[Paper Review] Bilevel Programming for Hyperparameter Optimization and Meta-Learning

Luca Franceschi, Paolo Frasconi+3 more|arXiv (Cornell University)|Jun 13, 2018

Domain Adaptation and Few-Shot Learning37 references110 citations

TL;DR

The paper presents a unified bilevel optimization framework that connects gradient-based hyperparameter optimization with meta-learning, and demonstrates its effectiveness for learning-to-learn in few-shot tasks. It instantiates a hyper-representation approach in deep networks and provides theoretical convergence guarantees for the approximate inner-outer problem.

ABSTRACT

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.

Motivation & Objective

• Motivate a unified mathematical framework for HO and ML as bilevel optimization.
• Show that approximate inner-outer problems can converge to the exact bilevel formulation under reasonable conditions.
• Instantiate the approach for meta-learning with shared representations across episodes in deep networks.
• Demonstrate empirical benefits on few-shot learning benchmarks (Omniglot and MiniImagenet).

Proposed method

• Formulate HO and ML as a bilevel problem with inner objective L_lambda and outer objective E.
• Solve the bilevel problem approximately by simulating inner optimization dynamics over T steps to obtain w_{T,λ}.
• Compute hypergradients via an extended reverse hypergradient algorithm to update hyperparameters λ.
• Instantiate ML by learning a shared representation h_λ across tasks and training task-specific classifiers g^j.
• Provide theoretical results ensuring existence and convergence of the approximate problem to the exact bilevel problem under mild assumptions.
• Experiment with representation learning in deep nets and analyze the impact of inner-iteration count T on performance.

Experimental results

Research questions

• RQ1 Can a bilevel formulation unify hyperparameter optimization and meta-learning in a single mathematical framework?
• RQ2 Under what conditions do the approximate inner-outer solutions (with finite T) converge to the exact bilevel solution?
• RQ3 Does learning a shared hyper-representation across tasks improve few-shot learning performance?
• RQ4 How does the number of inner-optimization steps T affect generalization and training time in few-shot settings?

Key findings

• The approximate bilevel approach converges to the exact problem as inner iterations T→∞ under suitable continuity and compactness assumptions.
• Early stopping (small T) can act as a regularizer and may yield better generalization than large-T solutions in some settings.
• Hyper-representation with shared representation layers improves few-shot accuracy on Omniglot and MiniImagenet compared to several baselines.
• Using a residual network as the representation mapping significantly boosts performance over a plain convolutional network in the hyper-representation setup.
• The proposed Hyper-representation method achieves competitive results with state-of-the-art few-shot learning approaches, highlighting the value of learned shared representations.