SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Influence of risk minimization principles. This paper is primarily devoted to studying the influence of the risk minimization principle on meta learning. The empirical risk minimization principle corresponds to reducing the Monte Carlo estimate of Average-case Meta Learning in this work. In comparison to TR-MAML (Collins et al., 2020), our approach has a couple of advantages as follows: (i) easier implementations. Note that min-max optimization is numerically unstable and requires a relaxation method for computationally intensive convex optimization, e.g., robust stochastic mirror-prox algorithm used in TR-MAML (Collins et al., 2020). (ii) more flexible in terms of robustness concept. Theoretically, the worst-case meta learning corresponds to the extreme case of the distributionally robust risk minimization principle. (iii) empirically better performance in
most cases. The expected tail risk minimization preserves a particular property that minimizing proportional worst-case fast adaptation seldom sacrifices the average performance. These advantages are also why we call our approach simple yet effective.
Comparison with Other Heuristic Optimization Strategies. To sum up the proposed optimization strategy, we empirically highlight the following points regarding the adopted heuristic strategies. There exist a couple of approximate algorithms for  $\mathrm{CVaR}_{\alpha}$  optimization. In comparison, the crude Monte Carlo one is the simplest for  $\mathrm{VaR}_{\alpha}$  estimates. It has an improvement guarantee under certain conditions, leaving it easier to analyze. Besides, we also compare ours to the risk reweighted method (Sagawa et al., 2020) in previously used benchmarks, and please take a closer look at that in Section (K).
B Pseudo Algorithms of DR-MAML & DR-CNPs
Algorithm 1: DR-MAML
Input: Task distribution  $p(\tau)$ ; Confidence level  $\alpha$ ; Task batch size  $\mathcal{B}$ ; Learning rates:  $\lambda_{1}$  and  $\lambda_{2}$ .  
Output: Meta-trained model parameter  $\vartheta$ .  
Randomly initialize the model parameter  $\vartheta$ ;  
while not converged do  
Sample a batch of tasks  $\{\tau_i\}_{i=1}^{\mathcal{B}} \sim p(\tau)$ ;  
// inner loop via gradient descent  
for  $i = 1$  to  $\mathcal{B}$  do  
Evaluate the gradient:  $\nabla_{\vartheta} \ell(\mathfrak{D}_{\tau_i}^C; \vartheta)$  in Eq. (7);  
Perform task-specific gradient updates:  $\vartheta_i \gets \vartheta - \lambda_1 \nabla_{\vartheta} \ell(\mathfrak{D}_{\tau_i}^C; \vartheta)$ ;  
end  
// estimate VaR $_{\alpha}$ [ $\ell(T, \vartheta)$ ]  $\approx \hat{\xi}_{\alpha}$   
Evaluate performance  $\mathcal{L}_{\mathcal{B}} = \{\ell(\mathfrak{D}_{\tau_i}^T; \vartheta_i)\}_{i=1}^{\mathcal{B}}$ ;  
Estimate VaR $_{\alpha}$ [ $\ell(T, \vartheta)$ ] and set  $\xi = \hat{\xi}_{\alpha}$  in Eq. (7) with either percentile rank or density estimators;  
// outer loop via gradient descent  
Screen the subset  $\mathcal{L}_{\hat{\mathcal{B}}} = \{\ell(\mathfrak{D}_{\hat{\tau}_i}^T; \vartheta_i)\}_{i=1}^K$  with  $\hat{\xi}_{\alpha}$  for meta initialization updates;  
 $\vartheta \gets \vartheta - \lambda_2 \nabla_{\vartheta} \sum_{i=1}^{K} \ell(\mathfrak{D}_{\hat{\tau}_i}^T; \vartheta_i)$  in Eq. (7);
Algorithm 2: DR-CNP
Input: Task distribution  $p(\tau)$ ; Confidence level  $\alpha$ ; Task batch size  $\mathcal{B}$ ; Learning rate  $\lambda$ .  
Output: Meta-trained model parameter  $\vartheta$ .  
