Role of two learning rates in convergence of model-agnostic meta-learning
Authors: Shiro Takagi
Yoshihiro Nagano
Yuki Yoshida
Masato Okada
Venue:  (2020)
Keywords: meta-learning
convergence

Abstract: Model-agnostic meta-learning (MAML) is known as a powerful meta-learning method. However, MAML is notorious for being hard to train because of the existence of two learning rates. Therefore, in this paper, we derive the conditions that inner learning rate $\alpha$ and meta-learning rate $\beta$ must satisfy for MAML to converge to minima with some simplifications. We find that the upper bound of $\beta$ depends on $ \alpha$, in contrast to the case of using the normal gradient descent method. Moreover, we show that the threshold of $\beta$ increases as $\alpha$ approaches its own upper bound. This result is verified by experiments on various few-shot tasks and architectures; specifically, we perform sinusoid regression and classification of Omniglot and MiniImagenet datasets with a multilayer perceptron and a convolutional neural network. Based on this outcome, we present a guideline for determining the learning rates: first, search for the largest possible $\alpha$; next, tune $\beta$ based on the chosen value of $\alpha$.

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ROLE OF TWo LEARNING RATES IN CONVERGENCE OF MODEL-AGNOSTIC META-LEARNING GENAAG
A Anonymous authors
Paper under double-blind review
ABSTRACT
Model-agnostic meta-learning (MAML) is known as a powerful meta-learning method. However, MAML is notorious for being hard to train because of the existence of two learning rates. Therefore, in this paper, we derived sufficinent condition of inner learning rate a and meta-learning rate 3 for a simplified MAML to locally converge to local minima from any point in the vicinity of the local minima. We find that the upper bound of depends on a. Moreover, we show that the threshold of 3 increases as a approaches its own upper bound. This result is verified by experiments on various few-shot tasks and architectures; specifically, we perform sinusoid regression and classification of Omniglot and Minilmagenet datasets with multilayer perceptron and a convolutional neural network. Based this outcome, we present guideline for determining the learning rates: first, search for the largest possible o;; next, tune 3 based on the chosen value of a.
INTRODUCTION
A pillar of human intelligence is the ability to learn and adapt to unseen tasks quickly and based on only limited quantity of data. Although machine learning has achieved remarkable results, many recent models require massive quantities of data and designed for solving particular tasks. Meta-learning, one of the ways of tackling this problem, tries to develop a model that can adapt to new tasks quickly by learning to learn new concepts from few data points (Schmidhuber, 1987; Thrun & Pratt, 1998).
Among meta-learning algorithms, model-agnostic meta-learning (MAML), gradient-based meta- learning method proposed by Finn et al. (2017), has recently been extensively studied. For example, MAML used for continual learning (Finn al., 2019; Jerfel et al., 2019; Spigler, 2019; Al-Shedivat et al., 2018), reinforcement learning (Finn et al., 2017; Al-Shedivat et al.. 2018; Gupta et al., 2018; Deleu Bengio, 2018; Liu Theodorou, 2019) and probablistic inference (Finn al., 2018; Yoon et al., 2018; Grant al., 2018). The reason why MAML is widely used because MAML is simple but efficient and applicable to a wide range of tasks independent of model architecture and the loss function. However, MAML is notorious for being hard to train (Antoniou et al., 2019). One of the reasons why training MAML is hard is the existence of two learning rates in MAML: the inner learning rate a and meta-learning rate B. A learning rate is known to be one of the most important parameters, and tuning this parameter may be challenging even if the simple gradient descent (GD) method used. Nevertheless, we do not yet know the relationship between these two learning rates and have little guidance on how to tune them. Hence, guidelines for choosing these parameters are urgently needed.
In this paper, we investigate the MAML algorithm and propose guideline for selecting the learning rates. First, in Section we briefly explain by using an approximation how MAML. can be regarded as optimization with the negative gradient penalty. Because the gradient norm is related to the shape of the loss surface, a bias towards a larger gradient norm can make training unstable. Next, based on the approximation explained in Section 2, in Section 3, we derive sufficinent condition of a and B for a simplified MAML to locally converge to local minima from any point in the neighborhood of the local minima. Furthermore, by removing a constraint, we derive sufficient condition for local convergence with fewer simplifications as well. We find that the upper bound 3, of meta-learning rate depends inner learning rate a. In particular, 3, of Or larger than that of 0.

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where ar is the upper bound of a. This is verified by experiments in Section 5. These results imply a guideline for selecting the learning rates: first, search for the largest possible a; next, tune B.
2
MAML AS OPTIMIZATION WITH NEGATIVE GRADIENT PENALTY
2.1 MAML
The goal of MAML is to find representation that can rapidly adapt to new tasks with small quantity of data. In other words, MAML performs optimization for parameters O E IRd that the optimizer can use to quickly reach the optimal parameter 0 for task with few data points. To this end, MAML takes the following steps to update First, it samples a batch of tasks from task distribution P(T) and updates 0 for each task with stochastic gradient descent (SGD). Although MAML allows multiple-step being taken to update 0, we will consider the case only one step being taken for simplicity. The update equation is follows:
0- aVoL (0),
where a is a step size referred as the inner learning rate, Ly (0) is the loss of T, and VoL, (0) is an estimate of the true gradient. The data used for this update is called training data. Next, MAML resamples data, referred to test data, from each r and computes the loss at the updated parameters 0',, obtaining L, (0',) for each task. Finally, to determine @ that can be adapted to 8 for all tasks, o updated with the gradient a sum of loss values L,7(0', over all tasks. In other words,
0 - V Σ Lr (0,).
