# Borel transforms of functions from a parameterized family of Hilbert spaces
of entire functions

K.P. Isaev, R.S. Yulmukhametov

##  1 Introduction

Let $D$ be a bounded convex domain on the complex plane. The Bergman space
$B_{2}(D)$ and the Smirnov space $E_{2}(D)$ are quite well studied due to
their importance in complex analysis problems. The space $B_{2}(D)=H(D)\bigcap
L_{2}(D)$, where $H(D)$ is the space of analytic functions on $D$ and
$L_{2}(D)$ is the space of functions with square-summable modulus. The space
$E_{2}(D)$ consists of functions $f\in H(D)$ having boundary values of the
class $L_{2}(\partial D)$. In [1], [2], in particular, it was found that the
Laplace transform of linear continuous functionals
$\mathcal{L}:S\longrightarrow\widehat{S}(\lambda)=S_{z}(e^{\lambda z})$
establishes an isomorphism between the space dual to $E_{2}(D)$ and the space
of entire functions

$\widehat{E}_{2}(D)=\left\\{F\in
H(\mathbb{C}):\int_{0}^{2\pi}\int_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}drd\Delta(\varphi)}{K_{1}(re^{i\varphi})}<\infty\right\\},$

where $h(\varphi)$ the support function of the domain $D$, that is,

$\displaystyle h(\varphi)=\max_{z\in\overline{D}}\textrm{Re
}ze^{i\varphi},\quad\varphi\in[0;2\pi],$
$\displaystyle\Delta(\varphi)=h(\varphi)+\int_{0}^{\varphi}h(\theta)d\theta,\quad\varphi\in[0;2\pi],$
$\displaystyle K_{1}(\lambda)=\|e^{\lambda z}\|^{2}_{L_{2}(\partial
D)}=\int_{D}|e^{\lambda z}|^{2}ds(z),\quad\lambda\in\mathbb{C},$

and $ds(z)$ is an element of length along the boundary of $D$. In [3] it is
shown that the Laplace transform establishes a continuous linear isomorphism
between the space dual to $B_{2}(D)$ and the space of entire functions

$\widehat{B}_{2}(D)=\left\\{F\in
H(\mathbb{C}):\int_{0}^{2\pi}\int_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}drd\Delta(\varphi)}{K(re^{i\varphi})}<\infty\right\\},$

where

$K(\lambda)=\|e^{\lambda z}\|^{2}_{L_{2}(D)}=\int_{D}|e^{\lambda
z}|^{2}dm(z),\quad\lambda\in\mathbb{C}.$

Considering that $K_{1}(\lambda)\cong|\lambda|K(\lambda)$,
$|\lambda|\longrightarrow\infty$, for $\beta\in\mathbb{R}$ it is natural to
introduce the spaces

$P_{\beta}(D)=\left\\{F\in H(\mathbb{C}):\
\|F\|^{2}:=\int_{0}^{2\pi}\int_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}drd\Delta(\varphi)}{K(re^{i\varphi})r^{2\beta}}<\infty\right\\}.$

The spaces $P_{\beta}(D)$ form a continuous scale of Hilbert spaces, and, as
stated above, $\widehat{P}_{0}(D)$ is isomorphic to $B_{2}(D)$ and
$\widehat{P}_{1/2}(D)$ is isomorphic $E_{2}(D)$. Thus, Bergman and Smirnov
spaces are embedded in the scale of Hilbert spaces, which, according to the
authors, should make it possible to apply the theory of Hilbert scales to
study of problems in these spaces.

The function associated with an entire function $F$ of exponential type
$\sigma$ in the sense of Borel is

$\gamma(z)=\sum_{k=0}^{\infty}\frac{F^{(k)}(0)}{z^{k+1}},\quad|z|>\sigma.$

Let $H_{0}(\mathbb{C}\setminus\overline{D})$ be the space of functions that
are analytic in $\mathbb{C}\setminus\overline{D}$ and vanish at infinity. Let
$G^{(\alpha)}(D)$ denote the space

$\left\\{h(\zeta)\in H_{0}(\mathbb{C}\setminus\overline{D}):\
\|h\|^{2}:=\int_{\mathbb{C}\setminus\overline{D}}|h^{\prime\prime}(\zeta)|^{2}{\textrm{
dist }}^{2\alpha}(D,\zeta)\,dm(\zeta)<\infty\right\\}$

for $\alpha>0$. In this paper we intend to prove the following theorem.

Main theorem. Let $F$ be an entire function with indicator diagram $D$,
$\gamma$ be the function associated with $F$ in the sense of Borel, and
$\beta\in(-\frac{1}{2};\frac{3}{2})$. Then for some constants
$c(\beta,D),C(\beta,D)>0$, depending on the domain $D$ and the parameter
$\beta$, the following relation holds:

$c(\beta,D)\|\gamma\|^{2}_{G^{\beta+1}}\leq\|F\|_{P_{\beta}}=\int_{0}^{\infty}\int_{0}^{2\pi}\frac{|F(re^{i\varphi})|^{2}}{K(re^{i\varphi})r^{2\beta}}d\Delta(\varphi)dr\leq
C(\beta,D)\|\gamma\|^{2}_{G^{\beta+1}}.$

The interest in these spaces is caused by the fact that the authors have a
reasonable assumption that the spaces $P_{\beta}(D)$ admit unconditional bases
of reproducing kernels (see [5] —[9]). Accordingly, a scale of Hilbert spaces
of functions that are analytic in the convex domain $D$ and admit
unconditional bases of exponentials will be found.

In Sections 2 and 3 we will prove the preparatory theorems. In Section 2 we
will estimate the integral

$\int_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}dr}{K(re^{i\varphi})r^{2\beta}}$

for $\beta>-\frac{1}{2}$. In Section 3, the localization of the norm in the
spaces $G^{\alpha}$ will be justified.

##  2 Estimation of the integral with respect to the radius

For a fixed $\varphi\in[0;2\pi]$, using the map $z\longrightarrow
w=ze^{i\varphi}-h(\varphi)$, we transform $D$ into a domain $D_{\varphi}$
which lies in the left half-plane and is tangent to the axis of ordinates. For
$t<0$ we denote by $s(t,\varphi)$ the area of the intersection of the domain
$D_{\varphi}$ and the strip $\\{z=x+iy:\ t<x<0\\}$.

Theorem 1 Let $F$ be an entire function satisfying the condition: for some
$\beta\in(-\frac{1}{2};+\infty)$ and $\varphi\in[0;2\pi]$

$I_{\varphi}=\int_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}}{K(re^{i\varphi})r^{2\beta}}dr<\infty,$

and $\gamma$ is the Borel transform of $F$. Then the inequalities

$a(\beta)I_{\varphi}\leq\int_{-\infty}^{\infty}\int_{-\infty}^{0}\frac{|\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-(x+iy))|^{2}|x|^{2\beta+3}}{s(x,\varphi)}dxdy\leq
A(\beta)I_{\varphi}$

hold for some constants $a(\beta),A(\beta)$ depending only on the parameter
$\beta$ (see the remark at the end of the section).

###### Proof.

The proof is essentially based on the reasoning of $\S 1$ in [3]. Therefore,
we tried to maintain the appropriate designations. In the half-plane
$P_{\varphi}=\\{\zeta:\ \textrm{Re }\zeta e^{i\varphi}>h(\varphi)\\}$ the
function $\gamma$ is restored by the formula

$\gamma(\zeta)=\int_{0}^{\infty}F(re^{i\varphi})e^{-\zeta
re^{i\varphi}}e^{i\varphi}dr.$

We write $\zeta=(h(\varphi)-\xi)e^{-i\varphi}$ for points in this half-plane,
where $\xi$ varies over the left half-plane $\textrm{Re }\xi<0$. Then

$\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-\xi))=\int_{0}^{\infty}\left(F(re^{i\varphi})r^{2}e^{3i\varphi}e^{-h(\varphi)r}\right)e^{\xi
r}dr.$ (1)

Let’s use the theorem from [4].

Theorem A Let $u(t)$ be a convex function on an interval $I$ and let
$\widetilde{u}$ be the Young conjugate of $u$,

$\widetilde{u}(x)=\sup_{t\in I}(xt-v(t)).$

We set

$K_{0}(x)=\int_{I}e^{2xt-2v(t)}dt,\quad x\in J=\\{x\in\mathbb{R}:\
\widetilde{v}(x)<\infty\\}.$

Then, for an arbitrary function $g$ on $I$ such that the integral

$\|g\|^{2}=\int_{I}|g(t)|^{2}e^{-2v(t)}dt,$

is finite, the function $\widehat{g}(z)=\int_{I}\overline{g}(t)e^{zt-2v(t)}dt$
satisfies the inequalities

$a\|g\|^{2}\leq\int_{-\infty}^{\infty}\int_{J}\frac{\widehat{g}(x+iy)|^{2}}{K_{0}(x)}d\widetilde{v}^{\prime}(x)dy\leq
A\|g\|^{2},$

where $a$ and $A$ are absolute positive constants, that is, they do not depend
on the function $g$ or the weight $v$.

Let

$\eta(r)=\frac{e^{2h(\varphi)r}}{K(re^{i\varphi})},\quad
u(r)=\frac{1}{2}\ln\frac{\eta(r)}{r^{4}},\quad v(r)=u(r)-\beta\ln r.$

Using these functions, we write formula (1) in the form

$\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-\xi))=\int_{0}^{\infty}\frac{F(re^{i\varphi})e^{3i\varphi}e^{h(\varphi)r}}{r^{2+2\beta}K(re^{i\varphi})}e^{\xi
r-2v(r)}dr.$

Next we intend to apply Theorem A, assuming $I=(0;+\infty)$,

$g(r)=\frac{\overline{F}(re^{i\varphi})e^{-3i\varphi}e^{h(\varphi)r}}{r^{2+2\beta}K(re^{i\varphi})},$

and taking $v(t)$ as the weight function. To do this, make sure that the
function $v(t)$ is convex.

Lemma 1

1) The function $v$ is convex on $I=(0;+\infty)$, and

$\left(\frac{1}{2}+\beta\right)\frac{1}{r^{2}}\leq
v^{\prime\prime}(r)\leq(2+\beta)\frac{1}{r^{2}},\quad r>0;$

2) If $\widetilde{v}(x)=\sup_{r>0}(xr-v(r))$, $x\in J=(-\infty;0)$, is the
Young conjugate of $v$, then

$\frac{(1+2\beta)^{2}}{4(2+\beta)}\frac{1}{x^{2}}\leq\widetilde{v}^{\prime\prime}(x)\leq\frac{2(2+\beta)^{2}}{1+2\beta}\frac{1}{x^{2}},\quad
x<0.$

###### Proof.

In the first item of Lemma 1 in [3], the following inequalities are proved

$\frac{1}{2r^{2}}\leq u^{\prime\prime}(r)\leq\frac{2}{r^{2}},\quad r>0.$

The estimates of item 1 follow from these inequalities. Let’s prove the second
item. The proof of Lemma 1 in [3] shows (see (5)) that

$-\frac{2}{r}\leq u^{\prime}(r)\leq-\frac{1}{2r},\quad r>0.$

Therefore

$-(2+\beta)\frac{1}{r}\leq
v^{\prime}(r)\leq-\left(\frac{1}{2}+\beta\right)\frac{1}{r}.$ (2)

The following relations are also proved there

$u(r)\longrightarrow-\infty,\quad\left|\frac{u(r)}{\ln r}\right|=O(1),\quad
r\longrightarrow+\infty.\quad\lim_{r\longrightarrow 0+}u(r)=+\infty,$

Hence, the function $\widetilde{v}(x)$ is defined on $(-\infty;0)$ and

$\lim_{r\longrightarrow
0+}(xr-v(r))=\lim_{r\longrightarrow+\infty}(xr-v(r))=-\infty.$

Thus, the supremum in the definition of $\widetilde{v}$ is attained at the
single point $r=r(x)>0$ for which $v^{\prime}(r)=x$. From estimates (2) we
obtain

$-\left(\frac{1}{2}+\beta\right)\frac{1}{x}\leq
r(x)\leq-(2+\beta)\frac{1}{x}.$ (3)

By the definition of $\widetilde{v}(x)$, we have the identity

$\widetilde{v}(x)\equiv xr(x)-v(r(x)),\quad x<0,$

or

$\widetilde{v}(v^{\prime}(r))\equiv v^{\prime}(r)r-u(r),\quad r>0.$

We differentiate the above identity twice:

$\widetilde{v}^{\prime\prime}(v^{\prime}(r))v^{\prime\prime}(r)\equiv 1,\quad
r>0.$

Hence, in view of formula (3) and assertion 1) of this Lemma, it follows that

$\frac{(1+2\beta)^{2}}{4(2+\beta)}\frac{1}{x^{2}}\leq\widetilde{v}^{\prime\prime}(x)=\frac{1}{v^{\prime\prime}(r(x))}\leq\frac{2(2+\beta)^{2}}{1+2\beta}\frac{1}{x^{2}},\quad
x<0.$

Lemma 1 is proved. ∎

We apply Theorem A to the function

$g(r)=\frac{\overline{F}(re^{i\varphi})e^{-3i\varphi}e^{h(\varphi)r}}{r^{2+2\beta}K(re^{i\varphi})},$

which shows that the expression

$\|g\|^{2}=\int_{0}^{\infty}|g(r)|^{2}e^{-2v(r)}dr=\int_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}dr}{K(re^{i\varphi})r^{2\beta}}=I_{\varphi}$

is equivalent to the integral

$\int_{-\infty}^{\infty}\int_{-\infty}^{0}\frac{|\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-(x+iy)))|^{2}}{K_{0}(x,\varphi)}d\widetilde{v}^{\prime}(x)dy.$

Estimating $d\widetilde{v}^{\prime}(x)$ according to item 2 of Lemma 1, we
obtain

$\frac{(1+2\beta)^{2}}{4(2+\beta)}aI_{\varphi}\leq\int_{-\infty}^{\infty}\int_{-\infty}^{0}\frac{|\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-(x+iy)))|^{2}}{K_{0}(x,\varphi)x^{2}}d\widetilde{v}^{\prime}(x)dy\leq$
$\leq\frac{2(2+\beta)^{2}}{1+2\beta}AI_{\varphi}.$ (4)

To complete the proof of Theorem 1, we need one more lemma.

