# secure_outofdistribution_task_generalization_with_energybased_models__c06dd309.pdf

Secure Out-of-Distribution Task Generalization with Energy-Based Models

Shengzhuang Chen1 Long-Kai Huang2 Jonathan Richard Schwarz3

Yilun Du4 Ying Wei1,5

1City University of Hong Kong 2Tencent AI Lab 3University College London 4Massachusetts Institute of Technology 5Nanyang Technological University szchen9-c@my.cityu.edu.hk {hlongkai, schwarzjn}@gmail.com yilundu@mit.edu ying.wei@ntu.edu.sg

The success of meta-learning on out-of-distribution (OOD) tasks in the wild has proved to be hit-and-miss. To safeguard the generalization capability of the metalearned prior knowledge to OOD tasks, in particularly safety-critical applications, necessitates detection of an OOD task followed by adaptation of the task towards the prior. Nonetheless, the reliability of estimated uncertainty on OOD tasks by existing Bayesian meta-learning methods is restricted by incomplete coverage of the feature distribution shift and insufficient expressiveness of the meta-learned prior. Besides, they struggle to adapt an OOD task, running parallel to the line of cross-domain task adaptation solutions which are vulnerable to overfitting. To this end, we build a single coherent framework that supports both detection and adaptation of OOD tasks, while remaining compatible with off-the-shelf meta-learning backbones. The proposed Energy-Based Meta-Learning (EBML) framework learns to characterize any arbitrary meta-training task distribution with the composition of two expressive neural-network-based energy functions. We deploy the sum of the two energy functions, being proportional to the joint distribution of a task, as a reliable score for detecting OOD tasks; during meta-testing, we adapt the OOD task to in-distribution tasks by energy minimization. Experiments on four regression and classification datasets demonstrate the effectiveness of our proposal.

1 Introduction

Meta-learning [48, 6] that builds general-purpose learners with limited data has been under constant investigation, recently demonstrating its potential to even advance few-shot learning of large language models [36, 44]. Analogous to the notorious domain shift [23] that degrades the performance of deep learning, meta-testing tasks that are out of the distribution of meta-training tasks (a.k.a. out-of-distribution (OOD) tasks) put the meta-learned prior knowledge at high risk of losing effectiveness [46]. In real-world applications, though, out-of-distribution tasks are highly prevalent, e.g., bin picking for a robot that has never been meta-trained on environments involving bins [55], MRI-based pancreas segmentation given a host of meta-training tasks with pathology images [35], and etc. Thus, it is imperative to secure the generalization ability of the meta-learned prior (i.e., meta-generalization) to OOD tasks, especially in safety-critical applications such as medical image analysis.

The first step to securing meta-generalization to a task is to develop awareness of whether the task is OOD or not, i.e., OOD task detection. Existing solutions in literature have pursued a variety of Bayesian meta-learning methods [7, 54, 41, 10, 43] that balance between flexibility and tractability of

Correspondence to Ying Wei

37th Conference on Neural Information Processing Systems (Neur IPS 2023).

7.5 5.0 2.5 0.0 2.5 5.0

Ground-truth

7.5 5.0 2.5 0.0 2.5 5.0

EBML-CNPs p(x,y| i)

7.5 5.0 2.5 0.0 2.5 5.0

CNPs p(y|x, i)

Likelihood ID OOD Boundary

(a) Incomplete OOD Coverage. EBML-CNPs (middle, ours) successfully identifies OOD input with low likelihood, whereas CNPs (right) trained by p(y|x, ϕi) makes overconfidently wrong prediction.

Learned Prior ID Task Region Learned Prior Prior Prediction Mean ID Task Region ID Task Mean

(b) Limited Expressiveness. The meta-training task distribution learned by EBML-CNPs (middle, ours) outperforms F-PACOH-GP [43] whose prior distribution is built on GP.

Figure 1: Comparison of EBML and Bayesian meta-learning baselines for OOD detection.

solving the hierarchical probabilistic model p(Yi|Xi) =

p(Yi|Xi, ϕi)p(ϕi|θ)p(θ) dϕidθ, where Ti = {Xi, Yi} represents the i-th task. θ and ϕi denote parameters of the meta-model and taskspecific model, respectively. Unfortunately, these methods present some limitations in their practical usage. (1) Incomplete OOD coverage: given that the Bayesian uncertainty is trained via maximizing the posterior p(Yi|Xi) above, it is not necessarily high when encountering an OOD task that shares the predictive function p(Yi|Xi) with some meta-training tasks but differs substantially in feature distributions p(Xi). We verify this in Figure 1a and Appendix D. (2) Limited expressiveness: for tractability purpose, the meta-learned prior p(ϕi|θ) predicates on simple known distributions, e.g., Maximum A Posterior (MAP) estimation [6, 53] and Gaussian [41, 7, 43], which may struggle to align with the complex probabilistic structure of the meta-training task distribution (see Figure 1b). This misalignment inevitably leads to unreliable estimation of OOD tasks.

Upon detection of an OOD task, secondly, adaptation of the meta-learned prior promotes its generalization to this OOD task. We dub this strategy during meta-testing as OOD task adaptation, which is closely related to cross-domain meta-learning [4, 25, 34, 49]. The core philosophy behind cross-domain meta-learning is the introduction of task-specific parameters which are inferred via either gradient descent [28, 29] or feed-forward amortized encoder [42, 8] on the support set of each OOD task. Learning task-specific parameters, however, is prone to overfitting given the usually very limited size of a support set (e.g., 5 examples only in 5-way 1-shot classification).

The limitations are further complicated by the detachment of the existing solution to OOD task detection from that to OOD task adaptation. An explicit prior model is absent in existing Bayesian meta-learning methods for OOD task detection, so that adapting the prior during meta-testing to accommodate an OOD task is ambitious to achieve. On the other hand, cross-domain meta-learning approaches by design do not offer uncertainty estimation, thereby being a risky OOD task detector. Pursuing a coherent framework that supports both detection and adaptation of OOD tasks remains an open question, which motivates our proposal of a novel probabilistic meta-learning framework.

By virtue of the flexibility and expressiveness of energy-based models [24] in modelling complex data distributions, we propose the Energy-Based Meta-Learning (EBML) framework that overcomes the above-mentioned limitations. Specifically, we derive an energy-based model to explicitly model any meta-training task distribution, resulting in the composition of an explicit prior energy function and a complexity energy function. The sum of the two energy functions, trained directly to meet the joint distribution p(Xi, Yi) and parameterized with neural networks, has completeness and expressiveness advantages that give it an edge in detection of OOD tasks. During meta-testing, we iteratively update the parameter for a task that has been identified as OOD by gradient descent of energy minimization, which eventually adapts the prior towards in-distribution tasks and maximally leverages the meta-learned prior for alleviating overfitting.

