SlowGuess/NeurIPS_2023_OCR huggingface dataset:
on the training set and observe the model performance on the validation set. Based on standard hyperparameter settings, we train for 100 epochs and use a multi-step scheduler to adjust the learning rate on 30, 60 and 90 epochs with a discount factor of size 0.1. We present our results in Table 2. We get similar conclusions on the ImageNet as on CIFAR: higher test set accuracy and stronger generalization performance, which coincides with the theoretical results in the previous section.
Table 2: Top-1 accuracy of ResNet50 on ImageNet under different optimization methods.
OPTIMIZER	TOP-1	TOP-5
SGDM	75.97	92.89
SWA	76.78	93.18
LOOKAHEAD	76.52	93.11
LOOKAROUND	77.32	93.29
4.2 Finetuning
The transfer learning technique, which involves pretraining a model on a large related dataset and then finetuning on a smaller target dataset of interest, has become a widely used and powerful approach for creating high-quality models across various tasks.
To explore the scalability of our Lookaround method, we apply our method to scenarios that require pre-training, including CNN or ViT [7] architecture. Unlike the previous experiments, we adopt more appropriate hyperparameters for its training in the ViT experiments, including the Adam optimizer and cosine annealing scheduler structure. Our results are shown in Table 3 and Figure 4. Under the ResNet architecture, our proposed Lookaround method remains stable and performs better, while the SGDM method shows greater fluctuations.
In contrast, the Lookaround methods based on weight averaging do not produce such a phenomenon, which partly demonstrates the hypothesis that average weight can reduce the convergence variance. In our comparison experiments with the other optimizations, we found that the results of ViT-B/16 under different optimizers were better than those reported in the original paper [7]. This is mainly due to our use of more data augmentation during the training process, aiming for a fair comparison with Lookaround. This result also indicates that Lookaround can still achieve better performance under improved baseline conditions.
Table 3: The test set accuracy and the NLL loss under training procedure with pretrained models. In this table, all models are fine-tuned for the same amount of time at  $224 \times 224$  resolution. The results of Adam† using ViT-B/16 are taken from reference [7].
Backbone	Method	CIFAR10	CIFAR100
Top-1	NLL	#param.	Top-1	NLL	#param.
ResNet50	SGDM	97.55	0.109	23.52M	84.50	0.692	23.71M
SWA	97.48	0.105	23.52M	84.81	0.668	23.71M
Lookahead	97.65	0.103	23.52M	84.78	0.676	23.71M
Ours	97.82	0.099	23.52M	85.20	0.658	23.71M
ViT-B/16	Adam†	98.13	-	85.65M	87.13	-	85.72M
Adam	98.34	0.060	85.65M	91.55	0.298	85.72M
SWA	98.47	0.049	85.65M	91.32	0.304	85.72M
Lookahead	98.51	0.047	85.65M	91.76	0.280	85.72M
Ours	98.71	0.041	85.65M	92.21	0.267	85.72M
Table 4: The test set accuracy under SAM and Lookaround optimization using ResNet50 or ViT-B/16. The results of  $\mathrm{SAM}^{\dagger}$  using ResNet50 are taken from [7].
Backbone	Method	resolution	pretrain	CIFAR10	CIFAR100
ResNet50	SAM†	32	-	-	79.10
Lookaround	32	-	96.59	81.60
Lookaround + SAM	32	-	96.38	79.79
ResNet50	SAM	224	✓	97.65	85.97
Lookaround	224	✓	97.82	85.20
Lookaround + SAM	224	✓	97.88	86.21
ViT-B/16	SAM	224	✓	98.51	92.39
Lookaround	224	✓	98.71	92.21
Lookaround + SAM	224	✓	98.67	92.54
4.3 Compared with Sharpness-Aware Minimization
We draw a comparison between Lookaround and Sharpness-Aware Minimization (SAM) [10], an algorithm designed to improve generalization by steering model parameters towards flatter loss regions. The comparative results are presented in Table 4.