Lemma 2 For arbitrary $\varphi\in[0;2\pi]$ and
$\beta\in(-\frac{1}{2};\infty)$, the inequalities

$2^{-(2\beta+5)}a_{0}(\beta)\frac{s(t,\varphi)}{|t|^{2\beta+5}}\leq
K_{0}(t,\varphi)\leq
a_{0}(\beta)\left(1+\frac{a_{+}(\beta)}{a_{-}(\beta)}\right)\frac{s(t,\varphi)}{|t|^{2\beta+5}},$

hold, where

$a_{0}(\beta)=\int_{0}^{\infty}t^{2\beta+4}e^{-2t}dt,\quad
a_{-}(\beta)=\int_{0}^{1}\frac{tdt}{(1+t)^{2\beta+5}},$
$a_{+}(\beta)=\int_{1}^{\infty}\frac{tdt}{(1+t)^{2\beta+5}}.$

###### Proof.

Let

$D_{\varphi}=\\{z=x+iy:\ f_{1}(x)<y<f_{2}(x),\quad R_{\varphi}<x<0\\}.$

Then $f(x)=f_{1}(x)-f_{2}(x)$ is a non-negative concave function on the
interval $[-R_{\varphi};0]$ and

$K_{0}(t,\varphi)=\int_{0}^{\infty}e^{2tr-2v(r)}dr=\int_{0}^{\infty}e^{2rt}\frac{r^{2\beta+4}dr}{\eta(r)}=$
$=\int_{0}^{\infty}e^{2xr}r^{2\beta+4}K(re^{i\varphi})e^{-2rh(\varphi)}dr=\int_{0}^{\infty}e^{2rt}r^{2\beta+4}\left(\int_{D_{\varphi}}e^{2rx}dxdy\right)dr=$
$=\int_{D_{\varphi}}\left(\int_{0}^{\infty}e^{2r(x+t)}r^{2\beta+4}dr\right)dxdy\quad\textrm{for}\quad
t<0.$

Therefore

$K_{0}(t,\varphi)=a_{0}(\beta)\int_{D_{\varphi}}\frac{dxdy}{|x+t|^{2\beta+5}},$

where

$a_{0}(\beta)=\int_{0}^{\infty}e^{-2\tau}\tau^{2\beta+4}d\tau.$

1\. Let $t\leq-D_{\varphi}$, then $|t|\leq|t+x|\leq 2|t|$ for values of $x$ in
the interval of integration, and therefore

$\frac{a_{0}(\beta)2^{-(2\beta+5)}}{t^{2\beta+5}}|D_{\varphi}|\leq
K_{0}(t,\varphi)\leq\frac{a_{0}(\beta)}{t^{2\beta+5}}|D_{\varphi}|,$

where $|D_{\varphi}|$ is the area of $D_{\varphi}$, which means that the
assertion of the lemma is true since $s(t,\varphi)=|D_{\varphi}|$.

2\. Let $0\geq t>-D_{\varphi}$ and $p=f(t)$. The concavity of $f$ implies the
inequalities

$f(x)\leq\frac{p}{t}x,\quad-R_{\varphi}\leq x\leq t,\quad
f(x)\geq\frac{p}{t}x,\quad t\leq x\leq 0.$

Therefore

$\int_{-R_{\varphi}}^{t}\frac{f(x)dx}{|t+x|^{2\beta+5}}\leq\frac{p}{|t|}\int_{|t|}^{\infty}\frac{rdr}{(r+|t|)^{2\beta+5}}=\frac{a_{+}(\beta)p}{|t|^{2\beta+4}},$
$\int_{t}^{0}\frac{f(x)dx}{|t+x|^{2\beta+5}}\geq\frac{p}{|t|}\int_{0}^{|t|}\frac{rdr}{(r+|t|)^{2\beta+5}}=\frac{a_{-}(\beta)p}{|t|^{2\beta+4}},$

where

$a_{+}(\beta)=\int_{1}^{\infty}\frac{\tau d\tau}{(1+\tau)^{2\beta+5}},\quad
a_{-}(\beta)=\int_{0}^{1}\frac{\tau d\tau}{(1+\tau)^{2\beta+5}},$

which means that

$K_{0}(t,\varphi)=a_{0}(\beta)\int_{-R_{\varphi}}^{0}\frac{f(x)dx}{|t+x|^{2\beta+5}}\leq
a_{0}(\beta)\left(1+\frac{a_{+}(\beta)}{a_{-}(\beta)}\right)\int_{t}^{0}\frac{f(x)dx}{|x+t|^{2\beta+5}}\leq$
$\leq
a_{0}(\beta)\left(1+\frac{a_{+}(\beta)}{a_{-}(\beta)}\right)\frac{1}{|t|^{2\beta+5}}\int_{t}^{0}f(x)dx=a_{0}(\beta)\left(1+\frac{a_{+}(\beta)}{a_{-}(\beta)}\right)\frac{s(t,\varphi)}{|t|^{2\beta+5}}.$

The lower estimate is obvious:

$K_{0}(t,\varphi)\geq
a_{0}(\beta)\int_{t}^{0}\frac{f(x)dx}{|t+x|^{2\beta+5}}\geq\frac{2^{-(2\beta+5)}a_{0}(\beta)}{|t|^{2\beta+5}}\int_{t}^{0}f(x)dx=$
$=\frac{2^{-(2\beta+5)}a_{0}(\beta)}{|t|^{2\beta+5}}s(t,\varphi).$

Lemma 2 is proved. ∎

To complete the proof of Theorem 1, it is enough to substitute the estimates
of Lemma 2 into relation (4). ∎

Remark.  We can take

$a(\beta)=aa_{0}(\beta)2^{-(2\beta+5)}\frac{(1+2\beta)^{2}}{4(2+\beta)},\quad
A(\beta)=Aa_{0}(\beta)\frac{2(2+\beta)^{2}}{(1+2\beta)}\left(1+\frac{a_{+}(\beta)}{a_{-}(\beta)}\right)$

as constants in Theorem 1. Constants $a_{0}(\beta),a_{\pm}(\beta)$ are defined
in Lemma 2 and $a,A$ are absolute constants in Theorem A.

##  3 Localization of the norm in the spaces $G^{\alpha}$

The next theorem enables one to localize the integrals over
$\mathbb{C}\setminus\overline{D}$, namely, to take them over
$\Omega\setminus\overline{D}$, where $\Omega$ is an arbitrary neighbourhood of
$\overline{D}$.

We denote by $B(z,\varepsilon)$ the circle of radius $\varepsilon$ with centre
at a point $z$. If $z=0$, then this point will not be indicated. Let
$D(\varepsilon)=D+B(\varepsilon)$ and

$R(D)=\inf\\{R>0:\quad D\subset\overline{B(R)}\\}.$

Theorem 2 Let $\gamma\in G^{(\alpha)}(D)$ and $\alpha\in[0;\frac{3}{2})$. Then

$\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta)\leq$
$\leq(1+B_{0}(\alpha))(1+B(\alpha,D))\int\limits_{D(\varepsilon)\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta),$

and

$\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha+1}(\zeta)\,dm(\zeta)\leq$
$\leq(1+5R(D)B_{1}(\alpha))(1+5R(D)B(\alpha,D))\int\limits_{D(\varepsilon)\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta),$

where

$B_{0}(\alpha)=4^{2\alpha}(4^{(2-\alpha)}-1)^{-1},\quad
B_{1}(\alpha)=4^{2\alpha}(2^{(3-2\alpha)}-1)^{-1},$
$B(\alpha,D)=256\frac{(20R)^{2\alpha}(|\partial
D|+\pi\varepsilon)^{2}}{\pi^{2}\varepsilon^{2(\alpha+1)}},$

for arbitrary $\varepsilon\in(0;R(D))$. If
$\alpha\in[\frac{3}{2};\frac{5}{2})$, then the same estimates are satisfied
under the additional condition
$\lim_{|z|\longrightarrow\infty}|z||\gamma(z)|=0$ with constants $B_{0},B_{1}$
replaced by

$B^{\prime}_{0}(\alpha)=4^{2\alpha}(2^{(3-\alpha)}-1)^{-1},\quad
B^{\prime}_{1}(\alpha)=4^{2\alpha}(2^{(5-2\alpha)}-1)^{-1}.$

###### Proof.

We will split the proof into two stages. In the first stage (Lemma 3), an
upper estimate for the integral of $|\gamma^{\prime\prime}(\zeta)|$ over the
exterior of the circle $\overline{B(4R(D))}$ will be found using the integral
over the set ${B(4R(D))}\setminus\overline{D}$. Then (Lemma 4) the integral
over ${B(4R(D))}\setminus\overline{D}$ will be bounded above by the integral
over $D(\varepsilon)\setminus\overline{D}$.

In what follows, the number $R(D)$ will be abbreviated to $R$.

Lemma 3 Let $\gamma\in G^{(\alpha)}D)$ and $\alpha\in[0;\frac{3}{2})$. Then
the relations

$\int\limits_{|\zeta|\geq 4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta)\leq
B_{0}(\alpha)\int\limits_{B(4R)\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta),$
$\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha+1}(\zeta)\,dm(\zeta)\leq
5RB_{1}(\alpha)\int\limits_{D(\varepsilon)\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta)$

hold. If $\alpha\in[\frac{3}{2};\frac{5}{2})$, then the same estimates are
satisfied under the additional condition
$\lim_{|z|\longrightarrow\infty}|z||\gamma(z)|=0$ with constants $B_{0},B_{1}$
replaced by $B_{0}^{\prime},B_{1}^{\prime}$.

###### Proof.

We represent the function $\gamma(\zeta)$ in the form of a Laurent series:

$\gamma(\zeta)=\sum\limits_{k=0}^{\infty}\frac{\gamma_{k}}{\zeta^{k+1}},\quad|\zeta|>R.$

By the hypotheses of the lemma, both this series and the series

$\gamma^{\prime\prime}(\zeta)=\sum\limits_{k=0}^{\infty}\frac{(k+2)(k+1)\gamma_{k}}{\zeta^{k+3}}=\sum\limits_{k=0}^{\infty}\frac{\gamma^{\prime\prime}_{k}}{\zeta^{k+3}},\quad|\zeta|>R,$

are uniformly convergent on $\mathbb{C}\setminus{B(2R)}$. Let’s take some
number $t\in[0;2)$. If $\zeta$ satisfies the condition $|\zeta|\geq 4R$, then
$\textrm{dist }(\zeta)\leq|\zeta|+R<2|\zeta|$, and therefore

$\int\limits_{|\zeta|\geq 4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\leq 2^{2t}\int\limits_{|\zeta|\geq
4R}|\gamma^{\prime\prime}(\zeta)|^{2}|\zeta|^{2t}\,dm(\zeta).$

Passing to polar coordinates in the right-hand integral and taking into
account the orthogonality of the system $e^{ik\varphi}$ with respect to the
measure $d\varphi$ on $[0;2\pi]$, we obtain

$\int\limits_{|\zeta|\geq 4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\leq
2^{2t+1}\pi\int\limits_{4R}^{\infty}\sum\limits_{k=0}^{\infty}\frac{|\gamma^{\prime\prime}_{k}|^{2}r^{2t+1}\,dr}{r^{2(k+3)}}=$
$=2^{2t+1}\pi\sum\limits_{k=0}^{\infty}\frac{|\gamma^{\prime\prime}_{k}|^{2}}{2(k+2-t)(4R)^{2(k+2-t)}}.$
(5)

Again using the Laurent series, we find a lower estimate for the integral over
the annulus $B(4R)\setminus B(2R)$ by taking into account the fact that the
inequality $\textrm{dist }(\zeta)\geq\frac{1}{2}|\zeta|$ holds for those
$\zeta$ with $|\zeta|\geq 2R$:

$\int\limits_{2R\leq|\zeta|<4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\geq
2^{-2t}\int\limits_{2R\leq|\zeta|<4R}|\gamma^{\prime\prime}(\zeta)|^{2}|\zeta|^{2t}\,dm(\zeta)=$
$=2^{-2t+1}\pi\sum\limits_{k=0}^{\infty}\frac{|\gamma^{\prime\prime}_{k}|^{2}}{2(k+2-t)(4R)^{2(k+2-t)}}(2^{2(k+2-t)}-1).$
(6)

Hence,

$\int\limits_{2R\leq|\zeta|<4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\geq$ $\geq
2^{-2t+1}\left(2^{2(2-t)}-1\right)\pi\sum\limits_{k=0}^{\infty}\frac{|\gamma^{\prime\prime}_{k}|^{2}}{2(k+2-t)(4R)^{2(k+2-t)}}.$
(7)

This, together with (5), implies the estimate

$\int\limits_{|\zeta|\geq 4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\leq$ $\leq
2^{4t}\left(2^{2(2-t)}-1\right)^{-1}\int\limits_{2R\leq|\zeta|<4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta).$

Assuming $t=\alpha<\frac{3}{2}$, we obtain the first estimate in Lemma 3 for
$\alpha\in[0;\frac{3}{2})$. Assuming $t=\alpha+\frac{1}{2}<2$ and taking into
account that $\textrm{dist }(\zeta)\leq 5R$ for $|\zeta|\leq 4R$, we obtain
the second estimate in Lemma 3 for $\alpha\in[0;\frac{3}{2})$.

If $\lim_{|z|\longrightarrow\infty}|z||\gamma(z)|=0$, then
$\gamma^{\prime\prime}_{k}=0$ and the sums in relations (5), (6) are taken
over $k\geq 1$. In this case we obtain the inequality

$\int\limits_{2R\leq|\zeta|<4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\geq$ $\geq
2^{-2t+1}\left(2^{2(3-t)}-1\right)\pi\sum\limits_{k=1}^{\infty}\frac{|\gamma^{\prime\prime}_{k}|^{2}}{2(k+2-t)(4R)^{2(k+2-t)}},\quad
t\in[0;3),$

instead of (7). Hence, and from relation (5) with $\gamma"_{0}=0$, we have

$\int\limits_{|\zeta|\geq 4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta)\leq$ $\leq
2^{4t}\left(2^{2(3-t)}-1\right)^{-1}\int\limits_{2R\leq|\zeta|<4R}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2t}(\zeta)\,dm(\zeta).$

Assuming $t=\alpha$, we obtain the first estimate in Lemma 3 for
$\alpha\in[\frac{3}{2};\frac{5}{2})$. Assuming $t=\alpha+\frac{1}{2}$ and
taking into account that $\textrm{dist }(\zeta)\leq 5R$ for $|\zeta|\leq 4R$,
we obtain the second estimate in Lemma 3 for
$\alpha\in[\frac{3}{2};\frac{5}{2})$.

