We consider a first-order logic for the integers with addition. This logic extends classical first-order logic by modulo-counting, threshold-counting and exact-counting quantifiers, all applied to tuples of variables (here, residues are given as terms while moduli and thresholds are given explicitly). Our main result shows that satisfaction for this logic is decidable in two-fold exponential space. If only threshold- and exact-counting quantifiers are allowed, we prove an upper bound of alternating two-fold exponential time with linearly many alternations. This latter result almost matches Berman's exact complexity of first-order logic without counting quantifiers. To obtain these results, we first translate threshold- and exact-counting quantifiers into classical first-order logic in polynomial time (which already proves the second result). To handle the remaining modulo-counting quantifiers for tuples, we first reduce them in doubly exponential time to modulo-counting quantifiers for single elements. For these quantifiers, we provide a quantifier elimination procedure similar to Reddy and Loveland's procedure for first-order logic and analyse the growth of coefficients, constants, and moduli appearing in this process. The bounds obtained this way allow to restrict quantification in the original formula to integers of bounded size which then implies the first result mentioned above. Our logic is incomparable with the logic considered by Chistikov et al. in 2022. They allow more general counting operations in quantifiers, but only unary quantifiers. The move from unary to non-unary quantifiers is non-trivial, since, e.g., the non-unary version of the H\"artig quantifier results in an undecidable theory.
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We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$. This problem is conjectured to have a sharp feasibility transition: for any $\varepsilon > 0$, if $n \leq (1 - \varepsilon) d^2 / 4$ then $(\mathrm{P})$ has a solution with high probability, while $(\mathrm{P})$ has no solutions with high probability if $n \geq (1 + \varepsilon) d^2 /4$. So far, only a trivial bound $n \geq d^2 / 2$ is known on the negative side, while the best results on the positive side assume $n \leq d^2 / \mathrm{polylog}(d)$. In this work, we improve over previous approaches using a key result of Bartl & Mendelson on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior. This allows us to give a simple proof that $(\mathrm{P})$ is feasible with high probability when $n \leq d^2 / C$, for a (possibly large) constant $C > 0$.
Conformal inference provides a general distribution-free method to rigorously calibrate the output of any machine learning algorithm for novelty detection. While this approach has many strengths, it has the limitation of being randomized, in the sense that it may lead to different results when analyzing twice the same data, and this can hinder the interpretation of any findings. We propose to make conformal inferences more stable by leveraging suitable conformal e-values instead of p-values to quantify statistical significance. This solution allows the evidence gathered from multiple analyses of the same data to be aggregated effectively while provably controlling the false discovery rate. Further, we show that the proposed method can reduce randomness without much loss of power compared to standard conformal inference, partly thanks to an innovative way of weighting conformal e-values based on additional side information carefully extracted from the same data. Simulations with synthetic and real data confirm this solution can be effective at eliminating random noise in the inferences obtained with state-of-the-art alternative techniques, sometimes also leading to higher power.
We investigate online classification with paid stochastic experts. Here, before making their prediction, each expert must be paid. The amount that we pay each expert directly influences the accuracy of their prediction through some unknown Lipschitz "productivity" function. In each round, the learner must decide how much to pay each expert and then make a prediction. They incur a cost equal to a weighted sum of the prediction error and upfront payments for all experts. We introduce an online learning algorithm whose total cost after $T$ rounds exceeds that of a predictor which knows the productivity of all experts in advance by at most $\mathcal{O}(K^2(\log T)\sqrt{T})$ where $K$ is the number of experts. In order to achieve this result, we combine Lipschitz bandits and online classification with surrogate losses. These tools allow us to improve upon the bound of order $T^{2/3}$ one would obtain in the standard Lipschitz bandit setting. Our algorithm is empirically evaluated on synthetic data
We study first-order logic over unordered structures whose elements carry a finite number of data values from an infinite domain. Data values can be compared wrt.\ equality. As the satisfiability problem for this logic is undecidable in general, we introduce a family of local fragments. They restrict quantification to the neighbourhood of a given reference point that is bounded by some radius. Our first main result establishes decidability of the satisfiability problem for the local radius-1 fragment in presence of one ``diagonal relation''. On the other hand, extending the radius leads to undecidability. In a second part, we provide the precise decidability and complexity landscape of the satisfiability problem for the existential fragments of local logic, which are parameterized by the number of data values carried by each element and the radius of the considered neighbourhoods. Altogether, we draw a landscape of formalisms that are suitable for the specification of systems with data and open up new avenues for future research.
