Diversity optimization seeks to discover a set of solutions that elicit diverse features. Prior work has proposed Novelty Search (NS), which, given a current set of solutions, seeks to expand the set by finding points in areas of low density in the feature space. However, to estimate density, NS relies on a heuristic that considers the k-nearest neighbors of the search point in the feature space, which yields a weaker stability guarantee. We propose Density Descent Search (DDS), an algorithm that explores the feature space via gradient descent on a continuous density estimate of the feature space that also provides stronger stability guarantee. We experiment with DDS and two density estimation methods: kernel density estimation (KDE) and continuous normalizing flow (CNF). On several standard diversity optimization benchmarks, DDS outperforms NS, the recently proposed MAP-Annealing algorithm, and other state-of-the-art baselines. Additionally, we prove that DDS with KDE provides stronger stability guarantees than NS, making it more suitable for adaptive optimizers. Furthermore, we prove that NS is a special case of DDS that descends a KDE of the feature space.

This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $O(n)$ and $O(n^2)$ respectively, as compared to the $O(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation and matrix square root problems. MATLAB code implementation is publicly available on GitHub : https://github.com/yogeshd-iitk/subspace_descent_over_SPD_manifold

Data valuation has wide use cases in machine learning, including improving data quality and creating economic incentives for data sharing. This paper studies the robustness of data valuation to noisy model performance scores. Particularly, we find that the inherent randomness of the widely used stochastic gradient descent can cause existing data value notions (e.g., the Shapley value and the Leave-one-out error) to produce inconsistent data value rankings across different runs. To address this challenge, we introduce the concept of safety margin, which measures the robustness of a data value notion. We show that the Banzhaf value, a famous value notion that originated from cooperative game theory literature, achieves the largest safety margin among all semivalues (a class of value notions that satisfy crucial properties entailed by ML applications and include the famous Shapley value and Leave-one-out error). We propose an algorithm to efficiently estimate the Banzhaf value based on the Maximum Sample Reuse (MSR) principle. Our evaluation demonstrates that the Banzhaf value outperforms the existing semivalue-based data value notions on several ML tasks such as learning with weighted samples and noisy label detection. Overall, our study suggests that when the underlying ML algorithm is stochastic, the Banzhaf value is a promising alternative to the other semivalue-based data value schemes given its computational advantage and ability to robustly differentiate data quality.

Stochastic gradient descent (SGD) performed in an asynchronous manner plays a crucial role in training large-scale machine learning models. However, the generalization performance of asynchronous delayed SGD, which is an essential metric for assessing machine learning algorithms, has rarely been explored. Existing generalization error bounds are rather pessimistic and cannot reveal the correlation between asynchronous delays and generalization. In this paper, we investigate sharper generalization error bound for SGD with asynchronous delay $\tau$. Leveraging the generating function analysis tool, we first establish the average stability of the delayed gradient algorithm. Based on this algorithmic stability, we provide upper bounds on the generalization error of $\tilde{\mathcal{O}}(\frac{T-\tau}{n\tau})$ and $\tilde{\mathcal{O}}(\frac{1}{n})$ for quadratic convex and strongly convex problems, respectively, where $T$ refers to the iteration number and $n$ is the amount of training data. Our theoretical results indicate that asynchronous delays reduce the generalization error of the delayed SGD algorithm. Analogous analysis can be generalized to the random delay setting, and the experimental results validate our theoretical findings.

The matrix completion problem aims to reconstruct a low-rank matrix based on a revealed set of possibly noisy entries. Prior works consider completing the entire matrix with generalization error guarantees. However, the completion accuracy can be drastically different over different entries. This work establishes a new framework of partial matrix completion, where the goal is to identify a large subset of the entries that can be completed with high confidence. We propose an efficient algorithm with the following provable guarantees. Given access to samples from an unknown and arbitrary distribution, it guarantees: (a) high accuracy over completed entries, and (b) high coverage of the underlying distribution. We also consider an online learning variant of this problem, where we propose a low-regret algorithm based on iterative gradient updates. Preliminary empirical evaluations are included.

