We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the estimation problem can be thought of as solving a zero-sum game between a modeler who is optimizing over the hypothesis space of the target model and an adversary who identifies violating moments over a test function space. We analyze the statistical estimation rate of the resulting estimator for arbitrary hypothesis spaces, with respect to an appropriate analogue of the mean squared error metric, for ill-posed inverse problems. We show that when the minimax criterion is regularized with a second moment penalty on the test function and the test function space is sufficiently rich, then the estimation rate scales with the critical radius of the hypothesis and test function spaces, a quantity which typically gives tight fast rates. Our main result follows from a novel localized Rademacher analysis of statistical learning problems defined via minimax objectives. We provide applications of our main results for several hypothesis spaces used in practice such as: reproducing kernel Hilbert spaces, high dimensional sparse linear functions, spaces defined via shape constraints, ensemble estimators such as random forests, and neural networks. For each of these applications we provide computationally efficient optimization methods for solving the corresponding minimax problem (e.g. stochastic first-order heuristics for neural networks). In several applications, we show how our modified mean squared error rate, combined with conditions that bound the ill-posedness of the inverse problem, lead to mean squared error rates. We conclude with an extensive experimental analysis of the proposed methods.

However, its violation can lead to models that learn to exploit spurious correlations in the training data, rendering them vulnerable to adversarial interventions, undermining their reliability, and limiting their practical application. To mitigate this problem, we present a method for learning multiple models, incorporating an objective that pressures each to learn a distinct way to solve the task. We propose a notion of diversity based on minimizing the conditional total correlation of final layer representations across models given the label, which we approximate using a variational estimator and minimize using adversarial training. To demonstrate our framework's ability to facilitate rapid adaptation to distribution shift, we train a number of simple classifiers from scratch on the frozen outputs of our models using a small amount of data from the shifted distribution. Under this evaluation protocol, our framework significantly outperforms a baseline trained using the empirical risk minimization principle.

With the rising number of machine learning competitions, the world has witnessed an exciting race for the best algorithms. However, the involved data selection process may fundamentally suffer from evidence ambiguity and concept drift issues, thereby possibly leading to deleterious effects on the performance of various models. This paper proposes a new Reinforced Data Sampling (RDS) method to learn how to sample data adequately on the search for useful models and insights. We formulate the optimisation problem of model diversification $\delta{-div}$ in data sampling to maximise learning potentials and optimum allocation by injecting model diversity. This work advocates the employment of diverse base learners as value functions such as neural networks, decision trees, or logistic regressions to reinforce the selection process of data subsets with multi-modal belief. We introduce different ensemble reward mechanisms, including soft voting and stochastic choice to approximate optimal sampling policy. The evaluation conducted on four datasets evidently highlights the benefits of using RDS method over traditional sampling approaches. Our experimental results suggest that the trainable sampling for model diversification is useful for competition organisers, researchers, or even starters to pursue full potentials of various machine learning tasks such as classification and regression. The source code is available at https://github.com/probeu/RDS.

Deep neural networks have successfully been trained in various application areas with stochastic gradient descent. However, there exists no rigorous mathematical explanation why this works so well. The training of neural networks with stochastic gradient descent has four different discretization parameters: (i) the network architecture; (ii) the size of the training data; (iii) the number of gradient steps; and (iv) the number of randomly initialized gradient trajectories. While it can be shown that the approximation error converges to zero if all four parameters are sent to infinity in the right order, we demonstrate in this paper that stochastic gradient descent fails to converge for rectified linear unit networks if their depth is much larger than their width and the number of random initializations does not increase to infinity fast enough.

End-to-end training of neural network solvers for combinatorial problems such as the Travelling Salesman Problem is intractable and inefficient beyond a few hundreds of nodes. While state-of-the-art Machine Learning approaches perform closely to classical solvers for trivially small sizes, they are unable to generalize the learnt policy to larger instances of practical scales. Towards leveraging transfer learning to solve large-scale TSPs, this paper identifies inductive biases, model architectures and learning algorithms that promote generalization to instances larger than those seen in training. Our controlled experiments provide the first principled investigation into such zero-shot generalization, revealing that extrapolating beyond training data requires rethinking the entire neural combinatorial optimization pipeline, from network layers and learning paradigms to evaluation protocols.

