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A Notes on notation and assumption
For any two real vectors  $x = (x_{1},\ldots ,x_{d})\in \mathbb{R}^{d}$  and  $y = (y_{1},\dots ,y_{d})\in \mathbb{R}^{d}$ ,  $\langle x,y\rangle$  denotes the inner product of  $x$  and  $y$ , i.e.,  $\langle x,y\rangle = \sum_{i = 1}^{d}x_{i}y_{i}$ . For any real vector  $x = (x_{1},\ldots ,x_{d})\in \mathbb{R}^{d}$ , let  $\| x\| _1$  denote the  $L_{1}$ -norm of  $x$ , i.e.,  $\| x\| _1 = \sum_{i = 1}^d |x_i|$ . For any real symmetric matrix  $X$ , let  $\operatorname {tr}(X)$  denote the trace of  $X$ . For any two real symmetric matrices  $X$  and  $Y$ , we denote  $X\preceq Y$  (or  $Y\succeq X$ ) if and only if  $Y - X$  is a positive semi-definite matrix. For any two real symmetric matrices  $X$  and  $Y$ ,  $X\bullet Y$  denotes the Frobenius inner product of  $X$  and  $Y$ , i.e.,  $X\bullet Y = \operatorname {tr}(AB)$ .
Without loss of generality, we may assume that  $\mathcal{A}$  spans all vectors in  $\mathbb{R}^d$ . Indeed, if not, we can make  $\mathcal{A}$  make full-dimensional by ignoring redundant dimensions. Under this assumption, distributions  $q$  given by exponential weight methods have non-singular variance-covariance matrices  $V(q)$  (and thus also  $H(q)$ ). This fact is used in some places in this paper without notice.
B Omitted proofs
B.1 Lemmas on log-concave distributions
Lemma 10 (Theorem 5.2 in [Lovasz and Vempala, 2007]). If  $X \in \mathbb{R}^n$  is a random variable following a log-concave distribution, for any constant matrix  $M \in \mathbb{R}^{m \times n}$ ,  $MX$  follows a log-concave distribution as well.
Lemma 11 (Lemmas 5.4 and 5.5 (a) in [Lovasz and Vempala, 2007]). If  $p$  is a one-dimensional log-concave distribution, we have
$$
\Pr_ {X \sim \alpha} [ X \leq \mu (p) ] \geq \frac {1}{\mathrm {e}}. \tag {16}
$$
Further, if  $V(p) = 1$ , we have
$$
\Pr_ {X \sim p} [ a \leq X \leq b ] \leq | b - a | \tag {17}
$$
for any  $a, b \in \mathbb{R}$ .
Lemma 12. For any one dimensional log-concave distribution  $p$  over a one-dimensional segment  $[L, R] \subseteq \mathbb{R}$ , we have
$$
V (p) \leq \mathrm {e} ^ {2} \min  \left\{\left(\mu (p) - L\right) ^ {2}, (R - \mu (p)) ^ {2} \right\}. \tag {18}
$$
Proof. From Lemma 11, we have
$$
\frac {1}{\mathrm {e}} \leq \Pr_ {X \sim p} [ X \leq \mu (p) ] = \Pr_ {X \sim p} \left[ \frac {L}{\sqrt {V (p)}} \leq \frac {X}{\sqrt {V (p)}} \leq \frac {\mu (p)}{\sqrt {V (p)}} \right] \leq \frac {\mu (p) - L}{\sqrt {V (p)}}, \tag {19}
$$
where the first and the second inequalities follow from (16) and (17), respectively. We hence have  $V(p) \leq \mathrm{e}^2 (\mu (p) - L)^2$ . By considering  $-X$  instead of  $X$ , we can show  $V(p) \leq \mathrm{e}^2 (R - \mu (p))^2$  in a similar way.
Lemma 13. Suppose  $\phi$  is defined as in (7). If  $y$  follows a log-concave distribution over  $\mathbb{R}$  and if  $s = \sqrt{\mathbf{E}[y^2]} \leq \sqrt{1/2}$ , we have
$$
\mathbf {E} [ \phi (y) ] \leq s ^ {2} + \frac {\exp \left(3 - s ^ {- 1}\right)}{1 - \exp \left(1 - s ^ {- 1}\right)} = O \left(s ^ {2}\right). \tag {20}
$$
Proof. See, e.g., the proof of [Ito et al., 2020, Lemma 5].
