We consider the problem of learning stabilizable systems governed by nonlinear state equation $h_{t+1}=\phi(h_t,u_t;\theta)+w_t$. Here $\theta$ is the unknown system dynamics, $h_t $ is the state, $u_t$ is the input and $w_t$ is the additive noise vector. We study gradient based algorithms to learn the system dynamics $\theta$ from samples obtained from a single finite trajectory. If the system is run by a stabilizing input policy, we show that temporally-dependent samples can be approximated by i.i.d. samples via a truncation argument by using mixing-time arguments. We then develop new guarantees for the uniform convergence of the gradients of empirical loss. Unlike existing work, our bounds are noise sensitive which allows for learning ground-truth dynamics with high accuracy and small sample complexity. Together, our results facilitate efficient learning of the general nonlinear system under stabilizing policy. We specialize our guarantees to entry-wise nonlinear activations and verify our theory in various numerical experiments

This paper revisits the celebrated temporal difference (TD) learning algorithm for the policy evaluation in reinforcement learning. Typically, the performance of the plain-vanilla TD algorithm is sensitive to the choice of stepsizes. Oftentimes, TD suffers from slow convergence. Motivated by the tight connection between the TD learning algorithm and the stochastic gradient methods, we develop the first adaptive variant of the TD learning algorithm with linear function approximation that we term AdaTD. In contrast to the original TD, AdaTD is robust or less sensitive to the choice of stepsizes. Analytically, we establish that to reach an $\epsilon$ accuracy, the number of iterations needed is $\tilde{O}(\epsilon^2\ln^4\frac{1}{\epsilon}/\ln^4\frac{1}{\rho})$, where $\rho$ represents the speed of the underlying Markov chain converges to the stationary distribution. This implies that the iteration complexity of AdaTD is no worse than that of TD in the worst case. Going beyond TD, we further develop an adaptive variant of TD($\lambda$), which is referred to as AdaTD($\lambda$). We evaluate the empirical performance of AdaTD and AdaTD($\lambda$) on several standard reinforcement learning tasks in OpenAI Gym on both linear and nonlinear function approximation, which demonstrate the effectiveness of our new approaches over existing ones.

We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of machines typically yields poor performance. It often leads to the dependence of convergence rate on maximum Lipschitz constant of gradients across the devices. In this paper, we propose a novel \emph{adaptive} sampling of machines specially catered to these settings. Our method relies on an adaptive estimate of local Lipschitz constants base on the information of past gradients. We show that the new way improves the dependence of convergence rate from maximum Lipschitz constant to \emph{average} Lipschitz constant across machines, thereby, significantly accelerating the convergence. Our experiments demonstrate that our method indeed speeds up the convergence of the standard SVRG algorithm in heterogeneous environments.

Estimating individual treatment effects from data of randomized experiments is a critical task in causal inference. The Stable Unit Treatment Value Assumption (SUTVA) is usually made in causal inference. However, interference can introduce bias when the assigned treatment on one unit affects the potential outcomes of the neighboring units. This interference phenomenon is known as spillover effect in economics or peer effect in social science. Usually, in randomized experiments or observational studies with interconnected units, one can only observe treatment responses under interference. Hence, how to estimate the superimposed causal effect and recover the individual treatment effect in the presence of interference becomes a challenging task in causal inference. In this work, we study causal effect estimation under general network interference using GNNs, which are powerful tools for capturing the dependency in the graph. After deriving causal effect estimators, we further study intervention policy improvement on the graph under capacity constraint. We give policy regret bounds under network interference and treatment capacity constraint. Furthermore, a heuristic graph structure-dependent error bound for GNN-based causal estimators is provided.

Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or Krylov methods. While there is classical and comprehensive analysis for subspace convergence guarantees with respect to the spectral norm, in many modern applications other notions of subspace distance are more appropriate. Recent theoretical work has focused on perturbations of subspaces measured in the $\ell_{2 \to \infty}$ norm, but does not consider the actual computation of eigenvectors. Here we address the convergence of subspace iteration when distances are measured in the $\ell_{2 \to \infty}$ norm and provide deterministic bounds. We complement our analysis with a practical stopping criterion and demonstrate its applicability via numerical experiments. Our results show that one can get comparable performance on downstream tasks while requiring fewer iterations, thereby saving substantial computational time.

