The Laplace approximation provides a closed-form model selection objective for neural networks (NN). Online variants, which optimise NN parameters jointly with hyperparameters, like weight decay strength, have seen renewed interest in the Bayesian deep learning community. However, these methods violate Laplace's method's critical assumption that the approximation is performed around a mode of the loss, calling into question their soundness. This work re-derives online Laplace methods, showing them to target a variational bound on a mode-corrected variant of the Laplace evidence which does not make stationarity assumptions. Online Laplace and its mode-corrected counterpart share stationary points where 1. the NN parameters are a maximum a posteriori, satisfying the Laplace method's assumption, and 2. the hyperparameters maximise the Laplace evidence, motivating online methods. We demonstrate that these optima are roughly attained in practise by online algorithms using full-batch gradient descent on UCI regression datasets. The optimised hyperparameters prevent overfitting and outperform validation-based early stopping.

Although gradient descent with momentum is widely used in modern deep learning, a concrete understanding of its effects on the training trajectory still remains elusive. In this work, we empirically show that momentum gradient descent with a large learning rate and learning rate warmup displays large catapults, driving the iterates towards flatter minima than those found by gradient descent. We then provide empirical evidence and theoretical intuition that the large catapult is caused by momentum "amplifying" the self-stabilization effect (Damian et al., 2023).B.1

We consider the generalization error associated with stochastic gradient descent on a smooth convex function over a compact set. We show the first bound on the generalization error that vanishes when the number of iterations $T$ and the dataset size $n$ go to zero at arbitrary rates; our bound scales as $\tilde{O}(1/\sqrt{T} + 1/\sqrt{n})$ with step-size $\alpha_t = 1/\sqrt{t}$. In particular, strong convexity is not needed for stochastic gradient descent to generalize well.

In recent years, multi-objective optimization (MOO) emerges as a foundational problem underpinning many multi-agent multi-task learning applications. However, existing algorithms in MOO literature remain limited to centralized learning settings, which do not satisfy the distributed nature and data privacy needs of such multi-agent multi-task learning applications. This motivates us to propose a new federated multi-objective learning (FMOL) framework with multiple clients distributively and collaboratively solving an MOO problem while keeping their training data private. Notably, our FMOL framework allows a different set of objective functions across different clients to support a wide range of applications, which advances and generalizes the MOO formulation to the federated learning paradigm for the first time. For this FMOL framework, we propose two new federated multi-objective optimization (FMOO) algorithms called federated multi-gradient descent averaging (FMGDA) and federated stochastic multi-gradient descent averaging (FSMGDA). Both algorithms allow local updates to significantly reduce communication costs, while achieving the {\em same} convergence rates as those of their algorithmic counterparts in the single-objective federated learning. Our extensive experiments also corroborate the efficacy of our proposed FMOO algorithms.

Training differentially private machine learning models requires constraining an individual's contribution to the optimization process. This is achieved by clipping the $2$-norm of their gradient at a predetermined threshold prior to averaging and batch sanitization. This selection adversely influences optimization in two opposing ways: it either exacerbates the bias due to excessive clipping at lower values, or augments sanitization noise at higher values. The choice significantly hinges on factors such as the dataset, model architecture, and even varies within the same optimization, demanding meticulous tuning usually accomplished through a grid search. In order to circumvent the privacy expenses incurred in hyperparameter tuning, we present a novel approach to dynamically optimize the clipping threshold. We treat this threshold as an additional learnable parameter, establishing a clean relationship between the threshold and the cost function. This allows us to optimize the former with gradient descent, with minimal repercussions on the overall privacy analysis. Our method is thoroughly assessed against alternative fixed and adaptive strategies across diverse datasets, tasks, model dimensions, and privacy levels. Our results indicate that it performs comparably or better in the evaluated scenarios, given the same privacy requirements.