Proof. From the definition of the regret, we have
$$
\begin{array}{l} \underset {a ^ {*} \sim p ^ {*}} {\mathbf {E}} \left[ R _ {T} (a ^ {*}) \right] = \underset {a ^ {*} \sim p ^ {*}} {\mathbf {E}} \left[ \sum_ {t = 1} ^ {T} \underset {a \sim p _ {t}} {\mathbf {E}} \left[ \langle \ell_ {t}, a - a ^ {*} \rangle \right] \right] \\ = \underset {a ^ {*} \sim p ^ {*}} {\mathbf {E}} \left[ \sum_ {t = 1} ^ {T} \left(\underset {x \sim q _ {t}} {\mathbf {E}} \left[ \left\langle \ell_ {t}, x - a ^ {*} \right\rangle \right] + \left\langle \ell_ {t}, \mu \left(q _ {t}\right) - \mu \left(p _ {t}\right) \right\rangle\right) \right]. \tag {21} \\ \end{array}
$$
From the definition of bias given in (6), the part of  $\mathbf{E}_{x\sim q_t}[\langle \ell_t,x - a^*\rangle ]$  can be bounded as follows:
$$
\begin{array}{l} \mathbf {E} _ {x \sim q _ {t}} [ \langle \ell_ {t}, x - a ^ {*} \rangle ] \\ = \mathbf {E} _ {a _ {t}, f _ {t}} \left[ \mathbf {E} _ {x \sim q _ {t}} \left[ \langle \ell_ {t}, x - a ^ {*} \rangle - g _ {t} \left(a ^ {*}; a _ {t}, f _ {t}\right) + g _ {t} \left(x; a _ {t}, f _ {t}\right) \right] + \mathbf {E} _ {x \sim q _ {t}} \left[ g _ {t} \left(x; a _ {t}, f _ {t}\right) - g _ {t} \left(a ^ {*}; a _ {t}, f _ {t}\right) \right] \right] \\ \leq \operatorname {b i a s} \left(g _ {t}, p _ {t}, q _ {t}, \ell_ {t}\right) + \mathbf {E} _ {a _ {t}, f _ {t}} \left[ \mathbf {E} _ {x \sim q _ {t}} \left[ g _ {t} \left(x; a _ {t}, f _ {t}\right) - g _ {t} \left(a ^ {*}; a _ {t}, f _ {t}\right) \right] \right]. \tag {22} \\ \end{array}
$$
From this and (21), we have
$$
\begin{array}{l} \mathbf {E} _ {a ^ {*} \sim p ^ {*}} \left[ R _ {T} \left(a ^ {*}\right) \right] \leq \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \left(\left\langle \ell_ {t}, \mu \left(q _ {t}\right) - \mu \left(p _ {t}\right) \right\rangle + \operatorname {b i a s} \left(g _ {t}, p _ {t}, q _ {t}, \ell_ {t}\right)\right) \right] \\ \left. \right. + \underset {a ^ {*} \sim p ^ {*}} {\mathbf {E}} \left[ \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \underset {x \sim q _ {t}} {\mathbf {E}} \left[ g _ {t} \left(x; a _ {t}, f _ {t}\right) - g _ {t} \left(a ^ {*}; a _ {t}, f _ {t}\right)\right]\right]\right]. \tag {23} \\ \end{array}
$$
By a standard analysis of the exponential weight method, we have
$$
\mathbf {E} _ {a ^ {*} \sim p ^ {*}} \left[ \sum_ {t = 1} ^ {T} \mathbf {E} _ {x \sim q _ {t}} \left[ g _ {t} \left(x; a _ {t}, f _ {t}\right) - g _ {t} \left(a ^ {*}; a _ {t}, f _ {t}\right) \right] \right] \leq \sum_ {t = 1} ^ {T} \frac {1}{\eta_ {t}} \operatorname {s t a b} \left(\eta_ {t} g _ {t} (\cdot ; a _ {t}, f _ {t}), q _ {t}\right) + \frac {1}{\eta_ {T + 1}} D _ {\mathrm {K L}} \left(p ^ {*}, q _ {0}\right). \tag {24}
