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  1. Book ; Online: Concentration without Independence via Information Measures

    Esposito, Amedeo Roberto / Mondelli, Marco

    2023  

    Abstract: We propose a novel approach to concentration for non-independent random variables. The main idea is to ``pretend'' that the random variables are independent and pay a multiplicative price measuring how far they are from actually being independent. This ... ...

    Abstract We propose a novel approach to concentration for non-independent random variables. The main idea is to ``pretend'' that the random variables are independent and pay a multiplicative price measuring how far they are from actually being independent. This price is encapsulated in the Hellinger integral between the joint and the product of the marginals, which is then upper bounded leveraging tensorisation properties. Our bounds represent a natural generalisation of concentration inequalities in the presence of dependence: we recover exactly the classical bounds (McDiarmid's inequality) when the random variables are independent. Furthermore, in a ``large deviations'' regime, we obtain the same decay in the probability as for the independent case, even when the random variables display non-trivial dependencies. To show this, we consider a number of applications of interest. First, we provide a bound for Markov chains with finite state space. Then, we consider the Simple Symmetric Random Walk, which is a non-contracting Markov chain, and a non-Markovian setting in which the stochastic process depends on its entire past. To conclude, we propose an application to Markov Chain Monte Carlo methods, where our approach leads to an improved lower bound on the minimum burn-in period required to reach a certain accuracy. In all of these settings, we provide a regime of parameters in which our bound fares better than what the state of the art can provide.
    Keywords Computer Science - Information Theory ; Mathematics - Probability
    Subject code 519
    Publishing date 2023-03-13
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  2. Book ; Online: From Generalisation Error to Transportation-cost Inequalities and Back

    Esposito, Amedeo Roberto / Gastpar, Michael

    2022  

    Abstract: In this work, we connect the problem of bounding the expected generalisation error with transportation-cost inequalities. Exposing the underlying pattern behind both approaches we are able to generalise them and go beyond Kullback-Leibler Divergences/ ... ...

    Abstract In this work, we connect the problem of bounding the expected generalisation error with transportation-cost inequalities. Exposing the underlying pattern behind both approaches we are able to generalise them and go beyond Kullback-Leibler Divergences/Mutual Information and sub-Gaussian measures. In particular, we are able to provide a result showing the equivalence between two families of inequalities: one involving functionals and one involving measures. This result generalises the one proposed by Bobkov and G\"otze that connects transportation-cost inequalities with concentration of measure. Moreover, it allows us to recover all standard generalisation error bounds involving mutual information and to introduce new, more general bounds, that involve arbitrary divergence measures.

    Comment: Submitted to ISIT 2022
    Keywords Computer Science - Information Theory ; Computer Science - Machine Learning ; Mathematics - Functional Analysis ; Mathematics - Probability
    Publishing date 2022-02-08
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  3. Book ; Online: Lower Bounds on the Bayesian Risk via Information Measures

    Esposito, Amedeo Roberto / Vandenbroucque, Adrien / Gastpar, Michael

    2023  

    Abstract: This paper focuses on parameter estimation and introduces a new method for lower bounding the Bayesian risk. The method allows for the use of virtually \emph{any} information measure, including R\'enyi's $\alpha$, $\varphi$-Divergences, and Sibson's $\ ... ...

    Abstract This paper focuses on parameter estimation and introduces a new method for lower bounding the Bayesian risk. The method allows for the use of virtually \emph{any} information measure, including R\'enyi's $\alpha$, $\varphi$-Divergences, and Sibson's $\alpha$-Mutual Information. The approach considers divergences as functionals of measures and exploits the duality between spaces of measures and spaces of functions. In particular, we show that one can lower bound the risk with any information measure by upper bounding its dual via Markov's inequality. We are thus able to provide estimator-independent impossibility results thanks to the Data-Processing Inequalities that divergences satisfy. The results are then applied to settings of interest involving both discrete and continuous parameters, including the ``Hide-and-Seek'' problem, and compared to the state-of-the-art techniques. An important observation is that the behaviour of the lower bound in the number of samples is influenced by the choice of the information measure. We leverage this by introducing a new divergence inspired by the ``Hockey-Stick'' Divergence, which is demonstrated empirically to provide the largest lower-bound across all considered settings. If the observations are subject to privatisation, stronger impossibility results can be obtained via Strong Data-Processing Inequalities. The paper also discusses some generalisations and alternative directions.
    Keywords Computer Science - Information Theory ; Computer Science - Machine Learning
    Subject code 000
    Publishing date 2023-03-22
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  4. Book ; Online: Generalization Error Bounds for Noisy, Iterative Algorithms via Maximal Leakage

