We introduce an efficient method for learning linear models from uncertain data, where uncertainty is represented as a set of possible variations in the data, leading to predictive multiplicity. Our approach leverages abstract interpretation and zonotopes, a type of convex polytope, to compactly represent these dataset variations, enabling the symbolic execution of gradient descent on all possible worlds simultaneously. We develop techniques to ensure that this process converges to a fixed point and derive closed-form solutions for this fixed point. Our method provides sound over-approximations of all possible optimal models and viable prediction ranges. We demonstrate the effectiveness of our approach through theoretical and empirical analysis, highlighting its potential to reason about model and prediction uncertainty due to data quality issues in training data.

Existing first-order optimizers mainly include two branches: classical optimizers represented by Stochastic Gradient Descent (SGD) and adaptive optimizers represented by Adam, along with their many derivatives. The debate over the merits and demerits of these two types of optimizers has persisted for a decade. In practical experience, it is generally considered that SGD is more suitable for tasks like Computer Vision(CV), while adaptive optimizers are widely used in tasks with sparse gradients, such as Large Language Models(LLM). Although adaptive optimizers always offer better convergence speeds in almost all tasks, they can lead to over-fitting in some cases, resulting in poorer generalization performance compared to SGD in certain tasks. Even in Large Language Models, Adam continues to face challenges, and its original strategy may not always have an advantage due to the introduction of improvements such as gradient clipping. With a wide variety of optimization methods available, it is essential to introduce a unified, interpretable theory. This paper will discuss under the framework of first-order optimizers and, through the intervention of the balance theory, will for the first time propose a unified strategy to integrate all first-order optimization methods.

Adaptive moment estimation (Adam), as a Stochastic Gradient Descent (SGD) variant, has gained widespread popularity in federated learning (FL) due to its fast convergence. However, federated Adam (FedAdam) algorithms suffer from a threefold increase in uplink communication overhead compared to federated SGD (FedSGD) algorithms, which arises from the necessity to transmit both local model updates and first and second moment estimates from distributed devices to the centralized server for aggregation. Driven by this issue, we propose a novel sparse FedAdam algorithm called FedAdam-SSM, wherein distributed devices sparsify the updates of local model parameters and moment estimates and subsequently upload the sparse representations to the centralized server. To further reduce the communication overhead, the updates of local model parameters and moment estimates incorporate a shared sparse mask (SSM) into the sparsification process, eliminating the need for three separate sparse masks. Theoretically, we develop an upper bound on the divergence between the local model trained by FedAdam-SSM and the desired model trained by centralized Adam, which is related to sparsification error and imbalanced data distribution. By minimizing the divergence bound between the model trained by FedAdam-SSM and centralized Adam, we optimize the SSM to mitigate the learning performance degradation caused by sparsification error. Additionally, we provide convergence bounds for FedAdam-SSM in both convex and non-convex objective function settings, and investigate the impact of local epoch, learning rate and sparsification ratio on the convergence rate of FedAdam-SSM. Experimental results show that FedAdam-SSM outperforms baselines in terms of convergence rate (over 1.1$\times$ faster than the sparse FedAdam baselines) and test accuracy (over 14.5\% ahead of the quantized FedAdam baselines).

Personalized Federated Learning (PFL) is proposed to find the greatest personalized models for each client. To avoid the central failure and communication bottleneck in the server-based FL, we concentrate on the Decentralized Personalized Federated Learning (DPFL) that performs distributed model training in a Peer-to-Peer (P2P) manner. Most personalized works in DPFL are based on undirected and symmetric topologies, however, the data, computation and communication resources heterogeneity result in large variances in the personalized models, which lead the undirected aggregation to suboptimal personalized performance and unguaranteed convergence. To address these issues, we propose a directed collaboration DPFL framework by incorporating stochastic gradient push and partial model personalized, called \textbf{D}ecentralized \textbf{Fed}erated \textbf{P}artial \textbf{G}radient \textbf{P}ush (\textbf{DFedPGP}). It personalizes the linear classifier in the modern deep model to customize the local solution and learns a consensus representation in a fully decentralized manner. Clients only share gradients with a subset of neighbors based on the directed and asymmetric topologies, which guarantees flexible choices for resource efficiency and better convergence. Theoretically, we show that the proposed DFedPGP achieves a superior convergence rate of $\mathcal{O}(\frac{1}{\sqrt{T}})$ in the general non-convex setting, and prove the tighter connectivity among clients will speed up the convergence. The proposed method achieves state-of-the-art (SOTA) accuracy in both data and computation heterogeneity scenarios, demonstrating the efficiency of the directed collaboration and partial gradient push.

Federated learning (FL) was recently proposed to securely train models with data held over multiple locations ("clients") under the coordination of a central server. Two major challenges hindering the performance of FL algorithms are long training times caused by straggling clients, and a decline in model accuracy under non-iid local data distributions ("client drift"). In this work, we propose and analyze AsynchRonous Exact Averaging (AREA), a new stochastic (sub)gradient algorithm that utilizes asynchronous communication to speed up convergence and enhance scalability, and employs client memory to correct the client drift caused by variations in client update frequencies. Moreover, AREA is, to the best of our knowledge, the first method that is guaranteed to converge under arbitrarily long delays, without the use of delay-adaptive stepsizes, and (i) for strongly convex, smooth functions, asymptotically converges to an error neighborhood whose size depends only on the variance of the stochastic gradients used with respect to the number of iterations, and (ii) for convex, non-smooth functions, matches the convergence rate of the centralized stochastic subgradient method up to a constant factor, which depends on the average of the individual client update frequencies instead of their minimum (or maximum). Our numerical results validate our theoretical analysis and indicate AREA outperforms state-of-the-art methods when local data are highly non-iid, especially as the number of clients grows.

