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    <title>Pinboard (mraginsky)</title>
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    <description>recent bookmarks from mraginsky</description>
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	<rdf:li rdf:resource="https://arxiv.org/abs/2303.07279"/>
	<rdf:li rdf:resource="https://projecteuclid.org/journals/annals-of-probability/volume-34/issue-5/A-theorem-on-majorizing-measures/10.1214/009117906000000241.full"/>
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	<rdf:li rdf:resource="https://arxiv.org/abs/2108.08198"/>
	<rdf:li rdf:resource="http://www.maths.manchester.ac.uk/~goran/lectures.pdf"/>
	<rdf:li rdf:resource="http://arxiv.org/abs/1201.2256"/>
	<rdf:li rdf:resource="http://arxiv.org/abs/1111.3486"/>
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  </channel><item rdf:about="https://arxiv.org/abs/2405.18055">
    <title>[2405.18055] Dimension-free uniform concentration bound for logistic regression</title>
    <dc:date>2024-06-01T18:52:37+00:00</dc:date>
    <link>https://arxiv.org/abs/2405.18055</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA[We provide a novel dimension-free uniform concentration bound for the empirical risk function of constrained logistic regression. Our bound yields a milder sufficient condition for a uniform law of large numbers than conditions derived by the Rademacher complexity argument and McDiarmid's inequality. The derivation is based on the PAC-Bayes approach with second-order expansion and Rademacher-complexity-based bounds for the residual term of the expansion. ]]></description>
<dc:subject>papers empirical-processes measure-concentration probability stochastic-analysis have-read</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:e60734333874/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:empirical-processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:measure-concentration"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:stochastic-analysis"/>
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<item rdf:about="https://arxiv.org/abs/2303.07279">
    <title>[2303.07279] Universal coding, intrinsic volumes, and metric complexity</title>
    <dc:date>2023-05-05T02:48:23+00:00</dc:date>
    <link>https://arxiv.org/abs/2303.07279</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA[We study sequential probability assignment in the Gaussian setting, where the goal is to predict, or equivalently compress, a sequence of real-valued observations almost as well as the best Gaussian distribution with mean constrained to a given subset of $\mathbf{R}^n$. First, in the case of a convex constraint set $K$, we express the hardness of the prediction problem (the minimax regret) in terms of the intrinsic volumes of $K$; specifically, it equals the logarithm of the Wills functional from convex geometry. We then establish a comparison inequality for the Wills functional in the general nonconvex case, which underlines the metric nature of this quantity and generalizes the Slepian-Sudakov-Fernique comparison principle for the Gaussian width. Motivated by this inequality, we characterize the exact order of magnitude of the considered functional for a general nonconvex set, in terms of global covering numbers and local Gaussian widths. This implies metric isomorphic estimates for the log-Laplace transform of the intrinsic volume sequence of a convex body. As part of our analysis, we also characterize the minimax redundancy for a general constraint set. We finally relate and contrast our findings with classical asymptotic results in information theory. ]]></description>
<dc:subject>papers to-read empirical-processes probability information-theory source-coding</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:9d5d5e5b6290/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:information-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:source-coding"/>
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<item rdf:about="https://projecteuclid.org/journals/annals-of-probability/volume-34/issue-5/A-theorem-on-majorizing-measures/10.1214/009117906000000241.full">
    <title>A theorem on majorizing measures</title>
    <dc:date>2022-09-05T21:15:40+00:00</dc:date>
    <link>https://projecteuclid.org/journals/annals-of-probability/volume-34/issue-5/A-theorem-on-majorizing-measures/10.1214/009117906000000241.full</link>
    <dc:creator>mraginsky</dc:creator><dc:subject>papers to-read probability empirical-processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:3eb1020056df/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:papers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:to-read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:empirical-processes"/>
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<item rdf:about="https://arxiv.org/abs/2012.13306">
    <title>[2012.13306] Majorizing Measures for the Optimizer</title>
    <dc:date>2022-08-02T16:36:31+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.13306</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA[The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes.
One of the crowning achievements of the theory is Talagrand's tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the generic chaining) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum.
