Meta-Learning with Implicit Processes
Authors: YIZHOU CHEN
DONG LI
NA LI
TONG LIANG
SHIZHUO ZHANG
Bryan Kian Hsiang Low
Venue:  (2021)

Abstract: This paper presents a novel implicit process-based meta-learning (IPML) algorithm that, in contrast to existing works, explicitly represents each task as a continuous latent vector and models its probabilistic belief within the highly expressive IP framework. Unfortunately, meta-training in IPML is computationally challenging due to its need to perform intractable exact IP inference in task adaptation. To resolve this, we propose a novel expectation-maximization algorithm based on the stochastic gradient Hamiltonian Monte Carlo sampling method to perform meta-training. Our delicate design of the neural network architecture for meta-training in IPML allows competitive meta-learning performance to be achieved. Unlike existing works, IPML offers the benefits of being amenable to the characterization of a principled distance measure between tasks using the maximum mean discrepancy, active task selection without needing the assumption of known task contexts, and synthetic task generation by modeling task-dependent input distributions. Empirical evaluation on benchmark datasets shows that IPML outperforms existing Bayesian meta-learning algorithms. We have also empirically demonstrated on an e-commerce company's real-world dataset that IPML outperforms the baselines and identifies outliers tasks which can potentially degrade meta-testing performance.

Under review as a conference paper at ICLR 2021
META- WITH IMPLICIT PROCESSES
Anonymous authors
Paper under double-blind review
ABSTRACT
This paper presents novel implicit process-based meta-learning (IPML) algo- rithm that, in contrast to existing works, explicitly represents each task as contin- uous latent vector and models its probabilistic belief within the highly expressive IP framework. Unfortunately, meta-training in IPML is computationally challeng- ing due to its need to perform intractable exact IP inference in task adaptation. To resolve this, we propose novel expectation-maxiiizaiiie maximizaaiinnn algorithm based the stochastic gradient Hamiltonian Monte Carlo sampling method to perform meta- training. Our delicate design of the neural network architecture for meta-training in IPML allows competitive meta-learning performance to be achieved. Unlike existing works, IPML offers the benefits of being amenable to the characteriza- tion of a principled distance measure between tasks using the maximum mean discrepancy, active task selection without needing the assumption of known task contexts, and synthetic task generation by modeling task-dependers input distribu- tions. Empirical evaluation on benchmark datasets shows that IPML outperforms existing Bayesian meta-learning algorithms. We have also empirically demon- strated on an e-commerce company's 's real-world dataset that IPML outperforms the baselines and identifies "outlier" tasks which can potentially degrade meta- testing performance.
INTRODUCTION
Few-shot learning (also known as meta-learning) is a defining characteristic of human intelligence. Its goal is to leverage the experiences from previous tasks to form a model (represented by meta- parameters) that can rapidly adapt to a new task using only a limited quantity of its training data. A number of meta- meta-learnigg algorithms (Finn et al. 2018 Jerfel et al. 2019 Ravi & Beatson 2018
Rusu et al. 2019; Yoon et al., 2018) have recently adopted a probabilistic perspective to characterize the uncertainty in the predictions via Bayesian treatment of the meta- meta-parameters. Though they can consequently represent different tasks with different values of meta-parameters, it is not clear how or whether they are naturally amenable (a) the characterization of principled similarity'distance measure between tasks (c.g.. for identifying outlier tasks that can potentially hurt training for the new task, procuring the most valuable/similar tasks/datasets to the new task, detecting task distribution shift, among others), (b) active task selection given limited budget of expensive task queries (see Appendix A.2.3] for an example of a real-world use case), and (c) synthetic task/dataset generation in privacy-aware applications without revealing the real data or for augmenting limited number of previous tasks to improve generalization performance.
To tackle the above challenge, this paper presents novel implicit process-based meta-learning (IPML) algorithm (Sec. 3 that, in contrast to existing works, explicitly represents each task as a continuous latent vector and models its probabilistic belief' within the highly expressive IF frame- work (Sec. Db. Unfortunately, meta-training in IPML is computationally challenging due to its need to perform intractable exact IP inference in task adaptation
To resolve this, we propose novel
'An IP (Ma et al. 2019) a stochastic process such that every finite collection of random variables has an implicitly defined joint prior distribution. Some typical examples of IP include Gaussian processes, Bayesian neural networks, neural processes Carnelo et al., 2018). among others. An IP is formally defined in Def. [
'The work of Ma et al. 2019 uses the well-studies Gaussian process as the variational family to perform variational inference in general applications of IP, which sacrifices the flexibility and expressivity of IP by constraining the distributions of the function outputs to be Gaussian. Such straightforward application of IP meta-learning has not yielded satisfactory results in our experiments (see Appendix A.4

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In the E step, we per- form task adaptation using the stochastic gradient Hamiltonian Monte Carlo sampling method (Chen et al. 2014) to draw samples from IP posterior beliefs for all meta-training tasks, which eliminates the need to learn a latent encoder (Garnelo et al. 2018). In the M step, we optimize the meta-learning objective w.r.t. the meta-parameters using these samples. Our delicate design of the neural network architecture for meta-training in IPML allows competitive meta-learning performance to be achieved (Sec. 3.2 Our IPML algorithm offers the benefits of being amenable to (a) the characterization of principled distance measure between tasks using maximum mean discrepancy (Gretton et al. 2012 (b) active task selection without needing the assumption of known task contexts in (Kaddour al.
