We propose a meta-learning method for semi-supervised learning that learns from multiple tasks with heterogeneous attribute spaces. The existing semi-supervised meta-learning methods assume that all tasks share the same attribute space, which prevents us from learning with a wide variety of tasks. With the proposed method, the expected test performance on tasks with a small amount of labeled data is improved with unlabeled data as well as data in various tasks, where the attribute spaces are different among tasks. The proposed method embeds labeled and unlabeled data simultaneously in a task-specific space using a neural network, and the unlabeled data's labels are estimated by adapting classification or regression models in the embedding space. For the neural network, we develop variable-feature self-attention layers, which enable us to find embeddings of data with different attribute spaces with a single neural network by considering interactions among examples, attributes, and labels. Our experiments on classification and regression datasets with heterogeneous attribute spaces demonstrate that our proposed method outperforms the existing meta-learning and semi-supervised learning methods.

We present an unbiased method for Bayesian posterior means based on kinetic Langevin dynamics that combines advanced splitting methods with enhanced gradient approximations. Our approach avoids Metropolis correction by coupling Markov chains at different discretization levels in a multilevel Monte Carlo approach. Theoretical analysis demonstrates that our proposed estimator is unbiased, attains finite variance, and satisfies a central limit theorem. It can achieve accuracy $\epsilon>0$ for estimating expectations of Lipschitz functions in $d$ dimensions with $\mathcal{O}(d^{1/4}\epsilon^{-2})$ expected gradient evaluations, without assuming warm start. We exhibit similar bounds using both approximate and stochastic gradients, and our method's computational cost is shown to scale logarithmically with the size of the dataset. The proposed method is tested using a multinomial regression problem on the MNIST dataset and a Poisson regression model for soccer scores. Experiments indicate that the number of gradient evaluations per effective sample is independent of dimension, even when using inexact gradients. For product distributions, we give dimension-independent variance bounds. Our results demonstrate that the unbiased algorithm we present can be much more efficient than the ``gold-standard" randomized Hamiltonian Monte Carlo.

One of the goals of causal inference is to generalize from past experiments and observational data to novel conditions. While it is in principle possible to eventually learn a mapping from a novel experimental condition to an outcome of interest, provided a sufficient variety of experiments is available in the training data, coping with a large combinatorial space of possible interventions is hard. Under a typical sparse experimental design, this mapping is ill-posed without relying on heavy regularization or prior distributions. Such assumptions may or may not be reliable, and can be hard to defend or test. In this paper, we take a close look at how to warrant a leap from past experiments to novel conditions based on minimal assumptions about the factorization of the distribution of the manipulated system, communicated in the well-understood language of factor graph models. A postulated $\textit{interventional factor model}$ (IFM) may not always be informative, but it conveniently abstracts away a need for explicitly modeling unmeasured confounding and feedback mechanisms, leading to directly testable claims. Given an IFM and datasets from a collection of experimental regimes, we derive conditions for identifiability of the expected outcomes of new regimes never observed in these training data. We implement our framework using several efficient algorithms, and apply them on a range of semi-synthetic experiments.

In the Wishart model for sparse PCA we are given $n$ samples $Y_1,\ldots, Y_n$ drawn independently from a $d$-dimensional Gaussian distribution $N({0, Id + \beta vv^\top})$, where $\beta > 0$ and $v\in \mathbb{R}^d$ is a $k$-sparse unit vector, and we wish to recover $v$ (up to sign). We show that if $n \ge \Omega(d)$, then for every $t \ll k$ there exists an algorithm running in time $n\cdot d^{O(t)}$ that solves this problem as long as \[ \beta \gtrsim \frac{k}{\sqrt{nt}}\sqrt{\ln({2 + td/k^2})}\,. \] Prior to this work, the best polynomial time algorithm in the regime $k\approx \sqrt{d}$, called \emph{Covariance Thresholding} (proposed in [KNV15a] and analyzed in [DM14]), required $\beta \gtrsim \frac{k}{\sqrt{n}}\sqrt{\ln({2 + d/k^2})}$. For large enough constant $t$ our algorithm runs in polynomial time and has better guarantees than Covariance Thresholding. Previously known algorithms with such guarantees required quasi-polynomial time $d^{O(\log d)}$. In addition, we show that our techniques work with sparse PCA with adversarial perturbations studied in [dKNS20]. This model generalizes not only sparse PCA, but also other problems studied in prior works, including the sparse planted vector problem. As a consequence, we provide polynomial time algorithms for the sparse planted vector problem that have better guarantees than the state of the art in some regimes. Our approach also works with the Wigner model for sparse PCA. Moreover, we show that it is possible to combine our techniques with recent results on sparse PCA with symmetric heavy-tailed noise [dNNS22]. In particular, in the regime $k \approx \sqrt{d}$ we get the first polynomial time algorithm that works with symmetric heavy-tailed noise, while the algorithm from [dNNS22]. requires quasi-polynomial time in these settings.

