# lookahead_meta_learning_for_continual_learning__b5405e14.pdf

La-MAML: Look-ahead Meta Learning for Continual Learning

Gunshi Gupta Mila, Ude M guptagun@mila.quebec

Karmesh Yadav * Carnegie Mellon University karmeshy@andrew.cmu.edu

Liam Paull Mila, Ude M paulll@iro.umontreal.ca

The continual learning problem involves training models with limited capacity to perform well on a set of an unknown number of sequentially arriving tasks. While meta-learning shows great potential for reducing interference between old and new tasks, the current training procedures tend to be either slow or offline, and sensitive to many hyper-parameters. In this work, we propose Look-ahead MAML (La-MAML), a fast optimisation-based meta-learning algorithm for onlinecontinual learning, aided by a small episodic memory. Our proposed modulation of per-parameter learning rates in our meta-learning update allows us to draw connections to prior work on hypergradients and meta-descent. This provides a more flexible and efficient way to mitigate catastrophic forgetting compared to conventional prior-based methods. La-MAML achieves performance superior to other replay-based, prior-based and meta-learning based approaches for continual learning on real-world visual classification benchmarks.

1 Introduction

Embodied or interactive agents that accumulate knowledge and skills over time must possess the ability to continually learn. Catastrophic forgetting [11, 18], one of the biggest challenges in this setup, can occur when the i.i.d. sampling conditions required by stochastic gradient descent (SGD) are violated as the data belonging to different tasks to be learnt arrives sequentially. Algorithms for continual learning (CL) must also use their limited model capacity efficiently since the number of future tasks is unknown. Ensuring gradient-alignment across tasks is therefore essential, to make shared progress on their objectives. Gradient Episodic Memory (GEM) [17] investigated the connection between weight sharing and forgetting in CL and developed an algorithm that explicitly tried to minimise gradient interference. This is an objective that meta-learning algorithms implicitly optimise for (refer to [20] for derivations of the effective parameter update made in first and second order meta learning algorithms). Meta Experience Replay (MER) [22] formalized the transferinterference trade-off and showed that the gradient alignment objective of GEM coincide with the objective optimised by the first order meta-learning algorithm Reptile [20].

Besides aligning gradients, meta-learning algorithms show promise for CL since they can directly use the meta-objective to influence model optimisation and improve on auxiliary objectives like generalisation or transfer. This avoids having to define heuristic incentives like sparsity [15] for better CL. The downside is that they are usually slow and hard to tune, effectively rendering them more suitable for offline continual learning [12, 22]. In this work, we overcome these difficulties and develop a gradient-based meta-learning algorithm for efficient, online continual learning. We first propose a base algorithm for continual meta-learning referred to as Continual-MAML (C-MAML) that utilizes a replay-buffer and optimizes a meta-objective that mitigates forgetting. Subsequently,

equal contribution

34th Conference on Neural Information Processing Systems (Neur IPS 2020), Vancouver, Canada.

we propose a modification to C-MAML, named La-MAML, which incorporates modulation of perparameter learning rates (LRs) to pace the learning of a model across tasks and time. Finally, we show that the algorithm is scalable, robust and achieves favourable performance on several benchmarks of varying complexity.

2 Related work

Relevant CL approaches can be roughly categorized into replay-based, regularisation (or prior-based) and meta-learning-based approaches.

In order to circumvent the issue of catastrophic forgetting, replay-based methods maintain a collection of samples from previous tasks in memory. Approaches utilising an episodic-buffer [5, 21] uniformly sample old data points to mimic the i.i.d. setup within continual learning. Generative-replay [27] trains generative models to be able to replay past samples, with scalability concerns arising from the difficulty of modeling complex non-stationary distributions. GEM [17] and A-GEM [6] take memory samples into account to determine altered low-interference gradients for updating parameters.

Regularisation-based methods avoid using replay at all by constraining the network weights according to heuristics intended to ensure that performance on previous tasks is preserved. This involves penalising changes to weights deemed important for old tasks [14] or enforcing weight or representational sparsity [3] to ensure that only a subset of neurons remain active at any point of time. The latter method has been shown to reduce the possibility of catastrophic interference across tasks [15, 26].

