SlowGuess/NeurIPS_2023_OCR huggingface dataset:
for various values of  $\gamma$ , and  $N_{Y}$  is the size of the test data  $\{y_{j}^{\mathrm{test}}\}$ . Note that ERR is the empirical approximation of the difference of optimality gap between the two-step naïve VB approach and RSVB( $\gamma$ ) approach. Recall, in the naïve VB method, we first compute the KL minimizer of the true posterior and then compute the optimal decision using this approximate posterior. Following [15], due to small size of the dataset in this example, the ERR is evaluated on the training data itself.
We modified the experiments in [15] by introducing  $\gamma = \{5,2.5,1,0.5\} = \{0.2^{-1},0.4^{-1},1^{-1},2^{-1}\}$  and obtain the results as summarized in Figure 2(a) and (b). In Figure 2(a), we observe that as  $\gamma$  increases, the ERR also increases, which implies that the decisions are more optimistic as empirical loss of the RSVB decision rule decreases as  $\gamma$  increases. Also, observe from Figure 2(b) that, as  $\gamma$  increases, the RSVB posterior (joint marginal posterior distribution for  $(\theta_3,\tau)$ ) approaches the true posterior, and the variance of the approximate posterior also reduces.
(a) ERR
(b)  $\{\theta_3,\tau \}$
Figure 2: a) Empirical risk reduction plot for different level of risk sensitivity, b) Joint RSVB posterior distribution of  $\{\theta_3,\tau \}$ , plotted for different level of risk sensitivity.
Notice that it is not obvious that the variance of the RSVB posterior will reduce as  $\gamma$  increases. We believe that it depends on the landscape of the expected risk function and the choice of the variational family. Intuitively, a possible explanation of this phenomenon can be provided using the equation (RSVB). Consider the RSVB formulation and note that  $\mathrm{KL} > 0$ , therefore as  $\gamma$  increases, there is more incentive to deviate from the true posterior and choose  $Q \in \mathcal{Q}$  that maximizes expected risk for a given  $a \in \mathcal{A}$ . Note that a  $Q$  that places more mass near the  $\theta$  that maximizes the risk will be preferred over the one with more spread.
The RSVB formulation as stated in this paper requires us to solve a stochastic minimax optimization problem to compute a decision rule. This is often difficult to solve, particularly in the nonconvex-nonconcave setting of RSVB, and indeed, we are not aware of any computationally efficient methods for solving such problems in all generality [8]. To circumvent this, the authors in the LCVB literature maximize utility instead of minimizing risk, and thus convert the whole problem to a much simpler max-max optimization problem. If we replace risk with utility in the RSVB derivation, we will also get a max-max problem. In this formulation, as you increase  $\gamma$ , the incentive to deviate from the posterior and maximize the maximum utility increases. Therefore, increasing  $\gamma$  corresponds to being more optimistic in making decisions than being more risk averse in the risk setting, as observed in the empirical results for the eight schools model in Section 4.3.
We note that we assumed the risk-function is lower-bounded, this is a natural assumption to place on risk functions. Converting our problem to a max-max problem would further require the risk to be upper bounded, which is also assumed for the methodologies presented (without theory) in [16, 15]. We emphasize that our theoretical results will continue to hold in this setting, but since our emphasis was on the statistical aspects of this problem, we chose to present our results with minimum assumptions. Developing an algorithm for solving general minimax optimization in RSVB without transforming it into a max-max problem is open and a part of our future work. Also an interesting topic for future work is identifying minimum additional assumptions to theoretically characterize computational aspects of this problem.
We thank the anonymous reviewers for their helpful comments and discussions. This work is supported by the National Science Foundation under grant DMS-1812197.
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A Additional definitions
We provide the definitions of important terms used throughout the paper. First, recall the definition of covering numbers:
Definition A.1 (Covering numbers). Let  $\mathcal{P} \coloneqq \{P_{\theta}, \theta \in \Theta\}$  be a parametric family of distributions and  $d: \mathcal{P} \times \mathcal{P} \mapsto [0, \infty)$  be a metric. An  $\epsilon$ -cover of a subset  $\mathcal{P}_K \coloneqq \{P_{\theta}: \theta \in K \subset \Theta\}$  of the parametric family of distributions is a set  $K' \subset K$  such that, for each  $\theta \in K$  there exists a  $\theta' \in K'$  that satisfies  $d(P_{\theta}, P_{\theta'}) \leq \epsilon$ . The  $\epsilon$ -covering number of  $\mathcal{P}_K$  is  $N(\epsilon, \mathcal{P}_K, d) = \min\{card(K'): K' \text{ is an } \epsilon$ -cover of  $K\}$ , where  $card(\cdot)$  represents the cardinality of the set.