Lemma 3 is proved. ∎

Lemma 4 If $\gamma\in G^{\alpha}(D)$ and $\alpha>0$, then

$\int\limits_{B(4R)\setminus\overline{D}(\varepsilon)}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta)\leq$ $\leq 256\frac{(20R)^{2\alpha}(|\partial
D|+\pi\varepsilon)^{2}}{\pi^{2}\varepsilon^{2(\alpha+1)}}\int\limits_{D(\varepsilon)\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\,dm(\zeta).$

###### Proof.

Let $\zeta\notin\overline{D}(\varepsilon)$. Since

$\textrm{dist }(\zeta,D)\leq\textrm{dist
}(\zeta,D(\varepsilon))+\varepsilon,\quad|\partial D(\varepsilon)|=|\partial
D|+2\pi\varepsilon,$

by Cauchy’s formula, the upper estimate

$|\gamma^{\prime\prime}(\zeta)|\leq\frac{1}{2\pi}\left|\int\limits_{\partial
D(\varepsilon/2)}\frac{\gamma^{\prime\prime}(z)\,dz}{z-\zeta}\right|\leq\frac{|\partial
D(\varepsilon/2)|}{2\pi\textrm{dist
}(\zeta,D(\varepsilon/2))}\max\limits_{z\in\partial
D(\varepsilon/2)}|\gamma^{\prime\prime}(z)|\leq$ $\leq\frac{|\partial
D|+\pi\varepsilon}{2\pi(\textrm{dist
}(\zeta,D)-\varepsilon/2)}\max\limits_{z\in\partial
D(\varepsilon/2)}|\gamma^{\prime\prime}(z)|$

holds. Clearly, we have $\textrm{dist }(\zeta,D)\geq\varepsilon$ for
$\zeta\notin\overline{D}(\varepsilon)$ and
$x/\left(x-\frac{\varepsilon}{2}\right)\leq 2$ for $x\geq\varepsilon$. In
addition, $\textrm{dist }(\zeta)\leq 5R$ for $\zeta\in B(4R)$. Therefore

$|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2\alpha}(\zeta)\leq\frac{5^{2\alpha}R^{2(\alpha-1)}(|\partial
D|+\pi\varepsilon)^{2}}{\pi^{2}}\max\limits_{z\in\partial
D(\varepsilon/2)}|\gamma^{\prime\prime}(z)|^{2}$ (8)

for $\zeta\in B(4R)\setminus\overline{D}(\varepsilon)$. If $z\in\partial
D(\varepsilon/2)$, then the circle $B(z,\varepsilon/4)$ lies in the domain
$D(3\varepsilon/4)\setminus\overline{D}$. In addition, if $w\in\partial
B(z,\frac{\varepsilon}{4})$, then $\textrm{dist
}(w)\geq\frac{\varepsilon}{4}$. Since $|\gamma^{\prime\prime}(z)|^{2}$ is a
subharmonic function, we obtain the upper estimate

$|\gamma^{\prime\prime}(z)|^{2}\leq\frac{16}{\pi\varepsilon^{2}}\int\limits_{B(z,\varepsilon/4)}|\gamma^{\prime\prime}(w)|^{2}\,dm(w)\leq$
$\leq\frac{16}{\pi\varepsilon^{2}}(\sup\limits_{B(z,\varepsilon/4)}{\textrm{dist
}}^{-2\alpha}(w))\int\limits_{B(z,\varepsilon/4)}|\gamma^{\prime\prime}(w)|^{2}{\textrm{dist
}}^{2\alpha}(w)\,dm(w)\leq$
$\leq\frac{4^{2(\alpha+1)}}{\pi\varepsilon^{2(\alpha+1)}}\int\limits_{D(\varepsilon)\setminus\overline{D}}|\gamma^{\prime\prime}(w)|^{2}{\textrm{dist
}}^{2\alpha}(w)\,dm(w).$

Substituting the above result in (8) and integrating over
$B(4R)\setminus\overline{D}(\varepsilon)$, we obtain the assertion of the
Lemma 4.

Lemma 4 is proved. ∎

The estimates in Theorem 2 follow directly from the assertions of Lemmas 3 and
4. ∎

##  4 Proof of the main theorem

It suffices to prove the main theorem for domains whose boundaries contain no
rectilinear segments and no corners. This follows from the fact that the
constants $c(\beta,D)$ and $C(\beta,D)$ depend continuously on $D$. Indeed,
let us assume that the theorem is true under these additional conditions on
the domain $D$. Let $\varepsilon>0$ be an arbitrary number.Let us inscribe in
the set $D(\varepsilon)$ a convex polygon containing $D$ and then replace each
side of the polygon by an arc of a sufficiently large circle in such a way
that the resulting domain $D^{\prime}$ remains convex and is contained in the
domain $D(2\varepsilon)$. The boundary of $D^{\prime}$ contains no rectilinear
segments but may have corners. To eliminate these we pass to the domain
$D^{\prime}+B(0,\varepsilon)$, and thus obtain a domain whose boundary
contains no corners or rectilinear segments, contains $D$, and is contained in
$D(3\varepsilon)$. Since $P_{\beta}(D)\subset P_{\beta}(D(\varepsilon))$, we
can apply the main theorem in the spaces $P_{\beta}(D(\varepsilon))$ for the
functions $F\in P_{\beta}(D)$ and then we can pass to the limit as
$\varepsilon\longrightarrow 0$.

For further proof we need some geometric objects. let $s(x,\varphi)$ denote
the area of the part

$D_{\varphi}=\\{w=x+iy:\ f_{1}(x)<y<f_{2}(x),\ -R_{\varphi}<x<0\\}$

of $D$ cut off by the line $\textrm{Re }w=x$. The domain $D_{\varphi}$ results
from the transformation $w=ze^{i\varphi}-h(\varphi)$ of the domain $D$. Hence,
$R_{\varphi}$ is the distance between the support lines $L(\varphi)$ and
$L(\varphi+\pi)$. Let $l(x,\varphi)$ denote the length of the part of the
boundary of $D_{\varphi}$ cut off by the same line, and $u(x,\varphi)$ denote
the length of the chord cut off by $D_{\varphi}$ on that line. We write

$\sigma(D)=\inf\limits_{\varphi\in[0;2\pi]}R_{\varphi}.$

It is clear that $\sigma(D)$ is the ”smallest” width of the domain.

By Theorem 1, the norm $\|F\|_{\beta}$ defined in the main theorem for an
entire function $F$ is equivalent to the triple integral

$\int\limits_{0}^{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{0}\frac{|\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-(x+iy)))|^{2}|x|^{2\beta+3}}{s(x,\varphi)}dxdyd\Delta(\varphi).$

We make the change of variables

$\zeta=e^{-i\varphi}(h(\varphi)-x-iy),\quad\theta=\varphi$

in this integral. Let us discuss the geometric meaning of the old and new
variables.

In what follows, $l(\varphi)$ will denote the directed line
$\\{te^{i\varphi},\ -\infty<t<\infty\\}$. We choose the anticlockwise
direction of motion along the boundary of the domain $D$ and thus define
directions along all tangents to the boundary. Let $L(\varphi)$ denote the
tangent parallel to and having the same direction as the line
$l(\frac{\pi}{2}-\varphi)$. If $te^{-i\varphi}$ is the point of intersection
of $L(\varphi)$ and $l(-\varphi)$, then it is easily verified that
$h(\varphi)=t$. If the variables $\varphi\in[0;2\pi)$, $x<0$ and $y$ are
given, then $\zeta$ is the point lying in the plane and having the coordinates
$(h(\varphi)-x;-y)$ in the coordinate system formed by the lines $l(-\varphi)$
(the axis of abcissae) and $l(\frac{\pi}{2}-\varphi)$ (the axis of ordinates).
In this case, the condition $x<0$ means that the support line $L(\varphi)$
separates the point $\zeta$ from the domain $D$. Let us determine the ranges
of the variables $\theta$ and $\zeta$. The point $\zeta$ obviously lies
outside $\overline{D}$. For a fixed $\zeta\in\mathbb{C}\setminus\overline{D}$,
the angle $\theta$ must satisfy the condition that the support line
$L(\theta)$ separates the point $\zeta$ from the domain $D$. We draw two
tangent lines from the point $\zeta$ to the boundary of $D$. Let their
directions coincide with the respective directions of the lines
$l(\varphi_{1})$ and $l(\varphi_{2})$ and let
$0\leq\varphi_{1}\leq\varphi_{2}$. Then the angle $\theta$ varies from
$\varphi_{-}(\zeta)=\frac{\pi}{2}-\varphi_{2}$ to
$\varphi_{+}(\zeta)=\frac{\pi}{2}-\varphi_{1}$. The Jacobian of the
transformation from the variables $\varphi,x,y$ to the variables
$\zeta,\theta$ is identically equal to 1, and we have $x=h(\theta)-\textrm{Re
}\zeta e^{i\theta}$. Thus,

$\int\limits_{0}^{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{0}\frac{|\gamma^{\prime\prime}(e^{-i\varphi}(h(\varphi)-\xi))|^{2}|x|^{2\beta+3}}{s(x,\varphi)}dxdyd\Delta(\varphi)=$
$=\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}\left(\int\limits_{\varphi_{-}(\zeta)}^{\varphi_{+}(\zeta)}\frac{(\textrm{Re
}\zeta e^{i\theta}-h(\theta))^{2\beta+3}}{s(h(\theta)-\textrm{Re }\zeta
e^{i\theta},\theta)}d\Delta(\theta)\right)dm(\zeta).$

The inner integral on the right-hand side of above equality will be denoted by
$p(\zeta)$. We have proven the following theorem.

Theorem 1′ Let $F=B(\gamma)$ be an entire function satisfying the condition

$\|F\|^{2}=\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}\frac{|F(re^{i\varphi})|^{2}}{K(re^{i\varphi})r^{2\beta}}\,drd\Delta(\varphi)<\infty$

and let

$p(\zeta)=\int\limits_{\varphi_{-}(\zeta)}^{\varphi_{+}(\zeta)}\frac{(\textrm{Re
}\zeta e^{i\theta}-h(\theta))^{2\beta+3}}{s(h(\theta)-\textrm{Re }\zeta
e^{i\theta},\theta)}\,d\Delta(\theta).$

Then

$a(\beta)\|F\|^{2}\leq\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{{}^{\prime\prime}}(\zeta)|^{2}p(\zeta)\,dm(\zeta)\leq
A(\beta)\|F\|^{2},$

where the constants $a(\beta),A(\beta)$ depend only on $\beta$ (see the remark
to Theorem 1).

Lemma 5 Let

$p_{0}(\zeta)=\int\limits_{\varphi_{-}(\zeta)}^{\varphi_{+}(\zeta)}\frac{(\textrm{Re
}\zeta e^{i\theta}-h(\theta))^{2\beta+2}}{u(h(\theta)-\textrm{Re }\zeta
e^{i\theta},\theta)}\,d\Delta(\theta).$

Then

1\. for points $\zeta$ such that $\textrm{dist }(\zeta)\leq\sigma(D)/2$, we
have

$\frac{2}{3}p_{0}(\zeta)\leq p(\zeta)\leq 2p_{0}(\zeta);$

2\. If $\textrm{dist }(\zeta)>\sigma(D)/2$, then we denote by $I_{0}$ the part
of $(\varphi_{-}(\zeta);\ \varphi_{+}(\zeta))$ in which the condition

$\textrm{Re }\zeta e^{i\varphi}-h(\varphi)\geq\frac{\sigma(D)}{2}$

holds and by $I$ the remaining part of the interval $(\varphi_{-}(\zeta);\
\varphi_{+}(\zeta))$. Then

$p(\zeta)\leq\frac{4\textrm{diam }^{2}(D)|\partial
D|}{\sigma^{2}(D)|D|}{\textrm{dist
}}^{2\beta+3}(\zeta)+2\int\limits_{I}\frac{(\textrm{Re }\zeta
e^{i\theta}-h(\theta))^{2\beta+2}}{u(h(\theta)-\textrm{Re }\zeta
e^{i\theta},\theta)}\,d\Delta(\theta).$

###### Proof.

1\. By the definition of $h(\varphi)$,

$\textrm{Re }\zeta
e^{i\varphi}-h(\varphi)=\min\limits_{z\in\overline{D}}(\textrm{Re }\zeta
e^{i\varphi}-\textrm{Re
}ze^{i\varphi})\leq\min\limits_{z\in\overline{D}}|\zeta-z|=\textrm{dist
}(\zeta).$ (9)

Therefore, for points $\zeta$ such that the condition $\textrm{dist
}(\zeta)<\sigma(D)/2$ holds and for all $\varphi\in(\varphi_{-}(\zeta);\
\varphi_{+}(\zeta))$, the expression $s(h(\varphi)-\textrm{Re }\zeta
e^{i\varphi},\ \varphi)$ can be estimated using Proposition 2 in [3], and this
immediately yields the estimates in the first part of the lemma.

2\. If the interval $I_{0}$ turns out to be empty, then for $\theta\in I_{0}$
the inequality in assertion (2) of Proposition 2 in [3] can be used to obtain

$s(h(\theta)-\textrm{Re }\zeta
e^{i\theta},\theta)\geq\frac{\sigma^{2}(D)|D|}{4\textrm{diam }^{2}(D)}.$

In view of (9) and the geometric meaning of the function $\Delta(\theta)$, the
second part of Lemma 5 follows.

Lemma 5 is proved. ∎

Let us describe in geometric terms the integral appearing in the definition of
the function $p_{0}(\zeta)$. The interval
$(\varphi_{-}(\zeta);\varphi_{+}(\zeta))$ of integration consists of the
angles $\varphi$ such that $\textrm{Re }\zeta e^{i\varphi}-h(\varphi)\geq 0$,
that is, the directions determined by $\varphi$ for which the support line
$L(\varphi)$ separates the point $\zeta$ from the domain $D$. In this case,
the expression $\textrm{Re }\zeta e^{i\varphi}-h(\varphi)$ is the distance
between $\zeta$ and $L(\varphi)$. If this support line undergoes parallel
translation by a distance of $\textrm{Re }\zeta e^{i\varphi}-h(\varphi)$, then
the domain $D$ cuts off the chord (whose length we denoted by
$u(h(\varphi)-\textrm{Re }\zeta e^{i\varphi},\varphi)$) from the translated
line. Finally, the geometric meaning of the function $\Delta(\varphi)$ is
that, for $\varphi_{1}\geq\varphi_{2}$, the difference
$\Delta(\varphi_{1})-\Delta(\varphi_{2})$ is equal to the length of the arc on
the boundary of $D$ between the points of tangency of the support lines
$L(\varphi_{1})$ and $L(\varphi_{2})$.

This description is unrelated to any coordinate system. In what follows, we
choose a coordinate system related to a fixed point $\zeta$, partially
describe the domain $D$ as the epigraph of a convex function $f$, and describe
the integral in the definition of the function $p_{0}(\zeta)$ using $f$.

Let the point $\zeta$ serve as the origin of the coordinate system. There is a
single point $z_{0}$ on the boundary of $D$ such that

$\textrm{dist }(\zeta)=\inf\limits_{z\in\partial D}|z-\zeta|=|z_{0}-\zeta|.$

We direct the axis of ordinates from $\zeta$ to $z_{0}$. In this coordinate
system, the domain $D$ is part of the epigraph of a convex function $f(x)$
defined on the interval $(X_{1};X_{2})$ with

$X_{1}=h(\pi),\quad X_{2}=h(0).$

Angles are naturally considered in the new coordinate system. The angles of
inclination with respect to the axis of abscissae for the two tangents to the
domain $D$ that pass through the origin will be denoted by $\varphi_{1}$ and
$\varphi_{2}$, where $\varphi_{1}\leq\varphi_{2}$. Then the upper and lower
limits of integration in the integral appearing in the definition of $p_{0}$
are $\varphi_{-}=\frac{\pi}{2}-\varphi_{2}$ and
$\varphi_{+}=\frac{\pi}{2}-\varphi_{1}$ respectively. The distance between the
point $\zeta$ and the domain $D$ in this coordinate system is expressed as
$f(0)$ or $-h(\pi/2)$.