$$
To show (24), we denote  $G_{t}(p) = \mathbf{E}_{a\sim p}[g_{t}(a;a_{t},f_{t})]$  in the following for notational simplicity. From (5), we have
$$
\begin{array}{l} \mathbf {E} _ {a ^ {*} \sim p ^ {*}} \left[ \sum_ {t = 1} ^ {T} g _ {t} \left(a ^ {*}; a _ {t}, f _ {t}\right) \right] + \frac {1}{\eta_ {T + 1}} D _ {\mathrm {K L}} \left(p ^ {*}, q _ {0}\right) \\ = \sum_ {t = 1} ^ {T} G _ {t} \left(p ^ {*}\right) + \frac {1}{\eta_ {T + 1}} D _ {\mathrm {K L}} \left(p ^ {*}, q _ {0}\right) \\ \geq \sum_ {t = 1} ^ {T} G _ {t} (q _ {T + 1}) + \frac {1}{\eta_ {T + 1}} D _ {\mathrm {K L}} (q _ {T + 1}, q _ {0}) \\ \geq \sum_ {t = 1} ^ {T - 1} G _ {t} \left(q _ {T + 1}\right) + \frac {1}{\eta_ {T}} D _ {\mathrm {K L}} \left(q _ {T + 1}, q _ {0}\right) + G _ {T} \left(q _ {T + 1}\right) \\ = \sum_ {t = 1} ^ {T - 1} G _ {t} \left(q _ {T}\right) + \frac {1}{\eta_ {T}} D _ {\mathrm {K L}} \left(q _ {T}, q _ {0}\right) + \frac {1}{\eta_ {T}} D _ {\mathrm {K L}} \left(q _ {T + 1}, q _ {T}\right) + G _ {T} \left(q _ {T + 1}\right) \\ \geq \dots \geq \sum_ {t = 1} ^ {T} \left(\frac {1}{\eta_ {t}} D _ {\mathrm {K L}} \left(q _ {t + 1}, q _ {t}\right) + G _ {t} \left(q _ {t + 1}\right)\right), \tag {25} \\ \end{array}
$$
where the first inequality follows from (5) and the second inequality follows from  $\eta_{T + 1}\leq \eta_T$  and the fact that the values of the KL divergence are nonnegative. The second equality follows from the definition of  $q_{T}(x)$ ; As  $q_{T}(x)$  can be expressed as  $q_{T}(x) = \kappa_{T}\exp (-\eta_{T}\tilde{g}_{T}(x))q_{0}(x)$  for some
$\kappa_{T} > 0$  , where we denote  $\tilde{g}_T(x) = \sum_{s = 1}^{T - 1}g_s(x;a_s,f_s)$  , we have
$$
\begin{array}{l} \sum_ {t = 1} ^ {T - 1} G _ {t} (q _ {T}) + \frac {1}{\eta_ {T}} D _ {\mathrm {K L}} (q _ {T}, q _ {0}) = \mathbf {E} _ {a \sim q _ {T}} \left[ \tilde {g} _ {T} (a) + \frac {1}{\eta_ {T}} \ln \frac {q _ {T} (a)}{q _ {0} (a)} \right] \\ = \mathbf {E} _ {a \sim q _ {T}} \left[ \tilde {g} _ {T} (a) + \frac {1}{\eta_ {T}} \left(\ln \kappa_ {T} - \eta_ {T} \tilde {g} _ {T} (x)\right) \right] = \frac {\ln \kappa_ {T}}{\eta_ {T}} \tag {26} \\ \end{array}
$$
$$
\begin{array}{l} \sum_ {t = 1} ^ {T - 1} G _ {t} \left(q _ {T + 1}\right) + \frac {1}{\eta_ {T}} D _ {\mathrm {K L}} \left(q _ {T + 1}, q _ {0}\right) = \mathbf {E} _ {a \sim q _ {T + 1}} \left[ \tilde {g} _ {T} (a) + \frac {1}{\eta_ {T}} \ln \frac {q _ {T + 1} (a)}{q _ {0} (a)} \right] \\ = \mathbf {E} _ {a \sim q _ {T + 1}} \left[ \tilde {g} _ {T} (a) + \frac {1}{\eta_ {T}} \ln \frac {q _ {T + 1} (a)}{q _ {T} (a)} + \frac {1}{\eta_ {T}} \ln \frac {q _ {T} (a)}{q _ {0} (a)} \right] \\ = \mathbf {E} _ {a \sim q _ {T + 1}} \left[ \tilde {g} _ {T} (a) + \frac {1}{\eta_ {T}} \ln \frac {q _ {T + 1} (a)}{q _ {T} (a)} + \frac {1}{\eta_ {T}} \left(\ln \kappa_ {T} - \eta_ {T} \tilde {g} _ {T} (x)\right) \right] \\ = \frac {\ln \kappa_ {T}}{\eta_ {T}} + \frac {1}{\eta_ {T}} \mathbf {E} _ {a \sim q _ {T + 1}} \left[ \ln \frac {q _ {T + 1} (a)}{q _ {T} (a)} \right] = \frac {\ln \kappa_ {T}}{\eta_ {T}} + D _ {\mathrm {K L}} \left(q _ {T + 1}, q _ {T}\right). \tag {27} \\ \end{array}
$$
These equalities together imply that the second equality in (25). The last inequality of (25) can be obtained by applying similar transformations repeatedly. From (25), we have
$$
\begin{array}{l} [ \text {L H S} (2 4) ] = \sum_ {t = 1} ^ {T} \left(G _ {t} \left(q _ {t}\right) - G _ {t} \left(p ^ {*}\right)\right) \\ \leq \sum_ {t = 1} ^ {T} \left(G _ {t} (q _ {t}) - G _ {t} (q _ {t + 1}) - \frac {1}{\eta_ {t}} D _ {\mathrm {K L}} (q _ {t + 1}, q _ {t})\right) + \frac {1}{\eta_ {T + 1}} D _ {\mathrm {K L}} (p ^ {*}, q _ {0}) \\ \leq \sum_ {t = 1} ^ {T} \frac {1}{\eta_ {t}} \operatorname {s t a b} \left(\eta_ {t} g _ {t} (\cdot ; a _ {t}, f _ {t}), q _ {t}\right) + \frac {1}{\eta_ {T + 1}} D _ {\mathrm {K L}} \left(p ^ {*}, q _ {0}\right), \tag {28} \\ \end{array}
$$
which implies that (24) holds. By combining (23) and (24), we obtain the bound of (6).
B.3 Proof of Theorem 1
Proof. For any  $a^* \in \mathcal{X}$ , let  $p^*$  be the uniform distribution over  $\{(1 - \frac{1}{T})a^* + \frac{1}{T}x \mid x \in \mathcal{X}\}$ . We then have  $R_T(a^*) \leq \mathbf{E}_{a \sim p^*}[R_T(a)] + 1$  and  $D_{\mathrm{KL}}(p^*, q_1) \leq d\log T$ . Define  $\eta_t$  by  $\eta_t = \min \left\{\eta_0, \sqrt{d\log T} / \sqrt{\alpha + \sum_{s=1}^{t-1}\eta_s^{-1}\Lambda_{q_s,\eta_s}^*}\right\}$ . Suppose that  $p_t$  and  $g_t$  are chosen so that  $\Lambda_{q_t,\eta_t}(p_t, g_t) = \Lambda_{q_t,\eta_t}^*$ . As we have  $\eta_t \leq \eta_0$ , from the assumption, we have  $\eta_t^{-1}\Lambda_{q_t,\eta_t} \leq \alpha$ . Hence, from (8), we have
$$
\begin{array}{l} R _ {T} \leq \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \Lambda_ {q _ {t}, \eta_ {t}} ^ {*} + \frac {d \log T}{\eta_ {T + 1}} \right] + 1 \leq \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \frac {\sqrt {d \log T} \eta_ {t} ^ {- 1} \Lambda_ {q _ {t} , \eta_ {t}} ^ {*}}{\sqrt {\alpha + \sum_ {s = 1} ^ {t - 1} \eta_ {s} ^ {- 1} \Lambda_ {q _ {s} , \eta_ {s}} ^ {*}}} + \frac {d \log T}{\eta_ {T + 1}} \right] + 1 \\ \leq 3 \mathbf {E} \left[ \sqrt {d \log T \left(\alpha + \sum_ {t = 1} ^ {T} \eta_ {t} ^ {- 1} \Lambda_ {q _ {t} , \eta_ {t}} ^ {*}\right)} \right] + \frac {d \log T}{\eta_ {0}} + 1 \\ \leq 3 \mathbf {E} \left[ \sqrt {d \log T \left(\alpha + \min  \left\{\alpha T , \beta \sum_ {t = 1} ^ {T} \Delta_ {\ell^ {*}} (\mu (q _ {t})) \right\}\right)} \right] + \frac {d \log T}{\eta_ {0}} + 1 =: R _ {T} ^ {\prime}, \tag {29} \\ \end{array}
