We prove concentration inequalities for the output of the exponential mechanism about the maximizer of the population objective function. This bound applies to objective functions that satisfy a mild regularity condition. To illustrate our result, we study the problem of differentially private multivariate median estimation. We present novel finite-sample performance guarantees for differentially private multivariate depth-based medians which are essentially sharp. Our results cover commonly used depth functions, such as the halfspace (or Tukey) depth, spatial depth, and the integrated dual depth. We show that under Cauchy marginals, the cost of heavy-tailed location estimation outweighs the cost of privacy. We demonstrate our results numerically using a Gaussian contamination model in dimensions up to $d = 100$, and compare them to a state-of-the-art private mean estimation algorithm.

Multi-task learning (MTL) aims to improve the generalization performance of multiple tasks by exploiting the shared factors among them. Various metrics (e.g., F-score, Area Under the ROC Curve) are used to evaluate the performances of MTL methods. Most existing MTL methods try to minimize either the misclassified errors for classification or the mean squared errors for regression. In this paper, we propose a method to directly optimize the evaluation metrics for a large family of MTL problems. The formulation of MTL that directly optimizes evaluation metrics is the combination of two parts: (1) a regularizer defined on the weight matrix over all tasks, in order to capture the relatedness of these tasks; (2) a sum of multiple structured hinge losses, each corresponding to a surrogate of some evaluation metric on one task. This formulation is challenging in optimization because both of its parts are non-smooth. To tackle this issue, we propose a novel optimization procedure based on the alternating direction scheme of multipliers, where we decompose the whole optimization problem into a sub-problem corresponding to the regularizer and another sub-problem corresponding to the structured hinge losses. For a large family of MTL problems, the first sub-problem has closed-form solutions. To solve the second sub-problem, we propose an efficient primal-dual algorithm via coordinate ascent. Extensive evaluation results demonstrate that, in a large family of MTL problems, the proposed MTL method of directly optimization evaluation metrics has superior performance gains against the corresponding baseline methods.

The cubic regularization method (CR) and its adaptive version (ARC) are popular Newton-type methods in solving unconstrained non-convex optimization problems, due to its global convergence to local minima under mild conditions. The main aim of this paper is to develop a momentum-accelerated adaptive cubic regularization method (ARCm) to improve the convergent performance. With the proper choice of momentum step size, we show the global convergence of ARCm and the local convergence can also be guaranteed under the \KL property. Such global and local convergence can also be established when inexact solvers with low computational costs are employed in the iteration procedure. Numerical results for non-convex logistic regression and robust linear regression models are reported to demonstrate that the proposed ARCm significantly outperforms state-of-the-art cubic regularization methods (e.g., CR, momentum-based CR, ARC) and the trust region method. In particular, the number of iterations required by ARCm is less than 10\% to 50\% required by the most competitive method (ARC) in the experiments.

In this paper, we consider a challenging but realistic continual learning (CL) problem, Few-Shot Continual Active Learning (FoCAL), where a CL agent is provided with unlabeled data for a new or a previously learned task in each increment and the agent only has limited labeling budget available. Towards this, we build on the continual learning and active learning literature and develop a framework that can allow a CL agent to continually learn new object classes from a few labeled training examples. Our framework represents each object class using a uniform Gaussian mixture model (GMM) and uses pseudo-rehearsal to mitigate catastrophic forgetting. The framework also uses uncertainty measures on the Gaussian representations of the previously learned classes to find the most informative samples to be labeled in an increment. We evaluate our approach on the CORe-50 dataset and on a real humanoid robot for the object classification task. The results show that our approach not only produces state-of-the-art results on the dataset but also allows a real robot to continually learn unseen objects in a real environment with limited labeling supervision provided by its user.

We study the problem of finding the Nash equilibrium in a two-player zero-sum Markov game. Due to its formulation as a minimax optimization program, a natural approach to solve the problem is to perform gradient descent/ascent with respect to each player in an alternating fashion. However, due to the non-convexity/non-concavity of the underlying objective function, theoretical understandings of this method are limited. In our paper, we consider solving an entropy-regularized variant of the Markov game. The regularization introduces structure into the optimization landscape that make the solutions more identifiable and allow the problem to be solved more efficiently. Our main contribution is to show that under proper choices of the regularization parameter, the gradient descent ascent algorithm converges to the Nash equilibrium of the original unregularized problem. We explicitly characterize the finite-time performance of the last iterate of our algorithm, which vastly improves over the existing convergence bound of the gradient descent ascent algorithm without regularization. Finally, we complement the analysis with numerical simulations that illustrate the accelerated convergence of the algorithm.

