SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Corollary C.7. Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt and assume that the optimal value  $p^{\star}$  is finite. Then there exists a solution  $(\gamma^{\star}, \phi^{\star})$  to DualOpt such that  $p^{\star} = g(\gamma^{\star}, \phi^{\star})$ .
Proof. This follows from the proof of Theorem C.3. When  $p^{\star}$  is finite, the parameter  $\tilde{\nu} >$  and, hence,  $\gamma^{\star} = \tilde{\gamma} / \tilde{\nu}, \phi^{\star} = \tilde{\phi} / \tilde{\nu}$  as defined in the proof of this result satisfy the property that  $g(\gamma^{\star}, \phi^{\star}) = p^{\star}$ .
We would now like to prove that, assuming that the optimal primal value is finite, the optimal dual variable  $\gamma^{\star}$  is non-zero. This allows us to write an explicit expression for an optimal solution to PrimalOpt. We first need to understand the properties of the following integral, which was defined in assumption (A5):
$$
I _ {f _ {\mathcal {D}}, \ell , \alpha} (x) = \int_ {\Omega} \ell_ {\alpha} (x, v) f _ {\mathcal {D}} (v) d \mu (v).
$$
When the parameters  $f_{\mathcal{D}}, \ell, \alpha$  will be clear from the context, we shall often abbreviate  $I_{f_{\mathcal{D}}, \ell, \alpha}(x)$  as  $I(x)$ .
C.4 Properties of the integral  $I(x)$
We study some of the key properties of the integral  $I(x)$ . We shall assume that assumptions (A0)-(A4) hold. The integral  $I(x)$  can be split into two parts:
$$
I (x) = \underbrace {\alpha \int_ {\Omega \cap (- \infty , x ]} \ell (x , v) f _ {\mathcal {D}} (v) d \mu (v)} _ {:= I ^ {L} (x)} + \underbrace {\int_ {\Omega \cap [ x , \infty ) ]} \ell (x , v) f _ {\mathcal {D}} (v) d \mu (v)} _ {:= I ^ {R} (x)}.
$$
Lemma C.8. The integral  $I^{L}(x)$  is a strictly increasing function of  $x$ . Further,  $I^{L}(x_{2}) - I^{L}(x_{1}) \geq \alpha (\ln (x_{2} - x_{1}) - \theta_{x_{1}})F_{\mathcal{D}}(x_{1})$ , for all  $x_{2}, x_{1} \in \Omega$  satisfying  $x_{2} - x_{1} \geq C$ . Here,  $F_{\mathcal{D}}$  denotes the cumulative distribution function (c.d.f.) of  $f_{\mathcal{D}}$ .
Recall that the parameter  $\theta_{x_1}$  appears in the assumption (A3).
Proof. Consider  $x_1, x_2 \in \Omega$  with  $x_1 < x_2$ . Then
$$
I ^ {L} (x _ {2}) - I ^ {L} (x _ {1}) = \alpha \int_ {- \infty} ^ {x _ {1}} (\ell (x _ {2}, v) - \ell (x _ {1}, v)) f _ {\mathcal {D}} (v) d \mu (v) + \alpha \int_ {[ x _ {1}, x _ {2} ] \cap \Omega} \ell (x _ {2}, v) f _ {\mathcal {D}} (v) d \mu (v).
$$
By assumption (A1),  $\ell(x_2, v) > \ell(x_1, v)$  for all  $v \leq x_1$  and  $\ell(x_2, v) \geq 0$  for all  $v \leq x_2$ , we see that  $I^L(x_2) > I^L(x_1)$ . Suppose  $x_2 - x_1 \geq C$ . Then  $\ell(x_2, v) - \ell(x_1, v) \geq \ln |x_2 - x_1| - \theta_{x_1}$ . Therefore, using A3, the first integral above is at least
$$
\alpha (\ln (| x _ {2} - x _ {1} | - \theta_ {x _ {1}}) \int_ {- \infty} ^ {x _ {1}} f _ {\mathcal {D}} (v) d \mu (v) = \alpha (\ln (x _ {2} - x _ {1}) - \theta_ {x _ {1}}) F _ {\mathcal {D}} (x _ {1}).
$$
Lemma C.9. The integral  $I^R(x)$  is a strictly increasing function of  $x$  when the loss function is TYPEN, and is a strictly decreasing function of  $x$  when the loss function is TYPEP.