Equipping a deep model the abaility of few-shot learning, i.e., learning quickly from only few examples, is a core challenge for artificial intelligence. Gradient-based meta-learning approaches effectively address the challenge by learning how to learn novel tasks. Its key idea is learning a deep model in a bi-level optimization manner, where the outer-loop process learns a shared gradient descent algorithm (i.e., its hyperparameters), while the inner-loop process leverage it to optimize a task-specific model by using only few labeled data. Although these existing methods have shown superior performance, the outer-loop process requires calculating second-order derivatives along the inner optimization path, which imposes considerable memory burdens and the risk of vanishing gradients. Drawing inspiration from recent progress of diffusion models, we find that the inner-loop gradient descent process can be actually viewed as a reverse process (i.e., denoising) of diffusion where the target of denoising is model weights but the origin data. Based on this fact, in this paper, we propose to model the gradient descent optimizer as a diffusion model and then present a novel task-conditional diffusion-based meta-learning, called MetaDiff, that effectively models the optimization process of model weights from Gaussion noises to target weights in a denoising manner. Thanks to the training efficiency of diffusion models, our MetaDiff do not need to differentiate through the inner-loop path such that the memory burdens and the risk of vanishing gradients can be effectvely alleviated. Experiment results show that our MetaDiff outperforms the state-of-the-art gradient-based meta-learning family in few-shot learning tasks.

Stochastic gradient descent (SGD) and its many variants are the widespread optimization algorithms for training deep neural networks. However, SGD suffers from inevitable drawbacks, including vanishing gradients, lack of theoretical guarantees, and substantial sensitivity to input. The Alternating Direction Method of Multipliers (ADMM) has been proposed to address these shortcomings as an effective alternative to the gradient-based methods. It has been successfully employed for training deep neural networks. However, ADMM-based optimizers have a slow convergence rate. This paper proposes an Anderson Acceleration for Deep Learning ADMM (AA-DLADMM) algorithm to tackle this drawback. The main intention of the AA-DLADMM algorithm is to employ Anderson acceleration to ADMM by considering it as a fixed-point iteration and attaining a nearly quadratic convergence rate. We verify the effectiveness and efficiency of the proposed AA-DLADMM algorithm by conducting extensive experiments on four benchmark datasets contrary to other state-of-the-art optimizers.

Planning for many manipulation tasks, such as using tools or assembling parts, often requires both symbolic and geometric reasoning. Task and Motion Planning (TAMP) algorithms typically solve these problems by conducting a tree search over high-level task sequences while checking for kinematic and dynamic feasibility. This can be inefficient as the width of the tree can grow exponentially with the number of possible actions and objects. In this paper, we propose a novel approach to TAMP that relaxes discrete-and-continuous TAMP problems into inference problems on a continuous domain. Our method, Stein Task and Motion Planning (STAMP) subsequently solves this new problem using a gradient-based variational inference algorithm called Stein Variational Gradient Descent, by obtaining gradients from a parallelized differentiable physics simulator. By introducing relaxations to the discrete variables, leveraging parallelization, and approaching TAMP as an Bayesian inference problem, our method is able to efficiently find multiple diverse plans in a single optimization run. We demonstrate our method on two TAMP problems and benchmark them against existing TAMP baselines.

Self-supervised learning (SSL) has empirically shown its data representation learnability in many downstream tasks. There are only a few theoretical works on data representation learnability, and many of those focus on final data representation, treating the nonlinear neural network as a ``black box". However, the accurate learning results of neural networks are crucial for describing the data distribution features learned by SSL models. Our paper is the first to analyze the learning results of the nonlinear SSL model accurately. We consider a toy data distribution that contains two features: the label-related feature and the hidden feature. Unlike previous linear setting work that depends on closed-form solutions, we use the gradient descent algorithm to train a 1-layer nonlinear SSL model with a certain initialization region and prove that the model converges to a local minimum. Furthermore, different from the complex iterative analysis, we propose a new analysis process which uses the exact version of Inverse Function Theorem to accurately describe the features learned by the local minimum. With this local minimum, we prove that the nonlinear SSL model can capture the label-related feature and hidden feature at the same time. In contrast, the nonlinear supervised learning (SL) model can only learn the label-related feature. We also present the learning processes and results of the nonlinear SSL and SL model via simulation experiments.

Stability analysis is an essential aspect of studying the generalization ability of deep learning, as it involves deriving generalization bounds for stochastic gradient descent-based training algorithms. Adversarial training is the most widely used defense against adversarial example attacks. However, previous generalization bounds for adversarial training have not included information regarding the data distribution. In this paper, we fill this gap by providing generalization bounds for stochastic gradient descent-based adversarial training that incorporate data distribution information. We utilize the concepts of on-average stability and high-order approximate Lipschitz conditions to examine how changes in data distribution and adversarial budget can affect robust generalization gaps. Our derived generalization bounds for both convex and non-convex losses are at least as good as the uniform stability-based counterparts which do not include data distribution information. Furthermore, our findings demonstrate how distribution shifts from data poisoning attacks can impact robust generalization.