The key contributions of this research are outlined below. (1) Coherence and generality: we provide a coherent probabilistic model that allows both detection and adaptation of OOD tasks. Also, EBML is agnostic to meta-learning backbones, being general to secure meta-generalization for arbitrary off-theshelf meta-learning approaches against OOD tasks. (2) Practical efficacy: we conduct our experiments on three regression and one classification datasets, on which EBML outperforms SOTA Bayesian meta-learning methods for OOD task detection with an improvement of up to 7% on AUROC and cross-domain meta-learning approaches for OOD task adaption with up to 1.5% improvement.

2 Related Work

Bayesian Meta-learning There has been a line of literature on Bayesian meta-learning algorithms with predictive uncertainty estimation for safeguarding safety-critical and few-shot applications. Grant

et al. [11] first recast gradient-based meta-learning as a tractable hierarchical Bayesian inference problem. Much of the subsequent research attempts to solve the problem with various approximations. Assuming a sufficient number of meta-training tasks, almost all works use a point estimate for the initialization [41, 19, 10]. However, estimates of exceptions including [54] rely on SVGD [32] for inference and require significant computation for an ensemble of task-specific weights. Several studies that estimate the uncertainty in task-specific parameters after inner-loop adaptation have explored MAP estimates [e.g. 47], sampling from a neural network [10, 53, 42], and variational inference [41, 7, 43]. The uncertainties considered in these methods are often modelled using isotropic Gaussians which suffer from limited expressiveness.

Meta-learning towards OOD Generalization Recent cross-domain meta-learning methods [e.g. 25, 4, 49, 34] deal with a distribution shift between meta-training and meta-testing tasks, by typically parameterizing deep networks with a large set of task-agnostic and a small set of task-specific weights that encode shared representations and task-specific representations for the training domains, respectively. The works of [42, 1, 34] augment a shared pre-trained backbone with task-specific Fi LM [40] layers whose parameters are estimated through an encoder network conditioned on the task s support set. TSA [28] and URL [29] propose to attach task-specific adaptors in matrix form to the pre-trained backbone at test time, inferring their parameters by gradient descent on the support set for each task from scratch. On the other hand, SUR [4] and URT [31] pre-train multiple backbones each for an ID training domain, and meta-learn an attention mechanism to selectively combine the pre-trained representations into task-specific ones for ID and OOD classification. While these methods generally have improved performance in the OOD domains of tasks, they nevertheless are not designed with any explicit mechanism for detecting OOD tasks, i.e., lacking OOD awareness.

EBMs for OOD Detection Recently, there has been increasing interest in leveraging EBMs for detecting testing samples that are OOD w.r.t. the training data distribution. Liu et al. [33] directly use the energy score for OOD input detection, while Grathwohl et al. in JEM [12] use gradient norm of the energy function as an alternative OOD score; both yield more superior OOD detection performance than traditional density-based detection methods. There are also a number of works that investigate the OOD detection capability of hybird and latent variable EBMs [38, 14, 13], and more advanced training techniques for improving the density modelling hence OOD detection performance of EBMs [5, 2, 57, 3]. While all aforementioned works focus on the standard supervised and unsupervised learning scenarios, Willette et al. in [52] study OOD detection in meta-learning. However, their work differs from EBML in that (a) EBML aims to detect a meta-testing task that is OOD of the meta-training tasks whereas [52] focuses on detecting a query sample that is OOD of the support samples in a meta-testing task, and (b) EBML explicitly meta-learns the distribution of meta-training tasks via the two proposed EBMs and develops the Energy Sum to flag those highenergy tasks as OOD tasks; while [52] resorts to post-hoc OOD detection via energy scaling (akin to temperature scaling in softmax output) without learning any EBM. Moreover, we offer EBML as a generic and flexible probabilistic meta-learning framework that supports both detection and adaptation of OOD tasks.

3 Preliminaries: Energy-based Models

An energy-based model (EBM) [24] expresses a probability density p(x) for x RD as

pθ(x) = exp( Eθ(x))

where Eθ(x) is the energy function parametrized by θ that maps each point x in the input space to a scalar value known as the energy. Z(θ) = R

x exp( Eθ(x))dx is the partition function that is a constant w.r.t. the variable x. Training pθ(x) to fit some data distribution p D(x) requires maximizing the log-likelihood L(θ) = Ex p D(x)[log pθ(x)] w.r.t. θ. Though an intractable integral in Zθ is involved in this objective, it is not a concern when computing the gradient [3, 12] θL = Ex pθ[ θEθ(x )] Ex p D[ θEθ(x)]. (2) Intuitively, Eqn. (2) encourages Eθ to assign low energy to the samples from the real data distribution p D while assigning high energy to those from the model distribution pθ. Computing Eqn. (2), thus, requires drawing samples from pθ, which is challenging. Recent approaches [12, 3] on training EBMs resort to stochastic gradient Langevin dynamics (SGLD) [51] which generates samples following

x0 p0(x), xk+1 = xk η2

xk + ηzk. (3)

The K-step sampling starts from an (typically uniform) initial distribution p0(x). zk N(0, I) RD is a perturbation, and η R+ controls the step size and noise magnitude. Denote the distribution qθ by Eqn. (3), which signifies x =x K qθ. When η 0 and K , then qθ pθ under some regularity conditions [51]. Consequently, the gradient of Eqn. (2) is approximated in practice [3, 12] by

θL = Ex stop_grad(qθ)[ θEθ(x )] Ex p D[ θEθ(x)], (4)

where the gradient does not back-propagate into SGLD sampling.

4 Energy-Based Meta-learning

For clarity, we use the notation PID to denote the unknown meta-training ID task distribution where the i-th training task is T i. We let Xi, Yi to denote sets of samples {xij, yij} in T i, and T s i, T q i to denote support and query sets, respectively. The size of T i, T s i, T q i is denoted by Ni, N s i , N q i , respectively. The subscript i denotes the task index, and j denotes the sample index.