Electronic health records (EHRs) are usually highly dimensional, heterogeneous, and multimodal. Besides, the random recording of clinical variables results in high missing rates and uneven time intervals between adjacent records in the multivariate clinical time-series data extracted from EHRs. Current works using clinical time-series data for patient representation regard the patients' physiological status as a discrete process described by sporadically collected records. However, changes in the patient's physiological condition are continuous and dynamic processes. The perception of time and velocity of change is crucial for patient representation learning. In this study, we propose a time- and velocity-aware gated recurrent unit model (GRU-TV) for patient representation learning of clinical multivariate time-series data in a time-continuous manner. The neural ordinary differential equations (ODEs) and velocity perception mechanism are applied to perceive the time interval between adjacent records and changing rate of the patient's physiological status, respectively. Our experiments on two real clinical EHR datasets (PhysioNet2012, MIMIC-III) establish that GRU-TV is a robust model on computer-aided diagnosis (CAD) tasks, especially on sequences with high-variance time intervals.

The pricing of housing properties is determined by a variety of factors. However, post-pandemic markets have experienced volatility in the Chicago suburb area, which have affected house prices greatly. In this study, analysis was done on the Naperville/Bolingbrook real estate market to predict property prices based on these housing attributes through machine learning models, and to evaluate the effectiveness of such models in a volatile market space. Gathering data from Redfin, a real estate website, sales data from 2018 up until the summer season of 2022 were collected for research. By analyzing these sales in this range of time, we can also look at the state of the housing market and identify trends in price. For modeling the data, the models used were linear regression, support vector regression, decision tree regression, random forest regression, and XGBoost regression. To analyze results, comparison was made on the MAE, RMSE, and R-squared values for each model. It was found that the XGBoost model performs the best in predicting house prices despite the additional volatility sponsored by post-pandemic conditions. After modeling, Shapley Values (SHAP) were used to evaluate the weights of the variables in constructing models.

Stock prices move as piece-wise trending fluctuation rather than a purely random walk. Traditionally, the prediction of future stock movements is based on the historical trading record. Nowadays, with the development of social media, many active participants in the market choose to publicize their strategies, which provides a window to glimpse over the whole market's attitude towards future movements by extracting the semantics behind social media. However, social media contains conflicting information and cannot replace historical records completely. In this work, we propose a multi-modality attention network to reduce conflicts and integrate semantic and numeric features to predict future stock movements comprehensively. Specifically, we first extract semantic information from social media and estimate their credibility based on posters' identity and public reputation. Then we incorporate the semantic from online posts and numeric features from historical records to make the trading strategy. Experimental results show that our approach outperforms previous methods by a significant margin in both prediction accuracy (61.20\%) and trading profits (9.13\%). It demonstrates that our method improves the performance of stock movements prediction and informs future research on multi-modality fusion towards stock prediction.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced by Lavenant et al. arXiv:2102.09204, and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schr\"odinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence to the desired estimator. Overall, this leads to an inference method with end-to-end theoretical guarantees that solves an interpretable model for trajectory inference. We also present how to adapt the method to deal with mass variations, a useful extension when dealing with single cell RNA-sequencing data where cells can branch and die.

Graph Convolutional Network (GCN) has exhibited strong empirical performance in many real-world applications. The vast majority of existing works on GCN primarily focus on the accuracy while ignoring how confident or uncertain a GCN is with respect to its predictions. Despite being a cornerstone of trustworthy graph mining, uncertainty quantification on GCN has not been well studied and the scarce existing efforts either fail to provide deterministic quantification or have to change the training procedure of GCN by introducing additional parameters or architectures. In this paper, we propose the first frequentist-based approach named JuryGCN in quantifying the uncertainty of GCN, where the key idea is to quantify the uncertainty of a node as the width of confidence interval by a jackknife estimator. Moreover, we leverage the influence functions to estimate the change in GCN parameters without re-training to scale up the computation. The proposed JuryGCN is capable of quantifying uncertainty deterministically without modifying the GCN architecture or introducing additional parameters. We perform extensive experimental evaluation on real-world datasets in the tasks of both active learning and semi-supervised node classification, which demonstrate the efficacy of the proposed method.