SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Qi Wang $^{1*}$  Yiqin Lv $^{1*}$  Yanghe Feng $^{2}$  Zheng Xie $^{1\dagger}$  Jincai Huang $^{2\dagger}$
<sup>1</sup>College of Science, National University of Defense Technology
$^{2}$ College of Systems Engineering, National University of Defense Technology {wangqi15, lvyiqin98, fengyanghe, xiezheng81, huangjincai} @nudt.edu.cn
Meta learning is a promising paradigm to enable skill transfer across tasks. Most previous methods employ the empirical risk minimization principle in optimization. However, the resulting worst fast adaptation to a subset of tasks can be catastrophic in risk-sensitive scenarios. To robustify fast adaptation, this paper optimizes meta learning pipelines from a distributionally robust perspective and meta trains models with the measure of expected tail risk. We take the two-stage strategy as heuristics to solve the robust meta learning problem, controlling the worst fast adaptation cases at a certain probabilistic level. Experimental results show that our simple method can improve the robustness of meta learning to task distributions and reduce the conditional expectation of the worst fast adaptation risk.
The past decade has witnessed the remarkable progress of deep learning in real-world applications (LeCun et al., 2015). However, training deep learning models requires an enormous dataset and intensive computational power. At the same time, these pre-trained models can frequently encounter deployment difficulties when the dataset's distribution drifts in testing time (Lesort et al., 2021).
As a result, the paradigm of meta learning or learning to learn is proposed and impacts the machine learning scheme (Finn et al., 2017), which leverages past experiences to enable fast adaptation to unseen tasks. Moreover, in the past few
years, there has grown a large body of meta learning methods to find plausible strategies to distill common knowledge into separate tasks (Finn et al., 2017; Duan et al., 2016; Garnelo et al., 2018a).
Notably, most previous work concentrates merely on the fast adaptation strategies and employs the standard risk minimization principle, e.g. the empirical risk minimization, ignoring the difference between tasks in fast adaptation. Given the sampled batch from the task distribution, the standard meta learning methods weight tasks equally in fast adaptation. Such an implementation raises concerns in some real-world scenarios, when worst fast adaptation is catastrophic in a range of risk-sensitive applications (Johannsmeier et al., 2019; JAAFra et al., 2019; Wang et al., 2023b). For example, in
Figure 1: Illustrations of Distributionally Robust Fast Adaptation. Shown are histograms of meta risk function values  $\ell (\mathfrak{D}_\tau^T,\mathfrak{D}_\tau^C;\vartheta)$  in the task distribution  $p(\tau)$ . Given a probability  $\alpha$ , we optimize meta learning model parameters  $\vartheta$  to decrease the risk quantity  $\mathrm{CVaR}_{\alpha}$  in Definition (3).
$^{\dagger}$ Zheng Xie and Jincai Huang are corresponding authors.
robotic manipulations, humanoid robots (Duan, 2017) can quickly leverage past motor primitives to walk on plain roads but might suffer from tribulation doing this on rough roads.
Research Motivations. Instead of seeking novel fast adaptation strategies, we take more interest in optimization principles for meta learning. Given the meta trained model, this paper stresses the performance difference in fast adaptation to various tasks as an indispensable consideration. As the concept of robustness in fast adaptation has not been sufficiently explored from the task distribution perspective, researching this topic has more practical significance and deserves more attention in meta learning. Naturally, we raise the question below:
Can we reconsider the meta learning paradigm through the lens of risk distribution, and are there plausible measures to enhance the fast adaptation robustness in some vulnerable scenarios?
Developed Methods. In an effort to address the above concerns and answer these questions, we reduce robust fast adaptation in meta learning to a stochastic optimization problem within the principle of minimizing the expected tail risk, e.g., conditional value-at-risk (CVaR) (Rockafellar et al., 2000). To tractably solve the problem, we adopt a two-stage heuristic strategy for optimization with the help of crude Monte Carlo methods (Kroese and Rubinstein, 2012) and give some theoretical analysis. In each optimization step, the algorithm estimates the value-at-risk (VaR) (Rockafellar et al., 2000) from a meta batch of tasks and screens and optimizes a percentile of task samples vulnerable to fast adaptation. As illustrated in Fig. (1), such an operation is equivalent to iteratively reshaping the task risk distribution to increase robustness. The consequence of optimizing the risk function distributions  $\vartheta_{k}\rightarrow \vartheta_{k + 1}$  is to transport the probability mass in high-risk regions to the left side gradually. In this manner, the distribution of risk functions in the task domain can be optimized toward the anticipated direction that controls the worst-case fast adaptation at a certain probabilistic level.