$$
I ^ {R} (x _ {2}) - I ^ {R} (x _ {1}) = - \int_ {x _ {1}} ^ {x _ {2}} \ell (x _ {1}, v) f _ {\mathcal {D}} (v) d \mu (v) + \int_ {[ x _ {2}, \infty) \cap \Omega} (\ell (x _ {2}, v) - \ell (x _ {1}, v)) f _ {\mathcal {D}} (v) d \mu (v).
$$
First, consider the case of TYPEN loss function. Then for all  $v \geq x_1$ ,  $\ell(x_1, v) \leq \ell(v, v) = 0$ , and hence, the first integrand above is positive. We also know that for all  $v$ ,  $\ell(x_2, v) \geq \ell(x_1, v)$ , and hence, the second integrand above is also positive. This shows that  $I^R(x)$  is an increasing function of  $x$  when the loss function is TYPEN.
Now consider the case when the loss function is TYPEP. For any value  $v \geq x_1, \ell(x_1, v) \geq \ell(v, v) = 0$ . Similarly, for  $v \geq x_2, \ell(x_2, v) < \ell(x_1, v)$ . Thus,  $I^R(x_2) < I^R(x_1)$  in this case.
Combining Lemma C.8 and Lemma C.9, we get the main result about the variation of  $I(x)$ .
Theorem C.10 (Monotonicity of  $I(x)$  with respect to  $x$ ). Assume that conditions (A0)-(A4) hold. The function  $I(x)$  is a continuous function of  $x$ . For a TYPEN loss function,  $I(x)$  is a monotonically strictly increasing function of  $x$ , with  $I(x)$  going to  $-\infty$  as  $x$  goes to  $-\infty$  (assuming  $\Omega$  is unbounded from below). For a TYPEP loss function,  $I(x)$  is the sum of an increasing and a decreasing function, and has a global minimum on  $\Omega$ . Further,  $I(x)$  goes to  $\infty$  as  $x$  goes to  $\infty$  or  $-\infty$  (assuming these values lie in  $\Omega$ ).
Proof. First, consider the case of TYPEN loss functions. It follows from Lemma C.8 and Lemma C.9 that  $I(x)$  is a strictly increasing function of  $x$ . The integrand in  $I^{L}(x)$  is  $\mathbf{I}[x \geq v]\ell(x,v)f_{\mathcal{D}}(v)$ , where  $\mathbf{I}[]$  denotes the indicator function. As shown in the proof of Lemma C.8, this is a monotonically decreasing function of  $x$ . Therefore, the monotone convergence theorem [126] implies that  $I^{L}(x)$  goes to 0 as  $x$  goes to  $-\infty$ . Similarly,  $I^{R}(x)$  goes to  $-\infty$  as  $x$  goes to  $-\infty$ . Similarly, in the case of TYPEP loss function,  $I^{L}(x)$  goes to 0 and  $I^{R}(x)$  goes to  $\infty$  as  $x$  goes to  $-\infty$ . Thus,  $I(x)$  goes to  $\infty$ . Similarly,  $I(x)$  goes to  $\infty$  as  $x$  goes to  $\infty$ . Since  $I(x)$  is the sum of an increasing and decreasing function, and is infinite as  $x$  goes to  $\infty$  or  $-\infty$ , it must have a global minimum on  $\Omega$ .
Corollary C.11. Consider an instance  $\mathcal{I} \coloneqq (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt. If the loss function is TYPEP, then the optimal value is always finite. If the loss function is TYPEN and the optimal value is finite, then  $\Omega$  is of the form  $[a, \infty)$  or  $[a, b]$  for some  $a, b \in \mathbb{R}$ .
Proof. First, consider the case when the loss function is TYPEP. We exhibit a solution  $f$  with finite objective value. Let  $f$  be the uniform distribution over an interval  $A$  of length  $e^{\tau}$  (by assumption (A0), such an interval always exists). Since  $I(x)$  is continuous, it achieves a maximum value on  $A$  - let this value be  $p$ . Then  $\mathrm{Err}_{\ell, \alpha}(f_{\mathcal{D}}, f) \leq p$ . Thus, we see that the optimal value for this instance is finite.