expectation-maxxiiaaatti (EM) algorithm to perform meta- veta-training (Sec. 3.1):
2020). and (c) synthetic task generation by modeling task-dependent input distributions (Sec. 3.3
2 BACKGROUND AND NOTATIONS
For simplicity, the inputs (outputs) for tasks are assumed to belong to the same input (output) space. Consider meta-learning on probabilistic regression tasks fl Each task is generated from a task distribution and associated with dataset yx) where the set X' and the vector yx (Ux))ex denote, respectively, the input vectors and the corresponding noisy outputs
1/x a f(x) + E(x)
(1)
which are outputs of an unknown underlying function corrupted by ii.d. Gaussian noise e(x) (0, o²) with variance o². Let f be distributed by an implicit process (IP), as follows:
Definition (Implicit process for meta-learning). Let the collection of random variables f() denote
an IP parameterized by meeaa-praaetters that is, every finite collection {x(x))xxx has a joint
prior distribution p(fx & (f(x)) implicitly defined by the following generative model:
~ p(z),
f(x) (x(
(2)
for all x E X where z is a latent task vector to be explained below and generator go can be an arbitrary model (e.g., deep neural network) parameterized by meta- reta-parameters O.
Definition I defines valid stochastic processes if is finite dimensional (Ma al. 2019 Though, in reality, a task may follow an unknown distribution, we assume the existence of an unknown function that maps each task to a latent task vector z satisfying the desired known distribution p(z). like in (Kaddour et al. 20200 Using piyx fxx = Nfxx, I ①I and the IP prior belief p(fx) from Def. we can derive the marginal likelihood p(yx) by marginalizing out fx-
Remark 1. Two sources of uncertainty exist in p(yx): Aleatoric uncertainty in p(yx (fx) reflects the noise (i.e., modeled in ( inherent the dataset, while epistemic uncertainty in the IP prior belief p(fx+ reflects the model uncertainty arising from the latent task prior belief p(z) in 2
Let the sets T and T. denote the meta-training and meta-testing tasks, respectively. Following the convention in (Finn et al. 2018) Gordon et al., 2019 Ravi & Beatson 2018: Yoon et al., 2018). for each meta-training task t T, we consider support-query (or train-test) split of its dataset yx,) into the support set (or training dataset) (2/2 (x;))x;;) and query set (or test/evaluation
dataset) (A''s xx) where X, = X2 U X2" and X n X; = 0. Specifically, for a N-way K-shot classification problem, the support set has K examples per class and N classes in total.
Meta-leaming can be defined as an optimization problem (Finn et al., 2017 2018) and its goal is to learn meta- peeta-prrmmters 0 that maximize the following objective defined over all meta-training tasks:
log [ p(y x; (xx;)) p( (fxg (xx)) df
Teeea log II p(yxx x (x;) =
ST
IET
Task adaptation pfEx xx;)) is performed via IP inference after observing the support set:
p(fxg = p( [xx (z) p(z)yx;) dz
(4)
We defer the discussion of meta-lcarning on probabilistic classification tasks using the robust-max likeli- hood [Hemáánee--Loball et al, 2011] to Appendix A.
"z(z) is often assumed to be simple distribution like multivariate Gaussian \/ I) Garnelo et al. 2018
Our work here considers point estimate of meta-parameters 0 instead of Bayesian treatment of 0 Finn
2018 Yoon al. 2018 This allows interpret the epistemic uncertainty in p(fx) via p(z) directly.
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The objective Imeta 5 is the "test" likelihood the query set, which reflects the idea of "learning to learn" " by assessing the effectiveness of "learning on the support set" through the query set. An alternative interpretation views p(Ex; yx²)) as "informative prior" after observing the support set. The objective Fmeu (3) is also known as the Bayesian held-out likelihood (Gordon et al.] 2019). In meta-testing task, adaptation is also performed via IP inference after observing its support set and evaluated on its query set. Similar to GP or any stochastic process, the input vectors of the dataset are assumed be known/fixed beforehand. We will relax this assumption by allowing them to be unknown when our IPML algorithm is exploited for synthetic task generation (Sec. 3.3
3 IMPLICIT PROCESS-BASEE META- (IPML)
3.1 EXPECTATION MAXIMIZATION (EM) ALGORITHM FOR IPML
From Def. if generator ge P can be an arbitrary model (e.g., deep neural network), then pfxx kyx; and p(xx²) cannot be eval- uated in closed form and have to be approximated by samples. Inspired by the Monte Carlo EM algorithm (Wei & Tanner, 1990) which utilizes posterior samples to obtain a maximum likelihood estimate of some hyperparameters,
Recall that task adaptation requires evaluating ((xx:1xx7)
propose EM algorithm for IPML: The E step the stochastic gradient Hamiltonian Monte Carlo (SGHMC) sampling method to draw samples from pfxx Xx; while the M step maximizes the meta-learning objective Jeeta w.r.t. meta-
parameters 0:
2 no uncertainty exists in pfxx (z) in 44. So, p(xxç xyx; can be evaluated using the same generator go belief p(z|yx; x; ), as follows:
Expectation (E) step. Note that since Exy = (go (x, z)) x
2 and the latent task posterior
Remark 2. Drawing samples from pffxg [xx; ) is thus equivalent to first drawing samples of z from p(z]yx;) and then passing them as inputs to generator ge to obtain samples of fx². Hence, given task t, adaptation pfxx; xx²) essentially reduces to a task identification problem by performing IP inference to obtain the latent task posterior belief p(zlyx;). ). This is direct consequence of epistemic uncertainty arising from p(z|yx;) and p(z) (Remark
In general, p(z(yx;) also cannot be evaluated in closed form. Instead of using variational infer- ence (VI) and approximating p(z|yx;) ) with potentially restrictive variational distribution Gar-
helo et al. 2018; Kaddour et al. 2020 Ma et al. 2019). we draw samples from p(z|yx; x; us- ing SGHMC (Chen et al.) 2014 SGHMC introduces an auxiliary random vector r and sam-
ples from joint distribution p(zz z, xx following the Hamiltonian dynamics (Brooks et al.
