Estimation of Spectral Risk Measures

1 Introduction

In the context of risk-sensitive optimization, Conditional value-at-risk (CVaR

) is a popular measure. CVaR is a conditional expectation of a random variable (r.v.) that usually models the losses in an application (e.g. finance), where the conditioning is based on value-at-risk

(VaR)

. The latter denotes the maximum loss that could be incurred, with high probability. The advantage of employing

CVaR instead of VaR in a risk-sensitive optimization setting is that CVaR is a coherent risk measure [3], while VaR is not, as it violates the sub-additivity assumption.

Spectral risk measures $(SRM)$ are a generalization of CVaR, and are defined as follows:

S (X) = \int_{0}^{1} φ (β) V_{β} (X) d β .

(1)

In the equation above, $φ (\cdot)$ is a risk-aversion function, which can be chosen to ensure that SRM is a coherent risk measure [1] and $V_{β}$ is the $β$

-quantile of the distribution of the r.v.

X

. In particular, a non-negative, increasing

φ

that integrates to

1

is sufficient for ensuring coherence. SRM can be seen as a weighted average of the quantiles (VaR) of the underlying distribution. Moreover, CVaR can be recovered by setting

φ(β)=11−α\indicβ>α,α∈(0,1)

. The latter choice translates to an equal weight for all tail-loss VaR values. In contrast, SRM can model a user’s risk aversion better, since the function

φ

can be chosen such that higher losses receive a higher weight, or at least, the same weight as lower losses [9].

In this paper, we consider the problem of estimating SRM of a random variable (r.v.), given independent and identically distributed (i.i.d.) samples from the underlying distribution. In this context, our contributions are as follows: First, we provide a natural estimation scheme for SRM that uses the empirical distribution function (EDF) to estimate quantiles, together with a trapezoidal rule-based approximation to the integral in (1). Second, we provide concentration bounds for our proposed SRM estimate for the following two cases: first, when the underlying distribution is assumed to have bounded support; and second, when the distribution is either Gaussian or exponential. To the best of our knowledge, no concentration bounds are available for SRM estimation. Third, we perform simulation experiments to show the efficacy of our proposed SRM estimation scheme. In particular, we consider a synthetic setup, and show that our scheme provides accurate estimates of SRM. Next, we incorporate our SRM estimation scheme in the inner loop of the successive rejects (SR) algorithm [4], which is a popular algorithm in the best arm identification framework for multi-armed bandits. We test the resulting SR algorithm variant in a vehicular traffic routing application using SUMO traffic simulator [5]. The application is motivated by the fact that, in practice, human road users may not always prefer the route with lowest mean delay. Instead, a route that minimized worst-case delay, while doing reasonably well on the average, is preferable, and such a preference can be encoded into the risk aversion function $φ (\cdot)$ in (1).

To the best of our knowledge, concentration bounds are not available for the SRM estimation. However, the bounds that we derive for SRM estimation could be specialized to the case of CVaR. In [7, 19] concentration bounds for the classic CVaR estimator are derived. Our bound, using a different estimator, exhibits a similar rate of exponential convergence around true CVaR. For the case of distributions with unbounded support, concentration bounds for empirical CVaR have been derived recently in [13, 14, 6]. In [13] (resp. [14, 6]), the authors derive an one-sided concentration bound (resp. two-sided bounds), when the underlying distributions are either sub-Gaussian or sub-exponential [18]. In comparison to [13]

, we derive two-sided concentration bounds for the special case of Gaussian and exponential distributions. Moreover, the bounds that we derive show an improved dependence on the number of samples, say

n

, and accuracy, say

ϵ

, when compared to the corresponding bounds in [14]. More precisely, the probability that the CVaR estimate is more than an

ϵ

away from the true CVaR is bounded above by

c_{1} exp (- c_{2} n ϵ^{2})

, for large enough

n

and some universal constants

c_{1}, c_{2}

, in our bound. On the other hand, the corresponding tail bound in [14] is

c_{1} exp (- c_{2} \sqrt{n} min (ϵ, ϵ^{2}))

for the sub-Gaussian case, and

c_{1} exp (- c_{2} n^{1 / 3} min (ϵ^{2 / 3}, ϵ^{2}))

for the sub-exponential case. Finally, in comparison to a recent result in [6], our bound exhibits exponential concentration, while the corresponding bound in [6] shows a polynomial decay for

ϵ > 1

. CVaR-based models have been explored in different contexts, for instance, in a bandit application [11], in a portfolio optimization problem [15], and in a general risk management setting [16]. In the simulation experiments, we consider each of these applications, and show the efficacy of our proposed estimation scheme in each application context.

