Stochastic gradient descent (SGD) is a widely used algorithm in machine learning, particularly for neural network training. Recent studies on SGD for canonical quadratic optimization or linear regression show it attains well generalization under suitable high-dimensional settings. However, a fundamental question -- for what kinds of high-dimensional learning problems SGD and its accelerated variants can achieve optimality has yet to be well studied. This paper investigates SGD with two essential components in practice: exponentially decaying step size schedule and momentum. We establish the convergence upper bound for momentum accelerated SGD (ASGD) and propose concrete classes of learning problems under which SGD or ASGD achieves min-max optimal convergence rates. The characterization of the target function is based on standard power-law decays in (functional) linear regression. Our results unveil new insights for understanding the learning bias of SGD: (i) SGD is efficient in learning ``dense'' features where the corresponding weights are subject to an infinity norm constraint; (ii) SGD is efficient for easy problem without suffering from the saturation effect; (iii) momentum can accelerate the convergence rate by order when the learning problem is relatively hard. To our knowledge, this is the first work to clearly identify the optimal boundary of SGD versus ASGD for the problem under mild settings.

Machine unlearning algorithms aim to efficiently remove data from a model without retraining it from scratch, in order to enforce data privacy, remove corrupted or outdated data, or respect a user's ``right to be forgotten." Certified machine unlearning is a strong theoretical guarantee that quantifies the extent to which data is erased from the model weights. Most prior works in certified unlearning focus on models trained on convex or strongly convex loss functions, which benefit from convenient convergence guarantees and the existence of global minima. For nonconvex objectives, existing algorithms rely on limiting assumptions and expensive computations that hinder practical implementations. In this work, we propose a simple first-order algorithm for unlearning on general nonconvex loss functions which unlearns by ``rewinding" to an earlier step during the learning process and then performs gradient descent on the loss function of the retained data points. Our algorithm is black-box, in that it can be directly applied to models pretrained with vanilla gradient descent with no prior consideration of unlearning. We prove $(\epsilon, \delta)$ certified unlearning and performance guarantees that establish the privacy-utility-complexity tradeoff of our algorithm, with special consideration for nonconvex functions that satisfy the Polyak-Lojasiewicz inequality.

Obtaining compositional mappings is important for the model to generalize well compositionally. To better understand when and how to encourage the model to learn such mappings, we study their uniqueness through different perspectives. Specifically, we first show that the compositional mappings are the simplest bijections through the lens of coding length (i.e., an upper bound of their Kolmogorov complexity). This property explains why models having such mappings can generalize well. We further show that the simplicity bias is usually an intrinsic property of neural network training via gradient descent. That partially explains why some models spontaneously generalize well when they are trained appropriately.

This paper investigates the task of tracking multiple objects in clutter under a distributed multi-sensor network with time-varying connectivity. Designed with the same objective as the centralised variational multi-object tracker, the proposed method achieves optimal decentralised fusion in performance with local processing and communication with only neighboring sensors. A key innovation is the decentralised construction of a locally maximised evidence lower bound, which greatly reduces the information required for communication. Our decentralised natural gradient descent variational multi-object tracker, enhanced with the gradient tracking strategy and natural gradients that adjusts the direction of traditional gradients to the steepest, shows rapid convergence. Our results verify that the proposed method is empirically equivalent to the centralised fusion in tracking accuracy, surpasses suboptimal fusion techniques with comparable costs, and achieves much lower communication overhead than the consensus-based variational multi-object tracker.