2011 Neal 1993): p(z, exp(-U 0.5r M MM"rr) where the negative log-probability
U(z) log 1ogp(zly,:;) resembles the potential energy and r resembles the momentum. SGHMC
updates and follows:
Az= = 'M -r=')v - (00, - B))
where a, C, M. and B are the step size, friction term, mass matrix, and Fisher informa- Vg log p(zlyx;) = V log yx;, vx flog [logglyy plyx; x [xx;- (x, x))xEx;; + log p(z)] can be evaluated tractably.
tion matrix, respectively Note that Vz(((z)
Maximization (M) step. We optimize Jmeta 1 w.r.t. 0 using samples of z. The original objective Imeta = ΣiET (op(( [p(Yx; x fxx age x, z)) xxx; )I) is not amenable to stochastic opti-
mization with data minibatches, which is usually not an issue in few-shot learning setting, When huge number of data points and samples of z are considered, we can resort to optimizing the lower
bound Is meta of Jmeta by applying the Jensen's inequality:
Imeta Js-meta EIST Ep(t.gloxg [logp(y>; [xx)) = EIET Eptelye; (lg bo [xx²)]
The sampler hyperparameters o, C, M, and B are set according to the auto-tuning method of Springenherg 2016) which has been verified to work well in our experiments; more details are given in Appendix A.2.1
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Xt
x
observed variable
10, z]
Φ
unobserved variable
Xt
x
z +
w
known constant
go
0a
hup
concatenation
Xt
Xt
z
Z
pointwise product
(a)
(b)
(c)
(d)
Figure 1: (a) Graphical model corresponding to IPML. (b) DNN implementation of generator go where 0 & (0a, 06) and 0₂ can be convolutions to obtain high-level representations of the input vector, while 0g is the last DNN layer's parameters which are masked by z during the forward passes. (c) Graphical model corresponding to input generation by X-Net. (d) CVAE implementation of X-Net (i.e., decoder neural network with parameters 6).
3.2 ARCHITECTURE DESIGN FOR
Our generator ge is implemented using deep neural network (DNN) parameterized by meta- parameters 0. Under this setup, we have empirically observed that the design of the coupling of z with the DNN go -x) )) is crucial to achieving competitive performance of our IPML algorithm. A naive design by concatenating z with x (or higher-level abstractions of x) as contextual input during forward passes has not worked well as the resulting gradients w.r.t z may not have provided enough guidance for SGHMC to learn sufficiently useful representation of z in meta-training,
To this end, inspired by the attention mechanism (Vaswani et al. 2017) and drepout method (Sn- vastava et al. 2014), introduce design of the coupling by applying layer's 's parameters: The last DNN layer's 's parameters are first masked by z (i.e., point-wise prod- be distinguished by different
mask the last DNN
uet with z), as illustrated in Figs. a and Ip. Different tasks masks, hence resembling different attentions on the last DNN layer's connections during forward propagation. We adopt soft masks (i.e., continuous values) instead of hard masks (i.e., either 0 or 1). Such a design of the coupling is empirically demonstrated to be effective in our experiments (Appendix A.4.3).
3.3 ARCHITECTURE DESIGN FOR SYNTHETIC TASK GENERATION
Recall the assumption of known/fixed input vectors in X₂ in the last paragraph of Sec. 2 which we will have relax here. Synthetic task generation can be performed by the following procedure if x is ausk-independent (e.g., p(x, z) = p(x)p(z)): After meta meeta-triiinng is completed (Sec. - draw a sample of latent task vector noisy outputs yx, = (x, z) + ((x)) xEx, to obtain the dataset (A,, yx.) ) for task
p(z), draw samples of x ~ p(x) to form X,, and then generate
When x is task-dependen (c.g., for image classifications of different objects, p(x, z) * p(x)p(z)), not modeling p(x|z) limits the ability generate t-dependent Xt. To resolve this, our IPML al- gorithm includes an X- X-generative network (X-Net): - ha(z, w) that learns to generate an input vector given samples of the latent task vector z and random vector w p(w) = N/ I) where w models the diversity of the input distribution given a fixed task represented by the sample of z. There are several options to implement X-Net: Note that during the training of X-Net. both X and the samples of z ~ p(zlyx;) xę) ) for all meta-training task E T are available. So, generative models such as the conditional variational autoencoder (CVAE) (Shnn et al., 2015) conditional generative adversarial networks (Mirza & Osindero 2014) are suitable for X-Net as they can utilize the contextual information. Our work here uses (the decoder of) CVAE implement X-Net. Figs. D and D illustrate such a design. We have empirically observed that a simple concatenation with Z suffices here as our delicate architecture design for meta-training (Sec. 3.2) can yield a useful representation of z for training X-Net well. Further details and method to ensure balanced data generation are given Appendix A.5 The training objective for synthetic task generation is the
"The latent task prior belief p(z) is thus assumed to be a multivariate Gaussian \/1, II. "This assumption reasonable for meta-training since only p(yx) (and p(x)) needs be modeled.