The rest of the paper is organized as follows: Section 2 introduces VaR, CVaR, SRM and their estimators from i.i.d. samples, Section 3 presents our estimate of SRM, together with concentration bounds for the case when the underlying distribution has bounded support. Section 4 presents a truncated SRM estimation scheme, and concentration bounds for the case when the underlying distribution is either Gaussian or exponential. Section 5 presents the simulation experiments, Section 6 provides the proofs of the concentration bounds in Sections 3–4, and finally, Section 7 concludes the paper.

2 Preliminaries

For a r.v. $X$ , VaR $V_{β} (X)$ and CVaR $C_{β} (X)$ at the level $β$ , $β \in (0, 1)$ , are defined as follows:

(2)

where $[x]^{+} = max (0, x)$ for a real number $x$ . $V_{β} (X)$ can be interpreted as the minimum loss that will not be exceeded with probability $β$ . Note that, if $X$

has a continuous and strictly increasing cumulative distribution function (CDF)

F

, then

V_{β} (X)

is a solution to the following:

P [X \leq ξ] = β, i.e., V_{β} (X) = F^{- 1} (β) .

Further, $C_{β} (X)$ can be interpreted as the expected loss, conditional on the event that the loss exceeds $V_{β} (X)$ , i.e., $C_{β} (X) = E [X | X \geq V_{β} (X)] .$

Let $X_{i}$ , $i = 1, \dots, n$ denote i.i.d. samples from the distribution of $X$ . Then, the estimate of $V_{β} (X)$ , denoted by ${ˆ V}_{n, β}$ , is formed as follows [17]:

{ˆ V}_{n, β}

= {ˆ F}_{n}^{- 1} (β) = inf {x : {ˆ F}_{n} (x) \geq β},

(3)

where ${ˆ F}_{n} (x) = \frac{1}{n} \sum_{i = 1}^{n} I [X_{i} \leq x]$ is the EDF of $X$ . Letting $X_{(1)}, \dots, X_{(n)}$ denote the order statistics, i.e., $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$ , we have ${ˆ V}_{n, β} = X_{(⌈ n β ⌉)}$ .

3 Distributions with bounded support

3.1 Estimation scheme

We estimate $S (X)$ , given i.i.d. samples $X_{1}, \dots, X_{n}$ from the distribution of $X$ , by approximating the integral in SRM definition (1). Notice that the integrand $V_{β} (X)$ in (1) has to be estimated using the samples. Recall that ${ˆ V}_{n, β}$ is the estimate of $V_{β} (X)$ , given by (3). We use the weighted VaR estimate to form a discrete sum to approximate the integral, an idea motivated by the trapezoidal rule [8]. The estimate ${ˆ S}_{n, m}$ of $S (X)$ is formed as follows:

{ˆ S}_{n, m} = m \sum k = 1 \frac{φ (β_{k - 1}) {ˆ V}_{n, β_{k - 1}} + φ (β_{k}) {ˆ V}_{n, β_{k}}}{2} Δ β .

(4)

In the equation above, ${β_{k}}_{k = 0}^{m}$ is a partition of $[0, 1]$ such that $β_{0} = 0$ and $β_{k} = β_{k - 1} + Δ β$ , where $Δ β = 1 / m$ is the length of each sub-interval.

In the next section, we present concentration bounds for the estimator presented above, assuming that the underlying distribution has bounded support.

3.2 Concentration bounds

For notational convenience, we shall use $V_{β}$ and $S$ to denote $V_{β} (X)$ and $S (X)$ , for any $β \in (0, 1)$ .

For all the results presented below, we let ${ˆ S}_{n, m}$ denote the SRM estimate formed from $n$ i.i.d. samples of $X$ and with m sub-intervals, using (4). Let $F$ and $f$ denote the distribution and density of $X$ , respectively.

For the sake of analysis, we make one of the following assumptions:
(A1) Let $φ (β)$ be a risk-aversion function such that $∣ φ (β) ∣\leq C_{1}$ and $∣ φ^{'} (β) ∣\leq C_{2}, \forall β \in [0, 1]$ .
(A1 $^{'}$ ) The conditions of (A1) hold. In addition, $∣ φ^{''} (β) ∣\leq C_{3}, \forall β \in [0, 1]$ .

[SRM concentration: bounded case] Let the r.v. $X$ be continuous and $X \leq B$ a.s. Fix $ϵ > 0$ .

(i) Assume (A1) holds and $f (x) \geq 1 / δ_{1} > 0$ , $x \leq B$ . If $∣ B C_{2} + δ_{1} C_{1} ∣\leq K_{1}$ , and $m \geq \frac{K_{1}}{2 ϵ}$ , then

P (∣ ∣ S - {ˆ S}_{n, m} ∣ ∣ > ϵ) \leq \frac{2 K_{1}}{ϵ} exp (\frac{- n c ϵ^{2}}{2 C_{1}^{2}}),

(5)

where $c = min {c_{0}, c_{1}, \dots, c_{m}}$ and $c_{k}, k \in {0, \dots, m}$ , is a constant that depends on the value of the density $f$ of the r.v. $X$ in a neighborhood of $V_{β_{k}}$ , with $β_{k}$ as in (4).

