SlowGuess/NeurIPS_2023_OCR huggingface dataset:
From Figure 3, we notice that BASS can train a meta-model that achieves good generalization performance at the early stage of meta-training. One application of this property is using BASS to assist meta-learning models under the Ensemble Inference settings, where separate models are combined to enhance the generalization ability. One renowned ensemble approach is the model-parameter ensemble [41, 33]. With a collection of  $N_E$  individual models  $\{\mathcal{F}(\cdot ;\Theta_i)\}_{i\in [N_E]}$  of the same architecture, the ensemble model will be  $\mathcal{F}_E(\cdot ;\Theta_E)$ , and its parameters  $\Theta_{E} = \frac{1}{N_{E}}\sum_{i\in [N_{E}]}\Theta_{i}$  are the averaged parameters across individual models. Then, the ensemble model  $\mathcal{F}_E(\cdot ;\Theta_E)$  will be applied as the inference model for downstream problems. Here, one natural way of obtaining the individual models  $\mathcal{F}(\cdot ;\Theta_i), i\in [N_E]$  is deeming the models from different training iterations as the individual models for ensemble [12, 16]. Here, we conduct experiments using the ensemble techniques with individual models  $\mathcal{F}(\cdot ;\Theta_i)$  trained by baselines and BASS. We choose the top  $N_{E} = 10$  models with the smallest validation loss across different meta-training iterations as the individual models  $\{\mathcal{F}(\cdot ;\Theta_i)\}_{i\in [N_E]}$  for ensemble. The results are shown in Table 3. We label the ensemble version of BASS as "BASS-E", and the non-ensemble version as "BASS-S".
Table 3: Ensemble case study [dataset (shot); results ± standard deviation].
Algo. \ Data	M-ImageNet (1)	M-ImageNet (5)	CIFAR (1)	CIFAR (5)
Uniform	0.231±0.014	0.313±0.027	0.270±0.014	0.534±0.012
SPL	0.218±0.006	0.298±0.004	0.219±0.005	0.363±0.038
FOCAL	0.204±0.005	0.347±0.030	0.235±0.015	0.499±0.003
DAML	0.222±0.011	0.326±0.031	0.261±0.008	0.432±0.019
GCP	0.226±0.006	0.297±0.011	0.268±0.019	0.512±0.015
PAML	0.213±0.024	0.232±0.009	0.223±0.009	0.336±0.029
ATS	0.202±0.002	0.334±0.052	0.313±0.081	0.517±0.017
BASS-S	0.198±0.004	0.351±0.012	0.272±0.025	0.553±0.008
BASS-E	0.242±0.004	0.366±0.003	0.327±0.010	0.551±0.004
Compared with the non-ensemble settings (Table 1), we see that the BASS-aided ensemble model can generally perform better. In particular, the ensemble model can improve the meta-model inference performance in significantly difficult cases, such as the Mini-ImageNet under the 1-shot setting (Subsec. 6.1). As a result, BASS can help generate high-quality ensemble model
with the meta-models trained in different iterations. While the ensemble inference technique can also benefit the other baselines, BASS still maintains decent performances.
In this paper, we formulate the task scheduling problem in meta-learning under the Contextual Bandits settings, and propose a novel bandit-based task scheduling framework named BASS. It directly optimizes the task sampling strategy based on the status of the meta-model rather than applying fixed task scheduling protocols. Instead of greedily making decisions, BASS can help deal with the insufficient knowledge problem at the early stage of meta-training, as well as plan for the future meta-training iterations with the adaptive exploration strategy. We include both theoretical analyses and a comprehensive set of experiments to demonstrate the effectiveness of our proposed framework as well as its key properties.
This work is supported by National Science Foundation under Award No. IIS-1947203, IIS-2117902, IIS-2137468, IIS-2002540, and Agriculture and Food Research Initiative (AFRI) grant no. 2020-67021-32799/project accession no.1024178 from the USDA National Institute of Food and Agriculture. The views and conclusions are those of the authors and should not be interpreted as representing the official policies of the funding agencies or the government.
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A Appendix: Experiments (Continue)
A.1 Further Details for the Experiment Settings
For the data partitioning, we have the Mini-ImageNet and CIFAR-100 data sets divided into the partitions  $64:16:20$ , which correspond to the training set, validation set and the testing set respectively. Each class is corresponding to a task. Then, for the Drug data set, we partition the tasks into  $4100:76:100$  representing the training set, validation set and the testing set.
