In this paper, we examine a class of non-convex stochastic optimization problems which we call variationally coherent, and which properly includes pseudo-/quasiconvex and star-convex optimization problems. To solve such problems, we focus on the widely used stochastic mirror descent (SMD) family of algorithms (which contains stochastic gradient descent as a special case), and we show that the last iterate of SMD converges to the problem's solution set with probability 1. This result contributes to the landscape of non-convex stochastic optimization by clarifying that neither pseudo-/quasi-convexity nor star-convexity is essential for (almost sure) global convergence; rather, variational coherence, a much weaker requirement, suffices. Characterization of convergence rates for the subclass of strongly variationally coherent optimization problems as well as simulation results are also presented.

Model-induced distribution shifts (MIDS) occur as previous model outputs pollute new model training sets over generations of models. This is known as model collapse in the case of generative models, and performative prediction or unfairness feedback loops for supervised models. When a model induces a distribution shift, it also encodes its mistakes, biases, and unfairnesses into the ground truth of its data ecosystem. We introduce a framework that allows us to track multiple MIDS over many generations, finding that they can lead to loss in performance, fairness, and minoritized group representation, even in initially unbiased datasets. Despite these negative consequences, we identify how models might be used for positive, intentional, interventions in their data ecosystems, providing redress for historical discrimination through a framework called algorithmic reparation (AR). We simulate AR interventions by curating representative training batches for stochastic gradient descent to demonstrate how AR can improve upon the unfairnesses of models and data ecosystems subject to other MIDS. Our work takes an important step towards identifying, mitigating, and taking accountability for the unfair feedback loops enabled by the idea that ML systems are inherently neutral and objective.

Neural networks have achieved remarkable success across various tasks, yet the precise mechanism driving this success remains theoretically elusive. The training of neural networks involve optimizing a non-convex loss function, where the training algorithm typically is a first-order methods such as gradient descent or its variants. A particularly puzzling aspect is how these training algorithms succeed in finding a solution with good generalization capabilities despite the non-convexity of the loss landscape. In addition to the choice of the training algorithm, the choice of initialization in these algorithms play a crucial role in determining the neural network performance. Indeed, recent works have made increasingly clear the benefit of small initializations, revealing that neural networks trained using (stochastic) gradient descent with small initializations exhibit feature learning [2] and also generalize better for various tasks [3, 4, 5]; see Section 2 for more details into the impact of initialization scale. However, for small initializations, the training dynamics of neural networks is extremely non-linear and not well understood so far. Our focus in this paper is on understanding the effect of small initialization on the training dynamics of neural networks. In pursuit of a deeper understanding of the training mechanism for small initializations, researchers have uncovered the phenomenon of directional convergence in the neural network weights during early phases of training [6, 7]. In [6], authors study the gradient flow dynamics of training two-layer Rectified Linear Unit (ReLU) neural networks.

Shuffling gradient methods, which are also known as stochastic gradient descent (SGD) without replacement, are widely implemented in practice, particularly including three popular algorithms: Random Reshuffle (RR), Shuffle Once (SO), and Incremental Gradient (IG). Compared to the empirical success, the theoretical guarantee of shuffling gradient methods was not well-understanding for a long time. Until recently, the convergence rates had just been established for the average iterate for convex functions and the last iterate for strongly convex problems (using squared distance as the metric). However, when using the function value gap as the convergence criterion, existing theories cannot interpret the good performance of the last iterate in different settings (e.g., constrained optimization). To bridge this gap between practice and theory, we prove last-iterate convergence rates for shuffling gradient methods with respect to the objective value even without strong convexity. Our new results either (nearly) match the existing last-iterate lower bounds or are as fast as the previous best upper bounds for the average iterate.