Next, recall the definition of a test function [26]:
Definition A.2 (Test function). Let  $\tilde{X}_n$  be a sequence of random variables on measurable space  $(\bigotimes_n\mathcal{X},\mathcal{S}^n)$ . Then any  $S^n$ -measurable sequence of functions  $\{\phi_n\}$ ,  $\phi_{n}:\tilde{X}_{n}\mapsto [0,1]\forall n\in \mathbb{N}$ , is a test of a hypothesis that a probability measure on  $S^n$  belongs to a given set against the hypothesis that it belongs to an alternative set. The test  $\phi_{n}$  is consistent for hypothesis  $P_0^n$  against the alternative  $P^n\in \{P_\theta^n:\theta \in \Theta \backslash \{\theta_0\}\}$  if  $\mathbb{E}_{P^n}[\phi_n]\to \mathbb{1}_{\{\theta \in \Theta \setminus \{\theta_0\}\}}(\theta),\forall \theta \in \Theta$  as  $n\to \infty$ , where  $\mathbb{1}_{\{\cdot \}}$  is an indicator function.
A classic example of a test function is  $\phi_n^{\mathrm{KS}} = \mathbb{1}_{\{\mathrm{KS}_n > K_\nu \}}(\theta)$  that is constructed using the Kolmogorov-Smirnov statistic  $\mathrm{KS}_n \coloneqq \sup_t |\mathbb{F}_n(t) - \mathbb{F}_\theta(t)|$ , where  $\mathbb{F}_n(t)$  and  $\mathbb{F}_\theta(t)$  are the empirical and true distribution respectively, and  $K_\nu$  is the confidence level. If the null hypothesis is true, the Glivenko-Cantelli theorem [29, Theorem 19.1] shows that the KS statistic converges to zero as the number of samples increases to infinity.
Furthermore, we define the Hellinger distance  $h(\theta_1, \theta_2)$  between the two probability distributions  $P_{\theta_1}$  and  $P_{\theta_2}$  is defined as  $d_H(\theta_1, \theta_2) = \left( \int \left( \sqrt{dP_{\theta_1}} - \sqrt{dP_{\theta_2}} \right)^2 \right)^{1/2}$ . We define the one-sided Hausdorff distance  $H(A \| B)$  between sets  $A$  and  $B$  in a metric space  $D$  with distance function  $d$  is defined as:
$$
H (A \| B) = \sup  _ {x \in A} d _ {h} (x, B), \text {w h e r e} d _ {h} (x, B) = \inf  _ {y \in B} d (x, y).
$$
Next, we define an arbitrary loss function  $L_{n}:\Theta \times \Theta \mapsto \mathbb{R}$  that measures the distance between models  $(P_{\theta_1}^n,P_{\theta_2}^n)\forall \{\theta_1,\theta_2\} \in \Theta$ . At the outset, we assume that  $L_{n}(\theta_{1},\theta_{2})$  is always positive. We define  $\{\epsilon_n\}$  as a sequence such that  $\epsilon_{n}\rightarrow 0$  as  $n\to \infty$  and  $n\epsilon_n^2\geq 1$ .
Definition A.3 ( $\Gamma$ -convergence). A sequence of functions  $F_{n}:\mathcal{U}\mapsto \mathbb{R}$ , for each  $n\in \mathbb{N}$ ,  $\Gamma$ -converges to  $F:\mathcal{U}\mapsto \mathbb{R}$ , if
- for every  $u \in \mathcal{U}$  and every  $\{u_n, n \in \mathbb{N}\}$  such that  $u_n \to u$ ,  $F(x) \leq \lim \inf_{n \to \infty} F_n(u_n)$ ;
- for every  $u \in \mathcal{U}$ , there exists some  $\{u_n, n \in \mathbb{N}\}$  such that  $u_n \to u$ ,  $F(x) \geq \lim_{n \to \infty} \sup_{n \to \infty} F_n(u_n)$ .
In addition, we define
Definition A.4 (Primal feasibility). For any two functions  $f: \mathcal{U} \mapsto \mathbb{R}$  and  $b: \mathcal{U} \mapsto \mathbb{R}$ , a point  $u^{*} \in \mathcal{U}$  is primal feasible to the following constraint optimization problem
$$
\inf  _ {u \in \mathcal {U}} f (u) \text {s u b j e c t} b (u) \leq c,
$$
if  $b(u^{*}) \leq c$ , for a given  $c \in \mathbb{R}$ .
