Graph Neural Networks (GNNs) have emerged as a powerful tool for data-driven learning on various graph domains. They are usually based on a message-passing mechanism and have gained increasing popularity for their intuitive formulation, which is closely linked to the Weisfeiler-Lehman (WL) test for graph isomorphism to which they have been proven equivalent in terms of expressive power. In this work, we establish new generalization properties and fundamental limits of GNNs in the context of learning so-called identity effects, i.e., the task of determining whether an object is composed of two identical components or not. Our study is motivated by the need to understand the capabilities of GNNs when performing simple cognitive tasks, with potential applications in computational linguistics and chemistry. We analyze two case studies: (i) two-letters words, for which we show that GNNs trained via stochastic gradient descent are unable to generalize to unseen letters when utilizing orthogonal encodings like one-hot representations; (ii) dicyclic graphs, i.e., graphs composed of two cycles, for which we present positive existence results leveraging the connection between GNNs and the WL test. Our theoretical analysis is supported by an extensive numerical study.

This paper presents a local energy distribution based hyperparameter determination for stochastic simulated annealing (SSA). SSA is capable of solving combinatorial optimization problems faster than typical simulated annealing (SA), but requires a time-consuming hyperparameter search. The proposed method determines hyperparameters based on the local energy distributions of spins (probabilistic bits). The spin is a basic computing element of SSA and is graphically connected to other spins with its weights. The distribution of the local energy can be estimated based on the central limit theorem (CLT). The CLT-based normal distribution is used to determine the hyperparameters, which reduces the time complexity for hyperparameter search from O(n^3) of the conventional method to O(1). The performance of SSA with the determined hyperparameters is evaluated on the Gset and K2000 benchmarks for maximum-cut problems. The results show that the proposed method achieves mean cut values of approximately 98% of the best-known cut values.

Stochastic gradient-based optimization is crucial to optimize neural networks. While popular approaches heuristically adapt the step size and direction by rescaling gradients, a more principled approach to improve optimizers requires second-order information. Such methods precondition the gradient using the objective's Hessian. Yet, computing the Hessian is usually expensive and effectively using second-order information in the stochastic gradient setting is non-trivial. We propose using Information-Theoretic Trust Region Optimization (arTuRO) for improved updates with uncertain second-order information. By modeling the network parameters as a Gaussian distribution and using a Kullback-Leibler divergence-based trust region, our approach takes bounded steps accounting for the objective's curvature and uncertainty in the parameters. Before each update, it solves the trust region problem for an optimal step size, resulting in a more stable and faster optimization process. We approximate the diagonal elements of the Hessian from stochastic gradients using a simple recursive least squares approach, constructing a model of the expected Hessian over time using only first-order information. We show that arTuRO combines the fast convergence of adaptive moment-based optimization with the generalization capabilities of SGD.

The growing size of available data has attracted increasing interest in solving minimax problems in a decentralized manner for various machine learning tasks. Previous theoretical research has primarily focused on the convergence rate and communication complexity of decentralized minimax algorithms, with little attention given to their generalization. In this paper, we investigate the primal-dual generalization bound of the decentralized stochastic gradient descent ascent (D-SGDA) algorithm using the approach of algorithmic stability under both convex-concave and nonconvex-nonconcave settings. Our theory refines the algorithmic stability in a decentralized manner and demonstrates that the decentralized structure does not destroy the stability and generalization of D-SGDA, implying that it can generalize as well as the vanilla SGDA in certain situations. Our results analyze the impact of different topologies on the generalization bound of the D-SGDA algorithm beyond trivial factors such as sample sizes, learning rates, and iterations. We also evaluate the optimization error and balance it with the generalization gap to obtain the optimal population risk of D-SGDA in the convex-concave setting. Additionally, we perform several numerical experiments which validate our theoretical findings.

Designing reward functions for efficiently guiding reinforcement learning (RL) agents toward specific behaviors is a complex task. This is challenging since it requires the identification of reward structures that are not sparse and that avoid inadvertently inducing undesirable behaviors. Naively modifying the reward structure to offer denser and more frequent feedback can lead to unintended outcomes and promote behaviors that are not aligned with the designer's intended goal. Although potential-based reward shaping is often suggested as a remedy, we systematically investigate settings where deploying it often significantly impairs performance. To address these issues, we introduce a new framework that uses a bi-level objective to learn \emph{behavior alignment reward functions}. These functions integrate auxiliary rewards reflecting a designer's heuristics and domain knowledge with the environment's primary rewards. Our approach automatically determines the most effective way to blend these types of feedback, thereby enhancing robustness against heuristic reward misspecification. Remarkably, it can also adapt an agent's policy optimization process to mitigate suboptimalities resulting from limitations and biases inherent in the underlying RL algorithms. We evaluate our method's efficacy on a diverse set of tasks, from small-scale experiments to high-dimensional control challenges. We investigate heuristic auxiliary rewards of varying quality -- some of which are beneficial and others detrimental to the learning process. Our results show that our framework offers a robust and principled way to integrate designer-specified heuristics. It not only addresses key shortcomings of existing approaches but also consistently leads to high-performing solutions, even when given misaligned or poorly-specified auxiliary reward functions.

