Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed distributions . Paper #2156 Prashanth L A ∗ , Krishna Jagannathan ∗ , Ravi Kolla † ∗ IIT Madras, Chennai, India † AB InBev
cn 2 One-Slide Summary tail . . tail bound for a truncated CVaR estimator. O exp 3) Heavy-tailed distributions with bounded variance: We derive an . . bound for an empirical CVaR estimator 2 cn min 2) Light-tailed distributions: We derive an O exp . but with better constants 1) Sub-Gaussian distributions: Our bounds match an existing result, . . r.v. X from n i.i.d. samples 4) Bandit application: Best CVaR arm identification and error bounds Objective: Estimate the Conditional Value-at-Risk (CVaR) c α ( X ) of a Our Contributions: Concentration bounds P [ | c n ,α − c α ( X ) | > ϵ ]
cn 2 One-Slide Summary tail . . tail bound for a truncated CVaR estimator. O exp 3) Heavy-tailed distributions with bounded variance: We derive an . . bound for an empirical CVaR estimator 2 cn min 2) Light-tailed distributions: We derive an O exp . . but with better constants 1) Sub-Gaussian distributions: Our bounds match an existing result, . . r.v. X from n i.i.d. samples 4) Bandit application: Best CVaR arm identification and error bounds Objective: Estimate the Conditional Value-at-Risk (CVaR) c α ( X ) of a Our Contributions: Concentration bounds P [ | c n ,α − c α ( X ) | > ϵ ]
cn 2 One-Slide Summary bound for an empirical CVaR estimator . . tail bound for a truncated CVaR estimator. O exp 3) Heavy-tailed distributions with bounded variance: We derive an . . 4) Bandit application: Best CVaR arm identification and error bounds . . but with better constants 1) Sub-Gaussian distributions: Our bounds match an existing result, . . r.v. X from n i.i.d. samples Objective: Estimate the Conditional Value-at-Risk (CVaR) c α ( X ) of a Our Contributions: Concentration bounds P [ | c n ,α − c α ( X ) | > ϵ ] 2) Light-tailed distributions: We derive an O ( exp ( − cn min ( ϵ, ϵ 2 ))) tail
One-Slide Summary bound for an empirical CVaR estimator . . 3) Heavy-tailed distributions with bounded variance: We derive an . . . . but with better constants 1) Sub-Gaussian distributions: Our bounds match an existing result, . . r.v. X from n i.i.d. samples 4) Bandit application: Best CVaR arm identification and error bounds Objective: Estimate the Conditional Value-at-Risk (CVaR) c α ( X ) of a Our Contributions: Concentration bounds P [ | c n ,α − c α ( X ) | > ϵ ] 2) Light-tailed distributions: We derive an O ( exp ( − cn min ( ϵ, ϵ 2 ))) tail O ( exp ( − cn ϵ 2 )) tail bound for a truncated CVaR estimator.
One-Slide Summary bound for an empirical CVaR estimator . . 3) Heavy-tailed distributions with bounded variance: We derive an . . . . but with better constants 1) Sub-Gaussian distributions: Our bounds match an existing result, . . r.v. X from n i.i.d. samples 4) Bandit application: Best CVaR arm identification and error bounds Objective: Estimate the Conditional Value-at-Risk (CVaR) c α ( X ) of a Our Contributions: Concentration bounds P [ | c n ,α − c α ( X ) | > ϵ ] 2) Light-tailed distributions: We derive an O ( exp ( − cn min ( ϵ, ϵ 2 ))) tail O ( exp ( − cn ϵ 2 )) tail bound for a truncated CVaR estimator.
What is Conditional Value-at-Risk (CVaR)?
VaR and CVaR are Risk Metrics c v X 1 1 X v . . X v X X X . • Widely used in financial portfolio optimization, credit risk . Conditional Value at Risk: X 1 F X v . . Value at Risk: • Let X be a continuous random variable assessment and insurance X • Fix a `risk level' α ∈ ( 0 , 1 ) (say α = 0 . 95)
VaR and CVaR are Risk Metrics X v X 1 1 X v . . X v X X c • Widely used in financial portfolio optimization, credit risk . . Conditional Value at Risk: . . Value at Risk: • Let X be a continuous random variable assessment and insurance X • Fix a `risk level' α ∈ ( 0 , 1 ) (say α = 0 . 95) v α ( X ) = F − 1 X ( α )
VaR and CVaR are Risk Metrics . assessment and insurance • Let X be a continuous random variable Value at Risk: . . • Widely used in financial portfolio optimization, credit risk Conditional Value at Risk: . 1 . • Fix a `risk level' α ∈ ( 0 , 1 ) (say α = 0 . 95) v α ( X ) = F − 1 X ( α ) c α ( X ) = E [ X | X > v α ( X )] 1 − α E [ X − v α ( X )] + = v α ( X ) +
CVaR Estimation and Concentration bounds
CVaR estimation X , estimate . . . . Nice to have : Sample complexity O Problem: Given i.i.d. samples X 1 , . . . , X n from the distribution F of r.v. c α ( X ) = E [ X | X > v α ( X )] ( 1 /ϵ 2 ) for accuracy ϵ
1 X i Empirical VaR and CVaR Estimates . i n n 1 1 v n c n . . CVaR estimate: v n . VaR estimate: n . . distribution F , Empirical distribution function (EDF): Given samples X 1 , . . . , X n from ˆ F n ( x ) = 1 ∑ n i = 1 I { X i ≤ x } , x ∈ R Using EDF and the order statistics X [ 1 ] ≤ X [ 2 ] ≤ . . . , X [ n ] , v n ,α = inf { x : ˆ ˆ F n ( x ) ≥ α } = X [ ⌈ n α ⌉ ] .
