Quantum reinforcement learning (QRL) has emerged as a framework to solve sequential decision-making tasks, showcasing empirical quantum advantages. A notable development is through quantum recurrent neural networks (QRNNs) for memory-intensive tasks such as partially observable environments. However, QRL models incorporating QRNN encounter challenges such as inefficient training of QRL with QRNN, given that the computation of gradients in QRNN is both computationally expensive and time-consuming. This work presents a novel approach to address this challenge by constructing QRL agents utilizing QRNN-based reservoirs, specifically employing quantum long short-term memory (QLSTM). QLSTM parameters are randomly initialized and fixed without training. The model is trained using the asynchronous advantage actor-aritic (A3C) algorithm. Through numerical simulations, we validate the efficacy of our QLSTM-Reservoir RL framework. Its performance is assessed on standard benchmarks, demonstrating comparable results to a fully trained QLSTM RL model with identical architecture and training settings.
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Student simulation presents a transformative approach to enhance learning outcomes, advance educational research, and ultimately shape the future of effective pedagogy. We explore the feasibility of using large language models (LLMs), a remarkable achievement in AI, to simulate student learning behaviors. Unlike conventional machine learning based prediction, we leverage LLMs to instantiate virtual students with specific demographics and uncover intricate correlations among learning experiences, course materials, understanding levels, and engagement. Our objective is not merely to predict learning outcomes but to replicate learning behaviors and patterns of real students. We validate this hypothesis through three experiments. The first experiment, based on a dataset of N = 145, simulates student learning outcomes from demographic data, revealing parallels with actual students concerning various demographic factors. The second experiment (N = 4524) results in increasingly realistic simulated behaviors with more assessment history for virtual students modelling. The third experiment (N = 27), incorporating prior knowledge and course interactions, indicates a strong link between virtual students' learning behaviors and fine-grained mappings from test questions, course materials, engagement and understanding levels. Collectively, these findings deepen our understanding of LLMs and demonstrate its viability for student simulation, empowering more adaptable curricula design to enhance inclusivity and educational effectiveness.
The rising popularity of deep learning (DL) methods and techniques has invigorated interest in the topic of SE4DL, the application of software engineering (SE) practices on deep learning software. Despite the novel engineering challenges brought on by the data-driven and non-deterministic paradigm of DL software, little work has been invested into developing AI-targeted SE tools. On the other hand, tools tackling more general engineering issues in DL are actively used and referred to under the umbrella term of ``MLOps tools''. Furthermore, the available literature supports the utility of conventional SE tooling in DL software development. Building upon previous MSR research on tool usage in open-source software works, we identify conventional and MLOps tools adopted in popular applied DL projects that use Python as the main programming language. About 70% of the GitHub repositories mined contained at least one conventional SE tool. Software configuration management tools are the most adopted, while the opposite applies to maintenance tools. Substantially fewer MLOps tools were in use, with only 9 tools out of a sample of 80 used in at least one repository. The majority of them were open-source rather than proprietary. One of these tools, TensorBoard, was found to be adopted in about half of the repositories in our study. Consequently, the use of conventional SE tooling demonstrates its relevance to DL software. Further research is recommended on the adoption of MLOps tooling by open-source projects, focusing on the relevance of particular tool types, the development of required tools, as well as ways to promote the use of already available tools.
This work presents a comparative study to numerically compute impulse approximate controls for parabolic equations with various boundary conditions. Theoretical controllability results have been recently investigated using a logarithmic convexity estimate at a single time based on a Carleman commutator approach. We propose a numerical algorithm for computing the impulse controls with minimal $L^2$-norms by adapting a penalized Hilbert Uniqueness Method (HUM) combined with a Conjugate Gradient (CG) method. We consider static boundary conditions (Dirichlet and Neumann) and dynamic boundary conditions. Some numerical experiments based on our developed algorithm are given to validate and compare the theoretical impulse controllability results.
