We consider the theory of regression on a manifold using reproducing kernel Hilbert space methods. Manifold models arise in a wide variety of modern machine learning problems, and our goal is to help understand the effectiveness of various implicit and explicit dimensionality-reduction methods that exploit manifold structure. Our first key contribution is to establish a novel nonasymptotic version of the Weyl law from differential geometry. From this we are able to show that certain spaces of smooth functions on a manifold are effectively finite-dimensional, with a complexity that scales according to the manifold dimension rather than any ambient data dimension. Finally, we show that given (potentially noisy) function values taken uniformly at random over a manifold, a kernel regression estimator (derived from the spectral decomposition of the manifold) yields minimax-optimal error bounds that are controlled by the effective dimension.

It is often critical for prediction models to be robust to distributional shifts between training and testing data. Viewed from a causal perspective, the challenge is to distinguish the stable causal relationships from the unstable spurious correlations across shifts. We describe a causal transfer random forest (CTRF) that combines existing training data with a small amount of data from a randomized experiment to train a model which is robust to the feature shifts and therefore transfers to a new targeting distribution. Theoretically, we justify the robustness of the approach against feature shifts with the knowledge from causal learning. Empirically, we evaluate the CTRF using both synthetic data experiments and real-world experiments in the Bing Ads platform, including a click prediction task and in the context of an end-to-end counterfactual optimization system. The proposed CTRF produces robust predictions and outperforms most baseline methods compared in the presence of feature shifts.

This paper presents a sublinear classical algorithm for principal component regression. The algorithm uses quantum-inspired linear algebra, an idea developed by Tang. Using this technique, her algorithm for recommendation systems achieved runtime only polynomially slower than its quantum counterpart. Her work was quickly adapted to solve many other problems in sublinear time complexity. In this work, we developed an algorithm for principal component regression that runs in time polylogarithmic to the number of data points, an exponential speed up over the state-of-the-art algorithm, under the mild assumption that the input is given in some data structure that supports a norm-based sampling procedure. This exponential speed up allows for potential applications in much larger data sets.

It was conjectured that any neural network of any structure and arbitrary differentiable transfer functions at the nodes cannot learn the following problem sample efficiently when trained with gradient descent: The instances are the rows of a $d$-dimensional Hadamard matrix and the target is one of the features, i.e. very sparse. We essentially prove this conjecture: We show that after receiving a random training set of size $k < d$, the expected square loss is still $1-\frac{k}{(d-1)}$. The only requirement needed is that the input layer is fully connected and the initial weight vectors of the input nodes are chosen from a rotation invariant distribution. Surprisingly the same type of problem can be solved drastically more efficient by a simple 2-layer linear neural network in which the $d$ inputs are connected to the output node by chains of length 2 (Now the input layer has only one edge per input). When such a network is trained by gradient descent, then it has been shown that its expected square loss is $\frac{\log d}{k}$. Our lower bounds essentially show that a sparse input layer is needed to sample efficiently learn sparse targets with gradient descent when the number of examples is less than the number of input features.

Feature selection often leads to increased model interpretability, faster computation, and improved model performance by discarding irrelevant or redundant features. While feature selection is a well-studied problem with many widely-used techniques, there are typically two key challenges: i) many existing approaches become computationally intractable in huge-data settings with millions of observations and features; and ii) the statistical accuracy of selected features degrades in high-noise, high-correlation settings, thus hindering reliable model interpretation. We tackle these problems by proposing Stable Minipatch Selection (STAMPS) and Adaptive STAMPS (AdaSTAMPS). These are meta-algorithms that build ensembles of selection events of base feature selectors trained on many tiny, (adaptively-chosen) random subsets of both the observations and features of the data, which we call minipatches. Our approaches are general and can be employed with a variety of existing feature selection strategies and machine learning techniques. In addition, we provide theoretical insights on STAMPS and empirically demonstrate that our approaches, especially AdaSTAMPS, dominate competing methods in terms of feature selection accuracy and computational time.

