The adapted Wasserstein distance controls the calibration errors of optimal values in various stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. Motivated by approximating the true underlying distribution by empirical data, we consider empirical measures of $\mathbb{R}^d$-valued stochastic process in finite discrete-time. It is known that the empirical measures do not converge under the adapted Wasserstein distance. To address this issue, we consider convolutions of Gaussian kernels and empirical measures as an alternative, which we refer to the Gaussian-smoothed empirical measures. By setting the bandwidths of Gaussian kernels depending on the number of samples, we prove the convergence of the Gaussian-smoothed empirical measures to the true underlying measure in terms of mean, deviation, and almost sure convergence. Although Gaussian-smoothed empirical measures converge to the true underlying measure and can potentially enlarge data, they are not discrete measures and therefore not applicable in practice. Therefore, we combine Gaussian-smoothed empirical measures and the adapted empirical measures in \cite{acciaio2022convergence} to introduce the adapted smoothed empirical measures, which are discrete substitutes of the smoothed empirical measures. We establish the polynomial mean convergence rate, the exponential deviation convergence rate and the almost sure convergence of the adapted smoothed empirical measures.
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Quantum Relative Entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy (SQRE). We also introduce a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package DDS. In addition to handling QRE constraints, DDS accepts any combination of several other conic and non-conic convex constraints. Our comprehensive numerical experiments encompass several parts including 1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, 2) using DDS for combining QRE constraints with various other constraint types, and 3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols.
Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and known nonlinear extensions of the fundamental lemma. Applying a transformation to the usual linear equation in Hankel matrices, we arrive at an alternative implicit kernel representation of the system trajectories while keeping the requirements on persistency of excitation. We show that this representation is equivalent to the solution of a specific kernel regression problem. We explore the possible structures of the underlying kernel as well as the system classes to which they correspond.
Despite the frequent use of agent-based models (ABMs) for studying social phenomena, parameter estimation remains a challenge, often relying on costly simulation-based heuristics. This work uses variational inference to estimate the parameters of an opinion dynamics ABM, by transforming the estimation problem into an optimization task that can be solved directly. Our proposal relies on probabilistic generative ABMs (PGABMs): we start by synthesizing a probabilistic generative model from the ABM rules. Then, we transform the inference process into an optimization problem suitable for automatic differentiation. In particular, we use the Gumbel-Softmax reparameterization for categorical agent attributes and stochastic variational inference for parameter estimation. Furthermore, we explore the trade-offs of using variational distributions with different complexity: normal distributions and normalizing flows. We validate our method on a bounded confidence model with agent roles (leaders and followers). Our approach estimates both macroscopic (bounded confidence intervals and backfire thresholds) and microscopic ($200$ categorical, agent-level roles) more accurately than simulation-based and MCMC methods. Consequently, our technique enables experts to tune and validate their ABMs against real-world observations, thus providing insights into human behavior in social systems via data-driven analysis.
The estimation of cumulative distribution functions (CDF) is an important learning task with a great variety of downstream applications, such as risk assessments in predictions and decision making. In this paper, we study functional regression of contextual CDFs where each data point is sampled from a linear combination of context dependent CDF basis functions. We propose functional ridge-regression-based estimation methods that estimate CDFs accurately everywhere. In particular, given $n$ samples with $d$ basis functions, we show estimation error upper bounds of $\widetilde O(\sqrt{d/n})$ for fixed design, random design, and adversarial context cases. We also derive matching information theoretic lower bounds, establishing minimax optimality for CDF functional regression. Furthermore, we remove the burn-in time in the random design setting using an alternative penalized estimator. Then, we consider agnostic settings where there is a mismatch in the data generation process. We characterize the error of the proposed estimators in terms of the mismatched error, and show that the estimators are well-behaved under model mismatch. Moreover, to complete our study, we formalize infinite dimensional models where the parameter space is an infinite dimensional Hilbert space, and establish a self-normalized estimation error upper bound for this setting. Notably, the upper bound reduces to the $\widetilde O(\sqrt{d/n})$ bound when the parameter space is constrained to be $d$-dimensional. Our comprehensive numerical experiments validate the efficacy of our estimation methods in both synthetic and practical settings.
In the evolving domain of cryptocurrency markets, accurate token valuation remains a critical aspect influencing investment decisions and policy development. Whilst the prevailing equation of exchange pricing model offers a quantitative valuation approach based on the interplay between token price, transaction volume, supply, and either velocity or holding time, it exhibits intrinsic shortcomings. Specifically, the model may not consistently delineate the relationship between average token velocity and holding time. This paper aims to refine this equation, enhancing the depth of insight into token valuation methodologies.