The results seem to indicate that, under the default parameters of SAM, this method is more suitable for higher resolutions, while Lookaround achieves the best performance at lower resolutions. For the finetuning, neither SAM nor Lookaround consistently emerges as the top choice when used individually. However, when combined, they often enhance model performance significantly. Thus, for higher accuracy, we recommend using both the SAM and Lookaround optimizers concurrently. Both are orthogonal to each other. By combining the two methods, we can achieve superior performance.
4.4 Compared with Ensemble Method
Ensemble methods are traditionally used to further improve inference performance by combining the outputs of multiple different models. We compare our Lookaround method with classical Ensemble methods (Logit Ensemble and Snapshot Ensemble [18]), and the results are shown in Table 5.
The performance of the single model obtained by our training exceeds the performance of the ensemble under multiple models. Moreover, the model obtained by our method is not a simple superposition of Ensemble models. We take the single model obtained by us with the three models obtained by Logit Ensemble to Logit Ensemble, and get a more significant performance improvement. This indi
Table 5: Performance compared with the ensemble methods (without pretrained weights) on CIFAR100 dataset. "Ours + Logit Ensemble" means that we take the model obtained by our method and the three models in Logit Ensemble to Logit Ensemble.
Method	ResNet50	ResNet101	ResNet152
Logit Ensemble	81.16	81.92	81.89
SnapShot Ensemble	79.88	80.63	80.23
Ours (Lookaround)	81.60	81.97	82.22
Ours + Logit Ensemble	83.43	83.35	83.53
cates that the model we obtained is orthogonal to the three models obtained by Logit Ensemble and can be combined to produce better performance.
In this subsection, we investigate the impact of different components of Lookaround on its performance, including data Augmentation methods and weight averaging. As shown in Table 6,
Table 6: Ablation Study of Data Augmentation (DA) and Weight Averaging (WA) in the proposed Lookaround using ResNet50.
Dataset	DA	WA	Acc (%)	Dataset	DA	WA	Acc (%)
CIFAR10	-	-	95.84	CIFAR100	-	-	78.81
✓	-	95.96	✓	-	79.61
-	✓	95.79	-	✓	79.49
✓	✓	96.59	✓	✓	81.60
Intriguingly, SAM is proved to exhibit dynamics akin to a decentralized weight averaging algorithm [51].
Figure 5: Top-1 accuracy of ResNet50 on CIFAR100 dataset with different steps  $k$ . (Left) Error bar of three different random seeds with random initialization weight. (Right) Top-1 accuracy of two different  $k$  values with pretrained weight.
Figure 6: Accuracy curve of sub-model state update on the test set with different learning rates using ResNet50 on CIFAR100. At position 0 on the horizontal axis, H-net (trained by randomly horizontally flipped data) and R-net (trained by RandAugment data) are averaged to Mean-net. Subsequently, during 160 weight iterations, H-net and R-net continue to be trained with their respective data, while Mean-net represents the model obtained by averaging the weights at each iteration.
when we consider Data Augmentation (DA) or Weight Averaging (WA) independently, their improvement effects may be relatively small. However, when we combine them, the synergistic effect between them can lead to more significant overall improvement. Specifically, we found that the improvement effects of DA and WA show a nonlinear trend in the interaction case, indicating a certain synergistic effect between them. Under the interaction, Data augmentation guides the model into different loss basins, and Weight Averaging guides different models into lower loss basins, resulting in higher performance improvement and a qualitative leap.
4.6 Additional Analysis
Robustness of the Parameters. For the number of data augmentations, we train the model with different numbers of data augmentations, and the results are shown in Table 7. The results indicate that an increase in the number of data augmentations results in improved network performance. In
Table 7: Top-1 accuracy (%) of different data augmentation (DA) number by using ResNet50 on CIFAR100 dataset.
# of DA	1	2	3	4	5	6
Top-1 (%)	78.20	80.82	81.60	81.19	81.74	82.02
Top-5 (%)	94.50	95.19	95.99	95.65	95.85	96.02
order to reduce the training time, only three data augmentation methods are selected in this paper for experimental verification at different levels. Note that the methods compared with ours use the same incremental data set, so the comparison is fair.