(ii) Assume (A1 $^{'}$ ) holds and $∣ f^{^{'}} (x) ∣ / f (x)^{3} \leq δ_{2}$ , $x \leq B$ . If $∣ B C_{3} + 2 δ_{1} C_{2} + δ_{2} C_{1} ∣\leq K_{2}$ , and $m \geq \sqrt{\frac{K_{2}}{6 ϵ}}$ , then

P (∣ ∣ S - {ˆ S}_{n, m} ∣ ∣ > ϵ) \leq \sqrt{\frac{8 K_{2}}{3 ϵ}} exp (\frac{- n c ϵ^{2}}{2 C_{1}^{2}}),

(6)

where $c$ is as in the case above.

Proof.

See Section 6.1. ∎

For small values of $ϵ$ , the bound in (6) is better than that in (5). However, the bound in (5) is derived under weaker assumptions on the r.v. $X$ and the risk-aversion function $φ$ , as compared to the bound in (6).

In part (i) of the theorem above, we assumed that the density $f$ of $X$ is bounded below by $\frac{1}{δ_{1}} > 0$ . This implies that the derivative of VaR is bounded above. The latter condition is required for the trapezoidal rule to provide a good approx to the integral in (1). Moreover, the assumption that the first derivative of VaR w.r.t the confidence level $β$ is bounded implies that the underlying r.v. $X$ is bounded. This claim can be made precise as follows: For any $β \in (0, 1)$ , it can be shown that (see Lemma B in the Appendix for a proof)

V_{β}^{^{'}}

(7)

Notice that the first derivative of VaR involves a $1 / f$ term, and if the random variable is unbounded, then for every $ϵ > 0$ , there is an $x$ such that $0 < f (x) < ϵ$ . This implies $1 / f$ cannot be bounded above uniformly w.r.t $x$ , and hence the derivative of VaR cannot be bounded either.

The stronger condition $∣ f^{^{'}} (x) ∣ / f (x)^{3} \leq δ_{2}$ used in part (ii) of Theorem 3.2, in conjunction with (7), implies that the second derivative of VaR is bounded. Now, as before, a bounded second derivative implies that the underlying r.v. $X$ is bounded. To see this, the expression for the second derivative of VaR involves a $f^{'} / f^{3}$ term, and if the r.v. $X$ is unbounded, then a uniform bound on $f^{'} / f^{3}$ would mean that, as $x \to \infty$ , $f$ decays too slowly to integrate to something finite, leading to a contradiction. More precisely, the differential inequality $| f^{'} | / f^{3} < K$ can be “solved” to get $f (x) > C / \sqrt{a + b x}$ for large $x$ and suitable constants $a$ , $b$ , and $C$ . However, the expression on the RHS integrates to infinity, and hence, no density $f$ with unbounded support can have $f^{'} / f^{3}$ bounded.

4 Gaussian and exponential distributions

4.1 Estimation scheme

Let $X_{1}, \dots, X_{n}$ denote i.i.d. samples from the distribution of $X$ . We form a truncated set of samples as follows:

¯Xi=Xi\indicXi≤Bn,

where $B_{n}$

is a truncation threshold that depends on the underlying distribution. For the case of Gaussian distribution with mean zero and variance

σ^{2}

, we set

B_{n} = \sqrt{2 σ^{2} log (n)}

, and for the case of exponential distribution with mean

1 / λ

, we set

B_{n} = \frac{log (n)}{λ}

We form an SRM estimate along the lines of (4), except that the samples used are truncated samples, i.e.,

{˜ S}_{n, m} = m \sum k = 1 \frac{φ (β_{k - 1}) {˜ V}_{n, β_{k - 1}} + φ (β_{k}) {˜ V}_{n, β_{k}}}{2} Δ β .

(8)

In the above, ${˜ V}_{n, β} = {˜ F}_{n}^{- 1} (β)$ , with ${˜ F}_{n} (x) = \frac{1}{n} \sum_{i = 1}^{n} I [{¯ X}_{i} \leq x]$ .

4.2 Concentration bounds

Next, we present concentration bounds for our SRM estimator assuming that the samples are either from a Gaussian distribution with mean zero and variance $σ^{2}$ , or from the exponential distribution with mean $1 / λ$ . Note that the estimation scheme is not provided this information about the underlying distribution. Instead ${˜ S}_{n, m}$ is formed from $n$ i.i.d. samples and with $m$ sub-intervals, using (8).