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empirical lower bound (Sohn et al. 2015) of VI on p(w]x,z):
Jx EieT (x-p(zixx))) X ExEx, (w)x))) (og Poog w)] - Pral/o 2))p(()))
where @ and 1 are, respectively, the parameters of X-Net (decoder neural network) and the encoder neural network, and DRi. denotes the KL distance. In the training of X-Net, we sample per update. We also sample one w per update to train with reparameterization tricks. Algorithms describe meta-training (with training of X-Net) and synthetic task generation, respectively.
||
and
Algorithm 1: IPML: Meta- Trainin
Algorithm 2: Synthetic Task Generation
while not converged do Sample task from T step: Sample (Z1,..... ......... Zn) with SGHMC
Sample pp(z)
Initialize synthetie task and X; = 0
for i I...., final size of Re do
M step: Sample z from {Z],.... ?
Sample w ~ (0.I)
+ 0 0+ V meeta Update X-Net with z and X2 Φ 3 o + V.Jx, 3 + S VeTx
Compute x ha(z, w)
Compute y/x = go
(x(x, 2) + E(x)
(x,) t (Ki U {x), (x(U(x))
return 8, o. vb
return (K,, yx,) for task
4 EXPERIMENTS AND DISCUSSION
Benchmark datasets: sinusoid regression and fhw-shot image classification. We first empirically evaluate the performance of our IPML algorithm against that of several Bayesian meta-learning baselines like the neural process (NP) (Grmelo et al.] 2018). Bayesian model-agnostic meta- learning (BMAML) (Yoon et al. 2018). PLATIPUS (Finn et al., 2018), and amorlized Bayesian meta-learning (ABML) (Ravi & Beatson 2018) on benchmark meta-learning datasets. For few- shot image classification, we also empirically compare IPML with a strong baseline: prototypical network (PN) (Snell et al. 2017). We run experiments on three datasets: sinusoid, Omniglot (Lake al. 20 and mini-ImageNet (Ravi Larochelle, 2017). Sinusoid is regression task of sine waves with uniformly sampled amplitude in [0.1, 5.0, phase in [0, πT, and input in [-5, 5]. The generator of IPML and the baseline regressors are neural networks with hidden layers of size 40 with ReLU nonlinearities. The Omniglot dataset consists of 20 instances of 1623 characters from 50 different alphabets. The mini- ImiilemgeeNet dataset involves 64 training classes. 12 validation classes, and 24 test classes. For Omniglot and mini- ImageNet, our implementation and baselines all use the same data pre-processing, same train-test split, and same data augmentation as that in (Finn et al., 2017]. The generator of IPML and the baseline classifiers are convolutional neural networks with modules of 3 x 3 convolutions and 64 filters, followed by batch normalization, ReLU nonlinearities, and strided convolutions (Omniglot) or experimental settings can be found in Appendix A.2.2
max-pooling (mini-lmageNet). More details of the
For sinusoid regression (Table D. IPML outperforms MAML and BMAML by fair margin. For Omniglot (Table 2 IPML is competitive with MAML and PN. For mini- Mami--mmgeNett (Table B IPML outperforms MAML and all tested Bayesian meta-learning algorithms while being competitive with PN. PN achieves a higher classification accuracy for s-ohot 20-way Omniglot and 5-shot way mini-ImageNee because PN utilizes more information from the extra classes during training (Snell al., 2017 Specifically, though meta-testing involves N-way classification for all tested algo- rithms, the training of PN requires more than N classes, that is, 60-way classification which is also the setting adopted in (Snell et al. 2017). As result, since PN utilizes more information from the extra classes during training. it is reasonable to expect that PN achieves a higher classification accuracy at times. Overall, IPML. is effective for benchmark datasets.
For both sinusoid regression (Table I and Omniglot (Table NP performs unsatisfactorily as compared IPML, likely because (a) it performs amortized variational inference of z through heavily parameterized encoder which may introduce optimization difficulties and overfitting during
meta-training, and (b) the encoder of NP takes in the simple concatenation of (x, 3x) and thus does not explicitly capture the x + 2/ relationship in the support set
10
9 "Some of the results are taken from Finn et al. 2018 Nguyen al. 2020
Youn et al. 2018 The 5-shot
5-way results for PLATIPUS and ABML are missing hecause there are no publcly available implementations.
An ablation study of the limitations of NP can be found in Appendix A.8
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Table 1: Mean square error (MSE) on few-shot
Table 2: Few-shot classification accuracy (%) on
sinusoid regression.
held-out Omniglot characters.
Sinusoid 5-shot Sinusoid 10-shot
Omniglot 1-shot 5-way
Omniglot 1-shot 20-way
NP MAML
0.460
0.264
0.712
0.287
NP
95.9
55.3
BMAML
0.409
0,200
MAML
98.7
92.5
(Our(
0.373
0.123
PN
98.8
96.0
(PuL(Ours)
98.8
94.0
Table 3: Few-shot classification accuracy (%) on mini-Imagenet test set.
IPML vct icaming
PHL
MAMIL
PM, + active learring PPM
mini- Mini--ImaeeNtt I-shot 5-way
mini-ImageNet
5-shot 5-way
MAML
48.6
65.9
1030
1500
2000
2000
4000
6000
8000
10000
PN
49.4
68.2
sampled (e)
sampled [b]
PLATIPUS
50.1
BMAML
49.1
64.2
Results of active task selection on (a) 5-shot sinusoid and (b) 1-shot 5-way mini-ImageNNt.
