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Continual Deep Learning by Functional Regularisation of Memorable Past

Pingbo Pan, Siddharth Swaroop, Alexander Immer, Runa Eschenhagen, Richard E. Turner, Mohammad Emtiyaz Khan

arXiv:2004.14070v4stat.MLcs.LG

TL;DR

Continual deep learning must adapt to new tasks without forgetting earlier skills, but weight regularisation may not preserve predictions and functional regularisation is costly. FROMP addresses this by selecting memorable past examples and using a Gaussian-process functional prior during weight-space training. It achieves state-of-the-art performance across benchmarks while combining regularisation with memory-based continual learning.

  • Problem

    Deep networks can catastrophically forget past skills, while existing weight- and functional-regularisation methods do not reliably preserve earlier predictions or scale efficiently.

  • Method

    FROMP selects influential memorable past examples and uses a Gaussian-process formulation to impose functional regularisation during weight-space training.

  • Results

    FROMP achieves state-of-the-art performance across benchmarks and consistently outperforms existing weight- and functional-regularisation methods.

  • Takeaways & Limitations

    The approach naturally combines regularisation and memory-based methods for continual and life-long learning.

  • Takeaways & Limitations

    The method lacks theoretical guarantees, so the paper advises against applying life-long learning methods to mission-critical problems until such guarantees are available.

Abstract

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Continually learning new skills is important for intelligent systems, yet standard deep learning methods suffer from catastrophic forgetting of the past. Recent works address this with weight regularisation. Functional regularisation, although computationally expensive, is expected to perform better, but rarely does so in practice. In this paper, we fix this issue by using a new functional-regularisation approach that utilises a few memorable past examples crucial to avoid forgetting. By using a Gaussian Process formulation of deep networks, our approach enables training in weight-space while identifying both the memorable past and a functional prior. Our method achieves state-of-the-art performance on standard benchmarks and opens a new direction for life-long learning where regularisation and memory-based methods are naturally combined.

1 Introduction

Continual learning requires retaining useful past experiences while adapting to new tasks, but deep networks can catastrophically forget when earlier data are unavailable. FROMP combines functional regularisation with memorable examples and a Gaussian-process formulation to address this limitation.

  • Motivation: Deep-learning methods can quickly forget previously acquired skills when learning new tasks, especially when earlier data are unavailable.This creates challenges for applications such as robotics, where tasks may appear during training.
  • Motivation: Weight regularisation may not preserve past predictions because network outputs depend on weights in a complex way.Functional regularisation instead directly constrains network outputs, but it is computationally costly.
  • FROMP: FROMP regularises outputs at a few memorable past examples and uses a Gaussian-process formulation to exploit their function-space correlations during weight training.The method requires only a slight modification of Adam and a minor increase in computation cost.
  • FROMP: FROMP connects functional regularisation with memory-based continual-learning approaches by imposing functional constraints at selected datapoints.This positions the method within hybrid approaches combining inference-based and memory-based ideas.
  • Related work: FROMP differs from related methods through its all-weights kernel, past-mean functional prior, and cheaper, interpretable memorable examples.The memorable examples avoid a separate discrete optimisation problem and can be interpreted intuitively.

2 Continual Learning with Weight/Functional Regularisation

Weight regularisation constrains parameters to reduce forgetting, but parameter similarity does not reliably preserve predictions because of neural-network symmetries. Functional regularisation directly constrains past outputs, yet remains expensive and often requires selecting a smaller set of past inputs.

  • Problem setting: Standard continual-learning settings may introduce new classes while old classes disappear, causing vanilla minibatch methods to catastrophically forget past information.The paper targets settings where unbiased gradients over all past data are unavailable.
  • Weight regularisation: Weight regularisation keeps important parameters near their values from previous tasks, but the resulting methods do not always produce satisfactory results.EWC and related approaches commonly use Fisher-information or variational precision matrices, often diagonally approximated.
  • Functional regularisation: Weight similarity may not preserve predictions because parametric symmetries make exact parameter values less important than network outputs.This motivates directly regularising function values on past-task examples.
  • Functional regularisation: Functional regularisation is expected to perform better by constraining outputs directly, but it can require storing all past data and evaluating functions over them.Existing approaches reduce this cost with working memories or inducing points, yet they do not consistently outperform weight regularisation.