Now we prove the second assertion. Suppose, for the sake of contradiction, that the loss function is  $\mathrm{TYPEN}$  and  $\Omega$  is unbounded from below. We claim that the optimal value for this instance is  $-\infty$ , which will be a contradiction. To see this, consider a density  $f \in \mathrm{Dom}$  which is uniform on an interval  $A = [s, t]$  of length  $e^{\tau}$  (by assumption (A0)). The entropy of this density is  $\tau$  and, hence, this is a feasible solution to the instance  $\mathcal{I}$ . However,
$$
\operatorname {E r r} _ {\ell , \alpha} (f _ {\mathcal {D}}, f) = \int_ {\Omega} I (x) f (x) d \mu (x) = \frac {1}{e ^ {\tau}} \int_ {A} I (x) d \mu (x) \leq I (t).
$$
Thus, we can keep moving the interval  $A$  to the left, which would mean that  $I(t)$  would tend to  $-\infty$  (using Theorem C.10). This shows that the optimal value of this instance is  $-\infty$ .
We shall use the following fact about the finiteness of optimal value when condition (A5) is also satisfied.
Claim C.12. Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt satisfying conditions (A0)-(A5). Then the optimal value is always finite.
Proof. If the loss function is TYPEP, this follows from Corollary C.11. In the case of a TYPEN loss function, the optimal value is finite unless  $\Omega$  is unbounded from below. In this case, Theorem C.10 shows that  $\operatorname{argmin}_{x\in \Omega}I(x)$  does not exist, and hence, assumption (A5) is violated.
C.5 Positivity of the optimal dual variable
We now prove that the optimal dual variable  $\gamma^{\star}$  is strictly positive. In this section, we assume that assumptions (A0)-(A5) hold. For this, we need certain technical results.
Lemma C.13. Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt with the loss function being TYPEP. Let  $x^{\star}$  be  $\operatorname*{argmin}_{x \in \Omega} I(x)$ . Then there is a value  $\eta > 0$ , such that any feasible solution  $f$  must have  $\mathrm{Err}_{\ell, \alpha}(f_{\mathcal{D}}, f) \geq I(x^{\star}) + \eta$ .
Proof. Let  $x^{\star}$  denote the unique global minimum of  $I(x)$  on  $\Omega$  (using assumption (A5)). We show that there is a small enough value  $\varepsilon > 0$  such that any feasible solution  $f$  must place at least  $\varepsilon$  amount of probability mass outside  $I_{\varepsilon} \coloneqq [x^{\star} - \varepsilon, x^{\star} + \varepsilon]$ . This suffices for the following reason. We know by the assumption (A5) that there is a positive value  $\zeta$  such that  $I(x) > I(x^{\star}) + \zeta$  for all  $x \notin I_{\varepsilon}$ . Thus,
$$
\begin{array}{l} \operatorname {E r r} _ {\ell , \alpha} (f _ {\mathcal {D}}, f) = \int_ {\Omega \cap I _ {\varepsilon}} I (x) f (x) d \mu (x) + \int_ {\Omega \setminus I _ {\varepsilon}} I (x) f (x) d \mu (x) \\ \geq I (x ^ {\star}) + \varepsilon \zeta . \\ \end{array}
$$
Hence, we can choose the desired value  $\eta$  to be  $\varepsilon \zeta$ .
It remains to find such a  $\varepsilon$ . We consider such a value  $\varepsilon$  and assume that a feasible solution  $f$  places strictly more than  $1 - \varepsilon$  probability mass inside  $I_{\varepsilon}$ . We need one notation: For an interval  $A$ , define
$$
\mathsf {E n t} _ {A} (f) := - \int_ {\Omega \cap A} f (x) \ln f (x) d \mu (x).
$$
Let  $m$  be the mean of  $f_{\mathcal{D}}$  and let  $R$  be the half-radius of  $f_{\mathcal{D}}$ , i.e.,  $f_{\mathcal{D}}$  places at least half of its mass in the interval  $[m - R, m + R] \cap \Omega$ . Let  $A_0$  be a large enough interval containing  $x^{\star}$  and the interval  $[m - R, m + R]$  – assume that the end-points of  $A_0$  are at least  $A_0 / 2$  away from any point in  $[m - R, m + R]$ . We consider intervals of growing size around  $A_0$ . Let  $A_0 \coloneqq [s_0, t_0]$ . Let  $A_1$  be the union of the intervals on both sides of  $A_0$ , each of length  $e^{2 / \varepsilon}$ . Similarly, having defined  $A_{i-1}$  (which will be a union of two intervals), define  $A_i$  to be the union of two intervals on both sides of  $A_{i-1}$ , each of length  $e^{2^i}$ .