Weight sharing is a fundamental concept in neural architecture search (NAS), enabling gradient-based methods to explore cell-based architecture spaces significantly faster than traditional blackbox approaches. In parallel, weight entanglement has emerged as a technique for intricate parameter sharing among architectures within macro-level search spaces. Since weight-entanglement poses compatibility challenges for gradient-based NAS methods, these two paradigms have largely developed independently in parallel sub-communities. This paper aims to bridge the gap between these sub-communities by proposing a novel scheme to adapt gradient-based methods for weight-entangled spaces. This enables us to conduct an in-depth comparative assessment and analysis of the performance of gradient-based NAS in weight-entangled search spaces. Our findings reveal that this integration of weight-entanglement and gradient-based NAS brings forth the various benefits of gradient-based methods (enhanced performance, improved supernet training properties and superior any-time performance), while preserving the memory efficiency of weight-entangled spaces. The code for our work is openly accessible here. The concept of weight-sharing in Neural Architecture Search (NAS) arose from the need to improve the efficiency of conventional blackbox NAS algorithms, which demand significant computational resources to evaluate individual architectures. Here, weight-sharing (WS) refers to the paradigm by which we represent the search space with a single large supernet, also known as the one-shot model, that subsumes all the candidate architectures in that space. Every edge of this supernet holds all the possible operations that can be assigned to that edge. Gradient-based NAS algorithms (or optimizers), such as DARTS (Liu et al., 2019), GDAS (Dong and Yang, 2019) and DrNAS (Chen et al., 2021b), assign an architectural parameter to every choice of operation on a given edge of the supernet.

Particle Swarm Optimization (PSO) has emerged as a powerful metaheuristic global optimization approach over the past three decades. Its appeal lies in its ability to tackle complex multidimensional problems that defy conventional algorithms. However, PSO faces challenges, such as premature stagnation in single-objective scenarios and the need to strike a balance between exploration and exploitation. Hybridizing PSO by integrating its cooperative nature with established optimization techniques from diverse paradigms offers a promising solution. In this paper, we investigate various strategies for synergizing gradient-based optimizers with PSO. We introduce different hybridization principles and explore several approaches, including sequential decoupled hybridization, coupled hybridization, and adaptive hybridization. These strategies aim to enhance the efficiency and effectiveness of PSO, ultimately improving its ability to navigate intricate optimization landscapes. By combining the strengths of gradient-based methods with the inherent social dynamics of PSO, we seek to address the critical objectives of intelligent exploration and exploitation in complex optimization tasks. Our study delves into the comparative merits of these hybridization techniques and offers insights into their application across different problem domains.

Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy.

In this article, the state estimation problems with unknown process noise and measurement noise covariances for both linear and nonlinear systems are considered. By formulating the joint estimation of system state and noise parameters into an optimization problem, a novel adaptive Kalman filter method based on conjugate-computation variational inference, referred to as CVIAKF, is proposed to approximate the joint posterior probability density function of the latent variables. Unlike the existing adaptive Kalman filter methods utilizing variational inference in natural-parameter space, CVIAKF performs optimization in expectation-parameter space, resulting in a faster and simpler solution. Meanwhile, CVIAKF divides optimization objectives into conjugate and non-conjugate parts of nonlinear dynamical models, whereas conjugate computations and stochastic mirror-descent are applied, respectively. Remarkably, the reparameterization trick is used to reduce the variance of stochastic gradients of the non-conjugate parts. The effectiveness of CVIAKF is validated through synthetic and real-world datasets of maneuvering target tracking.

We investigate theoretically how the features of a two-layer neural network adapt to the structure of the target function through a few large batch gradient descent steps, leading to improvement in the approximation capacity with respect to the initialization. We compare the influence of batch size and that of multiple (but finitely many) steps. For a single gradient step, a batch of size $n = \mathcal{O}(d)$ is both necessary and sufficient to align with the target function, although only a single direction can be learned. In contrast, $n = \mathcal{O}(d^2)$ is essential for neurons to specialize to multiple relevant directions of the target with a single gradient step. Even in this case, we show there might exist ``hard'' directions requiring $n = \mathcal{O}(d^\ell)$ samples to be learned, where $\ell$ is known as the leap index of the target. The picture drastically improves over multiple gradient steps: we show that a batch-size of $n = \mathcal{O}(d)$ is indeed enough to learn multiple target directions satisfying a staircase property, where more and more directions can be learned over time. Finally, we discuss how these directions allows to drastically improve the approximation capacity and generalization error over the initialization, illustrating a separation of scale between the random features/lazy regime, and the feature learning regime. Our technical analysis leverages a combination of techniques related to concentration, projection-based conditioning, and Gaussian equivalence which we believe are of independent interest. By pinning down the conditions necessary for specialization and learning, our results highlight the interaction between batch size and number of iterations, and lead to a hierarchical depiction where learning performance exhibits a stairway to accuracy over time and batch size, shedding new light on how neural networks adapt to features of the data.