Meta-Learning-based approaches are fairly recent and have shown impressive results on small benchmarks like Omniglot and MNIST. MER [22], inspired by GEM[17], utilises replay to incentivise alignment of gradients between old and new tasks. Online-aware Meta Learning (OML) [12] introduces a meta-objective for a pre-training algorithm to learn an optimal representation offline, which is subsequently frozen and used for CL. [2, 10, 19] investigate orthogonal setups in which a learning agent uses all previously seen data to adapt quickly to an incoming stream of data, thereby ignoring the problem of catastrophic forgetting. Our motivation lies in developing a scalable, online algorithm capable of learning from limited cycles through streaming data with reduced interference on old samples. In the following sections, we review background concepts and outline our proposed algorithm. We also note interesting connections to prior work not directly pertaining to CL.

3 Preliminaries

We consider a setting where a sequence of T tasks [τ1, τ2, ..τT ] is learnt by observing their training data [D1, D2, ..DT ] sequentially. We define Xi, Y i = {(xi n, yi n)}Ni n=0 as the set of Ni input-label pairs randomly drawn from Di. An any time-step j during online learning, we aim to minimize the empirical risk of the model on all the t tasks seen so far (τ1:t), given limited access to data (Xi, Y i) from previous tasks τi (i < t). We refer to this objective as the cumulative risk, given by:

i=1 E(Xi,Y i) ℓi fi Xi; θ , Y i = E(X1:t,Y 1:t) Lt f X1:t; θ , Y 1:t (1)

where ℓi is the loss on τi and fi is a learnt, possibly task-specific mapping from inputs to outputs using parameters θj 0. Lt = Pt i=1 ℓi is the sum of all task-wise losses for tasks τ1:t where t goes from 1 to T. Let ℓdenote some loss objective to be minimised. Then the SGD operator acting on parameters θj 0, denoted by U(θj 0) is defined as:

U θj 0 = θj 1 = θj 0 α θj 0ℓ(θj 0) = θj 0 αgj 0 (2)

where gj 0 = θj 0ℓ(θj 0). U can be composed for k updates as Uk θj 0 = U... U U(θj 0) = θj k. α is a scalar or a vector LR. U ( , x) implies gradient updates are made on data sample x. We now introduce the MAML [9] and OML [12] algorithms, that we build upon in Section 4.

Model-Agnostic Meta-Learning (MAML): Meta-learning [24], or learning-to-learn [29] has emerged as a popular approach for training models amenable to fast adaptation on limited data.

MAML [9] proposed optimising model parameters to learn a set of tasks while improving on auxiliary objectives like few-shot generalisation within the task distributions. We review some common terminology used in gradient-based meta-learning: 1) at a given time-step j during training, model parameters θj 0 (or θ0 for simplicity), are often referred to as an initialisation, since the aim is to find an ideal starting point for few-shot gradient-based adaptation on unseen data. 2) Fast or inner-updates, refer to gradient-based updates made to a copy of θ0, optimising some inner objective (in this case, ℓi for some τi). 3) A meta-update involves the trajectory of fast updates from θ0 to θk, followed by making a permanent gradient update (or slow-update) to θ0. This slow-update is computed by evaluating an auxiliary objective (or meta-loss Lmeta) on θk, and differentiating through the trajectory to obtain θ0Lmeta(θk). MAML thus optimises θj 0 at time j, to perform optimally on tasks in {τ1:t} after undergoing a few gradient updates on their samples. It optimises in every meta-update, the objective:

min θj 0 Eτ1:t h Lmeta Uk(θj 0) i = min θj 0 Eτ1:t h Lmeta(θj k) i . (3)

Equivalence of Meta-Learning and CL Objectives: The approximate equivalence of first and second-order meta-learning algorithms like Reptile and MAML was shown in [20]. MER [22] then showed that their CL objective of minimising loss on and aligning gradients between a set of tasks τ1:t seen till any time j (on the left), can be optimised by the Reptile objective (on the right), ie. :

ℓi(θj 0) α X

θj 0 ℓq θj 0

= min θj 0 Eτ1:t h Lt Uk(θj 0) i (4)

where the meta-loss Lt = Pt i=1 ℓi is evaluated on samples from tasks τ1:t. This implies that the procedure to meta-learn an initialisation coincides with learning optimal parameters for CL.