In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective.]]></description>
<dc:subject>papers have-read empirical-processes probability algorithms game-theory optimization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:c6e9efe1e3f1/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:game-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:optimization"/>
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<item rdf:about="https://arxiv.org/abs/2108.08198">
    <title>[2108.08198] Dimension-free Bounds for Sums of Independent Matrices and Simple Tensors via the Variational Principle</title>
    <dc:date>2022-03-28T01:45:12+00:00</dc:date>
    <link>https://arxiv.org/abs/2108.08198</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA[We consider the deviation inequalities for the sums of independent d by d random matrices, as well as rank one random tensors. Our focus is on the non-isotropic case and the bounds that do not depend explicitly on the dimension d, but rather on the effective rank. In an elementary and unified manner, we show the following results:
1) A deviation bound for the sums of independent positive-semi-definite matrices of any rank. This result generalizes the dimension-free bound of Koltchinskii and Lounici [Bernoulli, 23(1): 110-133, 2017] on the sample covariance matrix in the sub-Gaussian case. 2) A dimension-free version of the bound of Adamczak, Litvak, Pajor and Tomczak-Jaegermann [Journal Of Amer. Math. Soc,. 23(2), 535-561, 2010] on the sample covariance matrix in the log-concave case. 3) Dimension-free bounds for the operator norm of the sums of random tensors of rank one formed either by sub-Gaussian or by log-concave random vectors. This complements the result of Guédon and Rudelson [Adv. in Math., 208: 798-823, 2007]. 4) A non-isotropic version of the result of Alesker [Geom. Asp. of Funct. Anal., 77: 1-4, 1995] on the deviation of the norm of sub-exponential random vectors. 5) A dimension-free lower tail bound for sums of positive semi-definite matrices with heavy-tailed entries, sharpening the bound of Oliveira [Prob. Th. and Rel. Fields, 166: 1175-1194, 2016].
Our approach is based on the duality formula between entropy and moment generating functions. In contrast to the known proofs of dimension-free bounds, we avoid Talagrand's majorizing measure theorem, as well as generic chaining bounds for empirical processes. Some of our tools were pioneered by O. Catoni and co-authors in the context of robust statistical estimation. ]]></description>
<dc:subject>papers to-read probability random-matrices empirical-processes information-theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:654b5de716ce/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:random-matrices"/>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:information-theory"/>
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<item rdf:about="http://www.maths.manchester.ac.uk/~goran/lectures.pdf">
    <title>From Uniform Laws of Large Numbers to Uniform Ergodic Theorems</title>
    <dc:date>2015-08-27T02:27:53+00:00</dc:date>
    <link>http://www.maths.manchester.ac.uk/~goran/lectures.pdf</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA["The purpose of these lectures is to present three different approaches with their own methods for establishing uniform laws of large numbers and uni- form ergodic theorems for dynamical systems. The presentation follows the principle according to which the i.i.d. case is considered first in great de- tail, and then attempts are made to extend these results to the case of more general dependence structures. The lectures begin (Chapter 1) with a re- view and description of classic laws of large numbers and ergodic theorems, their connection and interplay, and their infinite dimensional extensions to- wards uniform theorems with applications to dynamical systems. The first approach (Chapter 2) is of metric entropy with bracketing which relies upon the Blum-DeHardt law of large numbers and Hoffmann-Jørgensen’s exten- sion of it. The result extends to general dynamical systems using the uniform ergodic lemma (or Kingman’s subadditive ergodic theorem). In this context metric entropy and majorizing measure type conditions are also considered. The second approach (Chapter 3) is of Vapnik and Chervonenkis. It relies upon Rademacher randomization (subgaussian inequality) and Gaussian ran- domization (Sudakov’s minoration) and offers conditions in terms of random entropy numbers. Absolutely regular dynamical systems are shown to sup- port the VC theory using a blocking technique and Eberlein’s lemma. The third approach (Chapter 4) is aimed to cover the wide sense stationary case which is not accessible by the previous two methods. This approach relies upon the spectral representation theorem and offers conditions in terms of the orthogonal stochastic measures which are associated with the underlying dynamical system by means of this theorem. The case of bounded variation is covered, while the case of unbounded variation is left as an open question. The lectures finish with a supplement in which the role of uniform conver- gence of reversed martingales towards consistency of statistical models is explained via the concept of Hardy’s regular convergence."]]></description>