Consider the feasible solution  $f$ . Let  $\beta_{i}$  be the total probability mass placed by  $f$  on  $A_{i}$ . Thus,
$$
\operatorname {E n t} (f) = \operatorname {E n t} _ {I _ {\varepsilon}} (f) + \operatorname {E n t} _ {A _ {0} \backslash I _ {\varepsilon}} (f) + \sum_ {i \geq 1} \operatorname {E n t} _ {A _ {i}} (f).
$$
We bound each of the terms above. Note that  $\mathsf{Ent}_{A_i}(f)$  is maximized when we distribute  $\beta_{i}$  mass uniformly over  $A_{i}$ . Similarly,  $\mathsf{Ent}_{I_{\varepsilon}}(f)$  is at most  $\ln (\varepsilon)$  and  $\mathsf{Ent}_{A_0\backslash I_\varepsilon}\leq \ln |A_0| = \ln D$ . Thus, we see that
$$
\tau \leq \mathsf {E n t} (f) \leq \ln (\varepsilon D) + \sum_ {i \geq 1} \beta_ {i} \ln \frac {| A _ {i} |}{\beta_ {i}} \leq \ln (\varepsilon D) + \sum_ {i \geq 1} \beta_ {i} \ln | A _ {i} |.
$$
$$
\sum_ {i \geq 1} \beta_ {i} \ln | A _ {i} | \geq \tau - \ln (\varepsilon D).
$$
Observe that we can still choose  $\varepsilon$  and, hence, we will choose it such that the r.h.s. above becomes as large as we want (ideally, much larger than  $p^{\star}$ ). Observe that any point in  $A_{i}$  is at least  $|A_{i - 1}|$
distance away from any point in  $[m - R, m + R]$ . Therefore, (and this is where we use non-negativity of the loss function, which has been assumed to be TYPEP)
$$
\operatorname {E r r} _ {\ell , \alpha} (f _ {\mathcal {D}}, f) \geq \sum_ {i} \frac {\beta_ {i}}{2} (\ln | A _ {i - 1} | - \theta_ {x}) \geq \sum_ {i} \frac {\beta_ {i} \ln | A _ {i} |}{4} - \theta_ {x},
$$
where we have used the fact that  $|\ln |A_i| = 2\ln |A_{i - 1}|$ . It follows from the above two inequalities that if we choose  $\varepsilon = \left(\frac{e^{\tau}}{4\theta_xp^\star D}\right)^4$ , then  $\mathrm{Err}_{\ell ,\alpha}(f_{\mathcal{D}},f)\geq 2p^{\star}$ . Hence, either  $f$  places less than  $1 - \varepsilon$  mass inside  $I_{\varepsilon}$  or its objective function value is more than  $2p^{\star}$ .
The above proof only worked for TYPEP loss functions. We need a similar result for TYPEN loss functions. The ideas are similar, but we need to use the function  $I(x)$  in a more subtle manner.
Lemma C.14. Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt with the loss function being  $\mathrm{TPEN}$ . Let  $x^{\star}$  be  $\operatorname*{argmin}_{x \in \Omega} I(x)$ . Then there is a value  $\eta > 0$  such that any feasible solution  $f$  must have  $\mathrm{Err}_{\ell, \alpha}(f_{\mathcal{D}}, f) \geq I(x^{\star}) + \eta$ .
Proof. As in the proof of Lemma C.13, we would like to argue that there is a positive  $\varepsilon > 0$  such that any feasible solution places at least  $\varepsilon$  amount of mass outside the interval  $I_{\varepsilon}$ . Since the optimum is finite, Corollary C.11 shows that  $\Omega$  is bounded from below, i.e., it is of the form  $[a, b]$  or  $[a, \infty)$ . Theorem C.10 now implies that  $x^{\star} = a$ .