Online-aware Meta-Learning (OML): [12] proposed to meta-learn a Representation-Learning Network (RLN) to provide a representation suitable for CL to a Task-Learning Network (TLN). The RLN s representation is learnt in an offline phase, where it is trained using catastrophic forgetting as the learning signal. Data from a fixed set of tasks (τval), is repeatedly used to evaluate the RLN and TLN as the TLN undergoes temporally correlated updates. In every meta-update s inner loop, the TLN undergoes fast updates on streaming task data with a frozen RLN. The RLN and updated TLN are then evaluated through a meta-loss computed on data from τval along with the current task. This tests how the performance of the model has changed on τval in the process of trying to learn the streaming task. The meta-loss is then differentiated to get gradients for slow updates to the TLN and RLN. This composition of two losses to simulate CL in the inner loop and test forgetting in the outer loop, is referred to as the OML objective. The RLN learns to eventually provide a better representation to the TLN for CL, one which is shown to have emergent sparsity.

4 Proposed approach

In the previous section, we saw that the OML objective can directly regulate CL behaviour, and that MER exploits the approximate equivalence of meta-learning and CL objectives. We noted that OML trains a static representation offline and that MER s algorithm is prohibitively slow. We show that optimising the OML objective online through a multi-step MAML procedure is equivalent to a more sample-efficient CL objective. In this section, we describe Continual-MAML (C-MAML), the base algorithm that we propose for online continual learning. We then detail an extension to C-MAML, referred to as Look-Ahead MAML (La-MAML), outlined in Algorithm 1.

C-MAML aims to optimise the OML objective online, so that learning on the current task doesn t lead to forgetting on previously seen tasks. We define this objective, adapted to optimise a model s parameters θ instead of a representation at time-step j, as:

min θj 0 OML(θj 0, t) = min θj 0

h Lt Uk(θj 0, Sj k) i (5)

where Sj k is a stream of k data tuples Xt j+l, Y t j+l k

l=1 from the current task τt that is seen by the

model at time j. The meta-loss Lt = Pt i=1 ℓi is evaluated on θj k = Uk(θj 0, Sj k). It evaluates the fitness of θj k for the continual learning prediction task defined in Eq. 1 until τt. We omit the implied data argument (xi, yi) (Xi, Y i) that is the input to each loss ℓi in Lt for any task τi. We will show in Appendix B that optimising our objective in Eq. 5 through the k-step MAML update in C-MAML also coincides with optimising the CL objective of AGEM [6]:

min θj 0 Eτ1:t h Lt Uk(θj 0) i = min θj 0

ℓi(θj 0) α ℓi θj 0

θj 0 ℓt θj 0

This differs from Eq. 4 s objective by being asymmetric: it focuses on aligning the gradients of τt and the average gradient of τ1:t instead of aligning all the pair-wise gradients between tasks τ1:t. In Appendix D, we show empirically that gradient alignment amongst old tasks doesn t degrade while a new task is learnt, avoiding the need to repeatedly optimise the inter-task alignment between them. This results in a drastic speedup over MER s objective (Eq. 4) which tries to align all τ1:t equally, thus resampling incoming samples s τt to form a uniformly distributed batch over τ1:t. Since each s then has 1

t -th the contribution in gradient updates, it becomes necessary for MER to take multiple passes over many such uniform batches including s.