    Issa, Ibrahim / Esposito, Amedeo Roberto / Gastpar, Michael

    2023  

    Abstract: We adopt an information-theoretic framework to analyze the generalization behavior of the class of iterative, noisy learning algorithms. This class is particularly suitable for study under information-theoretic metrics as the algorithms are inherently ... ...

    Abstract We adopt an information-theoretic framework to analyze the generalization behavior of the class of iterative, noisy learning algorithms. This class is particularly suitable for study under information-theoretic metrics as the algorithms are inherently randomized, and it includes commonly used algorithms such as Stochastic Gradient Langevin Dynamics (SGLD). Herein, we use the maximal leakage (equivalently, the Sibson mutual information of order infinity) metric, as it is simple to analyze, and it implies both bounds on the probability of having a large generalization error and on its expected value. We show that, if the update function (e.g., gradient) is bounded in $L_2$-norm and the additive noise is isotropic Gaussian noise, then one can obtain an upper-bound on maximal leakage in semi-closed form. Furthermore, we demonstrate how the assumptions on the update function affect the optimal (in the sense of minimizing the induced maximal leakage) choice of the noise. Finally, we compute explicit tight upper bounds on the induced maximal leakage for other scenarios of interest.

    Comment: Updated to fix an error in Theorem 4 (asymptotic analysis)
    Keywords Computer Science - Machine Learning ; Computer Science - Information Theory ; Statistics - Machine Learning
    Publishing date 2023-02-28
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  5. Book ; Online: Lower-bounds on the Bayesian Risk in Estimation Procedures via $f$-Divergences

    Vandenbroucque, Adrien / Esposito, Amedeo Roberto / Gastpar, Michael

    2022  

    Abstract: We consider the problem of parameter estimation in a Bayesian setting and propose a general lower-bound that includes part of the family of $f$-Divergences. The results are then applied to specific settings of interest and compared to other notable ... ...

    Abstract We consider the problem of parameter estimation in a Bayesian setting and propose a general lower-bound that includes part of the family of $f$-Divergences. The results are then applied to specific settings of interest and compared to other notable results in the literature. In particular, we show that the known bounds using Mutual Information can be improved by using, for example, Maximal Leakage, Hellinger divergence, or generalizations of the Hockey-Stick divergence.

    Comment: Submitted to ISIT 2022
    Keywords Computer Science - Information Theory ; Mathematics - Probability ; Mathematics - Statistics Theory
    Publishing date 2022-02-05
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  6. Book ; Online: On conditional Sibson's $\alpha$-Mutual Information

    Esposito, Amedeo Roberto / Wu, Diyuan / Gastpar, Michael

    2021  

    Abstract: In this work, we analyse how to define a conditional version of Sibson's $\alpha$-Mutual Information. Several such definitions can be advanced and they all lead to different information measures with different (but similar) operational meanings. We will ... ...

    Abstract In this work, we analyse how to define a conditional version of Sibson's $\alpha$-Mutual Information. Several such definitions can be advanced and they all lead to different information measures with different (but similar) operational meanings. We will analyse in detail one such definition, compute a closed-form expression for it and endorse it with an operational meaning while also considering some applications. The alternative definitions will also be mentioned and compared.

    Comment: Accepted for Publication by ISIT
    Keywords Computer Science - Information Theory ; Mathematics - Probability ; Mathematics - Statistics Theory
    Publishing date 2021-02-01
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  7. Book ; Online: Strengthened Information-theoretic Bounds on the Generalization Error

    Issa, Ibrahim / Esposito, Amedeo Roberto / Gastpar, Michael

    2019  

    Abstract: The following problem is considered: given a joint distribution $P_{XY}$ and an event $E$, bound $P_{XY}(E)$ in terms of $P_XP_Y(E)$ (where $P_XP_Y$ is the product of the marginals of $P_{XY}$) and a measure of dependence of $X$ and $Y$. Such bounds have ...

