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We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.

Efficiently representing source code is essential for various software engineering tasks such as code classification and code clone detection. Existing approaches for representing source code primarily use AST, and only a few works focus on semantic graphs such as CFG and PDG, which contain essential information about source code that AST does not have. Even though some works tried to utilize multiple representations, they do not provide any insights about the costs and benefits of using multiple representations against a single appropriate representation for the task. Moreover, they use hand-crafted program features to solve a specific task and have limited use cases. The primary goal of this paper is to discuss the implications of utilizing multiple code representations, specifically AST, CFG, and PDG, and how each of them affects the performance of a task. In this process, we use an approach that can use program features from multiple code graphs while not specifically coupling this approach to a specific task or a language. Our approach stems from the idea of modeling AST as a set of paths and using a learning model to capture program properties. We modify an existing AST path-based approach to accept multiple code representations as input. We do this since it allows us to measure the performance boost provided by additional representations over AST. We evaluate our approach on three tasks: Method Naming, Program Classification, and Code Clone Detection. Our approach increases the performance on these three tasks by 11% (F1), 15.7% (Accuracy), and 9.3% (F1), respectively, over the baseline. We discuss the impact of semantic features from the CFG and PDG paths on performance and the additional overheads incurred through our approach. We envision this work providing researchers with a lens to evaluate combinations of source code representations for various tasks.

The concept of extension-based proofs models the idea of a valency argument which is widely used in distributed computing. Extension-based proofs are limited in power: it has been shown that there is no extension-based proof of the impossibility of a wait-free protocol for $(n,k)$-set agreement among $n > k \geq 2$ processes. A discussion of a restricted type of reduction has shown that there are no extension-based proofs of the impossibility of wait-free protocols for some other distributed computing problems. We extend the previous result to general reductions that allow multiple instances of tasks. The techniques used in the previous work are designed for certain tasks, such as the $(n,k)$-set agreement task. We give a necessary and sufficient condition for general colorless tasks to have no extension-based proofs of the impossibility of wait-free protocols, and show that different types of extension-based proof are equivalent in power for colorless tasks. Using this necessary and sufficient condition, the result about reductions can be understood from a topological perspective.

We sometimes need to compute the most significant digits of the product of small integers with a multiplier requiring much storage: e.g., a large integer (e.g., $5^{100}$) or an irrational number ($\pi$). We only need to access the most significant digits of the multiplier-as long as the integers are sufficiently small. We provide an efficient algorithm to compute the range of integers given a truncated multiplier and a desired number of digits.

In this paper we study the finite sample and asymptotic properties of various weighting estimators of the local average treatment effect (LATE), several of which are based on Abadie's (2003) kappa theorem. Our framework presumes a binary treatment and a binary instrument, which may only be valid after conditioning on additional covariates. We argue that one of the Abadie estimators, which is weight normalized, is preferable in many contexts. Several other estimators, which are unnormalized, do not generally satisfy the properties of scale invariance with respect to the natural logarithm and translation invariance, thereby exhibiting sensitivity to the units of measurement when estimating the LATE in logs and the centering of the outcome variable more generally. On the other hand, when noncompliance is one-sided, certain unnormalized estimators have the advantage of being based on a denominator that is bounded away from zero. To reconcile these findings, we demonstrate that when the instrument propensity score is estimated using an appropriate covariate balancing approach, the resulting normalized estimator also shares this advantage. We use a simulation study and three empirical applications to illustrate our findings. In two cases, the unnormalized estimates are clearly unreasonable, with "incorrect" signs, magnitudes, or both.

During an infectious disease outbreak, public health decision-makers require real-time monitoring of disease transmission to respond quickly and intelligently. In these settings, a key measure of transmission is the instantaneous time-varying reproduction number, $R_t$. Estimation of this number using a Time-Since-Infection model relies on case-notification data and the distribution of the serial interval on the target population. However, in practice, case-notification data may contain measurement error due to variation in case reporting while available serial interval estimates may come from studies on non-representative populations. We propose a new data-driven method that accounts for particular forms of case-reporting measurement error and can incorporate multiple partially representative serial interval estimates into the transmission estimation process. In addition, we provide practical tools for automatically identifying measurement error patterns and determining when measurement error may not be adequately accounted for. We illustrate the potential bias undertaken by methods that ignore these practical concerns through a variety of simulated outbreaks. We then demonstrate the use of our method on data from the COVID-19 pandemic to estimate transmission and explore the relationships between social distancing, temperature, and transmission.

Empirical studies of the loss landscape of deep networks have revealed that many local minima are connected through low-loss valleys. Yet, little is known about the theoretical origin of such valleys. We present a general framework for finding continuous symmetries in the parameter space, which carve out low-loss valleys. Our framework uses equivariances of the activation functions and can be applied to different layer architectures. To generalize this framework to nonlinear neural networks, we introduce a novel set of nonlinear, data-dependent symmetries. These symmetries can transform a trained model such that it performs similarly on new samples, which allows ensemble building that improves robustness under certain adversarial attacks. We then show that conserved quantities associated with linear symmetries can be used to define coordinates along low-loss valleys. The conserved quantities help reveal that using common initialization methods, gradient flow only explores a small part of the global minimum. By relating conserved quantities to convergence rate and sharpness of the minimum, we provide insights on how initialization impacts convergence and generalizability.

Often in Software Engineering, a modeling formalism has to support scenarios of inconsistency in which several requirements either reinforce or contradict each other. Paraconsistent transition systems are proposed in this paper as one such formalism: states evolve through two accessibility relations capturing weighted evidence of a transition or its absence, respectively. Their weights come from a specific residuated lattice. A category of these systems, and the corresponding algebra, is defined as providing a formal setting to model different application scenarios. One of them, dealing with the effect of quantum decoherence in quantum programs, is used for illustration purposes.

The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example being instrumental variable regression. A standard approach reduces the problem to a finite set of marginal moment conditions and applies the optimally weighted generalized method of moments (OWGMM), but this requires we know a finite set of identifying moments, can still be inefficient even if identifying, or can be theoretically efficient but practically unwieldy if we use a growing sieve of moment conditions. Motivated by a variational minimax reformulation of OWGMM, we define a very general class of estimators for the conditional moment problem, which we term the variational method of moments (VMM) and which naturally enables controlling infinitely-many moments. We provide a detailed theoretical analysis of multiple VMM estimators, including ones based on kernel methods and neural nets, and provide conditions under which these are consistent, asymptotically normal, and semiparametrically efficient in the full conditional moment model. We additionally provide algorithms for valid statistical inference based on the same kind of variational reformulations, both for kernel- and neural-net-based varieties. Finally, we demonstrate the strong performance of our proposed estimation and inference algorithms in a detailed series of synthetic experiments.

Algorithm aversion occurs when humans are reluctant to use algorithms despite their superior performance. Studies show that giving users outcome control by providing agency over how models' predictions are incorporated into decision-making mitigates algorithm aversion. We study whether algorithm aversion is mitigated by process control, wherein users can decide what input factors and algorithms to use in model training. We conduct a replication study of outcome control, and test novel process control study conditions on Amazon Mechanical Turk (MTurk) and Prolific. Our results partly confirm prior findings on the mitigating effects of outcome control, while also forefronting reproducibility challenges. We find that process control in the form of choosing the training algorithm mitigates algorithm aversion, but changing inputs does not. Furthermore, giving users both outcome and process control does not reduce algorithm aversion more than outcome or process control alone. This study contributes to design considerations around mitigating algorithm aversion.