T~P(()
where 33 is the learning rate called the meta-learning rate and Vo ETTP(r) Ir (f') is an estimate of the true gradient by using the test data. Though learning rates a and / can be tuned during training or different for each task in practice, we will think them as fixed scalar hyperparameters.
2.2 NEGATIVE GRADIENT PENALTY
Unless otherwise noted, we will consider the case of only one step being made per update, and the data are not resampled to compute the loss for updating O. The case of multiple steps and that of training data and test data being separated are considered in Appendix A. The gradient of the loss at O' is gr (0²) = VoL, (0²) = VeO aL., 30, where g-)) is the gradient of L(-) with respect to O. If a small, we can assume that AL. = g- (0); this seems to hold since a is usually small (Finn al., 2017). Then,
ae;
VOL, (0,) = 000' OL,
OL, 00
= (I - (l-V2LL.)
(3)
00
(0) a HHH (0)-
(4)
The result but not the procedure with the approximation is the same as that with the well-known first- order approximation as long as data are not resampled and only one step being taken. The first order approximation has been mentioned by Finn et al. (2017) and extensively studied by Nichol et al. (2018) and Fallah et al. (2019). It is known that the error induced by the first-order approximation does not degrade the performance so much in practice (Finn et al., 2017). Also, Fallah et al. (2019) theoretically proved that the first-order approximation of MAML does not affect convergence result when each task is similar to each other or a is small enough.
For simplicity, will assume that only one task is considered during training, omitting task index т. Therefore, instead of ET>PG) L, (0,), we will consider L(0") as the loss the simplified MAML. Since the MAML loss is just a sum of task-specific loss, extension to the case of multiple tasks being considered is straightforward, as provided in Appendix A.I.I. Because (o(g(0)) g( 2H (H(0)g(e) (O), if we define i.(0)= L(O").
(( 2 L(Ø) -g((0)T g(0).
(5)
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The above means that the simplified MAML can be regarded optimization with the negative gradient penalty. We will analyze this simplified MAML loss in Section It can also be interpreted as Taylor series expansion of the simplified MAML loss for the first-order term, up scale:
((0) L(0 - aVVo(())
(6)
~ L(0) aVoL(0) VoL(0)
(Taylor series expansion)
T((0) og(0)' og()) 9(0).
The fact that the simplified MAML is optimization with the negative gradient penalty is worth keep- ing in mind. Because the goal of gradient- optimization to find a point where the gradient is zero, bias that favors a larger gradient is highly likely to make training unstable; this can be cause of instability of MAMI. (Antoniou et al., 2019). In fact, as shown in Fig. 1, the gradient norm becomes larger during training, as do the gradient inner products, as Guiroy et al. (2019) observed.
40
Joint training
Gradient Norm
MAML
30
10
0
10000 20000 30000 40000 50000 Iterations
Figure 1. Gradient norm during training. We compute the norm per task and subsequently compute their average. Joint training shows when 0, and MAML is when le-2. These results computed using training data, but those determined using test data behave similarly. The total number of iterations 50000, le-3 and the Adam optimizer is used. Other settings are same as those Section 5.2.
3 LEARNING RATE FOR LOCAL CONVERGENCE
In this section, we will derive a sufficient condition of learning rate a and 3 for local convergence from any point the vicinity the local minima. To this end, we will assume that only one step taken for update and training data and test data are not distinguished as we did in Section 2. Also, we do not consider SGD but steepest GD derive the condition. In other words, same training data are assumed to be used continuously for updating parameters during training. First, we will consider the case of only single task being considered. Next, we will consider the case of multiple tasks being considered.
3.1
SINGLE TASK
3.
CONDITION FOR INNER LEARNING RATE a
First, derive the sufficient condition of learning rate To this end, we will consider the sufficinet condition that a fixed point is a local minimum. Taking the Taylor series for the second-order term fixed point 0" the simplified MAML loss is
(( ~ L( 1 (000 0*).
(8)
where A H a (Tg + H2) is the Hessian matrix of i.(0) at 0* and g = VoL(O") E Rd. H (O-( E Raxd and T viL(ee) E Raxdxd The calculation of H is presented in Appendix B. We calculated the magnitudes Tg and H² numerically and observed that Tg was much smaller than 2 in practice. Hence, will ignore Tg while deriving the condition and will
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thus assume that H - - aH2 Further details provided in Appendix C, and the case of Tg being considered provided in Appendix D. Since PA = P - =A PT where A A is diagonal matrix with entries that are eigenvalues of H and P is a matrix with rows that are eigenvectors of H, the sufficinet condition of a for 0 to be local minimum is
Vi, X(H), - :HH); - = Vi, < X(H):
1
(10)
Note that X(A); represents the ith eigenvalue of matrix A. Hence, the sufficient condition of for O" to be local minimum is
1
Vi,
(11)
X(H):
Therefore, ag is the inverse of the largest eigenvalue of H.