    Abstract The following problem is considered: given a joint distribution $P_{XY}$ and an event $E$, bound $P_{XY}(E)$ in terms of $P_XP_Y(E)$ (where $P_XP_Y$ is the product of the marginals of $P_{XY}$) and a measure of dependence of $X$ and $Y$. Such bounds have direct applications in the analysis of the generalization error of learning algorithms, where $E$ represents a large error event and the measure of dependence controls the degree of overfitting. Herein, bounds are demonstrated using several information-theoretic metrics, in particular: mutual information, lautum information, maximal leakage, and $J_\infty$. The mutual information bound can outperform comparable bounds in the literature by an arbitrarily large factor.

    Comment: Submitted to ISIT 2019
    Keywords Computer Science - Information Theory
    Publishing date 2019-03-09
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  8. Book ; Online: A New Approach to Adaptive Data Analysis and Learning via Maximal Leakage

    Esposito, Amedeo Roberto / Gastpar, Michael / Issa, Ibrahim

    2019  

    Abstract: There is an increasing concern that most current published research findings are false. The main cause seems to lie in the fundamental disconnection between theory and practice in data analysis. While the former typically relies on statistical ... ...

    Abstract There is an increasing concern that most current published research findings are false. The main cause seems to lie in the fundamental disconnection between theory and practice in data analysis. While the former typically relies on statistical independence, the latter is an inherently adaptive process: new hypotheses are formulated based on the outcomes of previous analyses. A recent line of work tries to mitigate these issues by enforcing constraints, such as differential privacy, that compose adaptively while degrading gracefully and thus provide statistical guarantees even in adaptive contexts. Our contribution consists in the introduction of a new approach, based on the concept of Maximal Leakage, an information-theoretic measure of leakage of information. The main result allows us to compare the probability of an event happening when adaptivity is considered with respect to the non-adaptive scenario. The bound we derive represents a generalization of the bounds used in non-adaptive scenarios (e.g., McDiarmid's inequality for $c$-sensitive functions, false discovery error control via significance level, etc.), and allows us to replicate or even improve, in certain regimes, the results obtained using Max-Information or Differential Privacy. In contrast with the line of work started by Dwork et al., our results do not rely on Differential Privacy but are, in principle, applicable to every algorithm that has a bounded leakage, including the differentially private algorithms and the ones with a short description length.
    Keywords Statistics - Machine Learning ; Computer Science - Information Theory ; Computer Science - Machine Learning
    Subject code 005
    Publishing date 2019-03-05
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  9. Book ; Online: Generalization Error Bounds Via R\'enyi-, $f$-Divergences and Maximal Leakage

    Esposito, Amedeo Roberto / Gastpar, Michael / Issa, Ibrahim

    2019  

    Abstract: In this work, the probability of an event under some joint distribution is bounded by measuring it with the product of the marginals instead (which is typically easier to analyze) together with a measure of the dependence between the two random variables. ...

    Abstract In this work, the probability of an event under some joint distribution is bounded by measuring it with the product of the marginals instead (which is typically easier to analyze) together with a measure of the dependence between the two random variables. These results find applications in adaptive data analysis, where multiple dependencies are introduced and in learning theory, where they can be employed to bound the generalization error of a learning algorithm. Bounds are given in terms of $\alpha-$Divergence, Sibson's Mutual Information and $f-$Divergence. A case of particular interest is the Maximal Leakage (or Sibson's Mutual Information of order infinity) since this measure is robust to post-processing and composes adaptively. This bound can also be seen as a generalization of classical bounds, such as Hoeffding's and McDiarmid's inequalities, to the case of dependent random variables.

    Comment: arXiv admin note: text overlap with arXiv:1903.01777
    Keywords Computer Science - Information Theory ; Computer Science - Machine Learning ; Mathematics - Probability
    Subject code 519
    Publishing date 2019-12-01
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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