Herein the topics of (natural) gradient descent, data decorrelation, and approximate methods for backpropagation are brought into a dialogue. Natural gradient descent illuminates how gradient vectors, pointing at directions of steepest descent, can be improved by considering the local curvature of loss landscapes. We extend this perspective and show that to fully solve the problem illuminated by natural gradients in neural networks, one must recognise that correlations in the data at any linear transformation, including node responses at every layer of a neural network, cause a non-orthonormal relationship between the model's parameters. To solve this requires a solution to decorrelate inputs at each individual layer of a neural network. We describe a range of methods which have been proposed for decorrelation and whitening of node output, while providing a novel method specifically useful for distributed computing and computational neuroscience. Implementing decorrelation within multi-layer neural networks, we can show that not only is training via backpropagation sped up significantly but also existing approximations of backpropagation, which have failed catastrophically in the past, are made performant once more. This has the potential to provide a route forward for approximate gradient descent methods which have previously been discarded, training approaches for analogue and neuromorphic hardware, and potentially insights as to the efficacy and utility of decorrelation processes in the brain.

On-device training of DNNs allows models to adapt and fine-tune to newly collected data or changing domains while deployed on microcontroller units (MCUs). However, DNN training is a resource-intensive task, making the implementation and execution of DNN training algorithms on MCUs challenging due to low processor speeds, constrained throughput, limited floating-point support, and memory constraints. In this work, we explore on-device training of DNNs for Cortex-M MCUs. We present a method that enables efficient training of DNNs completely in place on the MCU using fully quantized training (FQT) and dynamic partial gradient updates. We demonstrate the feasibility of our approach on multiple vision and time-series datasets and provide insights into the tradeoff between training accuracy, memory overhead, energy, and latency on real hardware.

Ordinary differential equations (ODEs) are widely used to describe dynamical systems in science, but identifying parameters that explain experimental measurements is challenging. In particular, although ODEs are differentiable and would allow for gradient-based parameter optimization, the nonlinear dynamics of ODEs often lead to many local minima and extreme sensitivity to initial conditions. We therefore propose diffusion tempering, a novel regularization technique for probabilistic numerical methods which improves convergence of gradient-based parameter optimization in ODEs. By iteratively reducing a noise parameter of the probabilistic integrator, the proposed method converges more reliably to the true parameters. We demonstrate that our method is effective for dynamical systems of different complexity and show that it obtains reliable parameter estimates for a Hodgkin-Huxley model with a practically relevant number of parameters.

We consider nonparametric regression by an over-parameterized two-layer neural network trained by gradient descent (GD) or its variant in this paper. We show that, if the neural network is trained with a novel Preconditioned Gradient Descent (PGD) with early stopping and the target function has spectral bias widely studied in the deep learning literature, the trained network renders a particularly sharp generalization bound with a minimax optimal rate of $\cO({1}/{n^{4\alpha/(4\alpha+1)}})$, which is sharper the current standard rate of $\cO({1}/{n^{2\alpha/(2\alpha+1)}})$ with $2\alpha = d/(d-1)$ when the data is distributed uniformly on the unit sphere in $\RR^d$ and $n$ is the size of the training data. When the target function has no spectral bias, we prove that neural network trained with regular GD with early stopping still enjoys minimax optimal rate, and in this case our results do not require distributional assumptions in contrast with the current known results. Our results are built upon two significant technical contributions. First, uniform convergence to the NTK is established during the training process by PGD or GD, so that we can have a nice decomposition of the neural network function at any step of GD or PGD into a function in the RKHS and an error function with a small $L^{\infty}$-norm. Second, local Rademacher complexity is employed to tightly bound the Rademacher complexity of the function class comprising all the possible neural network functions obtained by GD or PGD. Our results also indicate that PGD can be another way of avoiding the usual linear regime of NTK and obtaining sharper generalization bound, because PGD induces a different kernel with lower kernel complexity during the training than the regular NTK induced by the network architecture trained by regular GD.