4.1 Energy-based Modelling of Task Distribution

As illustrated in Introduction, existing probabilistic meta-learning methods maximizing the predictive likelihood p(Y|X) suffer from incomplete OOD coverage. To this end, we model the metatraining task distribution by (1) formulating the joint distribution p(Xi, Yi) of each task T i and (2) maximizing the log-likelihood of all meta-training tasks. Concretely, by Kolmogorov s extension and de Finneti s theorems [22], we have the expected log-likelihood of the meta-training tasks as EPID[log p(T i)] = EPID[log p(Xi, Yi)] = EPID[log R

ϕi QNi j=1 p(xij, yij|ϕi)p(ϕi)dϕi]. Each p(T i) is written in a factorized form over Ni conditional independent distributions with ϕi being the task-specific latent variable. Due to the intractable integral over ϕi in high dimension, we resort to amortized inference [8, 41] and learn with a lower-bound instead. This gives the ELBO

EPID[log p(T i)] E Eϕi qψ(ϕi| T s i ) h log

j=1 p(xij, yij|ϕi) i KL qψ(ϕi| T s i)||p(ϕi) . (5)

Following the conventional wisdom [41, 28, 6], qψ is conditioned on the support set only during metatraining to align the inference procedure, i.e., ϕi qψ(ϕi| T s i), for meta-training and meta-testing. It remains now to parameterize the three distributions in Eqn. (5) including (a) the task-specific data distribution p(xij, yij|ϕi), (b) the prior latent distribution p(ϕi), and (c) the posterior latent distribution qψ(ϕi| T s i). Prior works parameterize these distributions in simple known forms, e.g., Gaussians [41, 7, 43] or MAP estimation [6, 53], which may be insufficient to match the complex probabilistic structure of the meta-training task distribution. To increase the expressiveness, we turn to EBMs for parameterizing the two distributions of p(xij, yij|ϕi) and p(ϕi). For one reason, EBMs are known to be sufficiently flexible and expressive for characterizing complex arbitrary density functions [3] not limiting to only uni-modal distributions like isotropic Gaussians and MAP estimation; for another, the energy function of an EBM is directly proportional to the negative log-likelihood, paving the way for OOD detection in Section 4.2.

(a) Task-specific data EBM We model p(xij, yij|ϕi) by an energy function parameterized with ω,

p(xij, yij|ϕi) = pω(xij, yij|ϕi) = exp( Eω(xij, yij, ϕi))

Z(ω, ϕi) , (6)

where Eω denotes the task-specific data energy function conditioned on the latent ϕi, and Z(ω, ϕi) is the corresponding partition function. Note that the parameter ω of this EBM is shared by all tasks.

(b) Latent prior EBM Inspired by [39], we model the prior latent distribution p(ϕi) as an unconditional EBM parameterized by λ; training such a EBM offers expressiveness benefits over a fixed non-informative prior distribution, e.g., isotropic Gaussian distribution. Specifically,

p(ϕi) = pλ(ϕi) = exp( Eλ(ϕi))

Z(λ) , i. (7)

(c) Latent posterior As many meta-learning algorithms have already carefully designated the posterior latent distribution qψ(ϕi| T s i), we simply follow the same implementation of qψ in the chosen base meta-learning algorithm, e.g., MAP estimation in [8, 42, 1, 53]. This design favorably empowers EBML to be a generic and flexible framework compatible with off-the-shelf meta-learning algorithms.

Grounded on the above parameterization, we are now ready to derive our EBML meta-training objective as below by plugging the two EBMs defined in Eqn. (6) and Eqn. (7) into Eqn. (5). The derivation shares the spirit with Eqn. (4), and more details can be found in Appendix A.1.

arg max ω,ψ,λ

Eϕi qψ(ϕi|T s i )[

j=1 Eω(xij, yij, ϕi) + Epω(x ,y |ϕi)[Eω(x ij, y ij, ϕi)]]

Eqψ(ϕi|T s i )[Eλ(ϕi)] + Epλ(ϕ i)[Eλ(ϕ i)] + H(qψ(ϕi| T s i))

Solving the above meta-training objective involves sampling of x , y from pω and ϕ i from pλ, in order to compute the expectations Epω(x ,y |ϕi) and Epλ(ϕ i) as Monte-Carlo averages. We follow the similar SGLD sampling procedure in Eqn. (3). Besides, since the majority of state-of-the-art meta-learning algorithms [8, 42, 1, 53] adopt the MAP estimation of the latent posterior qψ which is deterministic, the last entropy term of H essentially becomes zero and the expectations in the first and second terms are trivial to solve. For this reason, we focus on base meta-learning algorithms with MAP approximation in the following sections, which not only simplifies computation but also maintains the state-of-the-art performance. We left a discussion on EBML with distributional qψ in Appendix C.3. The complete pseudo codes for meta-training of EBML are available in Appendix E.

4.2 EBML for OOD Detection

0.0 0.2 0.4 0.6 0.8 1.0 Data Variance

Mean Data Energy

0.8 1.0 1.2 Euclidean Dist.

Prior Energy

Figure 2: The roles of Eω and Eλ in Energy Sum in detecting OOD tasks. Each dot denotes a task. Left: We perturb each support sample of a task T i by ηij Ni(0, σi) where we sample σi from [0, 1] uniformly. The y-axis shows the average energy Exs ij,ys ij T s i ,ηij Ni[Eω(xs i, ys i , ϕi)] and the xaxis plots the variance σ2 i . Right: We first compute the mean of the overall ID task latent prior as ϕID = Eϕi p ID[ϕi]. The y-axis shows the energy Eλ(ϕID + ηi) where ηi N(0, 1) for the i-th task and the x-axis plots the Euclidean distance of the perturbed latent from ϕID.

Detecting an OOD task w.r.t. the meta-training distribution constitutes an essential first step to guard successful meta-generalization. A straightforward solution is density-based OOD detection, for which the OOD score of a task following the Bayesian principle boils down to its log-likelihood log p(Xs i, Ys i )=log Eϕi pλ(ϕi)[pω(Xs i, Ys i |ϕi)]. Despite the meta-learned latent prior EBM pλ(ϕi) that is readily available, estimating this loglikelihood still presents daunting challenges. First, when the latent prior is expressed in the form of a distribution over model parameters in very high dimension, MCMC sampling from pλ(ϕi) is almost computationally infeasible. Second, especially when the latent prior exhibits multi-modality, drawing a considerable number of samples to achieve a low-variance MC estimation of the integral is prohibitively costly.