Figure 2:
ABML
45.0
IPML(Ours)
50.5
67.6
Active task selection. We can evaluate the efféctiveness of the uncertainty measure arising from la- tent task posterior belief p(z) yx²)) ) by performing active task selection. Unlike previous works Yoon et al.) 2018: Finn al., (0188) that can only perform active learning by querying data points, IPML can perform active learning by querying tasks and does not need the assumption of known task contexts in (Kaddour et al. 2020). In every iteration, set of tasks are proposed with only the sup- port set (At Xxp)) given; in image classification, it is usually one-shot. IPML will select among them the task with the maximum variance in p(z (xp) (with samples from the E step/SGHMC): arg max, Var(z) yx;), ). and request for its query set to perform meta-training. This corresponds to a variance-based active task selection criterion. We test on both sinusoid regression and mini- ImageNet classification. Fig. 2 shows that the performance of IPML with active task selection improves that of both MAML IPML without active task selection, that is, it reaches given MSE/accuracy with less training tasks. This shows that the uncertainty measure arising from p(zlyx;) can be exploited benefit meta-training.
Measuring distance between tasks using latent task representation. A most interesting question yet to be answered the following: Does IPML learn a useful latent task representation? IPML learns to model the task through z. If IPML learns the correct representation, then it can reflect patterns of task distribution in the latent space. While solid criterion for assessing the correctness of learned latent task representation is hard to define, we can resort to an oracle (e.g.. human expert with prior knowledge in designing the tasks). Our visualization of the latent task representation and quantitative evaluation of distance measure between tasks using maximum mean discrepancy (MMD) (Gretton et al. 2012) provide ways to assess the correctness of the learned task representa- tion. We denote the set of samples from p(z xx) ) as Z, The MMD between tasks tı and to can be calculated using
MMD/H, f1, t2] é supxcM 1 x(z) 1z, 1 I x<(z)
where is unit ball in the reproducing kernel Hilbert space with a radial basis function kernel.
We conduct experiments with the following 5-way 1-shot settings. Setting A: For subsampled Omniglot, we applied one rotation out of 4 possibilities (0. म, 37/22 uniformly across all the input images for each sampled task [ Setting B: For subsampled mini-ImageNNt, random artistic filter (normal, brighten, or darken) is applied for each sampled task. Setting C: For subsampled Net,-,,maeNNet random artistic filter (3 different types of hue) is applied for each sampled task. ImageNe,,,,,, a random zooming (no zooming, zooming 3 times, or
Setting D: For subsampled zooming 10 times) is applied for each sampled task. Setting E: On subsampled ming-ImageNet, a random artistic filter (normal, low contrast, or high contrast) is applied for each sampled task. Setting A has 4 types of tasks while settings B to E result in types of tasks.
In the previous experiment, the Omniglot dataset is augmented with rotations, but is random across the classes in single task.

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nommal brightnees .0.5
huel (red] (green)
normal zoom-in
normal contrast-1/10
brightness +0.5
hue3 (blue]
contrast-100
(A)
(0)
(C)
(0)
Figure 3: Visualization of latent task embeddings from settings A to
Table 4: Values of MMD metric between types tasks for
Table Results of meta-testing for training with real and generated
Omniglot (setting A). Larger value means larger dissimilarity,
tasks.
Rotations
0
T
7/2
3T /
0
1.166
0.594
1.134
Train real
Accuracy (%)
T n/2
1.166
0,913
0.596
73.83
0.594
0.913
0
0.917
generated
78.33
3T/ /
1.134
0.596
0,917
0
real generated
88.16
For each setting mentioned above, we first train our models in IPML to converge, and then sample tasks from their latent task posterior beliefs (i.e., one sample of per task). Finally, we visual- ize their latent task embeddings in the 2D space using TSNE (van der Maaten & Hinton 2008). Furthermore, for setting A, we evaluate the distance measure between tasks using the well-known MMD metric with radial basis function kernels on the samples. can be observed from Fig. and Table that IPML successfully distinguishes 4 types of rotations for Omniglot. Both Fig. 3 and Table contemporanccusly show that flipping upside down (i.c., either right half of the embedding 0 2 left half of the embedding /2 = 3T/ /2) are reckoned to be closer tasks compared with rotation of π. x/2, thus revealing that our visualization and evaluation of distance measure between tasks in accordance. From Fig. 3 to Fig- ED. IPML successfully distinguishes different types of transformations on the tasks while revealing interesting facts: for example, tasks of high brightness more isolated from that of low or normal brightness. Fig. 3 shows that tasks of low contrast are more distinct from that of normal or high contrast. The values of MMD metric for settings B E and more details of the experiments are provided in Appendix A.6 On the overall, both the visualization and evaluation of distance measure between tasks reveal that IPML successfull learns useful latent task representations and even provides interesting insights.
Synthetic task generation for Omniglot. We assess the usefulness of latent task representation z by performing synthetic task generation. The training tasks we consider are three types sub- sampled binary classifications: classification of characters A vs. B, B vs. C, and C vs. A, as in Fig. 4p. During meta-learning, we train a X-Net concurrently learn to generate task-related input The CVAE implementation of X-Net contains decoder neural network with 3
images (Sec. 3.3
2.0
Task type
Cp
1.5
1.0
2
0.5
Task type
V
9
-1.0
V
V
2
-1.5
type
-2.0
2
(a)
-2
-1
(b)
0
-2
(c)
Figure 4: (a) TSNE visualization of (samples of) 3 types of binary classification tasks; images of black/white hackground are black/white samples (1/x 1/9x = 0). (b) Visualization of latent em- bedding of real tasks in (normalized) space [-2, _ (c) Sampled generated task data by walking through the (normalized) z space 22, note that the inversion of color only for visualization to distinguish black and white samples. In training, NO images inverted.