$$
which implies the regret bound of  $R_{T} = O\left(\sqrt{\alpha d T \log T} + \frac{d \log T}{\eta_{0}}\right)$  that holds in adversarial environments. We next show the improved regret bounds for stochastically constrained adversarial environments. From the definition of  $\Lambda$ , we have
$$
\Lambda_ {q _ {t}, \eta_ {t}} ^ {*} = \Lambda_ {q _ {t}, \eta_ {t}} \left(p _ {t}, g _ {t}\right) \geq \sup  _ {\ell \in \mathcal {L}} \left\{\langle \ell , \mu \left(p _ {t}\right) - \mu \left(q _ {t}\right) \rangle \right\} \geq \langle \ell^ {*}, \mu \left(q _ {t}\right) - \mu \left(p _ {t}\right) \rangle , \tag {30}
$$
where the last inequality holds since  $-\ell^{*}\in \mathcal{L}$  for any  $\ell^{*}\in \mathcal{L}$ , which follows from the definition of  $\mathcal{L}$ . We then, hence, have
$$
\begin{array}{l} R _ {T} ^ {\prime \prime} := \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \Delta_ {\ell^ {*}} (\mu (q _ {t})) \right] = \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \left(\Delta_ {\ell^ {*}} (\mu (p _ {t})) + \langle \ell^ {*}, \mu (q _ {t}) - \mu (p _ {t}) \rangle\right) \right] \\ \leq \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \Delta_ {\ell^ {*}} (\mu (p _ {t})) \right] + \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \Lambda_ {q _ {t}, \eta_ {t}} ^ {*} \right] \leq R _ {T} + C + R _ {T} ^ {\prime} \leq 2 R _ {T} ^ {\prime} + C. \tag {31} \\ \end{array}
$$
From this, we have
$$
R _ {T} ^ {\prime} \leq 3 \mathbf {E} \left[ \sqrt {d \log T (\alpha + \beta R _ {T} ^ {\prime \prime})} \right] + \frac {d \log T}{\eta_ {0}} + 1 \leq 3 \mathbf {E} \left[ \sqrt {d \log T (\alpha + \beta (2 R _ {T} ^ {\prime} + C))} \right] + \frac {d \log T}{\eta_ {0}} + 1
$$
which implies that  $R_{T} = O\left(\left(\beta + \frac{1}{\eta_{0}}\right)d\log T + \sqrt{C\beta d\log T}\right)$  holds.
B.4 Proof of Lemma 2
Proof. We first see that  $\Lambda_{q,\eta}^{*} = O(1)$  holds for any  $q\in \mathcal{P}(\mathcal{X})$  and  $\eta >0$ . In fact, by setting  $p = q$  and letting  $g\in \mathcal{G}$  to be the zero function  $(g(x;a,f) = 0$  for all  $x,a$  and  $f)$ , we obtain  $\Lambda_{q,\eta}(p,g) = \sup_{\ell \in \mathcal{L}}\mathrm{bias}(g,p,q,\ell) = \sup_{\ell \in \mathcal{L},x\in \mathcal{X},a^{*}\in \mathcal{X}}\langle \ell ,x - a^{*}\rangle \leq 2$ .