3.1.2 CONDITION FOR META LEARRING RATE B
Next, we derive the sufficient condition of meta-learning rate 3 for the simplified MAML to locally converge to the local minima discussed above from any point in the vicinity of the local minimum. This is an extension of research of LeCun et al. (1998). Since PPT I, the simplified MAMI. loss can be written as
((0) ~ I(0)) + ! (0--0') (T00-''TPPPPP P)P HP(PT - 0*)).
(12)
By using the simplified loss defined in Eq. 8, the update equation of the parameter 0 with GD is
1) = (()) (Vo
(13)
((t) (0 0*)
(14)
where / (1 (t=000 0,..,, M) ) is iteration and M is the total number of iterations. Hence, 0(1 + 1) 0"
(I (0(()
0*). If we denote p (0 - 0*) by v, the simplified MAML loss in Eq. 12
((v) L(0) 1 UO u. Because the gradient of L(v) for v is v.Ī(v) = A
v, the update
2
equation of u is
1)++ v(t u() = (I BA
(15)
where ut) the value of during iteration t. Assuming that Eq. 11 holds, the sufficinet condition of 3 is as follows: for all i,
1 3A(H -aH 11- ((A(H); ax( <
(16)
-1 +(A(HH; a)(H))) <1 (cA(HH); aA( H 0 holds because of Eq. 11)
(17)
2
+ 3 X()): aX(H)
(18)
Consequently, the sufficient condition for the simplified MAML to locally converge to local minima from any point the vicinity of the local minima is as follows:
1
Vi, a X(H):
B X(H); HH
(19)
Vanilla GD with learning rate 3 corresponds MAML if a O. In this case, B < Amnoe is the condition of 3, where Amaz is the largest eigenvalue of H, because 2/2maz smaller than any other X; (LeCun et al., 1998). Though this holds for the simplified MAML as well, this is not the case if close to a.. The reason is that 8e diverges as a approaches X(H), or a as Eq. 18 indicates. Hence, for the simplified MAML we must consider not only the largest but also other eigenvalues and in particular, the second-larges, eigenvalue. In short, unlike when vanilla GD is employed, 3, depends on a in the case of the simplified MAML, and 8, expected to be larger is close to Oee as shown in Fig. 2. This finding validated by experiments presented in Section 5.
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3.2
MULTIPLE TASKS
We discussed the case of only one task being available during training in Section 3.1. In this sec- tion, derive upper bounds of and that apply if multiple tasks т P(r) considered. Assumptions, except that multiple tasks are considered, are the same as those in Section 3.1.
3.2.1 CONDITION FOR a
Now, we define the simplified meta-objective as a sum of task-specific objectives: L( Σr (0). Then, at fixed point 0"
((0) ~ L(0) + 1 -0 00) H
(20)
Σ
[0'(0") + 2 0 ( - aHH 8')
(21)
PwP(T)
Σ
L.(0") 2 0*) 0T) T. - - -0-0000000000 00)
(22)
PrP(T)
where A the Hessian matrix of (() at 0. Note that we ignore T,9; as we did Section 3.1.1. Since H,, HH is not necessarily simultaneously diagonalizable for each task cannot exactly express eigenvalues of H as function of ((H,) and a. Therefore, instead of the exact value of ae. we will derive an upper bound of Ck..
If 0 is a local minimum for all (0"), 0t is local minimum for (()) as well. Hence, if all eigenvalues of H, H, 2 are positive for all tasks, those of H are positive well. Therefore, if the inequality
1
Vr, a < X(H,),
(23)
holds, it guarantees that the condition that 0" is local minimum of i.(0) is satisfied, Note that this sufficient condition if 0" is local minimum for all (0*). Since 0" be a local minimum of i.(0) even when is not local minimum for all To*)),, the upper bound of a seems to be larger than that in Eq. 23 in practice.
3.2.2 CONDITION FOR |
The analysis for 3 is also basically the same as that performed in Section 3.1.2. Denoting P. (0 0²) by Vr, we obtain
[,,(++1)= (I- Σo((+t)=I(-B(AAe - -AA2.. (t).
(24)
Since H, is not always simultaneously diagonalizable for each task, as mentioned Section 3.2.1, P. differs from task task in general. Hence, we will consider upper bound of Be as we did for Oe in Section 3.2.1. Accordingly, if both Eq. 23 and
(({(H))) aA(H, )?) <
(25)
hold for any eigenvalue X; of any task v, it guarantees that 8 satisfies the condition for local conver- gence. Therefore, sufficient for the simplified MAML to locally converge to local minima from any point the neighborhood the local minima in the case of multiple tasks as follows:
2
VT, a X(H,);
B < X(H)); - HXHH)
(26)
4 RELATED WORKS
Sevral papers have investigated MAML and proposed various algorithms (Nichol et al., 2018; Guiroy al., 2019; Eshratifar et al., 2018; Antoniou al., 2019; Fallah al., 2019; Khodak al.,
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10'
Be A1
Az
10"
A
A
10"
10 10 10
Figure 2. Curves f. as a function for eigen- values of the Hessian, Xg <... < A4. Parameter | is supposed to be smaller than 3. for both Å and As. Hence, 8 should be chosen from the colored area. Since must satisfy a X, should also be in the colored region. The dashed line shows 3e if a 0.