Time-dependent scheduling with linear deterioration involves determining when to execute jobs whose processing times degrade as their beginning is delayed. Each job i is associated with a release time r_i and a processing time function p_i(s_i)=alpha_i + beta_i*s_i, where alpha_i, beta_i>0$ are constants and s_i is the job's start time. In this setting, the approximability of both single-machine minimum makespan and total completion time problems remains open. Here, we take a step forward by developing new bounds and approximation results for the interesting special case of the problems with uniform deterioration, i.e.\ beta_i=beta, for each i. The key contribution is a O(1+1/beta)-approximation algorithm for the makespan problem and a O(1+1/beta^2)-approximation algorithm for the total completion time problem. Further, we propose greedy constant-factor approximation algorithms for instances with beta=O(1/n) and beta=Omega(n), where n is the number of jobs. Our analysis is based on a new approach for comparing computed and optimal schedules via bounding pseudomatchings.
Federated Learning (FL) enables multiple clients to collaboratively learn a machine learning model without exchanging their own local data. In this way, the server can exploit the computational power of all clients and train the model on a larger set of data samples among all clients. Although such a mechanism is proven to be effective in various fields, existing works generally assume that each client preserves sufficient data for training. In practice, however, certain clients may only contain a limited number of samples (i.e., few-shot samples). For example, the available photo data taken by a specific user with a new mobile device is relatively rare. In this scenario, existing FL efforts typically encounter a significant performance drop on these clients. Therefore, it is urgent to develop a few-shot model that can generalize to clients with limited data under the FL scenario. In this paper, we refer to this novel problem as federated few-shot learning. Nevertheless, the problem remains challenging due to two major reasons: the global data variance among clients (i.e., the difference in data distributions among clients) and the local data insufficiency in each client (i.e., the lack of adequate local data for training). To overcome these two challenges, we propose a novel federated few-shot learning framework with two separately updated models and dedicated training strategies to reduce the adverse impact of global data variance and local data insufficiency. Extensive experiments on four prevalent datasets that cover news articles and images validate the effectiveness of our framework compared with the state-of-the-art baselines. Our code is provided at //github.com/SongW-SW/F2L.
In this paper, we describe and analyze the spectral properties of a number of exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the FGMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center (1,0) and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.
To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindel\"of iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers and Liouville-Bratu-Gelfand equations and comparing its performance with that of conventional time-stepping integrators.
Automated decision support systems promise to help human experts solve multiclass classification tasks more efficiently and accurately. However, existing systems typically require experts to understand when to cede agency to the system or when to exercise their own agency. Otherwise, the experts may be better off solving the classification tasks on their own. In this work, we develop an automated decision support system that, by design, does not require experts to understand when to trust the system to improve performance. Rather than providing (single) label predictions and letting experts decide when to trust these predictions, our system provides sets of label predictions constructed using conformal prediction$\unicode{x2014}$prediction sets$\unicode{x2014}$and forcefully asks experts to predict labels from these sets. By using conformal prediction, our system can precisely trade-off the probability that the true label is not in the prediction set, which determines how frequently our system will mislead the experts, and the size of the prediction set, which determines the difficulty of the classification task the experts need to solve using our system. In addition, we develop an efficient and near-optimal search method to find the conformal predictor under which the experts benefit the most from using our system. Simulation experiments using synthetic and real expert predictions demonstrate that our system may help experts make more accurate predictions and is robust to the accuracy of the classifier the conformal predictor relies on.
The counting task, which plays a fundamental role in numerous applications (e.g., crowd counting, traffic statistics), aims to predict the number of objects with various densities. Existing object counting tasks are designed for a single object class. However, it is inevitable to encounter newly coming data with new classes in our real world. We name this scenario as \textit{evolving object counting}. In this paper, we build the first evolving object counting dataset and propose a unified object counting network as the first attempt to address this task. The proposed model consists of two key components: a class-agnostic mask module and a class-incremental module. The class-agnostic mask module learns generic object occupation prior via predicting a class-agnostic binary mask (e.g., 1 denotes there exists an object at the considering position in an image and 0 otherwise). The class-incremental module is used to handle new coming classes and provides discriminative class guidance for density map prediction. The combined outputs of class-agnostic mask module and image feature extractor are used to predict the final density map. When new classes come, we first add new neural nodes into the last regression and classification layers of class-incremental module. Then, instead of retraining the model from scratch, we utilize knowledge distillation to help the model remember what have already learned about previous object classes. We also employ a support sample bank to store a small number of typical training samples of each class, which are used to prevent the model from forgetting key information of old data. With this design, our model can efficiently and effectively adapt to new coming classes while keeping good performance on already seen data without large-scale retraining. Extensive experiments on the collected dataset demonstrate the favorable performance.
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