Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern high-dimensional data and interactions thereof can be modeled as lying approximately in a low-dimensional subspace or manifold, possibly with additional structures, and its proper exploitations lead to significant reduction of costs in sensing, computation and storage. In recent years, there is a plethora of progress in understanding how to exploit low-rank structures using computationally efficient procedures in a provable manner, including both convex and nonconvex approaches. On one side, convex relaxations such as nuclear norm minimization often lead to statistically optimal procedures for estimating low-rank matrices, where first-order methods are developed to address the computational challenges; on the other side, there is emerging evidence that properly designed nonconvex procedures, such as projected gradient descent, often provide globally optimal solutions with a much lower computational cost in many problems. This survey article will provide a unified overview of these recent advances on low-rank matrix estimation from incomplete measurements. Attention is paid to rigorous characterization of the performance of these algorithms, and to problems where the low-rank matrix have additional structural properties that require new algorithmic designs and theoretical analysis.

Sparse subspace clustering (SSC) is one of the current state-of-the-art methods for partitioning data points into the union of subspaces, with strong theoretical guarantees. However, it is not practical for large data sets as it requires solving a LASSO problem for each data point, where the number of variables in each LASSO problem is the number of data points. To improve the scalability of SSC, we propose to select a few sets of anchor points using a randomized hierarchical clustering method, and, for each set of anchor points, solve the LASSO problems for each data point allowing only anchor points to have a non-zero weight (this reduces drastically the number of variables). This generates a multilayer graph where each layer corresponds to a different set of anchor points. Using the Grassmann manifold of orthogonal matrices, the shared connectivity among the layers is summarized within a single subspace. Finally, we use $k$-means clustering within that subspace to cluster the data points, similarly as done by spectral clustering in SSC. We show on both synthetic and real-world data sets that the proposed method not only allows SSC to scale to large-scale data sets, but that it is also much more robust as it performs significantly better on noisy data and on data with close susbspaces and outliers, while it is not prone to oversegmentation.

It is common practice to decay the learning rate. Here we show one can usually obtain the same learning curve on both training and test sets by instead increasing the batch size during training. This procedure is successful for stochastic gradient descent (SGD), SGD with momentum, Nesterov momentum, and Adam. It reaches equivalent test accuracies after the same number of training epochs, but with fewer parameter updates, leading to greater parallelism and shorter training times. We can further reduce the number of parameter updates by increasing the learning rate $\epsilon$ and scaling the batch size $B \propto \epsilon$. Finally, one can increase the momentum coefficient $m$ and scale $B \propto 1/(1-m)$, although this tends to slightly reduce the test accuracy. Crucially, our techniques allow us to repurpose existing training schedules for large batch training with no hyper-parameter tuning. We train ResNet-50 on ImageNet to $76.1\%$ validation accuracy in under 30 minutes.

Policy gradient methods have achieved remarkable successes in solving challenging reinforcement learning problems. However, it still often suffers from the large variance issue on policy gradient estimation, which leads to poor sample efficiency during training. In this work, we propose a control variate method to effectively reduce variance for policy gradient methods. Motivated by the Stein's identity, our method extends the previous control variate methods used in REINFORCE and advantage actor-critic by introducing more general action-dependent baseline functions. Empirical studies show that our method significantly improves the sample efficiency of the state-of-the-art policy gradient approaches.

We design a stochastic algorithm to train any smooth neural network to $\varepsilon$-approximate local minima, using $O(\varepsilon^{-3.25})$ backpropagations. The best result was essentially $O(\varepsilon^{-4})$ by SGD. More broadly, it finds $\varepsilon$-approximate local minima of any smooth nonconvex function in rate $O(\varepsilon^{-3.25})$, with only oracle access to stochastic gradients.

Recent progress in variational inference has paid much attention to the flexibility of variational posteriors. One promising direction is to use implicit distributions, i.e., distributions without tractable densities as the variational posterior. However, existing methods on implicit posteriors still face challenges of noisy estimation and computational infeasibility when applied to models with high-dimensional latent variables. In this paper, we present a new approach named Kernel Implicit Variational Inference that addresses these challenges. As far as we know, for the first time implicit variational inference is successfully applied to Bayesian neural networks, which shows promising results on both regression and classification tasks.

Implicit models, which allow for the generation of samples but not for point-wise evaluation of probabilities, are omnipresent in real-world problems tackled by machine learning and a hot topic of current research. Some examples include data simulators that are widely used in engineering and scientific research, generative adversarial networks (GANs) for image synthesis, and hot-off-the-press approximate inference techniques relying on implicit distributions. The majority of existing approaches to learning implicit models rely on approximating the intractable distribution or optimisation objective for gradient-based optimisation, which is liable to produce inaccurate updates and thus poor models. This paper alleviates the need for such approximations by proposing the Stein gradient estimator, which directly estimates the score function of the implicitly defined distribution. The efficacy of the proposed estimator is empirically demonstrated by examples that include meta-learning for approximate inference, and entropy regularised GANs that provide improved sample diversity.

Note: If you are looking for a review paper, this blog post is also available as an article on arXiv. In Machine Learning (ML), we typically care about optimizing for a particular metric, whether this is a score on a certain benchmark or a business KPI. In order to do this, we generally train a single model or an ensemble of models to perform our desired task. While we can generally achieve acceptable performance this way, by being laser-focused on our single task, we ignore information that might help us do even better on the metric we care about. Specifically, this information comes from the training signals of related tasks. By sharing representations between related tasks, we can enable our model to generalize better on our original task. This approach is called Multi-Task Learning (MTL) and will be the topic of this blog post. Multi-task learning has been used successfully across all applications of machine learning, from natural language processing [1] and speech recognition [2] to computer vision [3] and drug discovery [4]. MTL comes in many guises: joint learning, learning to learn, and learning with auxiliary tasks are only some names that have been used to refer to it. Generally, as soon as you find yourself optimizing more than one loss function, you are effectively doing multi-task learning (in contrast to single-task learning).

In regression analysis, curve fitting is the process of specifying the model that provides the best fit to the specific curves in your dataset. Curved relationships between variables are not as straightforward to fit and interpret as linear relationships.

Visual analytics systems combine machine learning or other analytic techniques with interactive data visualization to promote sensemaking and analytical reasoning. It is through such techniques that people can make sense of large, complex data. While progress has been made, the tactful combination of machine learning and data visualization is still under-explored. This state-of-the-art report presents a summary of the progress that has been made by highlighting and synthesizing select research advances. Further, it presents opportunities and challenges to enhance the synergy between machine learning and visual analytics for impactful future research directions.