Outline & Primary Contributions. We overview related meta learning and robust optimization work in Section (2). Section (3) introduces general notations and describes meta learning optimization objectives together with typical models. The distributionally robust meta learning problem is presented together with a heuristic optimization strategy in Section (4). We report experimental results and analysis in Section (5), followed by conclusions and limitations in Section (6). Our primary contribution is two-fold:
1. We recast the robustification of meta learning to a distributional optimization problem. The resulting framework minimizes the conditional expectation of task risks, namely the tail risk, which unifies vanilla meta-learning and worst-case meta learning frameworks.
2. To resolve the robust meta learning problem, we adopt the heuristic two-stage strategy and demonstrate its improvement guarantee. Experimental results show the effectiveness of our method, enhancing fast adaptation robustness and mitigating the worst-case performance.
2 Literature Review
Meta Learning Methods. In practice, meta learning enables fast learning (adaptation to unseen tasks) via slow learning (meta training in a collection of tasks). There exist different families of meta learning methods. The optimization-based methods, such as model agnostic meta learning (MAML) (Finn et al., 2017) and its variants (Finn et al., 2018; Rajeswaran et al., 2019; Grant et al., 2018; Vuorio et al., 2019; Abbas et al., 2022), try to find the optimal initial parameters of models and then execute gradient updates over them to achieve adaptation with a few examples. The context-based methods, e.g. conditional neural processes (CNPs) (Garnelo et al., 2018a), neural processes (NPs) (Garnelo et al., 2018b) and extensions (Gordon et al., 2019; Foong et al., 2020; Wang and Van Hoof, 2020; Gondal et al., 2021; Wang and van Hoof, 2022; Shen et al., 2021; Wang et al., 2023a), learn the representation of tasks in the function space and formulate meta learning models as exchangeable stochastic processes. The metrics-based methods (Snell et al., 2017; Allen et al., 2019; Bartunov and Vetrov, 2018) embed tasks in a metric space and can achieve competitive performance in few-shot classification tasks. Other methods like memory-augmented models (Santoro et al., 2016; Xiao et al., 2021, 2022), recurrent models (Duan et al., 2016) and hyper networks (Zhao et al., 2020; Beck et al., 2023) are also modified for meta learning purposes.
Besides, there exist several important works investigating the generalization capability of methods. Bai et al. (2021) conduct the theoretical analysis of the train-validation split and connect it to
optimality. In (Chen et al., 2021), a generalization bound is constructed for MAML through the lens of information theory. Denevi et al. (2019) study an average risk bound and estimate the bias for improving stochastic gradient optimization in meta learning.
Robust Optimization. When performing robust optimization for downstream tasks in deep learning, we can find massive work concerning the adversarial input noise (Goodfellow et al., 2018; Goel et al., 2020; Ren et al., 2021), or the perturbation on the model parameters (Goodfellow et al., 2014; Kurakin et al., 2016; Liu et al., 2018; Silva and Najafirad, 2020). In contrast, this paper studies the robustness of fast adaptation in meta learning. In terms of robust principles, the commonly considered one is the worst-case optimization (Olds, 2015; Zhang et al., 2020; Tay et al., 2022). For example, Collins et al. (2020) conducts the worst-case optimization in MAML to obtain the robust meta initialization. Considering the influence of adversarial examples, Goldblum et al. (2019) propose to adversarily meta train the model for few-shot image classification. Wang et al. (2020) adopt the worst-case optimization in MAML to increase the model robustness by injecting adversarial noise to the input. However, distributionally robust optimization (Rahimian and Mehrotra, 2019) is rarely examined in the presence of the meta learning task distribution.