We present a computationally efficient online kernel Cumulative Sum (CUSUM) method for change-point detection that utilizes the maximum over a set of kernel statistics to account for the unknown change-point location. Our approach exhibits increased sensitivity to small changes compared to existing kernel-based change-point detection methods, including Scan-B statistic, corresponding to a non-parametric Shewhart chart-type procedure. We provide accurate analytic approximations for two key performance metrics: the Average Run Length (ARL) and Expected Detection Delay (EDD), which enable us to establish an optimal window length to be on the order of the logarithm of ARL to ensure minimal power loss relative to an oracle procedure with infinite memory. Moreover, we introduce a recursive calculation procedure for detection statistics to ensure constant computational and memory complexity, which is essential for online implementation. Through extensive experiments on both simulated and real data, we demonstrate the competitive performance of our method and validate our theoretical results.

In this thesis, we develop theoretical, algorithmic and experimental contributions for Machine Learning with limited labels, and more specifically for the tasks of Image Classification and Object Detection in Computer Vision. In a first contribution, we are interested in bridging the gap between theory and practice for popular Meta-Learning algorithms used in Few-Shot Classification. We make connections to Multi-Task Representation Learning, which benefits from solid theoretical foundations, to verify the best conditions for a more efficient meta-learning. Then, to leverage unlabeled data when training object detectors based on the Transformer architecture, we propose both an unsupervised pretraining and a semi-supervised learning method in two other separate contributions. For pretraining, we improve Contrastive Learning for object detectors by introducing the localization information. Finally, our semi-supervised method is the first tailored to transformer-based detectors.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there were finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, many real-world data are not naturally posed in the setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions, and then convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is further developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications.

Contrastive learning has recently emerged as a promising approach for learning data representations that discover and disentangle the explanatory factors of the data. Previous analyses of such approaches have largely focused on individual contrastive losses, such as noise-contrastive estimation (NCE) and InfoNCE, and rely on specific assumptions about the data generating process. This paper extends the theoretical guarantees for disentanglement to a broader family of contrastive methods, while also relaxing the assumptions about the data distribution. Specifically, we prove identifiability of the true latents for four contrastive losses studied in this paper, without imposing common independence assumptions. The theoretical findings are validated on several benchmark datasets. Finally, practical limitations of these methods are also investigated.

The dominant approach in recent literature on causal discovery from interventional data is optimization of a model score. Although the scoring is straightforward in the sense that the optimization over DAGs refers to fully specified joint models for all observational and interventional data, the optimization landscape is very high-dimensional, making score-based algorithms infeasible for graphs with hundreds/thousands of nodes that are common in biological applications. This makes causal discovery difficult, in addition to many other challenges that remain such as departing from restrictive genericity assumptions on the underlying distributions and developing methodology for high-dimensional settings. In this article, to address these challenges, we depart from a score-based strategy and leverage special properties of polytrees to obtain a highly scalable "local" approach that learns from low-dimensional marginals. Our methods yield fast and consistent algorithms to learn linear Gaussian polytrees from interventional data.

In this paper, we develop data-dependent and algorithm-dependent generalization bounds for transductive learning algorithms in the context of information theory for the first time. We show that the generalization gap of transductive learning algorithms can be bounded by the mutual information between training labels and hypothesis. By innovatively proposing the concept of transductive supersamples, we go beyond the inductive learning setting and establish upper bounds in terms of various information measures. Furthermore, we derive novel PAC-Bayesian bounds and build the connection between generalization and loss landscape flatness under the transductive learning setting. Finally, we present the upper bounds for adaptive optimization algorithms and demonstrate the applications of results on semi-supervised learning and graph learning scenarios. Our theoretic results are validated on both synthetic and real-world datasets.