Given a data stream $\mathcal{A} = \langle a_1, a_2, \ldots, a_m \rangle$ of $m$ elements where each $a_i \in [n]$, the Distinct Elements problem is to estimate the number of distinct elements in $\mathcal{A}$.Distinct Elements has been a subject of theoretical and empirical investigations over the past four decades resulting in space optimal algorithms for it.All the current state-of-the-art algorithms are, however, beyond the reach of an undergraduate textbook owing to their reliance on the usage of notions such as pairwise independence and universal hash functions. We present a simple, intuitive, sampling-based space-efficient algorithm whose description and the proof are accessible to undergraduates with the knowledge of basic probability theory.
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The abundance of observed data in recent years has increased the number of statistical augmentations to complex models across science and engineering. By augmentation we mean coherent statistical methods that incorporate measurements upon arrival and adjust the model accordingly. However, in this research area methodological developments tend to be central, with important assessments of model fidelity often taking second place. Recently, the statistical finite element method (statFEM) has been posited as a potential solution to the problem of model misspecification when the data are believed to be generated from an underlying partial differential equation system. Bayes nonlinear filtering permits data driven finite element discretised solutions that are updated to give a posterior distribution which quantifies the uncertainty over model solutions. The statFEM has shown great promise in systems subject to mild misspecification but its ability to handle scenarios of severe model misspecification has not yet been presented. In this paper we fill this gap, studying statFEM in the context of shallow water equations chosen for their oceanographic relevance. By deliberately misspecifying the governing equations, via linearisation, viscosity, and bathymetry, we systematically analyse misspecification through studying how the resultant approximate posterior distribution is affected, under additional regimes of decreasing spatiotemporal observational frequency. Results show that statFEM performs well with reasonable accuracy, as measured by theoretically sound proper scoring rules.
In this paper, we provide a novel framework for the analysis of generalization error of first-order optimization algorithms for statistical learning when the gradient can only be accessed through partial observations given by an oracle. Our analysis relies on the regularity of the gradient w.r.t. the data samples, and allows to derive near matching upper and lower bounds for the generalization error of multiple learning problems, including supervised learning, transfer learning, robust learning, distributed learning and communication efficient learning using gradient quantization. These results hold for smooth and strongly-convex optimization problems, as well as smooth non-convex optimization problems verifying a Polyak-Lojasiewicz assumption. In particular, our upper and lower bounds depend on a novel quantity that extends the notion of conditional standard deviation, and is a measure of the extent to which the gradient can be approximated by having access to the oracle. As a consequence, our analysis provides a precise meaning to the intuition that optimization of the statistical learning objective is as hard as the estimation of its gradient. Finally, we show that, in the case of standard supervised learning, mini-batch gradient descent with increasing batch sizes and a warm start can reach a generalization error that is optimal up to a multiplicative factor, thus motivating the use of this optimization scheme in practical applications.
We consider the problem of Imitation Learning (IL) by actively querying noisy expert for feedback. While imitation learning has been empirically successful, much of prior work assumes access to noiseless expert feedback which is not practical in many applications. In fact, when one only has access to noisy expert feedback, algorithms that rely on purely offline data (non-interactive IL) can be shown to need a prohibitively large number of samples to be successful. In contrast, in this work, we provide an interactive algorithm for IL that uses selective sampling to actively query the noisy expert for feedback. Our contributions are twofold: First, we provide a new selective sampling algorithm that works with general function classes and multiple actions, and obtains the best-known bounds for the regret and the number of queries. Next, we extend this analysis to the problem of IL with noisy expert feedback and provide a new IL algorithm that makes limited queries. Our algorithm for selective sampling leverages function approximation, and relies on an online regression oracle w.r.t.~the given model class to predict actions, and to decide whether to query the expert for its label. On the theoretical side, the regret bound of our algorithm is upper bounded by the regret of the online regression oracle, while the query complexity additionally depends on the eluder dimension of the model class. We complement this with a lower bound that demonstrates that our results are tight. We extend our selective sampling algorithm for IL with general function approximation and provide bounds on both the regret and the number of queries made to the noisy expert. A key novelty here is that our regret and query complexity bounds only depend on the number of times the optimal policy (and not the noisy expert, or the learner) go to states that have a small margin.
The two-trials rule for drug approval requires "at least two adequate and well-controlled studies, each convincing on its own, to establish effectiveness". This is usually employed by requiring two significant pivotal trials and is the standard regulatory requirement to provide evidence for a new drug's efficacy. However, there is need to develop suitable alternatives to this rule for a number of reasons, including the possible availability of data from more than two trials. I consider the case of up to 3 studies and stress the importance to control the partial Type-I error rate, where only some studies have a true null effect, while maintaining the overall Type-I error rate of the two-trials rule, where all studies have a null effect. Some less-known $p$-value combination methods are useful to achieve this: Pearson's method, Edgington's method and the recently proposed harmonic mean $\chi^2$-test. I study their properties and discuss how they can be extended to a sequential assessment of success while still ensuring overall Type-I error control. I compare the different methods in terms of partial Type-I error rate, project power and the expected number of studies required. Edgington's method is eventually recommended as it is easy to implement and communicate, has only moderate partial Type-I error rate inflation but substantially increased project power.
