By enabling constraint-aware online model adaptation, model predictive control using Gaussian process (GP) regression has exhibited impressive performance in real-world applications and received considerable attention in the learning-based control community. Yet, solving the resulting optimal control problem in real-time generally remains a major challenge, due to i) the increased number of augmented states in the optimization problem, as well as ii) computationally expensive evaluations of the posterior mean and covariance and their respective derivatives. To tackle these challenges, we employ i) a tailored Jacobian approximation in a sequential quadratic programming (SQP) approach, and combine it with ii) a parallelizable GP inference and automatic differentiation framework. Reducing the numerical complexity with respect to the state dimension $n_x$ for each SQP iteration from $\mathcal{O}(n_x^6)$ to $\mathcal{O}(n_x^3)$, and accelerating GP evaluations on a graphical processing unit, the proposed algorithm computes suboptimal, yet feasible solutions at drastically reduced computation times and exhibits favorable local convergence properties. Numerical experiments verify the scaling properties and investigate the runtime distribution across different parts of the algorithm.

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal $L_2^2$-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

Hyperparameter tuning of deep learning models can lead to order-of-magnitude performance gains for the same amount of compute. Despite this, systematic tuning is uncommon, particularly for large models, which are expensive to evaluate and tend to have many hyperparameters, necessitating difficult judgment calls about tradeoffs, budgets, and search bounds. To address these issues and propose a practical method for robustly tuning large models, we present Cost-Aware Pareto Region Bayesian Search (CARBS), a Bayesian optimization algorithm that performs local search around the performance-cost Pareto frontier. CARBS does well even in unbounded search spaces with many hyperparameters, learns scaling relationships so that it can tune models even as they are scaled up, and automates much of the "black magic" of tuning. Among our results, we effectively solve the entire ProcGen benchmark just by tuning a simple baseline (PPO, as provided in the original ProcGen paper). We also reproduce the model size vs. training tokens scaling result from the Chinchilla project (Hoffmann et al. 2022), while simultaneously discovering scaling laws for every other hyperparameter, via an easy automated process that uses significantly less compute and is applicable to any deep learning problem (not just language models).

Stochastic min-max optimization has gained interest in the machine learning community with the advancements in GANs and adversarial training. Although game optimization is fairly well understood in the deterministic setting, some issues persist in the stochastic regime. Recent work has shown that stochastic gradient descent-ascent methods such as the optimistic gradient are highly sensitive to noise or can fail to converge. Although alternative strategies exist, they can be prohibitively expensive. We introduce Omega, a method with optimistic-like updates that mitigates the impact of noise by incorporating an EMA of historic gradients in its update rule. We also explore a variation of this algorithm that incorporates momentum. Although we do not provide convergence guarantees, our experiments on stochastic games show that Omega outperforms the optimistic gradient method when applied to linear players.

Federated Learning (FL) has emerged as a promising approach for collaborative model training without sharing private data. However, privacy concerns regarding information exchanged during FL have received significant research attention. Gradient Inversion Attacks (GIAs) have been proposed to reconstruct the private data retained by local clients from the exchanged gradients. While recovering private data, the data dimensions and the model complexity increase, which thwart data reconstruction by GIAs. Existing methods adopt prior knowledge about private data to overcome those challenges. In this paper, we first observe that GIAs with gradients from a single iteration fail to reconstruct private data due to insufficient dimensions of leaked gradients, complex model architectures, and invalid gradient information. We investigate a Temporal Gradient Inversion Attack with a Robust Optimization framework, called TGIAs-RO, which recovers private data without any prior knowledge by leveraging multiple temporal gradients. To eliminate the negative impacts of outliers, e.g., invalid gradients for collaborative optimization, robust statistics are proposed. Theoretical guarantees on the recovery performance and robustness of TGIAs-RO against invalid gradients are also provided. Extensive empirical results on MNIST, CIFAR10, ImageNet and Reuters 21578 datasets show that the proposed TGIAs-RO with 10 temporal gradients improves reconstruction performance compared to state-of-the-art methods, even for large batch sizes (up to 128), complex models like ResNet18, and large datasets like ImageNet (224*224 pixels). Furthermore, the proposed attack method inspires further exploration of privacy-preserving methods in the context of FL.