[SRM concentration: Gaussian case] Assume (A1). Suppose that the r.v. $X$ is Gaussian with mean zero and variance $σ^{2}$ , with $σ \leq σ_{max}$ . Fix $ϵ > 0$ . If $m \geq \frac{1}{5} \sqrt{\frac{σ_{max}}{ϵ}} exp (\frac{n c ϵ^{2}}{4 C_{1}^{2}})$ , then

P [∣ ∣ S - {˜ S}_{n, m} ∣ ∣ > ϵ] \leq \frac{2 σ (\sqrt{2 log (n)} C_{2} + \sqrt{2 π} n C_{1})}{(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})} exp ⎛ ⎜ ⎜ ⎜ ⎝ - \frac{n c {(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})}^{2}}{2 C_{1}^{2}} ⎞ ⎟ ⎟ ⎟ ⎠, \forall ϵ > \frac{2 σ C_{1}}{\sqrt{n}} .

where $c$ is as in Theorem 3.2 (i).

Proof.

See Section 6.2. ∎

[SRM concentration: Exponential case] Assume (A1). Suppose that the r.v. $X$ is exponentially distribution with parameter $λ$ , and $0 < λ_{min} \leq λ$ . Fix $ϵ > 0$ . If $m \geq \frac{1}{8} \sqrt{\frac{1}{λ_{min} ϵ}} exp (\frac{n c ϵ^{2}}{4 C_{1}^{2}})$ , then

P [∣ ∣ S - {˜ S}_{n, m} ∣ ∣ > ϵ] \leq \frac{2 (\frac{l o g (n) C_{2}}{λ} + n C_{1})}{(ϵ - \frac{C_{1} (n + 1)}{λ n})} exp ⎛ ⎜ ⎜ ⎜ ⎝ - \frac{n c {(ϵ - \frac{C_{1} (n + 1)}{λ n})}^{2}}{2 C_{1}^{2}} ⎞ ⎟ ⎟ ⎟ ⎠, \forall ϵ > \frac{C_{1} (n + 1)}{λ n} .

where $c$ is as in Theorem 3.2 (i).

Proof.

See Section 6.3. ∎

Note that concentration bounds for CVaR estimation can be derived using a completely parallel argument to that of the proof of the theorems above, together with following choice for risk aversion function $φ(β)=1/(1−α)\indicβ>α,α∈(0,1)$ . The CVaR-specific results are provided in Appendix D.

5 Simulation experiments

In this section, we demonstrate the efficacy of our proposed method for SRM estimation (4), which we shall refer to as SRM-Trapz. In our experiments, we set the risk aversion function as follows: $φ (β) = \frac{5 e^{- 5 (1 - β)}}{1 - e^{- 5}}, β \in [0, 1]$ . In the following sub-section, we consider a synthetic experimental setting to compare the accuracy of SRM estimators. Subsequently, we use SRM-Trapz as a subroutine in a vehicular traffic routing application (see section 5.2).

5.1 Synthetic setup

Figure 1 presents the estimation error as a function of the sample size for SRM-Trapz. The algorithm is run with two different sub-divisions. The samples are generated using a Gaussian distribution with mean $0.5$ and variance $25$ . We observe that SRM-Trapz with $500$ subdivisions performs on par with SRM-Trapz

with 150 subdivisions for every sample size. Further, as expected, increasing sample size leads to lower estimation error, while also increasing the confidence (demonstrated by the shrinkage in standard error).

Figure 1: Error in SRM estimation ( $∣$ True SRM - Empirical SRM $∣$ ) on different sample size. True SRM is calculated using definition 1. Empirical SRM is calculated by two methods, (i) SRM-Trapz method with $m = 150$ subdivisions (SRM-Trapz $150$ ), and (ii) SRM-Trapz method with $m = 500$ subdivisions (SRM-Trapz $500$ ). In both methods, SRM is estimated using (4). The underlying distribution considered for this simulation is $X \sim N (0.5, 5^{2})$ . The bars in the plot shows standard error averaged over $10^{3}$ iterations.

Table 1 presents the results obtained by SRM-Trapz with $1000$ subdivisions, for four different input distributions. We observe that SRM-Trapz is comparable to SRM-True (calculated using definition 1) under each input distribution.

5.2 Vehicular traffic routing

In the vehicular routing application, the traditional objective is to find a route with the lowest expected delay. However, such an objective ignores risk factors. An alternative is to consider the weighted-sum delay of each route, and we use SRM to quantify this objective. Thus, given a set of routes, the aim is to find the route (by adaptive sampling) with the lowest SRM of the delay. Simulation of Urban MObility (SUMO) [5]

is an open source, highly portable, microscopic road traffic simulation package designed to handle large road networks. Traffic Control Interface (TraCI)

[20] is a library, providing extensive commands to control the behavior of the simulation online, including vehicle state, road configuration, and traffic lights. We implement our routing algorithm using SUMO and TRACI.

Figure 2: Area of an urban city map, used for SUMO network.