Figure 3. Training loss linear regression. The area colored in black is when the loss is below le-2, and that in gray is when the loss is over 1e-2. Uncolored region is not considered. 8e
shows Be of A. where X1 X2. The dashed line is ae- Theoretical Se and a₂ correspond to empirical ones.
dias Be is larger than that at a
2019: Vuorio al., 2018; Finn al., 2019: Deleu & Bengio, 2018; Liu & Theodorou, 2019; Deleu & Bengio, 2018; Grant et al. 2018). Nichol et al. (2018) studied the first-order MAML fam- ily in detail and showed that the MAML gradient could be decomposed into two terms: term related to joint training and a term responsible for increasing the inner product between gradients for different tasks.
Guiroy et al. (2019) investigated the generalization ability of MAML. The researchers observed that generalization was correlated with the average gradient inner product and that flatness of the loss surface, often thought to be an indicator of strong generalizability in normal neural network train- ing, was not necessarily related to generalizability in the case MAML. Eshratifar et al. (2018) also noted that the average gradient inner product was important. Hence, the authors proposed an algorithm that considered the relative importance inner product between the task-specific gradient and the average gradient. Although the above stud- ies were cognizant of the importance of the inner product of the gradients, they did not explicitly insert the negative gradient inner product, which the negative squared gradient norm with sim- plifications, as regularization term. To consider the simplified MAML as optimization with a regularization term contribution of our study.
each parameter based on the magnitude of the
Antoniou et al. (2019) enumerated five factors that could cause training MAML to be difficult. Then, they authors proposed an algorithm address all of these problems and make training MAML easier and more stable. Behl et al. (2019), like us, pointed out that tuning the inner learning rate and meta-learning rate 3 was troublesome. The authors approached this problem by proposing an algorithm that tuned learning rates automatically during training.
Fallah et al. (2019) studied the convergence theory of MAML. They proposed a method selecting meta-learning rate by approximating smoothness of the loss. Based on this result, they proved that MAML can find an e-first-order stationary point after sufficient number of iterations. On the other hand, studied the relationship between the sufficient conditions of inner learning rate and meta-learning rate 3 and showed that how the largest possible / is affected by the value a.
EXPERIMENTS
In this section, we will present the results of experiments to confirm our expectation that MAML allows larger ( if is close to its upper bound. First, will show the result of linear regression with simplifications used Section 3.1. Because linear regression is convex optimization, the result is expected to exactly match the theory presented in Section 3.1. Second, to check if our expectation is confirmed in practice as well, we will present results of the practical case without any simplification. In particular, conducted sinusoid regression and classification of Omniglot and Minilmagenet

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datasets with multilayer perceptron and convolutional neural network (CNN). Note that the meta-objective used for experiments is not a sum of task-specific objectives but a mean of them.
766
7ge1
76
1n6
30c
422
6611
602
1e2
162
(a) Sinusoid
(b) Omniglot
(c) Minilmagenet
Figure 4. Training losses for (a) sinusoid regression, (b) Omniglot classification, and Minilmagenet classi- fication at various values a and / after a fixed number of iterations. The area with no color represents diverged losses, and the dashed line indicates the values of 3 above which the loss diverges for a = 0. The maximum possible / is larger if a is close to the value above which the losses diverge than that at 0.
5.1 LINEAR REGRESSION
We performed a linear regression, where the task is to regress a linear function with scale parameter in the range of [O, 5.0] and bias parameter in the range of [0, 5.0] based data points in the range of [-5.0, 5.0]. The true function has the same architecture as that of the model. We employed the steepest gradient descent method to minimize the mean squared loss, where step was taken during update. Only one task was considered during training and the same data was used to update the task-specific parameter and the meta parameter as did Section 3.1. Using these settings, we computed the training loss after 500 iterations with a in the range of [le-5, 9e-2] and 3 in the range of [5e-3, 9e+0J. The eigenvalues are those of the Hessian matrix of the training loss at the end of the training. where a 5e-2 and B 7--1. We chose this training loss because it was thought to be the closest to minima. Fig. 3 shows the training losses at various values of a and B. Horizontal axis indicates and vertical axis indicates B. The curves are Be of two eigenvalues and the dashed line shows ag-- In the case of linear regression with simplifications used in Section 1, the result of numerical experiment shows good agreement with upper bounds of Be and oa that derived in Section 3.1, as shown in Fig. 3.