B Applications
B.1 Single product newsvendor problem (cont.)
First, we fix the sieve set  $\Theta_{n}(\epsilon) = \Theta$ , which clearly implies that the restricted inverse-gamma prior  $\Pi (\theta)$ , places no mass on the complement of this set and therefore satisfies Assumption 2.2.
Second, under the condition that the true demand distribution is exponential with parameter  $\theta_0$  (and  $P_0 \equiv P_{\theta_0}$ ), we demonstrate the existence of test functions satisfying Assumption 2.1.
Lemma B.1. Fix  $n \geq 5$ . Then, for any  $\epsilon > \epsilon_n \coloneqq \frac{1}{\sqrt{n}}$  with  $\epsilon_n \to 0$ , and  $n\epsilon_n^2 \geq 1$ , there exists a test function  $\phi_n$  (depending on  $\epsilon$ ) such that  $L_n^{NV}(\theta, \theta_0) = n (\sup_{a \in \mathcal{A}} |R(a, \theta) - R(a, \theta_0)|)^2$  satisfies Assumption 2.1 with  $C_0 = 20$  and  $C = \frac{C_1}{2} (K_1^{NV})^{-2}$  for a constant  $C_1 > 0$  and  $K_1^{NV} = d_H(T, \theta_0)^{-1} \left[ \left( \frac{h}{\theta_0} - \frac{h}{T} \right)^2 + (b + h)^2 \left( \frac{e^{-\underline{a}T}}{T} - \frac{e^{-\underline{a}\theta_0}}{\theta_0} \right)^2 \right]^{1/2}$ .
The proof of the above result follows by showing that  $d_{L}^{NV} = n^{-1/2}\sqrt{L_{n}^{NV}(\theta, \theta_{0})}$  can be bounded above by the Hellinger distance between two exponential distributions on  $\Theta$  (under which a test function exists) in Lemma C.10 in the appendix.
Third, we show that there exist appropriate constants such that the inverse-gamma prior satisfies Assumption 2.3 when the demand distribution is exponential.
Lemma B.2. Fix  $n_2 \geq 2$  and any  $\lambda > 1$ . Let  $A_n := \{ \theta \in \Theta : D_{1 + \lambda}(P_0^n \| P_\theta^n) \leq C_3 n \epsilon_n^2 \}$ , where  $D_{1 + \lambda}(P_0^n \| P_\theta^n)$  is the Rényi divergence between  $P_0^n$  and  $P_\theta^n$ . Then for  $\epsilon_n^2 = \frac{\log n}{n}$  and any  $C_3 > 0$  such that  $C_2 = \alpha C_3 \geq 2$ , the truncated inverse-gamma prior  $Inv - \Gamma_{\Theta}(A; \alpha, \beta)$  satisfies  $\Pi(A_n) \geq \exp(-nC_2\epsilon_n^2), \forall n \geq n_2$ .
Fourth, it is straightforward to see that the newsvendor model risk  $R(a,\theta)$  is bounded below for a given  $a\in \mathcal{A}$ .
Lemma B.3. For any  $a \in \mathcal{A}$  and positive constants  $h$  and  $b$ , the newsvendor model risk  $R(a, \theta) = \left(ha - \frac{h}{\theta} + (b + h)\frac{e^{-a\theta}}{\theta}\right) \geq \left(\frac{ha^2\theta^*}{(1 + a\theta^*)}\right)$ , where  $\underline{a} := \min\{a \in \mathcal{A}\}$  and  $\theta^*$  satisfies  $h - (b + h)e^{-a\theta^*}(1 + a\theta^*) = 0$ .
This implies that  $R(a, \theta)$  satisfies Assumption 2.5. Finally, we also show that the newsvendor model risk satisfies Assumption 2.4.
Lemma B.4. Fix  $n \geq 1$  and  $\gamma > 0$ . For any  $\epsilon > \epsilon_n$  and any  $a \in \mathcal{A}$ ,  $R(a, \theta)$  satisfies  $\mathbb{E}_{\Pi}[\mathbb{1}_{\{R(a, \theta) \gamma > C_4(\gamma) n \epsilon^2\}} e^{\gamma R(a, \theta)}] \leq \exp(-C_5(\gamma) n \epsilon^2)$ , for any  $C_4(\gamma) > 2\gamma (h\overline{a} + \frac{b}{T})$  and  $C_5(\gamma) = C_4(\gamma) - 2\gamma (h\overline{a} + \frac{b}{T})$ , where  $\overline{a} := \max\{a \in \mathcal{A}\}$ .