Figure 1: The proposed La-MAML algorithm: For every batch of data, the initial weights undergo a series of k fast updates to obtain θj k (here j = 0), which is evaluated against a meta-loss to backpropagate gradients with respect to the weights θ0 0 and LRs α0. First α0 is updated to α1 which is then used to update θ0 0 to θ1 0 The blue boxes indicate fast weights while the green boxes indicate gradients for the slow updates. LRs and weights are updated in an asynchronous manner.

During training, a replay-buffer R is populated through reservoir sampling on the incoming data stream as in [22]. At the start of every metaupdate, a batch b is sampled from the current task. b is also combined with a batch sampled from R to form the meta-batch, bm, representing samples from both old and new tasks. θj 0 is updated through k SGD-based inner-updates by seeing the current task s samples from b one at a time. The outer-loss or meta-loss Lt(θj k) is evaluated on bm. It indicates the performance of parameters θj k on all the tasks τ1:t seen till time j. The complete training procedure is described in Appendix C.

4.2 La-MAML

Despite the fact that meta-learning incentivises the alignment of within-task and across-task gradients, there can still be some interference between the gradients of old and new tasks, τ1:t 1 and τt respectively. This would lead to forgetting on τ1:t 1, since its data is no longer fully available to us. This is especially true at the beginning of training a new task, when its gradients aren t necessarily aligned with the old ones. A mechanism is thus needed to ensure that meta-updates are conservative with respect to τ1:t 1, so as to avoid negative transfer on them. The magnitude and direction of the meta-update needs to be regulated, guided by how the loss on τ1:t 1 would be affected by the update.

We propose Lookahead-MAML (La-MAML), where we include a set of learnable per-parameter learning rates (LRs) to be used in the inner updates, as depicted in Figure 1. This is motivated by our observation that the expression for the gradient of Eq. 5 with respect to the inner loop s LRs directly reflects the alignment between the old and new tasks. The augmented learning objective is defined as

min θj 0,αj

h Lt Uk αj, θj 0, Sj k i , (7)

and the gradient of this objective at time j, with respect to the LR vector αj (denoted as g MAML(αj)) is then given as:

g MAML(αj) = αj Lt θj k =

θj k Lt θj k

θj k ℓt θj k !

We provide the full derivation in the Appendix A, and simply state the expression for a first-order approximation [9] of g MAML(α) here. The first term in g MAML(α) corresponds to the gradient of the meta-loss on batch bm: gmeta. The second term indicates the cumulative gradient from the inner-updates: gtraj. This expression indicates that the gradient of the LRs will be negative when the inner product between gmeta and gtraj is high, ie. the two are aligned; zero when the two are orthogonal (not interfering) and positive when there is interference between the two. Negative (positive) LR gradients would pull up (down) the LR magnitude. We depict this visually in Figure 2.

Algorithm 1 La-MAML : Look-ahead MAML

Input: Network weights θ, LRs α, inner objective ℓ, meta objective L, learning rate for α : η j 0, R {} Initialise Replay Buffer for t := 1 to T do for ep := 1 to numepochs do for batch b in (Xt, Y t) Dt do k sizeof(b) bm Sample(R) b for n = 0 to k 1 do Push b[k ] to R with reservoir sampling θj k +1 θj k αj θj k end for αj+1 αj η αj Lt(θj k, bm) (a) θj+1 0 θj 0 max(0, αj+1) θj 0Lt(θj k, bm) (b) j j + 1 end for end for end for

Figure 2: Different scenarios for the alignment of gtraj (blue dashed line) and gmeta, going from interference (left) to alignment (right). Yellow arrows denote the inner updates. The LR α increases (decreases) when gradients align (interfere).

We propose updating the network weights and LRs asynchronously in the meta-update. Let αj+1 be the updated LR vector obtained by taking an SGD step with the LR gradient from Eq. 8 at time j. We then update the weights as:

θj+1 0 θj 0 max(0, αj+1) θj 0Lt(θj k) (9)

where k is the number of steps taken in the inner-loop. The LRs αj+1 are clipped to positive values to avoid ascending the gradient, and also to avoid making interfering parameter-updates, thus mitigating catastrophic forgetting. The meta-objective thus conservatively modulates the pace and direction of learning to achieve quicker learning progress on a new task while facilitating transfer on old tasks. Algorithm 1 2 illustrates this procedure. Lines (a), (b) are the only difference between C-MAML and La-MAML, with C-MAML using a fixed scalar LR α for the meta-update to θj 0 instead of αj+1.