On this account, we define the OOD score of a task to be faithful to our proposed ELBO approximation of its log-likelihood in Eqn. (5), which gives

Eqψ(ϕi| T s i ) N s i X

j Eω(xs ij, ys ij, ϕi) + Eλ(ϕi) . (9)

We dub this OOD score tailored to EBML Energy Sum, whose full derivation is deferred to Appendix A.2. This energy sum enjoys not only the theoretical advantage, i.e., being provably proportional to the negative log-likelihood of a task, but also simple computation benefits. During meta-testing, evaluating the score of Eqn. (9) for each task requires only a single forward pass of the support set samples through the two energy functions.

More remarkably, the energy sum is intuitively appealing in the sense that it characterizes (1) how far a task is from the overall ID meta-training task distribution via the latent prior energy score Eλ and (2) how difficult it is to predict the observed support set conditioned on ϕi via the task-specific data energy score Eω. First, the terms in the last line of Eqn. (8) for learning the latent prior EBM altogether correspond to maximizing the likelihood ET i p(T ) Eqψ(ϕi|T s i )[log pλ(ϕi)], which enforces the latent prior energy score Eλ to capture the overall ID meta-training distribution. As illustrated in

Figure 2 (right), the further away a task is from the overall ID meta-training distribution measured in Euclidean distance, the larger the energy score Eλ is as expected. Second, conditioned on even the ID latent prior ϕi, those tasks with support samples as scattered as possible are especially difficult to predict. These tasks are considered to be OOD, as evidenced in higher values of Eω in Figure 2 (left).

4.3 EBML for OOD Generalization

The Energy Sum proposed in Section 4.2 develops OOD awareness of a meta-testing task, based on which we differentiate our meta-testing procedures for effective meta-generalization.

Meta-testing for ID tasks Given the support set T s of a meta-testing task, prediction of the label for its query xq j amounts to maximizing our approximated log-likelihood (see Eqn. (5)) of the task, i.e.,

yq j = arg min y Eϕ qψ(ϕ| T s) Eω(xq j, y, ϕ) + Eλ(ϕ) . (10)

Provided that the task has already been identified within the ID region, the second energy Eλ(ϕ) is negligibly small. Consequently, we reduce the above optimization problem to consider only the first term Eω(xq j, y, ϕ), and solve it via gradient descent. We provide the pseudo codes in Appendix E.

Meta-testing for OOD tasks For an OOD task, its meta-learned prior ϕ qψ(ϕ| T s) is located out of the ID meta-training task distribution and likely loses its effectiveness. We seek a solution that adapts this inadequate meta-learned prior back to the ID region, so as to make the most of the ID latent priors with guaranteed meta-generalization. This shares the idea with classifier editing in [45], where the editing parameters are trained to map an OOD image to an ID one for improving generalization. Therefore, we introduce task-specific parameters ζ which are optimized via the following,

arg min ζ Eϕ qψ ζ(ϕ| T s) N s X

j=1 Eω(xs j, ys j, ϕ) + max(Eλ(ϕ) m, 0) , (11)

where m is a hyper-parameter. We find that setting m as the empirical average of the latent prior energy over all ID training tasks works well in practice, i.e., m = Ep ID[Eϕi qψ(ϕi| T s i )[Eλ(ϕi)]].

1 0 1 2 3 4 5 6 0

Prior Energy -0.8 0.2

0 2 4 6 8 0.50

OOD Adaptation Step

Adaptation ID Task Region Prototypes Euclidean Dist.

Query Accuracy

Figure 3: Illustration of the OOD task adaptation process on OOD domains of the metadataset [49] where each dot in (a) represents an OOD task in latent space ϕ. Minimizing Eqn. (11) leads to (a) the latent ϕ of the OOD task moving to the ID region (contour plot), (b) the Euclidean distance between class prototypes enlarging, and consequently (c), the classification accuracy on query samples increasing.

As a result of optimizing the second term in Eqn. (11), the task-specific parameters ζ enable qψ ζ(ϕ| T s) to accommodate for OOD tasks by mapping the meta-learned prior back to ID metatraining tasks; while optimizing the first term preserves the data-level predictive ability of the model. We highlight that the task energy minimization approximates the minimization of a KL divergence between the task-specific posterior and the meta-learned prior, thereby inducing a metaregularization effect during adaptation. See Appendix A.3 for details. Eventually, we use the adapted task-specific parameters for final prediction on query samples as in Eqn. (10). Pseudo code for the EBML adaptation and inference algorithms described above can be found in Appendix E.

In Figure 3, we visualize the adaptation process when optimizing Eqn. (11) for OOD few-shot classification tasks in Meta-dataset [49]. As the prior energy of these OOD tasks decreases, their ϕi gradually shift towards to the ID region as desired. Within this region, minimizing the first term in Eqn. (11) continuously improves generalization. In contrast, given only a few support samples, existing SOTA methods that solely rely on feed-forward inference [1] and gradient-based optimization [28] for OOD task adaptation without a prior are both prone to overfitting. We provide more empirical evidence on this in Appendix C. On the other hand, meta-learning a BNN, which imposes a prior distribution on the parameter space during adaptation may be computationally cumbersome and often lead to sub-optimal performance in comparison to their non-Bayesian counterparts.

5 Experiments

In the experiments, we test EBML on both few-shot regression and image classification tasks in search for answers to the following key questions: RQ1: Whether the improved expressiveness of EBML over traditional Bayesian meta-learning methods can lead to a more accurate model of the meta-training ID task distribution, hence a more reliable OOD task detector. RQ2: Whether Energy Sum can be an effective score for detection of OOD meta-testing tasks. RQ3: Whether EBML instantiated with SOTA algorithms can exploit the meta-learned EBM prior in OOD task adaptation to achieve better prediction performance on OOD tasks.

5.1 Implementation Details

We now discuss two instantiations of the EBML framework with SOTA meta-learning algorithms for regression and classification. We illustrate our approach in Figure 4 below and defer a more detailed description for our models to Appendix B.

Figure 4: Overview of the EBML framework. The task latent variable ϕi is inferred from the support set T s i following the implementation of the base algorithm. The data and task energy scores are evaluated by the data and prior EBMs Eω1 and Eλ, respectively; while the query labels are predicted by the classifier pω2 of the base algorithm.

Regression. Take CNPs [8] as an example base model. CNPs implements qψ(ϕi| T s i) as a neural network encoder that outputs a function embedding in finite vector form, i.e., ϕi RD, from a given support set, T s i. That said, we let the prior EBM to model the empirical distribution over such finite-dimension function embedding, i.e., Eλ(ϕi) : RD R.