For our BASS, we apply two 2-layer FC networks for  $f_{1}(\cdot;\pmb{\theta}_{1})$ ,  $f_{2}(\cdot;\pmb{\theta}_{2})$  respectively, and set network width  $m = 200$ . For deriving approximated arm rewards, we let  $|\Omega_k^{\mathrm{valid}}| = 5$ . Recall that we apply the approximation approach mentioned in Remark 3 to reduce the space complexity and time complexity in practice for the experiments. Here, we tune the pooling step such that the inputs of  $f_{1}(\cdot;\pmb{\theta}_{1})$ ,  $f_{1}(\cdot;\pmb{\theta}_{1})$  are approximately 50 and 20 respectively. For the learning rate, we find the learning rate for BASS with grid search from {0.01, 0.001, 0.0001}, and choose the learning rates for the meta-model  $\eta_{1} = 0.01$ ,  $\eta_{2} = 0.001$ . The meta-model architecture as well as its learning rates will stay the same for all the baselines and our proposed BASS. For the CIFAR-100 and Mini-ImageNet data sets, we use the meta-model with four convolutional blocks where the network width of each block is 32, followed by an FC layer as the output layer. For the Drug data set, we apply a meta-model with two FC layers, where the network width is 500. All the experiments are performed on a Linux machine with Intel Xeon CPU, 128GB RAM, and Tesla V100 GPU. Code will be made available at https://github.com/yunzhe0306/Bandit_Task_Scheduler.
A.2 Effect of the Task Noise Magnitude
We conduct the experiments to show the effects of the noise magnitude factor  $\epsilon$  on the Drug and CIFAR-100 data sets. The experiment results are shown in Table 4.
Table 4: Comparison with baselines with different noise magnitude [data set (noise magnitude  $\epsilon$ ); final results  $\pm$  standard deviation].
Algo. \ Data	Drug (0.3)	Drug (0.5)	CIFAR100 (0.3)	CIFAR100 (0.5)
Uniform	0.218±0.007	0.220±0.001	0.655±0.009	0.526±0.011
SPL	0.243±0.008	0.236±0.004	0.625±0.017	0.367±0.039
FOCAL	0.224±0.019	0.223±0.003	0.638±0.010	0.485±0.006
DAML	0.182±0.025	0.177±0.003	0.543±0.017	0.414±0.025
GCP	N/A	N/A	0.653±0.005	0.508±0.009
PAML	0.186±0.006	0.205±0.009	0.537±0.009	0.316±0.022
ATS	0.239±0.011	0.237±0.014	0.651±0.001	0.505±0.015
BASS (Ours)	0.258±0.003	0.245±0.006	0.657±0.005	0.553±0.008
With increasing noise magnitude  $\epsilon$ , the performances of the meta-model trained by baselines and our BASS tend to drop, which is intuitive. In particular, for the CIFAR-100 data set, when we increase  $\epsilon$ , the performance difference between BASS and the other baselines tends to increase. This can be the reason that the greedy baselines with no exploration strategies can be more susceptible to the task noise perturbation, which can lead to the sub-optimal performances of the meta-model.
Table 5: Experiment results of noise-free settings on three real data sets (5-way, 5-shot).
Data\Algo.	Uniform	SPL	FOCAL-LOSS	DAML	GCP	PAML	ATS	BASS
Drug	0.206±0.012	0.234±0.006	0.240±0.003	0.190±0.002	N/A	0.220±0.010	0.233±0.001	0.256±0.003
M-ImageNet	0.576±0.016	0.554±0.004	0.582±0.005	0.437±0.015	0.564±0.002	0.467±0.007	0.561±0.004	0.586±0.008
CIFAR	0.681±0.010	0.681±0.008	0.692±0.023	0.662±0.027	0.681±0.016	0.640±0.011	0.695±0.035	0.697±0.029
From the Table 5, we can see that when there is no noise, the overall performance does not differ significantly across different methods. The reason could be that since the meta-learning backbone remains the same for all the methods, the meta-model performance upper bound can be similar for different scheduling algorithms, without the presence of other confounding factors (e.g., noise, task distribution skewness). In the practical application scenarios with noisy data, BASS-guided meta-models tend to perform well in presence of task noise and skewness compared with baselines, as presented by our experiments in the main body.