Again, define the sets  $A_{i}$  as in the proof of Lemma C.13, but now, we make  $A_{0}$  start from the lower limit  $a$  of  $\Omega$ . Thus, each of the  $A_{i}$  will now be a single interval, with  $A_{i}$  being to the right of  $A_{i-1}$ . Now we use the fact that when the loss function is TYPEN, both the functions  $I^{L}(x)$  and  $I^{R}(x)$  are monotonically increasing (Lemma C.8 and Lemma C.9). Further, for a point  $x \in A_{i}$ , Lemma C.8 shows that
$$
I ^ {L} (x) - I ^ {L} (m + R) \geq \frac {\alpha}{2} (\ln | A _ {i - 1} | - \theta_ {m + R}),)
$$
because the c.d.f. of  $f_{\mathcal{D}}$  at  $m + R$  is at least  $1/2$ . Hence,
$$
\int_ {A _ {i} \cap \Omega} (I ^ {L} (x) - I ^ {L} (x ^ {\star}) f (x) d \mu (x) \geq \frac {\alpha \beta_ {i}}{2} (\ln | A _ {i - 1} | - \theta_ {m + R}).
$$
Summing the above over all  $i \geq 1$ , and proceeding as in the proof of Lemma C.13, we see that by choosing a small enough  $\varepsilon$ , (here  $t_0$  denotes the right end-point of  $A_0$ )
$$
\sum_ {[ t _ {0}, \infty) \cap \Omega} I ^ {L} (x) f (x) d \mu (x) - I ^ {L} (x ^ {\star})
$$
can be made larger than a desired quantity  $Q$ , which depends only on the parameters of the input instance  $\mathcal{I}$ . Since  $I^R(x)$  is an increasing function of  $x$ , we also get
$$
\int_ {[ t _ {0}, \infty) \cap \Omega} I (x) f (x) d \mu (x) \geq I (x ^ {\star}) + Q.
$$
One remaining issue is that the integral on the l.h.s. does not include the terms for  $[a, t_0)$  (recall that  $I(x)$  can be negative). However, observe that  $I(x)$  is an increasing function of  $x$ , and hence  $I(x) \geq I(a)$  for all  $x$ . Therefore,
$$
\begin{array}{l} \operatorname {E r r} _ {\ell , \alpha} (f _ {\mathcal {D}}, f) = \int_ {\Omega} I (x) f (x) d \mu (x) \\ = \int_ {a} ^ {t _ {0}} I (x) f (x) d \mu (x) + \int_ {[ t _ {0}, \infty) \cap \Omega} I (x) f (x) d \mu (x) \\ \ge I (a) + I (a) + Q. \\ \end{array}
$$
Again, by choosing  $Q$  large enough, we can set the above to more than  $2I(a)$ .
As an immediate consequence of this result, we see that for an instance of PrimalOpt satisfying conditions (A0)-(A5), the optimal dual solution  $\gamma^{\star}$  is non-zero.
Theorem C.15. Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt satisfying conditions (A0)-(A5). Then there is an optimal dual solution  $(\gamma^{\star}, \phi^{\star})$  to DualOpt satisfying  $\gamma^{\star} > 0$ .
Proof. Let  $p^{\star}$  denote the optimal value of this instance (it is finite by Claim C.12). Corollary C.11 implies that  $I(x)$  has a (unique) global minimum  $x^{\star}$  in  $\Omega$ . Assume, for the sake of contradiction, that  $\gamma^{\star} = 0$ . We claim that  $g(\gamma^{\star}, \phi^{\star}) = I(x^{\star})$ . Indeed, consider a function  $f$  which places unit probability mass at  $x^{\star}$ . Then it is easy to verify that  $L(f, \gamma^{\star}, \phi^{\star}) = I(x^{\star})$ . However, Lemma C.13 and Lemma C.14 show that  $p^{\star} > I(x^{\star})$ , which is a contradiction (because by Theorem C.3, strong duality holds).
C.6 Optimality conditions and uniqueness of optimal solution
We now show that, under suitable conditions, there is an optimal solution to an instance of PrimalOpt.
Theorem C.16 (Optimality condition). Consider an instance  $\mathcal{I}$  of the optimization problem PrimalOpt defined on a space  $\Omega$ . Let  $\ell, \alpha, \tau, f_{\mathcal{D}}$  denote the loss function, risk-averseness parameter, resource-information parameter, and the density of the true utility respectively in  $\mathcal{I}$ . Assume that the optimal value for this instance is finite and assumptions (A0), (A3) hold. Let  $(\gamma^{\star}, \phi^{\star})$  be an optimal solution to the corresponding DualOpt with  $\gamma^{\star} > 0$ . Consider a function  $f^{\star} \in \mathrm{Dom}$  defined as follows:
$$
I _ {f _ {\mathcal {D}}, \ell , \alpha} (x) + \gamma^ {\star} (1 + \ln f ^ {\star} (x)) + \phi^ {\star} = 0, \quad \forall x \in \Omega . \tag {12}
$$
Then  $f^{\star}$  is an optimal solution to the instance  $\mathcal{I}$ .