Disentanglement is an enjoyable property in representation learning which increases the interpretability of generative models such as Variational Auto-Encoders (VAE), Generative Adversarial Models and their many variants. In the context of latent space models, this work presents a representation learning framework that explicitly promotes disentanglement thanks to the combination of an auto-encoder with Principal Component Analysis (PCA) in latent space. The proposed objective is the sum of an auto-encoder error term along with a PCA reconstruction error in the feature space. This has an interpretation of a Restricted Kernel Machine with an interconnection matrix on the Stiefel manifold. The construction encourages a matching between the principal directions in latent space and the directions of orthogonal variation in data space. The training algorithm involves a stochastic optimization method on the Stiefel manifold, which increases only marginally the computing time compared to an analogous VAE. Our theoretical discussion and various experiments show that the proposed model improves over many VAE variants along with special emphasis on disentanglement learning.

One fundamental problem in the learning treatment effect from observational data is confounder identification and balancing. Most of the previous methods realized confounder balancing by treating all observed variables as confounders, ignoring the identification of confounders and non-confounders. In general, not all the observed variables are confounders which are the common causes of both the treatment and the outcome, some variables only contribute to the treatment and some contribute to the outcome. Balancing those non-confounders would generate additional bias for treatment effect estimation. By modeling the different relations among variables, treatment and outcome, we propose a synergistic learning framework to 1) identify and balance confounders by learning decomposed representation of confounders and non-confounders, and simultaneously 2) estimate the treatment effect in observational studies via counterfactual inference. Our empirical results demonstrate that the proposed method can precisely identify and balance confounders, while the estimation of the treatment effect performs better than the state-of-the-art methods on both synthetic and real-world datasets.

A spectral mixture (SM) kernel is a flexible kernel used to model any stationary covariance function. Although it is useful in modeling data, the learning of the SM kernel is generally difficult because optimizing a large number of parameters for the SM kernel typically induces an over-fitting, particularly when a gradient-based optimization is used. Also, a longer training time is required. To improve the training, we propose an approximate Bayesian inference for the SM kernel. Specifically, we employ the variational distribution of the spectral points to approximate SM kernel with a random Fourier feature. We optimize the variational parameters by applying a sampling-based variational inference to the derived evidence lower bound (ELBO) estimator constructed from the approximate kernel. To improve the inference, we further propose two additional strategies: (1) a sampling strategy of spectral points to estimate the ELBO estimator reliably and thus its associated gradient, and (2) an approximate natural gradient to accelerate the convergence of the parameters. The proposed inference combined with two strategies accelerates the convergence of the parameters and leads to better optimal parameters.

We introduce Markov Random Geometric Graphs (MRGGs), a growth model for temporal dynamic networks. It is based on a Markovian latent space dynamic: consecutive latent points are sampled on the Euclidean Sphere using an unknown Markov kernel; and two nodes are connected with a probability depending on a unknown function of their latent geodesic distance. More precisely, at each stamp-time k we add a latent point X k sampled by jumping from the previous one X k--1 in a direction chosen uniformly Y k and with a length r k drawn from an unknown distribution called the latitude function. The connection probabilities between each pair of nodes are equal to the envelope function of the distance between these two latent points. We provide theoretical guarantees for the non-parametric estimation of the latitude and the envelope functions. We propose an efficient algorithm that achieves those non-parametric estimation tasks based on an ad-hoc Hierarchical Agglomerative Clustering approach, and we deploy this analysis on a real data-set given by exchange of messages on a social network.

In many machine learning applications one optimizes a non-convex loss function; this is often achieved using simple descending algorithms such as gradient descent or its stochastic variations. The positive results obtained in practice are often hard to justify from the theoretical point of view, and this apparent contradiction between non-convex landscapes and good performance of simple algorithms is a recurrent problem in machine learning. A successful line of research has studied the geometrical properties of the loss landscape, distinguishing between good minima - that lead to good generalization error - and spurious minima - associated with bad generalization error. The results showed that in some regimes, for several problems from matrix completion [1] to wide neural networks [2, 3], spurious minima disappear and consequently under weak assumptions [4] gradient descent will converge to good minima. However, these results do not justify numerous other results showing that good and spurious minima are present, but systematically gradient descent works [5, 6]. In [7] it was theoretically shown that in a toy model - the spiked matrix-tensor model - it is possible to find good minima with high probability in a regime where exponentially many spurious minima are provably present. In [8] it was shown that this is due to the presence of the so-called threshold states in the landscape, that play a key role in the dynamics of the gradient flow [9,10]: at first attracting it, and successively triggering the converge towards lower minima under certain conditions [11, 12]. However, the spiked matrix-tensor model is an unsupervised learning model and it remained open whether the picture put forward in [7, 8] happens also in learning with neural networks.