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1. An Exact Characterization of the Generalization Error for the Gibbs Algorithm

   Aminian, G; Bu, Y; Toni, L.; Rodrigues, M; Wornell, G. W. (December 2021, Advances in neural information processing systems)

   Various approaches have been developed to upper bound the generalization error of a supervised learning algorithm. However, existing bounds are often loose and lack of guarantees. As a result, they may fail to characterize the exact generalization ability of a learning algorithm.Our main contribution is an exact characterization of the expected generalization error of the well-known Gibbs algorithm (a.k.a. Gibbs posterior) using symmetrized KL information between the input training samples and the output hypothesis. Our result can be applied to tighten existing expected generalization error and PAC-Bayesian bounds. Our approach is versatile, as it also characterizes the generalization error of the Gibbs algorithm with data-dependent regularizer and that of the Gibbs algorithm in the asymptotic regime, where it converges to the empirical risk minimization algorithm. Of particular relevance, our results highlight the role the symmetrized KL information plays in controlling the generalization error of the Gibbs algorithm. 

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2. Tighter PAC-Bayes Bounds Through Coin-Betting

   Jang, Kyoungseok; Jun, Kwang-Sung; Kuzborskii, Ilja; Orabona, Francesco (July 2023, Conference on Learning Theory)

   We consider the problem of estimating the mean of a sequence of random elements f (θ, X_1) , . . . , f (θ, X_n) where f is a fixed scalar function, S = (X_1, . . . , X_n) are independent random variables, and θ is a possibly S-dependent parameter. An example of such a problem would be to estimate the generalization error of a neural network trained on n examples where f is a loss function. Classically, this problem is approached through concentration inequalities holding uniformly over compact parameter sets of functions f , for example as in Rademacher or VC type analysis. However, in many problems, such inequalities often yield numerically vacuous estimates. Recently, the PAC-Bayes framework has been proposed as a better alternative for this class of problems for its ability to often give numerically non-vacuous bounds. In this paper, we show that we can do even better: we show how to refine the proof strategy of the PAC-Bayes bounds and achieve even tighter guarantees. Our approach is based on the coin-betting framework that derives the numerically tightest known time-uniform concentration inequalities from the regret guarantees of online gambling algorithms. In particular, we derive the first PAC-Bayes concentration inequality based on the coin-betting approach that holds simultaneously for all sample sizes. We demonstrate its tightness showing that by relaxing it we obtain a number of previous results in a closed form including Bernoulli-KL and empirical Bernstein inequalities. Finally, we propose an efficient algorithm to numerically calculate confidence sequences from our bound, which often generates nonvacuous confidence bounds even with one sample, unlike the state-of-the-art PAC-Bayes bounds. 

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3. Efficient, Noise-Tolerant, and Private Learning via Boosting

   Bun, Mark; Carmosino, Marco L.; Sorrell, Jessica (January 2020, Proceedings of Thirty Third Conference on Learning Theory)

   null (Ed.)

   We introduce a simple framework for designing private boosting algorithms. We give natural conditions under which these algorithms are differentially private, efficient, and noise-tolerant PAC learners. To demonstrate our framework, we use it to construct noise-tolerant and private PAC learners for large-margin halfspaces whose sample complexity does not depend on the dimension. We give two sample complexity bounds for our large-margin halfspace learner. One bound is based only on differential privacy, and uses this guarantee as an asset for ensuring generalization. This first bound illustrates a general methodology for obtaining PAC learners from privacy, which may be of independent interest. The second bound uses standard techniques from the theory of large-margin classification (the fat-shattering dimension) to match the best known sample complexity for differentially private learning of large-margin halfspaces, while additionally tolerating random label noise. 

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4. On Leave-One-Out Conditional Mutual Information For Generalization

   Rammal, R; Achille, A; Golatkar, A; Diggavi, S N; Soatto, S (January 2022, Advances in Neural Information Processing Systems (NeurIPS))

   We derive information theoretic generalization bounds for supervised learning algorithms based on a new measure of leave-one-out conditional mutual information (loo-CMI). Contrary to other CMI bounds, which are black-box bounds that do not exploit the structure of the problem and may be hard to evaluate in practice, our loo-CMI bounds can be computed easily and can be interpreted in connection to other notions such as classical leave-one-out cross-validation, stability of the optimization algorithm, and the geometry of the loss-landscape. It applies both to the output of training algorithms as well as their predictions. We empirically validate the quality of the bound by evaluating its predicted generalization gap in scenarios for deep learning. In particular, our bounds are non-vacuous on large-scale image-classification tasks. 

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5. Tighter Expected Generalization Error Bounds via Convexity of Information Measures

   https://doi.org/10.1109/ISIT50566.2022.9834474  

   Aminian, Gholamali; Bu, Yuheng; Wornell, Gregory W.; Rodrigues, Miguel R. (June 2022, IEEE International Symposium on Information Theory)

   Generalization error bounds are essential to understanding machine learning algorithms. This paper presents novel expected generalization error upper bounds based on the average joint distribution between the output hypothesis and each input training sample. Multiple generalization error upper bounds based on different information measures are provided, including Wasserstein distance, total variation distance, KL divergence, and Jensen-Shannon divergence. Due to the convexity of the information measures, the proposed bounds in terms of Wasserstein distance and total variation distance are shown to be tighter than their counterparts based on individual samples in the literature. An example is provided to demonstrate the tightness of the proposed generalization error bounds. 

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