Parallel Continual Learning (PCL) tasks investigate the training methods for continual learning with multi-source input, where data from different tasks are learned as they arrive. PCL offers high training efficiency and is well-suited for complex multi-source data systems, such as autonomous vehicles equipped with multiple sensors. However, at any time, multiple tasks need to be trained simultaneously, leading to severe training instability in PCL. This instability manifests during both forward and backward propagation, where features are entangled and gradients are conflict. This paper introduces Stable Parallel Continual Learning (SPCL), a novel approach that enhances the training stability of PCL for both forward and backward propagation. For the forward propagation, we apply Doubly-block Toeplit (DBT) Matrix based orthogonality constraints to network parameters to ensure stable and consistent propagation. For the backward propagation, we employ orthogonal decomposition for gradient management stabilizes backpropagation and mitigates gradient conflicts across tasks. By optimizing gradients by ensuring orthogonality and minimizing the condition number, SPCL effectively stabilizing the gradient descent in complex optimization tasks. Experimental results demonstrate that SPCL outperforms state-of-the-art methjods and achieve better training stability.

We address the challenge of efficiently and reliably deleting data from machine learning models trained using Empirical Risk Minimization (ERM), a process known as machine unlearning. To avoid retraining models from scratch, we propose a novel algorithm leveraging Natural Gradient Descent (NGD). Our theoretical framework ensures strong privacy guarantees for convex models, while a practical Min/Max optimization algorithm is developed for non-convex models. Comprehensive evaluations show significant improvements in privacy, computational efficiency, and generalization compared to state-of-the-art methods, advancing both the theoretical and practical aspects of machine unlearning.

Recently, decentralized learning has emerged as a popular peer-to-peer signal and information processing paradigm that enables model training across geographically distributed agents in a scalable manner, without the presence of any central server. When some of the agents are malicious (also termed as Byzantine), resilient decentralized learning algorithms are able to limit the impact of these Byzantine agents without knowing their number and identities, and have guaranteed optimization errors. However, analysis of the generalization errors, which are critical to implementations of the trained models, is still lacking. In this paper, we provide the first analysis of the generalization errors for a class of popular Byzantine-resilient decentralized stochastic gradient descent (DSGD) algorithms. Our theoretical results reveal that the generalization errors cannot be entirely eliminated because of the presence of the Byzantine agents, even if the number of training samples are infinitely large. Numerical experiments are conducted to confirm our theoretical results.

We present a novel approach called differentially private stochastic block coordinate descent (DP-SBCD) for training neural networks with provable guarantees of differential privacy under the hidden state assumption. Our methodology incorporates Lipschitz neural networks and decomposes the training process of the neural network into sub-problems, each corresponding to the training of a specific layer. By doing so, we extend the analysis of differential privacy under the hidden state assumption to encompass non-convex problems and algorithms employing proximal gradient descent. Furthermore, in contrast to existing methods, we adopt a novel approach by utilizing calibrated noise sampled from adaptive distributions, yielding improved empirical trade-offs between utility and privacy.

Federated Learning (FL) offers innovative solutions for privacy-preserving collaborative machine learning (ML). Despite its promising potential, FL is vulnerable to various attacks due to its distributed nature, affecting the entire life cycle of FL services. These threats can harm the model's utility or compromise participants' privacy, either directly or indirectly. In response, numerous defense frameworks have been proposed, demonstrating effectiveness in specific settings and scenarios. To provide a clear understanding of the current research landscape, this paper reviews the most representative and state-of-the-art threats and defense frameworks throughout the FL service life cycle. We start by identifying FL threats that harm utility and privacy, including those with potential or direct impacts. Then, we dive into the defense frameworks, analyze the relationship between threats and defenses, and compare the trade-offs among different defense strategies. Finally, we summarize current research bottlenecks and offer insights into future research directions to conclude this survey. We hope this survey sheds light on trustworthy FL research and contributes to the FL community.

This paper presents a comprehensive study on the convergence rates of the stochastic gradient descent (SGD) algorithm when applied to overparameterized two-layer neural networks. Our approach combines the Neural Tangent Kernel (NTK) approximation with convergence analysis in the Reproducing Kernel Hilbert Space (RKHS) generated by NTK, aiming to provide a deep understanding of the convergence behavior of SGD in overparameterized two-layer neural networks. Our research framework enables us to explore the intricate interplay between kernel methods and optimization processes, shedding light on the optimization dynamics and convergence properties of neural networks. In this study, we establish sharp convergence rates for the last iterate of the SGD algorithm in overparameterized two-layer neural networks. Additionally, we have made significant advancements in relaxing the constraints on the number of neurons, which have been reduced from exponential dependence to polynomial dependence on the sample size or number of iterations. This improvement allows for more flexibility in the design and scaling of neural networks, and will deepen our theoretical understanding of neural network models trained with SGD.