Notations. Consider the distribution of tasks  $p(\tau)$  for meta learning and denote the task space by  $\Omega_{\tau}$ . Let  $\tau$  be a task sampled from  $p(\tau)$  with  $\mathcal{T}$  the set of all tasks. We denote the meta dataset by  $\mathfrak{D}_{\tau}$ . For example, in few-shot regression problems,  $\mathfrak{D}_{\tau}$  refers to a set of data points  $\{(x_i,y_i)\}_{i = 1}^m$  to fit.
Generally,  $\mathfrak{D}_{\tau}$  are processed into the context set  $\mathfrak{D}_{\tau}^{C}$  for fast adaptation, and the target set  $\mathfrak{D}_{\tau}^{T}$  for evaluating adaptation performance. As an instance, we process the dataset  $\mathfrak{D}_{\tau} = \mathfrak{D}_{\tau}^{C} \cup \mathfrak{D}_{\tau}^{T}$  with a fixed partition in MAML (Finn et al., 2017).  $\mathfrak{D}_{\tau}^{C}$  and  $\mathfrak{D}_{\tau}^{T}$  are respectively used for the inner loop and the outer loop in model optimization.
Definition 1 (Meta Risk Function) With the task  $\tau \in \mathcal{T}$  and the pre-processed dataset  $\mathfrak{D}_{\tau}$  and the model parameter  $\vartheta \in \Theta$ , the meta risk function is a map  $\ell : \mathfrak{D}_{\tau} \times \Theta \mapsto \mathbb{R}^{+}$ .
In meta learning, the meta risk function  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$ , e.g. instantiations in Example (1)/(2), is to evaluate the model performance after fast adaptation. Now we turn to the commonly-used risk minimization principle, which plays a crucial role in fast adaptation. To summarize, we include the vanilla and worst-case optimization objectives as follows.
Expected Risk Minimization for Meta Learning. The objective that applies to most vanilla meta learning methods can be formulated in Eq. (1), and the optimization executes in a distribution over tasks  $p(\tau)$ . The Monte Carlo estimate corresponds to the empirical risk minimization principle.
$$
\min  _ {\vartheta \in \Theta} \mathcal {E} (\vartheta) := \mathbb {E} _ {p (\tau)} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right] \tag {1}
$$
Here  $\vartheta$  is the parameter of meta learning models, which includes parameters for common knowledge shared across all tasks and for fast adaptation. Furthermore, the task distribution heavily influences the direction of optimization in meta training.
Worst-case Risk Minimization for Meta Learning. This also considers meta learning in the task distribution  $p(\tau)$ , but the worst case in fast adaptation is the top priority in optimization.
$$
\min  _ {\vartheta \in \Theta} \max  _ {\tau \in \mathcal {T}} \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \tag {2}
$$
The optimization objective is built upon the min-max framework, advancing the robustness of meta learning to the worst case. Approaches like TR-MAML (Collins et al., 2020) sample the worst task in a batch to meta train with gradient updates. Nevertheless, this setup might result in a highly conservative solution where the worst case only happens with an incredibly lower chance.
4 Distributionally Robust Fast Adaptation
This section starts with the concept of risk measures and the derived meta learning optimization objective. Then a heuristic strategy is designed to approximately solve the problem. Finally, we provide two examples of distributionally robust meta learning methods.
4.1 Meta Risk Functions as Random Variables
Assumption 1 The meta risk function  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$  is  $\beta_{\tau}$ -Lipschitz continuous w.r.t.  $\vartheta$ , which suggests: there exists a positive constant  $\beta_{\tau}$  such that  $\forall \{\vartheta, \vartheta'\} \in \Theta$ :
$$
\left| \ell (\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta) - \ell (\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta^ {\prime}) \right| \leq \beta_ {\tau} | | \vartheta - \vartheta^ {\prime} | |.
$$
Let  $(\Omega_{\tau},\mathcal{F}_{\tau},\mathbb{P}_{\tau})$  denote a probability measure over the task space, where  $\mathcal{F}_{\tau}$  corresponds to a  $\sigma$ -algebra on the subsets of  $\Omega_{\tau}$ . And we have  $(\mathbb{R}^{+},\mathcal{B})$  a probability measure over the non-negative real domain for the previously mentioned meta risk function  $\ell (\mathfrak{D}_\tau^T,\mathfrak{D}_\tau^C;\vartheta)$  with  $\mathcal{B}$  a Borel  $\sigma$ -algebra. For any  $\vartheta \in \Theta$ , the meta learning operator  $\mathcal{M}_{\vartheta}:\Omega_{\tau}\mapsto \mathbb{R}^{+}$  is defined as:
$$
\mathcal {M} _ {\vartheta}: \tau \mapsto \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right).