For the experiments, we use the street map of the area around IIT Madras, Chennai, India (see Figure 2) obtained from OpenStreetMap (OSM) [12], and then used Netconvert tool to load the map in SUMO. The network has 426 junctions and a total edge length of 123 km. We ran SUMO on this network for $30, 000$ time-steps, in which $7000$ cars, $500$ buses, $2000$ bikes, $1000$ cycles, and $1000$ pedestrians were added at different time-steps and in different lanes uniformly. We choose $K = 5$ routes between two fixed points, marked as S and D in Figure 2. On these selected routes, we add $n = 1000$ cars and track them. In Table 2, ${ˆ X}_{n, i}$ is the estimated average delay of the $i$ th route, and ${ˆ S}_{n, m, i}$ is the SRM estimate for the $i$ th route, $i = 1, \dots, K$ , using (4), and with $n$ samples. We set the number of subdivisions $m = 100$ .

	route $_{1}$	route $_{2}$	route $_{3}$	route $_{4}$	route $_{5}$
${^X}_{n, i}$	283.81	287.15	306.80	266.85	325.86
${ˆ S}_{n, m, i}$	431.28	361.81	455.83	378.68	390.95

Table 2: Results for the estimated average delay (

{^X}_{n, i}

) and estimated SRM (

{ˆ S}_{n, m, i}

), for

i

th route, where

i = 1, \dots, K

From Table 2 it is apparent that ROUTE $_{4}$ has the lowest expected delay, and ROUTE $_{2}$ has the lowest SRM. We consider a best-arm identification (BAI) bandit framework [4], where an algorithm is given a fixed budget. Here, the budget refers to the total number of samples across routes. After the sampling budget, the algorithm is expected to recommend a route, and is judged by the probability that the recommended route is correct (i.e. the best route).

We ran successive rejects (SR), which is a popular BAI algorithm, except that SR is modified to find the route with lowest SRM. Note that the regular SR algorithm finds the route with the lowest expected delay. Algorithm 1 presents the pseudocode for the SRM-SR-Trapz algorithm, with SRM-Trapz used to form SRM estimates for each route. The setting of SUMO is as noted above. We set the budget $n = 1000$ , number of routes $K = 5$ , and $m = 100$ subdivisions for SRM-Trapz. We observed that Algorithm 1 picks ROUTE $_{2}$ with probability $0.91$ .

6 Convergence proofs

6.1 Proof of Theorem 3.2

For establishing the bound in Theorem 3.2, we require a result concerning the error of a trapezoidal-rule-based approximation, and a concentration bound for the $V a R$ estimate in (3). We state these results below, and subsequently provide a proof of Theorem 3.2. Let $0 < a \leq b < 1$ , and ${β_{k}}_{k = 0}^{m}$ be a partition of $[a, b]$ such that $β_{0} = a$ and $β_{k} = β_{k - 1} + Δ β$ , $Δ β = (b - a) / m$ is length of each sub-interval.

(i) If $∣ ∣ {(φ (β) V_{β})}_{β}^{'} ∣ ∣ \leq K_{1}$ for $β \in [a, b]$ , then

∣ ∣ ∣ ∣ \int_{a}^{b} φ (β) V_{β} d β - m \sum k = 1 \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β ∣ ∣ ∣ ∣ \leq \frac{K_{1} (b - a)^{2}}{4 m} .

(9)

(ii) If $∣ ∣ {(φ (β) V_{β})}_{β}^{''} ∣ ∣ \leq K_{2}$ for $β \in [a, b]$ , then

∣ ∣ ∣ ∣ \int_{a}^{b} φ (β) V_{β} d β - m \sum k = 1 \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β ∣ ∣ ∣ ∣ \leq \frac{K_{2} (b - a)^{3}}{12 m^{2}} .

(10)

Proof.

See Appendix A. ∎

[VaR concentration] Let the r.v. X be continuous. Fix $ϵ > 0$ , then we have

P [∣ ∣ V_{β} - {ˆ V}_{n, β} ∣ ∣ \geq ϵ] \leq 2 exp (- 2 n ¯ c ϵ^{2})

where $¯ c$ is a constant that depends on the value of the density $f$ of the r.v. $X$ in a neighborhood of $V_{β}$ .

Proof:.

See Proposition 2 in [13]. ∎

Proof of Theorem 3.2..