Continuous-time approximation of Stochastic Gradient Descent (SGD) is a crucial tool to study its escaping behaviors from stationary points. However, existing stochastic differential equation (SDE) models fail to fully capture these behaviors, even for simple quadratic objectives. Built on a novel stochastic backward error analysis framework, we derive the Hessian-Aware Stochastic Modified Equation (HA-SME), an SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. Our analysis shows that HA-SME matches the order-best approximation error guarantee among existing SDE models in the literature, while achieving a significantly reduced dependence on the smoothness parameter of the objective. Further, for quadratic objectives, under mild conditions, HA-SME is proved to be the first SDE model that recovers exactly the SGD dynamics in the distributional sense. Consequently, when the local landscape near a stationary point can be approximated by quadratics, HA-SME is expected to accurately predict the local escaping behaviors of SGD.

We consider the distributed learning problem with data dispersed across multiple workers under the orchestration of a central server. Asynchronous Stochastic Gradient Descent (SGD) has been widely explored in such a setting to reduce the synchronization overhead associated with parallelization. However, the performance of asynchronous SGD algorithms often depends on a bounded dissimilarity condition among the workers' local data, a condition that can drastically affect their efficiency when the workers' data are highly heterogeneous. To overcome this limitation, we introduce the \textit{dual-delayed asynchronous SGD (DuDe-ASGD)} algorithm designed to neutralize the adverse effects of data heterogeneity. DuDe-ASGD makes full use of stale stochastic gradients from all workers during asynchronous training, leading to two distinct time lags in the model parameters and data samples utilized in the server's iterations. Furthermore, by adopting an incremental aggregation strategy, DuDe-ASGD maintains a per-iteration computational cost that is on par with traditional asynchronous SGD algorithms. Our analysis demonstrates that DuDe-ASGD achieves a near-minimax-optimal convergence rate for smooth nonconvex problems, even when the data across workers are extremely heterogeneous. Numerical experiments indicate that DuDe-ASGD compares favorably with existing asynchronous and synchronous SGD-based algorithms.

Estimating a policy that maps states to actions is a central problem in reinforcement learning. Traditionally, policies are inferred from the so called value functions (VFs), but exact VF computation suffers from the curse of dimensionality. Policy gradient (PG) methods bypass this by learning directly a parametric stochastic policy. Typically, the parameters of the policy are estimated using neural networks (NNs) tuned via stochastic gradient descent. However, finding adequate NN architectures can be challenging, and convergence issues are common as well. In this paper, we put forth low-rank matrix-based models to estimate efficiently the parameters of PG algorithms. We collect the parameters of the stochastic policy into a matrix, and then, we leverage matrix-completion techniques to promote (enforce) low rank. We demonstrate via numerical studies how low-rank matrix-based policy models reduce the computational and sample complexities relative to NN models, while achieving a similar aggregated reward.

Training data attribution (TDA) methods aim to quantify the influence of individual training data points on the model predictions, with broad applications in data-centric AI, such as mislabel detection, data selection, and copyright compensation. However, existing methods in this field, which can be categorized as retraining-based and gradient-based, have struggled with the trade-off between computational efficiency and attribution efficacy. Retraining-based methods can accurately attribute complex non-convex models but are computationally prohibitive, while gradient-based methods are efficient but often fail for non-convex models. Recent research has shown that augmenting gradient-based methods with ensembles of multiple independently trained models can achieve significantly better attribution efficacy. However, this approach remains impractical for very large-scale applications. In this work, we discover that expensive, fully independent training is unnecessary for ensembling the gradient-based methods, and we propose two efficient ensemble strategies, DROPOUT ENSEMBLE and LORA ENSEMBLE, alternative to naive independent ensemble. These strategies significantly reduce training time (up to 80%), serving time (up to 60%), and space cost (up to 80%) while maintaining similar attribution efficacy to the naive independent ensemble. Our extensive experimental results demonstrate that the proposed strategies are effective across multiple TDA methods on diverse datasets and models, including generative settings, significantly advancing the Pareto frontier of TDA methods with better computational efficiency and attribution efficacy.

Meta-learning has been proposed as a promising machine learning topic in recent years, with important applications to image classification, robotics, computer games, and control systems. In this paper, we study the problem of using meta-learning to deal with uncertainty and heterogeneity in ergodic linear quadratic regulators. We integrate the zeroth-order optimization technique with a typical meta-learning method, proposing an algorithm that omits the estimation of policy Hessian, which applies to tasks of learning a set of heterogeneous but similar linear dynamic systems. The induced meta-objective function inherits important properties of the original cost function when the set of linear dynamic systems are meta-learnable, allowing the algorithm to optimize over a learnable landscape without projection onto the feasible set. We provide a convergence result for the exact gradient descent process by analyzing the boundedness and smoothness of the gradient for the meta-objective, which justify the proposed algorithm with gradient estimation error being small. We also provide a numerical example to corroborate this perspective.