Market Microstructure

Adaptive gradient optimizers like Adam(W) are the default training algorithms for many deep learning architectures, such as transformers. Their diagonal preconditioner is based on the gradient outer product which is incorporated into the parameter update via a square root. While these methods are often motivated as approximate second-order methods, the square root represents a fundamental difference. In this work, we investigate how the behavior of adaptive methods changes when we remove the root, i.e. strengthen their second-order motivation. Surprisingly, we find that such square-root-free adaptive methods close the generalization gap to SGD on convolutional architectures, while maintaining their root-based counterpart's performance on transformers. The second-order perspective also has practical benefits for the development of non-diagonal adaptive methods through the concept of preconditioner invariance. In contrast to root-based methods like Shampoo, the root-free counterparts do not require numerically unstable matrix root decompositions and inversions, thus work well in half precision. Our findings provide new insights into the development of adaptive methods and raise important questions regarding the currently overlooked role of adaptivity for their success.

Non-convex minimization problems are universally considered hard, and even guaranteeing that a computed solution is locally minimizing is known to be NP-hard. In this general context, our paper focuses on the problem of finding stationary points that satisfy an approximate second-order optimality condition, which serves to exclude strict saddles and other non-minimizing stationary points. Our main result is that the problem of finding approximate second-order stationary points (SOSPs) is PLS-complete, i.e., of the same complexity as the problem of finding first-order stationary points (FOSPs), thus resolving an open question in the field. In particular, our results imply that, under the widely believed complexity conjecture that PLS $\neq$ FNP, finding approximate SOSPs in unconstrained domains is *easier* than in constrained domains, which is known to be NP-hard. This comes in stark contrast with earlier results which implied that, unless PLS = CLS, finding approximate FOSPs in unconstrained domains is *harder* than in constrained domains.

Discovering reliable and informative relationships among brain regions from functional magnetic resonance imaging (fMRI) signals is essential in phenotypic predictions in neuroscience. Most of the current methods fail to accurately characterize those interactions because they only focus on pairwise connections and overlook the high-order relationships of brain regions. We propose that these high-order relationships should be *maximally informative and minimally redundant* (MIMR). However, identifying such high-order relationships is challenging and under-explored due to the exponential search space and the absence of a tractable objective. In response to this gap, we propose a novel method named HyBRiD, which aims to extract MIMR high-order relationships from fMRI data. HyBRiD employs a Constructor to identify hyperedge structures, and a Weighter to compute a weight for each hyperedge, which avoids searching in exponential space. HyBRiD achieves the MIMR objective through an innovative information bottleneck framework named multi-head drop-bottleneck with theoretical guarantees. Our comprehensive experiments demonstrate the effectiveness of our model. Our model outperforms the state-of-the-art predictive model by an average of 11.2%, regarding the quality of hyperedges measured by CPM, a standard protocol for studying brain connections.

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

This paper studies the theoretical guarantees of the classical projected gradient and conditional gradient methods applied to constrained optimization problems with biased relative-error gradient oracles. These oracles are used in various settings, such as distributed optimization systems or derivative-free optimization, and are particularly common when gradients are compressed, quantized, or estimated via finite differences computations. Several settings are investigated: Optimization over the box with a coordinate-wise erroneous gradient oracle, optimization over a general compact convex set, and three more specific scenarios. Convergence guarantees are established with respect to the relative-error magnitude, and in particular, we show that the conditional gradient is invariant to relative-error when applied over the box with a coordinate-wise erroneous gradient oracle, and the projected gradient maintains its convergence guarantees when optimizing a nonconvex objective function.

Second-order Recurrent Neural Networks (2RNNs) extend RNNs by leveraging second-order interactions for sequence modelling. These models are provably more expressive than their first-order counterparts and have connections to well-studied models from formal language theory. However, their large parameter tensor makes computations intractable. To circumvent this issue, one approach known as MIRNN consists in limiting the type of interactions used by the model. Another is to leverage tensor decomposition to diminish the parameter count. In this work, we study the model resulting from parameterizing 2RNNs using the CP decomposition, which we call CPRNN. Intuitively, the rank of the decomposition should reduce expressivity. We analyze how rank and hidden size affect model capacity and show the relationships between RNNs, 2RNNs, MIRNNs, and CPRNNs based on these parameters. We support these results empirically with experiments on the Penn Treebank dataset which demonstrate that, with a fixed parameter budget, CPRNNs outperforms RNNs, 2RNNs, and MIRNNs with the right choice of rank and hidden size.