First, we prove the claim in part (i). Notice that

	$P [∣ ∣ S - {ˆ S}_{n, m} ∣ ∣ > ϵ]$
	$= P ⎡ ⎣ ∣ ∣ ∣ ∣ \int_{0}^{1} φ (β) V_{β} d β - m \sum k = 1 \frac{φ (β_{k - 1}) {ˆ V}_{n, β_{k - 1}} + φ (β_{k}) {ˆ V}_{n, β_{k}}}{2} Δ β ∣ ∣ ∣ ∣ > ϵ ⎤ ⎦$
	$= P [∣ ∣ ∣ ∣ \int_{0}^{1} φ (β) V_{β} d β - m \sum k = 1 \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β$
	$+ m \sum k = 1 \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β - m \sum k = 1 \frac{φ (β_{k - 1}) {ˆ V}_{n, β_{k - 1}} + φ (β_{k}) {ˆ V}_{n, β_{k}}}{2} Δ β ∣ ∣ ∣ ∣ > ϵ ⎤ ⎦$
	$\leq P ⎡ ⎣ ∣ ∣ ∣ ∣ m \sum k = 1 \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β - m \sum k = 1 \frac{φ (β_{k - 1}) {ˆ V}_{n, β_{k - 1}} + φ (β_{k}) {ˆ V}_{n, β_{k}}}{2} Δ β ∣ ∣ ∣ ∣ > \frac{ϵ}{2} ⎤ ⎦,$

where the final inequality follows by using Lemma 6.1(i) to infer that for $m \geq \frac{K_{1}}{2 ϵ}$ , we have $∣ ∣ ∣ \int_{0}^{1} φ (β) V_{β} d β - \sum_{k = 1}^{m} \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β ∣ ∣ ∣ < \frac{ϵ}{2}$ . Now, we have

	$P [∣ S - {ˆ S}_{n, m} ∣> ϵ]$
	$\leq P ⎡ ⎣ ∣ ∣ ∣ ∣ m \sum k = 1 \frac{φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}}{2} Δ β - m \sum k = 1 \frac{φ (β_{k - 1}) {ˆ V}_{n, β_{k - 1}} + φ (β_{k}) {ˆ V}_{n, β_{k}}}{2} Δ β ∣ ∣ ∣ ∣ > \frac{ϵ}{2} ⎤ ⎦$
	$= P [∣ ∣ ∣ ∣ m \sum k = 1 ((φ (β_{k - 1}) V_{β_{k - 1}} + φ (β_{k}) V_{β_{k}}) - (φ (β_{k - 1}) {ˆ V}_{n, β_{k - 1}} + φ (β_{k}) {ˆ V}_{n, β_{k}})) ∣ ∣ ∣ ∣ > \frac{ϵ}{Δ β}]$
	$= P [∣ ∣ φ (β_{0}) V_{β_{0}} - φ (β_{0}) {ˆ V}_{n, β_{0}} + 2 (φ (β_{1}) V_{β_{1}} - φ (β_{1}) {ˆ V}_{n, β_{1}})$
	$+ \dots + 2 (φ (β_{m - 1}) V_{β_{m - 1}} - φ (β_{m - 1}) {ˆ V}_{n, β_{m - 1}}) + φ (β_{m}) V_{β_{m}} - φ (β_{m}) {ˆ V}_{n, β_{m}} ∣ ∣ > \frac{ϵ}{Δ β}]$
	$\leq P [∣ ∣ φ (β_{0}) V_{β_{0}} - φ (β_{0}) {ˆ V}_{n, β_{0}} ∣ ∣ > \frac{ϵ}{2 m Δ β}] + 2 P [∣ ∣ φ (β_{1}) V_{β_{1}} - φ (β_{1}) {ˆ V}_{n, β_{1}} ∣ ∣ > \frac{ϵ}{2 m Δ β}]$
	$+ \dots + 2 P [∣ ∣ φ (β_{m - 1}) V_{β_{m - 1}} - φ (β_{m - 1}) {ˆ V}_{n, β_{m - 1}} ∣ ∣ > \frac{ϵ}{2 m Δ β}]$
	$+ P [∣ ∣ φ (β_{m}) V_{β_{m}} - φ (β_{m}) {ˆ V}_{n, β_{m}} ∣ ∣ > \frac{ϵ}{2 m Δ β}]$

We now apply Lemma 6.1 to bound each of the terms on the RHS above, to obtain

	$P [∣ ∣ S - {ˆ S}_{n, m} ∣ ∣ > ϵ]$
	$\leq 2 exp ⎛ ⎝ - 2 n c_{0} {(\frac{ϵ}{2 m φ (β_{0}) Δ β})}^{2} ⎞ ⎠ + 4 exp ⎛ ⎝ - 2 n c_{1} {(\frac{ϵ}{2 m φ (β_{1}) Δ β})}^{2} ⎞ ⎠$
	$+ \dots + 4 exp ⎛ ⎝ - 2 n c_{m - 1} {(\frac{ϵ}{2 m φ (β_{m - 1}) Δ β})}^{2} ⎞ ⎠ + 2 exp ⎛ ⎝ - 2 n c_{m} {(\frac{ϵ}{2 m φ (β_{m}) Δ β})}^{2} ⎞ ⎠,$

where $c_{i}$ is a constant that depends on the value of the density $f$ in the neighborhood of $V_{β_{i}}$ , for i = $0 \dots m$ . Thus,

	$P [∣ ∣ S - {ˆ S}_{n, m} ∣ ∣ > ϵ]$	$\leq 4 m exp (- 2 n c {(\frac{ϵ}{2 m C_{1} Δ β})}^{2})$		(11)
		$= 4 m exp (- \frac{n c ϵ^{2}}{2 C_{1}^{2}}) = \frac{2 K_{1}}{ϵ} exp (- \frac{n c ϵ^{2}}{2 C_{1}^{2}}) .$

Note that $c = min {c_{0}, c_{1}, \dots, c_{m}}$ in (11). The claim in part (i) follows.