Proof. Let  $p^{\star}$  denote the value of the optimal solution to  $\mathcal{I}$ . Since  $p^{\star}$  is finite, Theorem C.15 shows that there is an optimal dual solution satisfying  $\gamma^{\star} > 0$ . Recall that
$$
g \left(\gamma^ {\star}, \phi^ {\star}\right) := \min  _ {f \in \operatorname {D o m}} L \left(f, \gamma^ {\star}, \phi^ {\star}\right).
$$
Let  $f^{\star}$  be the function defined by (12)  $(f^{\star}(x)$  is well-defined since  $\gamma^{\star} > 0$ , this is where we need strict positivity of  $\gamma^{\star}$ . We argue that  $L(f^{\star}, \gamma^{\star}, \phi^{\star}) = g(\gamma^{\star}, \phi^{\star})$ .
Indeed, consider any other function  $h \in \mathrm{Dom}$ . Define a function  $\theta : [0, \infty) \to \mathbb{R}$  as follows:
$$
\theta (t) := L \left((1 - t) f ^ {\star} (x) + t h (x), \gamma^ {\star}, \phi^ {\star}\right) = L \left(f ^ {\star} (x) + t e (x), \gamma^ {\star}, \phi^ {\star}\right),
$$
where  $e(x) = h(x) - f^{\star}(x)$ . We first claim that  $\theta(t)$  is a convex function of  $t$ .
Claim C.17. The function  $\theta :[0,1]\to \mathbb{R}$  is a convex function.
Proof. We observe that  $\mathrm{Err}(f^{\star} + t\cdot e,f_{\mathcal{D}})$  is a linear function of  $t$ . The function  $\gamma (\tau -\mathsf{Ent}(f^{\star} + t\cdot e))$  is a convex function of  $t$ , and  $\phi \left(\int_{\Omega}(f(x) + te(x))d\mu (x) - 1\right)$  is a linear function of  $t$ . Therefore,  $\theta (t)$  which is the sum of these three functions, is a convex function of  $t$ .
We shall show that  $\left.\frac{d\theta(t)}{dt}\right|_{t = 0^{+}} = 0$ . Along with the convexity of  $\theta (t)$ , this implies that  $\theta (0)\leq \theta (1)$  and hence  $L(f^{\star},\gamma^{\star},\phi^{\star})\leq L(h,\gamma^{\star},\phi^{\star})$ . A routine calculation shows that  $\left.\frac{d\theta(t)}{dt}\right|_{t = 0^{+}}$  is equal to
$$
\int_ {\Omega} \left(I _ {f _ {\mathcal {D}}, \ell , \alpha} (x) + \gamma^ {\star} \left(1 + \ln f ^ {\star} (x)\right) + \phi^ {\star}\right) e (x) d \mu (x).
$$
Inequality (12) implies that the above expression is 0. This proves the desired result. Thus, we have shown that  $L(f^{\star}, \gamma^{\star}, \phi^{\star}) = g(\gamma^{\star}, \phi^{\star})$ , and therefore,  $f^{\star}$  is an optimal solution to the instance  $\mathcal{I}$ .
We now show the uniqueness of the optimal solution to PrimalOpt. Consider an instance of this optimization problem specified by  $\Omega$ , loss function  $\ell$ , density  $f_{\mathcal{D}}$  and parameters  $\alpha, \tau$ . Let  $\mathcal{F}$  denote the set of feasible densities, i.e.,
$$
\mathcal {F} := \left\{f: f \in \operatorname {D o m}, f \text {i s a d e n s i t y o n} \Omega \text {a n d} \int_ {\Omega} f (x) \ln f (x) d \mu (x) \leq - \tau \right\}.
