Learning where to learn: Gradient sparsity in meta and continual learning
Authors: Johannes von Oswald
Dominic Zhao
Seijin Kobayashi
Simon Schug
Massimo Caccia
Nicolas Zucchet
Joao Sacramento
Venue: NeurIPS 2021 Poster (2021)
Keywords: meta-learning
continual learning
few-shot learning
deep learning
sparsity
sparse gradients

Abstract: Finding neural network weights that generalize well from small datasets is difficult. A promising approach is to learn a weight initialization such that a small number of weight changes results in low generalization error. We show that this form of meta-learning can be improved by letting the learning algorithm decide which weights to change, i.e., by learning where to learn. We find that patterned sparsity emerges from this process, with the pattern of sparsity varying on a problem-by-problem basis. This selective sparsity results in better generalization and less interference in a range of few-shot and continual learning problems. Moreover, we find that sparse learning also emerges in a more expressive model where learning rates are meta-learned. Our results shed light on an ongoing debate on whether meta-learning can discover adaptable features and suggest that learning by sparse gradient descent is a powerful inductive bias for meta-learning systems.

Learning where to learn: Gradient sparsity in meta and continual learning
Johannes von Oswald*.1 Dominic Zhao Seijin Kobayashi', Simon Schug', Massimo Caccia", Nicolas Zucchet', João Sacramento'
Equal contribution
Institute of Neuroinformatiis, University of Zürich and ETH Zürich Mila,, University of Montreal & ServiceNow {voswaldj, dozhao)@eehhz.dl ch
Abstract
Finding neural network weights that generalize well from small datasets is difficult. A promising approach is to learn a weight initialization such that a small number of weight changes results in low generalization error. We show that this form of meta-learning can be improved by letting the learning algorithm decide which weights to change, i.e., by learning where learn. We find that patterned sparsity emerges from this process, with the pattern of sparsity varying on problem-by- problem basis. This selective sparsity results in better generalization and less interference in a range of few-shot and continual learning problems. Moreover, we find that sparse learning also emerges more expressive model where learning rates are meta-lcarned Our results shed light on an ongoing debate on whether meta-learning can discover adaptable features and suggest that learning by sparse gradient descent is powerful inductive bias for meta-learning systems.
Introduction
Meta-learning holds the promise of discovering inductive biases that improve the performance of primary learning process. Such set of assumptions can materialize in various elements of the learner. The well- koolnn model-agnostic meta-learning [MAML; algorithm aims to learn a neural network initialization that generalizes well to new learning tasks. More sophisticated meta- augment
this procedure by additionally modulating the inner-loop learning dynamics 35 32
12
It has been recently shown that applying MAML while adapting only last-layer weights leads to almost no decrease performance standard few-shot learning benchmarks. Our study builds upon the surprising effectiveness of this form of meta-learning, known as almost no inner-loop training [ANIL: 44). Here, instead deciding which weights to freeze a priori, we endow the meta-learner with the possibility to explicitly stop changing certain weights in the inner-loop learning process. We do this by introducing an adjustable binary mask which elementwise multiplied with gradient updates. This can be understood as simple form of learned gradient modulation that induces sparsity. Overlitting can thus be prevented and learning sped up by focusing adaptation to a sparse parameter subset, discovered by meta-learning.
We find that our sparse -MMM algorithm recovers a behavior that is reminiscent of ANIL. It induces high gradient sparsity in earlier layers of the network while allowing for adaptation in deeper layers including the network's output. Despite this reduction the number of adaptable parameters, the sparse learning patterns formed by sparse-MAML do not overly specialize to the family of tasks
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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observed during meta-learning: both ANIL and MAML are outperformed by the resulting sparse learners in cross-adaptation problems involving shift in task distribution g 40) Furthermore. sparsity adapts intuitively to the number of inner-loop gradient steps as well as its learning rate, few-shot dataset size, and network specifications. This leads to a robust and interpretable variant of MAML, that improves generalization by self-regularizing the parameters that the model should learn.
An exciting avenue of meta-learning research concerns continual learning. Learning tasks sequentially by gradient descent generally leads to poor results, as past tasks tend to be rapidly forgotten due to interfering weight updates. Such interference can be reduced with online meta-learmin methods which optimize the base learning algorithm using both present and past data, kept in replay buffer [5]. Our findings translate to this setting. We analyze the state-of-the thettrrrrt look-ahead MAML algorithm [La-MAML; [3] which introduces per-paune parameter meta-learned learning rates and find that sparse learning emerges, as a large fraction of learning rates drops to zero. Notably, similarly high performance can be reached when meta- meta-learning binary gradient masks only. Moreover, performance improves after endowing a version of MAMI. adapted for online learning 7 with binary gradient masks. Thus, sparse learning can improve generalization, accelerate future learning, and reduce forgetting, and these benefits can be realized within online meta-learning.
2 From MAML to sparse-MAML
The MAML algorithm seeks neural network weights @ from which only few gradient descent steps suffice to reach high performance on a given task т, that is assumed to be drawn from a certain distribution p(r). Formally, a task is defined by an outer loss function L and an inner loss function in We will later make explicit the form the two loss functions can take depending on the problem being solved. The result of the inner loss minimization is evaluated by the outer loss leading to the following optimization problem:
MAMI.
min E,)p(rr [Iout (0+, (0))] s.t. 0T,K+1 Φr.6 V (d+,k) and Φ-,0 0
(1)
with o,,* denoting the task-specific weights after k steps of gradient descent, a the inner-loop learning rate and K the inner-loop length. The initialization 0 is then obtained by iterative updating, using
- T BT=-(r) de L (or,x(0))]
with Da the outer-loop learning rate. Note that we need the total derivative de in Eq. 2 and not the partial derivative Va due the complex relationship between Φr.A and 0. In practice, the expectations over the task distribution that appear above are estimated by Monte Carlo integration. The updates in 0 therefore correspond to stochastic gradient descent on the expected outer loss.
In MAML, the total derivative w.r.t. to 0 is obtained by backpropagating through the inner opti- mization, resource-intensive procedure. First-order MAML (FOMAML) drastically reduces the computational cost by setting to zero the second-order derivatives that appear when differentiating the inner-loop update.
Learning the learning rates. Some variants of MAML focus on learning the learning rate and consider inner-loop updates of the following form:
Φr, = Φr.k a M V a I. (dr,)),
for some learnable preconditioning matrix M, that is optimized similarly to the initialization A. Through M, these algorithms learn some information on the geometry of the loss with the hope of faster inner-loop optimization. Meta-SGD [35] considers diagonal M, i.e., learnable learning rates, meta-curvature [4 considers a block matrix, while vanilla MAML corresponds to the M case.
Sparse-MAML In line with these approaches, we introduce sparse- MAM.. Together with an initial set of weights 0, our algorithm dynamically learns the parameters which will be updated and the ones that will not. Hence, sparse-MAML learns where to learn. To do so, we use a vector m (instead of matrix M) that modulates the gradient in the inner-loop update in the following way:
+ Φr, - a m20 V (
(4)
with 1:00 R. -> {0, 1)2 the step function that is applied elementwise to the underlying parameter vector E R and a the pointwise multiplication. We differentiate the step function by considering

