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A PAC-Bayesian bound for Lifelong Learning
Anastasia Pentina, Christoph H. Lampert
TL;DR
Lifelong learning seeks to transfer information to future tasks for which no data have yet been observed, but its theoretical properties remain limited. The paper develops a PAC-Bayesian generalization bound, uses it to derive principled algorithms for parameter and representation transfer, and reports results comparable with existing methods.
Problem
Lifelong learning must use observed tasks to prepare for future tasks without access to data from those tasks, limiting the direct applicability of ordinary prior-learning techniques.
Method
The paper develops a PAC-Bayesian lifelong-learning bound and optimizes it to derive algorithms for transferring parameters or low-dimensional representations.
Results
The bound unifies parameter and subspace transfer and yields principled algorithms with results comparable to existing manually designed methods.
Takeaways & Limitations
The bound can serve as a quality measure for transferred information and as a basis for principled lifelong-learning algorithms.
Takeaways & Limitations
The framework assumes tasks are sampled i.i.d. from an environment, while future work proposes relaxing this assumption and using hierarchical or multimodal hyperposteriors.
Abstract
from arXiv · showhide
Transfer learning has received a lot of attention in the machine learning community over the last years, and several effective algorithms have been developed. However, relatively little is known about their theoretical properties, especially in the setting of lifelong learning, where the goal is to transfer information to tasks for which no data have been observed so far. In this work we study lifelong learning from a theoretical perspective. Our main result is a PAC-Bayesian generalization bound that offers a unified view on existing paradigms for transfer learning, such as the transfer of parameters or the transfer of low-dimensional representations. We also use the bound to derive two principled lifelong learning algorithms, and we show that these yield results comparable with existing methods.
1. Introduction
The paper frames lifelong learning as transferring prior knowledge to future tasks without observed data, and develops a PAC-Bayesian basis for principled transfer algorithms.
- Motivation: Machine learning systems typically need large training datasets, unlike humans who can learn new concepts from few examples using previously acquired knowledge.
- Transfer-learning setting: Lifelong learning aims to perform well on future tasks using information from observed tasks, even though no data are available for those future tasks.
- Contribution: The paper proves a PAC-Bayesian bound relating expected loss on a future task to average loss on observed tasks.
- Contribution: The bound depends on data representation and the task-learning algorithm, allowing transferred information to be evaluated and optimized.
- Algorithms: Two cases yield principled algorithms: transferring parameter vectors with task-specific perturbations and transferring a shared low-dimensional feature subspace.
- PAC-Bayesian framework: The PAC-Bayesian framework analyzes randomized Gibbs predictors and balances empirical loss against divergence from a prior distribution.
2. PAC-Bayesian Lifelong Learning
The paper develops a PAC-Bayesian framework for lifelong learning that bounds future-task transfer risk using observed tasks, then uses the bound to learn task-adapted priors and representations.
- Lifelong learning targets future tasks with no observed data, so the agent must infer useful prior knowledge from previously observed tasks.The setting is harder than multitask learning and domain adaptation because future-task data are unavailable when decisions are made.
- The framework treats the prior P as a random variable and learns a hyperposterior Q over priors from observed tasks.The resulting hyperposterior is intended to provide promising priors for future tasks while retaining PAC-Bayesian guarantees.
- The main theorem bounds the difference between transfer risk on a randomly sampled future task and empirical multi-task risk on observed tasks.Its proof separates uncertainty at the task-environment level from uncertainty within observed tasks and uses two corresponding complexity terms.
- The bound's environment-level complexity decreases as 1/√n with more tasks, whereas task-level uncertainty can remain when each task has finite data.Conversely, unlimited data per task reduces the task-level term as 1/√¯m but leaves environment-level uncertainty when the number of tasks is finite.
- Minimizing the observable bound yields priors or representations adapted to the task environment, including parameter-transfer and low-dimensional-subspace instantiations.The representation formulation favors subspaces that support low loss and small weight norms across observed tasks.
3. Experiments
The experiments evaluate principled parameter- and representation-transfer algorithms on three datasets, using held-out future tasks and repeated transfer-risk estimates. PL-G and PL-L achieve performance comparable to existing methods, with improvements depending on observed-task count and hyperparameter choices.
- Experimental setup: Experiments compare PL-G and PL-L with adaptive ridge regression and ELLA across three public datasets.PL-G implements parameter transfer, while PL-L implements representation transfer.
- Experimental setup: The evaluation holds out future tasks, varies the number of observed tasks, and repeats each procedure 100 times to estimate mean errors and standard errors.Held-out tasks are excluded from training and used only for evaluation.
- Evaluation measures: Results use AUC for unbalanced Landmine classification, mean 0/1 error for balanced Animals classification, and MSE for Schools regression.Higher AUC is better, whereas lower MSE is better.
- Parameter Transfer: For parameter transfer, PL-G is comparable to manually designed techniques and improves over the baseline given sufficiently many observed tasks.With σ = 10, PL-G converges faster and achieves results comparable with ARR or slightly better.
- Representation Transfer: For representation transfer, PL-L achieves improvements comparable to ELLA, while performance on Schools depends more strongly on k and σ.On Schools, PL-L with σ = 10 nearly matches ELLA, whereas σ = 1 yields smaller improvements.
4. Conclusion
The paper develops a PAC-Bayesian generalization bound for lifelong learning and uses its observable quantities to derive principled algorithms. The bound recovers parameter and representation transfer as special cases, while future work targets broader task assumptions.
- The main result is a generalization bound in a PAC-Bayesian framework for lifelong learning.
- The bound unifies transfer of classifier parameters and transfer of subspaces or representations as special cases.
- Because the bound uses only observable quantities, it can derive principled lifelong-learning algorithms with results comparable to existing manually designed methods.
- A unimodal hyperposterior assumes that all tasks are related to each other.
- Future work includes hierarchical or multimodal hyperposteriors and relaxing the assumption that tasks are sampled independently and identically from an environment.
A. Proof of Lemma 2
The proof combines a Kullback-Leibler divergence property with independence and concentration arguments. Hoeffding’s lemma and Markov’s inequality yield the probability statement needed to obtain the target result.
- The proof invokes Hoeffding’s lemma for bounded real-valued random variables.
- It also uses a Kullback-Leibler divergence property valid for every λ > 0 and provable by convex duality.
- Independence of the factors allows Hoeffding’s lemma to be applied to each factor.
- Since π is fixed independently of the samples, expectations can be exchanged before applying Markov’s inequality.
- Combining the resulting inequalities yields equation (2).