We propose theoretical analyses of a modified natural gradient descent method in the neural network function space based on the eigendecompositions of neural tangent kernel and Fisher information matrix. We firstly present analytical expression for the function learned by this modified natural gradient under the assumptions of Gaussian distribution and infinite width limit. Thus, we explicitly derive the generalization error of the learned neural network function using theoretical methods from eigendecomposition and statistics theory. By decomposing of the total generalization error attributed to different eigenspace of the kernel in function space, we propose a criterion for balancing the errors stemming from training set and the distribution discrepancy between the training set and the true data. Through this approach, we establish that modifying the training direction of the neural network in function space leads to a reduction in the total generalization error. Furthermore, We demonstrate that this theoretical framework is capable to explain many existing results of generalization enhancing methods. These theoretical results are also illustrated by numerical examples on synthetic data.

This paper revisits Deep Mutual Learning (DML), a simple yet effective computing paradigm. We propose using R\'{e}nyi divergence instead of the KL divergence, which is more flexible and tunable, to improve vanilla DML. This modification is able to consistently improve performance over vanilla DML with limited additional complexity. The convergence properties of the proposed paradigm are analyzed theoretically, and Stochastic Gradient Descent with a constant learning rate is shown to converge with $\mathcal{O}(1)$-bias in the worst case scenario for nonconvex optimization tasks. That is, learning will reach nearby local optima but continue searching within a bounded scope, which may help mitigate overfitting. Finally, our extensive empirical results demonstrate the advantage of combining DML and R\'{e}nyi divergence, leading to further improvement in model generalization.

Gradient-based methods for value estimation in reinforcement learning have favorable stability properties, but they are typically much slower than Temporal Difference (TD) learning methods. We study the root causes of this slowness and show that Mean Square Bellman Error (MSBE) is an ill-conditioned loss function in the sense that its Hessian has large condition-number. To resolve the adverse effect of poor conditioning of MSBE on gradient based methods, we propose a low complexity batch-free proximal method that approximately follows the Gauss-Newton direction and is asymptotically robust to parameterization. Our main algorithm, called RANS, is efficient in the sense that it is significantly faster than the residual gradient methods while having almost the same computational complexity, and is competitive with TD on the classic problems that we tested.

Due to its speed and simplicity, subgradient descent is one of the most used optimization algorithms in convex machine learning algorithms. However, tuning its learning rate is probably its most severe bottleneck to achieve consistent good performance. A common way to reduce the dependency on the learning rate is to use implicit/proximal updates. One such variant is the Importance Weight Aware (IWA) updates, which consist of infinitely many infinitesimal updates on each loss function. However, IWA updates' empirical success is not completely explained by their theory. In this paper, we show for the first time that IWA updates have a strictly better regret upper bound than plain gradient updates in the online learning setting. Our analysis is based on the new framework, generalized implicit Follow-the-Regularized-Leader (FTRL) (Chen and Orabona, 2023), to analyze generalized implicit updates using a dual formulation. In particular, our results imply that IWA updates can be considered as approximate implicit/proximal updates.

Each round in Differential Private Stochastic Gradient Descent (DPSGD) transmits a sum of clipped gradients obfuscated with Gaussian noise to a central server which uses this to update a global model which often represents a deep neural network. Since the clipped gradients are computed separately, which we call Individual Clipping (IC), deep neural networks like resnet-18 cannot use Batch Normalization Layers (BNL) which is a crucial component in deep neural networks for achieving a high accuracy. To utilize BNL, we introduce Batch Clipping (BC) where, instead of clipping single gradients as in the orginal DPSGD, we average and clip batches of gradients. Moreover, the model entries of different layers have different sensitivities to the added Gaussian noise. Therefore, Adaptive Layerwise Clipping methods (ALC), where each layer has its own adaptively finetuned clipping constant, have been introduced and studied, but so far without rigorous DP proofs. In this paper, we propose {\em a new ALC and provide rigorous DP proofs for both BC and ALC}. Experiments show that our modified DPSGD with BC and ALC for CIFAR-$10$ with resnet-$18$ converges while DPSGD with IC and ALC does not.

Optimal Transport has sparked vivid interest in recent years, in particular thanks to the Wasserstein distance, which provides a geometrically sensible and intuitive way of comparing probability measures. For computational reasons, the Sliced Wasserstein (SW) distance was introduced as an alternative to the Wasserstein distance, and has seen uses for training generative Neural Networks (NNs). While convergence of Stochastic Gradient Descent (SGD) has been observed practically in such a setting, there is to our knowledge no theoretical guarantee for this observation. Leveraging recent works on convergence of SGD on non-smooth and non-convex functions by Bianchi et al. (2022), we aim to bridge that knowledge gap, and provide a realistic context under which fixed-step SGD trajectories for the SW loss on NN parameters converge. More precisely, we show that the trajectories approach the set of (sub)-gradient flow equations as the step decreases. Under stricter assumptions, we show a much stronger convergence result for noised and projected SGD schemes, namely that the long-run limits of the trajectories approach a set of generalised critical points of the loss function.

