Generative models, such as large language models or text-to-image diffusion models, can generate relevant responses to user-given queries. Response-based vector embeddings of generative models facilitate statistical analysis and inference on a given collection of black-box generative models. The Data Kernel Perspective Space embedding is one particular method of obtaining response-based vector embeddings for a given set of generative models, already discussed in the literature. In this paper, under appropriate regularity conditions, we establish high probability concentration bounds on the sample vector embeddings for a given set of generative models, obtained through the method of Data Kernel Perspective Space embedding. Our results tell us the required number of sample responses needed in order to approximate the population-level vector embeddings with a desired level of accuracy. The algebraic tools used to establish our results can be used further for establishing concentration bounds on Classical Multidimensional Scaling embeddings in general, when the dissimilarities are observed with noise.

In Figure 1 we visualise the normalized effective sample size (ESS) using 10 Monte Carlo samples from our approximate posterior evaluated on the test data throughout PSVI inference. ESS takes consistently non-trivial values (larger than 0.1) even for 200-dimensional data, with a visible decreasing trend as data--and hence model--parameters dimensionality increases.

Due to the computational complexity of evaluating interatomic forces from first principles, the creation of interatomic machine learning force fields has become a highly active field of research. However, the generation of training datasets of sufficient size and sample diversity itself comes with a computational burden that can make this approach impractical for modeling rare events or systems with a large configuration space. Fine-tuning foundation models that have been pre-trained on large-scale material or molecular databases offers a promising opportunity to reduce the amount of training data necessary to reach a desired level of accuracy. However, even if this approach requires less training data overall, creating a suitable training dataset can still be a very challenging problem, especially for systems with rare events and for end-users who don't have an extensive background in machine learning. In on-the-fly learning, the creation of a training dataset can be largely automated by using model uncertainty during the simulation to decide if the model is accurate enough or if a structure should be recalculated with classical methods and used to update the model. A key challenge for applying this form of active learning to the fine-tuning of foundation models is how to assess the uncertainty of those models during the fine-tuning process, even though most foundation models lack any form of uncertainty quantification. In this paper, we overcome this challenge by introducing a fine-tuning approach based on Bayesian neural network methods and a subsequent on-the-fly workflow that automatically fine-tunes the model while maintaining a pre-specified accuracy and can detect rare events such as transition states and sample them at an increased rate relative to their occurrence.

Analyses of causal mediation often involve exposure-induced confounders or, relatedly, multiple mediators. In such applications, researchers aim to estimate a variety of different quantities, including interventional direct and indirect effects, multivariate natural direct and indirect effects, and/or path-specific effects. This study introduces a general approach to estimating all these quantities by simulating potential outcomes from a series of distribution models for each mediator and the outcome. Building on similar methods developed for analyses with only a single mediator (Imai et al. 2010), we first outline how to implement this approach with parametric models. The parametric implementation can accommodate linear and nonlinear relationships, both continuous and discrete mediators, and many different types of outcomes. However, it depends on correct specification of each model used to simulate the potential outcomes. To address the risk of misspecification, we also introduce an alternative implementation using a novel class of nonparametric models, which leverage deep neural networks to approximate the relevant distributions without relying on strict assumptions about functional form. We illustrate both methods by reanalyzing the effects of media framing on attitudes toward immigration (Brader et al. 2008) and the effects of prenatal care on preterm birth (VanderWeele et al. 2014).