Our meta-learning based algorithm incorporates concepts from both prior-based and replay-based approaches. The LRs modulate the parameter updates in an data-driven manner, guided by the interplay between gradients on the replay samples and the streaming task. However, since LRs evolve with every meta-update, their decay is temporary. This is unlike many prior-based approaches, where penalties on the change in parameters gradually become so high that the network capacity saturates [14]. Learnable LRs can be modulated to high and low values as tasks arrive, thus being a simpler, flexible and elegant way to constrain weights. This asynchronous update resembles trust-region optimisation [31] or look-ahead search since the step-sizes for each parameter are adjusted based on

2The code for our algorithm can be found at: https://github.com/montrealrobotics/La-MAML

the loss incurred after applying hypothetical updates to them. Our LR update is also analogous to the heuristic uncertainty-based LR update in UCB [8], BGD [32], which we compare to in Section 5.3.

4.3 Connections to Other Work

Stochastic Meta-Descent (SMD): When learning over a non-stationary data distribution, using decaying LR schedules is not common. Strictly diminishing LR schedules aim for closer convergence to a fixed mimima of a stationary distribution, which is at odds with the goal of online learning. It is also not possible to manually tune these schedules since the extent of the data distribution is unknown. However, adaptivity in LRs is still highly desired to adapt to the optimisation landscape, accelerate learning and modulate the degree of adaptation to reduce catastrophic forgetting. Our adaptive LRs can be connected to work on meta-descent [4, 25] in offline supervised learning (OSL). While several variations of meta-descent exist, the core idea behind them and our approach is gain adaptation. While we adapt the gain based on the correlation between old and new task gradients to make shared progress on all tasks, [4, 25] use the correlation between two successive stochastic gradients to converge faster. We rely on the meta-objective s differentiability with respect to the LRs, to obtain LR hypergradients automatically.

Learning LRs in meta-learning: Meta-SGD [16] proposed learning the LRs in MAML for few-shot learning. Some notable differences between their update and ours exist. They synchronously update the weights and LRs while our asynchronous update to the LRs serves to carry out a more conservative update to the weights. The intuition for our update stems from the need to mitigate gradient interference and its connection to the transfer-interference trade-off ubiquitous in continual learning. α-MAML [28] analytically updates the two scalar LRs in the MAML update for more adaptive few-shot learning. Our per-parameter LRs are modulated implicitly through back-propagation, to regulate change in parameters based on their alignment across tasks, providing our model with a more powerful degree of adaptability in the CL domain.

5 Experiments

In this section, we evaluate La-MAML in settings where the model has to learn a set of sequentially streaming classification tasks. Task-agnostic experiments, where the task identity is unknown at training and test-time, are performed on the MNIST benchmarks with a single-headed model. Taskaware experiments with known task identity, are performed on the CIFAR and Tiny Imagenet [1] datasets with a multi-headed model. Similar to [22], we use the retained accuracy (RA) metric to compare various approaches. RA is the average accuracy of the model across tasks at the end of training. We also report the backward-transfer and interference (BTI) values which measure the average change in the accuracy of each task from when it was learnt to the end of the last task. A smaller BTI implies lesser forgetting during training.

Efficient Lifelong Learning (LLL): Formalized in [6], the setup of efficient lifelong learning assumes that incoming data for every task has to be processed in only one single pass: once processed, data samples are not accessible anymore unless they were added to a replay memory. We evaluate our algorithm on this challenging (Single-Pass) setup as well as the standard (Multiple-Pass) setup, where

Table 1: RA, BTI and their standard deviation on MNIST benchmarks. Each experiment is run with 5 seeds.

