We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional Euclidean space by learning a coordinate chart, but it requires that the entire manifold can be embedded in a single chart. This is impossible for manifolds with holes. In such cases, it is necessary to learn an atlas: a collection of charts that collectively cover the entire manifold. We begin with many small charts, and combine them in a bottom-up approach, where charts are only combined if doing so will not introduce problematic topological features. When it is no longer possible to combine any charts, each chart is individually embedded with standard manifold learning techniques, completing the construction of the atlas. We show the efficacy of our method by constructing atlases for challenging synthetic manifolds; learning human motion embeddings from motion capture data; and learning kinematic models of articulated objects.

In image retrieval, standard evaluation metrics rely on score ranking, e.g. average precision (AP). In this paper, we introduce a method for robust and decomposable average precision (ROADMAP) addressing two major challenges for end-to-end training of deep neural networks with AP: non-differentiability and non-decomposability. Firstly, we propose a new differentiable approximation of the rank function, which provides an upper bound of the AP loss and ensures robust training. Secondly, we design a simple yet effective loss function to reduce the decomposability gap between the AP in the whole training set and its averaged batch approximation, for which we provide theoretical guarantees. Extensive experiments conducted on three image retrieval datasets show that ROADMAP outperforms several recent AP approximation methods and highlight the importance of our two contributions. Finally, using ROADMAP for training deep models yields very good performances, outperforming state-of-the-art results on the three datasets.

Extreme multi-label text classification (XMC) seeks to find relevant labels from an extreme large label collection for a given text input. Many real-world applications can be formulated as XMC problems, such as recommendation systems, document tagging and semantic search. Recently, transformer based XMC methods, such as X-Transformer and LightXML, have shown significant improvement over other XMC methods. Despite leveraging pre-trained transformer models for text representation, the fine-tuning procedure of transformer models on large label space still has lengthy computational time even with powerful GPUs. In this paper, we propose a novel recursive approach, XR-Transformer to accelerate the procedure through recursively fine-tuning transformer models on a series of multi-resolution objectives related to the original XMC objective function. Empirical results show that XR-Transformer takes significantly less training time compared to other transformer-based XMC models while yielding better state-of-the-art results. In particular, on the public Amazon-3M dataset with 3 million labels, XR-Transformer is not only 20x faster than X-Transformer but also improves the Precision@1 from 51% to 54%.

We say an algorithm is batch size-invariant if changes to the batch size can largely be compensated for by changes to other hyperparameters. Stochastic gradient descent is well-known to have this property at small batch sizes, via the learning rate. However, some policy optimization algorithms (such as PPO) do not have this property, because of how they control the size of policy updates. In this work we show how to make these algorithms batch size-invariant. Our key insight is to decouple the proximal policy (used for controlling policy updates) from the behavior policy (used for off-policy corrections). Our experiments help explain why these algorithms work, and additionally show how they can make more efficient use of stale data.

Supervised Causal Learning (SCL) aims to learn causal relations from observational data by accessing previously seen datasets associated with ground truth causal relations. This paper presents a first attempt at addressing a fundamental question: What are the benefits from supervision and how does it benefit? Starting from seeing that SCL is not better than random guessing if the learning target is non-identifiable a priori, we propose a two-phase paradigm for SCL by explicitly considering structure identifiability. Following this paradigm, we tackle the problem of SCL on discrete data and propose ML4C. The core of ML4C is a binary classifier with a novel learning target: it classifies whether an Unshielded Triple (UT) is a v-structure or not. Starting from an input dataset with the corresponding skeleton provided, ML4C orients each UT once it is classified as a v-structure. These v-structures are together used to construct the final output. To address the fundamental question of SCL, we propose a principled method for ML4C featurization: we exploit the vicinity of a given UT (i.e., the neighbors of UT in skeleton), and derive features by considering the conditional dependencies and structural entanglement within the vicinity. We further prove that ML4C is asymptotically perfect. Last but foremost, thorough experiments conducted on benchmark datasets demonstrate that ML4C remarkably outperforms other state-of-the-art algorithms in terms of accuracy, robustness, tolerance and transferability. In summary, ML4C shows promising results on validating the effectiveness of supervision for causal learning.