Averaging techniques such as Ruppert--Polyak averaging and exponential movering averaging (EMA) are powerful approaches to accelerate optimization procedures of stochastic gradient descent (SGD) optimization methods such as the popular ADAM optimizer. However, depending on the specific optimization problem under consideration, the type and the parameters for the averaging need to be adjusted to achieve the smallest optimization error. In this work we propose an averaging approach, which we refer to as parallel averaged ADAM (PADAM), in which we compute parallely different averaged variants of ADAM and during the training process dynamically select the variant with the smallest optimization error. A central feature of this approach is that this procedure requires no more gradient evaluations than the usual ADAM optimizer as each of the averaged trajectories relies on the same underlying ADAM trajectory and thus on the same underlying gradients. We test the proposed PADAM optimizer in 13 stochastic optimization and deep neural network (DNN) learning problems and compare its performance with known optimizers from the literature such as standard SGD, momentum SGD, Adam with and without EMA, and ADAMW. In particular, we apply the compared optimizers to physics-informed neural network, deep Galerkin, deep backward stochastic differential equation and deep Kolmogorov approximations for boundary value partial differential equation problems from scientific machine learning, as well as to DNN approximations for optimal control and optimal stopping problems. In nearly all of the considered examples PADAM achieves, sometimes among others and sometimes exclusively, essentially the smallest optimization error. This work thus strongly suggest to consider PADAM for scientific machine learning problems and also motivates further research for adaptive averaging procedures within the training of DNNs.

Many inverse problems in nuclear fusion and high-energy astrophysics research, such as the optimization of tokamak reactor geometries or the inference of black hole parameters from interfer-ometric images, necessitate high-dimensional parameter scans and large ensembles of simulations to be performed. Such inverse problems typically involve large uncertainties, both in the measurement parameters being inverted and in the underlying physics models themselves. Monte Carlo sampling, when combined with modern non-linear dimensionality reduction techniques such as autoencoders and manifold learning, can be used to reduce the size of the parameter spaces considerably. However, there is no guarantee that the resulting combinations of parameters will be physically valid, or even mathematically consistent. In this position paper, we advocate adopting a hybrid approach that leverages our recent advances in the development of formal verification methods for numerical algorithms, with the goal of constructing parameter space restrictions with provable mathematical and physical correctness properties, whilst nevertheless respecting both experimental uncertainties and uncertainties in the underlying physical processes.

Additional Feedback: Introduction: The focus of this work seems to be on cases where the inputs are stochastic and the simulator are stochastic. This would be equally applicable to scenarios which were deterministic but otherwise non differentiable right? One related work that comes to mind is related to sobolev training. It's a bit different in motivation and setup but might be nice to cite. The introduction is generally well motivated and concise.

The neural process (NP) is a family of computationally efficient models for learning distributions over functions. However, it suffers from under-fitting and shows suboptimal performance in practice. Researchers have primarily focused on incorporating diverse structural inductive biases, \textit{e.g.} attention or convolution, in modeling. The topic of inference suboptimality and an analysis of the NP from the optimization objective perspective has hardly been studied in earlier work. To fix this issue, we propose a surrogate objective of the target log-likelihood of the meta dataset within the expectation maximization framework. The resulting model, referred to as the Self-normalized Importance weighted Neural Process (SI-NP), can learn a more accurate functional prior and has an improvement guarantee concerning the target log-likelihood. Experimental results show the competitive performance of SI-NP over other NPs objectives and illustrate that structural inductive biases, such as attention modules, can also augment our method to achieve SOTA performance. Our code is available at \url{https://github.com/hhq123gogogo/SI_NPs}.

The main challenge of applying Wiener-Kallianpur innovation Whereas standard probabilistic forecasting aims to estimate representation for inference and decision-making is the conditional probability distribution of the time series at twofold. First, obtaining a causal encoder to extract the a future time, GPF obtains a generative model capable of innovation process requires knowing the marginal and joint producing arbitrarily many Monte Carlo samples of future distributions of the time series, which is rarely possible without time series realizations according to the conditional probability imposing some parametric structure. Furthermore, even when distribution of the time series given past observations. As the probability distribution is known, there is no known computationally a nonparametric probabilistic forecasting technique, GPF is tractable way to construct the causal encoder to essential for decision-making under uncertainty where datadriven obtain an innovation process. Second, the Wiener-Kallianpur risk-sensitive optimization requires conditional samples innovation representation may not exist for a broad class of of future randomness. The Monte Carlo samples generated random processes, including some of the important cases of from GPF can be used to produce any form of point forecast.