While access to large amounts of labeled data has enabled the training of big models with great successes in applied machine learning, it remains a key bottleneck. In numerous settings (e.g., scientific measurements, experiments, medicine) obtaining a large number of labels can be prohibitively expensive, error prone, or otherwise infeasible. When labels are scarce, a common alternative is to use additional sources of information: "weak labels" that contain information about the "strong" target task and are more readily available, e.g., a related task, or noisy versions of strong labels from non-experts or cheaper measurements. Such a setting is called weakly supervised learning, and given its great practical relevance it has received much attention [11, 25, 34, 43, 67]. A prominent example that enabled breakthrough results in computer vision and is now standard, is to pre-train a complex model on a related, large data task, and to then use the learned features for fine-tuning for instance the last layer on the small-data target task [15, 21, 49, 63].

Federated learning is a technique that enables distributed clients to collaboratively learn a shared machine learning model while keeping their training data localized. This reduces data privacy risks, however, privacy concerns still exist since it is possible to leak information about the training dataset from the trained model's weights or parameters. Setting up a federated learning environment, especially with security and privacy guarantees, is a time-consuming process with numerous configurations and parameters that can be manipulated. In order to help clients ensure that collaboration is feasible and to check that it improves their model accuracy, a real-world simulator for privacy-preserving and secure federated learning is required. In this paper, we introduce PrivacyFL, which is an extensible, easily configurable and scalable simulator for federated learning environments. Its key features include latency simulation, robustness to client departure, support for both centralized and decentralized learning, and configurable privacy and security mechanisms based on differential privacy and secure multiparty computation. In this paper, we motivate our research, describe the architecture of the simulator and associated protocols, and discuss its evaluation in numerous scenarios that highlight its wide range of functionality and its advantages. Our paper addresses a significant real-world problem: checking the feasibility of participating in a federated learning environment under a variety of circumstances. It also has a strong practical impact because organizations such as hospitals, banks, and research institutes, which have large amounts of sensitive data and would like to collaborate, would greatly benefit from having a system that enables them to do so in a privacy-preserving and secure manner.

Multi-output regression seeks to infer multiple latent functions using data from multiple groups/sources while accounting for potential between-group similarities. In this paper, we consider multi-output regression under a weakly-supervised setting where a subset of data points from multiple groups are unlabeled. We use dependent Gaussian processes for multiple outputs constructed by convolutions with shared latent processes. We introduce hyperpriors for the multinomial probabilities of the unobserved labels and optimize the hyperparameters which we show improves estimation. We derive two variational bounds: (i) a modified variational bound for fast and stable convergence in model inference, (ii) a scalable variational bound that is amenable to stochastic optimization. We use experiments on synthetic and real-world data to show that the proposed model outperforms state-of-the-art models with more accurate estimation of multiple latent functions and unobserved labels.

Gaussian mixture reduction (GMR) is the problem of approximating a finite Gaussian mixture by one with fewer components. It is widely used in density estimation, nonparametric belief propagation, and Bayesian recursive filtering. Although optimization and clustering-based algorithms have been proposed for GMR, they are either computationally expensive or lacking in theoretical supports. In this work, we propose to perform GMR by minimizing the entropic regularized composite transportation distance between two mixtures. We show our approach provides a unified framework for GMR that is both interpretable and computationally efficient. Our work also bridges the gap between optimization and clustering-based approaches for GMR. A Majorization-Minimization algorithm is developed for our optimization problem and its theoretical convergence is also established in this paper. Empirical experiments are also conducted to show the effectiveness of GMR. The effect of the choice of transportation cost on the performance of GMR is also investigated.

Random Feature (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with $P$ features, $N$ data points, and a ridge $\lambda$, we show that the average (i.e. expected) RF predictor is close to a KRR predictor with an effective ridge $\tilde{\lambda}$. We show that $\tilde{\lambda} > \lambda$ and $\tilde{\lambda} \searrow \lambda$ monotonically as $P$ grows, thus revealing the implicit regularization effect of finite RF sampling. We then compare the risk (i.e. test error) of the $\tilde{\lambda}$-KRR predictor with the average risk of the $\lambda$-RF predictor and obtain a precise and explicit bound on their difference. Finally, we empirically find an extremely good agreement between the test errors of the average $\lambda$-RF predictor and $\tilde{\lambda}$-KRR predictor.