<dc:subject>lecture-notes to-read probability ergodic-theory empirical-processes via:cshalizi</dc:subject>
<dc:identifier>https://pinboard.in/u:mraginsky/b:8e70b5d2b67f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:lecture-notes"/>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:ergodic-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:empirical-processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:via:cshalizi"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1201.2256">
    <title>[1201.2256] Empirical Processes of Markov Chains and Dynamical Systems Indexed by Classes of Functions</title>
    <dc:date>2012-01-18T19:05:54+00:00</dc:date>
    <link>http://arxiv.org/abs/1201.2256</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA[We study weak convergence of empirical processes of dependent data, indexed by classes of functions. We obtain results that are especially suitable for data arising from dynamical systems and Markov chains, where the Central Limit Theorem for partial sums is commonly derived via the spectral gap technique. Our results apply, e.g. to the empirical process of ergodic torus automorphisms.]]></description>
<dc:subject>papers to-read empirical-processes dynamical-systems markov-chains re:adaptive_control_project</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:ca7cadb183f0/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:empirical-processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:markov-chains"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:re:adaptive_control_project"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1111.3486">
    <title>[1111.3486] New Concentration Inequalities for Suprema of Empirical Processes</title>
    <dc:date>2011-11-16T16:59:20+00:00</dc:date>
    <link>http://arxiv.org/abs/1111.3486</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA[While effective concentration inequalities for suprema of empirical processes exist under boundedness or strict tail assumptions, no comparable results have been available under considerably weaker assumptions. In this paper, we derive concentration inequalities assuming only low moments for an envelope of the empirical process. These concentration inequalities are beneficial even when the envelope is much larger than the single functions under consideration.]]></description>
<dc:subject>papers to-read probability empirical-processes measure-concentration</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:mraginsky/b:464f5eb9e4d3/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:empirical-processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:measure-concentration"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1103.3188">
    <title>[1103.3188] &quot;Exact&quot; deviations in Wasserstein distance for empirical and occupation measures</title>
    <dc:date>2011-03-17T02:52:28+00:00</dc:date>
    <link>http://arxiv.org/abs/1103.3188</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA["We study the problem of so-called "exact" or non-asymptotic deviations between a reference measure $\mu$ and its empirical version $L_n$, in the $p$-Wasserstein metric, $1 \leq p \leq 2$, under the standing assumption that $\mu$ satisfies a transport-entropy inequality. This work is a generalization of an article by F.Bolley, A.Guillin and C.Villani, where the case of measures with support in $\R^d$ was studied. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a contracting Markov chain in $W_1$ distance are also given. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processes."
]]></description>
<dc:subject>papers to-read probability measure-concentration empirical-processes</dc:subject>
<dc:identifier>https://pinboard.in/u:mraginsky/b:35f32ce279f8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:papers"/>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:measure-concentration"/>
	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:empirical-processes"/>
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<item rdf:about="http://arxiv.org/abs/1008.2697">
    <title>[1008.2697] A CLT for Empirical Processes Involving Time Dependent Data</title>
    <dc:date>2010-08-17T15:37:47+00:00</dc:date>
    <link>http://arxiv.org/abs/1008.2697</link>
    <dc:creator>mraginsky</dc:creator><description><![CDATA["For stochastic processes $\{X_t: t \in E\}$, we establish sufficient conditions for the empirical process based on $\{ I_{X_t \le y} - P(X_t \le y): t \in E, y \in \mathbb{R}\}$ to satisfy the CLT uniformly in $ t \in E, y \in \mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E= [0,1]$."
]]></description>
<dc:subject>papers to-read probability empirical-processes dependent-data</dc:subject>
<dc:identifier>https://pinboard.in/u:mraginsky/b:5bf9f9e1f5aa/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:mraginsky/t:dependent-data"/>
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