$$
In this way,  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$  can be viewed as a random variable to induce the distribution  $p(\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta))$ . Further, the cumulative distribution function can be formulated as  $F_{\ell}(l; \vartheta) = \mathbb{P}(\{\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta) \leq l; \tau \in \Omega_{\tau}, l \in \mathbb{R}^{+}\})$  w.r.t. the task space. Note that  $F_{\ell}(l; \vartheta)$  implicitly depends on the model parameter  $\vartheta$ , and we cannot access a closed-form in practice.
Definition 2 (Value-at-Risk) Given the confidence level  $\alpha \in [0,1]$ , the task distribution  $p(\tau)$  and the model parameter  $\vartheta$ , the VaR (Rockafellar et al., 2000) of the meta risk function is defined as:
$$
V a R _ {\alpha} \left[ \ell (\mathcal {T}, \vartheta) \right] = \inf  _ {l \in \mathbb {R} ^ {+}} \left\{l \mid F _ {\ell} (l; \vartheta) \geq \alpha , \tau \in \mathcal {T} \right\}.
$$
Definition 3 (Conditional Value-at-Risk) Given the confidence level  $\alpha \in [0,1]$ , the task distribution  $p(\tau)$  and the model parameter  $\vartheta$ , we focus on the constrained domain of the random variable  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$  with  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta) \geq VaR_{\alpha}[\ell(\mathcal{T}, \vartheta)]$ . The conditional expectation of this is termed as conditional value-at-risk (Rockafellar et al., 2000):
$$
C V a R _ {\alpha} \left[ \ell (\mathcal {T}, \vartheta) \right] = \int_ {0} ^ {\infty} l d F _ {\ell} ^ {\alpha} (l; \vartheta),
$$
where the normalized cumulative distribution is as follows:
$$
F _ {\ell} ^ {\alpha} (l; \vartheta) = \left\{ \begin{array}{l l} 0, & l <   V a R _ {\alpha} [ \ell (\mathcal {T}, \vartheta) ] \\ \frac {F _ {\ell} (l ; \vartheta) - \alpha}{1 - \alpha}, & l \geq V a R _ {\alpha} [ \ell (\mathcal {T}, \vartheta) ]. \end{array} \right.
$$
This results in the normalized probability measure  $(\Omega_{\alpha,\tau},\mathcal{F}_{\alpha,\tau},\mathbb{P}_{\alpha,\tau})$  over the task space, where  $\Omega_{\alpha,\tau} := \bigcup_{\ell \geq \mathrm{VaR}_{\alpha}[\ell(\mathcal{T},\vartheta)]}\left[\mathcal{M}_{\vartheta}^{-1}(\ell)\right]$ . For ease of presentation, we denote the corresponding task distribution constrained in  $\Omega_{\alpha,\tau}$  by  $p_{\alpha}(\tau;\vartheta)$ .
Rather than optimizing  $\mathrm{VaR}_{\alpha}$ , a quantile, in meta learning, we take more interest in  $\mathrm{CVaR}_{\alpha}$  optimization, a type of the expected tail risk. Such risk measure regards the conditional expectation and has more desirable properties for meta learning: more adequate in handling adaptation risks in extreme tails, more accessible sensitivity analysis w.r.t.  $\alpha$ , and more efficient optimization.
Remark 1  $CVaR_{\alpha}[\ell(\mathcal{T}, \vartheta)]$  to minimize is respectively equivalent with the vanilla meta learning optimization objective in Eq. (1) when  $\alpha = 0$  and the worst-case meta learning optimization objective in Eq. (3) when  $\alpha$  is sufficiently close to 1.
4.2 Meta Learning via Controlling the Expected Tail Risk
As mentioned in Remark (1), the previous two meta learning objectives can be viewed as special cases within the  $\mathrm{CVaR}_{\alpha}$  principle. Furthermore, we turn to a particular distributionally robust fast
adaptation with the adjustable confidence level  $\alpha$  to control the expected tail risk in optimization as follows.