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Table 6: Averaged meta-testing
Persccal running and Coemwirs.
performance on 10 meta-testing categories.
Dornedss and Callyssss
Coticar
Costume and accessotics
Accuracy (%)
F1
Network oquipeent
IPML Multi-task 84,1
84,3
70.5
Waterss and glazees
60.5
In-game curmecces
Garing
Gaming tarra
Table 7: Averaged meta-testing performance on 5 desired cate- gories (IDs 19, 21, 23, 36, 44).
Figure 5: (a) TSNE visualization of latent task embedding of 10 meta-testing categories and (b) their analysis (see main text). Legend shows IDs and names of categories.
Accuracy (%)
F1
Settngg A Setting B
87.4
75.8
86.4 74.4
hidden layers of size [128, 128, 256) and ReLU nonlinearity, and symmetric design of the encoder. After meta-training is completed, we continue to train the X-Net to converge. In this experiment, the dimension of z set as which further allows walking through such latent space/embedding to visualize how the generated tasks map to their latent representations. Fig. P shows the latent embedding of real tasks. Fig. 4 shows the sampled synthetic tasks by walking through the latent space. It can be observed that X-Net successfully captures the task-dependent input distributions and can generate high-quality data of task type 1, 2, and 3 when sampled from their corresponding latent clusters (see samples of task type 1, 2, and 3 in the colored bounding boxes in Fig. Fb).
We further evaluate the quality of generated tasks by training on it. We hold out half of the images for each character during meta-training to construct the meta-testing tasks. The results are presented in Table 5 When training on both real and generated tasks, we first train on the generated tasks to converge and then train on the real tasks for another 30 iterations. It can be observed that compared to only using real tasks, higher accuracy is achieved with training merely using generated tasks. When training on both real and generated tasks, huge boost in accuracy is observed. We conjecture that due to their diversity, generated tasks (i.e., sometimes containing more ambiguous tasks) alleviate overfitting and provide promising direction meta-task augmentation.
Real-world risk detection. We perform experiments on a real-world risk detection dataset provided by an anonymous e-commerce company. The task classify whether an item in the online shop has risks (e.g., fraud, pornography, contraband). Such risks appear in different forms and in different categories (of items). It is hard to detect risks in different categories by training models separately for each category because some categories have only very limited amounts of black samples (i.e., < 50). The similarities of the detected risks in different categories, if discovered, can help improve the performance. Meta-learning is thus suitable algorithm for its ability to perform (a) detection of risks across different categories of items and (b) adaptation to new categories. The input x of the dataset is the text (title and descriptions) embedding obtained from self-supervised learning. while its label is a binary variable indicating whether it contains risks (i.c., //x = for black samples and Vx = 0 for white samples). The data separated by categories of items to yield 47 categories in total. Initially, we hold out 10 categories for meta-testing 12 while the rest are used for meta tratt-traiiing..
Table shows results comparing the performance of IPML a multi-task learning baseline It can be observed that IPMI. outperforms multi-task learning, which indicates its stronger ability to generalize to unseen categories. Fig. 5 visualizes the latent task embedding of the 10 meta-testing categories for analysis. IPML learns useful latent task representations: For example, from Fig. Dh. gaming-related categories with IDs 46 and 47 are mapped closely in the latent task space/embedding.
The individual meta-testing performance on the 10 meta-testing categories, which are given in Ap- pendix A.3 can be further examined: For the five categories with IDs 19, 21, 23, 36, and 44 covered by the shaded light green zone in Fig. 55. IPML outperforms multi-task learning by a large margin. They are mapped the center of the latent task space (Fig, 5b)). which may imply that IPML's adap- tations to them can largely build on previous experiences of the meta-training categories and IPML's exploitation of such similarities allows their performance to improve over multi-task learning. For
Their category names and IDs are given in Fig.
13 When testing an unseen category, multi-task learning performs adaptation by randomly initializing its untied parameters for retraining on the few-shot support data.
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Under review as a conference paper at ICLR 2021
the three categories with IDs 3. 9, and 39 covered by the shaded light orange zone, IPML does not have a performance advantage over multi-task learning. For the two categories with IDs 46 and 47 covered by the shaded light pink zone, both IPML and multi-task learning perform unsatisfactorily. As a matter of fact, for IPMI, the categories with unsatisfactory performance (i.e., either covered by the shaded light orange or pink zone) are all mapped to be some distance away from the center, which indicates that they are likely considered by IPML as "outsier"dissimile tasks.
We further compare meta-learning on (A) the same setting as before by holding out the 10 meta- testing categories vs. (B) training on all categories in setting A as well as the dissimilar ones with IDs 3, 9, 39, 46, and 47. Table shows results on the desired categories with IDs 19, 21, 23, 36, and 44. It can be observed that when a meta-learning model is trained to perform well (during meta- testing) on the desired categories/tasks, training alongside with dissimilar ones can compromise its performance. More details of the experimental settings and data preparation, experimental results, and analysis are given in Appendix A.3 We have also empirically compared the time efficiency of IPML against that of several meta-learning baselines and reported the results in Appendix A.7
5 RELATED WORK
A number of meta-leariing algorithms (Finn et al. have proposed a Bayesian extension of the MAML framework (Finn et al. 2017
2018 Ravi & Beatson
2018
Yoon et al. 2018
Their difference with IPML is that they model the uncertainty in the predictions with a set of particles (Yoon et al.)