Classification Many cross-domain few-shot classification algorithms [28, 42, 1] rely on a metricbased classifier for prediction, which assigns query sample to the class with nearest prototype to the query representation based on some distance measure. In these cases, it is natural to specify the task-specific latent ϕi as the set of class prototypes in each ID training task. Since ϕi is a set of variables, we build the prior EBM model as a permutation-invariant neural network function. Suitable choices include Deep Sets [56] and set transformer [26].

To align with the state-of-the-art prediction performance, we follow the practice in [50, 37] to train another decoder ω2 with the loss function (e.g., cross entropy) in the base meta-learning model, which serves as a surrogate for Eω(xq j, y, ϕ) in Eqn. (10) and Eqn. (11). We use this decoder for prediction.

Baseline Models For regression, we compare against: 1) MAML [6] which is a deterministic metalearning method, and 2) Bayesian meta-learning methods that use Gaussians for prediction or prior, including ABML [41], Meta Fun [53], CNPs [8] and F-PACOH-GP [43]. For classification, we consider Simple-CNAPs [1] and TSA [28], which respectively resort to amortized variational inference and gradient-based optimization for estimating the task-specific parameters from the support set. Both are SOTA cross-domain few-shot classification approach on the Meta-dataset [49] benchmark. For more experimental details, hyper-parameter configurations, and additional experimetal results, please refer to Appendix B and C.

5.2 Datasets and Evaluation Metrics

Sinusoids Few-shot Regression We consider 1D sinusoids regression tasks in the form y(x) = Asin(B(x+C)). For ID meta-training, we consider frequency B = 1, while sample amplitude A and phase C uniformly from a set of equally-spaced points {1, 1.1, 1.2, ..., 4} and {0, 0.1, 0.2, ..., 0.5π}, respectively. Each training task consists of 2 to 5 support and 10 query points with x uniformly sampled from X [ 5.0, 5.0]. During testing, we evaluate the models on 500 ID and OOD tasks each with 512 equal-distant query points in X. For ID testing, we expand the range of the tasks by uniformly sampling A [1, 4] and C [0, 0.5π]. For OOD tasks, we randomly change either the phase distribution to C [0.6π, 0.75π], amplitude to A [0.1, 0.8] [4.2, 5.0] or frequency to B [1.1, 1.25]. Details for the multi-sinusoids regression experiment can be found in C.1. We use MSE and negative log-likelihood on query samples to evaluate the regression performance.

Drug Activity Prediction Few-shot Regression In each task, we aim to predict the drug-target binding affinity of query molecular compounds given 10 to 50 labelled examples from the same domain defined by molecular size. We use the lbap-general-ic50-size ID/OOD task split in the Drug OOD [21] benchmark, which divides the molecules into 222/145/23 domains by molecular size for ID Train / ID Test / OOD Test, respectively. The regression performance is evaluated by the square of Pearson coefficient (R2) between predictions and the ground-truth values. We report the mean and median R2 on 500 tasks sampled from ID and OOD testing domains.

Meta-dataset [49] 5-way 1-shot Classification This experiment considers image classification problems on Meta-dataset [49]. Each task contains up to 10 query images per class from the same domain. Following the current state-of-the-art practice [28, 1], we use Aircraft, dtd, cub, vgg-flower, fungi, quickdraw and omniglot as the ID datasets for meta-training and meta-testing, while traffic, mscoco, cifar10, cifar100 and mnist are treated as OOD datasets for meta-testing only.

OOD Task Detection Evaluation We compare the OOD task detection performance of Energy Sum against several model-agnostic OOD detection baselines. Concretely, for classification, we compare against max-softmax score [16], ODIN [30], MAH [27], and max-logits score [15]; for regression, we consider Averaged Bayesian prediction uncertainty in standard deviation (Std) on support samples, and Averaged Support samples Negative Log-Likelihood (SNLL) under model s task-specific predictive probability, i.e., Eϕi qψ(ϕi|T s i )[Ej[log pω(ys ij|xs ij, ϕi)]] for baselines and Eϕi qψ(ϕi|T s i )[Ej[Eω(ys ij, xs ij, ϕi)]] for EBML. Following common practice [17, 16], we report AUROC, AUPR and FPR95 for OOD detection performance. Details for these metrics can be found in Appendix B.1.

5.3 OOD Detection Results

Energy sum performs best in OOD task detection. Table 1 and 8. The proposed energy sum further improves our SNLL-only results in all three OOD detection metrics - with 15.2% and 11.8% significant reduction in FPR95, outperforming the best baseline methods by 20.0% and 39.1%, in single and multi-sinusoids situations respectively. In Table 2 for OOD classification task detection, Energy Sum consistently results in superior OOD detection performance, outperforming the best baselines by large margins of 36.84% and 20.19% in FPR95 for Simple-CNAPs and TSA, respectively.

Table 1: OOD task detection performance on single-sine and Drug OOD [21] few-shot regression tasks.

OOD Scores Models Sinusoids Drug OOD

AUROC AUPR FPR95 AUROC AUPR FPR95

ABML [41] 50.14 54.80 97.20 57.82 50.31 74.80 F-PACOH-GP [43] 49.52 51.30 94.20 81.74 71.99 32.00 CNPs [8] 22.72 35.34 99.60 93.56 89.58 13.00 Metafun [53] 76.57 80.33 82.40 85.68 80.55 58.18

ABML [41] 82.48 81.31 61.00 80.99 79.12 47.60 F-PACOH-GP [43] 91.78 93.23 52.40 37.73 45.01 85.21 CNPs [8] 95.63 96.46 34.22 17.25 34.07 91.40 Metafun [53] 96.25 97.11 32.00 83.54 85.54 65.17 EBML-CNPs (Ours) 96.46 97.41 29.40 99.71 99.71 2.20

Energy Sum EBML-CNPs (Ours) 97.74 98.31 14.20 99.79 99.78 1.40

Modelling the joint distribution improves OOD detection under Domain-shift. In Table 1 Drug OOD regression tasks, using either our SNLL or Energy Sum as OOD scores can achieve better detection performance than baselines. In particular, our method outperforms the best OOD detection results obtained using Gaussian SNLL and Std by 43.84% and 11.6% in FPR95, respectively.

Figure 5: Predictive distribution of Middle an data EBM vs Right a Gaussian for an ID task.