Randomly initialize the model parameter  $\vartheta$ ;  
while not converged do  
    Sample a batch of tasks  $\{\tau_i\}_{i=1}^{\mathcal{B}} \sim p(\tau)$ ;  
// estimate VaR $_{\alpha}[\ell(\mathcal{T}, \vartheta)]$   
Evaluate performance  $\mathcal{L}_{\mathcal{B}} = \{\ell(\mathfrak{D}_{\tau_i}^T; z, \vartheta_i)\}_{i=1}^{\mathcal{B}}$ ;  
Estimate VaR $_{\alpha}[\ell(\mathcal{T}, \vartheta)] \approx \hat{\xi}_{\alpha}$  with either percentile rank or density estimators;  
// execute gradient descent  
Screen the subset  $\mathcal{L}_{\mathcal{B}} = \{\ell(\mathfrak{D}_{\hat{\tau}_i}^T; z, \vartheta)\}_{i=1}^K$  with  $\hat{\xi}_{\alpha}$  for meta initialization updates;  
 $\vartheta \gets \vartheta - \lambda \nabla_{\vartheta} \sum_{i=1}^{K} \ell(\mathfrak{D}_{\hat{\tau}_i}^T; z, \vartheta)$  in Eq. (8);  
end
C Properties of Risk Minimization Principles
C.1 Stochastic Optimization in the Constrained Form
In the section above, meta learning optimization objectives are discussed within three different principles, respectively the average-case in Eq. (1), the worst-case in Eq. (2) and  $\mathrm{CVaR}_{\alpha}$  worst-case in Eq. (5). This subsection continues this topic and introduces the relaxation variable  $\xi$  in the optimization objective. In this way, the robust fast adaptation can be reframed in the case of stochastic optimization with the probabilistic constraint.
We can equivalently express the minimization of the worst-case problem (Shalev-Shwartz and Wexler, 2016) within the following constrained stochastic optimization framework as follows.
$$
\begin{array}{c c} \min  & \xi \end{array}
$$
$$
\vartheta \in \Theta , \xi \in \mathbb {R} ^ {+} \tag {10}
$$
$$
\text {s . t .} \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \leq \xi \quad \text {w i t h} \tau \in \mathcal {T}
$$
C.2 SGD Intractability of Meta Learning  $\mathbf{CVaR}_{\alpha}$  Optimization Objective
This subsection shows that directly stochastic gradient descent is intractable for meta learning to optimize  $\mathrm{CVaR}_{\alpha}$ . Note that the normalized density distribution function  $p_{\alpha}(\tau; \vartheta)$  implicitly depends on  $\vartheta$  and  $\alpha$ , so we cannot access the exact form of such a distribution.
$$
\begin{array}{l} \nabla_ {\vartheta} \mathcal {E} _ {\alpha} (\vartheta) = \int p _ {\alpha} (\tau ; \vartheta) \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \nabla_ {\vartheta} \ln p _ {\alpha} (\tau ; \vartheta) + \nabla_ {\vartheta} \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right] d \tau (11a) \\ \approx \frac {1}{K} \sum_ {k = 1} ^ {K} \left[ \underbrace {\ell \left(\mathfrak {D} _ {\tau_ {k}} ^ {T} , \mathfrak {D} _ {\tau_ {k}} ^ {C} ; \vartheta\right) \nabla_ {\vartheta} \ln p _ {\alpha} \left(\tau_ {k} ; \vartheta\right)} _ {\text {S c o r e F u n c t i o n}} + \nabla_ {\vartheta} \ell \left(\mathfrak {D} _ {\tau_ {k}} ^ {T}, \mathfrak {D} _ {\tau_ {k}} ^ {C}; \vartheta\right) \right] (11b) \\ \end{array}
$$
As illustrated in Eq. (11), the stochastic gradient estimate is not plausible since  $p_{\alpha}(\tau; \vartheta)$  has no closed form. Hence, heuristic or convex programming algorithms for specific cases are mostly used as this domain's optimization strategy. However, designing optimization strategies for non-convex cases is non-trivial in this domain and requires more consideration of theoretical guarantees.
