We propose a new method for the problem of controlling linear dynamical systems under partial observation and adversarial disturbances. Our new algorithm, Double Spectral Control (DSC), matches the best known regret guarantees while exponentially improving runtime complexity over previous approaches in its dependence on the system's stability margin. Our key innovation is a two-level spectral approximation strategy, leveraging double convolution with a universal basis of spectral filters, enabling efficient and accurate learning of the best linear dynamical controllers.

With large neural models becoming increasingly accurate and powerful, they have raised privacy and transparency concerns on data usage. Therefore, data platforms, regulations and user expectations are rapidly evolving leading to enforcing privacy via aggregation. We focus on the use case of online advertising where the emergence of aggregate data is imminent and can significantly impact the multi-billion dollar industry. In aggregated datasets, labels are assigned to groups of data points rather than individual data points. This leads to a formulation of a weakly supervised task - Learning from Label Proportions where a model is trained on groups (a.k.a bags) of instances and their corresponding label proportions to predict labels for individual instances. While learning on aggregate data due to privacy concerns is becoming increasingly popular there is no large scale benchmark for measuring performance and guiding improvements on this important task. We propose LLP-Bench - a web scale benchmark with ~ 70 datasets and 45 million datapoints. To the best of our knowledge, LLP-Bench is the first large scale tabular LLP benchmark with an extensive diversity in constituent datasets, realistic in terms of the sponsored search datasets used and aggregation mechanisms followed. Through more than 3000 experiments we compare the performance of 9 SOTA methods in detail. To the best of our knowledge, this is the first study that compares diverse approaches in such depth.

Learning from label proportions (LLP) is a generalization of supervised learning in which the training data is available as sets or bags of feature-vectors (instances) along with the average instance-label of each bag. The goal is to train a good instance classifier. While most previous works on LLP have focused on training models on such training data, computational learnability of LLP was only recently explored by Saket (2021, 2022) who showed worst case intractability of properly learning linear threshold functions (LTFs) from label proportions. However, their work did not rule out efficient algorithms for this problem for natural distributions. In this work we show that it is indeed possible to efficiently learn LTFs using LTFs when given access to random bags of some label proportion in which feature-vectors are, conditioned on their labels, independently sampled from a Gaussian distribution $N(µ, Σ)$. Our work shows that a certain matrix – formed using covariances of the differences of feature-vectors sampled from the bags with and without replacement – necessarily has its principal component, after a transformation, in the direction of the normal vector of the LTF. Our algorithm estimates the means and covariance matrices using subgaussian concentration bounds which we show can be applied to efficiently sample bags for approximating the normal direction. Using this in conjunction with novel generalization error bounds in the bag setting, we show that a low error hypothesis LTF can be identified. For some special cases of the $N(0, I)$ distribution we provide a simpler mean estimation based algorithm. We include an experimental evaluation of our learning algorithms along with a comparison with those of Saket (2021, 2022) and random LTFs, demonstrating the effectiveness of our techniques.