Qualitative Illustration. In Figure 5, we visualize the predictive distribution p(yij|xij, ϕi) learned using an EBM decoder and a Gaussian decoder on a sampled ID multi-sinusoids task. The EBM clearly shows two prediction modes at all non-overlapping positions, whereas the Gaussian decoder is unable to model the multi-modality, resulting in a blurry prediction.

Computational Complexity Analysis. We conduct a computational complexity analysis for EBML by comparing its wall-clock training time and convergence to baselines in Figure 6 below. EBMLCNPs eventually achieves better OOD detection performance than baseline CNPs meanwhile matching its regression performance at all training epochs. In Table 15 Appendix C.4, we show EBMLCNPs is computationally cheaper and faster than traditional Bayesian methods, namely, F-PACOHGP [43] which requires matrix inversion for inference with Gaussian processes prior, and ABML [41] which imposes a Gaussian prior over the entire parameter space of the model.

0 1000 2000 3000 4000 seconds

CNPs EBML-CNPs

0 1000 2000 3000 4000 seconds

(1-AUROC)+(1-AUPR)+FPR95

500 1000 1500 2000 2500 3000 epoch

500 1000 1500 2000 2500 3000 epoch

(1-AUROC)+(1-AUPR)+FPR95

Figure 6: Left : Wall-clock convergence in seconds, and Right: performance vs number of training epochs, for EBML-CNPs vs CNPs in single-sinosoid few-shot regression tasks. The plots show the regression (MSE ) and combined OOD tasks detection (1-AUROC)+(1-AUPR)+FPR95 performance on single sine few-shot regression tasks during training. Curves are moving averages with window size 3. EBML-CNPs achieves better final performance than CNPs.

Energy sum achieves better OOD detection results with EMB prior than Gaussian. In Table 3 and 9, we investigate the contribution of the prior EBM in improving the modelling of meta-training task distribution. We train CNPs and ABML using diagonal Gaussian distribution as the prior in ELBO, and compute OOD scores as (a) SNLL, and (b) the sum between SNLL and the NLL of task-specific latent evaluated under the learned Gaussian prior (indicated by +Gauss Prior). The results show that energy sum using an EBM prior outperforms all ablated models. The OOD detection performance of our model benefits from adding the prior EBM energy to the data EBM energy (SNLL), resulting in the most reduction in FPR95 on both single and multi-sinusoids tasks (15.2% and 11.8%, respectively). This suggests the improved expressive of EBM over simple distributions can indeed lead to learning a more accurate model of the meta-training ID task distribution.

Energy sum achieves better OOD detection results when learning the joint distribution In Table 16, we compare EBML-joint, which is exactly our proposed training procedure in the paper,

Table 2: OOD task detection performance on Meta-dataset 5-way 1-shot classification tasks.

OOD Scores Simple-CNAPs [1] TSA [28]

AUROC AUPR FPR95 AUROC AUPR FPR95

max-softmax [16] 85.50 85.54 65.43 89.25 87.14 46.02 max-logits [15] 50.00 70.83 95.00 50.14 44.64 95.28 ODIN [30] 90.49 89.42 43.57 92.02 90.18 37.36 MAH [27] 71.18 69.76 90.52 94.54 93.95 23.83 Domain Classifier 83.10 73.18 53.17 n/a n/a n/a

EBML Energy Sum 97.01 94.92 6.74 99.10 98.48 3.64

Table 3: Ablation study on Energy Sum for OOD detection on single-sinusoids.

Models OOD Scores Sinusoids

AUROC AUPR FPR95

ABML [41] SNLL 82.48 81.31 61.00 +Gauss Prior 86.95 86.64 52.20

CNPs [8] SNLL 94.81 96.34 38.40 +Gauss Prior 94.61 96.10 34.40

EBML-CNPs SNLL 96.46 97.41 29.40 +EBM Prior 97.74 98.31 14.20

Table 4: Few-shot regression performance on single-sinusoids and Drug OOD [21] tasks.

Models Sinusoids Drug OOD

ID MSE ID Mean R2 ID Median R2 OOD Mean R2 OOD Median R2

F-PACOH-GP [43] 0.068 0.016 0.492 0.454 0.055 0.027 Metafun[53] 0.009 0.002 0.537 0.541 0.054 0.027 CNPs [8] 0.009 0.002 0.540 0.549 0.066 0.046 ABML [41] 0.127 0.013 0.452 0.443 0.051 0.029 MAML [6] 0.119 0.013 0.462 0.475 0.055 0.024

EBML-CNPs 0.009 0.002 0.533 0.553 0.071 0.043

and EBML-conditional, which follows the same training with EBML-joint but models p(Y | X) instead of p(X, Y). With all other factors being the same, EBML-joint significantly outperform EBML-conditional in OOD detection on Drug OOD regression tasks with domain shift in X. This supports our motivation for using the joint distribution instead of the conditional distribution for training a potentially better OOD detector. Detail of this ablation study can be found in Appendix D.

5.4 OOD Generalization Results

Table 5: Classification performance on 5-way 1-shot tasks for both ID and OOD domains in Meta-dataset.

Datasets TSA EBML-TSA [28] (Ours)

Omniglot 98.63 0.26 98.67 0.26 Textures 51.93 0.87 52.35 0.88 Aircraft 78.91 0.86 78.47 0.86 Birds 75.02 0.90 75.52 0.90 VGG Flower 80.37 0.80 80.30 0.83 Fungi 70.89 0.93 72.29 0.94 Quickdraw 79.02 0.84 80.27 0.85

MSCOCO 52.28 0.94 53.03 0.97 Traffic Sign 57.40 0.94 58.85 1.01 CIFAR10 49.16 0.82 50.04 0.89 CIFAR100 62.25 1.01 62.77 1.05 MNIST 74.72 0.83 76.08 0.88

Avg ID 76.40 76.84 Avg OOD 59.16 60.15 Avg All 69.22 69.89

EBML achieves SOTA regression performance. In Table 4, for single-sinusoids, EBML is able to match the MSE of the best-performing baseline methods; while on multi-sinusoids in Table 7, EBML obtains the lowest ID NLL, specifically 0.58 lower than the best baseline, thanks to our energy-based decoder which is sufficiently expressive for modelling the multimodality at each input.