This study addresses the challenges in parameter estimation of stochastic differential equations driven by non-Gaussian noises, which are critical in understanding dynamic phenomena such as price fluctuations and the spread of infectious diseases. Previous research highlighted the potential of LSTM networks in estimating parameters of alpha stable Levy driven SDEs but faced limitations including high time complexity and constraints of the LSTM chaining property. To mitigate these issues, we introduce the PEnet, a novel CNN-LSTM-based three-stage model that offers an end to end approach with superior accuracy and adaptability to varying data structures, enhanced inference speed for long sequence observations through initial data feature condensation by CNN, and high generalization capability, allowing its application to various complex SDE scenarios. Experiments on synthetic datasets confirm PEnet significant advantage in estimating SDE parameters associated with noise characteristics, establishing it as a competitive method for SDE parameter estimation in the presence of Levy noise.
Human forecasting accuracy in practice relies on the 'wisdom of the crowd' effect, in which predictions about future events are significantly improved by aggregating across a crowd of individual forecasters. Past work on the forecasting ability of large language models (LLMs) suggests that frontier LLMs, as individual forecasters, underperform compared to the gold standard of a human crowd forecasting tournament aggregate. In Study 1, we expand this research by using an LLM ensemble approach consisting of a crowd of twelve LLMs. We compare the aggregated LLM predictions on 31 binary questions to that of a crowd of 925 human forecasters from a three-month forecasting tournament. Our preregistered main analysis shows that the LLM crowd outperforms a simple no-information benchmark and is not statistically different from the human crowd. In exploratory analyses, we find that these two approaches are equivalent with respect to medium-effect-size equivalence bounds. We also observe an acquiescence effect, with mean model predictions being significantly above 50%, despite an almost even split of positive and negative resolutions. Moreover, in Study 2, we test whether LLM predictions (of GPT-4 and Claude 2) can be improved by drawing on human cognitive output. We find that both models' forecasting accuracy benefits from exposure to the median human prediction as information, improving accuracy by between 17% and 28%: though this leads to less accurate predictions than simply averaging human and machine forecasts. Our results suggest that LLMs can achieve forecasting accuracy rivaling that of human crowd forecasting tournaments: via the simple, practically applicable method of forecast aggregation. This replicates the 'wisdom of the crowd' effect for LLMs, and opens up their use for a variety of applications throughout society.
The effectiveness of clopidogrel, a widely used antiplatelet medication, varies significantly among individuals, necessitating the development of precise predictive models to optimize patient care. In this study, we leverage federated learning strategies to address clopidogrel treatment failure detection. Our research harnesses the collaborative power of multiple healthcare institutions, allowing them to jointly train machine learning models while safeguarding sensitive patient data. Utilizing the UK Biobank dataset, which encompasses a vast and diverse population, we partitioned the data based on geographic centers and evaluated the performance of federated learning. Our results show that while centralized training achieves higher Area Under the Curve (AUC) values and faster convergence, federated learning approaches can substantially narrow this performance gap. Our findings underscore the potential of federated learning in addressing clopidogrel treatment failure detection, offering a promising avenue for enhancing patient care through personalized treatment strategies while respecting data privacy. This study contributes to the growing body of research on federated learning in healthcare and lays the groundwork for secure and privacy-preserving predictive models for various medical conditions.
Co-evolving time series appears in a multitude of applications such as environmental monitoring, financial analysis, and smart transportation. This paper aims to address the following challenges, including (C1) how to incorporate explicit relationship networks of the time series; (C2) how to model the implicit relationship of the temporal dynamics. We propose a novel model called Network of Tensor Time Series, which is comprised of two modules, including Tensor Graph Convolutional Network (TGCN) and Tensor Recurrent Neural Network (TRNN). TGCN tackles the first challenge by generalizing Graph Convolutional Network (GCN) for flat graphs to tensor graphs, which captures the synergy between multiple graphs associated with the tensors. TRNN leverages tensor decomposition to model the implicit relationships among co-evolving time series. The experimental results on five real-world datasets demonstrate the efficacy of the proposed method.
Deep neural networks (DNNs) are successful in many computer vision tasks. However, the most accurate DNNs require millions of parameters and operations, making them energy, computation and memory intensive. This impedes the deployment of large DNNs in low-power devices with limited compute resources. Recent research improves DNN models by reducing the memory requirement, energy consumption, and number of operations without significantly decreasing the accuracy. This paper surveys the progress of low-power deep learning and computer vision, specifically in regards to inference, and discusses the methods for compacting and accelerating DNN models. The techniques can be divided into four major categories: (1) parameter quantization and pruning, (2) compressed convolutional filters and matrix factorization, (3) network architecture search, and (4) knowledge distillation. We analyze the accuracy, advantages, disadvantages, and potential solutions to the problems with the techniques in each category. We also discuss new evaluation metrics as a guideline for future research.
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