C.3 Risk-Sensitive Applications & Optimization Strategies
Related Applications. There are a number of applications concerning robust optimization with probabilistic constraints. Most of them are for the sake of safety. The risk principle  $\mathrm{CVaR}_{\alpha}$  firstly occurs in the financial domain as a coherent principle (Rockafellar et al., 2000). It enjoys much popularity in portfolio optimization (Quaranta and Zaffaroni, 2008). Gagne and Dayan (2021) adopt  $\mathrm{CVaR}_{\alpha}$  principles to improve the distributional reinforcement learning performance. To robustify robotic control, Pinto et al. (2017) varies hyper-parameters of Markov decision processes and optimizes proportional worst-case trajectories in policy optimization within the principle of  $\mathrm{CVaR}_{\alpha}$ . In work (Wilder, 2018), a  $\mathrm{CVaR}_{\alpha}$  related strategy is devised to solve submodular optimization problems. Such a risk measure is also included in producing robust options (Hiraoka et al., 2019). Regarding probabilistic robust meta learning within  $\mathrm{CVaR}_{\alpha}$  principles, there exists scarce related work until now.
**Optimization Strategies.** Concerning the constrained stochastic optimization problem, we particularly overview related work in this subsection. In the past few decades, there emerge substantial optimization strategies together with theoretical analysis for convex risk functions in  $\mathrm{CVaR}_{\alpha}$  optimization (Nemirovski et al., 2009; Fan et al., 2017; Meng et al., 2020; Levy et al., 2020; Duchi and Namkoong, 2021; Wang et al., 2022). As for the min-max risk minimization principle, which focuses on the worst case instead of a propositional worst cases, some researchers have designed relaxation or other heuristic algorithms to handle convex or non-convex risk functions (Chen et al., 2017; Collins et al., 2020; Jiang et al., 2021; Hsieh et al., 2021). Nevertheless, it remains challenging to design algorithms with convergence guarantees for non-convex risk function cases. One of the latest  $\mathrm{CVaR}_{\alpha}$  work on non-convex risk functions is (Sagawa et al., 2020), where a risk reweighted algorithm for robust neural networks is proposed to handle distributional shifts. Moreover, most  $\mathrm{CVaR}_{\alpha}$  optimization in non-convex risk functions follows this type of risk reweighted strategies in applications.
Remarks on Literature Work: Risk functions for robust fast adaptation cases are mostly non-convex, and it is non-trivial to design optimization strategies with a convergence guarantee. Meanwhile, we also conduct comparison experiments between work in (Sagawa et al., 2020) and ours in meta learning downstream tasks. We refer the reader to Appendix (K) for more details and analysis.
D Computational Complexity
The primary difference between distributionally robust meta learning and expected risk based meta learning lies in that a fixed probabilistic portion of task gradients are considered in meta updates. Such an operation drives the optimization procedure to focus more on proportional vulnerable scenarios, increasing the robustness in worst cases.
However, the iteration of the surrogate function  $\varphi (\hat{\xi}_{\alpha_t};\vartheta)$  involves the screening of proportional worst cases since our strategies require extra evaluation of fast adaptation. This brings additional computational cost with complexity  $\mathcal{O}\Big(\mathcal{B}\log (\mathcal{B})\Big)$ , where  $\mathcal{B}$  is the number of tasks in each batch. On the other hand, the proposed strategy performs sub-gradient updates instead of complete gradient updates. This helps reduce computational cost with complexity  $\mathcal{O}\big(\alpha \mathcal{B}|\mathcal{M}|\big)$  in each iteration, where  $|\mathcal{M}|$  corresponds to the scale of parameters of meta learning models and  $\alpha$  is the confidence level in  $\mathrm{CVaR}_{\alpha}$  optimization.
Universally analyzing the computational complexity in meta learning is intractable since various meta learning methods exist. Some are gradient-based ones, while some are non-parametric ones. The exact number of iterations for convergence heavily relies on specific methods. In summary, given the same number of iterations and a fixed confidence level, the computational complexity difference for  $\mathrm{CVaR}_{\alpha}$  optimization in meta learning scenarios is  $\mathcal{O}\Big(\mathcal{B}(\alpha |\mathcal{M}| - \log (\mathcal{B}))\Big)$ .