METHOD ROTATIONS PERMUTATIONS MANY

RA BTI RA BTI RA BTI

ONLINE 53.38 1.53 -5.44 1.70 55.42 0.65 -13.76 1.19 32.62 0.43 -19.06 0.86 EWC 57.96 1.33 -20.42 1.60 62.32 1.34 -13.32 2.24 33.46 0.46 -17.84 1.15 GEM 67.38 1.75 -18.02 1.99 55.42 1.10 -24.42 1.10 32.14 0.50 -23.52 0.87 MER 77.42 0.78 -5.60 0.70 73.46 0.45 -9.96 0.45 47.40 0.35 -17.78 0.39

C-MAML 77.33 0.29 -7.88 0.05 74.54 0.54 -10.36 0.14 47.29 1.21 -20.86 0.95 SYNC 74.07 0.58 -6.66 0.44 70.54 1.54 -14.02 2.14 44.48 0.76 -24.18 0.65 LA-MAML 77.42 0.65 -8.64 0.403 74.34 0.67 -7.60 0.51 48.46 0.45 -12.96 0.073

ideally offline training-until-convergence is performed for every task, once we have access to the data.

5.1 Continual learning benchmarks

Table 2: Running times for MER and La-MAML on MNIST benchmarks for one epoch

METHOD ROTATIONS PERMUTATIONS

LA-MAML 45.95 0.38 46.13 0.42 MER 218.03 6.44 227.11 12.12

First, we carry out experiments on the toy continual learning benchmarks proposed in prior CL works. MNIST Rotations, introduced in [17], comprises tasks to classify MNIST digits rotated by a different common angle in [0, 180] degrees in each task. In MNIST Permutations, tasks are generated by shuffling the image pixels by a fixed random permutation. Unlike Rotations, the input distribution of each task is unrelated here, leading to less positive transfer between tasks. Both MNIST Permutation and MNIST Rotation have 20 tasks with 1000 samples per task. Many Permutations, a more complex version of Permutations, has five times more tasks (100 tasks) and five times less training data (200 images per task). Experiments are conducted in the low data regime with only 200 samples for Rotation and Permutation and 500 samples for Many, which allows the differences between the various algorithm to become prominent (detailed in Appendix G). We use the same architecture and experimental settings as in MER [22], allowing us to compare directly with their results. We use the cross-entropy loss as the inner and outer objectives during meta-training. Similar to [20], we see improved performance when evaluating and summing the meta-loss at all steps of the inner updates as opposed to just the last one.

We compare our method in the Single-Pass setup against multiple baselines including Online, Independent, EWC [14], GEM [17] and MER [22] (detailed in Appendix H), as well as different ablations (discussed in Section 5.3). In Table 1, we see that La-MAML achieves comparable or better performance than the baselines on all benchmarks. Table 2 shows that La-MAML matches the performance of MER in less than 20% of the training time, owing to its sample-efficient objective which allows it to make make more learning progress per iteration. This also allows us to scale it to real-world visual recognition problems as described next.

5.2 Real-world classification

While La-MAML fares well on the MNIST benchmarks, we are interested in understanding its capabilities on more complex visual classification benchmarks. We conduct experiments on the CIFAR-100 dataset in a task-incremental manner [17] where, 20 tasks comprising of disjoint 5way classification problems are streamed. We also evaluate on the Tiny Imagenet-200 dataset by partitioning its 200 classes into 40 5-way classification tasks. Experiments are carried out in both the Single-Pass and Multiple-Pass settings, where in the latter we allow all CL approaches to train up to a maximum of 10 epochs. Each method is allowed a replay-buffer, containing upto 200 and 400 samples for CIFAR-100 and Tiny Imagenet respectively. We provide further details about the baselines in Appendix H and about the architectures, evaluation setup and hyper-parameters in Appendix G.