Furthermore, let us assume that the boundary of $D$ contains no corners or
rectilinear segments (see the remark in the beginning of this section). For
the function $f$, this means that its derivative $f^{\prime}$ is a strictly
increasing continuous function.

As $\theta$ varies from $\varphi_{-}$ to $\varphi_{+}$, the expression
$\frac{\pi}{2}-\theta$ varies monotonically from $\varphi_{2}$ to
$\varphi_{1}$, that is, from the angle of inclination of the tangent to the
graph of the function $f$ at the point $X_{1}$ to that of the tangent at the
point $X_{2}$. Consequently, if $x(\theta)$ is given by the relation

$f^{\prime}(x(\theta))=\tan\left(\frac{\pi}{2}-\theta\right)=\cot(\theta),$
(10)

then the point corresponding to $x(\theta)$ will move monotonically from
$X_{1}$ to $X_{2}$, and $(x(\theta);\ f(x(\theta)))$ is the support point of
the support line $L(\theta)$. The geometrical meaning of $\Delta(\theta)$
implies that

$d\Delta(\theta)=d\left(\int\limits^{x(\theta)}\sqrt{1+f^{\prime}(s)^{2}}\,ds\right)=\sqrt{1+f^{\prime}(x(\theta))^{2}}\,dx(\theta)=\frac{1}{|\sin\theta|}\,dx(\theta).$

We have the following representation for the function $p_{0}$:

$p_{0}(\zeta)=\int\limits_{\varphi_{-}}^{\varphi_{+}}\frac{|h(\theta)|^{2\beta+2}}{u(h(\theta),\theta)}\frac{1}{|\sin\theta|}\,dx(\theta),$
(11)

where the expressions involved in the integral on the right-hand side are
written in the coordinate system related to the point $\zeta$.

Next, it is necessary to represent $u(h(\theta),\theta)$ using $f$. For this,
we introduce some new functions. We take an arbitrary point $x_{0}\in[X_{1};\
X_{2}]$ and a positive number $\delta$ and set

$\rho_{+}(f,x_{0},\delta)=\sup\left\\{\rho:\ \rho\leq X_{2}-x_{0},\
\int\limits_{0}^{\rho}(f^{\prime}(x_{0}+y)-f^{\prime}(x_{0}))\,dy\leq\delta\right\\},$
$\rho_{-}(f,x_{0},\delta)=\sup\left\\{\rho:\ \rho\leq x_{0}-X_{1},\
\int\limits_{0}^{\rho}(f^{\prime}(x_{0})-f^{\prime}(x_{0}-y))\,dy\leq\delta\right\\}.$
$\widetilde{\rho}(f,x_{0},\delta)=\rho_{-}(f,x_{0},\delta)+\rho_{+}(f,x_{0},\delta).$

The expression

$g(t)=\sup_{x\in[X_{1};X_{2}]}(xt-f(x)),\quad-\infty<t<\infty,$

is called the Young conjugate of $f$. Let $T_{1}=f^{\prime}(X_{1})$ and
$T_{2}=f^{\prime}(X_{2})$. If $t\in(T_{1};T_{2})$, then the supremum in the
definition of $g$ is attained at the single stationary point $x=x(t)$
determined by the condition $f^{\prime}(x)=t$, that is, $g(t)\equiv
x(t)t-f(x(t)),\ t\in(T_{1};T_{2})$. We differentiate this identity with
respect to $t$ and obtain $g^{\prime}(t)\equiv x(t)$ or
$g^{\prime}(f^{\prime}(x))\equiv x$. Assuming $x=x(\theta)$, from (10) we have

$x(\theta)=g^{\prime}(\cot(\theta)).$ (12)

For a convex function $g$ on $\mathbb{R}$ and an arbitrary positive number
$\delta$, we define the expression $\rho=\rho(g,t_{0},\delta)$ by the
condition

$\rho=\sup\left\\{s>0:\
\int\limits_{-s}^{s}|g^{\prime}(t_{0}+t)-g^{\prime}(t_{0})|\,dt\leq\delta\right\\}.$

From Lemma 5 and from (11), (12) in this paper and from Lemmas 3, 4 in [3] we
obtain the following statement.

Lemma 6 For points $\zeta$ such that $\textrm{dist }(\zeta)\leq\sigma(D)/2$,
the inequalities

$\frac{2}{9}\int\limits_{\varphi_{-}}^{\varphi_{+}}|h(\theta)|^{2\beta+1}|\sin\theta|\rho\left(g,\cot\theta,\frac{|h(\theta)|}{|\sin\theta|}\right)\,dg^{\prime}(\cot\theta)\leq
p(\zeta)\leq$ $\leq\frac{48\textrm{diam
}^{4}(D)}{|D|^{2}}\int\limits_{\varphi_{-}}^{\varphi_{+}}|h(\theta)|^{2\beta+1}|\sin\theta|\rho\left(g,\cot\theta,\frac{|h(\theta)|}{|\sin\theta|}\right)\,dg^{\prime}(\cot\theta)$

hold.

If $\textrm{dist }(\zeta)>\sigma(D)/2$, then we denote by $I_{0}$ the part of
$(\varphi_{-};\ \varphi_{+})$ where the condition

$-h(\theta)>\frac{\sigma(D)}{2}$

holds and let $I$ be the remaining part of the interval $(\varphi_{-};\
\varphi_{+})$. Then

$p(\zeta)\leq\frac{4\textrm{diam }^{2}(D)|\partial
D|}{\sigma^{2}(D)|D|}{\textrm{dist }}^{2\beta+3}(\zeta)+$
$+\frac{48\textrm{diam
}^{4}(D)}{|D|^{2}}\int\limits_{I}|h(\theta)|^{2\beta+1}|\sin\theta|\rho\left(g,\cot\theta,\frac{|h(\theta)|}{|\sin\theta|}\right)\,dg^{\prime}(\cot\theta).$

We make the change of variable $\theta=\frac{\pi}{2}-\varphi$ in the integrals
and set

$p_{1}(\zeta)=\int\limits_{\theta_{-}}^{\theta_{+}}\left|h\left(\frac{\pi}{2}-\varphi\right)\right|^{2\beta+1}|\cos\varphi|\rho\left(g,\tan\varphi,\frac{|h(\frac{\pi}{2}-\varphi)|}{\cos\varphi}\right)\,dg^{\prime}(\tan\varphi),$
(13)

where $\theta_{\pm}=\frac{\pi}{2}-\varphi$ are the angles of inclination with
respect to the axis of abscissae of the tangents to the graph of $f(x)$ that
pass through the origin. In the above coordinate system, the distance
$\textrm{dist }(\zeta)$ is equal to $-h(\frac{\pi}{2})$, and we will denote it
by $d$. Let us define a point $x=x(\varphi)\in[X_{1};\ X_{2}]$ using the
condition

$f^{\prime}(x(\varphi))=\tan\varphi.$

Then the supremum $\sup_{x}(xt-f(x))$ is attained at $x(\varphi)$ for
$t=\tan\varphi$, that is,

$g(\tan\varphi)=\tan\varphi\cdot x(\varphi)-f(x(\varphi)).$ (14)

Besides, the support line $L(\varphi)$ of $D$ is the tangent to the graph of
$f(x)$ at the point $x(\varphi)$, and $-h(\frac{\pi}{2}-\varphi)$ is the
distance between this tangent line and the origin. It can easily be seen that,
in this case, $-h(\frac{\pi}{2}-\varphi)/\cos\varphi$ is equal to the ordinate
of the point of intersection of the support line with the axis of ordinates.
The equation of the tangent line through the point $x(\varphi)$ has the form

$y=\tan\varphi\cdot(x-x(\varphi))+f(x(\varphi)),$

and therefore

$\frac{-h(\frac{\pi}{2}-\varphi)}{\cos\varphi}=f(x(\varphi))-\tan\varphi\cdot
x(\varphi).$

From (14), we obtain that

$\frac{-h(\frac{\pi}{2}-\varphi)}{\cos\varphi}=-g(\tan\varphi).$

Thus, if $\varphi$ monotonically increases from $0$ to $\theta_{+}$ or
decreases from $0$ to $\theta_{-}$, then $-g(\tan\varphi)$ monotonically
decreases from $d$ to $0$. We set $\varphi_{0}=0$ and find the angles
$\varphi_{n}$ from the conditions

$\frac{-h(\frac{\pi}{2}-\varphi_{n})}{\cos\varphi_{n}}=\frac{d}{2^{|n|}}$

or, which is the same, from the relations $-g(\tan\varphi_{n})=2^{-|n|}d$. The
interval of integration splits into the subintervals $(\varphi_{n};\
\varphi_{n+1}]$, $n\in\mathbb{Z}$, and the integral in (13) itself can thus be
represented as the sum of integrals over these subintervals. We note that
$\rho(g,t,\delta)$ in a non-decreasing function of the argument $\delta$, and
therefore

$\rho(g,\tan\varphi,2^{-n-1}d)\leq\rho\left(g,\tan\varphi,\frac{-h(\frac{\pi}{2}-\varphi)}{\cos\varphi}\right)\leq\rho(g,\tan\varphi,2^{-n}d),\quad
n\geq 0,$
$\rho(g,\tan\varphi,2^{-|n|}d)\leq\rho\left(g,\tan\varphi,\frac{-h(\frac{\pi}{2}-\varphi)}{\cos\varphi}\right)\leq\rho(g,\tan\varphi,2^{-|n|+1}d),\quad
n<0,$

for $\varphi\in(\varphi_{n};\ \varphi_{n+1}]$. Hence, the change of variables
$t=\tan\varphi$ results in

$p_{1}(\zeta)\leq\sum\limits_{n=0}^{\infty}\left(\frac{d}{2^{n}}\right)^{2\beta+1}\int\limits_{\tan\varphi_{n}}^{\tan\varphi_{n+1}}\rho\left(g,t,\frac{d}{2^{n}}\right)\,dg^{\prime}(t)+$
$+\sum\limits_{n=-1}^{-\infty}\left(\frac{d}{2^{|n|-1}}\right)^{2\beta+1}\int\limits_{\tan\varphi_{n}}^{\tan\varphi_{n+1}}\rho\left(g,t,\frac{d}{2^{|n|-1}}\right)\,dg^{\prime}(t),$
(15)
$p_{1}(\zeta)\geq\left(\frac{d}{2}\right)^{2\beta+1}\int\limits_{\tan\varphi_{-1}}^{\tan\varphi_{1}}\frac{1}{(1+t^{2})^{\beta+1}}\rho\left(g,t,\frac{d}{2}\right)\,dg^{\prime}(t).$
(16)

Let us find upper bounds for $p_{1}(\zeta)$. We set

$t_{n}=\frac{\tan\varphi_{n}+\tan\varphi_{n+1}}{2}.$

Then, obviously,

$g(t_{n})\geq g(\tan\varphi_{n})=-\frac{d}{2^{n}}$

for $n\geq 0$ and

$g(t_{n})\geq g(\tan\varphi_{n+1})=-\frac{d}{2^{|n|-1}}$

for $n<0$. Therefore

$g(\tan\varphi_{n})+g(\tan\varphi_{n+1})-2g(t_{n})\leq-\frac{d}{2^{n+1}}-\frac{d}{2^{n}}+\frac{2d}{2^{n}}=\frac{d}{2^{n+1}}<\frac{d}{2^{n}},\quad
n\geq 0,$
$g(\tan\varphi_{n})+g(\tan\varphi_{n+1})-2g(t_{n})\leq\frac{d}{2^{|n|}}<\frac{d}{2^{|n|-1}},\quad
n<0.$

Comparing these estimates with definition of $\rho(g,t,\delta)$, we obtain

$\tan\varphi_{n+1}-t_{n}=t_{n}-\tan\varphi_{n}<\rho\left(g,t_{n},\frac{d}{2^{n}}\right),\quad
n\geq 0,$
$\tan\varphi_{n+1}-t_{n}=t_{n}-\tan\varphi_{n}<\rho\left(g,t_{n},\frac{d}{2^{|n|-1}}\right),\quad
n<0.$

We write

$\rho_{n}=\rho\left(g,t_{n},\frac{d}{2^{n}}\right),\quad n\geq
0,\quad\rho_{n}=\rho\left(g,t_{n},\frac{d}{2^{|n|-1}}\right),\quad n<0.$

Formula (15) now implies the inequality

$p_{1}(\zeta)\leq\sum\limits_{0}^{\infty}\left(\frac{d}{2^{n}}\right)^{2\beta+1}\int\limits_{t_{n}-\rho_{n}}^{t_{n}+\rho_{n}}\rho\left(g,t,\frac{d}{2^{n}}\right)\,dg^{\prime}(t)+$
$+\sum\limits_{-1}^{-\infty}\left(\frac{d}{2^{|n|-1}}\right)^{2\beta+1}\int\limits_{t_{n}-\rho_{n}}^{t_{n}+\rho_{n}}\rho\left(g,t,\frac{d}{2^{|n|-1}}\right)\,dg^{\prime}(t).$

According to item 3 of Lemma 5 in [3] we obtain

$p_{1}(\zeta)\leq\sum_{0}^{\infty}\left(\frac{d}{2^{n}}\right)^{2\beta+1}\cdot\frac{4d}{2^{n}}+\sum_{-1}^{-\infty}\left(\frac{d}{2^{|n|-1}}\right)^{2\beta+1}\cdot\frac{4d}{2^{|n|-1}}=2\frac{4^{\beta+2}}{4^{\beta+1}-1}{\textrm{dist
}}^{2\beta+2}(\zeta).$

Let $I^{\prime}=\\{\frac{\pi}{2}-\theta,\ \theta\in I\\}$, where
$I\subset(\varphi_{-};\varphi_{+})$ is the interval defined in Lemma 6. Then
$I^{\prime}\subset(\theta_{-};\theta_{+})$ and

$\int\limits_{I}|h(\theta)|^{2\beta+1}|\sin\theta|\rho\left(g,\cot\theta,\frac{|h(\theta)|}{|\sin\theta|}\right)\,dg^{\prime}(\cot\theta)=$
$=\int\limits_{I^{\prime}}\left|h\left(\frac{\pi}{2}-\varphi\right)\right|^{2\beta+1}|\cos\varphi|\rho\left(g,\tan\varphi,\frac{|h(\frac{\pi}{2}-\varphi)|}{|\cos\varphi|}\right)\,dg^{\prime}(\tan\varphi)\leq
p_{1}(\zeta)\leq$ $\leq 2\frac{4^{\beta+2}}{4^{\beta+1}-1}{\textrm{dist
}}^{2\beta+2}(\zeta).$