To deepen understanding of our method, we explain more via specific examples. The main idea is to execute the sub-gradient operation over the batch of task gradients. Here we take the DR-MAML in Example (1) to show the operation and how the distributionally robust meta initialization is obtained:
$$
\vartheta_ {t + 1} ^ {\mathrm {m e t a}} \leftarrow \vartheta_ {t} ^ {\mathrm {m e t a}} - \lambda_ {1} \left[ \sum_ {i = 1} ^ {\mathcal {B}} \nabla_ {\vartheta} \left[ \delta \left(\tau_ {i}\right) \cdot \ell \left(\mathfrak {D} _ {\tau_ {i}} ^ {T}; \vartheta_ {t} ^ {\tau_ {i}}\right) \right] \right], \tag {12}
$$
$$
\text {w i t h} \vartheta_ {t} ^ {\tau_ {i}} = \vartheta_ {t} ^ {\mathrm {m e t a}} - \lambda_ {2} \nabla_ {\vartheta} \ell \left(\mathfrak {D} _ {\tau_ {i}} ^ {C}; \vartheta\right), \tau_ {i} \sim p (\tau),
$$
where  $\lambda_{1}$  is the outer loop learning rate,  $\lambda_{2}$  is the inner loop learning rate, and  $\delta (\tau_i)$  is the indicator variable. Here  $\delta (\tau_i) = 1$  in the case when the  $\ell (\mathfrak{D}_{\tau_i}^T;\vartheta_t^{\tau_i})$  falls into the  $(1 - \alpha)$ -probabilistic worst-case region otherwise  $\delta (\tau_i) = 0$ .
As for the optimal rate for convergence or the generalization bound, it is up to specific meta learning methods and risk minimization principles. For worst-case risk minimization for meta learning, there already exists theoretical analysis in previous work, e.g., optimization-based meta learning (Collins et al., 2020) when fast adaptation functions hold the convexity property. When it comes to more universal cases, considering diverse meta learning methods and optimization strategies, it is still challenging to estimate the optimal rate for convergence. Also, our considered scenarios exist distributional drift between meta training and meta testing task distributions, which makes it tough to derive the generalization bound.
Finally, note that our developed optimization strategies for meta learning are regardless of meta learning methods, and DR-MAMLs and DR-CNPs are merely two examples. For the sake of convenience, we only implement DR-MAML to compare with TR-MAML in this work.
E Proof of Remark (2)
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 convex programming algorithms (Fan et al., 2017; Meng et al., 2020; Levy et al., 2020).
Proof: We at first show that  $\left[\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta) - \xi\right]^{+} := \max \{\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta) - \xi, 0\}$  is convex w.r.t  $\vartheta$  if  $\ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta)$  is convex w.r.t.  $\vartheta$ : For ease of derivation, let us redenote two functions  $f_{1}(\xi; \vartheta) := \ell(\mathfrak{D}_{\tau}^{T}, \mathfrak{D}_{\tau}^{C}; \vartheta) - \xi$  and  $f_{2}(\xi; \vartheta) := 0$ .
With any constant  $\lambda \in [0,1]$  and any two parameters  $\vartheta_{1}\in \Theta$  and  $\vartheta_{2}\in \Theta$ , we can have:
$$
\left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \lambda \vartheta_ {1} + (1 - \lambda) \vartheta_ {2}\right) - \xi \right] ^ {+} = f _ {i} (\xi ; \lambda \vartheta_ {1} + (1 - \lambda) \vartheta_ {2}) [ \text {f o r s o m e} i \in \{1, 2 \} ] \tag {13a}
$$
$$
\leq \lambda f _ {i} (\xi ; \vartheta_ {1}) + (1 - \lambda) f _ {i} (\xi ; \vartheta_ {2}) \tag {13b}
$$
$$
\leq \lambda \max  \left\{f _ {i} (\xi ; \vartheta_ {1}) \right\} _ {i = 1, 2} + (1 - \lambda) \max  \left\{f _ {i} (\xi ; \vartheta_ {2}) \right\} _ {i = 1, 2} \tag {13c}
$$
$$
\leq \lambda \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta_ {1}\right) - \xi \right] ^ {+} + (1 - \lambda) \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta_ {2}\right) - \xi \right] ^ {+}, \tag {13d}
$$
which shows that the risk function with slack variables is convex w.r.t.  $\vartheta$
As  $\varphi_{\alpha}(\xi ;\vartheta) = \xi +\frac{1}{1 - \alpha}\mathbb{E}_{p(\tau)}\left[\left[\ell (\mathfrak{D}_\tau^T,\mathfrak{D}_\tau^C;\vartheta) - \xi \right]^+\right]$  is the convex combination of the above mentioned convex function, the resulting  $\varphi_{\alpha}(\xi ;\vartheta)$  is naturally convex w.r.t.  $\vartheta$
F Proof of Proposition (1)
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  $\forall \{\vartheta, \vartheta'\} \in \Theta$ :
$$
\left| \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) - \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta^ {\prime}\right) \right| \leq \beta_ {\tau} | | \vartheta - \vartheta^ {\prime} | |.