5.2 SINUSOID REGRESSION
We conducted sinusoid regression, where each task regress a sine wave with amplitude in the range of [0.1, 1, 5.0] and phase in the range of [0, π] based on data points in the range of [-5.0, 5.0]. A multilayer perceptron with two hidden units of size 40 and ReLU was trained with SGD. The batch size of data was 10, the number of tasks was 100, and step was taken for update. Using these settings, we computed the training loss after 500 iterations with a in the range of [le-4, 9c-1] and 3 in the range of [le-2, 9c-0]. Fig. 4 (a) shows the training losses with various values of a and 3. The dashed line indicates 3 a = 0 over which training loss diverges. According to Fig. (a), is close to the value above which the losses diverge, larger 3 can be used. As explained above, we did not put any simplification as we did in Section 3 and 5.1 in the experiment, meaning that used different data for updating task-specific parameter and meta parameter, considered multiple tasks, and employed not steepest GD but SGD as optimizer. Despite simplifications, surprisingly, this result confirms the expectation that MAML allows larger / if close to ag.
5.3 CLASSIFICATION
We performed classification of the Omniglot and Minilmagenet datasets (Lake et al., 2011 ravi Larochelle, 2017), which are benchmark datasets for few-shot learning. The model used was essentially the same that Finn et al. (2017), and hence, Vinyals et al. (2016) used. The task a five-way one-shot classification, where the query size is 15, the number of update steps is two, and the task batch size is 32 for Omniglot and four for Minilmagenet. In this setup, we computed
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the training losses after 100 iterations for the Omniglot dataset and one epoch for the Minilmagenet dataset with various values of a and 3; for Omniglot, a was in the range of [1e-3, 9c-0] and 3 was in the range of [le-1, 1, 9c+1], and for Minilmagenet, was in the range of [lc-4, 9c-1] and 3 was in the range of [1e-2, 9e-0]. Fig. 4 (b) and (c) show the training losses of classification task at various values of and (3. The dashed line indicates 8 of
O above which training loss diverges. As shown in Fig. 4 (b) and (c), the maximum 3 is larger at large a. Even though the model architecture composed of convolutional layer, max-pooling, and batch normalization (loffe & Szegedy, 2015) and practical dataset is used for training, our expectation is confirmed in this case as well. This result confirms that our theory is applicable in practice.
Our experimental result confirms that larger a is good for stabilizing MAML training. According to Fig. (a), (b) and (c), moreover, while large /3 does not necessarily make training loss smaller, employing large leads to smaller training loss comparatively. These result has a practical impli- cation for tuning the learning rates: first, the largest possible a should be identified, and | may be subsequently tuned based on the value of a. Once you identify ate, MAML is likely to work well even meta-learning rate is roughly chosen. Taking large a is desirable for the goal of MAML as well. The aim of MAML is to find a good initial parameter which quickly adapt to new tasks and the quickness is determined by inner learning rate a because a determines the step size from initial parameter task-specific parameter when model is fine-tuned. Therefore, identifying or is not only good for robustifying the model against divergence but also good for finding good initial parameter.
6 CONCLUSIONS
We regard simplified MAML as training with the negative gradient penalty. Based on this formu- lation, we derived the sufficient condition of the inner learning rate and the meta-learning rate B for the simplified MAML to locally converge local minima from any point in the vicinity of the local minima. We showed that the upper bound of 3 required for the simplified MAML to locally converge to local minima depends on a. Moreover, we found that if a is close to its upper bound ac. the maximum possible meta-learning rate Be is larger than that used while training with ordinary SGD. This finding is validated by experiments, confirming that our theory is applicable in practice. According to this result, we propose guideline for determining and 3; first, search for close to agi next, tune 3 based on the selected value of a.
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Jaesik Yoon, Taesup Kim, Ousmane Dia. Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian Model-Agnostic Meta-Learning. Neural Information Processing Systems 2018, 2018.

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Under review as a conference paper at ICLR 2020
A NEGATIVE GRADIENT PENALTY: GENERAL CASE
In Section 2, we explained the simplest case of training data and test data being the same, only one task being considered and only one step being taken for update. In this section, analyze the cases of waiving one of these simplifications. Here, we will interpret the simplified MAML loss as a Taylor series expansion of the MAML loss for the first-order term.
A.1 WHEN THE TRAINING DATA AND TEST DATA ARE DIFFERENT
When using different data to compute the losses for updating the task-specific parameters and the meta parameters as done in practical applications, the simplified loss as follows:
(() = L test (0 a VoL train (0)
(27)
~ L test (0) (0) test ((0)
(28)
where T,train and T,teat represent the loss values for training and test data, respectively, and grrann and glest are gradients of these losses.
A.1.1 WHEN MULTIPLE TASKS ARE CONSIDERED
When multiple tasks are considered, as MAML does, the meta-loss L()) is just a sum of losses (f) for all tasks T:
i(0)= = Σ (0) = E
(I, (0) test 9- (r)r))))..
(29)
P()((T
TNP(T)
A.2 WHEN STEPS ARE TAKEN DURING THE UPDATE
If task-specific parameters are updated with k-step SGD, the loss can be written as follows;
(0) (0) [
(30)
Note that (0) is the loss computed with the data at the ith step, and (0) is the gradient of the loss.