In this paper, we propose an algorithm for registering sequential bounding boxes with point cloud streams. Unlike popular point cloud registration techniques, the alignment of the point cloud and the bounding box can rely on the properties of the bounding box, such as size, shape, and temporal information, which provides substantial support and performance gains. Motivated by this, we propose a new approach to tackle this problem. Specifically, we model the registration process through an overall objective function that includes the final goal and all constraints. We then optimize the function using gradient descent. Our experiments show that the proposed method performs remarkably well with a 40\% improvement in IoU and demonstrates more robust registration between point cloud streams and sequential bounding boxes

Distributed learning has become the standard approach for training large-scale machine learning models across private data silos. While distributed learning enhances privacy preservation and training efficiency, it faces critical challenges related to Byzantine robustness and communication reduction. Existing Byzantine-robust and communication-efficient methods rely on full gradient information either at every iteration or at certain iterations with a probability, and they only converge to an unnecessarily large neighborhood around the solution. Motivated by these issues, we propose a novel Byzantine-robust and communication-efficient stochastic distributed learning method that imposes no requirements on batch size and converges to a smaller neighborhood around the optimal solution than all existing methods, aligning with the theoretical lower bound. Our key innovation is leveraging Polyak Momentum to mitigate the noise caused by both biased compressors and stochastic gradients, thus defending against Byzantine workers under information compression. We provide proof of tight complexity bounds for our algorithm in the context of non-convex smooth loss functions, demonstrating that these bounds match the lower bounds in Byzantine-free scenarios. Finally, we validate the practical significance of our algorithm through an extensive series of experiments, benchmarking its performance on both binary classification and image classification tasks.

In this paper, we present a cross-entropy optimization method for hyperparameter optimization in stochastic gradient-based approaches to train deep neural networks. The value of a hyperparameter of a learning algorithm often has great impact on the performance of a model such as the convergence speed, the generalization performance metrics, etc. While in some cases the hyperparameters of a learning algorithm can be part of learning parameters, in other scenarios the hyperparameters of a stochastic optimization algorithm such as Adam [5] and its variants are either fixed as a constant or are kept changing in a monotonic way over time. We give an in-depth analysis of the presented method in the framework of expectation maximization (EM). The presented algorithm of cross-entropy optimization for hyperparameter optimization of a learning algorithm (CEHPO) can be equally applicable to other areas of optimization problems in deep learning. We hope that the presented methods can provide different perspectives and offer some insights for optimization problems in different areas of machine learning and beyond.

The performance of mini-batch stochastic gradient descent (SGD) strongly depends on setting the batch size and learning rate to minimize the empirical loss in training the deep neural network. In this paper, we present theoretical analyses of mini-batch SGD with four schedulers: (i) constant batch size and decaying learning rate scheduler, (ii) increasing batch size and decaying learning rate scheduler, (iii) increasing batch size and increasing learning rate scheduler, and (iv) increasing batch size and warm-up decaying learning rate scheduler. We show that mini-batch SGD using scheduler (i) does not always minimize the expectation of the full gradient norm of the empirical loss, whereas it does using any of schedulers (ii), (iii), and (iv). Furthermore, schedulers (iii) and (iv) accelerate mini-batch SGD. The paper also provides numerical results of supporting analyses showing that using scheduler (iii) or (iv) minimizes the full gradient norm of the empirical loss faster than using scheduler (i) or (ii).

The number of free parameters, or dimension, of a model is a straightforward way to measure its complexity: a model with more parameters can encode more information. However, this is not an accurate measure of complexity: models capable of memorizing their training data often generalize well despite their high dimension. Effective dimension aims to more directly capture the complexity of a model by counting only the number of parameters required to represent the functionality of the model. Singular learning theory (SLT) proposes the learning coefficient $ \lambda $ as a more accurate measure of effective dimension. By describing the rate of increase of the volume of the region of parameter space around a local minimum with respect to loss, $ \lambda $ incorporates information from higher-order terms. We compare $ \lambda $ of models trained using natural gradient descent (NGD) and stochastic gradient descent (SGD), and find that those trained with NGD consistently have a higher effective dimension for both of our methods: the Hessian trace $ \text{Tr}(\mathbf{H}) $, and the estimate of the local learning coefficient (LLC) $ \hat{\lambda}(w^*) $.

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernel Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is the observation that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant `negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller `positive part'. This observation leads to $\mathsf{KSD}$ rates of order $1/\sqrt{N}$, providing a near optimal double exponential improvement over the recent result by~\cite{shi2024finite}. Under mild assumptions on the kernel and potential, these bounds also grow linearly in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence. For the case of `bilinear + Mat\'ern' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.