$$
Assumption (2): For meta risk function values, the risk 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  $\mathcal{L}_{\mathrm{max}}$ :
$$
\sup  _ {\tau \in \Omega_ {\tau} ^ {\alpha}} \ell \left(\mathfrak {D} _ {\tau_ {i}} ^ {T}, \mathfrak {D} _ {\tau_ {i}} ^ {C}; \vartheta\right) \leq \mathcal {L} _ {\max }.
$$
Proposition (1): Under Assumptions (1)/(2)/(3), the meta learning optimization objective  $\mathcal{E}_{\alpha}(\vartheta)$  in Eq. (3) is continuous w.r.t.  $\vartheta$ .
Proof: Suppose that  $\forall \{\vartheta, \vartheta'\} \in \Theta$  and  $||\vartheta - \vartheta'|| < \delta$ , we can have the following inequality based on Assumption (2):
$$
\left| p _ {\alpha} (\tau ; \vartheta) - p _ {\alpha} (\tau ; \vartheta^ {\prime}) \right| \leq \beta_ {\theta} | | \vartheta - \vartheta^ {\prime} | |.
$$
Meanwhile, we can have the following inequality based on Assumption (1):
Together with the boundness Assumption (3) of meta risk function value:
$$
\sup  _ {\tau \in \Omega_ {\tau} ^ {\alpha}} \ell (\mathfrak {D} _ {\tau_ {i}} ^ {T}, \mathfrak {D} _ {\tau_ {i}} ^ {C}; \vartheta) \leq \mathcal {L} _ {\max },
$$
we can roughly estimate the probabilistic constrained expected meta risk function values as follows:
$$
\left| \mathcal {E} _ {\alpha} (\vartheta) - \mathcal {E} _ {\alpha} \left(\vartheta^ {\prime}\right) \right| = \left| \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta)} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right] - \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta^ {\prime})} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta^ {\prime}\right) \right] \right| \tag {14a}
$$
$$
= \left| \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta)} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right] - \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta^ {\prime})} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right] \right. \tag {14b}
$$
$$
\left. + \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta^ {\prime})} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) - \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta^ {\prime}\right) \right] \right| \tag {14c}
$$
$$
\begin{array}{l} \leq \int \left| p _ {\alpha} (\tau ; \vartheta) - p _ {\alpha} (\tau ; \vartheta^ {\prime}) \right| \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) d \tau + \mathbb {E} _ {p _ {\alpha} (\tau ; \vartheta^ {\prime})} \left[ \left| \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) - \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta^ {\prime}\right) \right| \right] (14d) \\ \leq \beta_ {\theta} \| \vartheta - \vartheta^ {\prime} \| \sup  _ {\tau \in \Omega_ {\tau} ^ {\alpha}} \left\{\ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \right\} + \sup  _ {\tau \in \Omega_ {\tau} ^ {\alpha}} \left\{\beta_ {\tau} \right\} \| \vartheta - \vartheta^ {\prime} \| (14e) \\ = \left(\beta_ {\theta} \mathcal {L} _ {\max } + \beta_ {\max }\right) \left\| \vartheta - \vartheta^ {\prime} \right\|. (14f) \\ \end{array}
$$
From the above inequality, we can see that  $\mathcal{E}_{\alpha}(\vartheta) - \mathcal{E}_{\alpha}(\vartheta^{\prime})$  is a  $(\beta_{\theta}\mathcal{L}_{\mathrm{max}} + \beta_{\mathrm{max}})$ -Lipschitz continuous w.r.t.  $\vartheta$ . As a result, we demonstrate Proposition (1).