Table 3 reports the results of these experiments. We consistently observe superior performance of La-MAML as compared to other CL baselines on both datasets across setups. While the i CARL baseline attains lower BTI in some setups, it achieves that at the cost of much lower performance throughout learning. Among the high-performing approaches, La-MAML has the lowest BTI. Recent work [7, 22] noted that Experience Replay (ER) is often a very strong baseline that closely matches the performance of the proposed algorithms. We highlight the fact that meta-learning and LR modulation combined show an improvement of more than 10 and 18% (as the number of tasks increase from CIFAR to Tiny Imagenet) over the ER baseline in our case, with limited replay. Overall, we see that our method is robust and better-performing under both the standard and LLL setups of CL which come with different kinds of challenges. Many CL methods [8, 26] are suitable for only one of the two setups. Further, as explained in Figure 3, our model evolves to become resistant to forgetting as training progresses. This means that beyond a point, it can keep making gradient updates on a small window of incoming samples without needing to do meta-updates.

5.3 Evaluation of La-MAML s learning rate modulation

To capture the gains from learning the LRs, we compare La-MAML with our base algorithm, C-MAML. We ablate our choice of updating LRs asynchronously by constructing a version of C-MAML where per-parameter learnable LRs are used in the inner updates while the meta-update still uses a constant scalar LR during training. We refer to it as Sync-La-MAML or Sync since it has synchronously updated LRs that don t modulate the meta-update. We also construct an ablation referred to as La-ER, where the parameter updates are carried out as in ER but the LRs are modulated using the La-MAML objective s first-order version. This tells us what the gains of LR modulation are over ER, since there is no meta-learning to encourage gradient alignment of the model parameters. While only minor gains are seen on the MNIST benchmarks from asynchronous LR modulation, the performance gap increases as the tasks get harder. On CIFAR-100 and Tiny Imagenet, we see a trend in the RA of our variants with La-MAML performing best followed by Sync. This shows that optimising the LRs aids learning and our asynchronous update helps in knowledge consolidation by enforcing conservative updates to mitigate interference.

To test our LR modulation against an alternative bayesian modulation scheme proposed in BGD [32], we define a baseline called Meta-BGD where per-parameter variances are modulated instead of LRs. This is described in further detail in Appendix H. Meta-BGD emerges as a strong baseline and matches the performance of C-MAML given enough Monte Carlo iterations m, implying m times more computation than C-MAML. Additionally, Meta-BGD was found to be sensitive to hyperparameters and required extensive tuning. We present a discussion of the robustness of our approach in Appendix E, as well as a discussion of the setups adopted in prior work, in Appendix I.

We also compare the gradient alignment of our three variants along with ER in Table 4 by calculating the cosine similarity between the gradients of the replay samples and newly arriving data samples. As previously stated, the aim of many CL algorithms is to achieve high gradient alignment across tasks to allow parameter-sharing between them. We see that our variants achieve an order of magnitude higher cosine similarity compared to ER, verifying that our objective promotes gradient alignment.

6 Conclusion

We introduced La-MAML, an efficient meta-learning algorithm that leverages replay to avoid forgetting and favor positive backward transfer by learning the weights and LRs in an asynchronous manner. It is capable of learning online on a non-stationary stream of data and scales to vision tasks. We presented results that showed better performance against the state-of-the-art in the setup of efficient lifelong learning (LLL) [6], as well as the standard continual learning setting. In the future, more work on analysing and producing good optimizers for CL is needed, since many of our standard go-to optimizers like Adam [13] are primarily aimed at ensuring faster convergence in stationary supervised learning setups. Another interesting direction is to explore how the connections to meta-descent can lead to more stable training procedures for meta-learning that can automatically adjust hyper-parameters on-the-fly based on training dynamics.

Table 3: Results on the standard continual (Multiple) and LLL (Single) setups with CIFAR-100 and Tiny Imagenet-200. Experiments are run with 3 seeds. * indicates result omitted due to high instability.

