Organization: This supplementary material is presented in a format parallel to the main paper. The section numbers and titles are consistent with the main paper. But, here we also add one new section: Section 10 where we describe the societal impacts and possible negative impacts of the paper. Similarly, the Theorem numbers are consistent with the main paper, but we also have several additional theorems and lemmas which were not included in the main paper. GAN-type Objective for KLEstimation Let f be a discriminator, f: X IR. Let p(x) and q(x) be two probability density functions defined over the space X.

Financial statement auditing is conducted under a risk-based evidence approach to obtain reasonable assurance. In practice, auditors often perform additional sampling or related procedures when an initial sample does not provide a sufficient basis for a conclusion. Across jurisdictions, current standards and practice manuals acknowledge such extensions, while the statistical design of sequential audit procedures has not been fully explored. This study formulates audit sampling with additional, sequentially collected items as a sequential testing problem for a finite population under sampling without replacement. We define null and alternative hypotheses in terms of a tolerable deviation rate, specify stopping and decision rules, and formulate exact sequential boundary conditions in terms of finite-population error probabilities. For practical implementation, we calibrate those boundaries by Monte Carlo simulation at least-favorable deviation rates. The exact design yields ex ante control of decision error probabilities, and the simulation-based implementation approximates that design while allowing the computation of expected stopping times. The framework is most naturally suited to attribute auditing and deviation-rate auditing, especially tests of controls, and it can be extended to one-sided, two-stage, and truncated designs.

We study the fundamental problem of sequential probability assignment, also known as online learning with logarithmic loss, with respect to an arbitrary, possibly nonparametric hypothesis class. Our goal is to obtain a complexity measure for the hypothesis class that characterizes the minimax regret and to determine a general, minimax optimal algorithm. Notably, the sequential $\ell_{\infty}$ entropy, extensively studied in the literature (Rakhlin and Sridharan, 2015, Bilodeau et al., 2020, Wu et al., 2023), was shown to not characterize minimax regret in general. Inspired by the seminal work of Shtarkov (1987) and Rakhlin, Sridharan, and Tewari (2010), we introduce a novel complexity measure, the \emph{contextual Shtarkov sum}, corresponding to the Shtarkov sum after projection onto a multiary context tree, and show that the worst case log contextual Shtarkov sum equals the minimax regret. Using the contextual Shtarkov sum, we derive the minimax optimal strategy, dubbed \emph{contextual Normalized Maximum Likelihood} (cNML). Our results hold for sequential experts, beyond binary labels, which are settings rarely considered in prior work. To illustrate the utility of this characterization, we provide a short proof of a new regret upper bound in terms of sequential $\ell_{\infty}$ entropy, unifying and sharpening state-of-the-art bounds by Bilodeau et al. (2020) and Wu et al. (2023).

In this paper we consider the case of generativemodels. In particular,we investigate generativeadversarialnetworks (GANs) in the task of learning new categories in a sequential fashion.

In this section, we provide a detailed description of the dataset following'datasheet for datasets' [

The remarkable capability of over-parameterised neural networks to generalise effectively has been explained by invoking a "simplicity bias": neural networks prevent overfitting by initially learning simple classifiers before progressing to