Perfect synchronization in distributed machine learning problems is inefficient and even impossible due to the existence of latency, package losses and stragglers. We propose a Robust Fully-Asynchronous Stochastic Gradient Tracking method (R-FAST), where each device performs local computation and communication at its own pace without any form of synchronization. Different from existing asynchronous distributed algorithms, R-FAST can eliminate the impact of data heterogeneity across devices and allow for packet losses by employing a robust gradient tracking strategy that relies on properly designed auxiliary variables for tracking and buffering the overall gradient vector. More importantly, the proposed method utilizes two spanning-tree graphs for communication so long as both share at least one common root, enabling flexible designs in communication architectures. We show that R-FAST converges in expectation to a neighborhood of the optimum with a geometric rate for smooth and strongly convex objectives; and to a stationary point with a sublinear rate for general non-convex settings. Extensive experiments demonstrate that R-FAST runs 1.5-2 times faster than synchronous benchmark algorithms, such as Ring-AllReduce and D-PSGD, while still achieving comparable accuracy, and outperforms existing asynchronous SOTA algorithms, such as AD-PSGD and OSGP, especially in the presence of stragglers.

This work proposes a new analysis technique for gradient descent, establishing provably better convergence rates for smooth, convex optimization than the prior state-of-art textbook proofs. Our theory allows for nonconstant stepsize policies, periodically taking larger steps that may violate the monotone decrease in objective value typically needed by analysis. In fact, contrary to the common intuition, we show periodic long steps, which may increase the objective value in the short term, provably speed up convergence in the long term, with increasingly large gains as longer and longer steps are periodically included. This bears a similarity to accelerated momentum methods, which also depart from ensuring a monotone objective decrease at every iteration. Establishing this requires a proof technique capable of analyzing the overall effect of many iterations at once rather than the typical (naive) one-iteration inductions used in most first-order method analyses. Our proofs are based on the Performance Estimation Problem (PEP) ideas of [1-3], which cast computing/bounding the worst-case problem instance of a given algorithm as a Semidefinite Program (SDP). We show that the existence of a feasible solution to a related SDP proves a descent guarantee after applying a corresponding pattern of nonconstant stepsizes, from which faster convergence guarantees follow.

Uncertainty sampling is a prevalent active learning algorithm that queries sequentially the annotations of data samples which the current prediction model is uncertain about. However, the usage of uncertainty sampling has been largely heuristic: (i) There is no consensus on the proper definition of "uncertainty" for a specific task under a specific loss; (ii) There is no theoretical guarantee that prescribes a standard protocol to implement the algorithm, for example, how to handle the sequentially arrived annotated data under the framework of optimization algorithms such as stochastic gradient descent. In this work, we systematically examine uncertainty sampling algorithms under both stream-based and pool-based active learning. We propose a notion of equivalent loss which depends on the used uncertainty measure and the original loss function and establish that an uncertainty sampling algorithm essentially optimizes against such an equivalent loss. The perspective verifies the properness of existing uncertainty measures from two aspects: surrogate property and loss convexity. Furthermore, we propose a new notion for designing uncertainty measures called \textit{loss as uncertainty}. The idea is to use the conditional expected loss given the features as the uncertainty measure. Such an uncertainty measure has nice analytical properties and generality to cover both classification and regression problems, which enable us to provide the first generalization bound for uncertainty sampling algorithms under both stream-based and pool-based settings, in the full generality of the underlying model and problem. Lastly, we establish connections between certain variants of the uncertainty sampling algorithms with risk-sensitive objectives and distributional robustness, which can partly explain the advantage of uncertainty sampling algorithms when the sample size is small.

Data augmentation is essential to achieve state-of-the-art performance in many deep learning applications. However, the most effective augmentation techniques become computationally prohibitive for even medium-sized datasets. To address this, we propose a rigorous technique to select subsets of data points that when augmented, closely capture the training dynamics of full data augmentation. We first show that data augmentation, modeled as additive perturbations, improves learning and generalization by relatively enlarging and perturbing the smaller singular values of the network Jacobian, while preserving its prominent directions. This prevents overfitting and enhances learning the harder to learn information. Then, we propose a framework to iteratively extract small subsets of training data that when augmented, closely capture the alignment of the fully augmented Jacobian with labels/residuals. We prove that stochastic gradient descent applied to the augmented subsets found by our approach has similar training dynamics to that of fully augmented data. Our experiments demonstrate that our method achieves 6.3x speedup on CIFAR10 and 2.2x speedup on SVHN, and outperforms the baselines by up to 10% across various subset sizes. Similarly, on TinyImageNet and ImageNet, our method beats the baselines by up to 8%, while achieving up to 3.3x speedup across various subset sizes. Finally, training on and augmenting 50% subsets using our method on a version of CIFAR10 corrupted with label noise even outperforms using the full dataset. Our code is available at: https://github.com/tianyu139/data-efficient-augmentation