3 Functional-Regularisation of Memorable Past (FROMP)

FROMP converts a deep network into a Gaussian-process representation, identifies influential past examples using a cheap relevance measure, and trains weights with a functional prior over those examples. Its approximation preserves function-space correlations while keeping optimisation compatible with Adam.

  • GP functional prior: FROMP uses DNN2GP to convert a neural-network weight posterior into a Gaussian-process posterior that serves as a functional prior for the next task.The resulting GP posterior is used to regularise future training in function space.
  • Identifying memorable past: Memorable past examples are selected by ranking each example’s loss-derived noise precision and choosing the most influential examples.The relevance measure is cheap to compute and tends to select difficult examples near the decision boundary.
  • Algorithm: The training procedure adds functional-prior computations to Adam, updates the current task’s weights, and identifies a new memorable past after convergence.The algorithm maintains diagonal weight uncertainty while retaining memorable examples across previous tasks.
  • Functional regularisation: The functional regulariser forces network outputs at memorable examples to remain similar to their past values while weighting them with a kernel matrix.The kernel accounts for uncertainty and the weighting of memorable examples from earlier tasks.
  • Weight-space training: FROMP exploits correlations among memorable examples through a full kernel matrix while optimising a computationally cheap weight-space objective with Adam.This combines function-space correlations with weight-space training.

4 Experiments

Experiments across MNIST and Split CIFAR show that FROMP generally outperforms competing continual-learning methods, with careful memory selection especially valuable when memory is small. Its kernel and transfer properties provide additional benefits, although these vary by benchmark.

  • Permuted and Split MNIST: FROMP outperforms weight-regularisation methods and FRCL on MNIST, while also exceeding Improved VCL with a random coreset.The comparison uses standard errors over 5 runs.
  • Permuted and Split MNIST: FROMP achieves 99.2±0.1% accuracy on Split MNIST using a smaller network architecture.This result is reported over 5 runs.
  • Split CIFAR: On Split CIFAR, FROMP is close to the joint-training upper limit and outperforms all other methods across tasks.It matches joint training on tasks 4–6 and achieves the best average performance.
  • Split CIFAR: With 10 memorable examples, careful selection raises average accuracy to 70% from 45% with FRORP.The kernel does not significantly improve over FROMP-L2 in this experiment.
  • Split CIFAR: FROMP achieves 6.1 ± 0.7% forward transfer, compared with 0.17 ± 0.9% for EWC and 1.8 ± 3.1% for VCL+coresets.Its backward-transfer score is −2.6 ± 0.9%, comparable to EWC’s −2.3±1.4% and better than VCL+coresets’ −9.2±1.8%.

5 Discussion

The discussion presents FROMP as a functional-regularisation approach that combines Gaussian-process modelling with memorable examples while retaining simple weight-space training. It reports state-of-the-art performance but identifies scalability, modelling, memory-selection, and theoretical questions for future work.

  • Discussion: FROMP converts weight-space distributions into function-space using a Gaussian Process formulation of neural networks.It uses this formulation to identify memorable examples and functionally regularise neural-network weights.
  • Discussion: The method combines functional constraints at datapoints with memory-based approaches while maintaining simple training methods.The paper describes this as an initial direction connecting neural-network and GP communities.
  • Discussion: The paper leaves open whether FROMP will scale to large datasets such as ImageNet and how to obtain better memory selection, task-boundary detection, and theoretical guarantees.It also raises questions about variational inference, assumptions, and Bayesian formulations.

Broader Impact

The paper situates continual deep learning within lifelong learning and develops GP-based posterior approximations for neural networks across common loss settings. It also identifies practical limitations, including approximation error and the absence of theoretical guarantees for mission-critical use.

  • Broader Impact: Lifelong learning methods could extend deep learning to settings where data is limited and collected gradually, including robotics, medicine, healthcare, and climate science.
  • Broader Impact: A central limitation is the lack of theoretical guarantees, so the paper advises against applying lifelong-learning methods to mission-critical problems.
  • GP Posterior Formulations: The paper develops GP posterior formulations for binary and multiclass neural-network classification, including sigmoid- and softmax-based predictive distributions.The binary formulation uses a sigmoid-transformed mean and covariance terms involving the Jacobian and loss curvature; the multiclass formulation extends this through softmax outputs.
  • Practical Approximation: The optimizer implementation uses RMSprop/Adam for convenience, which may make intermediate variance estimates inaccurate, followed by covariance-diagonal correction after convergence.The fixed point is unchanged despite replacing the theoretically derived update, and the covariance is recomputed at the converged solution.