A.3 Parameter Study for Exploration Coefficient
As in Eq. 7 and Eq. 9, BASS involves an exploration coefficient  $\alpha$  to balance the exploitation-exploration and the two exploration objectives. Here, we conduct the parameter study for the exploration coefficient  $\alpha$ , and include the results with no exploration (i.e., removing  $f_{2}$ ).
Table 6: Comparison among different  $\alpha$  values [dataset (shot); final results  $\pm$  standard deviation].
Algo. \ Data	Drug (1)	Drug (5)	CIFAR100 (1)	CIFAR100 (5)
No Exploration	0.234±0.003	0.239±0.012	0.256±0.027	0.537±0.012
α = 0.1	0.231±0.005	0.233±0.013	0.264±0.051	0.522±0.024
α = 0.3	0.228±0.013	0.231±0.008	0.268±0.047	0.528±0.014
α = 0.5	0.236±0.004	0.245±0.006	0.272±0.025	0.553±0.008
α = 0.7	0.242±0.012	0.227±0.006	0.241±0.005	0.543±0.021
α = 1.0	0.236±0.002	0.235±0.013	0.266±0.006	0.537±0.005
From the results in Table 6, we see that the exploration module can indeed improve the performance of BASS compared with the performance with no exploration. This also fits our initial argument that the greedy algorithm alone can lead to sub-optimal performances of meta-model. By properly choosing the  $\alpha$  value, we will be able to achieve a good balance between exploitation and exploration, as well as between the two exploration objectives. Here, setting  $\alpha \in [0.5, 0.7]$  will be good enough to achieve satisfactory performances. Meanwhile, we also note that even with no exploration, our BASS still achieves good performances by directly learning the correlation between the adapted meta-parameter and the generalization score, and refining the scheduling strategy based on the status of the meta-model.
A.4 Running Time Comparison
In Figure 5, we include the running time comparison with baselines. We can see that BASS can achieve significant improvement in terms of the running time, and can take as little as  $50\%$  of ATS's running time. The intuition is that our proposed BASS only needs one round of the optimization process to update the meta-model and BASS. On the other hand, from Algorithm 1 of the ATS paper [51], we see that ATS requires two optimization rounds for each meta-training iteration to (1) update the scheduler with the temporal meta-model, and (2) update the actual meta-model respectively. Based on the figure on the RHS, we also see that BASS can achieve a relatively good balance between computational cost and performance.
Figure 5: Running time results (including training both the scheduler and the meta-model). "D", "M" and "C" refer to the "Drug", "Mini-ImageNet", "CIFAR-100" data sets respectively. BASS can take as little as approximately  $50\%$  of ATS's running time. On the RHS, we have the scatter plot in terms of running time vs. performance on the Mini-ImageNet dataset.
A.5 Performances with Different Task Skewness Settings
In Table 7, we include the experiments with different levels of skewness. Here, we see that with less skewness levels (the skewness level reduces from Setting 1 to Setting 3), the accuracy of BASS as well as the baselines will continue to improve, while BASS still maintains decent performances.
Skewness Setting \ Algo.	Uniform	ATS	BASS
Skewness Setting 1	0.375±0.009	0.382±0.007	0.408±0.008
Skewness Setting 2	0.429±0.012	0.448±0.006	0.460±0.013
Skewness Setting 3	0.497±0.008	0.502±0.010	0.539±0.009
Table 7: Results for different skewness levels on CIFAR-100 data set (5-shot). (1) Setting 1 is the original setting in paper Subsec. 5.2. (2) For Setting 2, we assign 5 tasks with  $8\%$  sampling probability, 5 tasks with  $3\%$ , and the rest of the tasks equally share the  $45\%$  probability. (3) For Setting 3, we assign 5 tasks with  $5\%$ , 5 tasks with  $2\%$ , while the rest of the tasks equally share the  $65\%$  probability.
A.6 Performances with Different Batch Size
With Table 8, we include additional experiments with different batch sizes  $B$ , in comparison with the ATS and the uniform sampling approach. Here, we see that with larger  $B$  values, the accuracy of BASS as well as the baselines will generally improve.
