Physics-informed neural networks (PINN) have recently emerged as a promising application of deep learning in a wide range of engineering and scientific problems based on partial differential equation (PDE) models. However, evidence shows that PINN training by gradient descent displays pathologies that often prevent convergence when solving PDEs with irregular solutions. In this paper, we propose the use of a particle swarm optimization (PSO) approach to train PINNs. The resulting PSO-PINN algorithm not only mitigates the undesired behaviors of PINNs trained with standard gradient descent but also presents an ensemble approach to PINN that affords the possibility of robust predictions with quantified uncertainty. We also propose PSO-BP-CD (PSO with Back-Propagation and Coefficient Decay), a hybrid PSO variant that combines swarm optimization with gradient descent, putting more weight on the latter as training progresses and the swarm zeros in on a good local optimum. Comprehensive experimental results show that PSO-PINN with the proposed PSO-BP-CD algorithm outperforms PINN ensembles trained with other PSO variants or with pure gradient descent.

The best neural architecture for a given machine learning problem depends on many factors: not only the complexity and structure of the dataset, but also on resource constraints including latency, compute, energy consumption, etc. Neural architecture search (NAS) for tabular datasets is an important but under-explored problem. Previous NAS algorithms designed for image search spaces incorporate resource constraints directly into the reinforcement learning (RL) rewards. However, for NAS on tabular datasets, this protocol often discovers suboptimal architectures. This paper develops TabNAS, a new and more effective approach to handle resource constraints in tabular NAS using an RL controller motivated by the idea of rejection sampling. TabNAS immediately discards any architecture that violates the resource constraints without training or learning from that architecture. TabNAS uses a Monte-Carlo-based correction to the RL policy gradient update to account for this extra filtering step. Results on several tabular datasets demonstrate the superiority of TabNAS over previous reward-shaping methods: it finds better models that obey the constraints.

In the machine learning literature stochastic gradient descent has recently been widely discussed for its purported implicit regularization properties. Much of the theory, that attempts to clarify the role of noise in stochastic gradient algorithms, has widely approximated stochastic gradient descent by a stochastic differential equation with Gaussian noise. We provide a novel rigorous theoretical justification for this practice that showcases how the Gaussianity of the noise arises naturally.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods (SMM) for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.

Decentralized optimization is increasingly popular in machine learning for its scalability and efficiency. Intuitively, it should also provide better privacy guarantees, as nodes only observe the messages sent by their neighbors in the network graph. But formalizing and quantifying this gain is challenging: existing results are typically limited to Local Differential Privacy (LDP) guarantees that overlook the advantages of decentralization. In this work, we introduce pairwise network differential privacy, a relaxation of LDP that captures the fact that the privacy leakage from a node $u$ to a node $v$ may depend on their relative position in the graph. We then analyze the combination of local noise injection with (simple or randomized) gossip averaging protocols on fixed and random communication graphs. We also derive a differentially private decentralized optimization algorithm that alternates between local gradient descent steps and gossip averaging. Our results show that our algorithms amplify privacy guarantees as a function of the distance between nodes in the graph, matching the privacy-utility trade-off of the trusted curator, up to factors that explicitly depend on the graph topology. Finally, we illustrate our privacy gains with experiments on synthetic and real-world datasets.

Originally published on Towards AI the World's Leading AI and Technology News and Media Company. If you are building an AI-related product or service, we invite you to consider becoming an AI sponsor. At Towards AI, we help scale AI and technology startups. Let us help you unleash your technology to the masses. Most deep learning algorithms train by optimizing some kind of objective (or loss) function. We typically try to either maximize or minimize these functions.

Optimizing expensive-to-evaluate black-box functions of discrete (and potentially continuous) design parameters is a ubiquitous problem in scientific and engineering applications. Bayesian optimization (BO) is a popular, sample-efficient method that leverages a probabilistic surrogate model and an acquisition function (AF) to select promising designs to evaluate. However, maximizing the AF over mixed or high-cardinality discrete search spaces is challenging standard gradient-based methods cannot be used directly or evaluating the AF at every point in the search space would be computationally prohibitive. To address this issue, we propose using probabilistic reparameterization (PR). Instead of directly optimizing the AF over the search space containing discrete parameters, we instead maximize the expectation of the AF over a probability distribution defined by continuous parameters. We prove that under suitable reparameterizations, the BO policy that maximizes the probabilistic objective is the same as that which maximizes the AF, and therefore, PR enjoys the same regret bounds as the original BO policy using the underlying AF. Moreover, our approach provably converges to a stationary point of the probabilistic objective under gradient ascent using scalable, unbiased estimators of both the probabilistic objective and its gradient. Therefore, as the number of starting points and gradient steps increase, our approach will recover of a maximizer of the AF (an often-neglected requisite for commonly used BO regret bounds). We validate our approach empirically and demonstrate state-of-the-art optimization performance on a wide range of real-world applications. PR is complementary to (and benefits) recent work and naturally generalizes to settings with multiple objectives and black-box constraints.

Variance-reduced stochastic gradient methods have gained popularity in recent times. Several variants exist with different strategies for the storing and sampling of gradients and this work concerns the interactions between these two aspects. We present a general proximal variance-reduced gradient method and analyze it under strong convexity assumptions. Special cases of the algorithm include SAGA, L-SVRG and their proximal variants. Our analysis sheds light on epoch-length selection and the need to balance the convergence of the iterates with how often gradients are stored. The analysis improves on other convergence rates found in the literature and produces a new and faster converging sampling strategy for SAGA. Problem instances for which the predicted rates are the same as the practical rates are presented together with problems based on real world data.

In response to the growing demand, differential-private (DP) machine learning [10] provides a computational framework for privacy protection and has been widely studied in various settings, including both convex and non-convex optimization [15, 33, 34]. One widely used procedure for privacy-preserving learning is the (Differentially) Private Gradient Descent (PGD) [1, 3]. A typical gradient descent procedure updates its model by gradients of the loss evaluated on a training dataset. When the data is sensitive, the gradients should be privatized to prevent excess privacy leakage. The PGD privatizes a gradient by adding controlled noise. As such, the models from PGD is expected to have a lower utility as compared to those from unprotected algorithms. In the cases where strict privacy control is exercised, or equivalently, a tight privacy budget, accumulating effects from highly-noised gradients may lead to unacceptable model performance. It is thus critical to design effective privatization procedures for PGD to maintain a great balance between utility and privacy. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.