This paper explores the connection between learning trajectories of Deep Neural Networks (DNNs) and their generalization capabilities when optimized using (stochastic) gradient descent algorithms. Instead of concentrating solely on the generalization error of the DNN post-training, we present a novel perspective for analyzing generalization error by investigating the contribution of each update step to the change in generalization error. This perspective allows for a more direct comprehension of how the learning trajectory influences generalization error. Building upon this analysis, we propose a new generalization bound that incorporates more extensive trajectory information. Our proposed generalization bound depends on the complexity of learning trajectory and the ratio between the bias and diversity of training set. Experimental findings reveal that our method effectively captures the generalization error throughout the training process. Furthermore, our approach can also track changes in generalization error when adjustments are made to learning rates and label noise levels. These results demonstrate that learning trajectory information is a valuable indicator of a model's generalization capabilities.

We develop generalization error bounds for stochastic gradient descent (SGD) with label noise in non-convex settings under uniform dissipativity and smoothness conditions. Under a suitable choice of semimetric, we establish a contraction in Wasserstein distance of the label noise stochastic gradient flow that depends polynomially on the parameter dimension $d$. Using the framework of algorithmic stability, we derive time-independent generalisation error bounds for the discretized algorithm with a constant learning rate. The error bound we achieve scales polynomially with $d$ and with the rate of $n^{-2/3}$, where $n$ is the sample size. This rate is better than the best-known rate of $n^{-1/2}$ established for stochastic gradient Langevin dynamics (SGLD) -- which employs parameter-independent Gaussian noise -- under similar conditions. Our analysis offers quantitative insights into the effect of label noise.

Using backward error analysis, we compute implicit training biases in multitask and continual learning settings for neural networks trained with stochastic gradient descent. In particular, we derive modified losses that are implicitly minimized during training. They have three terms: the original loss, accounting for convergence, an implicit flatness regularization term proportional to the learning rate, and a last term, the conflict term, which can theoretically be detrimental to both convergence and implicit regularization. In multitask, the conflict term is a well-known quantity, measuring the gradient alignment between the tasks, while in continual learning the conflict term is a new quantity in deep learning optimization, although a basic tool in differential geometry: The Lie bracket between the task gradients.

We study the optimisation problem associated with Gaussian process regression using squared loss. The most common approach to this problem is to apply an exact solver, such as conjugate gradient descent, either directly, or to a reducedorder version of the problem. Recently, driven by successes in deep learning, stochastic gradient descent has gained traction as an alternative. In this paper, we show that when done right--by which we mean using specific insights from the optimisation and kernel communities--this approach is highly effective. We thus introduce a particular stochastic dual gradient descent algorithm, that may be implemented with a few lines of code using any deep learning framework. We explain our design decisions by illustrating their advantage against alternatives with ablation studies and show that the new method is highly competitive. Our evaluations on standard regression benchmarks and a Bayesian optimisation task set our approach apart from preconditioned conjugate gradients, variational Gaussian process approximations, and a previous version of stochastic gradient descent for Gaussian processes. Gaussian processes are a probabilistic framework for learning unknown functions. They are the de facto standard model of choice in areas like Bayesian optimisation, where uncertainty-aware decision making is required to gather data in an efficient manner.

We analyze asynchronous-type algorithms for distributed SGD in the heterogeneous setting, where each worker has its own computation and communication speeds, as well as data distribution. In these algorithms, workers compute possibly stale and stochastic gradients associated with their local data at some iteration back in history and then return those gradients to the server without synchronizing with other workers. We present a unified convergence theory for non-convex smooth functions in the heterogeneous regime. The proposed analysis provides convergence for pure asynchronous SGD and its various modifications. Moreover, our theory explains what affects the convergence rate and what can be done to improve the performance of asynchronous algorithms. In particular, we introduce a novel asynchronous method based on worker shuffling. As a by-product of our analysis, we also demonstrate convergence guarantees for gradient-type algorithms such as SGD with random reshuffling and shuffle-once mini-batch SGD. The derived rates match the best-known results for those algorithms, highlighting the tightness of our approach. Finally, our numerical evaluations support theoretical findings and show the good practical performance of our method.