B (batch size) \ Algo.	Uniform	ATS	BASS
1	0.459±0.009	0.449±0.010	0.472±0.012
2	0.526±0.011	0.515±0.015	0.553±0.008
3	0.570±0.012	0.563±0.007	0.588±0.010
5	0.581±0.005	0.571±0.007	0.586±0.009
Table 8: Results for different  $B$  values (batch sizes) on CIFAR-100 data set (5-shot).
A.7 Performances with Different Embedding Approaches of Arm Contexts
In Table 9, we include additional experiments with different levels of average pooling, such that after the average pooling, the dimensionality of the pooled vector representation will fall into  $\{20,100,500\}$ . We see that overly small dimensionality of the average-pooled vector representation (e.g., 20) can lead to sub-optimal performance of the BASS framework. In addition, we see that setting the dimensionality to 50 can generally lead to good enough performance.
Dimensionality	20	50	100	500
Accuracy	0.541±0.008	0.553±0.008	0.558±0.006	0.555±0.010
Table 9: With CIFAR-100 (5-shot), different dimensionality of the average-pooled vector representation (Remark 3) of the meta-parameters.
Here, we also include additional experimental results using MLP to map the original context into the lower dimensional space instead of using our proposed average pooling (Remark 3). Results are shown in Table 10. Here, we use the one-layer MLP with the ReLU activation to embed the original meta-parameters to the low-dimensional vector representations. We can see that the MLP-based method can indeed lead to some performance improvement. But in general, the performance difference between MLP-based embedding and the average-pooling vector representation is subtle. We also note that the MLP-based mapping approach is more time consuming compared with the average pooling approach, since we also need to train the additional embedding layer, which has a considerable number of trainable parameters.
Dimensionality	Original avg-pooled (50)	50	100	200
Accuracy	0.553±0.008	0.558±0.013	0.560±0.012	0.553±0.015
Table 10: With CIFAR-100 (5-shot), different dimensionality of the one-layer MLP(with ReLU)-embedded vector representation of the meta-parameters. "original avg-pooled (50)" refers to the average-pooled vector representation (Remark 3) with dimensionality of 50.
A.8 Additional Experiments on the "DomainNet" data set
In Table 11, we include additional experiments on the new "DomainNet" data set [37]. Within the "real" domain, we filter 100 classes that have at least 600 images. In this way, with each class being a task with 600 images, we will have a total of 100 tasks. Compared with image data sets in our paper (Mini-ImageNet and CIFAR-100), we increase the image resolution of "DomainNet" by resizing its images to  $128 \times 128$  pixels. Following the settings in our paper, we divide tasks into  $64:16:20$  portions that correspond to the training set, validation set and the test set respectively. For the few-shot settings, we formulate the problem to be 5-shot, 5-way / 7-way with uniform sampling and ATS as baselines. With a higher image resolution of the "DomainNet" data set, BASS can still maintain the good performance compared with the baselines.
Setting\Algo.	Uniform	ATS	BASS
5-way	0.475±0.002	0.483±0.006	0.511±0.012
7-way	0.411±0.005	0.372±0.009	0.435±0.008
Table 11: Results for the "DomainNet" data set (noise level 0.5, 5-shot settings).
B Appendix: Additional Discussion on the Necessity of Assumption 5.1
We would like to mention that in order to finish the convergence and generalization analysis for the neural Contextual Bandit works (e.g., [54, 2, 9]), the separateness assumption of the arm context is the minimum requirement of the data set. This is because the training data needs to be non-degenerate (i.e., every pairs of samples are distinct) to ensure that the neural network can consistently converge, as indicated by Assumption 2.1 in [3]. Therefore, our Assumption 5.1 regarding the arm separateness aims to ensure that the BASS is able to adequately learn the underlying reward mapping function with sufficient information. Comparing with the existing works, in the convergence analysis works on meta-learning [47, 48], they measure the arm separateness in terms of the minimum eigenvalue  $\lambda_0$  (with  $\lambda_0 > 0$ ) of the Neural Tangent Kernel (NTK) [23] matrix, which is comparable with our Euclidean separateness  $\rho$ . For existing neural bandit works, Assumption 5.1 in [9] is similar to our separateness assumption. Meanwhile, Assumption 4.2 in [54] and Assumption 3.4 from [53] also imply that no two arms are the same in terms of the minimum NTK matrix eigenvalue  $\lambda_0 > 0$ .