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix X with gradient descent on a factorization of X. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

We propose a federated version of adaptive gradient methods, particularly AdaGrad and Adam, within the framework of over-the-air model training. This approach capitalizes on the inherent superposition property of wireless channels, facilitating fast and scalable parameter aggregation. Meanwhile, it enhances the robustness of the model training process by dynamically adjusting the stepsize in accordance with the global gradient update. We derive the convergence rate of the training algorithms, encompassing the effects of channel fading and interference, for a broad spectrum of nonconvex loss functions. Our analysis shows that the AdaGrad-based algorithm converges to a stationary point at the rate of $\mathcal{O}( \ln{(T)} /{ T^{ 1 - \frac{1}{\alpha} } } )$, where $\alpha$ represents the tail index of the electromagnetic interference. This result indicates that the level of heavy-tailedness in interference distribution plays a crucial role in the training efficiency: the heavier the tail, the slower the algorithm converges. In contrast, an Adam-like algorithm converges at the $\mathcal{O}( 1/T )$ rate, demonstrating its advantage in expediting the model training process. We conduct extensive experiments that corroborate our theoretical findings and affirm the practical efficacy of our proposed federated adaptive gradient methods.

With the widespread adoption of Non-Intrusive Load Monitoring (NILM) in building energy management, ensuring the high quality of NILM data has become imperative. However, practical applications of NILM face challenges associated with data loss, significantly impacting accuracy and reliability in energy management. This paper addresses the issue of NILM data loss by introducing an innovative tensor completion(TC) model- Proportional-Integral-Derivative (PID)-incorporated Non-negative Latent Factorization of Tensors (PNLFT) with twofold ideas: 1) To tackle the issue of slow convergence in Latent Factorization of Tensors (LFT) using Stochastic Gradient Descent (SGD), a Proportional-Integral-Derivative controller is introduced during the learning process. The PID controller utilizes historical and current information to control learning residuals. 2) Considering the characteristics of NILM data, non-negative update rules are proposed in the model's learning scheme. Experimental results on three datasets demonstrate that, compared to state-of-the-art models, the proposed model exhibits noteworthy enhancements in both convergence speed and accuracy.

We show that deep reinforcement learning can maintain its ability to learn without resetting network parameters in settings where the number of gradient updates greatly exceeds the number of environment samples. Under such large update-to-data ratios, a recent study by Nikishin et al. (2022) suggested the emergence of a primacy bias, in which agents overfit early interactions and downplay later experience, impairing their ability to learn. In this work, we dissect the phenomena underlying the primacy bias. We inspect the early stages of training that ought to cause the failure to learn and find that a fundamental challenge is a long-standing acquaintance: value overestimation. Overinflated Q-values are found not only on out-of-distribution but also in-distribution data and can be traced to unseen action prediction propelled by optimizer momentum. We employ a simple unit-ball normalization that enables learning under large update ratios, show its efficacy on the widely used dm_control suite, and obtain strong performance on the challenging dog tasks, competitive with model-based approaches. Our results question, in parts, the prior explanation for sub-optimal learning due to overfitting on early data.

Advanced machine learning methods, and more prominently neural networks, have become standard to solve inverse problems over the last years. However, the theoretical recovery guarantees of such methods are still scarce and difficult to achieve. Only recently did unsupervised methods such as Deep Image Prior (DIP) get equipped with convergence and recovery guarantees for generic loss functions when trained through gradient flow with an appropriate initialization. In this paper, we extend these results by proving that these guarantees hold true when using gradient descent with an appropriately chosen step-size/learning rate. We also show that the discretization only affects the overparametrization bound for a two-layer DIP network by a constant and thus that the different guarantees found for the gradient flow will hold for gradient descent.

We study Markov potential games under the infinite horizon average reward criterion. Most previous studies have been for discounted rewards. We prove that both algorithms based on independent policy gradient and independent natural policy gradient converge globally to a Nash equilibrium for the average reward criterion. To set the stage for gradient-based methods, we first establish that the average reward is a smooth function of policies and provide sensitivity bounds for the differential value functions, under certain conditions on ergodicity and the second largest eigenvalue of the underlying Markov decision process (MDP). We prove that three algorithms, policy gradient, proximal-Q, and natural policy gradient (NPG), converge to an $\epsilon$-Nash equilibrium with time complexity $O(\frac{1}{\epsilon^2})$, given a gradient/differential Q function oracle. When policy gradients have to be estimated, we propose an algorithm with $\tilde{O}(\frac{1}{\min_{s,a}\pi(a|s)\delta})$ sample complexity to achieve $\delta$ approximation error w.r.t~the $\ell_2$ norm. Equipped with the estimator, we derive the first sample complexity analysis for a policy gradient ascent algorithm, featuring a sample complexity of $\tilde{O}(1/\epsilon^5)$. Simulation studies are presented.