2018 or a variational distribution (Finn et al. 2018; Ravi & Beatson 2018 which does not allow latent task modeling. The work of Rusu et al. (2019) introduces a generative model that decodes latent vectors into the meraa-parameters but does not scale well in the dimension of meta-parameters.. In comparison, IPML explicitly represents each task as a latent continuous vector and models its probabilistic belief and is hence scalable in the dimension of meta-parameters. Moreover, MAML- based algorithms usually require evaluating computationally-inee intennivv second-order derivatives of the meta- pattaaatrrettrrss during meta-training because they approximate the Bayesian inference through an inner loop of gradient descent. Although this issue can be addressed by methods such as first- order approximations (e.g.. first-order MAML (Finn et al. 2017), Reptile (Nichol et al. 2018)) or implicit MAML (Rajeswaran et al., 2019) using implicit gradient, these works are not Bayesian. In contrast, our IPML algorithm naturally utilizes Bayes' rule to perform sampling during Bayesian inference and does not need second-order derivatives.
The work of Kaddour et al. (220)) uses latent information to perform active task selection, but assumes known -esc-desr (lask context) which is usually unknown. The work Garneo al. (2018) introduces the first use of stochastic processes (i.c., neural processes) in meta-learning and learns heavily parameterized encoder to encode dataset into its latent representation, which might introduce optimization difficulties and overfitting and can only output Gaussian posterior beliefs. In comparison, our IPML algorithm the first to consider SGHMC task adaptation/inferrnce of meta-learning, which can capture a non-Gaussian posterior belief to achieve a better performance (Appendix A.4 Our IPML algorithm is also the first to explicitly model task-dependen input distributions, which is lacking in the literature. Such modeling enables synthetic task generation of complex image classification tasks for the first time.
CONCLUSION
This paper describes novel IPML algorithm that, in contrast to existing works, explicitly repre- sents each task as a continuous latent vector and models its probabilistic belief within the highly expressive framework. Unlike existing works, IPML. offers the benefits of being amenable to (a) the characterization of a principled distance measure between tasks using MMD. (b) active task selection without needing the assumption of known task contexts in (Kaddour et al. 2020), and (c) synthetic task generation of complicated image classifications via modeling of dependenndd input distributions using our X-Net. Empirical evaluation on benchmark datasets shows that IPML out- performs existing Bayesian meta-learnig algorithms. We have also empirically demonstrated on an anonymous e-commerce company's 's real-world dataset that IPML outperforms the multi-task learn- ing baseline and identifies Jdisiier"dissiiilar tasks which degrade meta-testing performance.

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Under review as a conference paper at ICLR 2021
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Reviews:
[{"review_id": "soePrsd2he", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes Implicit Process Meta-Learning (IPML) where each task is represented as a continuous latent vector $\\mathbf{z}$, and corresponding data points are described as function values evaluated at an implicit process conditioned on the task latent vector $\\mathbf{z}$. To conduct the intractable inference, a stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm is employed. A VAE-like network called X-Net is trained simultaneously to generate synthetic tasks from the task latent vectors. The experimental results demonstrate that the proposed algorithm shows decent performances on few-shot classification tasks, and the task latent vectors indeed represent a meaningful space of the tasks on which measuring distances between tasks and detecting outlier tasks are possible.\n\nOverall I find this paper is clearly written. Various experiments are conducted to demonstrate the benefits of the proposed model. Personally, I like the implicit process framework, and I guess this is a nice application of the idea to meta-learning. \n\nSince the performance of the proposed algorithm is not overwhelming, I presume the main advantage of the proposed algorithm is its ability to model a proper latent space of tasks.  My first concern is that whether this is a property only for the proposed IPML. For instance, consider a prototypical net. Although not being as principled as IPML, a prototypical net can also represent a task via prototypical vectors and consider them as a latent representation of the task (e.g., just take the average of all prototype vectors to get a single representative of a task).  Is this space completely meaningless so measuring the distance metric doesn't reflect the semantic relationship between tasks at all?\n\nAlso, as far as I understood, the ability to construct a meaningful latent space of tasks is not solely from the implicit process itself but aided by the additional X-Net learning a generative model over the data points in the tasks. What if we conduct measuring the MMD between tasks for the model without X-Net (not between the generated tasks and an existing task, but between existing tasks semantically apart)? Moreover, how the prototypical-like model with a generative model attached compare to the proposed one on synthetic task generation? The reason why I'm concerned about this is training a generative model and generating an image from it is usually quite expensive. For instance, one can easily imagine that training X-Net for miniImageNet or tiredImageNet and generating images from it might be effective.\n\nSome minor questions\n- Figure 3 shows the shift in the latent space when tasks data are manipulated. But is it really demonstrate the semantic difference between tasks? I guess a representation from an ordinary classifier might also show this because the input \"images\" were shifted using non-trivial operations. I guess this point may be made clear by having a look at the representations of semantically different tasks (e.g., classification of different classes) without being modified at the image level.\n- For a binary classification task (Figure 4), when generating a task from a task vector $\\mathbf{z}$,  as far as I understood, each data $\\mathbf{x}$ is generated conditioned on $\\mathbf{z}$ and then corresponding label $y$ is generated. Are the generated data is well balanced between positive and negative classes?\n- Have you considered using SGLD instead of SGHMC? If so, did you observe any empirical degrade in the performance?