Note that Lemma B.1 implies that  $C = \frac{C_1}{2(K_1^{NV})^2}$  for any constant  $C_1 > 0$ . Fixing  $\alpha = 1$  and using Lemma B.2 we can choose  $C_2 = C_3 = 2$ . Now,  $C_1$  can be chosen large enough such that  $C > C_4(\gamma) + C_5(\gamma)$  for a given risk sensitivity  $\gamma > 0$ . Therefore, the condition on constants in Theorem 3.1 reduces to  $C_5(\gamma) > 2 + C_2 + C_3 = 5$ , and it can be satisfied easily by fixing  $C_5(\gamma) = 5.1$  (say).
These lemmas show that when the demand distribution is exponential and with a non-conjugate truncated inverse-gamma prior, our result in Theorem 3.2 can be used for RSVB method to bound the optimality gap in decisions and values for various values of the risk-sensitivity parameter  $\gamma$ . Recall that the bound obtained in Theorem 3.2 depends on  $\epsilon_n^2$  and  $\eta_n^R (\gamma)$ .
Lemma B.2 implies that  $\epsilon_n^2 = \frac{\log n}{n}$ , but in order to get the complete bound we further need to characterize  $\eta_n^R (\gamma)$ . Recall that, as a consequence of Assumption 3.1 in Proposition 3.1, for a given  $C_8 = -\inf_{Q\in \mathcal{Q}}\inf_{a\in \mathcal{A}}\mathbb{E}_Q[R(a,\theta)]$  that  $C_9 > 0$  and  $\eta_n^R (\gamma)\leq \gamma n^{-1}C_8 + C_9\epsilon_n'^2$ .
Therefore, in our next result, we show that in the newsvendor setting, we can construct a sequence  $\{Q_n(\theta)\} \subset \mathcal{Q}$  that satisfies Assumption 3.1, and thus identify  $\epsilon_{n}^{\prime}$  and the constant  $C_9$ . We fix  $\mathcal{Q}$  to be the family of shifted gamma distributions with support  $[T,\infty)$ .
Lemma B.5. Let  $\{Q_n(\theta)\}$  be a sequence of shifted gamma distributions with shape parameter  $a = n$  and rate parameter  $b = \frac{n}{\theta_0}$ , then for truncated inverse gamma prior and exponentially distributed likelihood model
$$
\frac {1}{n} \left[ \mathrm {K L} \left(Q _ {n} (\theta) \| \Pi (\theta)\right) + \mathbb {E} _ {Q _ {n} (\theta)} \left[ \mathrm {K L} \left(d P _ {0} ^ {n} (\tilde {X} _ {n}) \| d P _ {\theta} ^ {n} (\tilde {X} _ {n})\right) \right] \right] \leq C _ {9} \epsilon_ {n} ^ {\prime 2},
$$
where  $\epsilon_n^{\prime 2} = \frac{\log n}{n}$  and  $C_9 = \frac{1}{2} +\max \left(0,2 + \frac{2\beta}{\theta_0} -\log \sqrt{2\pi} -\log \left(\frac{\beta^\alpha}{\Gamma(\alpha)}\right) + \alpha \log \theta_0\right)$  and prior parameters are chosen such that  $C_9 > 0$
B.2 Multi-product newsvendor problem
Analogous to the one-dimensional newsvendor loss function, the loss function in its multi-product version is defined as
$$
\ell (a, \xi) := h ^ {T} (a - \xi) ^ {+} + b ^ {T} (\xi - a) ^ {+}
$$
where  $h$  and  $b$  are given vectors of underage and overage costs respectively for each product and mapping  $(\cdot)^{+}$  is defined component-wise. We assume that there are  $d$  items or products and  $\xi \in \mathbb{R}^{d}$  denotes the random vector of demands. Let  $a \in \mathcal{A} \subset \mathbb{R}_{+}^{d}$  be the inventory or decision variable, typically assumed to take values in a compact decision space  $\mathcal{A}$  with  $\underline{a} := \{\{\min\{a_{i}: a_{i} \in \mathcal{A}_{i}\}\}_{i=1}^{d}$  and  $\bar{a} := \{\{\max\{a_{i}: a_{i} \in \mathcal{A}_{i}\}\}_{i=1}^{d}, \text{and } \underline{a} > 0\}$ , where  $\mathcal{A}_{i}$  is the marginal set of  $i^{th}$  component of  $\mathcal{A}$ . The random demand is assumed to be multivariate Gaussian, with unknown mean parameter  $\theta \in \mathbb{R}^{d}$  but with known covariance matrix  $\Sigma$ . We also assume that  $\Sigma$  is a symmetric positive definite matrix and can be decomposed as  $Q^{T}\Lambda Q$ , where  $Q$  is an orthogonal matrix and  $\Lambda$  is a diagonal matrix consisting of respective eigenvalues of  $\Sigma$ . We also define  $\overline{\Lambda} = \max_{i \in \{1,2,\ldots,d\}} \Lambda_{ii}$  and  $\underline{\Lambda} = \min_{i \in \{1,2,\ldots,d\}} \Lambda_{ii}$ . The model risk
