Robust Bounds on Risk-Sensitive Functionals via Rényi Divergence

We extend the duality between exponential integrals and relative entropy to a variational formula for exponential integrals involving the Renyi divergence. This formula characterizes the dependence of risk-sensitive functionals and related quantities determined by tail behavior to perturbations in the underlying distributions, in terms of the Renyi divergence. The characterization gives rise to upper and lower bounds that are meaningful for all values of a large deviation scaling parameter, allowing one to quantify in explicit terms the robustness of risk-sensitive costs. As applications we consider problems of uncertainty quantification when aspects of the model are not fully known, as well their use in bounding tail properties of an intractable model in terms of a tractable one.

[1]  I. Vajda,et al.  Convex Statistical Distances , 2018, Statistical Inference for Engineers and Data Scientists.

[2]  Jing Li,et al.  Computation of Failure Probability Subject to Epistemic Uncertainty , 2012, SIAM J. Sci. Comput..

[3]  Djalil CHAFAÏ.  Entropies, convexity, and functional inequalities , 2002 .

[4]  Alan Weiss,et al.  Large Deviations For Performance Analysis: Queues, Communication and Computing , 1995 .

[5]  I. Vajda.  Distances and discrimination rates for stochastic processes , 1990 .

[6]  Djalil CHAFAÏ.  Entropies, convexity, and functional inequalities : On Phi-entropies and Phi-Sobolev inequalities , 2004 .

[7]  P. Glynn,et al.  Efficient rare-event simulation for the maximum of heavy-tailed random walks , 2008, 0808.2731.

[8]  Paul Dupuis,et al.  Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling , 2005, Math. Oper. Res..

[9]  P. Dupuis,et al.  Large deviations for Markov processes with discontinuous statistics, II: random walks , 1992 .

[10]  Paul Dupuis,et al.  Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification , 2011, 1103.1861.

[11]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[12]  H. Kushner.  Robustness and Approximation of Escape Times and Large Deviations Estimates for Systems with Small Noise Effects , 1984 .

[13]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[14]  Martin J. Wainwright,et al.  Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization , 2008, IEEE Transactions on Information Theory.

[15]  Ian R. Petersen,et al.  Robust Properties of Risk-Sensitive Control , 2000, Math. Control. Signals Syst..

[16]  Peter Harremoës,et al.  Rényi divergence and majorization , 2010, 2010 IEEE International Symposium on Information Theory.

[17]  Venkat Anantharam,et al.  How large delays build up in a GI/G/1 queue , 1989, Queueing Syst. Theory Appl..

[18]  B. Hajek,et al.  On large deviations of Markov processes with discontinuous statistics , 1998 .

[19]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[20]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[21]  E. Todorov,et al.  A UNIFIED THEORY OF LINEARLY SOLVABLE OPTIMAL CONTROL , 2012 .

[22]  Emanuel Todorov,et al.  A Unifying Framework for Linearly Solvable Control , 2011, UAI.

[23]  P. Whittle.  Risk-sensitive linear/quadratic/gaussian control , 1981, Advances in Applied Probability.

[24]  Ward Whitt,et al.  Functional large deviation principles for first-passage-time processes , 1997 .

[25]  G. Crooks.  On Measures of Entropy and Information , 2015 .

[26]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[27]  Mark D. Reid,et al.  Tighter Variational Representations of f-Divergences via Restriction to Probability Measures , 2012, ICML.

[28]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[29]  Gholamhossein Yari,et al.  Some properties of Rényi entropy and Rényi entropy rate , 2009, Inf. Sci..