METHOD CIFAR-100 TINYIMAGENET MULTIPLE SINGLE MULTIPLE SINGLE RA BTI RA BTI RA BTI RA BTI

IID 85.60 0.40 - - - 77.1 1.06 - - - ER 59.70 0.75 -16.50 1.05 47.88 0.73 -12.46 0.83 48.23 1.51 -19.86 0.70 39.38 0.38 -14.33 0.89 ICARL 60.47 1.09 -15.10 1.04 53.55 1.69 -8.03 1.16 54.77 0.32 -3.93 0.55 45.79 1.49 -2.73 0.45 GEM 62.80 0.55 -17.00 0.26 48.27 1.10 -13.7 0.70 50.57 0.61 -20.50 0.10 40.56 0.79 -13.53 0.65 AGEM 58.37 0.13 -17.03 0.72 46.93 0.31 -13.4 1.44 46.38 1.34 -19.96 0.61 38.96 0.47 -13.66 1.73 MER - - 51.38 1.05 -12.83 1.44 - - 44.87 1.43 -12.53 0.58

META-BGD 65.09 0.77 -14.83 0.40 57.44 0.95 -10.6 0.45 * * 50.64 1.98 -6.60 1.73 C-MAML 65.44 0.99 -13.96 0.86 55.57 0.94 -9.49 0.45 61.93 1.55 -11.53 1.11 48.77 1.26 -7.6 0.52 LA-ER 67.17 1.14 -12.63 0.60 56.12 0.61 -7.63 0.90 54.76 1.94 -15.43 1.36 44.75 1.96 -10.93 1.32 SYNC 67.06 0.62 -13.66 0.50 58.99 1.40 -8.76 0.95 65.40 1.40 -11.93 0.55 52.84 2.55 -7.3 1.93 LA-MAML 70.08 0.66 -9.36 0.47 61.18 1.44 -9.00 0.2 66.99 1.65 -9.13 0.90 52.59 1.35 -3.7 1.22

Figure 3: Retained Accuracy (RA) for La-MAML plotted every 25 meta-updates up to Task 5 on CIFAR-100. RA at iteration j (with j increasing along the x-axis) denotes accuracy on all tasks seen uptil then. Red denotes the RA computed during the inner updates (at θj k). Blue denotes RA computed at θj+1 0 right after a meta-update. We see that in the beginning, inner updates lead to catastrophic forgetting (CF) since the weights are not suitable for CL yet, but eventually become resistant when trained to retain old knowledge while learning on a stream of correlated data. We also see that RA maintains its value even as more tasks are added indicating that the model is successful at learning new tasks without sacrificing performance on old ones.

Table 4: Gradient Alignment on CIFAR-100 and Tiny Imagenet dataset (values lie in [-1,1], higher is better)

DATASET ER C-MAML SYNC LA-MAML

CIFAR-100 0.22 10 2 0.0017 1.84 10 2 0.0003 2.28 10 2 0.0004 1.86 10 2 0.0027 TINYIMAGENET 0.27 10 2 0.0005 1.74 10 2 0.0005 2.17 10 2 0.0020 2.14 10 2 0.0023

Broader Impact

This work takes a step towards enabling deployed models to operate while learning online. This would be very relevant for online, interactive services like recommender systems or home robotics, among others. By tackling the problem of catastrophic forgetting, the proposed approach goes some way in allowing models to add knowledge incrementally without needing to be re-trained from scratch. Training from scratch is a compute intensive process, and even requires access to data that might not be available anymore. This might entail having to navigate a privacy-performance trade-off since many techniques like federated learning actually rely on not having to share data across servers, in order to protect user-privacy.

The proposed algorithm stores and replays random samples of prior data, and even with the higher alignment of the samples within a task under the proposed approach, there will eventually be some concept drift. While the proposed algorithm itself does not rely on or introduce any biases, any bias in the sampling strategy itself might influence the distribution of data that the algorithm remembers and performs well on.

Acknowledgments and Disclosure of Funding

The authors are grateful to Matt Riemer, Sharath Chandra Raparthy, Alexander Zimin, Heethesh Vhavle and the anonymous reviewers for proof-reading the paper and suggesting improvements. This research was enabled in part by support provided by Compute Canada (www.computecanada.ca).

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