B CALCULATION OF H
Because H is the Hessian matrix L. at 0", we derive the Hessian of Eq. 8. Then,
H = Vii(
(31)
V [((0) 2 gg( gg()))(()) 9(0))
(32)
H(0) aV. (H())(())
(33)
H(0)) a (V (V)H()))(0) H())H(())
(34)
-H -H-a H
(35)
C MAGNITUDE OF Tg AND 2
We conducted sinusoid regression with essentially the same condition that we explain Section 5.2 except that the total number of iterations is 50000 and learnig rates are fixed. Parameters and are le-2 and le-3 respectively. We calculated Tg numerically with the training error at the end of the training. As we showed Section 3, especially large eigenvalues upper bounds of learning rates. Therefore. if >(Tg + H ) ss 2) mazs can ignore Tg when deriving the condition. We calculate the maximum and the second- -argest eigenvalues Tg, H² and Tg 2 of the trained model. As shown in Fig. 5 (a), X(Tg H mar is almost equal to
H are important for the
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Under review as a conference paper at ICLR 2020
XHH2) mar. and A(TT) maz is by far smaller than them. Therefore, ignoring A(Tg) maz is reasonable when the conditions are derived. Furthermore, we calculate the Frobenius norm of Tg and H As Fig. 5 (b) indicates, the Frobenius norm of Tg is much smaller than that of H meaning that Tg negligible the sense of the magnitude of the norm as well. These results confirm that we can neglect Tg when considering H.
3x10
2x10"
2x10"
10"
79
HP
Tg
HP
To+++
HP
(a) Maximum eigenvalue
(b) Second largest eigenvalue
Frohenius
Figure 5. (a): The maximum eigenvalues of Tg. H and Tg + H² is clear that the maximum eigenvalue of
Tg + H is almost the same as that of H while that of Tg is much smaller than them. (b): The second-largest eigenvalues of Tg. H and Tg H Like (a), the second-largest eigenvalue Tg H is almost equal to that of H (c): The Frobenius norm of Tg and H The Frobenius norm Tg much smaller than that of H
D CONVERGENCE CONDITION WHEN Tg IS CONSIDERED
We assumed that Tg was negligible in Section 3. In this section, we derive sufficient condition of and for the simplified MAML to locally converge local minima from any point in the vicinity of the local minima under some assumptions when Tg is considered.
D.1 CONDITION INNER LEARNING a SATISFY
When Tg is considered. the Hessian matrix of the simplified MAML loss i H (Tg + H²). Because H (Tg + 2 is a real symmetric matrix, it can be diagonalized. Then, the sufficient condition that fixed point 0* is local minimum
Vi, X(E), = A(H (Tg H²)); > 0.
(36)
Note that Tg and H are not simultancously diagonalizable, so we cannot decompose X(H); y into did in Section 3. Therefore, have to consider the relationship
eigenvalues of each matrix
among XH)), A(H); and X(Tg), In general, it is known that x Hermitian matrices A and B satisfy the following equation (Bhatia, 2001):
X ( Xt(B) < X(A
(37)
where A(A) represents a vector with elements that are cigenvalues of A, 1 indicates the operation of sorting vector in the ascending order, and 1 indicates that in the descending order. two real vectors I, E Rd are related in the following way, x is said to be majorized by and the relationship written as x < y:
for 1 1sk5d 2
(38)
1=1
(39)
1
11

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Under review as a conference paper at ICLR 2020
We define A = H aH and B -agg; then,
(A) + (B) < X(A B)
(40)
(A) aA (Tg) < B)
(41)
k
Σ (A)) ax - Σ (A+++B); for 1 S d.
(42)
Let us suppose that for top k eigenvalues of A, Tg. and + B, conditions X(A + B); R CA+B.
X(Tg): & (Tg and X(A); s ca hold, where CA+B. CA and CTg are some constant values. Then,
XA). (Tgg)) < A(+BB, for Sisk
(43)
X H oH (ggT); S A(H a: [Tg + H²)); for 1sisk k.
(44)
The sufficient condition for a fixed point 0" to be a local minimum is
Vi, X(H aH (Tg))) >
(45)
Now, we assume that all eigenvalues of H below a threshold are O; X(H); for k < n. In fact, many eigenvalues of the loss computed by a trained model such as a deep neural network are known to be very small. Thus, the following condition is satisfied, is enough say that the condition is satisfied:
A(H - xHH); ax((TT))))))))) 00 for isisk k
(46)
(<((T)); <0
for k siid
(MHH): aA(H)² ) a>(Tg)) 0 for isisk k: (((T)) <0
(47)
for k s d
((H); a({(H) mmn + A(Tg).) for isisk
(48)
(((Tg): <0 0
for ksidd
A(H); aA(Tg); > 0 for 1 1 i k ((Tg))
(: X(H) min = 0)
(49)
for h $ i d
In other words, if the smaller d k eigenvalues of Tg are not negative values, the fixed point is not guaranteed to be a minimum, and if they are negative, satisfying Eq. 49 is sufficient.
D.2 CONDITION FOR META LEARNING RATE 33 TO SATISFY
Next, we derive a sufficient condition of meta-learning rate B. Our analysis will be based on the assumptions identical to those mentioned above. If the inequalities in Eq. 49 hold, it is sufficient for 33 to satisfy
Vi, -1 B a)HH); a>(Tg):) S -1+ 3A(H aTg + H²)), <
(50)
2
Vi, XHH) - a\(Tg); (:: (:A(H) min 0)
(51)
for the condition of local convergence.