We study the complexity of producing $(\delta,\epsilon)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $\Omega(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(d\delta^{-1}\epsilon^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $\delta,\epsilon$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful geometric lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex optimization.
Electroencephalogram (EEG) signals reflect brain activity across different brain states, characterized by distinct frequency distributions. Through multifractal analysis tools, we investigate the scaling behaviour of different classes of EEG signals and artifacts. We show that brain states associated to sleep and general anaesthesia are not in general characterized by scale invariance. The lack of scale invariance motivates the development of artifact removal algorithms capable of operating independently at each scale. We examine here the properties of the wavelet quantile normalization algorithm, a recently introduced adaptive method for real-time correction of transient artifacts in EEG signals. We establish general results regarding the regularization properties of the WQN algorithm, showing how it can eliminate singularities introduced by artefacts, and we compare it to traditional thresholding algorithms. Furthermore, we show that the algorithm performance is independent of the wavelet basis. We finally examine its continuity and boundedness properties and illustrate its distinctive non-local action on the wavelet coefficients through pathological examples.
The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference that govern the use of the connective. However, what if we go a step further and ask about the meaning of a proof as a whole? In this paper we address this question and lay out a framework to distinguish sense and denotation of proofs. Two questions are central here. First of all, if we have two (syntactically) different derivations, does this always lead to a difference, firstly, in sense, and secondly, in denotation? The other question is about the relation between different kinds of proof systems (here: natural deduction vs. sequent calculi) with respect to this distinction. Do the different forms of representing a proof necessarily correspond to a difference in how the inferential steps are given? In our framework it will be possible to identify denotation as well as sense of proofs not only within one proof system but also between different kinds of proof systems. Thus, we give an account to distinguish a mere syntactic divergence from a divergence in meaning and a divergence in meaning from a divergence of proof objects analogous to Frege's distinction for singular terms and sentences.
Objectives: Present a novel deep learning-based skull stripping algorithm for magnetic resonance imaging (MRI) that works directly in the information rich k-space. Materials and Methods: Using two datasets from different institutions with a total of 36,900 MRI slices, we trained a deep learning-based model to work directly with the complex raw k-space data. Skull stripping performed by HD-BET (Brain Extraction Tool) in the image domain were used as the ground truth. Results: Both datasets were very similar to the ground truth (DICE scores of 92\%-98\% and Hausdorff distances of under 5.5 mm). Results on slices above the eye-region reach DICE scores of up to 99\%, while the accuracy drops in regions around the eyes and below, with partially blurred output. The output of k-strip often smoothed edges at the demarcation to the skull. Binary masks are created with an appropriate threshold. Conclusion: With this proof-of-concept study, we were able to show the feasibility of working in the k-space frequency domain, preserving phase information, with consistent results. Future research should be dedicated to discovering additional ways the k-space can be used for innovative image analysis and further workflows.
Variational quantum algorithms (VQAs) prevail to solve practical problems such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. For variational quantum machine learning, a variational algorithm with model interpretability built into the algorithm is yet to be exploited. In this paper, we construct a quantum regression algorithm and identify the direct relation of variational parameters to learned regression coefficients, while employing a circuit that directly encodes the data in quantum amplitudes reflecting the structure of the classical data table. The algorithm is particularly suitable for well-connected qubits. With compressed encoding and digital-analog gate operation, the run time complexity is logarithmically more advantageous than that for digital 2-local gate native hardware with the number of data entries encoded, a decent improvement in noisy intermediate-scale quantum computers and a minor improvement for large-scale quantum computing Our suggested method of compressed binary encoding offers a remarkable reduction in the number of physical qubits needed when compared to the traditional one-hot-encoding technique with the same input data. The algorithm inherently performs linear regression but can also be used easily for nonlinear regression by building nonlinear features into the training data. In terms of measured cost function which distinguishes a good model from a poor one for model training, it will be effective only when the number of features is much less than the number of records for the encoded data structure to be observable. To echo this finding and mitigate hardware noise in practice, the ensemble model training from the quantum regression model learning with important feature selection from regularization is incorporated and illustrated numerically.
Car pooling is expected to significantly help in reducing traffic congestion and pollution in cities by enabling drivers to share their cars with travellers with similar itineraries and time schedules. A number of car pooling matching services have been designed in order to efficiently find successful ride matches in a given pool of drivers and potential passengers. However, it is now recognised that many non-monetary aspects and social considerations, besides simple mobility needs, may influence the individual willingness of sharing a ride, which are difficult to predict. To address this problem, in this study we propose GoTogether, a recommender system for car pooling services that leverages on learning-to-rank techniques to automatically derive the personalised ranking model of each user from the history of her choices (i.e., the type of accepted or rejected shared rides). Then, GoTogether builds the list of recommended rides in order to maximise the success rate of the offered matches. To test the performance of our scheme we use real data from Twitter and Foursquare sources in order to generate a dataset of plausible mobility patterns and ride requests in a metropolitan area. The results show that the proposed solution quickly obtain an accurate prediction of the personalised user's choice model both in static and dynamic conditions.
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