Substituting the above two inequalities in the relations of Lemma 6, we find
upper estimates for $p(\zeta)$: if $\textrm{dist
}(\zeta)\leq\frac{\sigma(D)}{2}$, then

$p(\zeta)\leq\frac{48\textrm{diam
}^{4}(D)}{|D|^{2}}p_{1}(\zeta)\leq\frac{6\cdot 4^{\beta+4}\textrm{diam
}^{4}(D)}{(4^{\beta+1}-1)|D|^{2}}{\textrm{dist }}^{2\beta+2}(\zeta),$ (17)

and if $\textrm{dist }(\zeta)>\frac{\sigma(D)}{2}$, then

$p(\zeta)\leq\frac{4\textrm{diam }^{2}(D)|\partial
D|}{\sigma^{2}(D)|D|}{\textrm{dist }}^{2\beta+3}(\zeta)+\frac{6\cdot
4^{\beta+4}\textrm{diam }^{4}(D)}{(4^{\beta+1}-1)|D|^{2}}{\textrm{dist
}}^{2\beta+2}(\zeta).$ (18)

Furthermore, using (16), we can find lower bounds for $p_{1}(\zeta)$ and thus
for $p(\zeta)$. In what follows, we shall assume without loss of generality
that

$\min(-\tan\ \varphi_{-1},\tan\ \varphi_{1})=\tan\ \varphi_{1}.$

We write $\rho=\rho\left(g,0,\frac{d}{2}\right)$ for brevity. By the
definition of $\rho\left(g,0,\frac{d}{2}\right)$,

$g(\rho)+g(-\rho)-2g(0)=\frac{d}{2},$

On the other hand,

$g(0)=\max\limits_{X_{1}\leq x\leq X_{2}}(-f(x))=-\min\limits_{X_{1}\leq x\leq
X_{2}}f(x)=-f(0)=-d.$

Therefore the relation

$g(\rho)+g(-\rho)=-\frac{3}{2}d$

must hold. By the definition of the angles $\varphi_{\pm 1}$, we have

$g(\tan\ \varphi_{1})+g(-\tan\ \varphi_{1})=-\frac{d}{2}+g(-\tan\
\varphi_{1})\geq-\frac{d}{2}+g(0)=-\frac{3}{2}d.$

Consequently, the inequality

$\tan\ \varphi_{1}\geq\rho=\rho\left(g,0,\frac{d}{2}\right)$

holds, which, together with (16), implies that

$p_{1}(\zeta)\geq\left(\frac{d}{2}\right)^{2\beta+1}\int\limits_{-\tan\
\varphi_{1}}^{\tan\
\varphi_{1}}\frac{1}{(1+t^{2})^{\beta+1}}\rho\left(g,t,\frac{d}{2}\right)\,dg^{\prime}(t)\geq$
$\geq\left(\frac{d}{2}\right)^{2\beta+1}\frac{1}{(1+\tan^{2}\
\varphi_{1})^{\beta+1}}\int\limits_{-\rho}^{\rho}\rho\left(g,t,\frac{d}{2}\right)\,dg^{\prime}(t).$

Proposition 3 from [3] can now be used to find the upper bound for $\tan\
\varphi_{1}$, and the item 3 of Lemma 5 from [3] can be applied to obtain the
lower estimate for the integral:

$p_{1}(\zeta)\geq\left(\frac{d}{2}\right)^{2\beta+1}\cdot\left(1+\frac{25\textrm{diam
}^{2}(D)}{4\sigma^{2}(D)}\right)^{-(\beta+1)}\cdot\frac{d}{2}=$
$=4^{-(\beta+1)}\left(1+\frac{25\textrm{diam
}^{2}(D)}{4\sigma^{2}(D)}\right)^{-(\beta+1)}{\textrm{dist
}}^{2\beta+2}(\zeta).$

This estimate, together with formulas (17) and (18), enables us to deduce from
Lemma 6 the lemma stated below.

Lemma 7 The inequalities

$m(\beta,D){\textrm{dist }}^{2\beta+2}(\zeta)\leq p(\zeta)\leq
M(\beta,D){\textrm{dist }}^{2\beta+2}(\zeta)$

hold for points $\zeta$ such that $\textrm{dist }(\zeta)\leq\sigma(D)/2$.

For points $\zeta$ such that $\textrm{dist }(\zeta)>\sigma(D)/2$, the estimate

$p(\zeta)\leq M_{0}(\beta,D){\textrm{dist
}}^{2\beta+3}(\zeta)+M(\beta,D){\textrm{dist }}^{2\beta+2}(\zeta)$

holds.

Here

$m(\beta,D)=\frac{2}{9}\cdot 4^{-(\beta+1)}\left(1+\frac{25\textrm{diam
}^{2}(D)}{4\sigma^{2}(D)}\right)^{-(\beta+1)},$ $M(\beta,D)=\frac{6\cdot
4^{\beta+4}\textrm{diam }^{4}\ (D)}{(4^{\beta+1}-1)|D|^{2}},\quad
M_{0}(\beta,D)=\frac{4\textrm{diam }^{2}(D)|\partial D|}{\sigma^{2}(D)|D|},$

$\sigma(D)$ is the smallest width of the domain $D$ over all directions,
$\textrm{dist }(\zeta)$ is the distance between the point $\zeta$ and the
domain $D$, $|D|$ is the area of $D$, $|\partial D|$ is the length of the
boundary of $D$ and $\textrm{diam }(D)$ is the diameter of $D$.

To complete the proof of the main theorem, all we have to do is put together
all the proven estimates.

1\. Let $\beta\in(-\frac{1}{2};\frac{1}{2})$.

1.1. Let us prove the lower bound in the main theorem. Using Theorem
$1^{\prime}$ and Lemma 7, we have

$\|F\|_{P_{\beta}}^{2}\geq\frac{1}{A(\beta)}\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}p(\zeta)\,dm(\zeta)\geq$
$\geq\frac{m(\beta,D)}{A(\beta)}\int\limits_{D(\sigma(D)/2)\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2(\beta+1)}\,dm(\zeta).$

Next we apply the first part of Theorem 2 with $\varepsilon=\sigma(D)/2$ and
$\alpha=\beta+1\in[0;\frac{3}{2})$:

$\|F\|_{P_{\beta}}^{2}\geq\frac{m(\beta,D)}{A(\beta)(1+B_{0}(\beta,D))(1+B(\beta,D))}\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{\prime\prime}(\zeta)|^{2}{\textrm{dist
}}^{2(\beta+1)}\,dm(\zeta).$

Thus, the left inequality in the main theorem in this case is proved with the
constant

$c(\beta,D)=\frac{m(\beta,D)}{A(\beta)}(1+B_{0}(\beta,D))^{-1}(1+B(\beta,D))^{-1}.$

1.2. To prove the upper bound, we apply Theorem $1^{\prime}$:

$\|F\|_{P_{\beta}}^{2}\leq\frac{1}{a(\beta)}\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{{}^{\prime\prime}}(\zeta)|^{2}p(\zeta)\,dm(\zeta).$

Then we apply the second estimate in Lemma 7:

$\|F\|_{P_{\beta}}^{2}\leq\frac{M(\beta,D)}{a(\beta)}\int\limits_{\mathbb{C}\setminus\overline{D}}|\gamma^{{}^{\prime\prime}}(\zeta)|^{2}{\textrm{dist
}}^{2(\beta+1)}(\zeta)\,dm(\zeta)+$
$+\frac{M_{0}(\beta,D)}{a(\beta)}\int\limits_{\mathbb{C}\setminus
D(\sigma(D)/2)}|\gamma^{{}^{\prime\prime}}(\zeta)|^{2}{\textrm{dist
}}^{2\beta+3}(\zeta)\,dm(\zeta).$

Let us estimate the second integral using the second inequality of the first
part of Theorem 2 for $\alpha=\beta+1\in[0;\frac{3}{2})$. The right inequality
in the main theorem in this case is proved with the constant

$C(\beta,D)=\frac{M(\beta,D)}{a(\beta)}+\frac{M_{0}(\beta,D)}{a(\beta)}(1+5RB_{0}(\beta+1,D))^{-1}(1+B(\beta+1,D))^{-1}.$

2\. Let $\beta\in[\frac{1}{2};\frac{3}{2})$. If $F\in P_{\beta}(D)$, then by
the definition of $\|F\|_{P_{\beta}}$, for $\beta\geq\frac{1}{2}$ we have
$F(0)=\lim_{|z|\longrightarrow\infty}|z||\gamma(z)|=0$. Therefore we can use
the second part of Theorem 2. As a result, estimates of the main theorem will
be proved with the same constants as in the case of $\beta<\frac{1}{2}$ with
the constant $B_{0}(\beta+1,D)$ replaced by $B^{\prime}_{0}(\beta+1,D)$.

## References

  * [1] Levin B. Ya., Lyubarskii Yu. I., Interpolation by means of special classes of entire functions and related expansions in series of exponentials, Mathematics of the USSR-Izvestiya, 9 (3) (1975) 621–662, https://doi.org/10.1070/im1975v009n03abeh001493
  * [2] Lutsenko V. I., Yulmukhametov R. S., A generalization of Wiener-Paley theorem to functionals in Smirnov spaces, Proceedings of the Steklov Institute of Mathematics, 200 (1993) 271–280, https://zbmath.org/0802.46037
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  * [6] Baranov A., Belov Yu., Borichev A., Fock type spaces with Riesz bases of reproducing kernels and de Branges spaces, Studia Mathematica, 236 (2) (2017), 127–142, https://doi.org/10.4064/sm8504-9-2016
  * [7] Isaev K. P., Yulmukhametov R. S., Unconditional bases in radial Hilbert spaces, Izvestiya: Mathematics, 86 (1) (2022), 150–168, https://doi.org/10.1070/IM9071
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  * [9] Isaev K. P., Lutsenko A. V., Yulmukhametov R. S., On a sufficient condition for the existence of unconditional bases of reproducing kernels in Fock type spaces with nonradial weights, Analysis and Mathematical Physics, 13 (6) (2023), 83, 11 pp., https://doi.org/10.1007/s13324-023-00848-0


1pt 0.8pt

# Multi-Objective SPIBB: Seldonian Offline Policy Improvement with Safety
Constraints in Finite MDPs

Harsh Satija
McGill University, Mila
<EMAIL_ADDRESS>
Philip S. Thomas
University of Massachusetts
<EMAIL_ADDRESS>
Joelle Pineau
McGill University, Mila, Facebook AI Research
<EMAIL_ADDRESS>
Romain Laroche
Microsoft Research
<EMAIL_ADDRESS>
###### Abstract

We study the problem of Safe Policy Improvement (SPI) under constraints in the
offline Reinforcement Learning (RL) setting. We consider the scenario where:
(i) we have a dataset collected under a known baseline policy, (ii) multiple
reward signals are received from the environment inducing as many objectives
to optimize. We present an SPI formulation for this RL setting that takes into
account the preferences of the algorithm’s user for handling the trade-offs
for different reward signals while ensuring that the new policy performs at
least as well as the baseline policy along each individual objective. We build
on traditional SPI algorithms and propose a novel method based on Safe Policy
Iteration with Baseline Bootstrapping (SPIBB, laroche2017safe) that provides
high probability guarantees on the performance of the agent in the true
environment. We show the effectiveness of our method on a synthetic grid-world
safety task as well as in a real-world critical care context to learn a policy
for the administration of IV fluids and vasopressors to treat sepsis.

##  1 Introduction

Reinforcement Learning (RL) as a paradigm for sequential decision-making
(sutton1988learning) has shown tremendous success in a variety of simulated
domains (mnih2015human; silver2017mastering; OpenAI_dota). However, there are
still quite a few challenges between the traditional RL research and real-
world tasks. Most of these challenges stem from assumptions that are rarely
satisfied in practice (dulac2019challenges), or the inability of the
algorithm’s user to specify the desired behavior of the agent without being a
domain expert (Thomas2019). We focus on the real-world application point of
view and posit the following requirements:

  * •

Multiple reward functions: Traditional RL methods assume a single scalar
reward is present in the environment. However, most real-world tasks, have
multiple (possibly conflicting) objectives or constraints that need to be
taken into consideration together, such as the signals related to the safety
(physical well-being of the agent or the environment), budget utilization
(energy or maintenance costs), etc.

  * •

Stakeholder control of the trade-off: The ML practitioners should have the
ability to control the different trade-offs the agent is making and choose the
one they consider best for the task at hand.

  * •

Offline setting: In many real-world domains (e.g., healthcare, finance or
autonomous vehicles), there is an abundance of data, collected under a sub-
optimal policy, but training the agent directly via interactions with the
environment is expensive and risky. We assume that we only have access to a
dataset of past trajectories that can be used for training (lange2012batch).

  * •

Preventing unintended behavior: We want the agent to be robust to both
extrapolation errors from offline RL and misaligned objectives that are poor
proxy of the user’s intentions and algorithm’s actual performance
(ng1999policy; amodei2016concrete). We consider the case where the user can
specify undesirable behavior in the context of the performance observed in the
batch.

  * •

Practical guarantees: We want guarantees about the undesirable behavior that
might be caused by the agent in the real-world. We care about the results that
can be obtained using the finite amount of samples we have in the batch, and
aim to provide some measure of confidence in deploying the agents in the
environment.

To achieve this set of properties, we adopt the Seldonian framework
(Thomas2019), which is a general algorithm design framework that allows high-
confidence guarantees for constraint satisfaction in a multi-objective
setting. Based on the above specifications, we seek to answer the question: if
we are given a batch of data collected under some (suboptimal) behavioral
policy and some user preference, can we build a policy improvement algorithm
that returns a policy with practical high-confidence guarantees on the
performance of the policy w.r.t. the behavioral policy?

We acknowledge that there are other important challenges in RL, such as
partial observability, safe exploration, non-stationary environments and
function approximation in high-dimensional spaces, that also stand in the way
of making RL a more applicable paradigm. These challenges are beyond the scope
of this work, which should rather be thought of as taking a step towards this
broader goal.

In Section 2, we present our contribution positioned with respect to other
related work. In Section 3, we formalize the setting and then extend
traditional SPI algorithms to this setting. We then show it is possible to
extend the previous work on Safe Policy Iteration (SPI), particularly Safe
Policy Iteration with Baseline Bootstrapping (SPIBB, laroche2017safe), for the
design of agents that satisfy the above requirements. We show that the
resulting algorithm is theoretically-grounded and provides practical high-
probability guarantees. We extensively test our approach on a synthetic
safety-gridworld task in LABEL:sec:synthetic-experiments and show that the
proposed algorithm achieves better data efficiency than the existing
approaches. Finally, we show its benefits on a critical-care task in
LABEL:sec:sepsis-experiments. The accompanying codebase is available at
https://github.com/hercky/mo-spibb-codebase.