The proof of the result in part (ii) follows in a similar manner. In particular, using part (ii) in Lemma 6.1, with $m \geq \sqrt{\frac{K_{2}}{6 ϵ}}$ , we obtain

P [∣ S - {ˆ S}_{n, m} ∣> ϵ]

\leq 4 m exp (- \frac{n c ϵ^{2}}{2 C_{1}^{2}}) = \sqrt{8 K_{2} / 3 ϵ} . exp (- \frac{n c ϵ^{2}}{2 C_{1}^{2}})

∎

6.2 Proof of Theorem 4.2

Proof..

Recall that the truncation threshold $B_{n} = \sqrt{2 σ^{2} log (n)}$ . Letting $η = F (B_{n})$ , we have

	$P [S - {˜ S}_{n, m} > ϵ]$
	$\leq P [\int_{0}^{1} φ (β) V_{β} d β - m \sum k = 1 \frac{φ (β_{k - 1}) {˜ V}_{n, β_{k - 1}} + φ (β_{k}) {˜ V}_{n, β_{k}}}{2} Δ β > ϵ]$
	$= P [\int_{0}^{η} φ (β) V_{β} d β - m \sum k = 1 \frac{φ (β_{k - 1}) {˜ V}_{n, β_{k - 1}} + φ (β_{k}) {˜ V}_{n, β_{k}}}{2} Δ β + \int_{η}^{1} φ (β) V_{β} d β > ϵ]$
	$= P [I_{1} + I_{2} > ϵ],$		(12)

where $I_{1} = \int_{0}^{η} φ (β) V_{β} d β - \sum_{k = 1}^{m} \frac{φ (β_{k - 1}) {˜ V}_{n, β_{k - 1}} + φ (β_{k}) {˜ V}_{n, β_{k}}}{2} Δ β$ , and $I_{2} = \int_{η}^{1} φ (β) V_{β} d β$ .
We bound $I_{2}$ as follows:

1 - β = P (X > V_{β}) \leq exp (- \frac{{V_{β}}^{2}}{2 σ^{2}}),

(13)

since $X$ is Gaussian with mean zero, and variance $σ^{2}$ . Using $log x \leq \frac{x}{e} \forall x > 0$ , we obtain

V_{β} \leq \sqrt{2 σ^{2} log (\frac{1}{1 - β})} \leq \sqrt{\frac{2 σ^{2}}{e (1 - β)}},

leading to

$\int_{η}^{1} V_{β} d β$	$\leq \sqrt{\frac{2 σ^{2}}{e}} \int_{η}^{1} \frac{d β}{\sqrt{1 - β}} = 2 \sqrt{\frac{2 σ^{2}}{e}} \sqrt{1 - η}$
	$\leq 2 \sqrt{\frac{2 σ^{2}}{e}} exp (- \frac{{V_{η}}^{2}}{4 σ^{2}})$	(using (13))
	$= 2 \sqrt{\frac{2 σ^{2}}{e}} exp (- \frac{{B_{n}}^{2}}{4 σ^{2}})$	(since $V_{η} = B_{n}$ )

Hence,

I_{2} = \int_{η}^{1} φ (β) V_{β} d β \leq C_{1} \int_{η}^{1} φ (β) V_{β} d β \leq \frac{2 σ C_{1}}{\sqrt{n}} .

(14)

s Applying the bound in the Theorem 3.2 to the truncated r.v. $Z=X\indicX≤Bn$ , we bound $I_{1}$ as follows:

P [I_{1} > ϵ]

\leq \frac{K_{1}}{ϵ} exp (- \frac{n c ϵ^{2}}{} 2 C_{1}^{2})

(15)

