Theoretical analysis confirms that the proposed algorithm achieves sub-linear regret in relation to the number of rounds and arms.

State aggregation is a popular model reduction method rooted in optimal control. It reduces the complexity of engineering systems by mapping the system's states into a small number of meta-states. The choice of aggregation map often depends on the data analysts' knowledge and is largely ad hoc. In this paper, we propose a tractable algorithm that estimates the probabilistic aggregation map from the system's trajectory. We adopt a soft-aggregation model, where each meta-state has a signature raw state, called an anchor state.

This is completely different from previous analyses of Oja's algorithm and matrix products, which

Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning. However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings. In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted.

We study and provide instance-optimal algorithms in differential privacy by extending and approximating the inverse sensitivity mechanism. We provide two approximation frameworks, one which only requires knowledge of local sensitivities, and a gradient-based approximation for optimization problems, which are efficiently computable for a broad class of functions. We complement our analysis with instance-specific lower bounds for vector-valued functions, which demonstrate that our mechanisms are (nearly) instance-optimal under certain assumptions and that minimax lower bounds may not provide an accurate estimate of the hardness of a problem in general: our algorithms can significantly outperform minimax bounds for well behaved instances. Finally, we use our approximation framework to develop private mechanisms for unbounded-range mean estimation, principal component analysis, and linear regression. For PCA, our mechanisms give an efficient (pure) differentially private algorithm with near-optimal rates.

Variational Autoencoders for multimodal data hold promise for many tasks in data analysis, such as representation learning, conditional generation, and imputation. Current architectures either share the encoder output, decoder input, or both across modalities to learn a shared representation. Such architectures impose hard constraints on the model. In this work, we show that a better latent representation can be obtained by replacing these hard constraints with a soft constraint. We propose a new mixture-of-experts prior, softly guiding each modality's latent representation towards a shared aggregate posterior. This approach results in a superior latent representation and allows each encoding to preserve information better from its uncompressed original features. In extensive experiments on multiple benchmark datasets and two challenging real-world datasets, we show improved learned latent representations and imputation of missing data modalities compared to existing methods.

Synthesizing larger texture images from a smaller exemplar is an important task in graphics and vision. The conventional CNNs, recently adopted for synthesis, require to train and test on the same set of images and fail to generalize to unseen images. This is mainly because those CNNs fully rely on convolutional and upsampling layers that operate locally and not suitable for a task as global as texture synthesis. In this work, inspired by the repetitive nature of texture patterns, we find that texture synthesis can be viewed as (local) upsampling in the Fast Fourier Transform (FFT) domain. However, FFT of natural images exhibits high dynamic range and lacks local correlations. Therefore, to train CNNs we design a framework to perform FFT upsampling in feature space using deformable convolutions. Such design allows our framework to generalize to unseen images, and synthesize textures in a single pass. Extensive evaluations confirm that our method achieves state-of-the-art performance both quantitatively and qualitatively.

Dimension reduction (DR) is an important and widely studied technique in exploratory data analysis. However, traditional DR methods are not applicable to datasets with a contrastive structure, where data are split into a foreground group of interest (case or treatment group), and a background group (control group). This type of data, common in biomedical studies, necessitates contrastive dimension reduction (CDR) methods to effectively capture information unique to or enriched in the foreground group relative to the background group. Despite the development of various CDR methods, two critical questions remain underexplored: when should these methods be applied, and how can the information unique to the foreground group be quantified? In this work, we address these gaps by proposing a hypothesis test to determine the existence of contrastive information, and introducing a contrastive dimension estimator (CDE) to quantify the unique components in the foreground group. We provide theoretical support for our methods and validate their effectiveness through extensive simulated, semi-simulated, and real experiments involving images, gene expressions, protein expressions, and medical sensors, demonstrating their ability to identify the unique information in the foreground group.

Dynamic Time Warping (DTW) is widely used as a similarity measure in various domains. Due to its invariance against warping in the time axis, DTW provides more meaningful discrepancy measurements between two signals than other distance measures. In this paper, we propose a novel component in an artificial neural network. In contrast to the previous successful usage of DTW as a loss function, the proposed framework leverages DTW to obtain a better feature extraction. For the first time, the DTW loss is theoretically analyzed, and a stochastic backpropogation scheme is proposed to improve the accuracy and efficiency of the DTW learning. We also demonstrate that the proposed framework can be used as a data analysis tool to perform data decomposition.

Matrix sketching is a powerful tool for reducing the size of large data matrices. Yet there are fundamental limitations to this size reduction when we want to recover an accurate estimator for a task such as least square regression. We show that these limitations can be circumvented in the distributed setting by designing sketching methods that minimize the bias of the estimator, rather than its error. In particular, we give a sparse sketching method running in optimal space and current matrix multiplication time, which recovers a nearly-unbiased least squares estimator using two passes over the data. This leads to new communication-efficient distributed averaging algorithms for least squares and related tasks, which directly improve on several prior approaches. Our key novelty is a new bias analysis for sketched least squares, giving a sharp characterization of its dependence on the sketch sparsity. The techniques include new higher-moment restricted Bai-Silverstein inequalities, which are of independent interest to the non-asymptotic analysis of deterministic equivalents for random matrices that arise from sketching.