##  2 Related work

Multi-Objective RL (MORL): Traditional multi-objective approaches
(mannor2004geometric; roijers2013survey; liu2014multiobjective) focus on
finding the Pareto-frontier of optimal reward functions that gives all the
possible trade-offs between different objectives. The user can then select a
policy from the solution set according to their arbitrary preferences. In
practice, an alternate trial and error based approach of scalarization is used
to transform the multiple reward functions into a scalar reward based on
preferences across objectives (usually, by taking a linear combination). Most
traditional MORL approaches have focused on the online, interactive settings
where the agent has access to the environment. While some recent approaches
are based on off-policy learning methods (lizotte2012linear; van2014multi;
yang2019generalized; abdolmaleki2020distributional), they lack guarantees. In
contrast, our work focuses exclusively on learning in the offline setting and
gives high-probability guarantees on the performance in the environment.

Constrained-RL: RL under constraints frameworks, such as Constrained MDPs
(CMDPs, altman1999constrained), present an alternative way to define
preferences in the form of constraints over policy’s returns. Here, the user
assigns a single reward function to be the primary objective (to maximize) and
hard constraints are specified for the others. The major limitation of this
setting is that it assumes the thresholds for the constraints are known a
priori. le2019batch study offline policy learning under constraints and
provide performance guarantees w.r.t. the optimal policy, but their work
relies on the concentrability assumption (munos2003error).

Concentrability is a strong assumption that upper bounds the ratio between the
future state-action distributions of any non-stationary policy and the
baseline policy under which the dataset was generated by some constant. From a
practical perspective, it is unclear how to get a tractable estimate of this
constant, as the space of future state-action distributions of non-stationary
policies is vast. Thus, this constant can be arbitrarily huge, potentially
even infinite when the baseline policy fails to cover the support of the space
of all non-stationary policies (such as in the low-data regime), leading to
the performance bounds given by these methods to blow up (and even be
unbounded). Additionally, the guarantees in le2019batch are only valid with
respect to the performance of the optimal policy. In this work, we instead
focus on the performance guarantees based on returns observed in the dataset,
as it does not require making any of the above assumptions.

Reward design: Reward-design (sorg2010internal) and reward-modelling
approaches (christiano2017deep; littman2017environment; leike2018scalable)
focus on designing suitable reward functions that are consistent with the
user’s intentions. These approaches rely heavily on the human or simulator
feedback, and thus do not carry over easily to the offline setting.

Seldonian-RL (and Safe Policy Improvement): The Seldonian framework
(Thomas2019) is a general algorithm design framework that allows the user to
design ML algorithms that can avoid undesirable behavior with high-probability
guarantees. In the context of RL, the Seldonian framework allows to design
policy optimization problems with multiple constraints, where the solution
policies satisfy the constraints with high-probability. In the offline-RL
setting, SPI refers to the objective of guaranteeing a performance improvement
over the baseline with high-probability guarantees (thomas2015highImprovement;
petrik2016safe; laroche2017safe). Therefore, SPI algorithms are a specific
setting that falls in the general category of Seldonian-RL algorithms.

We focus on two categories of SPI algorithms that provide practical error
bounds on safety: SPIBB (laroche2017safe) that provides Bayesian bounds and
HCPI (thomas2015highImprovement; thomas2015highEvaluation) that provides
frequentist bounds. SPIBB methods constrain the change in the policy according
to the local model uncertainty. SPIBB has been formulated in the context of a
single reward function, and as such does not handle multiple rewards and by
extension also lacks the ability for the user to specify preferences. Our
primary focus is to provide a construction for extending the SPIBB methodology
to the multi-objective setting that handles user preferences and provides
high-probability guarantees.

Instead of relying on model uncertainty, HCPI methods utilize the high-
confidence lower bounds on the Importance Sampling (IS) estimates of a target
policy’s performance to ensure safety guarantees. HCPI has been applied to
solve Seldonian optimization problems for constrained-RL setting using an
enumerable policy class. Thomas2019 suggested using HCPI for the MORL setting,
and we build on that idea. Particularly, we show how HCPI can be implemented
with stochastic policies in the context of our setting with user preferences
and baseline constraints.

##  3 Methodology

###  3.1 Setting

We consider the setting where the agent’s interactions with the environment
can be modelled as a Markov Decision Process (MDP, bellman1957markovian). Let
$\mathcal{X}$ and $\mathcal{A}$ respectively be the (finite) state and action
spaces. Let
$p^{\star}:\mathcal{X}\times\mathcal{A}\rightarrow\mathscr{P}(\mathcal{X})$
denote the true (unknown) transition probability function, where
$\mathscr{P}(\mathcal{X})$ denotes the set of probability distributions on
$\mathcal{X}$. Without loss of generality, we assume that the process
deterministically begins in the state $x_{0}$. We define $[N]$ to be the set
$\\{0,1,\dots,N-1\\}$ for any positive integer $N$. Let there be $d$ different
reward signals and
$\bm{r}^{\star}=\left\\{r_{k}\right\\}_{k\in[d]}:\mathcal{X}\times\mathcal{A}\rightarrow[-r_{\mathrel{\scalebox{0.5}{$\top$}}},r_{\mathrel{\scalebox{0.5}{$\top$}}}]^{d}$
be the true (unknown) stochastic multi-reward signal.111Costs, which are meant
to be minimized, can be expressed as negative rewards. Finally,
$\bm{\gamma}=\left\\{\gamma_{k}\right\\}_{k\in[d]}\in[0,1)^{d}$ is the multi-
discount-factor.

The MDP, $m^{\star}$, can now be defined with the tuple
$(\mathcal{X},\mathcal{A},p^{\star},\bm{r}^{\star},\bm{\gamma},x_{0})$. A
policy $\pi:\mathcal{X}\rightarrow\mathscr{P}(\mathcal{A})$ maps a state to a
distribution over actions. We denote by $\Pi$ the set of stochastic policies.
We consider the infinite horizon discounted return setting. For any $k\in[d]$,
the $k$th reward value function
$v^{\pi}_{m,k}(x):\mathcal{X}\rightarrow\mathbb{R}$ denotes the expected
discounted sum of rewards when when following policy $\pi$ in an MDP $m$
starting from state $x$. Analogously, we define the state-action value
functions for performing action $a$ in state $x$ in MDP $m$ under $\pi$ for
rewards as $q^{\pi}_{m,k}(x,a)$. Let
$\operatorname*{Adv}^{\pi}_{m,k}(x,a)=q^{\pi}_{m,k}(x,a)-v^{\pi}_{m,k}(x)$
denote the corresponding advantage function. The expected return of policy
$\pi$ w.r.t. the $k$th reward in the true MDP $m^{\star}$ is denoted by
$\mathcal{J}^{\pi}_{m^{\star},k}=v^{\pi}_{m^{\star},k}(x_{0})=\mathop{\mathbb{E}}_{\pi,m^{\star}}[\sum_{t=0}^{\infty}\gamma_{k}^{t}R_{k,t}\mid
X_{0}=x_{0}]$, where action $A_{t}\sim\pi(\cdot\mid X_{t})$, immediate reward
$R_{k,t}\sim r^{\star}_{k}(\cdot\mid X_{t},A_{t})$, and state $X_{t+1}\sim
p^{\star}(\cdot\mid X_{t},A_{t})$.

We consider the offline setting, where instead of having access to the
environment we have a pre-collected dataset of trajectories denoted by
$\mathcal{D}=\\{\tau_{i}\\}_{i\in[|\mathcal{D}|]}$, where $|\mathcal{D}|$
denotes the number of trajectories in the dataset. A trajectory $\tau$ of
length $T$ is an ordered set of transition tuples of the form
$\tau=\\{x_{i},a_{i},x^{\prime}_{i},\bm{r}_{i}\\}_{i\in[T]}$, where
$x^{\prime}_{i}$ denotes the state at the next time-step. We denote the
Maximum Likelihood Estimation (MLE) of the MDP with
$\hat{m}=(\mathcal{X},\mathcal{A},\hat{p},\hat{\bm{r}},\bm{\gamma},x_{0})$,
where $\hat{p}$ and $\hat{\bm{r}}$ denote the transition and reward models
estimated from the dataset’s statistics.

######  Assumption 3.1 (Baseline policy).

We assume that we have access to the policy that generated the dataset. We
call such policy the baseline policy and denote it by $\pi_{b}$. 222simao2020
proved that SPIBB/Soft-SPIBB bounds may be obtained with an estimate of
$\pi_{b}$.

###  3.2 Problem formulation

We consider safe policy improvement with respect to the baseline according to
the $d$ dimensions of the multi-objective setting. Therefore, under a Bayesian
approach, we search for target policies such that they perform better (up to a
precision error $\zeta$) than the baseline along every objective function with
high probability $1-\delta$, where $\zeta$ and $\delta$ are hyper-parameters
controlled by the user, denoting the risk that the practitioner is willing to
take. We denote by $\Pi_{\textsc{a}}$ the set of admissible policies that
satisfy:

$\displaystyle\mathbb{P}\left(\forall
k\in[d],\mathcal{J}^{\pi}_{m^{\star},k}-\mathcal{J}^{\pi_{b}}_{m^{\star},k}>-\zeta\Big{|}\mathcal{D}\right)>1-\delta$

In the multi-objective case, there does not exist a single optimal value, but
a Pareto frontier of optimal values. One way to evaluate the MORL problems is
via the multiple-policy approaches (vamplew2011empirical; roijers2013survey)
that compute the policies that approximate the true optimal Pareto-frontier.
However, note that optimality and safety are contradicting objectives. It is
not clear how (and if) one can make claims about optimality in the offline
setting without bringing in additional unrealistic assumptions (Section 2,
MORL). Instead, we take an alternate approach inspired by another category of
MORL methods called single-policy (roijers2013survey; van2014multi) where the
trade-offs between different objectives are explicitly controlled by the user
via providing a scalarization or preferences over objectives. We assume the
user preference $\bm{\lambda}=\left\\{\lambda_{k}\right\\}_{k\in[d]}$ is given
as an input to our algorithms, and is used for scalarization of the
objectives, where $\lambda_{k}\in\mathbb{R}^{+}$. Our objective therefore
becomes

$\displaystyle\operatorname*{arg\,max}_{\pi\in\Pi_{\textsc{a}}}$
$\displaystyle\quad\sum_{k\in[d]}\lambda_{k}\mathcal{J}^{\pi}_{m^{\star},k}$

The above formulation gives freedom to the user in terms of what particular
quantity they want to optimize via $\bm{\lambda}$, but still ensures that the
solution policy performs as well as the baseline policy across all $d$
objectives. Note that our explicit goal is to maximize the objective specified
by the user. However, the user might make mistakes in specifying this
objective (Section 2, Reward design), and the above formulation offers
guarantees that prevent deteriorating the performance of the policy across any
of the $d$ objectives. This allows the user to to experiment with different
reward design strategies in safety-critical settings without worrying about
the risks of ill-defined scalarizations. A naïve approach would be applying
the user scalarization to also define the safety constraints. However, this
construction fails to prevent undesirable behavior for the individual
objectives (shown in LABEL:app:naive-construction).

###  3.3 Multi-Objective SPIBB (MO-SPIBB)

Robust MDPs (Iyengar2005; Nilim2005) can be regarded as an approximation of
the Bayesian formulation by partitioning the MDP space $\mathcal{M}$ into two
subsets: the subset of plausible MDPs $\Xi$ and the subset of implausible
MDPs. The plausible set is classically constructed from concentration bounds
over the reward and transition function:

$\displaystyle\Xi=$ $\displaystyle\left\\{m,\textnormal{ s.t. }\forall
x,a,\begin{array}[]{ll}\lVert p(\cdot|x,a)-\hat{p}(\cdot|x,a)\rVert_{1}\leq
e(x,a),\\\ \lVert\bm{r}(x,a)-\hat{\bm{r}}(x,a)\rVert_{\infty}\leq
e(x,a)r_{\mathrel{\scalebox{0.5}{$\top$}}}\end{array}\right\\},$

where $e$ is an upper bound on the state-action error function of the model
that are classically obtained with concentration bounds, such that the true
environment $m^{\star}\in\Xi$ with high probability $1-\delta$. In the single
objective framework, laroche2017safe empirically show that optimising the
worst-case performance policy in $\Xi$ provides policies that are too
conservative. petrik2016safe prove that it is NP-hard to find the policy $\pi$
that maximises the worst-case policy improvement over $\Xi$.

Instead, the SPIBB methodology (laroche2017safe) consists in searching for a
policy that maximizes the safe policy improvement in the MLE MDP, under some
policy constraints: SPIBB and Soft-SPIBB (nadjahi2019safe) policy search
constraints both revolve around the idea that we must only consider policies
for which the policy improvement may be accurately estimated. Using $\pi_{b}$
as reference, SPIBB allows policy changes only in state-action pairs for which
more than $n_{\wedge}$ samples have been collected. Soft-SPIBB extends this by
applying soft constraints that allow slight changes in the policy for the
uncertain state-action pairs, which are controlled by an error bound related
to model uncertainty. As such, on low-confidence transitions, this class of
methods provides a mechanism that prevents the agent from deviating too much
from $\pi_{b}$. In this work, we build on Soft-SPIBB because it has yielded
better empirical results. Formally, its constraint on the policy class is
defined by:

$\Pi_{\textsc{s}}=\left\\{\pi,\text{ s.t. }\forall x,\sum_{a}e(x,a)\
|\pi(a|x)-\pi_{b}(a|x)|\leq\epsilon\right\\},$

where $\epsilon$ is a hyper-parameter that controls the deviation from the
baseline policy.

We define
$q^{\pi}_{m,\bm{\lambda}}(x,a)=\sum_{k\in[d]}\lambda_{k}q^{\pi}_{m,k}(x,a)$ to
be the state-action value function associated with the linearized
$\bm{\lambda}$ parameters. The same notation extension is used for
$v^{\pi}_{m,\bm{\lambda}}(x,a)$ and $\mathcal{J}^{\pi}_{m,\bm{\lambda}}$. The
application of Soft-SPIBB to multi-objective safe policy improvement is
therefore direct: Figure 7:  Results on 100 random CMDPs for different
$\bm{\lambda}$ and $\rho$ combinations with $\delta=0.1$. The different agents
are represented by different markers and colored lines. Each point on the plot
denotes the mean (with standard error bars) for 12 different
$\bm{\lambda},\rho$ combinations for the 100 randomly generated CMDPs (1200
datapoints). The x-axis denotes the amount of data the agents were trained on.
The y-axis for left subplot in each sub-figure represents the improvement over
baseline and the right subplot denotes the failure rate. The dotted black line
in the right subplots represents the high-confidence parameter $\delta=0.1$.
LABEL:fig:lag-only-single-sopt denotes the case when MO-SPIBB (LABEL:eq:s-opt)
is run with $\epsilon=0.1$, MO-HCPI (LABEL:eq:h-opt) with $\text{IS}=$ Doubly
Robust (DR) estimator with student’s t-test concentration inequality, and
Lagrangian (le2019batch) with $\eta=0.01$ . Figure 7(b) shows how MO-SPIBB and
Lagrangian perform across different hyper-parameters.

