Theoretical Investigations and Practical Enhancements on Tail Task Risk Minimization in Meta Learning

TL;DR

This work reduces the distributionally robust strategy to a max-min optimization problem, constitute the Stackelberg equilibrium as the solution concept, and estimate the convergence rate, and derives the generalization bound in the presence of tail risk.

Abstract

Meta learning is a promising paradigm in the era of large models and task distributional robustness has become an indispensable consideration in real-world scenarios. Recent advances have examined the effectiveness of tail task risk minimization in fast adaptation robustness improvement \citep{wang2023simple}. This work contributes to more theoretical investigations and practical enhancements in the field. Specifically, we reduce the distributionally robust strategy to a max-min optimization problem, constitute the Stackelberg equilibrium as the solution concept, and estimate the convergence rate. In the presence of tail risk, we further derive the generalization bound, establish connections with estimated quantiles, and practically improve the studied strategy. Accordingly, extensive evaluations demonstrate the significance of our proposal and its scalability to multimodal large models in boosting robustness.

Figures (15)

• Figure 1: Illustration of optimization stages in distributionally robust meta learning from a Stackelberg game. Given the DR-MAML example, the pipeline can be interpreted as bi-level optimization: the leader's move for characterizing tail task risk and the follower's move for robust fast adaptation.
• Figure 2: The sketch of theoretical and empirical contributions in two-stage robust strategies. On the left side is the two-stage distributionally robust strategy wang2023simple. The contributed theoretical understanding is right-down, with the right-up the empirical improvement. Arrows show connections between components.
• Figure 3: Meta testing performance in sinusoid regression problems (5 runs). The charts report testing mean square errors (MSEs) over 490 unseen tasks collins2020task with $\alpha=0.7$, where black vertical lines indicate standard error bars.
• Figure 4: Meta testing performance in Pendulum 10-shot and 20-shot problems (5 runs). Reported are testing MSEs over 529 unseen tasks with $\alpha=0.5$, where black vertical lines indicate standard error bars.
• Figure 5: $\text{VaR}_{\alpha}$ approximation errors with the crude MC and KDE. We compute the difference between the estimated $\hat{\text{VaR}}_{\alpha}$ and the Oracle $\text{VaR}_{\alpha}$ in the absolute value $|\hat{\text{VaR}}_{\alpha}-\text{VaR}_{\alpha}|$.

Theorems & Definitions (11)

• Example 1: DR-MAML wang2023simple
• Proposition 1
• Definition 1: Global Stackelberg Equilibrium
• Proposition 2: Existence of Equilibrium
• Definition 2: Local Stackelberg Equilibrium
• Theorem 4.1: Convergence Rate for the Second Player
• Theorem 4.2: Asymptotic Performance Gap in Tail Task Risk
• Theorem 4.3: Generalization Bound in the Tail Risk Cases
• Theorem 4.4
• Remark 1