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it linear: this method is called the straight-throghh estimator [3 and it was recently used for similar large-scale masking [4]] Following FOMAML, we ignore second-order derivatives. This leads to the update
K E
- m + I' Brrp(r)
v VoL:" k))
VoL
(5)
detailed derivation of the mask update, alongside the presentation of the initialization update, can be found in the supplementary material (SM).
Our mask update depends on the alignment between the outer-loss gradient g:th V I.g (or, k and the inner loss gradient g.th = Kー VoL." (Øx) accumulated over the inner loop trajectory. Learning tends to be shut off on coordinates for which these two quantities are of opposing sign, E, [ 9r,99 < O. Such freezing of learning when parameter updates are conflicting on the training and validation sets can decrease negative interference across tasks, which can turn improve generalization performance (39 49
3 Few-shot learning
Finding a network that performs well when trained on few samples of unseen data can be formulated meta-learning problem. We study here the supervised few-shot learning setting where tasks comprise small labelled datasets. A loss function C(ó, D) measures how much the predictions of network parameterized by & deviate from the ground truth labels dataset D. During meta-learning, the data of given task r is split into training and validation datasets, D and D., respectively. The sparse-MAML MAML formulation of few-shot learning then consists in optimizing the meta-parameen and m that, given the training set, in turn yield parameters & that improve validation set performance:
sparse-
min Exppp(r) (C( (0, m), 2")]
(6)
s.t. 0r = Q. - a m20 V VoL(or,k: Dz. and Φ,.0 = 8,
This corresponds to setting the outer- and inner-loop loss functions introduced in Section 2 (o"(() = ( D2) and (o) [( D.).
to
We apply sparse-MAML to the standard few-shot learning benchmark based on the minilmageNet dataset [47]. Our main purpose is to understand whether our meta-learning algorithm gives rise to sparse learning by shutting off weight updates, and if the resulting sparse learners achieve better generalization performance. Furthermore, we analyze the patterns of sparsity discovered by sparse- MAMI. over range of hyperparameter settings governing the meta-learning process.
Our experimental setup follows refs. 11 56 unless stated otherwise. particular, by default, our experimental results are obtained using the standard 4-convolutional-lan neural network (ConvNet)
model that has been intensively used to benchmark meta-learning algorithms. As is also conventional,
Source code available at: httt::///ghhbbb..c//ooaaaallaaad
citere_to_leary
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Iik: 100%
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Outpur
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Figure 1: Gradient sparsity emerges in 5-shot, 5-way minilmageNet classification, standard ConvNet model. Results averaged over 5 seeds + std. Left: Averaged gradient sparsity adapts for different sparsity initializations. Center: Different levels of gradient sparsity for convolutional and output layer weights emerge, with gradually less sparsity from earlier to deeper layers, while all being initialized 50% sparsity. Right: Sparse-MAML reaches higher test set accuracy for higher initial levels of gradient sparsity.