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "slW-8C_haIa", "reviewer": "AnonReviewer2", "summary": "", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes an efficient meta-learning approach using implicit processes. Specifically, authors represent each task as a continuous latent vector and use expectation-maximization algorithm to perform meta-learning. The E step performs task adaption using stochastic gradient Hamiltonian Monte Carlo sampling method, while the M step optimizes meta-learning objective using these samples. Their framework can measure principled distance between tasks by maximum mean discrepancy (MMD) and generate synthetic tasks by task-dependent distribution. Finally, the authors validate their proposed framework on several benchmark datasets and real-world datasets. The novelty and originality of this paper is good by proposing new ideas and methods. In addition, the paper is well-organized and clearly written. We can quickly get to know what problem they are trying to solve, how they solve and what their results are.\n\nPros: \n1.Representing each task as a continuous latent vector is very suitable when ones need to measure the task similarity, synthesize new tasks or actively select a task without the assumption of known task contexts.\n2.The expectation-maximization algorithm based on the stochastic gradient Hamiltonian Monte Carlo sampling method can mitigate the enormous computation overhead by using first-order approximation without second-order derivative.\n3.Authors demonstrate the effectiveness of this IPML on benchmark datasets and real-world tasks.\n\nCons:\n1.The results of IPML in Table 2 do not outperform the baseline methods. I think authors could provide more analysis or reasons on results.\n2.Authors should add more ablation study and analysis on why the proposed method would work. For example, how much is the contribution of representing each task as a continuous latent vector? How much is the contribution of the expectation-maximization algorithm?\n3.Apart from the performance in the experiment, authors should also mention the efficiency of the IPML compared with other baseline methods. For example, the run time of the algorithm on computational expensive tasks.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "cq9gdX2FSne", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "=== Summary ===\n\nThe paper proposes a meta-learning method based on implicit process (IP) framework in which each task is represented by a latent vector. The IP setup for meta-learning seems identical to that of Neural processes [1]. In that, the key challenge for adaptation to a task based on a context/support set $(X_c, Y_c)$ is inferring the distribution $p(z|X_c, Y_c)$ over latent vectors. While [1] use amortized (variational) inference with a variational Gaussian distribution to approximate the true $p(z|X_c, Y_c)$, the paper proposes to use stochastic gradient Hamiltonian Monte Carlo (SG-HMC) for sampling latent state vectors from $p(z|X_c, Y_c)$.\n\nIn the experiment section, the paper compares the proposed method (IMPL) against various previous methods, mostly gradient-based meta-learners, on common benchmarks, demonstrating competitive results. In addition, the paper provides a detailed inspection of the latent space and argues that IMPL could be used for active task selection and synthetic task generation.\n\n=== Main argument ===\n\nThe IP meta-learning setup of the paper is well-known in the meta-learning community (e.g. [1]) and thus not a contribution. Unlike to previous work (e.g. [1]) which uses the IP framework for meta-learning, the paper proposes to use SG-HMC instead of amortized (variational) inference for approximating p(z|X_c, Y_c), resulting in a new, sound meta-learning algorithm. This can be considered the methodological main contribution of the paper.\n\nThe experiment section features standard, well-accepted meta-learning benchmarks and demonstrates that IMPL is not only theoretically sound but also performs well in practice. The analysis and visualization of the learned latent task representation is interesting and provides nice qualitative evidence that proximity of task representation vectors in the latent space reflects the human-perceived task similarity to a fair degree. \n\nUnfortunately, the paper’s experiment section does not address some of the core questions: How does IMPL compare to Neural Processes (NPs)? What are the pros and cons of replacing the amortized (variational) inference with SG-HMC? E.g. How much performance increase does the arguably much more complex and computationally demanding approximate inference method buy us? Since, the key contribution of the paper is replacing the approximate inference method of NPs with SG-HMC, an empirical in-detail comparison with NPs would strengthen the paper a lot!\n\nIn the related work section, the paper argues that a limitation of NPs is the Gaussian variational distribution in the latent space. However, since the latent space is meta-learned and thus quite arbitrary - this is not a very convincing argument. Empirical evidence for the hypothesized shortcomings of NPs would be much better.\n\nFurthermore, it remains unclear why/how active task selection could be relevant in any real-world setting. Motivating the setup with a real-world use case would strengthen this aspect of the paper. The real-world risk detection experimental results in the appendix are nice and interesting. Putting more of the results in the main part (there is still some space) and briefly discussing them could give a nice argument for the benefits of the latent task representation inherent in IMPL - i.e. it provides a straightforward way for identifying tasks that are very dissimilar from the rest - excluding such outlier tasks may increase performance.\n\n=== Overall assessment === \n\nIn the current form, I see the paper slightly above the acceptance threshold. The proposed meta-learning algorithm based on IPs is sound and the extensive experiments demonstrate that it works well in practice. However, the only innovation of IMPL over Neural Processes (NPs) is replacing variational inference by SG-HMC. Thus, the algorithmic contribution is limited. What the advantage of IMPL over NPs is, has not been addressed sufficiently. If a proper experimental comparison to NPs is added to the paper, I am happy to increase my score.\n\n[1] Garnelo, Marta, et al. \"Conditional neural processes.\" arXiv preprint arXiv:1807.01613 (2018).\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper sits at the borderline: the reviewers agree it is a well-written and interesting paper, but have concerns about efficiency as well as a comparison with the neural process (the authors did include a revision with this comparison, though the numbers they report are worse than in the original neural processes paper on the same experiment). Ultimately, this paper probably requires another round of reviews before it is ready for publication.
Decision: Reject