G Proof of Proposition (2)
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
$$
\varphi_ {\alpha} (\hat {\xi} _ {\alpha} (\vartheta); \vartheta) - \kappa_ {\alpha} \delta <   \mathcal {E} _ {\alpha} (\vartheta) \leq \varphi_ {\alpha} (\hat {\xi} _ {\alpha} (\vartheta); \vartheta).
$$
Proof: Based on the direct deduction in work (Rockafellar et al., 2000), we know that for any  $\vartheta \in \Theta$ , the inequality holds:  $\varphi_{\alpha}(\xi_{\alpha};\vartheta) \leq \varphi_{\alpha}(\hat{\xi}_{\alpha};\vartheta)$ .
In the case when  $\hat{\xi}_{\alpha} = \xi_{\alpha} + \delta$  with  $\delta \in \mathbb{R}^{+}$ , the probability space of the task  $\Omega_{\tau}$  can be respectively partitioned into three disjoint probability space:
$$
\mathbb {P} ^ {-} (\tau) = \mathbb {P} \left(\left\{\mathcal {M} _ {\vartheta} ^ {- 1} (\ell) \mid \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \leq \xi_ {\alpha}, \tau \in \Omega_ {\tau} \right\}\right) \tag {15a}
$$
$$
\mathbb {P} ^ {+} (\tau) = \mathbb {P} \left(\left\{\mathcal {M} _ {\vartheta} ^ {- 1} (\ell) \mid \xi_ {\alpha} <   \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \leq \xi_ {\alpha} + \delta , \tau \in \Omega_ {\tau} \right\}\right) \tag {15b}
$$
$$
\mathbb {P} ^ {+ +} (\tau) = \mathbb {P} \left(\left\{\mathcal {M} _ {\vartheta} ^ {- 1} (\ell) \mid \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) > \xi_ {\alpha} + \delta , \tau \in \Omega_ {\tau} \right\}\right). \tag {15c}
$$
As a result, we can estimate the difference between the two terms as follows:
$$
\varphi_ {\alpha} \left(\hat {\xi} _ {\alpha}; \vartheta\right) - \varphi_ {\alpha} \left(\xi_ {\alpha}; \vartheta\right) = \hat {\xi} _ {\alpha} - \xi_ {\alpha} \tag {16a}
$$
$$
+ \frac {1}{1 - \alpha} \mathbb {E} _ {\tau \sim \mathbb {P} ^ {+} (\tau)} \left[ \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) - \xi_ {\alpha} \right] + \frac {1}{1 - \alpha} \mathbb {E} _ {\tau \sim \mathbb {P} ^ {+ +} (\tau)} \left[ \xi_ {\alpha} - \hat {\xi} _ {\alpha} \right] \tag {16b}
$$
$$
\leq \delta + \frac {1}{1 - \alpha} \mathbb {E} _ {\tau \sim \mathbb {P} ^ {+} (\tau)} [ \delta ] - \frac {1}{1 - \alpha} \mathbb {E} _ {\tau \sim \mathbb {P} ^ {+ +} (\tau)} [ \delta ] \leq \delta + \frac {1}{1 - \alpha} \delta = \frac {2 - \alpha}{1 - \alpha} \delta . \tag {16c}
$$
Similarly, in the case when  $\hat{\xi}_{\alpha} = \xi_{\alpha} - \delta$  with  $\delta \in \mathbb{R}^{+}$ , the probability space of the task  $\Omega_{\tau}$  can be partitioned into three disjoint space with the probability respectively:
$$
\mathbb {P} ^ {- -} (\tau) = \mathbb {P} \left(\left\{\mathcal {M} _ {\vartheta} ^ {- 1} (\ell) \mid \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) \leq \xi_ {\alpha} - \delta , \tau \in \Omega_ {\tau} \right\}\right) \tag {17a}
$$
$$
\mathbb {P} ^ {-} (\tau) = \mathbb {P} \left(\left\{\mathcal {M} _ {\vartheta} ^ {- 1} (\ell) \mid - \delta <   \ell \left(\mathfrak {D} _ {\tau} ^ {T}, \mathfrak {D} _ {\tau} ^ {C}; \vartheta\right) - \xi_ {\alpha} <   0, \tau \in \Omega_ {\tau} \right\}\right) \tag {17b}
$$