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we consider two data regimes: 5-shot 5-way, and 1-shot 5 5-way (the term 'shot" denotes the number of examples per class, and 'way" the number of classes). As we vary the hyperparameters of our algorithms, we monitor few-shot learning performance and the gradient sparsity level, defined for parameter group or the entire network as m [Immal?/di((mm) All experimental details can found in the SM.
3.1 Gradient sparsity decreases with layer depth
Our first finding validates and extends the phenomena described by Raghu et al. [4] and Chen et al. [ As shown Figure
Table 1: 5-way few-shot classification accuracy (%) on minilmageNeet, standard ConvNet model. We report mean + std. over 5 seeds. All results except ours taken from the respective papers (we use the symbol --- to indicate missing results). The results for meta-curvature (MC) are not directly comparable as additional data augmentation was used.
MAML. dynamically adjusts gradi- ent sparsity across the network, with very different values over the layers. As an ex- ample, we show the average gradient spar- sity of the four convolutional weight ma- trices and the output layer during training. The same trend is observed for other pa- rameter groups in the network except the output bias (for which sparsity is always high; see SM). Sparsity clearly correlates with depth and gradually increases towards the early layers of the network, despite the similar value before training (around 50%), i.e. MAALL suppresses inner-loop updates of weights in earlier layers while allowing deeper layers to adjust to new tasks. This effect is robust across differ- ent sparsity initializations, with final few- shot learning performance correlating with sparsity, cf. Figure] |
Method
1-shot
5-shot
MAML
48. 48.07=1.75
63.15=00.91
ANIL
46.70=0.40
61.50
BOIL
49.61 16
66.45-t0.37
Meta-SGD 335
50.47#1.87
64.03
MT- 32
51.70 1.84
MC (+data aug.)
41
54.23 1.888
68.47:10.69
Shrinkage p
47.7
exp-MANE MAML
48.38-t0.45
60.01
sparse-ReL ReL.UMMML
40.84
66.80+0.43
sparse-MAMI. sparse-MAML"
50.35
67. 67.03
51.04 10.59
68.05 1.0.54 10.84
These findings validate that our method can discover sparse learning algorithms. Moreover, they show that the level of sparsity is anti-correlated with depth. This result can be interpreted in the light of neural network models with human-engineered patterns of frozen features, which freeze layers of features based on their depth (in combination with MAML, see e.g. ANIL and BOIL, [44] 40)). Our method justifies these approaches, while outperforming them, cf. Table T. suggesting that might be preferable to meta-leamy which features to freeze. We note that another related method for automatic discovery of task-shared weights based on learning per-parantn per-parameter L2 regularization strengths [Shrinkage. [9] yields a similar trend of high freezing for lower-level features, without however improving performance against standard MAML.. Our findings hold when applying our method to deeper and wider residual neural network (ResNet-12 12) model, see Tables 2 and ST (SM). where we observe the same trend of decreasing gradient sparsity with depth emerge.
Sparse-MMLL
Final sparsity (%)
80
set (%)
65
MAMI.
50
Test set (56)
66
sparse-MAML
Final sparsity (%))
63
45
64
40
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Sparaily
15
25
20
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10
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10a
Inner loop learning rate
Figure 2: Sparse learning algorithms found by sparse-MAML work best combination with highly- plastic models. Both gradient sparsity and generalization performance increase with number of inner-loop learning steps (left) and learning rate (right). Standard MAML, which does not employ sparse learning, requires more careful learning rate tuning and does not benefit much from large learning rates. In all experiments, gradient sparsity is initially ~ 50%. The inner-loop learning rate is set to 0.1 when varying the number of steps. Results are for 5-shot, 5-way minilmageNet, averaged over 5 seeds + std.

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3.2 Sparse learning prefers highly-plastic models
We hypothesize that restricting learning to an appropriate parameter subset allows for longer training and larger changes without overfitting, beyond meta-learning initial parameter values. To verify this hypothesis we scan over different inner-loop learning rates and lengths and compare the resulting test set performances of MAML and sparse-MAML.
First, we test three different inner-loop durations (5, 15 or 25 gradient steps, see Figure 2 left). We find that neither MAML nor sparse-MAML exhibit overfitting for the duration range considered here (for reference, the original study of MAML applied 5 inner-loop steps during meta-training). In contrast to MAML, the solutions found by sparse-MAML generalize significantly better for longer adaptation phases. This improvement in generalization performance is accompanied by an increase in gradient sparsity. Furthermore, applying sparse-MAML. in the very-low data regime of 1-shot learning results in higher levels of gradient sparsity, even though the exact same model and training setup is used for both 1- and 5-shot learning experiments.
We further investigate increasing the learning rate can result in improved generalization performance in combination with sparse learning. We scan the inner-loop learning rate over a large range, cf. Figure (right), and find clear trend towards gradient sparsity going along with better test-set accuracy for larger learning rates. Interestingly, similar effects have been reported in standard (non- meta-learned) neural network training where both freezing layers throughout training [5] and the use of large learning rates 33 seem improve generalization performance.
3.3 Sparse learning vs. more expressive gradient modulation methods
Sparse-MAML can be understood as a bi-
Table 2: 5-way few-shot classification accuracy (%) on minilmageNee with a ResNet- model. We report mean + std. over 3 seeds. We report MetaOptNet [31 figures when no additional regularization techniques are applied. Results from BOIL and MetaOptNet are taken from the respective papers.
nary gradient modulation method. Second- order methods such as meta-curvature [4]
modulate gradients by meta- meta-learning pre- conditioning matrices; in meta-SGI) [35],
these matrices are restricted to be diagonal; sparse-MAMI. further restricts the diago- nal values to be binary. From this point of view, sparse-MAML the least expressive form of gradient modulation.
Method
1-shot
5-shot
MetaOptNet
51.13
70.88
Surprisingly, we find that despite its re- duced expressiveness, sparse-MAML re- covers the performance improvements achieved by the more sophisticated alter- natives, significantly improving the perfor- mance of standard MAML (cf. Table [ We point out that sparse-MAMi MAMML uses a first-order update (Eq, 5), while all three gradient modulation methods we compare to (meta-SGD, meta-curvature and MT-nets) use second-order derivatives that are more costly to evaluate.
MAML
53.91 53.9140.61
#:0611
69.36:11.23
ANIL BOIL
55.25 10.33
70.03
70.50 =0.28
sparse-MAML
55.02
70.02
sparse-eeLU-MAML.
56.39-0033
73.01 +0.24
Sparse learning emerges when meta- learning learning rates. We also im- plement variant of meta-SGD which uses rectified learning rates (sparse-Rel.U) MAML). Concretely, we replace the step function 2 in the inner-loop dynamics (Eq. 4 by the positive part of m, (m)+ := 1 m2 m. Then, we learn the underlying learning rate parameter m using our first- order straight-through update of Eq. to prevent learning rates from getting stuck at zero. Besides standard meta-SGD, which allows learning rates to go negative, we compare this method to an alternative exponential learning rate param- eterization (53), exp m, which like sparse-Rel Re-e-RRUUUMMMM........ enforces non-negativity while avoiding permanently frozen updates (cXp-MAML, Table It is, however, harder to reach sparse learning rate
Table 3: Average gradient sparsity levels (%) after meta- learning on 5-way minilmageNet few-shot tasks, stan- dard ConvNet model. Mean + std. over 5 seeds.
Method
1-shot
5-shot
Oparse-ReLU-MAI sparse-MAML
77.5 77.53=0073
73.53
19.04
74.98
sparse-MAML +
78.05 1.67
76.66 11.13 16.66
5