(b) MO-SPIBB (LABEL:eq:s-opt) and Lagrangian (le2019batch) comparisons across
different hyper-parameters.

We test the method by le2019batch (henceforth referred to as Lagrangian) in
the synthetic navigation CMDP task described in LABEL:sec:synthetic-
experiments. In LABEL:fig:lag-only-single-sopt, we present the results for the
best performing Lagrangian baseline on 100 random CMDPs for different
$\bm{\lambda}$ and $\rho$ combinations with $\delta=0.1$. Similar to
LABEL:fig:delta-params-mean, we provide a more detailed plot of how the
Lagrangian baseline performs with different hyper-parameters in the above
setting in Figure 7(b).

Results: As expected, we observe that the Lagrangian baseline has a high
failure rate, particularly in the low-data setting. This makes sense as the
guarantees provided by le2019batch are of the form
$\mathcal{J}^{\pi}_{k,m^{\star}}-\mathcal{J}^{\pi_{b}}_{k,m^{\star}}\geq-\frac{C}{(1-\gamma)^{3/2}}$
(Theorem 4.4 of le2019batch), where $C$ is a term that depends on a constant
that comes from the Concentrability assumption (Assumption 1 of le2019batch).
This assumption upper bounds the ratio between the future state-action
distributions of any non-stationary policy and the baseline policy under which
the dataset was generated by some constant. In other words, it makes
assumptions on the quality of the data gathered under the baseline policy.
Unfortunately, this assumption cannot be verified in practice, and it is
unclear how to get a tractable estimate of this constant. As such, this
constant can be arbitrarily large (even infinite) when the baseline policy
fails to cover the support of all non-stationary policies, for instance, when
the baseline policy is not exploratory enough or when the size of the dataset
is small. Hence, we observe a high failure rate of le2019batch in the
experiments, especially in the low data setting. Compared to le2019batch, our
performance guarantees do not make any assumptions on the quality of the
dataset or the baseline. Therefore, our approach can ensure a low failure rate
even in the low-data regime.

Implementation details and Hyper-parameters: We build on top of the publicly
available code of le2019batch released by the authors and extend it to our
setting. We are confident that our implementation is correct as we made sure
it passes various sanity tests such as convergence of the primal-dual gap and
feasibility on access to true MDP parameters.

The algorithm in le2019batch (Algorithm 2, Constrained Batch Policy Learning)
requires the following hyper-parameters:

  * *

Online Learning Subroutine: We use the same online learning algorithm as used
by the authors in their experiments, i.e. Exponentiated Gradient
(kivinen1997exponentiated).

  * *

Duality gap $\omega$ : This denotes the primal-dual gap or the early
termination condition. We tried the values in $\\{0.01,0.001\\}$ and fix the
value to $0.01$.

  * *

Number of iterations: This parameter denotes the number of iterations for
which the Lagrange coefficients should be updated. We experimented in the
range $\\{100,250,500\\}$ and set this to $250$.

  * *

Norm bound $B$: The bound on the norm of Lagrange coefficients vector. We
tried the values in $\\{1,10,50,100\\}$ and fixed it $10$.

  * *

Learning rate $\eta$: This parameter denotes the learning rate for the update
of the Lagrange coefficients via the online learning subroutine. We found that
this is the most sensitive variable and we tried with values in
$\\{0.005,0.01,0.05,0.1,0.5,1.0,5.0\\}$. For the final experiments, we
benchmark with three different values $(0.01,0.1,1.0)$ as mentioned in the
Figure 7(b).

We would like to point out that the hyper-parameter tuning for the Lagrangian
baseline can be particularly challenging as in the low-data setting none of
the combinations of the above hyper-parameters can ensure a low failure rate
even though the duality gap has converged.

The above experiments show the advantage of our approach over le2019batch,
particularly in the low-data safety-critical tasks, where our methods can
improve over the baseline policy while ensuring a low failure rate.

###  D.5 Scaling experiments with number of objectives $d$

We experimented with the different number of objectives $d$ to validate if the
trends we observed for LABEL:eq:s-opt and LABEL:eq:h-opt in
LABEL:sec:synthetic-experiments also extend to $d>2$. In the CMDP formulation,
as there can only be one primary reward, we extend the CMDP to include more
than 1 type of pits. The extended CMDP now has $d-1$ different kinds of pits
and corresponding reward functions, where the agent gets a pit reward of $-1$
if the agent steps into a cell containing that particular kind of pit. We
relax the CMDP threshold to $c_{i}=-10.0$ as the CMDP problem gets harder with
more number of pits, and a lower threshold makes the problem of finding
$\pi^{*}$ of a random CMDP easier. Therefore, the task objective for the agent
in the extended CMDP is to reach the goal in the least amount of steps, such
that it can only step into at most 10 pits of every different type.

We use the same experiment methodology from LABEL:sec:synthetic-experiments.
As the focus is to see how the trends scale with $d$, we fix the
$\bm{\lambda}$, with $\lambda_{0}=1.0$ and the rest of $\lambda_{\geq 1}=0.0$.
We compare LABEL:eq:s-opt and LABEL:eq:h-opt over different
$|\mathcal{D}|\in\\{10,50,500,2000\\}$, $\rho\in\\{0.1,0.4,0.7,0.9\\}$, the
fixed set of parameters: IS=DR, $\text{CI}=$student’s t-test,
$\epsilon\in\\{0.001,0.01,0.1,1.0\\}$, and $\delta=0.1$.

The results over 10 random CMDPs with fixed parameters can be found in Figure
8. We notice that the trends from LABEL:sec:synthetic-experiments case still
carry till $d\leq 1+16$, where for some value of $\epsilon$, LABEL:eq:s-opt
can lead to better improvement in $\pi_{b}$ while still having failure rate
$<\delta$. However, $d>1+16$ we see there are no obvious trends and both
LABEL:eq:s-opt and LABEL:eq:h-opt tend to become very conservative and
returning the baseline becomes the best solution choice.

Figure 8: Scaling with $d$ with results on a 10 random CMDPs and $\delta=0.1$.
The different agents are represented by different markers and color. Each
point on the graph denotes the mean for 100 runs, the standard errors is
denoted by the error bars. The x-axis denotes the amount of data the agents
were trained on. They y-axis for the top plot in a grid represents the
improvement over baseline and the y-axis for bottom plot denotes the failure
rate.

###  D.6 Additional details

For the experiments in LABEL:sec:synthetic-experiments, on an Intel(R) Xeon(R)
Gold 6230 CPU (2.10GHz), the baselines take around 3 seconds to run, and both
LABEL:eq:s-opt and LABEL:eq:h-opt take about 5 seconds.

##  Appendix E Additional details for sepsis experiments

###  E.1 Sepsis data and cohort details

We followed the pre-processing methodology from tang2020clinician;
komorowski2018artificial and we refer the reader to the original work for more
details.

The dosage of prescribed IV fluids and vasopressors is converted into discrete
variables to be used as actions for the constructed MDP. Each type of action
(IV or vasopressor) is divided into 4 bins (each representing one quantile),
and an additional action for "No drug" (0 dose) is also introduced. As such,
the $|\mathcal{A}|=5\times 5$. The cohort statistics can be found in Table 2.
The patient data consists of 48 dimensional time-series with features
representing attributes such as demographics, vitals and lab work results
(Table 3). The patient data is discretized into 4-hour windows, each of which
is pre-processed to be treated as a single time-step. The state-space and is
discretized using a k-means based clustering algorithm to map the states to
$750$ clusters. Two additional absorbing states are added for death and
survival ($|\mathcal{X}|=752$).

Table 2: Cohort statistics after following the data pre-processing methodology from tang2020clinician; komorowski2018artificial. Survivors | N | % Female | Mean Age | Hours in ICU
---|---|---|---|---
Survivors | 18066 | 44.5% | 64.1 | 56.6
Non-survivors | 2888 | 42.9% | 68.8 | 60.9
Table 3: Summary of the patient state features from (tang2020clinician, Table 3). Demographics/Static |  Age, Gender, SOFA, Shock Index, Elixhauser, SIRS, Re-admission, GCS - Glasgow Coma Scale
---|---
Lab values |  Albumin, Arterial pH, Calcium, Glucose, Hemoglobin, Magnesium, PTT - Partial Thromboplastin Time, Potassium, SGPT - Serum Glutamic-Pyruvic Transaminase, Arterial Blood Gas, Blood Urea Nitrogen, Chloride, Bicarbonate, International Normalized Ratio, Sodium, Arterial Lactate, CO2, Creatinine, Ionised Calcium, Prothrombin Time, Platelets Count, SGOT - Serum Glutamic-Oxaloacetic Transaminase, Total bilirubin, White Blood Cell Count
Vital signs |  Diastolic Blood Pressure, Systolic Blood Pressure, Mean Blood Pressure, PaCO2, PaO2, FiO2, PaO/FiO2 ratio, Respiratory Rate, Temperature (Celsius), Weight (kg), Heart Rate, SpO2
Intake and output events |  Fluid Output - 4 hourly period, Total Fluid Output, Mechanical Ventilation

###  E.2 Performance on changing hyper-parameters

For the experiments in LABEL:sec:sepsis-experiments, we treat $\delta$ as a
hyper-parameter. For LABEL:eq:s-opt instead of searching over both $\delta$
and $\epsilon$, we follow the strategy proposed in Soft-SPIBB: fix the
$\delta=1.0$ and only search over $\epsilon$. For LABEL:eq:h-opt, we found
that only DR and WDR gave reliable off-policy estimates so report the results
with both of them with different $\delta$. As in previous sections, we used
student’s t-test as the choice of concentration inequality for LABEL:eq:h-opt.

####  E.2.1 LABEL:eq:s-opt parameters

Here, we fix $\delta=1.0$ and try with different values of the hyper-parameter
$\epsilon$ and directly report the results directly on the test set. The
results are presented in Table 4.

####  E.2.2 LABEL:eq:h-opt parameters

We run with different values of the hyper-parameter $\delta$ and directly
report the results directly on the test set for different IS estimators. The
results for DR estimator are presented in Table 4 and for WDR estimator are
presented in Table 4.

###  E.3 Additional qualitative Analysis

We calculate how many rare-actions are recommended by different solution
policies and compare them with the most common actions taken by the
clinicians. For each state, for the action recommended by a solution policy,
we calculate the frequency with which that state-action was observed in the
training data and calculate the percentage of time that state-action pair was
observed among all the possible actions taken from that state. Across all the
states, the actions suggested by the traditional single-objective RL baseline
are observed only 3% of the time on average (5.3 observations per state).
Whereas, the actions most commonly chosen by the clinicians are observed 51.4%
of the time on average (138.2 observations per state). We study this behavior
for two of the policies returned by MO-SPIBB that deviate the most from the
baseline: for the policy returned by LABEL:eq:s-opt ($\bm{\lambda}=[1,1]$) the
recommended actions are observed 24.8% of time on average (61.0 observations
per state) and for LABEL:eq:s-opt ($\bm{\lambda}=[0,1]$) the recommended
actions are observed 23.4% of times (56.14 observations per state).

###  E.4 Additional details

For the experiments in LABEL:sec:sepsis-experiments, on an Intel(R) Xeon(R)
Gold 6230 CPU (2.10GHz), running the Linearized baseline takes around 30
seconds, Adv-Linearized takes around 60 seconds, LABEL:eq:s-opt take about
90-120 seconds and LABEL:eq:h-opt takes about 90 seconds.

Table 4:  Performance of various LABEL:eq:s-opt policy candidates (with
different $\epsilon$) using DR and WDR estimation with standard errors on 10
random splits of the TEST dataset. The red cells denote the corresponding
safety constraint violation, i.e, either
$\mathcal{J}_{0}^{\pi}<\mathcal{J}_{0}^{\pi_{b}}$ or
$-\mathcal{J}_{1}^{\pi}>-\mathcal{J}_{1}^{\pi_{b}}$.