Task adaption using Eqn. (11) improves few-shot classification performance. In Table 5, we report the average classification accuracy computed over 600 test tasks per ID and OOD domains. In meta-testing, we obtain classification results for EBML-TSA by running gradient descent on the objective in Eqn. (11) to optimize the task-specific modules in TSA from scratch. With this addition of prior energy in the OOD adaption objective, EBML-TSA further improves TSA results in 5/7 ID domains and all 5 OOD domains. Additional OOD classification results in Table 11 Appendix C further confirm the superiority of our proposed OOD task adaptation strategy in Eqn. (11) over prior baselines.

6 Conclusion and Limitation

This paper proposes a new energy-based meta-learning (EBML) framework for the first time, which directly characterizes any arbitrary meta-training task distribution using two data and prior energy functions. EBML is compatible with many existing SOTA meta-learning algorithms and allows both detection and adaption of OOD tasks. The sum of the two learned energy functions gives an unnormalized probability distribution proportional to the underlying task likelihood, deployable as OOD scores. The experiment results show the superiority of Energy Sum over traditional methods in detecting both OOD regression and classification tasks, and the possibility of achieving improved OOD adaptation performance with EBML through minimizing the task energy. One limitation of EBML is that our current OOD task adaptation strategy does not consider the effect of negative transfer, as some OOD tasks may benefit from adaptating from scratch without ID energy prior regularization. Thus, in future works, we are interested in designing task-specific adaptation strategies for EBML that can selectively adapt OOD tasks for better performance.

[1] Peyman Bateni, Raghav Goyal, Vaden Masrani, Frank Wood, and Leonid Sigal. Improved few-shot visual classification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.

[2] Yilun Du, Shuang Li, Joshua Tenenbaum, and Igor Mordatch. Improved Contrastive Divergence Training of Energy-Based Models. In Proceedings of the 38th International Conference on Machine Learning, pages 2837 2848. PMLR, July 2021. ISSN: 2640-3498.

[3] Yilun Du and Igor Mordatch. Implicit Generation and Modeling with Energy Based Models. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.

[4] Nikita Dvornik, Cordelia Schmid, and Julien Mairal. Selecting relevant features from a multidomain representation for few-shot classification. In European Conference on Computer Vision, 2020.

[5] Sven Elflein, Bertrand Charpentier, Daniel Zügner, and Stephan Günnemann. On Out-ofdistribution Detection with Energy-based Models. page 13.

[6] Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks. In Proceedings of the 34th International Conference on Machine Learning, pages 1126 1135. PMLR, July 2017. ISSN: 2640-3498.

[7] Chelsea Finn, Kelvin Xu, and Sergey Levine. Probabilistic model-agnostic meta-learning. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS 18, page 9537 9548, Red Hook, NY, USA, 2018. Curran Associates Inc.

[8] Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and S. M. Ali Eslami. Conditional neural processes. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 1704 1713. PMLR, 10 15 Jul 2018.

[9] Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J. Rezende, S. M. Ali Eslami, and Yee Whye Teh. Neural Processes. Technical Report ar Xiv:1807.01622, ar Xiv, July 2018. ar Xiv:1807.01622 [cs, stat] type: article.

[10] Jonathan Gordon, John Bronskill, Matthias Bauer, Sebastian Nowozin, and Richard Turner. Meta-learning probabilistic inference for prediction. In International Conference on Learning Representations, 2019.

[11] Erin Grant, Chelsea Finn, Sergey Levine, Trevor Darrell, and Thomas Griffiths. Recasting gradient-based meta-learning as hierarchical bayes. In International Conference on Learning Representations, 2018.

[12] Will Grathwohl, Kuan-Chieh Wang, Joern-Henrik Jacobsen, David Duvenaud, Mohammad Norouzi, and Kevin Swersky. Your classifier is secretly an energy based model and you should treat it like one. In International Conference on Learning Representations, 2020.

[13] Will Sussman Grathwohl, Jacob Jin Kelly, Milad Hashemi, Mohammad Norouzi, Kevin Swersky, and David Duvenaud. No {mcmc} for me: Amortized sampling for fast and stable training of energy-based models. In International Conference on Learning Representations, 2021.

[14] Tian Han, Erik Nijkamp, Linqi Zhou, Bo Pang, Song-Chun Zhu, and Ying Nian Wu. Joint Training of Variational Auto-Encoder and Latent Energy-Based Model. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 7975 7984, Seattle, WA, USA, June 2020. IEEE.

[15] Dan Hendrycks, Steven Basart, Mantas Mazeika, Mohammadreza Mostajabi, Jacob Steinhardt, and Dawn Xiaodong Song. Scaling out-of-distribution detection for real-world settings. In International Conference on Machine Learning, 2022.

[16] Dan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. Proceedings of International Conference on Learning Representations, 2017.

[17] Dan Hendrycks, Mantas Mazeika, and Thomas Dietterich. Deep anomaly detection with outlier exposure. In International Conference on Learning Representations, 2019.

[18] Shion Honda, Shoi Shi, and Hiroki R. Ueda. Smiles transformer: Pre-trained molecular fingerprint for low data drug discovery. 2019.

[19] Ekaterina Iakovleva, Jakob Verbeek, and Karteek Alahari. Meta-learning with shared amortized variational inference. In ICML, pages 4572 4582. PMLR, 2020.

[20] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In Francis Bach and David Blei, editors, Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 448 456, Lille, France, 07 09 Jul 2015. PMLR.

[21] Yuanfeng Ji, Lu Zhang, Jiaxiang Wu, Bingzhe Wu, Long-Kai Huang, Tingyang Xu, Yu Rong, Lanqing Li, Jie Ren, Ding Xue, Houtim Lai, Shaoyong Xu, Jing Feng, Wei Liu, Ping Luo, Shuigeng Zhou, Junzhou Huang, Peilin Zhao, and Yatao Bian. Drug OOD: Out-of-Distribution (OOD) Dataset Curator and Benchmark for AI-aided Drug Discovery A Focus on Affinity Prediction Problems with Noise Annotations. ar Xiv:2201.09637 [cs, q-bio], January 2022. ar Xiv: 2201.09637.

[22] Achim Klenke. Probability theory: a comprehensive course. Springer Science Business Media, 2013.

[23] Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Balsubramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A benchmark of in-the-wild distribution shifts. In International Conference on Machine Learning, pages 5637 5664. PMLR, 2021.

[24] Yann Le Cun, Sumit Chopra, Raia Hadsell, Marc Aurelio Ranzato, and Fu Jie Huang. A Tutorial on Energy-Based Learning. page 59, 2006.