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distributions under this parameterization, as the meta- deta-gradient dm L becomes exponentially small as m approaches zero.
We analyze the distributions of learning rates that Ceanse-Rel.U-MMA
yields on minilmageNet and observe that gradient sparsity once more emerges, cf. Table 3 We find that the levels of gradient sparsity and generalization performance when meta-learning binary (sparse-MAML) or rectified learning rates DiparseReLLUMMMMLL are approximately the same, with both methods outperforming exp-MAML on both I-shot and 5-shot tasks. These results support the hypothesis that shutting off weight updates one of the essential gradient modulation operations few-shot learning. We note that while sparse-MAML MAML and sparse- sparse-ReLUs ReLL-MMMAAAN quickly disable learning in a large fraction of weights, exp- 1MMM tends to push learning rates down, in particular for layers close the input, but at much slower pace; increasing the meta-lcarning rate Y'm cannot compensate for this slowdown as learning becomes unstable (data not shown).
This picture changes for the the deeper and larger ResNet-12 model, cf. Table 2 When using this more complex architecture, we find that sparse rectified learning rates Apparse-ReLU-MMNN beneficial over binary gradient masks (sparse-MAML). In particular, the combination of sparse learning (see Table [ST for additional details) with learning rate modulation found by sparse-ReLU- MAML outperforms all other methods, including standard (dense-learning) MAML, as well as methods based on manually freezing layers in the inner-loop: BOIL 40), ANIL 44 and the closely- related MetaOptNet [3]] method. Like ANIL. MetaOptNet only adapts the final classification layer in the inner-loop. but uses more sophisticated solver instead of a few steps of gradient descent to learn task-specific solvers. Thus, once more, learning by sparse gradient descent is an effective strategy to improve the generalization performance of few-shot learner.
are
Stochastic gradient masking. We further investigate whether stochastic binary gradient masks can improve few-shot learning performance. Our interest in studying stochastic masks is two-fold: as way improve meta-optimization based on our straight-through estimator; and to determine if stochastic masking is beneficial at meta-test time. We thus investigate sparse-MAML.- a variant of algorithm in which gradient masks are generated from low-dimensional Gaussian vector, with noise intensity determined by meta-learning (see SM), As before, we adjust meta-parameters using a first-order update. We find that this mask generation method does result in improved performance, cf. Table I Interestingly, mask randomness is entirely suppressed by meta-learning; eventually, + and we recover single deterministic mask m. The performance improvements observed on few-shot learning therefore stem from improvements to the meta- ootrmoizttiiiionnnn process, likely related to the challenges of optimizing binary variables with grradditgrddeentt.. based methods.
3.4 Sparse learning improves performance in cross-domain adaptation tasks
We now investigate whether the patterns of gradient sparsity discovered by our method overfit to the particular task family where they were obtained, namely, to few-shot minilmageNet classification
Table 4: Few-shot classification accuracy (%) when meta-learning minilmageNet but meta-testing on TeeredImageNet. CUB and Cars. Mean + std. over seeds.
Problem
Method
TirredImageNet
CUB
Cars
MAML
51.61 20.20
40.01
33.57#0.14
ANIL
52.82
41.12=0.15
34.77+0.31
BOIL
53.23 10.41
14.20-L..1
36.12
1-shot
sparse- Reuse-RRLU-MMMLL
53. +3.12
41.86=0.95
35.46+0.67
sparse-MAML
53.47
41.37=20778
35.90
sparse-MAML+
53.91 +0.67 #0.67
43.43=1.01
37.14
MAML
65.76
53.09
44.56-10.21
ANIL
66.52
55,82
46.55_10.29
BOIL
69.37
60.92+0.11
50.64+0022
5-shot
sparse-ReLU-MAML
69.06
69.06+t0.28 +0.28
19,55
51.21 $1.21 50.89
sparse-MAML
60.65
60.58+1.10
52.63
sparsc-MAML
69.92 10.21
62.02
53.18 L0.44

