayesian model selection, in particular, within the variational Bayes framework, and frequentist estimation, including a general penalized estimation setting. We investigate adaptive optimal property of the ESA approach across three learning paradigms. We first show that ESA achieves optimal adaptive contraction rates in the variational Bayes setting under mild conditions. We extend this result to variational empirical Bayes, where prior hyperparameters are chosen in a data-dependent manner. In addition, we apply the ESA approach to frequentist aggregation including both penalization-based and sample-splitting implementations, and establish corresponding theory. As we demonstrate, there is a clear unification between early-stopped Bayes and frequentist penalized aggregation, with a common "energy" functional comprising a data-fitting term and a complexity-control term that drives both procedures. We further present several applications and numerical studies that highlight the efficiency and strong performance of the proposed approach.

Appendix for "Episodic Multi-T ask Learning with Heterogeneous Neural Processes" In this section, we list frequently asked questions from researchers who help proofread this manuscript. As shown in Table 1, we use "Heterogeneous tasks" to distinguish the different branches of multi-task Meanwhile, "Episodic training" is used to describe the data-feeding strategy. Thus, "Heterogeneous tasks" is not available here (-). In episodic multi-task learning, we restrict the scope of the problem to the case where tasks in the same episode are related and share the same target space. This also implies that tasks with the same target space are related.

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