User preferences $(\bm{\lambda})$ Policy Survival return ($\mathcal{J}_{0}$)
Rare-treatment return ($-\mathcal{J}_{1}$) DR WDR DR WDR Clinician’s
($\pi_{b}$) 64.78 $\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$
0.19 $[\lambda_{0}=1.0,\lambda_{1}=0.0]$ Linearized 97.68 $\pm$ 0.22 97.58
$\pm$ 0.20 27.64 $\pm$ 1.11 27.84 $\pm$ 1.09 LABEL:eq:s-opt, $\epsilon=0.0$
64.78 $\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$ 0.19
LABEL:eq:s-opt, $\epsilon=0.001$ 64.91 $\pm$ 0.90 64.91 $\pm$ 0.90 13.56 $\pm$
0.19 13.56 $\pm$ 0.19 LABEL:eq:s-opt, $\epsilon=0.01$ 66.11 $\pm$ 0.87 66.05
$\pm$ 0.86 13.42 $\pm$ 0.20 13.46 $\pm$ 0.20 LABEL:eq:s-opt, $\epsilon=0.1$
73.70 $\pm$ 0.84 71.96 $\pm$ 0.69 12.30 $\pm$ 0.39 13.80 $\pm$ 0.33
LABEL:eq:s-opt, $\epsilon=0.5$ 78.19 $\pm$ 0.54 81.01 $\pm$ 0.36 16.21 $\pm$
0.49 13.10 $\pm$ 0.31 LABEL:eq:s-opt, $\epsilon=1.0$ 84.03 $\pm$ 0.48 87.11
$\pm$ 0.33 15.54 $\pm$ 0.59 12.17 $\pm$ 0.59 LABEL:eq:s-opt, $\epsilon=2.5$
90.05 $\pm$ 0.25 91.37 $\pm$ 0.20 15.35 $\pm$ 0.72 13.53 $\pm$ 0.56
LABEL:eq:s-opt, $\epsilon=5.0$ 91.58 $\pm$ 0.49 92.66 $\pm$ 0.28 15.39 $\pm$
0.59 13.71 $\pm$ 0.38 LABEL:eq:s-opt, $\epsilon=10.0$ 91.64 $\pm$ 0.47 92.68
$\pm$ 0.23 15.19 $\pm$ 0.59 13.56 $\pm$ 0.42 LABEL:eq:s-opt, $\epsilon=\infty$
91.62 $\pm$ 0.46 92.68 $\pm$ 0.23 15.18 $\pm$ 0.59 13.56 $\pm$ 0.42
$[\lambda_{0}=1.0,\lambda_{1}=1.0]$ Linearized 87.17 $\pm$ 0.48 89.11 $\pm$
0.37 2.41 $\pm$ 0.47 1.52 $\pm$ 0.41 LABEL:eq:s-opt, $\epsilon=0.0$ 64.78
$\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$ 0.19 LABEL:eq:s-opt,
$\epsilon=0.001$ 64.90 $\pm$ 0.90 64.90 $\pm$ 0.90 13.53 $\pm$ 0.19 13.54
$\pm$ 0.19 LABEL:eq:s-opt, $\epsilon=0.01$ 66.02 $\pm$ 0.88 65.94 $\pm$ 0.87
13.15 $\pm$ 0.20 13.20 $\pm$ 0.20 LABEL:eq:s-opt, $\epsilon=0.1$ 74.34 $\pm$
0.78 72.04 $\pm$ 0.87 9.32 $\pm$ 0.29 10.48 $\pm$ 0.45 LABEL:eq:s-opt,
$\epsilon=0.5$ 76.47 $\pm$ 0.50 78.42 $\pm$ 0.41 7.61 $\pm$ 0.44 5.02 $\pm$
0.17 LABEL:eq:s-opt, $\epsilon=1.0$ 81.39 $\pm$ 0.46 84.54 $\pm$ 0.36 4.64
$\pm$ 0.40 2.38 $\pm$ 0.22 LABEL:eq:s-opt, $\epsilon=2.5$ 86.26 $\pm$ 0.33
88.09 $\pm$ 0.24 1.98 $\pm$ 0.28 1.14 $\pm$ 0.27 LABEL:eq:s-opt,
$\epsilon=5.0$ 86.76 $\pm$ 0.47 88.55 $\pm$ 0.22 2.52 $\pm$ 0.48 1.55 $\pm$
0.41 LABEL:eq:s-opt, $\epsilon=10.0$ 86.77 $\pm$ 0.49 88.58 $\pm$ 0.25 2.53
$\pm$ 0.50 1.57 $\pm$ 0.43 LABEL:eq:s-opt, $\epsilon=\infty$ 86.77 $\pm$ 0.49
88.58 $\pm$ 0.25 2.53 $\pm$ 0.50 1.57 $\pm$ 0.43
$[\lambda_{0}=0.0,\lambda_{1}=0.0]$ Linearized -89.39 $\pm$ 0.43 -90.90 $\pm$
0.29 22.99 $\pm$ 0.40 22.81 $\pm$ 0.30 LABEL:eq:s-opt, $\epsilon=0.0$ 64.78
$\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$ 0.19 LABEL:eq:s-opt,
$\epsilon=0.001$ 64.80 $\pm$ 0.90 64.80 $\pm$ 0.90 13.57 $\pm$ 0.19 13.57
$\pm$ 0.19 LABEL:eq:s-opt, $\epsilon=0.01$ 64.92 $\pm$ 0.90 64.92 $\pm$ 0.90
13.50 $\pm$ 0.19 13.51 $\pm$ 0.19 LABEL:eq:s-opt, $\epsilon=0.1$ 65.78 $\pm$
0.89 65.70 $\pm$ 0.88 13.20 $\pm$ 0.20 13.25 $\pm$ 0.20 LABEL:eq:s-opt,
$\epsilon=0.5$ 67.73 $\pm$ 0.82 67.22 $\pm$ 0.88 13.24 $\pm$ 0.24 13.55 $\pm$
0.33 LABEL:eq:s-opt, $\epsilon=1.0$ 69.12 $\pm$ 0.75 67.90 $\pm$ 0.84 13.57
$\pm$ 0.27 14.39 $\pm$ 0.44 LABEL:eq:s-opt, $\epsilon=2.5$ 71.00 $\pm$ 0.63
68.28 $\pm$ 0.46 14.27 $\pm$ 0.30 15.73 $\pm$ 0.40 LABEL:eq:s-opt,
$\epsilon=5.0$ 71.95 $\pm$ 0.54 69.27 $\pm$ 0.63 15.29 $\pm$ 0.39 16.12 $\pm$
0.70 LABEL:eq:s-opt, $\epsilon=10.0$ 72.73 $\pm$ 0.64 71.17 $\pm$ 0.65 16.59
$\pm$ 0.37 16.21 $\pm$ 0.41 LABEL:eq:s-opt, $\epsilon=\infty$ 60.27 $\pm$ 0.49
61.44 $\pm$ 0.85 18.40 $\pm$ 0.27 15.36 $\pm$ 0.58
$[\lambda_{0}=0.0,\lambda_{1}=1.0]$ Linearized 58.27 $\pm$ 2.18 60.52 $\pm$
2.07 0.04 $\pm$ 0.03 0.02 $\pm$ 0.01 LABEL:eq:s-opt, $\epsilon=0.0$ 64.78
$\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$ 0.19 LABEL:eq:s-opt,
$\epsilon=0.001$ 64.83 $\pm$ 0.90 64.83 $\pm$ 0.90 13.52 $\pm$ 0.19 13.52
$\pm$ 0.19 LABEL:eq:s-opt, $\epsilon=0.01$ 65.36 $\pm$ 0.88 65.27 $\pm$ 0.88
12.96 $\pm$ 0.19 13.01 $\pm$ 0.19 LABEL:eq:s-opt, $\epsilon=0.1$ 71.35 $\pm$
0.96 69.29 $\pm$ 0.92 7.75 $\pm$ 0.19 8.30 $\pm$ 0.18 LABEL:eq:s-opt,
$\epsilon=0.5$ 71.01 $\pm$ 0.72 71.30 $\pm$ 0.68 2.54 $\pm$ 0.37 1.50 $\pm$
0.11 LABEL:eq:s-opt, $\epsilon=1.0$ 74.19 $\pm$ 0.57 76.11 $\pm$ 0.57 0.90
$\pm$ 0.14 0.34 $\pm$ 0.09 LABEL:eq:s-opt, $\epsilon=2.5$ 76.42 $\pm$ 0.61
77.20 $\pm$ 0.72 0.10 $\pm$ 0.06 0.06 $\pm$ 0.04 LABEL:eq:s-opt,
$\epsilon=5.0$ 76.08 $\pm$ 0.65 76.87 $\pm$ 0.74 0.07 $\pm$ 0.05 0.05 $\pm$
0.03 LABEL:eq:s-opt, $\epsilon=10.0$ 76.07 $\pm$ 0.65 76.87 $\pm$ 0.73 0.07
$\pm$ 0.05 0.04 $\pm$ 0.03 LABEL:eq:s-opt, $\epsilon=\infty$ 76.05 $\pm$ 0.65
76.85 $\pm$ 0.72 0.07 $\pm$ 0.05 0.04 $\pm$ 0.03

Table 4:  Performance of various LABEL:eq:h-opt policy candidates (with
different $\delta$) using $\text{IS}=$ DR estimator with standard errors on 10
random splits of the TEST dataset. The red cells denote the corresponding
safety constraint violation, i.e, either
$\mathcal{J}_{0}^{\pi}<\mathcal{J}_{0}^{\pi_{b}}$ or
$-\mathcal{J}_{1}^{\pi}>-\mathcal{J}_{1}^{\pi_{b}}$.

User preferences $(\bm{\lambda})$ Policy Survival return ($\mathcal{J}_{0}$)
Rare-treatment return ($-\mathcal{J}_{1}$) DR WDR DR WDR Clinician’s
($\pi_{b}$) 64.78 $\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$
0.19 $[\lambda_{0}=1.0,\lambda_{1}=0.0]$ Linearized 97.68 $\pm$ 0.22 97.58
$\pm$ 0.20 27.64 $\pm$ 1.11 27.84 $\pm$ 1.09 LABEL:eq:h-opt, $\delta=0.1$
65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.3$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$
0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.5$ 65.95 $\pm$ 0.00 65.95
$\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$
65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.9$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$
0.00 13.37 $\pm$ 0.00 $[\lambda_{0}=1.0,\lambda_{1}=1.0]$ Linearized 87.17
$\pm$ 0.48 89.11 $\pm$ 0.37 2.41 $\pm$ 0.47 1.52 $\pm$ 0.41 LABEL:eq:h-opt,
$\delta=0.1$ 86.37 $\pm$ 0.00 88.03 $\pm$ 0.00 2.58 $\pm$ 0.00 1.43 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.3$ 86.37 $\pm$ 0.00 88.03 $\pm$ 0.00 2.58 $\pm$ 0.00
1.43 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.5$ 86.37 $\pm$ 0.00 88.03 $\pm$ 0.00
2.58 $\pm$ 0.00 1.43 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$ 86.37 $\pm$ 0.00
88.03 $\pm$ 0.00 2.58 $\pm$ 0.00 1.43 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.9$
86.37 $\pm$ 0.00 88.03 $\pm$ 0.00 2.58 $\pm$ 0.00 1.43 $\pm$ 0.00
$[\lambda_{0}=0.0,\lambda_{1}=0.0]$ Linearized -89.39 $\pm$ 0.43 -90.90 $\pm$
0.29 22.99 $\pm$ 0.40 22.81 $\pm$ 0.30 LABEL:eq:h-opt, $\delta=0.1$ 65.95
$\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt,
$\delta=0.3$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$
0.00 LABEL:eq:h-opt, $\delta=0.5$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37
$\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$ 68.28 $\pm$ 0.00
63.25 $\pm$ 0.00 14.16 $\pm$ 0.00 16.41 $\pm$ 0.00 LABEL:eq:h-opt,
$\delta=0.9$ 68.28 $\pm$ 0.00 63.25 $\pm$ 0.00 14.16 $\pm$ 0.00 16.41 $\pm$
0.00 $[\lambda_{0}=0.0,\lambda_{1}=1.0]$ Linearized 58.27 $\pm$ 2.18 60.52
$\pm$ 2.07 0.04 $\pm$ 0.03 0.02 $\pm$ 0.01 LABEL:eq:h-opt, $\delta=0.1$ 76.54
$\pm$ 0.00 77.55 $\pm$ 0.00 0.09 $\pm$ 0.00 0.05 $\pm$ 0.00 LABEL:eq:h-opt,
$\delta=0.3$ 76.54 $\pm$ 0.00 77.55 $\pm$ 0.00 0.09 $\pm$ 0.00 0.05 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.5$ 76.54 $\pm$ 0.00 77.55 $\pm$ 0.00 0.09 $\pm$ 0.00
0.05 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$ 76.54 $\pm$ 0.00 77.55 $\pm$ 0.00
0.09 $\pm$ 0.00 0.05 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.9$ 76.54 $\pm$ 0.00
77.55 $\pm$ 0.00 0.09 $\pm$ 0.00 0.05 $\pm$ 0.00

Table 4:  Performance of various LABEL:eq:h-opt policy candidates (with
different $\delta$) using $\text{IS}=$Weighed DR (WDR) estimator with standard
errors on 10 random splits of the TEST dataset. The red cells denote the
corresponding safety constraint violation, i.e, either
$\mathcal{J}_{0}^{\pi}<\mathcal{J}_{0}^{\pi_{b}}$ or
$-\mathcal{J}_{1}^{\pi}>-\mathcal{J}_{1}^{\pi_{b}}$.

User preferences $(\bm{\lambda})$ Policy Survival return ($\mathcal{J}_{0}$)
Rare-treatment return ($-\mathcal{J}_{1}$) DR WDR DR WDR Clinician’s
($\pi_{b}$) 64.78 $\pm$ 0.90 64.78 $\pm$ 0.90 13.58 $\pm$ 0.19 13.58 $\pm$
0.19 $[\lambda_{0}=1.0,\lambda_{1}=0.0]$ Linearized 97.68 $\pm$ 0.22 97.58
$\pm$ 0.20 27.64 $\pm$ 1.11 27.84 $\pm$ 1.09 LABEL:eq:h-opt, $\delta=0.1$
65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.3$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$
0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.5$ 65.95 $\pm$ 0.00 65.95
$\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$
65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.9$ 91.39 $\pm$ 0.00 92.61 $\pm$ 0.00 15.41 $\pm$
0.00 13.89 $\pm$ 0.00 $[\lambda_{0}=1.0,\lambda_{1}=1.0]$ Linearized 87.17
$\pm$ 0.48 89.11 $\pm$ 0.37 2.41 $\pm$ 0.47 1.52 $\pm$ 0.41 LABEL:eq:h-opt,
$\delta=0.1$ 86.37 $\pm$ 0.00 88.03 $\pm$ 0.00 2.58 $\pm$ 0.00 1.43 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.3$ 86.37 $\pm$ 0.00 88.03 $\pm$ 0.00 2.58 $\pm$ 0.00
1.43 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.5$ 86.37 $\pm$ 0.00 88.03 $\pm$ 0.00
2.58 $\pm$ 0.00 1.43 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$ 86.37 $\pm$ 0.00
88.03 $\pm$ 0.00 2.58 $\pm$ 0.00 1.43 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.9$
86.37 $\pm$ 0.00 88.03 $\pm$ 0.00 2.58 $\pm$ 0.00 1.43 $\pm$ 0.00
$[\lambda_{0}=0.0,\lambda_{1}=0.0]$ Linearized -89.39 $\pm$ 0.43 -90.90 $\pm$
0.29 22.99 $\pm$ 0.40 22.81 $\pm$ 0.30 LABEL:eq:h-opt, $\delta=0.1$ 65.95
$\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt,
$\delta=0.3$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$
0.00 LABEL:eq:h-opt, $\delta=0.5$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37
$\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$ 65.95 $\pm$ 0.00
65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$ 0.00 LABEL:eq:h-opt,
$\delta=0.9$ 65.95 $\pm$ 0.00 65.95 $\pm$ 0.00 13.37 $\pm$ 0.00 13.37 $\pm$
0.00 $[\lambda_{0}=0.0,\lambda_{1}=1.0]$ Linearized 58.27 $\pm$ 2.18 60.52
$\pm$ 2.07 0.04 $\pm$ 0.03 0.02 $\pm$ 0.01 LABEL:eq:h-opt, $\delta=0.1$ 76.54
$\pm$ 0.00 77.55 $\pm$ 0.00 0.09 $\pm$ 0.00 0.05 $\pm$ 0.00 LABEL:eq:h-opt,
$\delta=0.3$ 76.54 $\pm$ 0.00 77.55 $\pm$ 0.00 0.09 $\pm$ 0.00 0.05 $\pm$ 0.00
LABEL:eq:h-opt, $\delta=0.5$ 76.54 $\pm$ 0.00 77.55 $\pm$ 0.00 0.09 $\pm$ 0.00
0.05 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.7$ 76.54 $\pm$ 0.00 77.55 $\pm$ 0.00
0.09 $\pm$ 0.00 0.05 $\pm$ 0.00 LABEL:eq:h-opt, $\delta=0.9$ 76.54 $\pm$ 0.00
77.55 $\pm$ 0.00 0.09 $\pm$ 0.00 0.05 $\pm$ 0.00