B Detailed Derivation of FROMP Algorithm

The detailed derivation constructs a GP predictive posterior from neural-network weights and uses it to form a computationally cheaper functional regularizer. A sequence of approximations reduces cost through mean substitution, derivative simplification, task factorization, deterministic updates, diagonal covariance, and multiclass decomposition.

  • GP Posterior Construction: The method defines GP mean and covariance functions from neural-network outputs, Jacobians, and an approximate weight covariance, then forms predictive posteriors for sampled weights.For binary classification, the predictive distribution is Gaussian with sigmoid mean and covariance involving Λ, J, and Σ.
  • Functional Regularization: The functional regularizer penalizes differences between current and previous GP means at memorable examples, weighted by inverse covariance matrices from past tasks.Its analytical form decomposes across previous tasks and classes in the multiclass setting.
  • Computational Approximations: Five approximations reduce computational cost by replacing sampled weights with posterior means, omitting covariance derivatives, factorizing task kernels, using deterministic updates, and adopting diagonal covariance.These substitutions simplify gradient computation and kernel inversion while producing a cheaper approximation of the original objective.
  • Optimizer Approximation: The final optimizer uses an RMSprop-like update with diagonal covariance and a regularizer δ to avoid division by zero.The resulting covariance has diagonal entries 1/(s + δ1), and the objective is computationally cheaper to optimize.
  • Multiclass Extension: In multiclass classification, independent GPs for each class make kernel inversion linear in the number of classes per task.Each class-task block has size M × M, and the decomposition reduces the computation relative to the full multiclass GP.
  • Memorable Past and Approximation Limits: Memorable examples are selected by sorting the trace of each classification Hessian matrix, while the function-space integral remains an approximation when the prior cannot be expressed solely through network outputs.The latter limitation arises because log q_{t−1}(w) is not always a function of f := J_{w_t}w.

E Further details on continual learning metrics reported

The paper evaluates forward and backward transfer on Split CIFAR using accuracy-based metrics, alongside benchmark comparisons and sensitivity analyses. FROMP shows stronger forward transfer than EWC and VCL, while backward transfer is comparable to EWC.

  • Transfer metrics: Forward transfer measures the average improvement in accuracy on a new task over a model trained only on that task.The evaluation reports higher values as better for both transfer metrics.
  • Forward transfer: FROMP achieves 6.1 ± 0.7% forward transfer, compared with 0.17 ± 0.9% for EWC and 1.8 ± 3.1% for VCL+coresets.These results indicate stronger forward transfer for FROMP on Split CIFAR.
  • Backward transfer: FROMP obtains −2.6 ± 0.9% backward transfer, comparable to EWC’s −2.3 ± 1.4% and better than VCL+coresets’ −9.2 ± 1.8%.Backward transfer roughly captures the difference between a task’s initial accuracy and its accuracy after the final task.
  • Additional evaluation: A smaller-network comparison reports FROMP at (99.2 ± 0.1)%, improving over the larger-network result.The authors attribute this improvement to a pruning effect described in prior work.
  • Hyperparameter sensitivity: Changing τ by an order of magnitude does not significantly affect final average accuracy, whereas larger changes cause more than a 0.1% loss.This sensitivity analysis concerns FROMP and FROMP-L2.

F.3 Split CIFAR

On Split CIFAR, FROMP is evaluated against regularisation baselines and variants across standard and extended task sequences. Its advantage over random memorable examples is clearest when memory is small, while its improvement over FRORP is not always significant.

  • Standard Split CIFAR: FROMP reaches (76.2 ± 0.4)% final average validation accuracy on Split CIFAR, exceeding EWC at (71.6 ± 0.9)% and VCL + random coreset at (67.4 ± 1.4)%.FROMP-L2 obtains (75.6 ± 0.4)%, while SI obtains (73.5 ± 0.5)%.
  • Memory selection: Careful memorable-example selection in FROMP gives better or more consistent results than random examples in FRORP, particularly when memory size is small.The comparison concerns Split CIFAR with 11 tasks.
  • Method illustration: In the toy illustration, memorable examples support the decision boundary, and matching previous outputs preserves performance across five tasks.The new decision boundary classifies both the first and second tasks well.
  • Longer task sequence: With 11 Split CIFAR tasks, FROMP typically outperforms FRORP, especially at smaller memorable-past sizes, but is similar to FROMP-L2.The extended experiment compares different memorable-past sizes.
  • Experimental setting: When fewer memorable examples are used, τ is increased to compensate for the reduced number of datapoints.For Split CIFAR, 40 rather than 200 points changes τ from 10N to 50N.

