Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.

Bayesian deep learning plays an important role especially for its ability evaluating epistemic uncertainty (EU). Due to computational complexity issues, approximation methods such as variational inference (VI) have been used in practice to obtain posterior distributions and their generalization abilities have been analyzed extensively, for example, by PAC-Bayesian theory; however, little analysis exists on EU, although many numerical experiments have been conducted on it. In this study, we analyze the EU of supervised learning in approximate Bayesian inference by focusing on its excess risk. First, we theoretically show the novel relations between generalization error and the widely used EU measurements, such as the variance and mutual information of predictive distribution, and derive their convergence behaviors. Next, we clarify how the objective function of VI regularizes the EU. With this analysis, we propose a new objective function for VI that directly controls the prediction performance and the EU based on the PAC-Bayesian theory. Numerical experiments show that our algorithm significantly improves the EU evaluation over the existing VI methods.

Combining skeleton structure with graph convolutional networks has achieved remarkable performance in human action recognition. Since current research focuses on designing basic graph for representing skeleton data, these embedding features contain basic topological information, which cannot learn more systematic perspectives from skeleton data. In this paper, we overcome this limitation by proposing a novel framework, which unifies 15 graph embedding features into the graph convolutional network for human action recognition, aiming to best take advantage of graph information to distinguish key joints, bones, and body parts in human action, instead of being exclusive to a single feature or domain. Additionally, we fully investigate how to find the best graph features of skeleton structure for improving human action recognition. Besides, the topological information of the skeleton sequence is explored to further enhance the performance in a multi-stream framework. Moreover, the unified graph features are extracted by the adaptive methods on the training process, which further yields improvements. Our model is validated by three large-scale datasets, namely NTU-RGB+D, Kinetics and SYSU-3D, and outperforms the state-of-the-art methods. Overall, our work unified graph embedding features to promotes systematic research on human action recognition.

Proving algorithm-dependent generalization error bounds for gradient-type optimization methods has attracted significant attention recently in learning theory. However, most existing trajectory-based analyses require either restrictive assumptions on the learning rate (e.g., fast decreasing learning rate), or continuous injected noise (such as the Gaussian noise in Langevin dynamics). In this paper, we introduce a new discrete data-dependent prior to the PAC-Bayesian framework, and prove a high probability generalization bound of order $O(\frac{1}{n}\cdot \sum_{t=1}^T(\gamma_t/\varepsilon_t)^2\left\|{\mathbf{g}_t}\right\|^2)$ for Floored GD (i.e. a version of gradient descent with precision level $\varepsilon_t$), where $n$ is the number of training samples, $\gamma_t$ is the learning rate at step $t$, $\mathbf{g}_t$ is roughly the difference of the gradient computed using all samples and that using only prior samples. $\left\|{\mathbf{g}_t}\right\|$ is upper bounded by and and typical much smaller than the gradient norm $\left\|{\nabla f(W_t)}\right\|$. We remark that our bound holds for nonconvex and nonsmooth scenarios. Moreover, our theoretical results provide numerically favorable upper bounds of testing errors (e.g., $0.037$ on MNIST). Using a similar technique, we can also obtain new generalization bounds for certain variants of SGD. Furthermore, we study the generalization bounds for gradient Langevin Dynamics (GLD). Using the same framework with a carefully constructed continuous prior, we show a new high probability generalization bound of order $O(\frac{1}{n} + \frac{L^2}{n^2}\sum_{t=1}^T(\gamma_t/\sigma_t)^2)$ for GLD. The new $1/n^2$ rate is due to the concentration of the difference between the gradient of training samples and that of the prior.

Learning high-level causal representations together with a causal model from unstructured low-level data such as pixels is impossible from observational data alone. We prove under mild assumptions that this representation is however identifiable in a weakly supervised setting. This involves a dataset with paired samples before and after random, unknown interventions, but no further labels. We then introduce implicit latent causal models, variational autoencoders that represent causal variables and causal structure without having to optimize an explicit discrete graph structure. On simple image data, including a novel dataset of simulated robotic manipulation, we demonstrate that such models can reliably identify the causal structure and disentangle causal variables.

Author summary Modern experimental techniques such as mass cytometry (CyTOF) make it possible to quickly make high-dimensional measurements on upwards of tens of millions of cells with single-cell resolution. An important problem in biology is to use these measurements to group together similar cells to identify biologically meaningful cell types that can be used to study disease progression, drug responses, and other clinical outcomes. However, the size and complexity of experimental data sets makes this problem computationally and theoretically extremely difficult. Here, we present a new algorithm HAL-X that accurately and quickly identifies cell clusters from biological data. Importantly, our algorithm does not require large amounts of memory. This eliminates the need for specialized high-end computing resources, allowing biologists to quickly analyze their data using a standard laptop or desktop computer.

You're looking for a complete Classification modeling course that teaches you everything you need to create a Classification model in Python, right? You've found the right Classification modeling course! How this course will help you? A Verifiable Certificate of Completion is presented to all students who undertake this Machine learning basics course. Why should you choose this course?

You're looking for a complete Linear Regression course that teaches you everything you need to create a Linear Regression model in Python, right? You've found the right Linear Regression course! A Verifiable Certificate of Completion is presented to all students who undertake this Machine learning basics course. How this course will help you? Why should you choose this course?

Social media apps have become very promising and omnipresent in daily life. Most social media apps are used to deliver vital information to those nearby and far away. As our lives become more hectic, many of us strive to limit our usage of social media apps because they are too addictive, and the majority of us have gotten preoccupied with our daily lives. Because of this, we frequently overlook crucial information, such as invitations to weddings, interviews, birthday parties, etc., or find ourselves unable to attend the event. In most cases, this happens because users are more likely to discover the invitation or information only before the event, giving them little time to prepare. To solve this issue, in this study, we created a system that will collect social media chat and filter it using Natural Language Processing (NLP) methods like Tokenization, Stop Words Removal, Lemmatization, Segmentation, and Named Entity Recognition (NER). Also, Machine Learning Algorithms such as K-Nearest Neighbor (KNN) Algorithm are implemented to prioritize the received invitation and to sort the level of priority. Finally, a customized notification will be delivered to the users where they acknowledge the upcoming event. So, the chances of missing the event are less or can be planned.

Modeling multivariate time series as temporal signals over a (possibly dynamic) graph is an effective representational framework that allows for developing models for time series analysis. In fact, discrete sequences of graphs can be processed by autoregressive graph neural networks to recursively learn representations at each discrete point in time and space. Spatiotemporal graphs are often highly sparse, with time series characterized by multiple, concurrent, and long sequences of missing data, e.g., due to the unreliable underlying sensor network. In this context, autoregressive models can be brittle and exhibit unstable learning dynamics. The objective of this paper is, then, to tackle the problem of learning effective models to reconstruct, i.e., impute, missing data points by conditioning the reconstruction only on the available observations. In particular, we propose a novel class of attention-based architectures that, given a set of highly sparse discrete observations, learn a representation for points in time and space by exploiting a spatiotemporal propagation architecture aligned with the imputation task. Representations are trained end-to-end to reconstruct observations w.r.t. the corresponding sensor and its neighboring nodes. Compared to the state of the art, our model handles sparse data without propagating prediction errors or requiring a bidirectional model to encode forward and backward time dependencies. Empirical results on representative benchmarks show the effectiveness of the proposed method.