The dynamical stability of optimization methods at the vicinity of minima of the loss has recently attracted significant attention. For gradient descent (GD), stable convergence is possible only to minima that are sufficiently flat w.r.t. the step size, and those have been linked with favorable properties of the trained model. However, while the stability threshold of GD is well-known, to date, no explicit expression has been derived for the exact threshold of stochastic GD (SGD). In this paper, we derive such a closed-form expression. Specifically, we provide an explicit condition on the step size $\eta$ that is both necessary and sufficient for the stability of SGD in the mean square sense. Our analysis sheds light on the precise role of the batch size $B$. Particularly, we show that the stability threshold is a monotonically non-decreasing function of the batch size, which means that reducing the batch size can only hurt stability. Furthermore, we show that SGD's stability threshold is equivalent to that of a process which takes in each iteration a full batch gradient step w.p. $1-p$, and a single sample gradient step w.p. $p$, where $p \approx 1/B $. This indicates that even with moderate batch sizes, SGD's stability threshold is very close to that of GD's. Finally, we prove simple necessary conditions for stability, which depend on the batch size, and are easier to compute than the precise threshold. We demonstrate our theoretical findings through experiments on the MNIST dataset.

Despite extraordinary progress, current machine learning systems have been shown to be brittle against adversarial examples: seemingly innocuous but carefully crafted perturbations of test examples that cause machine learning predictors to misclassify. Can we learn predictors robust to adversarial examples? and how? There has been much empirical interest in this contemporary challenge in machine learning, and in this thesis, we address it from a theoretical perspective. In this thesis, we explore what robustness properties can we hope to guarantee against adversarial examples and develop an understanding of how to algorithmically guarantee them. We illustrate the need to go beyond traditional approaches and principles such as empirical risk minimization and uniform convergence, and make contributions that can be categorized as follows: (1) introducing problem formulations capturing aspects of emerging practical challenges in robust learning, (2) designing new learning algorithms with provable robustness guarantees, and (3) characterizing the complexity of robust learning and fundamental limitations on the performance of any algorithm.

Since 2006, the MaxSAT Evaluation (MSE) Bacchus et al. [2022] has been held annually with the primary objective of advancing MaxSAT technology and assessing its current state-of-the-art. The evaluation consists of multiple solvers being tested on various benchmarks across different evaluation tracks. This event has undeniably spurred the MaxSAT community to create more cutting-edge solvers and enhance their competitiveness. It is not surprising that solver performance depends on several factors, including the power of the algorithm implemented by the solver, proper configuration of solver parameters to unleash its full potential, and implementation issues. Therefore, we must interpret the MaxSAT Evaluation ranking results carefully and derive conclusions according to the goal of our analysis.

Moment restrictions and their conditional counterparts emerge in many areas of machine learning and statistics ranging from causal inference to reinforcement learning. Estimators for these tasks, generally called methods of moments, include the prominent generalized method of moments (GMM) which has recently gained attention in causal inference. GMM is a special case of the broader family of empirical likelihood estimators which are based on approximating a population distribution by means of minimizing a $\varphi$-divergence to an empirical distribution. However, the use of $\varphi$-divergences effectively limits the candidate distributions to reweightings of the data samples. We lift this long-standing limitation and provide a method of moments that goes beyond data reweighting. This is achieved by defining an empirical likelihood estimator based on maximum mean discrepancy which we term the kernel method of moments (KMM). We provide a variant of our estimator for conditional moment restrictions and show that it is asymptotically first-order optimal for such problems. Finally, we show that our method achieves competitive performance on several conditional moment restriction tasks.

This paper addresses the communication issues when estimating hyper-gradients in decentralized federated learning (FL). Hyper-gradients in decentralized FL quantifies how the performance of globally shared optimal model is influenced by the perturbations in clients' hyper-parameters. In prior work, clients trace this influence through the communication of Hessian matrices over a static undirected network, resulting in (i) excessive communication costs and (ii) inability to make use of more efficient and robust networks, namely, time-varying directed networks. To solve these issues, we introduce an alternative optimality condition for FL using an averaging operation on model parameters and gradients. We then employ Push-Sum as the averaging operation, which is a consensus optimization technique for time-varying directed networks. As a result, the hyper-gradient estimator derived from our optimality condition enjoys two desirable properties; (i) it only requires Push-Sum communication of vectors and (ii) it can operate over time-varying directed networks. We confirm the convergence of our estimator to the true hyper-gradient both theoretically and empirically, and we further demonstrate that it enables two novel applications: decentralized influence estimation and personalization over time-varying networks.