Let us consider the case in which  $\eta \leq 1 / d$  holds. As  $\Lambda_{q,\eta}^{*} = O(1)$  for any  $\eta$  and  $q$ , we have  $\Lambda_{q,\eta}^{*} = O(\eta \omega(q))$  if  $\eta \omega(q) = \Omega(1)$ . In the following, we assume  $\eta \omega(q) \leq 1 / 8$ . Let  $\tilde{p} \in \mathcal{P}(\mathcal{X})$  be a distribution such that  $\mu(\tilde{p}) = \mu(q)$  and  $w(q) = H(\tilde{p})^{-1} \bullet V(q)$ . Let  $\pi_0 \in \mathcal{P}(\mathcal{X})$  be a distribution such that
$$
a ^ {\top} H \left(\pi_ {0}\right) ^ {- 1} a \leq d \tag {32}
$$
holds for any  $a \in \mathcal{A}$ . Note that such a distribution always exists (see, e.g., [Lattimore and Szepesvári, 2020a, Theorem 21.1]). Define  $p$  by
$$
p = (1 - \gamma) \tilde {p} + \gamma \pi_ {0} \tag {33}
$$
with  $\gamma = 4\eta \omega(q) \leq 1/2$ . With this  $p$ , we define  $g$  by (4). Then, as we have  $\mathrm{bias}(g, p, q, \ell) = 0$ , we have
$$
\Lambda_ {q, \eta} (p, g) \leq \sup  _ {\ell \in \mathcal {A}} \left\{\langle \ell , \mu (p) - \mu (q) \rangle \right\} + \frac {1}{\eta} \sup  _ {f \in [ - 1, 1 ]} \mathbf {E} _ {a \sim p} [ \operatorname {s t a b} (\eta g (\cdot ; a, f), q) ]. \tag {34}
$$
As the definition of  $p$  implies  $\mu(p) = (1 - \gamma)\mu(\tilde{p}) + \gamma\mu(\pi_0) = (1 - \gamma)\mu(q) + \gamma\mu(\pi_0)$ , it holds for any  $\ell \in \mathcal{L}$  that
$$
\langle \ell , \mu (p) - \mu (q) \rangle = \langle \ell , (1 - \gamma) \mu (q) + \gamma \mu (\pi_ {0}) - \mu (q) \rangle = \gamma \langle \ell , \gamma \mu (\pi_ {0}) - \mu (q) \rangle \leq 2 \gamma . \tag {35}
$$
Further, from (7) and (4), we have
$$
\begin{array}{l} \operatorname {s t a b} (\eta g (\because a, f), q) \leq \mathbf {E} _ {x \sim q} [ \phi (\eta g (x; a, f) - \eta g (\mu (q); a, f)) ] \\ = \mathbf {E} _ {x \sim q} \left[ \phi (\eta f \cdot \langle (H (p)) ^ {- 1} a, x - \mu (q) \rangle) \right] \tag {36} \\ \end{array}
$$
for any fixed  $a \in \mathcal{A}$  and  $f \in [-1, 1]$ . Let  $r$  represent the distribution of  $f \cdot \left\langle (H(p))^{-1} a, x - \mu(q) \right\rangle$  for  $x \sim q$  with any fixed  $a \in \mathcal{A}$  and  $f \in [-1, 1]$ . Then, as  $q$  is a log-concave distribution, from Lemma 10,  $r$  is a log-concave distribution as well. Further, we have
$$
\begin{array}{l} H (r) = \underset {y \sim r} {\mathbf {E}} [ y ^ {2} ] = f ^ {2} \underset {x \sim q} {\mathbf {E}} \left[ \left(\left\langle (H (p)) ^ {- 1} a, x - \mu (q) \right\rangle\right) ^ {2} \right] \\ = f ^ {2} a ^ {\top} (H (p)) ^ {- 1} \underset {x \sim q} {\mathbf {E}} \left[ (x - \mu (q)) (x - \mu (q)) ^ {\top} \right] (H (p)) ^ {- 1} a \\ = f ^ {2} a ^ {\top} (H (p)) ^ {- 1} V (q) (H (p)) ^ {- 1} a \leq a ^ {\top} (H (p)) ^ {- 1} V (q) (H (p)) ^ {- 1} a. \tag {37} \\ \end{array}