Convolutional neural networks often dominate fully-connected counterparts in generalization performance, especially on image classification tasks. This is often explained in terms of 'better inductive bias'. However, this has not been made mathematically rigorous, and the hurdle is that the fully connected net can always simulate the convolutional net (for a fixed task). Thus the training algorithm plays a role. The current work describes a natural task on which a provable sample complexity gap can be shown, for standard training algorithms. We construct a single natural distribution on $\mathbb{R}^d\times\{\pm 1\}$ on which any orthogonal-invariant algorithm (i.e. fully-connected networks trained with most gradient-based methods from gaussian initialization) requires $\Omega(d^2)$ samples to generalize while $O(1)$ samples suffice for convolutional architectures. Furthermore, we demonstrate a single target function, learning which on all possible distributions leads to an $O(1)$ vs $\Omega(d^2/\varepsilon)$ gap. The proof relies on the fact that SGD on fully-connected network is orthogonal equivariant. Similar results are achieved for $\ell_2$ regression and adaptive training algorithms, e.g. Adam and AdaGrad, which are only permutation equivariant.

We prove a new upper bound on the generalization gap of classifiers that are obtained by first using self-supervision to learn a representation $r$ of the training data, and then fitting a simple (e.g., linear) classifier $g$ to the labels. Specifically, we show that (under the assumptions described below) the generalization gap of such classifiers tends to zero if $\mathsf{C}(g) \ll n$, where $\mathsf{C}(g)$ is an appropriately-defined measure of the simple classifier $g$'s complexity, and $n$ is the number of training samples. We stress that our bound is independent of the complexity of the representation $r$. We do not make any structural or conditional-independence assumptions on the representation-learning task, which can use the same training dataset that is later used for classification. Rather, we assume that the training procedure satisfies certain natural noise-robustness (adding small amount of label noise causes small degradation in performance) and rationality (getting the wrong label is not better than getting no label at all) conditions that widely hold across many standard architectures. We show that our bound is non-vacuous for many popular representation-learning based classifiers on CIFAR-10 and ImageNet, including SimCLR, AMDIM and MoCo.

Computing the discrepancy between time series of variable sizes is notoriously challenging. While dynamic time warping (DTW) is popularly used for this purpose, it is not differentiable everywhere and is known to lead to bad local optima when used as a "loss". Soft-DTW addresses these issues, but it is not a positive definite divergence: due to the bias introduced by entropic regularization, it can be negative and it is not minimized when the time series are equal. We propose in this paper a new divergence, dubbed soft-DTW divergence, which aims to correct these issues. We study its properties; in particular, under conditions on the ground cost, we show that it is non-negative and minimized when the time series are equal. We also propose a new "sharp" variant by further removing entropic bias. We showcase our divergences on time series averaging and demonstrate significant accuracy improvements compared to both DTW and soft-DTW on 84 time series classification datasets.

Gaussian processes are employed for non-parametric regression in a Bayesian setting. They generalize linear regression, embedding the inputs in a latent manifold inside an infinite-dimensional reproducing kernel Hilbert space. We can augment the inputs with the observations of low-fidelity models in order to learn a more expressive latent manifold and thus increment the model's accuracy. This can be realized recursively with a chain of Gaussian processes with incrementally higher fidelity. We would like to extend these multi-fidelity model realizations to case studies affected by a high-dimensional input space but with low intrinsic dimensionality. In this cases physical supported or purely numerical low-order models are still affected by the curse of dimensionality when queried for responses. When the model's gradient information is provided, the presence of an active subspace can be exploited to design low-fidelity response surfaces and thus enable Gaussian process multi-fidelity regression, without the need to perform new simulations. This is particularly useful in the case of data scarcity. In this work we present a multi-fidelity approach involving active subspaces and we test it on two different high-dimensional benchmarks.

Supervised learning requires a large amount of training data, limiting its application where labeled data is scarce. To compensate for data scarcity, one possible method is to utilize auxiliary tasks to provide additional supervision for the main task. Assigning and optimizing the importance weights for different auxiliary tasks remains an crucial and largely understudied research question. In this work, we propose a method to automatically reweight auxiliary tasks in order to reduce the data requirement on the main task. Specifically, we formulate the weighted likelihood function of auxiliary tasks as a surrogate prior for the main task. By adjusting the auxiliary task weights to minimize the divergence between the surrogate prior and the true prior of the main task, we obtain a more accurate prior estimation, achieving the goal of minimizing the required amount of training data for the main task and avoiding a costly grid search. In multiple experimental settings (e.g. semi-supervised learning, multi-label classification), we demonstrate that our algorithm can effectively utilize limited labeled data of the main task with the benefit of auxiliary tasks compared with previous task reweighting methods. We also show that under extreme cases with only a few extra examples (e.g. few-shot domain adaptation), our algorithm results in significant improvement over the baseline.