Hence,

$P [I_{1} + I_{2} > ϵ]$	$\leq \frac{K_{1}}{(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})} exp ⎛ ⎜ ⎜ ⎜ ⎝ - \frac{n c {(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})}^{2}}{2 C_{1}^{2}} ⎞ ⎟ ⎟ ⎟ ⎠$	(using (14) and (15))
	$= \frac{(B_{n} C_{2} + δ_{1} C_{1})}{(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})} exp ⎛ ⎜ ⎜ ⎜ ⎝ - \frac{n c {(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})}^{2}}{2 C_{1}^{2}} ⎞ ⎟ ⎟ ⎟ ⎠$
	$\leq \frac{(\sqrt{2 σ^{2} log (n)} C_{2} + \sqrt{2 π} σ n C_{1})}{(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})} exp ⎛ ⎜ ⎜ ⎜ ⎝ - \frac{n c {(ϵ - \frac{2 σ C_{1}}{\sqrt{n}})}^{2}}{2 C_{1}^{2}} ⎞ ⎟ ⎟ ⎟ ⎠,$

where the final inequality follows from the fact that $δ_{1} = \sqrt{2 π σ^{2}} exp (\frac{B_{n}^{2}}{2 σ^{2}}) = \sqrt{2 π σ^{2}} n$ , which holds since the underlying Gaussian distribution is truncated at $B_{n}$ .

By using a parallel argument, a concentration result for bounding the lower semi-deviations can be derived, and we omit the details. ∎

6.3 Proof of Theorem 4.2

Proof..

The proof for the exponential case follows in a similar manner as that of the proof of Theorem 4.2. For the sake of completness, we provide the detailed proof in Appendix C. ∎

7 Conclusions

We proposed a novel SRM estimation scheme that is based on numerical integration, and derived concentration bounds for our SRM estimator for the case of distributions with bounded support, Gaussian and exponential. As future work, it would be interesting to generalize the bounds for Gaussian/exponential distributions to the class of sub-Gaussian/sub-exponential distributions. An orthogonal direction for future work is to derive a lower bound for SRM estimation, and close the gap (if any) in the upper bound that we have derived.

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Appendix A Proof of Lemma 6.1

Proof..

The proof follows in a similar fashion as a result in [8], and we provide the details below for the sake of completeness. Let $h = Δ β = \frac{b - a}{m}$ and $β_{k} = a + k h$ . We look at a single interval and operate integrate by parts twice:

$\int_{β_{k}}^{β_{k + 1}} φ (β) V_{β} d β$	$= \int_{0}^{h} φ (t + β_{k}) V_{(t + β_{k})} d t$
	$= {[(t + A) φ (t + β_{k}) V_{(t + β_{k})}]}_{k}^{h} - \int_{0}^{h} (t + A) {(φ (t + β_{k}) V_{(t + β_{k})})}_{k}^{'} d t$	(16)
	$= {[(t + A) φ (t + β_{k}) V_{(t + β_{k})}]}_{k}^{h} - {[(}$

Estimation of Spectral Risk Measures

Concentration bounds for empirical conditional value-at-risk: The unbounded case

Online Estimation and Optimization of Utility-Based Shortfall Risk

Improved Concentration Bounds for Conditional Value-at-Risk and Cumulative Prospect Theory using Wasserstein distance

On rate optimal private regression under local differential privacy

Finite-sample concentration of the empirical relative entropy around its mean

Local Dvoretzky-Kiefer-Wolfowitz confidence bands

Learning Sparse Additive Models with Interactions in High Dimensions

1 Introduction

2 Preliminaries

3 Distributions with bounded support

3.1 Estimation scheme

3.2 Concentration bounds

Proof.

4 Gaussian and exponential distributions

4.1 Estimation scheme

4.2 Concentration bounds

Proof.

Proof.

5 Simulation experiments

5.1 Synthetic setup

5.2 Vehicular traffic routing

6 Convergence proofs

6.1 Proof of Theorem 3.2

Proof.

Proof:.

Proof of Theorem 3.2..

6.2 Proof of Theorem 4.2

Proof..

6.3 Proof of Theorem 4.2

Proof..

7 Conclusions

References

Appendix A Proof of Lemma 6.1

Proof..

Estimation of Spectral Risk Measures

Related Research

Concentration bounds for empirical conditional value-at-risk: The unbounded case

Online Estimation and Optimization of Utility-Based Shortfall Risk

Improved Concentration Bounds for Conditional Value-at-Risk and Cumulative Prospect Theory using Wasserstein distance

On rate optimal private regression under local differential privacy

Finite-sample concentration of the empirical relative entropy around its mean

Local Dvoretzky-Kiefer-Wolfowitz confidence bands

Learning Sparse Additive Models with Interactions in High Dimensions

1 Introduction

2 Preliminaries

3 Distributions with bounded support

3.1 Estimation scheme

3.2 Concentration bounds

Proof.

4 Gaussian and exponential distributions

4.1 Estimation scheme

4.2 Concentration bounds

Proof.

Proof.

5 Simulation experiments

5.1 Synthetic setup

5.2 Vehicular traffic routing

6 Convergence proofs

6.1 Proof of Theorem 3.2

Proof.

Proof:.

Proof of Theorem 3.2..

6.2 Proof of Theorem 4.2

Proof..

6.3 Proof of Theorem 4.2

Proof..

7 Conclusions

References

Appendix A Proof of Lemma 6.1

Proof..