$$
Since we have
$$
\begin{array}{l} \operatorname {t r} (H (p) ^ {- 1 / 2} V (q) H (p) ^ {- 1 / 2}) = V (q) \bullet H (p) ^ {- 1} = V (q) \bullet ((1 - \gamma) H (\tilde {p}) + \gamma H (\pi_ {0})) ^ {- 1} \\ \leq V (q) \bullet ((1 - \gamma) H (\tilde {p})) ^ {- 1} = \frac {1}{1 - \gamma} \omega (q), \tag {38} \\ \end{array}
$$
$$
H (p) ^ {- 1 / 2} V (q) H (p) ^ {- 1 / 2} \preceq \frac {\omega (q)}{1 - \gamma} I _ {d} \tag {39}
$$
where  $I_{d}$  represents the identity matrix of size  $d$ . Hence, we have
$$
\begin{array}{l} H (r) \leq a ^ {\top} (H (p)) ^ {- 1 / 2} \left(\frac {\omega (q)}{1 - \gamma} I _ {d}\right) (H (p)) ^ {- 1 / 2} a \leq \frac {\omega (q)}{1 - \gamma} a ^ {\top} (H (p)) ^ {- 1} a \\ = \frac {\omega (q)}{1 - \gamma} a ^ {\top} ((1 - \gamma) H (\tilde {p}) + \gamma H (\pi_ {0})) ^ {- 1} a \leq \frac {\omega (q)}{1 - \gamma} a ^ {\top} (\gamma H (\pi_ {0})) ^ {- 1} a \\ = \frac {\omega (q)}{(1 - \gamma) \gamma} a ^ {\top} H \left(\pi_ {0}\right) ^ {- 1} a \leq \frac {\omega (q) d}{(1 - \gamma) \gamma} \leq 2 \frac {\omega (q) d}{\gamma} = \frac {d}{2 \eta}. \tag {40} \\ \end{array}
$$
This implies that  $\eta^2 H(r) \leq d\eta / 2 \leq 1 / 2$ . Hence, from Lemma 13, we have
$$
\operatorname {s t a b} \left(\eta g (\because a, f), q\right) \leq \mathbf {E} _ {y \sim r} [ \phi (\eta y) ] = O \left(\eta^ {2} H (r)\right) = O \left(\eta^ {2} a ^ {\top} (H (p)) ^ {- 1} V (q) (H (p)) ^ {- 1} a\right). \tag {41}
$$
$$
\begin{array}{l} \sup  _ {f \in [ - 1, 1 ]} \mathbf {E} _ {a \sim p} \left[ \operatorname {s t a b} \left(\eta g (\cdot ; a, f), q\right) \right] = O \left(\eta^ {2} \mathbf {E} _ {a \sim p} \left[ a ^ {\top} (H (p)) ^ {- 1} V (q) (H (p)) ^ {- 1} a \right]\right) \\ = O \left(\eta^ {2} \operatorname {t r} \left(\mathbf {E} _ {a \sim p} \left[ a a ^ {\top} \right] (H (p)) ^ {- 1} V (q) (H (p)) ^ {- 1}\right)\right) = O \left(\eta^ {2} V (q) \bullet (H (p)) ^ {- 1}\right) \\ \leq O \left(\frac {\eta^ {2}}{1 - \gamma} V (q) \bullet (H (\tilde {p})) ^ {- 1}\right) = O \left(\frac {\eta^ {2} \omega (q)}{1 - \gamma}\right) = O \left(\eta^ {2} \omega (q)\right). \tag {42} \\ \end{array}
$$
Combining this with (34) and (35), we obtain
$$
\Lambda_ {q, \eta} (p, g) \leq 2 \gamma + \frac {1}{\eta} \sup  _ {f \in [ - 1, 1 ]} \mathbf {E} _ {a \sim p} [ \operatorname {s t a b} (\eta g (\because a, f), q) ] = O (\gamma + \eta \omega (q)) = O (\eta \omega (q)), \tag {43}
$$
which implies that  $\Lambda_{q,\eta}^{*} = O(\eta \omega (q))$