E EXPERIMENT WITH ADAM
We conducted the same experiment as that explained in Section 5.2 and 5.3, but this time, we used Adam optimizer instead of SGD for the meta-optimizer (Kingma & Ba, 2014). The results are different from those we showed in Section 5. For training with Adam, MAML no longer allows us to use larger 3c.. Because Adam has more parameters that we have to consider than SGD does, this result is not at all surprising. Nonetheless, it is important to keep these facts in mind when Adam is used for MAML optimization.
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Under review as a conference paper at ICLR 2020
360
560
760
5e0
461
561
52
Sorl
3p-1
345
8.8
-6
36-3 06- 0e-3
36-2
6e-2
3e-
3erl
5e-1 96-1
(a) Sinusoid
(b) Omniglot
(c) Minilmagenet
Figure 6. Training losses for (a) the sinusoid regression, (b) Omniglot classification and (c) Minilmagenet classification for various values of the inner learning rate and meta-learning rate after a fixed number of iterations. The with no color represents the diverged losses.
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Reviews:
[{"review_id": "rJx4PXPnFr", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors study a method to help tuning the two learning rates used in the MAML training algorithm. First, they derive a necessary condition for the convergence of the gradient descent in the single task setting. The condition relies on the eigenvalues of the Hessian of the task loss. This condition is reminiscent of convergence criteria for gradient descent on quadratic objectives, and the authors make the interesting observation that the criteria for the exterior learning rate beta depends on the interior learning rate alpha. \nHowever, in the setting of interest that is the multitask learning, the trick used to analyse the eigenvalues doesn’t work anymore. To circumvent this issue, the authors provides a sufficient condition for the multitask equivalent of the necessary condition to hold.\n\nThe derivations are correct (few typos, see nitpick below) and the paper is nicely presented. I’m not a MAML expert so I’ll let the other reviewers judge how this paper compares to the current literature. However, I think that the empirical work can be pushed further. The experiments on Omniglot and MiniImagenet are coherent with the theory, but I am not completely sure of their impact. Indeed, the learning rate domain on which MAML converge seems coherent with what the authors are predicting, but it doesn’t allow us to choose the learning rates before training the model. It doesn’t help that the criteria relies on the spectrum of the Hessian evaluated at a critical point, which of course is not known before convergence in non quadratic setting.\nThe paper would be way more convincing if the authors could use their findings to describe some more precise heuristics to tune the learning rates, and conduct a proper comparison of its performance with other methods.\n\nAs a result, I think the paper is exploring an interesting direction, but that the empirical work might be too preliminary for publication.\n\nNitpick:\n\n- Notations in 2.2 are a bit confused.The gradient of L wrt theta is denoted first with nabla, then with partial derivatives. Nabla is then used to denote the Jacobian of theta’ wrt theta. There is a small mistake in the 4th line of 2.2. Chain rule for gradient of L wrt theta’ does not give (d L / d theta) times gradient but instead give gradient times (d L / d theta)^T. The error is fixed when passing from (3) to (4) so it doesn’t impact the paper.\n- Also, a transpose is missing in the taylor expansion on page 3.\n- \"A multilayer perceptron with two hidden units of size 40\" -> layers?\n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}, {"review_id": "SJloDQiSFB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In this paper, the authors tackled one important research problem in MAML, i.e., the optimization instability, by investigating the two learning rates. Though I appreciate the theoretic contribution this work makes, I am not sure with the practical significance of it.  Below please find my detailed comments. \n\nPros:\n-\tIn this work, the authors focused on an important problem – training MAML is kind of unstable and tricky, so that developing guidelines that stabilize MAML or its first-order approximations is of significance. \n-\tThis work theoretically discusses the relationship between the inner loop learning rate and the outer one, under a set of assumptions. \n-\tThe paper is well written and easy to follow.\n\nCons:\n-\tSome of the simplifications for proving are empirical, so that the proof itself is not that rigorous. \n  o\tFor example, In Section 3.1.1, the authors ignored Tg based on an observation that Tg is small. However, even in the Appendix, the authors did not explicate the experimental setting where they reach such a conclusion. Will Tg be always small then, in any task and any dataset?\n-\tThe take-away of this work is not clear, in other words, it makes small contribution to the practical training of MAML. \n  o\tPractically, we often train MAML in 5 or 10 steps instead of only one step. Will the conclusion apply to such setups? Or will the model itself diverge during the inner loop as more steps are taken, provided with a larger value of \\alpha?\n  o\tPractically, we merely use vanilla SGD. The figure in the Appendix shows that the main conclusion actually does not apply to Adam, which disempower the practicability of the theoretical guideline. \n  o\tHow can we search the “largest possible” inner loop learning rate? Should we still use grid-search or heuristic search? In that case, what is the implication/shortcut that this proposed guideline brings? We still reach a stable training process by tuning the hyperparameters in a brute-force fashion.  \n-\tThe baselines should be compared, to support the effectiveness of this proposed algorithm. For example, Behl et al. (2019) automatically tuning the learning rates during training definitely needs to be compared, since both aim to stabilize the training of MAML.