$$
\begin{array}{l} R (a, \theta) = \mathbb {E} _ {P _ {\theta}} [ \ell (a, \xi) ] = \sum_ {i = 1} ^ {d} \mathbb {E} _ {P _ {\theta_ {i}}} [ h _ {i} (a _ {i} - \xi_ {i}) ^ {+} + b _ {i} (\xi_ {i} - a _ {i}) ^ {+} ] \\ = \sum_ {i = 1} ^ {d} \left[ (h _ {i} + b _ {i}) a _ {i} \Phi \left(\frac {(a _ {i} - \theta_ {i})}{\sigma_ {i i}}\right) - b _ {i} a _ {i} + \theta_ {i} (b _ {i} - h _ {i}) \right. \\ + \sigma_ {i i} \left[ h \frac {\phi \left(\frac {\left(a _ {i} - \theta_ {i}\right)}{\sigma_ {i i}}\right)}{\Phi \left(\frac {\left(a _ {i} - \theta_ {i}\right)}{\sigma_ {i i}}\right)} + b \frac {\phi \left(\frac {\left(a _ {i} - \theta_ {i}\right)}{\sigma_ {i i}}\right)}{1 - \Phi \left(\frac {\left(a _ {i} - \theta_ {i}\right)}{\sigma_ {i i}}\right)} \right], \tag {11} \\ \end{array}
$$
which is convex in  $a$ . Here  $P_{\theta_i}$  is the marginal distribution of  $\xi$  for  $i^{th}$  product,  $\phi(\cdot)$  and  $\Phi(\cdot)$  are probability and cumulative distribution function of the standard Normal distribution. We also assume that the true mean parameter  $\theta_0$  lies in a compact subspace  $\Theta \subset \mathbb{R}^d$ . We fix the prior to be uniformly distributed on  $\Theta$  with no correlation across its components, that is  $\pi(A) = \frac{m(A \cap \Theta)}{m(\Theta)} = \prod_{i=1}^{d} \frac{m(A_i \cap \Theta_i)}{m(\Theta_i)}$ , where  $m(B)$  is the Lebesgue measure (or volume) of  $B \subset \mathbb{R}^d$ . As in the previous example, we fix the sieve set  $\Theta_n(\epsilon) = \Theta$ , which clearly implies that  $\Pi(\theta)$  places no mass on the complement of this set and therefore satisfies Assumption 2.2.
Then under the condition that the true demand distribution has a multivariate Gaussian distribution (with known  $\Sigma$ ) and mean  $\theta_0$  ( $P_0 \equiv P_{\theta_0}$ ), we demonstrate the existence of test functions satisfying Assumption 2.1 by constructing a test function unlike the single-product newsvendor problem with exponential demand.
Lemma B.6. Fix  $n \geq 1$ . Then, for any  $\epsilon > \epsilon_{n} \coloneqq \frac{1}{\sqrt{n}}$  with  $\epsilon_{n} \to 0$ , and  $n\epsilon_{n}^{2} \geq 1$  and test function  $\phi_{n,\epsilon} \coloneqq \mathbb{1}\left\{\tilde{X}_{n}:\|\hat{\theta}_{n} - \theta_{0}\| > \sqrt{\tilde{C}\epsilon^{2}}\right\}$ ,  $L_{n}^{MNV}(\theta, \theta_{0}) = n$  ( $\sup_{a \in \mathcal{A}} |R(a, \theta) - R(a, \theta_{0})|$ )
satisfies Assumption 2.1 with  $C_0 = 1$ ,  $C_1 = 4K^2C$  and  $C = 1/8\left(\frac{\tilde{C}}{d\bar{\Lambda}} - 1\right)$  for sufficiently large  $\tilde{C}$ . Such that  $C > 1$  and  $\overline{\Lambda} = \max_{i \in \{1,2,\ldots,d\}} \Lambda_{ii}$ , where  $K = \sup_{A,\Theta}\|\partial_{\theta}R(a,\theta)\|$ .
In the following result, we show that there exist appropriate constants such that prior distribution satisfies Assumption 2.3 when the demand distribution is a multivariate Gaussian with unknown mean.