Distributionally Robust Meta Learning Objective. With the previously introduced normalized probability density function  $p_{\alpha}(\tau; \vartheta)$ , minimizing  $\mathrm{CVaR}_{\alpha}[\ell(\mathcal{T}, \vartheta)]$  can be rewritten as Eq. (3).
$$
\min  _ {\vartheta \in \Theta} \mathcal {E} _ {\alpha} (\vartheta) := \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta)} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right] \tag {3}
$$
Even though  $\mathrm{CVaR}_{\alpha}[\ell(\mathcal{T},\vartheta)]$  is a function of the model parameter  $\vartheta$ , the integral in Eq. (3) is intractable due to the involvement of  $p_{\alpha}(\tau;\vartheta)$  in a non-closed form.
Assumption 2 For meta risk function values, the cumulative distribution function  $F_{\ell}(l;\vartheta)$  is  $\beta_{\ell}$ -Lipschitz continuous w.r.t.  $l$ , and the implicit normalized probability density function of tasks  $p_{\alpha}(\tau ;\vartheta)$  is  $\beta_{\theta}$ -Lipschitz continuous w.r.t.  $\vartheta$ .
Assumption 3 For any valid  $\vartheta \in \Theta$  and corresponding implicit normalized probability density function of tasks  $p_{\alpha}(\tau ;\vartheta)$ , the meta risk function value can be bounded by a positive constant  $\mathcal{L}_{\mathrm{max}}$ :
$$
\sup  _ {\tau \in \Omega_ {\alpha , \tau}} \ell \left(\mathfrak {D} _ {\tau_ {i}} ^ {T}, \mathfrak {D} _ {\tau_ {i}} ^ {C}; \vartheta\right) \leq \mathcal {L} _ {\max }.
$$
Proposition 1 Under assumptions (1)/(2)/(3), the meta learning optimization objective  $\mathcal{E}_{\alpha}(\vartheta)$  in Eq. (3) is continuous w.r.t.  $\vartheta$ .
Further, we use  $\xi_{\alpha}(\vartheta)$  to denote the  $\mathrm{VaR}_{\alpha}[\ell(\mathcal{T},\vartheta)]$  for simple notations. The same as that in (Rockafellar et al., 2000), we introduce a slack variable  $\xi \in \mathbb{R}$  and the auxiliary risk function  $[\ell(\mathfrak{D}_{\tau}^{T},\mathfrak{D}_{\tau}^{C};\vartheta)-\xi]^{+}:= \max \{\ell(\mathfrak{D}_{\tau}^{T},\mathfrak{D}_{\tau}^{C};\vartheta)-\xi,0\}$ . To circumvent directly optimizing the nonanalytical  $p_{\alpha}(\tau;\vartheta)$ , we can convert the probability constrained function  $\mathcal{E}_{\alpha}(\vartheta)$  to the below unconstrained one after optimizing  $\xi$ :
$$
\varphi_ {\alpha} (\xi ; \vartheta) = \xi + \frac {1}{1 - \alpha} \mathbb {E} _ {p (\tau)} \left[ \left[ \ell (\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta) - \xi \right] ^ {+} \right].
$$
It is demonstrated that  $\mathcal{E}_{\alpha}(\vartheta) = \min_{\xi \in \mathbb{R}}\varphi_{\alpha}(\xi ;\vartheta)$  and  $\xi_{\alpha}\in \arg \min_{\xi \in \mathbb{R}}\varphi_{\alpha}(\xi ;\vartheta)$  in (Rockafellar et al., 2000), and also note that  $\mathrm{CVaR}_{\alpha}$  is the upper bound of  $\xi_{\alpha}$ , implying
$$
\xi_ {\alpha} \leq \varphi_ {\alpha} (\xi_ {\alpha}; \vartheta) \leq \varphi_ {\alpha} (\xi ; \vartheta), \quad \forall \xi \in \mathbb {R} \text {a n d} \forall \vartheta \in \Theta . \tag {4}
$$
With the deductions from Eq.s (3)/(4), we can resort the distributionally robust meta learning optimization objective with the probability constraint into a unconstrained optimization objective as Eq.(5).