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "BygN9L1ycS", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper studies the two learning rates \\alpha and \\beta used in Model-Agnostic Meta-Learning (MAML) algorithm by [Finn et al., 2017]. MAML is known to be difficult to train, and part of the reason why is the need to tune the two learning rates. Under simplifications, the paper derives some necessary conditions on \\alpha and \\beta for the MAML iterates to converge to local minima, and then verifies the theory by experiments on synthetic and real-world data.\n\nOverall, I think this paper should be rejected. In my opinion, this paper analyzes a simplified setting which is far from the original MAML setting and yet derives conditions that are not very meaningful or useful. Although it is interesting to see some match between theory and experiments (especially Fig 4), the theory part seems to have room for improvement, and I will detail the reasons in the following.\n\nI believe that the analysis is done on an overly simplified setting and easily breaks without such simplifications.\n- The paper assumes that training set is equal to test set and these sets do not change over iterations, which is essentially equivalent to assuming full gradient access; this is different from the stochastic setting of MAML. \n- The \"necessary condition\" is derived in the case where there is only one task, and there is no discussion on generalizing to some other tasks. This means that the setting is too simplified so that it is not even a meta-learning setup.\n- In the extension to multiple tasks, the paper derives \"sufficient conditions\" to a set of \"necessary conditions\" for convergence of MAML to local minima. This means that, if we consider the sets of events A = {convergence of MAML to local min}, B = {necessary conditions for A}, and C = {sufficient conditions for B}, A is a subset of B and C is a subset of B and nothing can be specified between A and C. In the extreme case, A and C may be even disjoint subsets of B. Thus, at least in theory, the conditions for multiple tasks do not tell us anything about convergence.\n- The analysis relies on a number of \"A is approximately equal to B\" arguments without careful handling of errors, e.g., (3) and (4), and Tg = 0.\n\nI also have concerns about the correctness of the analysis in the single task case.\n- In Section 3.1.2 and eq (12), the paper reparametrizes \\theta to v and analyzes updates on v. How is (12) obtained from the original update rule of \\theta? In fact, the paper analyzes steepest GD but doesn't provide the explicit update rule; for example, is it steepest GD with respect to which norm? It'd be helpful to have the details in the main text.\n- Even if (12) is true, there are pathological cases lying in a set of measure zero, where (12) converges even when (13) is not satisfied. For simplicity, consider v(t+1) = G v(t), where G is a real symmetric matrix. Assume all but one eigenvalue \\lambda_1 are greater than 1 and |\\lambda_1| < 1. Let v_1 be the corresponding eigenvalue for \\lambda_1. Then, if v(0) = v_1, the sequence v(t) converges to zero. This means that the \"necessary condition\" derived in (16) may not actually be a necessary condition for convergence.\n\nThere are also some points in the main text that doesn't describe the setting clearly; if the reader has no prior knowledge of MAML algorithm, they may get confused.\n- In the update rule (1) and (2), it'd be better to mention that \\nabla_\\theta L_\\tau (\\theta) is an *estimate* of the true gradient using the training data and test data, respectively. In their current status, the gradients in (1) and (2) will read as the full gradients of the population where the training/test data points are sampled from.\n- Section 2.1 only introduces the case where the update (1) is done only once in the inner loop. However, later Section 2.2 and Section 3, the paper says \"only one step is taken for update...\". Without the prior knowledge that there are versions in which multiple updates (1) are done, the readers can easily get confused.\n\nMinor comments\n- The word \"minima\" used throughout should better be corrected to \"local minima,\" as mere minima may be understood as \"global minima.\"\n- In the abstract, there is a phrase \"in contrast to the case of using the normal gradient descent method.\" Which setting do you mean by the \"normal gradient descent\"?\n- After equations (3) and (4), the paper says \"The above is known as the first-order approximation...\", but as far as I'm concerned, the first-order approximation version of MAML is (3) without the hessian term, not (4). This statement can potentially be misleading.\n- Column vectors / row vectors are mixed up. In (3), the partial derivatives are row vectors, but in (4) g_\\tau(\\theta)'s are column vectors. However, right above (3), g_\\tau(\\theta) is a row vector this time.\n- Did (5) come from the approximation (4)? If so, as \\tilde L(\\theta) is defined to be L(\\theta'), the two sides of (5) are \"approximately equal\" not \"equal.\"\n- In Eq (19): P_\\tau (\\theta - \\theta^*) -> (\\theta-\\theta^*).\n- At the end of page 6, what do you mean by \"not a sum but a mean\"? Isn't sum of n things equal to the mean, except for a factor of 1/n?", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper theoretically and empirically studies the inner and outer learning rate of the MAML algorithm and their role in convergence. While the paper presents some interesting ideas and add to our theoretical understanding of meta-learning algorithms, the reviewers raised concerns about the relevance of the theory. Further the empirical study is somewhat preliminary and doesn't compare to prior works that also try to stabilize the MAML algorithm, further bringing into question its usefulness. As such, the current form of the paper doesn't meet the bar for ICLR.
Decision: Reject