In this work, we analyze alternative effective sample size (ESS) metrics for importance sampling algorithms, and discuss a possible extended range of applications. We show the relationship between the ESS expressions used in the literature and two entropy families, the Rényi and Tsallis entropy. The Rényi entropy is connected to the Huggins-Roy's ESS family introduced in \cite{Huggins15}. We prove that that all the ESS functions included in the Huggins-Roy's family fulfill all the desirable theoretical conditions. We analyzed and remark the connections with several other fields, such as the Hill numbers introduced in ecology, the Gini inequality coefficient employed in economics, and the Gini impurity index used mainly in machine learning, to name a few. Finally, by numerical simulations, we study the performance of different ESS expressions contained in the previous ESS families in terms of approximation of the theoretical ESS definition, and show the application of ESS formulas in a variable selection problem.

We study the dynamics and implicit bias of gradient flow (GF) on univariate ReLU neural networks with a single hidden layer in a binary classification setting. We show that when the labels are determined by the sign of a target network with $r$ neurons, with high probability over the initialization of the network and the sampling of the dataset, GF converges in direction (suitably defined) to a network achieving perfect training accuracy and having at most $\mathcal{O}(r)$ linear regions, implying a generalization bound. Unlike many other results in the literature, under an additional assumption on the distribution of the data, our result holds even for mild over-parameterization, where the width is $\tilde{\mathcal{O}}(r)$ and independent of the sample size.

The residual pathway of Eq. (10) was originally in Finzi et al. [2021a] for general groups Similar to the original residual pathway paper [Finzi et al., 2021a], we could also write linear Further, we give an extension for group convolutional layers. First, we split up the definition from Eq. (5): y ( c For sparsified factored layers S-FC, we extend the K-FAC approximation for factored layers. Similarly to above, we use the existing derivation to approximate KFAC in terms of Jacobians w.r.t. Table 4: Analysis of learned layer types after training FC+CONV model on CIFAR-10. Prior precisions learned for the CONV+FC model are shown on the left in Table 4.

Hyperparameters greatly impact models' capabilities; however, modern models are too large for extensive search. Instead, researchers design recipes that train well across scales based on their understanding of the hyperparameters. Despite this importance, few tools exist for understanding the hyperparameter loss surface. We discover novel structure in it and propose a new theory yielding such tools. The loss surface is complex, but as you approach the optimum simple structure emerges. It becomes characterized by a few basic features, like its effective dimension and the best possible loss. To uncover this asymptotic regime, we develop a novel technique based on random search. Within this regime, the best scores from random search take on a new distribution we discover. Its parameters are exactly the features defining the loss surface in the asymptotic regime. From these features, we derive a new asymptotic law for random search that can explain and extrapolate its convergence. These new tools enable new analyses, such as confidence intervals for the best possible performance or determining the effective number of hyperparameters. We make these tools available at https://github.com/nicholaslourie/opda .

Risk categorization in 10-K risk disclosures matters for oversight and investment, yet no public benchmark evaluates unsupervised topic models for this task. We present GRAB, a finance-specific benchmark with 1.61M sentences from 8,247 filings and span-grounded sentence labels produced without manual annotation by combining FinBERT token attention, YAKE keyphrase signals, and taxonomy-aware collocation matching. Labels are anchored in a risk taxonomy mapping 193 terms to 21 fine-grained types nested under five macro classes; the 21 types guide weak supervision, while evaluation is reported at the macro level. GRAB unifies evaluation with fixed dataset splits and robust metrics--Accuracy, Macro-F1, Topic BERTScore, and the entropy-based Effective Number of Topics. The dataset, labels, and code enable reproducible, standardized comparison across classical, embedding-based, neural, and hybrid topic models on financial disclosures.