$$
\min  _ {\vartheta \in \Theta , \xi \in \mathbb {R}} \xi + \frac {1}{1 - \alpha} \mathbb {E} _ {p (\tau)} \left[ \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) - \xi \right] ^ {+} \right] \tag {5}
$$
Sample Average Approximation. For the stochastic programming problem above, it is mostly intractable to derive the analytical form of the integral. Hence, we need to perform Monte Carlo estimates of Eq. (5) to obtain Eq. (6) for optimization.
$$
\min  _ {\vartheta \in \Theta , \xi \in \mathbb {R}} \xi + \frac {1}{(1 - \alpha) \mathcal {B}} \sum_ {i = 1} ^ {\mathcal {B}} \left[ \ell \left(\mathfrak {D} _ {\tau_ {i}} ^ {T}, \mathfrak {D} _ {\tau_ {i}} ^ {C}; \vartheta\right) - \xi \right] ^ {+} \tag {6}
$$
Remark 2 If  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$  is convex w.r.t.  $\vartheta$ , then Eq.s (5)/(6) are also convex functions. In this case, the optimization objective Eq. (6) of our interest can be resolved with the help of several convex programming algorithms (Fan et al., 2017; Meng et al., 2020; Levy et al., 2020).
Figure 2: Optimization Diagram of Distributionally Robust Meta Learning with Surrogate Functions. From left to right: the meta model parameters  $\vartheta$  in the middle block are optimized w.r.t. the constructed surrogate function  $\varphi(\hat{\xi}_{\alpha_t}; \vartheta)$  marked in blue in  $t$ -th iteration. Under certain conditions in Theorem (1), the distributionally robust meta learning objective  $\varphi(\xi_{\alpha}; \vartheta)$  marked in pink can be decreased monotonically until it reaches the convergence in the  $H$ -th iteration.
4.3 Heuristic Algorithms for Optimization
Unfortunately, most existing meta learning models' risk functions (Finn et al., 2017; Garnelo et al., 2018a; Santoro et al., 2016; Li et al., 2017; Duan et al., 2016),  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$  are non-convex w.r.t.  $\vartheta$ , bringing difficulties in optimization of Eq.s (5)/(6).
To this end, we propose a simple yet effective optimization strategy, where the sampled task batch is used to approximate the  $\mathrm{VaR}_{\alpha}$  and the integral in Eq. (5) for deriving Eq. (6). In detail, two stages are involved in iterations: (i) approximate  $\mathrm{VaR}_{\alpha}[\ell(\mathcal{T},\vartheta)] \approx \hat{\xi}_{\alpha}$  with the meta batch values, which can be achieved via either a quantile estimator (Dong and Nakayama, 2018) or other density estimators; (ii) optimize  $\vartheta$  in Eq. (6) via stochastic updates after replacing  $\xi_{\alpha}$  by the estimated  $\hat{\xi}_{\alpha}$ .
Proposition 2 Suppose there exists  $\delta \in \mathbb{R}^{+}$  such that  $|\xi_{\alpha}(\vartheta) - \hat{\xi}_{\alpha}(\vartheta)| < \delta$  with  $\hat{\xi}_{\alpha}(\vartheta)$  an estimate of  $\xi_{\alpha}(\vartheta)$ . Then there exists a constant  $\kappa_{\alpha} = \max \left\{\frac{2 - \alpha}{1 - \alpha}, \frac{\alpha}{1 - \alpha}\right\}$  such that
$$
\varphi_ {\alpha} \big (\hat {\xi} _ {\alpha} (\vartheta); \vartheta \big) - \kappa_ {\alpha} \delta <   \mathcal {E} _ {\alpha} (\vartheta) \leq \varphi_ {\alpha} \big (\hat {\xi} _ {\alpha} (\vartheta); \vartheta \big).
$$
The performance gap resulting from  $\mathrm{VaR}_{\alpha}$  approximation error is estimated in Proposition (2). For ease of implementation, we adopt crude Monte Carlo methods (Kroese and Rubinstein, 2012) to obtain a consistent estimator of  $\xi_{\alpha}$ .