We next consider the case in which  $\eta \leq 1$  holds. As we have  $\Lambda_{q,\eta}^{*} = O(\eta \omega'(q))$  if  $\eta \omega'(q) > 1/8$  (since  $\Lambda_{q,\eta}^{*} = O(1)$  for any  $q$  and  $\eta$ ), we assume  $\eta \omega'(q) \leq 1/8$ . We set  $p$  by (33) where  $\tilde{p}$  is the minimizer in the right-hand side of (12) and  $\gamma = 4\eta \omega'(q) \leq 1/2$ . We set  $g$  by (4). Then, (34) and (35) can be shown in a similar way to in the first half of this proof. As we have
$$
d V (q) \preceq \omega^ {\prime} (q) H (\tilde {p}) \preceq \frac {\omega^ {\prime} (q)}{1 - \gamma} H (p), \tag {44}
$$
$$
H (p) ^ {- 1 / 2} V (q) H (p) ^ {- 1 / 2} \preceq \frac {\omega^ {\prime} (q)}{(1 - \gamma) d} I _ {d}. \tag {45}
$$
From this, if we define  $r$  in a similar way to in the first half of this proof, we have  $H(r) \leq 1 / (2\eta)$ . We hence have  $\eta^2 H(r) \leq \eta / 2 \leq 1 / 2$  under the assumption of  $\eta \leq 1$ . We can then apply Lemma 13 to obtain the following:
$$
\begin{array}{l} \sup  _ {f \in [ - 1, 1 ]} \mathbf {E} _ {a \sim p} \left[ \operatorname {s t a b} \left(\eta g (\cdot ; a, f), q\right) \right] \leq O \left(\frac {\eta^ {2}}{1 - \gamma} V (q) \bullet \left(H (\tilde {p})\right) ^ {- 1}\right) \\ \leq O \left(\frac {\eta^ {2}}{1 - \gamma} \frac {\omega^ {\prime} (q)}{d} H (\tilde {p}) \bullet \left(H (\tilde {p})\right) ^ {- 1}\right) = O \left(\frac {\eta^ {2} \omega^ {\prime} (q)}{1 - \gamma}\right) = O \left(\eta^ {2} \omega^ {\prime} (q)\right), \tag {46} \\ \end{array}
$$
where the first inequality follows from a similar argument as in (42). Combining this with (34), (35), and  $\gamma = 4\eta \omega'(q)$ , we obtain  $\Lambda_{q,\eta}(p,g) = O(\eta \omega'(q))$ , which implies  $\Lambda_{q,\eta}^* = O(\eta \omega'(q))$ .
B.5 Proof of Lemma 3
Proof of Lemma 3. We will show that
$$
\text {t h e r e} \quad p \in \mathcal {P} (\mathcal {X}) \quad \text {s u c h t h a t} \quad \mu (p) = \mu (q) \quad \text {a n d} \quad V (q) \preceq z \cdot V (p) \tag {47}
$$
with  $z = O(1, \Delta_{\ell^{*}}(\mu(q)) / \Delta_{\ell^{*},\min})$ . As we have  $H(p) = V(p) + \mu(p)\mu(p)^{\top} \succeq V(p)$ , (47) leads to  $d \cdot V(q) \preceq dz \cdot V(p) \preceq dz \cdot H(p)$ , which implies  $\omega'(q) \leq dz$ . Hence, to prove the lemma, it is sufficient to show that (47) holds with  $z = O(1, \Delta_{\ell^{*}}(\mu(q)) / \Delta_{\ell^{*},\min})$ . By considering the case of  $p = q$ , we can easily see that (47) holds with  $z = O(1)$ . For any  $q$ . In the following, we show that (47) holds with  $z = O\left(\frac{\Delta_{\ell^{*}}(\mu(q))}{\Delta_{\ell^{*},\min}}\right)$ .