The efficiency of Hamiltonian Monte Carlo (HMC) can suffer when sampling a distribution with a wide range of length scales, because the small step sizes needed for stability in high-curvature regions are inefficient elsewhere. To address this we present a delayed rejection variant: if an initial HMC trajectory is rejected, we make one or more subsequent proposals each using a step size geometrically smaller than the last. We extend the standard delayed rejection framework by allowing the probability of a retry to depend on the probability of accepting the previous proposal. We test the scheme in several sampling tasks, including multiscale model distributions such as Neal's funnel, and statistical applications. Delayed rejection enables up to five-fold performance gains over optimally-tuned HMC, as measured by effective sample size per gradient evaluation. Even for simpler distributions, delayed rejection provides increased robustness to step size misspecification. Along the way, we provide an accessible but rigorous review of detailed balance for HMC. Keywords: delayed rejection, Hamiltonian Monte Carlo, detailed balance, multiscale.

Two-level stochastic optimization formulations have become instrumental in a number of machine learning contexts such as neural architecture search, continual learning, adversarial learning, and hyperparameter tuning. Practical stochastic bilevel optimization problems become challenging in optimization or learning scenarios where the number of variables is high or there are constraints. The goal of this paper is twofold. First, we aim at promoting the use of bilevel optimization in large-scale learning and we introduce a practical bilevel stochastic gradient method (BSG-1) that requires neither lower level second-order derivatives nor system solves (and dismisses any matrix-vector products). Our BSG-1 method is close to first-order principles, which allows it to achieve a performance better than those that are not, such as DARTS. Second, we develop bilevel stochastic gradient descent for bilevel problems with lower level constraints, and we introduce a convergence theory that covers the unconstrained and constrained cases and abstracts as much as possible from the specifics of the bilevel gradient calculation.

Unlike its intercept, a linear classifier's weight vector cannot be tuned by a simple grid search. Hence, this paper proposes weight vector tuning of a generic binary linear classifier through the parameterization of a decomposition of the discriminant by a scalar which controls the trade-off between conflicting informative and noisy terms. By varying this parameter, the original weight vector is modified in a meaningful way. Applying this method to a number of linear classifiers under a variety of data dimensionality and sample size settings reveals that the classification performance loss due to non-optimal native hyperparameters can be compensated for by weight vector tuning. This yields computational savings as the proposed tuning method reduces to tuning a scalar compared to tuning the native hyperparameter, which may involve repeated weight vector generation along with its burden of optimization, dimensionality reduction, etc., depending on the classifier. It is also found that weight vector tuning significantly improves the performance of Linear Discriminant Analysis (LDA) under high estimation noise. Proceeding from this second finding, an asymptotic study of the misclassification probability of the parameterized LDA classifier in the growth regime where the data dimensionality and sample size are comparable is conducted. Using random matrix theory, the misclassification probability is shown to converge to a quantity that is a function of the true statistics of the data. Additionally, an estimator of the misclassification probability is derived. Finally, computationally efficient tuning of the parameter using this estimator is demonstrated on real data. Alouni, and T. Y. Al-Naffouri are with the Electrical and Computer Engineering Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia; emails: {lama.niyazi,

Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA Many datasets give partial information about an ordering or ranking by indicating which team won a game, which item a user prefers, or who infected whom. We define a continuous spin system whose Gibbs distribution is the posterior distribution on permutations, given a probabilistic model of these interactions. Using the cavity method we derive a belief propagation algorithm that computes the marginal distribution of each node's position. In addition, the Bethe free energy lets us approximate the number of linear extensions of a partial order and perform model selection. Ranking or ordering objects is a natural problem in In this case, the energy H(π) is the number of violations, many contexts.

This transformation of food delivery businesses to online platforms has gained high attention in recent years. This due to the availability of customizing ordering experiences, easy payment methods, fast delivery, and others. The competition between online food delivery providers has intensified to attain a wider range of customers. Hence, they should have a better understanding of their customers' needs and predict their purchasing decisions. Machine learning has a significant impact on companies' bottom line. They are used to construct models and strategies in industries that rely on big data and need a system to evaluate it fast and effectively. Predictive modeling is a type of machine learning that uses various regression algorithms, analytics, and statistics to estimate the probability of an occurrence. The incorporation of predictive models helps online food delivery providers to understand their customers. In this study, a dataset collected from 388 consumers in Bangalore, India was provided to predict their purchasing decisions. Four prediction models are considered: CART and C4.5 decision trees, random forest, and rule-based classifiers, and their accuracies in providing the correct class label are evaluated. The findings show that all models perform similarly, but the C4.5 outperforms them all with an accuracy of 91.67%.