C Appendix: Limitation
One potential limitation of BASS is that its improvement over baselines may not be significant when dealing with noise-free settings and non-skewed task distributions (Table 5). Meanwhile, the non-adaptive FOCAL-LOSS [30] tends to achieve a similar performance comparing with the adaptive method ATS [51], while enjoying an advantage in terms of the computational cost. In practical terms, although BASS can generally achieve the decent performance and enjoys a smaller computational cost than ATS, the practitioner still needs to consider whether their task distribution is noisy or skewed in order to strike a good balance between the computational resource needed and the meta-model performance, as BASS can achieve a more significant advantage over baselines given the noisy or skewed task distribution.
D Appendix: Theoretical Analysis
In this section, we present the proof for Theorem 5.2. Here, instead of directly going for the batch setting where we adopt training task batch  $\Omega_{k}$  for each iteration  $k\in [K](|\Omega_k| = |\Omega_k^* | = B)$ , we first introduce the results of the single-task setting (Subsec. D.1), i.e.,  $|\Omega_k| = |\Omega_k^* | = 1$ . Then, the results will be extended to the batch settings as in Subsec. D.2. Recall that for the meta-model, we first consider it to be a  $L_{\mathcal{F}}$ -layer fully-connected (FC) network (of width  $m_{\mathcal{F}}$  for the theoretical analysis (lines 237-239). In particular, we follow the settings in [3] for the Gaussian initialization of weight matrices. For the weight matrix elements in meta-model's first  $(L_{\mathcal{F}} - 1)$  layers, we draw each of them from the Gaussian distribution  $\mathcal{N}(0,2 / m_{\mathcal{F}})$ . Then, for the weight matrix elements of the last layer  $(L_{\mathcal{F}}$ -th layer), we draw each of them from the Gaussian distribution  $\mathcal{N}(0,1)$ .
D.1 Single-task settings
For the brevity of notation, we denote the scheduler output  $f(\Theta^{(K - 1)}[\mathcal{T}_{k,i}];\pmb{\theta}^{(k - 1)}) = f_1(\pmb{\chi}_{k,i}^q;\pmb{\theta}_1^{(k - 1)}) + f_2\big([ \nabla_\theta f_1(\pmb{\chi}_{k,i}^s);\nabla_\theta f_1(\pmb{\chi}_{k,i}^q)];\pmb{\theta}_2^{(k - 1)}\big)$ , which corresponds to the definition in Eq. 7. In this case,  $\mathcal{T}(K) = \{\mathcal{T}_1,\dots ,\mathcal{T}_K\}$  refer to the chosen tasks and  $\mathcal{T}^{*}(K) = \{\mathcal{T}_{1}^{*},\dots ,\mathcal{T}_{K}^{*}\}$  are the optimal ones. Based on the problem definition, we will have
where the first inequality is due to the arm pulling mechanism, i.e.,  $f(\Theta^{(K - 1)}[\mathcal{T}_K^* ];\pmb{\theta}^{(K - 1)}) \leq f(\Theta^{(K - 1)}[\mathcal{T}_K];\pmb{\theta}^{(K - 1)})$ . Here,  $f(\cdot ;\tilde{\pmb{\theta}}^{(K - 1)})$  is defined as the "shadow" bandit model that are trained on optimal tasks  $\{\mathcal{T}_1^*,\mathcal{T}_2^*,\dots ,\mathcal{T}_{K - 1}^*\}$  and the corresponding meta-model parameters. Here, denote  $\chi_{K} = \Theta^{(K - 1)}[\mathcal{T}_{K}]\in \mathbb{R}^{p}$  as the arm context given the arm  $\mathcal{T}_K$  and the meta-model parameter  $\Theta^{(K - 1)}$ ; similarly, we have  $\chi_K^* = \Theta^{(K - 1),*}[\mathcal{T}_K^* ]\in \mathbb{R}^p$  being the arm context given the arm  $\mathcal{T}_K^*$  and the meta-model parameter  $\Theta^{(K - 1),*}$ . Thus, for the term  $I_0$  on the RHS, we have
$$
I _ {0} = \left| f \left(\chi^ {*}; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) - f \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} ^ {*} \right]; \boldsymbol {\theta} ^ {(K - 1)}\right) \right|
$$