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Can the $\lambda$-calculus be considered a reasonable computational model? Can we use it for measuring the time $\textit{and}$ space consumption of algorithms? While the literature contains positive answers about time, much less is known about space. This paper presents a new reasonable space cost model for the $\lambda$-calculus, based on a variant over the Krivine abstract machine. For the first time, this cost model is able to accommodate logarithmic space. Moreover, we study the time behavior of our machine and show how to transport our results to the call-by-value $\lambda$-calculus.

The adversarial vulnerability of neural nets, and subsequent techniques to create robust models have attracted significant attention; yet we still lack a full understanding of this phenomenon. Here, we study adversarial examples of trained neural networks through analytical tools afforded by recent theory advances connecting neural networks and kernel methods, namely the Neural Tangent Kernel (NTK), following a growing body of work that leverages the NTK approximation to successfully analyze important deep learning phenomena and design algorithms for new applications. We show how NTKs allow to generate adversarial examples in a ``training-free'' fashion, and demonstrate that they transfer to fool their finite-width neural net counterparts in the ``lazy'' regime. We leverage this connection to provide an alternative view on robust and non-robust features, which have been suggested to underlie the adversarial brittleness of neural nets. Specifically, we define and study features induced by the eigendecomposition of the kernel to better understand the role of robust and non-robust features, the reliance on both for standard classification and the robustness-accuracy trade-off. We find that such features are surprisingly consistent across architectures, and that robust features tend to correspond to the largest eigenvalues of the model, and thus are learned early during training. Our framework allows us to identify and visualize non-robust yet useful features. Finally, we shed light on the robustness mechanism underlying adversarial training of neural nets used in practice: quantifying the evolution of the associated empirical NTK, we demonstrate that its dynamics falls much earlier into the ``lazy'' regime and manifests a much stronger form of the well known bias to prioritize learning features within the top eigenspaces of the kernel, compared to standard training.

Industrial Control Systems (ICS) are often built from geographically distributed components and often use programmable logic controllers for localized processes. Since verification of such systems is challenging because of both time sensitivity of the system specifications and the inherent asynchrony in distributed components, developing runtime assurance that verifies not just the correctness of different components, but also generates aggregated statistics of the systems is of interest. In this paper, we first present a general technique for runtime monitoring of distributed applications whose behavior can be modeled as input/output {\em streams} with an internal computation module in the partially synchronous semantics, where an imperfect clock synchronization algorithm is assumed. Second, we propose a generalized stream-based decentralized runtime verification technique. We also rigorously evaluate our algorithm on extensive synthetic experiments and several ICS and aircraft SBS message datasets.

This work proposes a novel perspective on adversarial attacks by introducing the concept of sample attackability and robustness. Adversarial attacks insert small, imperceptible perturbations to the input that cause large, undesired changes to the output of deep learning models. Despite extensive research on generating adversarial attacks and building defense systems, there has been limited research on understanding adversarial attacks from an input-data perspective. We propose a deep-learning-based method for detecting the most attackable and robust samples in an unseen dataset for an unseen target model. The proposed method is based on a neural network architecture that takes as input a sample and outputs a measure of attackability or robustness. The proposed method is evaluated using a range of different models and different attack methods, and the results demonstrate its effectiveness in detecting the samples that are most likely to be affected by adversarial attacks. Understanding sample attackability can have important implications for future work in sample-selection tasks. For example in active learning, the acquisition function can be designed to select the most attackable samples, or in adversarial training, only the most attackable samples are selected for augmentation.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

In this article, we propose a one-sample test to check whether the support of the unknown distribution generating the data is homologically equivalent to the support of some specified distribution or not OR using the corresponding two-sample test, one can test whether the supports of two unknown distributions are homologically equivalent or not. In the course of this study, test statistics based on the Betti numbers are formulated, and the consistency of the tests is established under the critical and the supercritical regimes. Moreover, some simulation studies are conducted and results are compared with the existing methodologies such as Robinson's permutation test and test based on mean persistent landscape functions. Furthermore, the practicability of the tests is shown on two well-known real data sets also.

We present a novel method for the safety verification of nonlinear dynamical models that uses neural networks to represent abstractions of their dynamics. Neural networks have extensively been used before as approximators; in this work, we make a step further and use them for the first time as abstractions. For a given dynamical model, our method synthesises a neural network that overapproximates its dynamics by ensuring an arbitrarily tight, formally certified bound on the approximation error. For this purpose, we employ a counterexample-guided inductive synthesis procedure. We show that this produces a neural ODE with non-deterministic disturbances that constitutes a formal abstraction of the concrete model under analysis. This guarantees a fundamental property: if the abstract model is safe, i.e., free from any initialised trajectory that reaches an undesirable state, then the concrete model is also safe. By using neural ODEs with ReLU activation functions as abstractions, we cast the safety verification problem for nonlinear dynamical models into that of hybrid automata with affine dynamics, which we verify using SpaceEx. We demonstrate that our approach performs comparably to the mature tool Flow* on existing benchmark nonlinear models. We additionally demonstrate and that it is effective on models that do not exhibit local Lipschitz continuity, which are out of reach to the existing technologies.

We study deviation of U-statistics when samples have heavy-tailed distribution so the kernel of the U-statistic does not have bounded exponential moments at any positive point. We obtain an exponential upper bound for the tail of the U-statistics which clearly denotes two regions of tail decay, the first is a Gaussian decay and the second behaves like the tail of the kernel. For several common U-statistics, we also show the upper bound has the right rate of decay as well as sharp constants by obtaining rough logarithmic limits which in turn can be used to develop LDP for U-statistics. In spite of usual LDP results in the literature, processes we consider in this work have LDP speed slower than their sample size $n$.

We consider a multi-agent multi-armed bandit setting in which $n$ honest agents collaborate over a network to minimize regret but $m$ malicious agents can disrupt learning arbitrarily. Assuming the network is the complete graph, existing algorithms incur $O( (m + K/n) \log (T) / \Delta )$ regret in this setting, where $K$ is the number of arms and $\Delta$ is the arm gap. For $m \ll K$, this improves over the single-agent baseline regret of $O(K\log(T)/\Delta)$. In this work, we show the situation is murkier beyond the case of a complete graph. In particular, we prove that if the state-of-the-art algorithm is used on the undirected line graph, honest agents can suffer (nearly) linear regret until time is doubly exponential in $K$ and $n$. In light of this negative result, we propose a new algorithm for which the $i$-th agent has regret $O( ( d_{\text{mal}}(i) + K/n) \log(T)/\Delta)$ on any connected and undirected graph, where $d_{\text{mal}}(i)$ is the number of $i$'s neighbors who are malicious. Thus, we generalize existing regret bounds beyond the complete graph (where $d_{\text{mal}}(i) = m$), and show the effect of malicious agents is entirely local (in the sense that only the $d_{\text{mal}}(i)$ malicious agents directly connected to $i$ affect its long-term regret).

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.