In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting decay behavior of the observation in time, we establish a novel H{\"o}lder type stability estimate for a large terminal time $T$. This is achieved by novel decay estimates of the (fractional) time derivative of the solution. To numerically recover the diffusion coefficient, we employ the standard output least-squares formulation with an $H^1(\Omega)$-seminorm penalty, and discretize the regularized problem by the Galerkin finite element method with continuous piecewise linear finite elements in space and backward Euler convolution quadrature in time. Further, we provide an error analysis of discrete approximations, and prove a convergence rate that matches the stability estimate. The derived $L^2(\Omega)$ error bound depends explicitly on the noise level, regularization parameter and discretization parameter(s), which gives a useful guideline of the \textsl{a priori} choice of discretization parameters with respect to the noise level in practical implementation. The error analysis is achieved using the conditional stability argument and discrete maximum-norm resolvent estimates. Several numerical experiments are also given to illustrate and complement the theoretical analysis.
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In this work, we propose using mechanistic interpretability -- techniques for reverse engineering model weights into human-interpretable algorithms -- to derive and compactly prove formal guarantees on model performance. We prototype this approach by formally proving lower bounds on the accuracy of 151 small transformers trained on a Max-of-$K$ task. We create 102 different computer-assisted proof strategies and assess their length and tightness of bound on each of our models. Using quantitative metrics, we find that shorter proofs seem to require and provide more mechanistic understanding. Moreover, we find that more faithful mechanistic understanding leads to tighter performance bounds. We confirm these connections by qualitatively examining a subset of our proofs. Finally, we identify compounding structureless noise as a key challenge for using mechanistic interpretability to generate compact proofs on model performance.
This work presents a procedure to solve the Euler equations by explicitly updating, in a conservative manner, a generic thermodynamic variable such as temperature, pressure or entropy instead of the total energy. The presented procedure is valid for any equation of state and spatial discretization. When using complex equations of state such as Span-Wagner, choosing the temperature as the generic thermodynamic variable yields great reductions in the computational costs associated to thermodynamic evaluations. Results computed with a state of the art thermodynamic model are presented, and computational times are analyzed. Particular attention is dedicated to the conservation of total energy, the propagation speed of shock waves and jump conditions. The procedure is thoroughly tested using the Span-Wagner equation of state through the CoolProp thermodynamic library and the Van der Waals equation of state, both in the ideal and non-ideal compressible fluid-dynamics regimes, by comparing it to the standard total energy update and analytical solutions where available.
Dialogue policies play a crucial role in developing task-oriented dialogue systems, yet their development and maintenance are challenging and typically require substantial effort from experts in dialogue modeling. While in many situations, large amounts of conversational data are available for the task at hand, people lack an effective solution able to extract dialogue policies from this data. In this paper, we address this gap by first illustrating how Large Language Models (LLMs) can be instrumental in extracting dialogue policies from datasets, through the conversion of conversations into a unified intermediate representation consisting of canonical forms. We then propose a novel method for generating dialogue policies utilizing a controllable and interpretable graph-based methodology. By combining canonical forms across conversations into a flow network, we find that running graph traversal algorithms helps in extracting dialogue flows. These flows are a better representation of the underlying interactions than flows extracted by prompting LLMs. Our technique focuses on giving conversation designers greater control, offering a productivity tool to improve the process of developing dialogue policies.
In this work, we design and analyze an asymptotic preserving (AP), semi-implicit finite volume scheme for the scaled compressible isentropic Euler system with a singular pressure law known as the congestion pressure law. The congestion pressure law imposes a maximal density constraint of the form $0\leq \varrho <1$, and the scaling introduces a small parameter $\varepsilon$ in order to control the stiffness of the density constraint. As $\varepsilon\to 0$, the solutions of the compressible system converge to solutions of the so-called free-congested Euler equations that couples compressible and incompressible dynamics. We show that the proposed scheme is positivity preserving and energy stable. In addition, we also show that the numerical densities satisfy a discrete variant of the constraint. By means of extensive numerical case studies, we verify the efficacy of the scheme and show that the scheme is able to capture the two dynamics in the limiting regime, thereby proving the AP property.