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tasks. This is an important question, since excessive parameter freezing may prevent adaptation to tasks that are too different from those presented during meta-learning
We therefore move our analysis of few-shot learning to a cross-domain adaptation setting. In evaluate
cross-domain adaptation problems, the family of tasks presented post-meta-learning algorithms is shifted by sampling classes from a different dataset. In particular, we train our meta- learner on the minilmageNee dataset and then evaluate learning performance on the TieredImageNet, CUB and Cars datasets. It has previously been demonstrated that manually freezing either the head (BOIL) or the body (ANIL) during meta-testing improves performance in this setting [40], compared to letting all weights adapt (MAML). In Table we compare the performance of our method to these baselines. We find that meta-learning the freezing pattern with sparse- -AMM as opposed to manually selecting it consistently improves cross- doss-n adaptation.
4 Continual learning
We now turn to a continual learning setting, where tasks must be learned sequentially. A successful continual learner is able to learn similar tasks faster, as in the few-shot learning case, while retaining high performance on previously seen tasks. We conjecture that sparse learning can improve memory retention and accelerate future learning by reducing interference with past updates.
4.1 Gradient sparsity emerges when learning continually with Look-ahead MAMI.
We investigate the benefits of sparse gradients the recently proposed La-MAML algorithm [5]. This algorithm combines online meta-learning in conjunction with small replay buffer which holds representative examples from the past in memory. Standard replay methods (51), define joint objective using present and buffered data and directly optimize this objective. La-MAML follows a technique known as meta-experience replay [49] and introduces a bi-level optimization problem. The outer loss L is the multi-task objective optimized with standard replay methods, while the inner loss i is evaluated on the new incoming data only. Riemer et al. [49] have shown that such meta-learning promotes gradient alignment over tasks, which is a way reduce interference [36].
Like the variants of MAML reviewed Section 2 La-MAML introduces meta-learned per-parameter learning rates. We now briefly review single iteration of the algorithm; complete pseudocode provided in the SM. Each iteration of La-MAML consists of processing a new batch data B as follows: (i) starting from do 0 taking inner-loop step on each sample in B, with (Øx) = L(dx, B&); (ii) defining an outer-loss (Øx,, B U R) on both new data B and a batch of past data R sampled from the replay buffer;
(ii)) taking outer-loop step on the learning rate parameter using a first-order update followed by an outer-loop step (using the newly updated learning rate) on the neural network parameters 0; (iv) re-populating the replay buffer with data in The
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Figure 3: Gradient sparsity when learning MNIST rotations with the La-MAML and sparse-La- MAML algorithms. Results averaged over 3 seeds + std. Left: Sparsity emerges on the original La- La-MAML algorithm across the three layer network and monotonically increases with the number of tasks and with depth for both weight (W1, W2, W3) and bias parameters (61, b2, by). Center: A similar behavior is observed when replacing meta-learned learning rates by meta-learned binary gradient masks (sparse-La-MAMM)
Right: Overall sparsity of Larrs-La-MMAL. decreases with increased network capacity accompanied with higher retained accuracy (RA). Network capacity is varied by changing the number of neurons in the two hidden layers simultaneously.
7

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sequence of inner-loop updates is given by
@ σ2 - (a)+ V VoLK((nn)
s.t. Φ = 0,
where is a vector of learning rates whose components are constrained be non-negative by elementwise application of the positive part function. We note that while the main text of ref. [T5] presents an inner-loop learning rate parameter that is allowed to go negative, the implementation for the experiments reported in ref. [16] uses rectified learning rates. In this implementation. a learning rate that is updated below zero will never recover, which can lead to dead coordinates and promote sparsity. The inner loss T.f (Φ) is defined on different data sample on each step k. A first-order update is applied to again modulated by the adaptive learning rate:
0 0 - (a)+ VoL (Φx).
Sparse-La-MAMI. Our sparse- -AMML can be readily applied to continual learning problems by modifying the inner- and outer-loop updates of La-MAML. We replace the meta-lcarned learning rates in equations 7 By by meta- learned binary gradient masks, a au m with og E R+ some scalar (fixed) learning rate value. To learn the underlying parameter m, we again resort to our first-order update (equation 5)
Spaadd (%5)
50
6
W
6
0
1
2
3 Tasks
4
5
Figure 4: Structured sparsity emerges and tends to converge within a task in multi-pass continual learning. Results shown for La-MAML after training on MNIST rotations, averaged over 3 seeds std., for weight layers (W1, W2, W3) and bias parameters (b1, b2, by). Sparsity increases with depth.
Sparse learning improves continual learning. We hy- pothesize that large fraction of learning rates approaches zero when the hyperparameters of La-MAML and sparse- La-MAML are tuned for best continual learning perfor- mance. To test this hypothesis, we follow the exact same setup as in the original study of La-MAML [1S. We perform grid search over the learning rates (ig and T'm and search for best continual learning performance, not sparsity (cf. SM). The remaining hyperparameters are kept to the values provided in [5].
We study the three MNIST 30 continual learning problems rotations, permutations and many permutations using a singlc-headed network using the code accompanying ref. [1]]. Task information is not given to the network, and each data point is seen only once, unless noted otherwise. Full details as well as additional experiments using the CIFAR-10 10 (28 dataset are provided in the SM.
We verify that our initial hypothesis is correct: La-MAML shuts off learning in many coordinates (cf. Figure 3 full results may be found in the SM, Table S5 reaching even higher levels of sparsity than Lparse-La-MAML. This can be explained by the fact that dead coordinates can arise in La- MAML, which can lead to excess sparsity. By contrast, our straight-through update dynamically and continually adjusts the pattern of sparsity allowing previously frozen parameters to be unfrozen. This results in matching or slightly improved performances when using our binary gradient mask across all three MNIST variants, see Table 5) both in terms of final retained accuracy (RA) and backward- transfer and interference (BTI; the change in accuracy measured at the end of the experiment minus just after learning a task, averaged over tasks). Moreover, the patterns of sparsity adjust to the capacity of the network, decreasing and eventually vanishing for larger models (Figure B as retained accuracy goes up, indicating that the task is not sufficiently difficult to create interference on large capacity models.
As in our few-shot learning experiments, structured sparsity emerges across the different parameter groups of the network (cf. Figure B}. We observe that now sparsity is highest closest to the output layer, the exact opposite of the trend found in our few-shot learning experiments. This provides evidence that online meta-learning can discover how to rewire low-level features without interference in order to accommodate different tasks that share high-level structure. We further investigate a multi-pass setting, where the examples from each task are visited multiple times (10 epochs instead of 1) before proceeding to the next task. In this setting. it can be seen that sparsity levels (displayed in Figure 4 tend to converge within tasks and then raise again when tasks switch, presumably to preserve past memories via gradient sparsification. Taken together, our results support the hypothesis that gradient sparsity is beneficial for continual learning and that appropriate patterns of sparsity can be discovered by simple online basedd meta-learning
8