G Toy data experiments

Toy 2D experiments examine the brittleness of weight-space regularisation and the consistency of FROMP. Across visualisations and repeated runs, FROMP maintains performance more reliably than VCL-based alternatives.

  • Toy-data comparison: Weight-regularisation methods such as VCL with coresets show lower accuracy, higher variance across random seeds, and visually poor decision boundaries than functional regularisation.The experiments use a small multilayer perceptron on a 2D binary classification dataset.
  • Repeated runs: FROMP is very consistent, whereas VCL with coresets performs well in only 1 of 5 runs and usually fails in the other 4.Without coresets, VCL forgets many past tasks and has very low performance.
  • Coreset selection: VCL remains inconsistent regardless of whether its coreset points are random or selected from FROMP’s memorable past.The selection of coreset points does not explain the observed inconsistency.
  • Visualisation: Three middle-performing VCL-MP runs make VCL’s inconsistent behaviour visually clear.The figure selects three runs from five random seeds.

G.2 Dataset variations

Additional toy-data experiments test FROMP across dataset variations and compare it with batch-trained Adam. FROMP performs well across these variations, while batch Adam serves as an upper-bound reference.

  • Dataset variations: FROMP performs well across variations of the toy 2D binary-classification dataset.The reported means and standard deviations are based on 10 FROMP runs and 3 Adam runs.
  • Reference model: Batch-trained Adam is presented as an upper bound on performance for the dataset-variation experiments.The comparison uses visualisations of selected FROMP and batch Adam runs.
  • Dataset scale: The visual comparisons include FROMP on a dataset 10× smaller, with 400 points per task, and on a dataset 10× larger, with 40,000 points per task.These figures compare FROMP with batch Adam.

G.3 VCL and FROMP hyperparameter settings for toy datasets

The toy-dataset experiments fix FROMP’s hyperparameters across variations and examine why its kernel should include uncertainty from all network layers.

  • FROMP settings: FROMP uses 50 epochs, batch size 20, and learning rate 0.01, selected from five-run toy-dataset hyperparameter searches.These settings were fixed across toy-data experimental runs, with epochs scaled for dataset size.
  • VCL+coresets settings: VCL+coresets hyperparameters were likewise selected through five-run searches using mean training accuracy.The search varied epochs, coreset epochs, learning rate, batch size, and prior variance.
  • All-layer kernel: All network layers exhibit high weight entropy, especially during the first tasks, indicating uncertainty is not confined to the last layer.The analysis uses entropy histograms for Gaussian weight distributions across layers and tasks.
  • Toy-dataset variations: Figure 9 compares the middle-performing FROMP run with batch Adam on a dataset containing a new, easy sixth task.The figure is one of the toy-dataset variations used to visualize experimental behavior.
  • All-layer kernel: Considering uncertainties across weights in all layers, rather than only the last layer, is expected to improve performance.This motivates defining FROMP’s kernel over all network weights.

H Task boundary detection

The task-boundary experiment detects changes without task IDs by comparing predictions across minibatches and applying a thresholded Welch’s t-test.

  • Experimental setting: The experiment uses 10 Permuted MNIST tasks whose minibatches arrive without task-boundary information.The goal is to identify when each new minibatch belongs to a new task.
  • Detection principle: New-task data is expected to produce similar predictions under current and prior networks, unlike data from the current task.The method uses this prediction contrast before training on each minibatch.
  • Detection procedure: For each minibatch, the method computes squared prediction-mean differences, calculates Welch’s t-test statistics, and flags a task change above a threshold.In the multiclass setting, the statistic is repeated across functions and averaged.
  • Results: The method recognised every task change without mistakes across a wide threshold range, although testing was skipped during the first 10 iterations of each new task.The first-10-iterations exclusion is an explicit experimental condition.
  • Results: A threshold range of approximately 0.9 to 1.8 successfully recognised task changes in the reported run.The range was limited by the requirement to detect the first task change.
  • Scope and limitation: Mean predictions were sufficient in this setting, while more complicated scenarios might require comparing full GP predictive distributions.The authors suggest using a divergence against the GP prior for that extension.
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