In an era where scientific experiments can be very costly, multi-fidelity emulators provide a useful tool for cost-efficient predictive scientific computing. For scientific applications, the experimenter is often limited by a tight computational budget, and thus wishes to (i) maximize predictive power of the multi-fidelity emulator via a careful design of experiments, and (ii) ensure this model achieves a desired error tolerance with some notion of confidence. Existing design methods, however, do not jointly tackle objectives (i) and (ii). We propose a novel stacking design approach that addresses both goals. A multi-level reproducing kernel Hilbert space (RKHS) interpolator is first introduced to build the emulator, under which our stacking design provides a sequential approach for designing multi-fidelity runs such that a desired prediction error of $\epsilon > 0$ is met under regularity assumptions. We then prove a novel cost complexity theorem that, under this multi-level interpolator, establishes a bound on the computation cost (for training data simulation) needed to achieve a prediction bound of $\epsilon$. This result provides novel insights on conditions under which the proposed multi-fidelity approach improves upon a conventional RKHS interpolator which relies on a single fidelity level. Finally, we demonstrate the effectiveness of stacking designs in a suite of simulation experiments and an application to finite element analysis.
Bilevel learning is a powerful optimization technique that has extensively been employed in recent years to bridge the world of model-driven variational approaches with data-driven methods. Upon suitable parametrization of the desired quantities of interest (e.g., regularization terms or discretization filters), such approach computes optimal parameter values by solving a nested optimization problem where the variational model acts as a constraint. In this work, we consider two different use cases of bilevel learning for the problem of image restoration. First, we focus on learning scalar weights and convolutional filters defining a Field of Experts regularizer to restore natural images degraded by blur and noise. For improving the practical performance, the lower-level problem is solved by means of a gradient descent scheme combined with a line-search strategy based on the Barzilai-Borwein rule. As a second application, the bilevel setup is employed for learning a discretization of the popular total variation regularizer for solving image restoration problems (in particular, deblurring and super-resolution). Numerical results show the effectiveness of the approach and their generalization to multiple tasks.
Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks. We consider the transductive setting, where decisions are made on a test sample of $m$ new points, giving rise to $m$ conformal $p$-values. {While classical results only concern their marginal distribution, we show that their joint distribution follows a P\'olya urn model, and establish a concentration inequality for their empirical distribution function.} The results hold for arbitrary exchangeable scores, including {\it adaptive} ones that can use the covariates of the test+calibration samples at training stage for increased accuracy. We demonstrate the usefulness of these theoretical results through uniform, in-probability guarantees for two machine learning tasks of current interest: interval prediction for transductive transfer learning and novelty detection based on two-class classification.
In many applications of machine learning, a large number of variables are considered. Motivated by machine learning of interacting particle systems, we consider the situation when the number of input variables goes to infinity. First, we continue the recent investigation of the mean field limit of kernels and their reproducing kernel Hilbert spaces, completing the existing theory. Next, we provide results relevant for approximation with such kernels in the mean field limit, including a representer theorem. Finally, we use these kernels in the context of statistical learning in the mean field limit, focusing on Support Vector Machines. In particular, we show mean field convergence of empirical and infinite-sample solutions as well as the convergence of the corresponding risks. On the one hand, our results establish rigorous mean field limits in the context of kernel methods, providing new theoretical tools and insights for large-scale problems. On the other hand, our setting corresponds to a new form of limit of learning problems, which seems to have not been investigated yet in the statistical learning theory literature.
The fundamental computational issues in Bayesian inverse problems (BIPs) governed by partial differential equations (PDEs) stem from the requirement of repeated forward model evaluations. A popular strategy to reduce such cost is to replace expensive model simulations by computationally efficient approximations using operator learning, motivated by recent progresses in deep learning. However, using the approximated model directly may introduce a modeling error, exacerbating the already ill-posedness of inverse problems. Thus, balancing between accuracy and efficiency is essential for the effective implementation of such approaches. To this end, we develop an adaptive operator learning framework that can reduce modeling error gradually by forcing the surrogate to be accurate in local areas. This is accomplished by fine-tuning the pre-trained approximate model during the inversion process with adaptive points selected by a greedy algorithm, which requires only a few forward model evaluations. To validate our approach, we adopt DeepOnet to construct the surrogate and use unscented Kalman inversion (UKI) to approximate the solution of BIPs, respectively. Furthermore, we present rigorous convergence guarantee in the linear case using the framework of UKI. We test the approach on several benchmarks, including the Darcy flow, the heat source inversion problem, and the reaction diffusion problems. Numerical results demonstrate that our method can significantly reduce computational costs while maintaining inversion accuracy.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.
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