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Table 5: Retained accuracy (RA) and backward-transfer and interference (BTI) for three different MNIST continual learning problems: rotations, permutations and many permutations. We report mean std. over 5 seeds. Negative BTI values closer to zero imply less forgetting and are therefore better. Results of related work are taken from [[]; for completeness we include the GEM [36 and MER 44 methods next to a stochastic gradient descent baseline. Although sparse-La-MMAM (sp-LaMM) is strictly less expressive than the original La-MAMI. algorithm, it shows competitive performance across all variants and both metrics. The lower baseline BTI values can be explained by
lower overall accuracies achieved by La-MAML.
Method
Rotations
Permutations
Many permutations
RA
BTI
RA
BTI
RA
BTI
Baseline
53,38 1.53
1..4
55.42
-13.76
32.62 #044
-19.66 00.86
GEM
67.38 -18.02=1.00
55.42=
-24.42=t1.10
32. 32.14 10.50
-23.52=0.87
MER
77.42
-5.60
73.46-b0.45
-9.96=0.455
47.40 0.335
-17.78
La-M
77.4
-8.64 #0.40
74.34+
-7.60=0.51
48.46
-12.96+0.07
sp-LaM
77.77=0.68
-8.16 #0.61
76.88+00.72
-8.39=0.63
50.81 70.79
-13.73#0073
4.2
Sparse online learning
We finally consider another online learning setting in which the underlying task is con- cealed from the learner and can randomly change at each step. potentially going back
Table 6: Sparse learning improves continualMMAML performance. Cumulative online accuracy on Omniglot- MNIST-FFshhonMMNIST benchmark. Tasks switch with probability p. Results from previous work taken from 77. Mean + std. over 5 seeds.
to previously seen tasks [50 19 7. At each time step t, the data D, is an i.ddd sam- ple from a stationary distribution that only depends on the current task. The learner, whose current state is denoted by ot. is evaluated whenever new data is presented and modifies its behavior accordingly. The goal is then to minimize the cumulative loss Σi= L( De) measuring the perfor- mance before adaptation takes place.
Method
p 0.98
p 0.9
Online Adam 25
73.9
23,8+112
Fine-tuning
72.7
22.1 #11
MAML 11
84.5#1.7
75.5+0.7
ANIL 44
75.3+2.0
69.1
BGD
660
87.8
63,4
MetaCOG 19
88.0+1.0
63.6
MetaBGD
91.1
74.
This online learning protocol differs from the one adopted in the previous section, where tasks were visited only once and only the final loss ΣT L(Ør, Dr) evaluated ér mattered. The cumulative loss criterion emphasizes fast learning and adaptability while memory is still needed avoid re-learning, since tasks can be re-encountered. Recently, it has been shown that a simple modification of MAML Continual-MAMM.. 7 can outperform a number of algorithms specifically tailored for this setting as well as plain stochastic gradient descent 4 Briefly, Continual-MAMi extends MAML by introducing task-switer detection mechanism based on changes in loss; data is buffered until switch is detected. When this occurs, the buffered data is used to perform meta-parameter update; the buffer is reset; and the inner-loop optimization restarts. Here, we merge continual-MAMI. with MAArse-MAML, and modulate inner-loop gradients according to equation 4 We present complete pseudocode for the algorithm in the SM.
C-MAML
92,8=t0.6
83.3+0.1
sparse-C- sparse-C-MAMI. C-MAMI
94.2=004
86.3+0.1
Dparse-ReLU-C--MNN
93.5
86.1
We reproduce the experiments of 7 in which a sequence of 10000 examples from the Omniglot [29], MNIST [30] and FaShionMNIST [59] datasets presented for online learning single-headed neural network, using the code provided by the authors. We carry out grid search to tune the inner-loop learning rate ag and the mask learning rate "m introduced by sparse-MAML for best performance, not sparsity (see SM). We observe again structured (layer-dependent) gradient sparsity emerge when using this algorithm (sparse-C-MAMI)
as shown Figure SS5 and an increase in cumulative online learning accuracy over the original continual-MAMI. algorithm (cf. Table (). Finally, we observe the same qualitative behavior (see SM for sparsity levels) and obtain similar performance when replacing our binary masks by rectified learning rates (sparse- Sparse-RRLL-CCE ReLU-C-MALL