$$
Proof. Let  $f_1, f_2 \in \mathcal{F}$  and  $\lambda$  be a parameter in  $[0, 1]$ . We first show that the density  $f$  defined by  $f(x) \coloneqq \lambda f_1(x) + (1 - \lambda)f_2(x)$ ,  $x \in \Omega$ , is also in  $\mathcal{F}$ . Clearly,  $f$  is a density, because  $f(x) \geq 0$  and
$$
\int_ {\Omega} f (x) d \mu (x) = \lambda \int_ {\Omega} f _ {1} (x) d \mu (x) + (1 - \lambda) \int_ {\Omega} f _ {2} (x) d \mu (x) = \lambda + (1 - \lambda) = 1.
$$
Hence, the fact that  $g(y) \coloneqq y \ln y$  is a strongly convex function on  $\mathbb{R}_{\geq 0}$  implies that
$$
f (x) \ln f (x) \leq \lambda f _ {1} (x) \ln f _ {1} (x) + (1 - \lambda) f _ {2} (x) \ln f _ {2} (x)),
$$
with equality if and only if  $f_{1}(x) = f_{2}(x)$ . Integrating both sides, and using the fact that  $f_{1}, f_{2}$  belong to  $\mathcal{F}$ , we get
$$
\begin{array}{l} \int_ {\Omega} f (x) \ln f (x) d \mu (x) \leq - \lambda \tau - (1 - \lambda) \tau \\ = - \tau . \\ \end{array}
$$
Thus, if  $f_{1}(x)$  and  $f_{2}(x)$  differ on a set of positive measure, then  $f(x)\ln f(x) < \lambda f_{1}(x) + (1 - \lambda)f_{2}(x)$  on a set of positive measure. Integrating both sides, we get  $\mathsf{Ent}(f) > \tau$ . This shows that the set  $\mathcal{F}$  is strictly convex.
Strict convexity of  $\mathcal{F}$  now allows us to show the uniqueness of the optimal solution.
Theorem C.19 (Uniqueness of optimal solution). Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of PrimalOpt satisfying the property that the optimal value is finite and there is an optimal dual solution with  $\gamma^{\star} > 0$ . Then there is a unique optimal solution to this instance.
Proof. Let  $p^{\star}$  denote the optimal value of this instance. Theorem C.16 already shows the existence of an optimal solution. We show the uniqueness of an optimal solution. We have assumed that there is an optimal dual solution  $(\gamma^{\star},\phi^{\star})$  such that  $\gamma^{\star} > 0$ . The complementary slackness condition shows that for any optimal solution  $f$  to the instance  $\mathcal{I}$ ,  $\operatorname{Ent}(f)$  must equal  $\tau$ .
Suppose, for the sake of contradiction, that there are two distinct optimal solutions  $f_1^\star$  and  $f_2^\star$  to  $\mathcal{I}$  (i.e.,  $f_1^\star(x) \neq f_2^\star$  on a subset of positive measure). Consider a solution  $f^\star = tf_1^\star + (1 - t)f_2^\star$  for some  $t \in (0, 1)$ . Linearity of  $\operatorname{Err}_{\ell, \alpha}(f_{\mathcal{D}}, f)$  shows that  $\operatorname{Err}_{\ell, \alpha}(f_{\mathcal{D}}, f^\star) = p^\star$  as well. Now  $f^\star$  is also a feasible solution by Lemma C.18; in fact, this lemma shows that  $\operatorname{Ent}(f^\star) > \tau$ . But this contradicts the fact that every optimal solution must have entropy equal to  $\tau$ . Thus, we see that there must be a unique optimal solution to the instance  $\mathcal{I}$ .
Combining Lemma F.2, Theorem C.16 and Theorem C.19, we see that Theorem C.1 holds.
C.7 Conditions under which optimal solution is  $f_{\mathcal{D}}$
We show that our optimization framework can output the true utility  $f_{\mathcal{D}}$  for a choice of the parameters  $\alpha, \tau$ , and the loss function  $\ell$ .
Theorem C.20 (Parameters that recover the true utility). Consider an instance  $\mathcal{I} = (\Omega, f_{\mathcal{D}}, \ell, \alpha, \tau)$  of the optimization problem PrimalOpt. Assume that the density  $f_{\mathcal{D}}$  has finite entropy. Suppose,  $\alpha := 1, \tau := \operatorname{Ent}(f_{\mathcal{D}})$ , and the loss function  $\ell(x, v) := \ln f_{\mathcal{D}}(v) - \ln f_{\mathcal{D}}(x)$ . Then the density  $f_{\mathcal{D}}$  is the unique optimal solution to PrimalOpt for  $\mathcal{I}$ .