In practice, users of a Recommender System (RS) fall into a few clusters based on their preferences. In this work, we conduct a systematic study on user-cluster targeted data poisoning attacks on Matrix Factorisation (MF) based RS, where an adversary injects fake users with falsely crafted user-item feedback to promote an item to a specific user cluster. We analyse how user and item feature matrices change after data poisoning attacks and identify the factors that influence the effectiveness of the attack on these feature matrices. We demonstrate that the adversary can easily target specific user clusters with minimal effort and that some items are more susceptible to attacks than others. Our theoretical analysis has been validated by the experimental results obtained from two real-world datasets. Our observations from the study could serve as a motivating point to design a more robust RS.
Software engineering (SE) is a dynamic field that involves multiple phases all of which are necessary to develop sustainable software systems. Machine learning (ML), a branch of artificial intelligence (AI), has drawn a lot of attention in recent years thanks to its ability to analyze massive volumes of data and extract useful patterns from data. Several studies have focused on examining, categorising, and assessing the application of ML in SE processes. We conducted a literature review on primary studies to address this gap. The study was carried out following the objective and the research questions to explore the current state of the art in applying machine learning techniques in software engineering processes. The review identifies the key areas within software engineering where ML has been applied, including software quality assurance, software maintenance, software comprehension, and software documentation. It also highlights the specific ML techniques that have been leveraged in these domains, such as supervised learning, unsupervised learning, and deep learning. Keywords: machine learning, deep learning, software engineering, natural language processing, source code
In this paper, we study the offline RL problem with linear function approximation. Our main structural assumption is that the MDP has low inherent Bellman error, which stipulates that linear value functions have linear Bellman backups with respect to the greedy policy. This assumption is natural in that it is essentially the minimal assumption required for value iteration to succeed. We give a computationally efficient algorithm which succeeds under a single-policy coverage condition on the dataset, namely which outputs a policy whose value is at least that of any policy which is well-covered by the dataset. Even in the setting when the inherent Bellman error is 0 (termed linear Bellman completeness), our algorithm yields the first known guarantee under single-policy coverage. In the setting of positive inherent Bellman error ${\varepsilon_{\mathrm{BE}}} > 0$, we show that the suboptimality error of our algorithm scales with $\sqrt{\varepsilon_{\mathrm{BE}}}$. Furthermore, we prove that the scaling of the suboptimality with $\sqrt{\varepsilon_{\mathrm{BE}}}$ cannot be improved for any algorithm. Our lower bound stands in contrast to many other settings in reinforcement learning with misspecification, where one can typically obtain performance that degrades linearly with the misspecification error.
In this paper, we present a quantum property testing algorithm for recognizing a context-free language that is a concatenation of two palindromes $L_{REV}$. The query complexity of our algorithm is $O(\frac{1}{\varepsilon}n^{1/3}\log n)$, where $n$ is the length of an input. It is better than the classical complexity that is $\Theta^*(\sqrt{n})$. At the same time, in the general setting, the picture is different a little. Classical query complexity is $\Theta(n)$, and quantum query complexity is $\Theta^*(\sqrt{n})$. So, we obtain polynomial speed-up for both cases (general and property testing).
In this paper we consider the problem of obtaining sharp bounds for the performance of temporal difference (TD) methods with linear function approximation for policy evaluation in discounted Markov decision processes. We show that a simple algorithm with a universal and instance-independent step size together with Polyak-Ruppert tail averaging is sufficient to obtain near-optimal variance and bias terms. We also provide the respective sample complexity bounds. Our proof technique is based on refined error bounds for linear stochastic approximation together with the novel stability result for the product of random matrices that arise from the TD-type recurrence.
This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.
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