---

meta-learned with our throughhtth update. These findings once more support the hypothesis that sparse learning, and not learning rate modulation, lead the improved performance reported here.
5 Discussion
We studied gradient- based meta-learning systems with the ability of learning where to learn. This was modeled by adding binary variables which masked gradients on per-parameter basis, therefore determining which parameters are allowed to change. We observed gradient sparsity emerge stan- dard few-shot and continual learning problems, without introducing an explicit bias towards sparsity. This form of sparse learning, which may be understood as sparse gradient descent, was accompanied by overall improvements in generalization, as well as reduced interference and forgetting.
Previous work on gradient modulation has focused on estimating task-shared loss geometry pre- condition the optimization procedure [35 62 12
32 61 In addition, stochastic variant of gradient masking was featured in the MT-net algorithm 32 as part of more complex model. Our approach differs from these previous studies in its simplicity. We restrict gradient modulation to be binary and deterministic and use an inexpensive first-order update to learn the gradient masks. In contrast to traditional methods for inducing sparsity via regularization [54] (here, gradient regularization) our approach does not require evaluating second derivatives, which would result from differentiating gra- dient regularizers. Despite these simplifications, we find competitive performance our experiments. These results point towards sparse gradient descent as a powerful learning principle.
The idea of meta-learning learning rates can be traced back to the seminal work of Sutton [53 who proposed to estimate learning rate meta- grata-ggaiiets online using forward mode automatic differen- tiation, and to use consecutive batches of data to define inner- and outer-loop loss functions. This approach, known as stochastic meta-descent (SMD), was extended to nonlinear models by Schrau- dolph 52 using fast Hessian-vector product techniques. Using SMD to optimize neural network is an interesting question whether gradient sparsity emerges when applying SMD to online learning problems that are not clearly structured in tasks, as considered here. Furthermore, this line of work suggests that it might be possible to obtain finer binary gradient mask updates in online fashion using forward-mode automatic differentiation.
models an ongoing area of research [55 58
22 24].
recent study has put into question whether any useful adaptation still takes place when MAML few-shot learners are presented with a novel task after meta-learning 44 Our findings shed light on this question, by demonstrating that few-shot learning performance can be improved when learning an adequate small subset of parameters. The additional plasticity of our meta-learner sparse learners led to a significant performance increase over handwired schemes based on frozen layers, in particular when encountering tasks drawn from a different family of problems than that used for meta-learning This finding complements recent work showing that modular recurrent networks with sparse updating mechanisms hold great promise in improving out-of- diitr---ittiiitiion generalization performance [14] 38
Our results may be of special interest the design of neuromorphic hardware. Updating weights on-chip implies a significant power overhead whose cost scales with the number of plastic weights Reducing the number of plastic weights can therefore result in immediate improvements in energy efficiency and scalability. Likewise, synaptic plasticity is costly in biological neural networks. Given the high energy demands of the brain there has likely been selective pressure reduce costs associated with synaptic change (34. It is therefore conceivable that the brain developed mechanisms to restrict learning to an appropriate subset of synapses to save energy. Our study presents further evidence in favor of sparse synaptic change, given its potential benefits in the biologically-relevant scenarios of fho-shot and continual learning investigated here.
42
Limitations. To arrive at our simple mask update (cquation 5 we introduced two important approximations. First, we dropped all terms involving second-order derivatives, and second, we used straight-through estimation to differentiate through the step function. Despite their frequent use in previous work, both approximations remain poorly understood. For this reason it possible that our algorithm fails unexpectedly outside the experiments considered here; potential problems stemming from the con-differentiability of the step function are likely unavoidable in our approach. Our update is also likely inappropriate for long inner loops (large K) [58]. Finally, scaling our few-shot learning experiments to more complex neural network models is potentially difficult, a challenge that our approach shares with other methods based on MAML.
10

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Acknowledgments and Disclosure of Funding
This work was supported by an Ambizione grant (000002771177)) awarded to João Sacramento by the Swiss National Science Foundation. Johannes von Oswald is funded by the Swiss Data Science Center (J.v... P18-03). Dominic Zhao is supported by AlayaLabs (Montreal, Canada). Massimo Caccia was supported through MITACS during his part time employment with Element Al the ServiceNow company, and by Amazon, during his part time employment there. We thank Charlotte Frenkel, Frederik Benzing, Angelika Steger and Laura Sainz for helpful discussions.
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14