Empirical VaR and CVaR Estimates . CVaR estimate: 1 . . VaR estimate: . n . . distribution F , Empirical distribution function (EDF): Given samples X 1 , . . . , X n from ˆ F n ( x ) = 1 ∑ n i = 1 I { X i ≤ x } , x ∈ R Using EDF and the order statistics X [ 1 ] ≤ X [ 2 ] ≤ . . . , X [ n ] , v n ,α = inf { x : ˆ ˆ F n ( x ) ≥ α } = X [ ⌈ n α ⌉ ] . v n ,α ) + ˆ c n ,α = ˆ i = 1 ( X i − ˆ v n ,α + ∑ n n ( 1 − α )
Empirical CVaR concentration: What is known ? [Kolla et al. ORL 2019] This work Sub-exponential/ Wasserstein [S. Bhat & P. L.A. NeurIPS 2019] Sub-Gaussian One-sided CVaR sub-exponential VaR conc. Sub-Gaussian/ Goal: Bound [Wang et al. ORL 2010] Bounded support Salient Feature Reference Distribution type . Heavy-tailed P [ | ˆ c n ,α − c α ( X ) | > ϵ ] exp ( − cn ϵ 2 )
2 n 2 2 Proof uses DKW inequality; no tail assumptions required. Assumption (A1): X is a continuous r.v. with a CDF F that satisfies a Lemma ( VaR concentration ) Suppose that (A1) holds. We have for all 0 v n v 2 exp 1 Concentration bounds for empirical conditional value-at-risk: The unbounded case; R. Kolla, L.A. Prashanth, S. P. Bhat, K. Jagannathan; Operations Research Letters, 2019 VaR Concentration 1 condition of sufficient growth around the VaR v α : There exists constants δ, η > 0 such that min ( F ( v α + δ ) − F ( v α ) , F ( v α ) − F ( v α − δ )) ≥ ηδ.
Assumption (A1): X is a continuous r.v. with a CDF F that satisfies a Lemma ( VaR concentration ) 1 Concentration bounds for empirical conditional value-at-risk: The unbounded case; R. Kolla, L.A. Prashanth, S. P. Bhat, K. Jagannathan; Operations Research Letters, 2019 VaR Concentration 1 condition of sufficient growth around the VaR v α : There exists constants δ, η > 0 such that min ( F ( v α + δ ) − F ( v α ) , F ( v α ) − F ( v α − δ )) ≥ ηδ. Suppose that (A1) holds. We have for all ϵ ∈ ( 0 , δ ) , P [ | ˆ ( − 2 n η 2 ϵ 2 ) v n ,α − v α | ≥ ϵ ] ≤ 2 exp . Proof uses DKW inequality; no tail assumptions required.
(iii) we use a truncated CVaR estimator • Need to make some assumptions on the tail distribution • We work with three progressive broader distribution classes . . (i) X is sub-Gaussian or . (ii) X is sub-exponential (i.e., light-tailed) or . (iii) X has a bounded second moment • For (i) and (ii), we use the empirical CVaR estimator; for Concentration for CVaR α Estimator • Obtaining concentration for CVaR α estimator is more involved than for VaR α
(iii) we use a truncated CVaR estimator . • For (i) and (ii), we use the empirical CVaR estimator; for (iii) X has a bounded second moment . . (ii) X is sub-exponential (i.e., light-tailed) or . (i) X is sub-Gaussian or . . classes • We work with three progressive broader distribution • Need to make some assumptions on the tail distribution Concentration for CVaR α Estimator • Obtaining concentration for CVaR α estimator is more involved than for VaR α
(iii) we use a truncated CVaR estimator . • For (i) and (ii), we use the empirical CVaR estimator; for (iii) X has a bounded second moment . . (ii) X is sub-exponential (i.e., light-tailed) or . (i) X is sub-Gaussian or . . classes • We work with three progressive broader distribution • Need to make some assumptions on the tail distribution Concentration for CVaR α Estimator • Obtaining concentration for CVaR α estimator is more involved than for VaR α
e X e X c 1 exp 2 b 0 s.t. . . e 2 Sub-Gaussian and Sub-Exponential distributions 2 Or equivalently, b Or . . X c 2 0 1 c 0 .Tail dominated by an exponential r.v . . . 0 s.t. c 0 A random variable is X is sub-exponential if .Tail dominated by a Gaussian . . . . A random variable is X is sub-Gaussian if ∃ σ > 0 s.t. [ e λ X ] σ 2 λ 2 2 , ∀ λ ∈ R . ≤ e E Or equivalently, letting Z ∼ N ( 0 , σ 2 ) , P [ X > ϵ ] ≤ c P [ Z > ϵ ] , ∀ ϵ > 0 .
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