# PACMAN: PAC-style bounds accounting for the Mismatch between Accuracy and Negative log-loss

arXiv: https://arxiv.org/abs/2112.05547 · HTML: https://mlsanity.com/paper/2112.05547/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 51 references that are in the map.
- First posted on arXiv 10 Dec 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Compression Approach](https://mlsanity.com/paper/1804.05862/) (2018)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors](https://mlsanity.com/paper/1712.09376/) (2017)
- [Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle](https://mlsanity.com/paper/1802.09766/) (2018)
- [On the role of data in PAC-Bayes bounds](https://mlsanity.com/paper/2006.10929/) (2020)
- [Deterministic PAC-Bayesian generalization bounds for deep networks via generalizing noise-resilience](https://mlsanity.com/paper/1905.13344/) (2019)
- [Recent advances in deep learning theory](https://mlsanity.com/paper/2012.10931/) (2020)
- [Geometry of Optimization and Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1705.03071/) (2017)
- [PAC-Bayes with Backprop](https://mlsanity.com/paper/1908.07380/) (2019)
- [On Margins and Derandomisation in PAC-Bayes](https://mlsanity.com/paper/2107.03955/) (2021)
- [On the Estimation of Information Measures of Continuous Distributions](https://mlsanity.com/paper/2002.02851/) (2020)
- [Robust Generalization via α-Mutual Information](https://mlsanity.com/paper/2001.06399/) (2020)

## Built on by (most cited first, 1 of 1)

- [Don't stop me now: How Validation Criteria Affect Checkpoint Selection and Early Stopping](https://mlsanity.com/paper/2602.22107/) (2026)

## Abstract (arXiv)

The ultimate performance of machine learning algorithms for classification tasks is usually measured in terms of the empirical error probability (or accuracy) based on a testing dataset. Whereas, these algorithms are optimized through the minimization of a typically different--more convenient--loss function based on a training set. For classification tasks, this loss function is often the negative log-loss that leads to the well-known cross-entropy risk which is typically better behaved (from a numerical perspective) than the error probability. Conventional studies on the generalization error do not usually take into account the underlying mismatch between losses at training and testing phases. In this work, we introduce an analysis based on point-wise PAC approach over the generalization gap considering the mismatch of testing based on the accuracy metric and training on the negative log-loss. We label this analysis PACMAN. Building on the fact that the mentioned mismatch can be written as a likelihood ratio, concentration inequalities can be used to provide some insights for the generalization problem in terms of some point-wise PAC bounds depending on some meaningful information-theoretic quantities. An analysis of the obtained bounds and a comparison with available results in the literature are also provided.