Reviews:
[{"review_id": "tr_Z5rybHLw", "reviewer": "Reviewer_WL2f", "summary": "- This paper introduces a sparsity regularizer for gradient-based meta learners. A binary mask for each layer is used to decide whether a parameter should be adapted or not during the inner loop phase.\n- The authors provide an analysis of sparse-MAML on standard few-shot learning   (FSL) datasets in terms of performance (accuracy) and gradient sparsity. They also verify the performance of sparse-MAML in the cross-domain setting comparing it with BOIL and ANIL, other work manually freezing the head or the body of the classifier, rather than learning what to freeze/adapt as sparse-MAML does. The second part of the paper revolves around look-ahead MAML, an algorithm for continual learning investigating the gradient sparsity caused by meta-learning the inner learning rate (with rectifier, non-negative LR, or binary mask). Finally, experiments on online learning meta/multitask learning are provided.\n", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "- The paper is clearly written and well explained. Every concept is introduced with particular attention and the rationale of each section is explained step by step making.\n- The main limitation of this approach is the necessity of having a binary mask to cover all parameters. It is definitely challenging (interesting) scaling the proposed method to other architectures. I believe this is the reason why the authors do not provide experiments with other architectures but always use the same conv4 vgg-like model (also for easier comparison with existing approaches). Authors leave this issue for future work. \n- **Questions/considerations on experiments**\n    1. It would have been great to have a comparison with other sparsity regularizers. A simple differentiable L1/L0 constraint on the weights could be an interesting experiment, although much more difficult to train and requiring to tune a sparsity coefficient hyperparameter.\n    2. A comparison with Shrinkage meta-learning [6] algorithm is also missing.\n    3. I guess my main question is: \"Is sparsity really necessary for meta-learning?\" I'm not sure the paper really wants to answer this question, but I find it interesting. Since the proposed approach claim sparsity to be important for generalization in meta/continual learning it would have been nice to see experiments with masks being free to vary, rather than being constrained to be binary. This would be important to reinforce the claim about the importance of sparsity.  I believe that would correspond to something similar to MT-Net [20] or warpgrad[9].\n    4. It is important to notice that when you add per-parameters LR or binary masks you double the number of parameters. Considering this point, how is your comparison fair wrt MAML and other methods? \n    5. I see in the Supp. Mat. that training is performed following the transductive usage of BatchNorm. I would also include experiments with non-transductive settings to disentangle the effect of bn statistics.\n- Pros\n    - Code, ablation studies, and information for reproducibility are present in the supplementary material\n\n\n——————————————-\nAfter the rebuttal and discussions, I decided to increase my score from 5 to 7 and recommend acceptance.\nThe paper is an important empirical contribution to the fields of Meta and Continual Learning that might be useful for some researchers.\nThe authors have agreed to include new results and important changes/clarifications to the final version of the paper.\nIf accepted, these changes should be checked by the AC and reviewers when the camera-ready is uploaded.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Some limitations are discussed. No potential negative impact in my opinion.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4.5"}, "extra_text": {}}, {"review_id": "dlt7ThuvpO", "reviewer": "Reviewer_giJs", "summary": "This paper proposes learning to where to learn to decide which neural network weights to change.  The proposed sparse-MAML introduces an adjustable binary mask that masked gradients on a per-parameter basis, therefore determining which parameters are allowed to change.  The sparse-MAML is also used for continual learning, which could improve generalization and reduce forgetting. The experiment results suggest that learning by sparse gradient descent can learn a powerful inductive bias for meta-learning systems.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Quality: The paper is generally well written and easy to follow., and the experiments are thorough and well-executed.  There are extensive experiments demonstrating the empirical effectiveness of the proposed approach.  The proposed learning where to learn decide which weights to change. The selective sparsity achieves better generalization in a range of few-shot and continual learning problems. The versatility experiment is a bonus.\n\nClarity: Most of my concerns about clarity, which are very small, i.e., better exposition of why the binary mask is better than the 'soft-mask' methods (Meta-SGD) works. \n\nOriginality: As far as I am aware, this work presents a novel method (sparse-MAML) to do the few-shot learning and continual learning.  The code and the experimental details are provided in the supplementary, which would be helpful for reproduction.\n\nSignificance: This work seems significant to researchers interested in meta-learning, cross-domain few-shot learning and continual learning. The proposed sparse-MAML is so promising and further testing.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "(1) Perhaps missing some additional details that would help the reader better understand why and how the method works. See the clarity.\n\n(2)  As the limitations of the paper say,  this paper introduced two important approximations, which may result in some small gaps from the real objective.  \n\n(3) One small question: Is the sparse-MAML based on transductive batch normalization? It needs to be clarified in the paper.\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": ["I don’t know"], "time_spent_reviewing": "12 hours"}, "extra_text": {}}, {"review_id": "UX5dD7bt_2v", "reviewer": "Reviewer_ah52", "summary": "This paper proposes a method to selectively update parameters of a model in the paradigm of meta learning, which introduces  sparsity into the updates and can find a better initialization for learning further tasks. It can be applied in few-shot learning, domain adaptation, continual learning, and online learning. \n\n", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper is well written and the proposed method is clearly justified. Comparing with related methods, the difference is not significant but there are some interesting experimental results showing the sparsity is highly related to the model structure and brings some improvements based on MAML.  \n\nThe authors try to exhibit the potential of this method in different learning paradigms, however, the page limitation wouldn't allow adequate experiments and analysis in all of these paradigms, which makes some claims are not convincing enough.  \nEq.5 shows that only parameters with non-negative product of  \\nabla L^{out} and \\sum \\nabla L^{in} will be updated. It is even more restrictive than GEM in continual learning which just requires the inner product of the two gradient vectors is non-negative. Such restriction  is likely to reduce the plasticity in the model. However, the experiments in continual learning only include Domain-IL tasks. How the proposed method performs in Class-IL tasks in not clear.  And it is strange that this method has higher BTI than baseline and MER in Ratation MNIST tasks, which is not explained or  analyzed. \n\nSome minor issues:\n1. How does the 100% initial sparsity work? Does it mean there is no update at all in the beginning?\n2. In Tab.3, bias parameters tend to be frozen? Why is that? Would it be better without the bias parameters?\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "\nThe limitation of experiments are stated above. No societal impact in my understanding.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3 hours"}, "extra_text": {}}, {"review_id": "FteLvJtLPU_", "reviewer": "Reviewer_C1hv", "summary": "The paper investigates how gradient sparsity emerges and adapts in meta-learning and continual learning scenario. The authors suggest to investigate the effects and behavior of sparsity, for example, layer-wise sparsity pattern for few-shot learning, how the sparsity rates adjust to the number of inner-gradient steps, inner-learning rates, and hidden layer size. The authors further perform empirical analysis with continual learning setting, where  combined with La-MAML, the performance improves with proper rate of sparsity.", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "== Pros ==\nThe paper involves extensive empirical study, which may be useful for some readers.\n\n== Cons ==\nWhile reading the paper, I really felt that each of the experiments are poorly motivated, without sufficient explanation of why each experiment should appear at a certain point and why each experiment and the corresponding message is important. It seems that the whole paper is simply enumerating many experimental results without proper structures. Taking a few examples,\n1. Why is the observation of section 3.1 and Figure 1 important? I see that sparsity levels can be different for different layers and for different initial sparsity ratio, then so what? Such a discussion has not been provided with any proper context, so it is really confusing what we should do next and what will appear next in the paper.\n2.  Same for the section 3.2. Personally, this section was a bit informative, such as in Figure 2 and 3, sparsity allows much larger learning rates and longer gradient steps, which is not available for standard MAML. However, I'm still confused why we should investigate this phenomena at this point. What is the context? The paper simply repeats \"We next study ...\" without properly motivating the readers.\n3. Same for the section 3.3. Why do we need to compare with gradient modulation methods at this point? I roughly see that the role of gradient modulation can be overlapping with gradient sparsity, but I'm not sure because there is no such discussion. Furthermore, in L158, why suddenly discuss about stochastic gradient masks? Is the Section 3.3 about stochasticity vs. performance? \n4. Same for the section 3.4. I understand that sparsity is beneficial for cross-domain adaptation by looking at the table, but again, what is the context of this experiments? Why the performance improves?\n\nSame for all the other sections. I strongly think that the paper is poorly structured and should be rewritten with clear motivation. I understand that the paper is analysis-styled one, but it should be structured with clear motivation everywhere.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6 hours"}, "extra_text": {}}]

Author rebuttal:

Meta-review: Echoing the reviewers, I think this work is of interest to the community, and I think the experimental section is thoroughly done. Taking into account the engagement from the authors, and the correction they proposed and showed in this comments (which I hope will be reflected in the camera ready) I think the paper matches the requirements for being accepted at the conference. 
Decision: Accept (Poster)