Biased Stochastic First-Order Methods for Conditional Stochastic Optimization and Applications in Meta Learning
This work addresses a bottleneck in gradient estimation for conditional stochastic optimization, which is incremental as it builds on existing methods by introducing bias to improve efficiency in applications like meta-learning.
The paper tackles the challenge of constructing gradient estimators for conditional stochastic optimization problems, such as meta-learning, by proposing a biased stochastic gradient descent (BSGD) algorithm and analyzing its bias-variance tradeoff. It establishes sample complexities for various objective types and introduces an accelerated algorithm, BSpiderBoost, that matches lower bounds in specific settings, with numerical experiments validating performance on tasks like invariant logistic regression and meta-learning.
Conditional stochastic optimization covers a variety of applications ranging from invariant learning and causal inference to meta-learning. However, constructing unbiased gradient estimators for such problems is challenging due to the composition structure. As an alternative, we propose a biased stochastic gradient descent (BSGD) algorithm and study the bias-variance tradeoff under different structural assumptions. We establish the sample complexities of BSGD for strongly convex, convex, and weakly convex objectives under smooth and non-smooth conditions. Our lower bound analysis shows that the sample complexities of BSGD cannot be improved for general convex objectives and nonconvex objectives except for smooth nonconvex objectives with Lipschitz continuous gradient estimator. For this special setting, we propose an accelerated algorithm called biased SpiderBoost (BSpiderBoost) that matches the lower bound complexity. We further conduct numerical experiments on invariant logistic regression and model-agnostic meta-learning to illustrate the performance of BSGD and BSpiderBoost.