[25] Hae Beom Lee, Hayeon Lee, Donghyun Na, Saehoon Kim, Minseop Park, Eunho Yang, and Sung Ju Hwang. Learning to balance: Bayesian meta-learning for imbalanced and out-ofdistribution tasks. In International Conference on Learning Representations, 2020.

[26] Juho Lee, Yoonho Lee, Jungtaek Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh. Set transformer: A framework for attention-based permutation-invariant neural networks. In Proceedings of the 36th International Conference on Machine Learning, pages 3744 3753, 2019.

[27] Kimin Lee, Kibok Lee, Honglak Lee, and Jinwoo Shin. A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks. In Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.

[28] Wei-Hong Li, Xialei Liu, and Hakan Bilen. Cross-domain few-shot learning with task-specific adapters. 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 7151 7160, 2021.

[29] Wei-Hong Li, Xialei Liu, and Hakan Bilen. Universal representation learning from multiple domains for few-shot classification. 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 9506 9515, 2021.

[30] Shiyu Liang, Yixuan Li, and R. Srikant. Enhancing the reliability of out-of-distribution image detection in neural networks. In International Conference on Learning Representations, 2018.

[31] Lu Liu, William Hamilton, Guodong Long, Jing Jiang, and H. Larochelle. A universal representation transformer layer for few-shot image classification. Ar Xiv, abs/2006.11702, 2020.

[32] Qiang Liu and Dilin Wang. Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm. In Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016.

[33] Weitang Liu, Xiaoyun Wang, John Owens, and Yixuan Li. Energy-based Out-of-distribution Detection. In Advances in Neural Information Processing Systems, volume 33, pages 21464 21475. Curran Associates, Inc., 2020.

[34] Yanbin Liu, Juho Lee, Linchao Zhu, Ling Chen, Humphrey Shi, and Yi Yang. A Multi Mode Modulator for Multi-Domain Few-Shot Classification. In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 8433 8442, Montreal, QC, Canada, October 2021. IEEE.

[35] Shuai Luo, Yujie Li, Pengxiang Gao, Yichuan Wang, and Seiichi Serikawa. Meta-seg: A survey of meta-learning for image segmentation. Pattern Recognition, page 108586, 2022.

[36] Sewon Min, Mike Lewis, Luke Zettlemoyer, and Hannaneh Hajishirzi. Metaicl: Learning to learn in context. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 2791 2809, 2022.

[37] Yifei Ming, Ying Fan, and Yixuan Li. Poem: Out-of-distribution detection with posterior sampling. In International Conference on Machine Learning. PMLR, 2022.

[38] Bo Pang, Tian Han, Erik Nijkamp, Song-Chun Zhu, and Ying Nian Wu. Learning Latent Space Energy-Based Prior Model. In Advances in Neural Information Processing Systems, volume 33, pages 21994 22008. Curran Associates, Inc., 2020.

[39] Bo Pang, Tian Han, Erik Nijkamp, Song-Chun Zhu, and Ying Nian Wu. Learning Latent Space Energy-Based Prior Model. In Advances in Neural Information Processing Systems, volume 33, pages 21994 22008. Curran Associates, Inc., 2020.

[40] Ethan Perez, Florian Strub, Harm de Vries, Vincent Dumoulin, and Aaron C. Courville. Film: Visual reasoning with a general conditioning layer. In AAAI, 2018.

[41] Sachin Ravi and Alex Beatson. Amortized bayesian meta-learning. In International Conference on Learning Representations, 2019.

[42] James Requeima, Jonathan Gordon, John Bronskill, Sebastian Nowozin, and Richard E Turner. Fast and flexible multi-task classification using conditional neural adaptive processes. In H. Wallach, H. Larochelle, A. Beygelzimer, F. dÁlché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 7957 7968. Curran Associates, Inc., 2019.

[43] Jonas Rothfuss, Dominique Heyn, Jinfan Chen, and Andreas Krause. Meta-learning reliable priors in the function space. In Advances in Neural Information Processing Systems, volume 34, 2021.

[44] Victor Sanh, Albert Webson, Colin Raffel, Stephen Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Arun Raja, Manan Dey, et al. Multitask prompted training enables zero-shot task generalization. In International Conference on Learning Representations, 2022.

[45] Shibani Santurkar, Dimitris Tsipras, Mahalaxmi Elango, David Bau, Antonio Torralba, and Aleksander Madry. Editing a classifier by rewriting its prediction rules. Advances in Neural Information Processing Systems, 34:23359 23373, 2021.

[46] Amrith Setlur, Oscar Li, and Virginia Smith. Two sides of meta-learning evaluation: In vs. out of distribution. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021.

[47] Zhuo Sun, Jijie Wu, Xiaoxu Li, Wenming Yang, and Jing-Hao Xue. Amortized bayesian prototype meta-learning: A new probabilistic meta-learning approach to few-shot image classification. In International Conference on Artificial Intelligence and Statistics, pages 1414 1422. PMLR, 2021.

[48] Sebastian Thrun and Lorien Pratt. Learning to learn. Springer Science & Business Media, 2012.

[49] Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Utku Evci, Kelvin Xu, Ross Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, and Hugo Larochelle. Meta-dataset: A dataset of datasets for learning to learn from few examples. In International Conference on Learning Representations, 2020.

[50] Haotao Wang, Aston Zhang, Yi Zhu, Shuai Zheng, Mu Li, Alex J Smola, and Zhangyang Wang. Partial and asymmetric contrastive learning for out-of-distribution detection in long-tailed recognition. In International Conference on Machine Learning, pages 23446 23458, 2022.

[51] Max Welling and Yee Whye Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th International Conference on International Conference on Machine Learning, ICML 11, page 681 688, Madison, WI, USA, 2011. Omnipress.

[52] Jeffrey Ryan Willette, Hae Beom Lee, Juho Lee, and Sung Ju Hwang. Meta learning low rank covariance factors for energy based deterministic uncertainty. In International Conference on Learning Representations, 2022.

[53] Jin Xu, Jean-Francois Ton, Hyunjik Kim, Adam R Kosiorek, and Yee Whye Teh. Metafun: Meta-learning with iterative functional updates. In International Conference on Machine Learning (ICML), 2020.

[54] Jaesik Yoon, Taesup Kim, Ousmane Dia, Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian Model-Agnostic Meta-Learning. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.

[55] Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Co RL, pages 1094 1100, 2020.

[56] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep Sets. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.

[57] Yang Zhao, Jianwen Xie, and Ping Li. Learning Energy-Based Generative Models via Coarseto-Fine Expanding and Sampling. February 2022.