For different choices of  $k$ , the number of around steps, we select different intervals for full verification, and the results are shown in Figure 5 (Left). Let us look at Large-k (391) versus Small-k(5) in Figure 5 (Right) (391 is determined by the number of batches in a single epoch). We find that large steps (green line) can achieve higher accuracy and stability in the early stage of training, while small steps can achieve higher accuracy in the later stage. In future research, we may explore a learning rate scheduler suitable for Lookaround to help it achieve the best performance.
We visualize the training trajectory accuracy of the sub-models in Lookaround in Figure 6. Under different learning rates, the model, after averaging the weights, can maintain a good baseline and continuously improve the performance of the model under the low variance, and the sub-models also obtain more beneficial training effects due to the weight averaging.
In this paper, we present the Lookaround optimizer, an SGD-based optimizer that employs a diversity-in-locality exploration strategy to find solutions with high generalization ability in the loss basin. We theoretically prove that Lookaround can obtain lower variance and faster convergence speed. Experimental results on CIFAR and ImageNet and across various network backbones show that, our proposed Lookaround significantly improves the training stability and achieves better generalization performance comparable to the single-trajectory WA optimizers and ensemble methods.
Limitation. Lookaround is limited by the additional cost of network searching using different data augmentation, resulting in a longer training time proportional to the number of trained networks. Notably, Lookaround incurs the same inference time as other optimization methods. Considering the significantly superior performance, we believe the extra time consumption is worthwhile in many real-world scenarios where the training time is not so concerned as the inference time. Moreover, training time can be reduced due to the parallelizability of the "around step" mechanism in Lookaround.
This work is supported by the National Key Research and Development Project (Grant No: 2022YFB2703100), National Natural Science Foundation of China (62106220, 61976186, U20B2066), and Ningbo Natural Science Foundation (2021J189) and the advanced computing resources provided by the Supercomputing Center of Hangzhou City University. We sincerely thank the anonymous reviewers for their valuable comments on the work.
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A Exploration in the Loss Landscape
To demonstrate the effectiveness of Lookaround during training, we set up an experiment and visualize the loss landscape of the models under different data augmentations in Figure 7. By observing the loss landscape, we can gain a clearer understanding of the role played by the weight averaging at different stages during the training process.
Training Process and Parameter Settings. We first train a ResNet50 on the CIFAR100 dataset. The learning rate is initialized to 0.1 and decay at 60, 120, and 160 epochs using a MultiStepLR scheduler with a decay factor 0.2. The batch size is set to 128, we use stochastic gradient descent with momentum to optimize the model and use random crops and random vertical flips augmentation to enhance the training datasets. We use model checkpoints at epochs 50, 110, and 150 as our three pretrained models(V-network). The three pretrained models correspond to learning rates of 0.1, 0.02 and 0.004, respectively. Using these pretrained models as the starting point, we finetune the model with 1, 10, 100, 1000, 10000 iterations under the corresponding learning rate and the setting of random horizontal flipping (H-network) or RandAugment (R-network).
Visualization Method. We use the visualization method in Garipov et al. [12] to plot the loss landscape. In this method, the weights of the three models are flattened respectively as one-dimensional vectors  $w_{v}, w_{h}, w_{r}$ , and then two orthogonal vectors are calculated between the three vectors as the X-axis direction and the Y-axis direction:  $u = (w_{h} - w_{v})$ ,  $v = (w_{r} - w_{v}, w_{h} - w_{v}) / \| w_{h} - w_{v} \|^2 \cdot (w_{h} - w_{v})$ . Then the normalized vectors  $\hat{u} = u / \| u \|$ ,  $\hat{v} = v / \| v \|$  foam an orthonormal basis in the plane contain  $w_{v}, w_{h}, w_{r}$ . Then a point P with coordinates  $(x, y)$  in the plane would be given by  $P = w_{v} + x \cdot \hat{u} + y \cdot \hat{v}$ .
Discussion and Inspiring. Under different learning rates and different around steps  $k$ , Lookaround has the tendency to lead the model trained on different data augmentation to the near-constant loss manifold. In such circumstances, the "average step" can lead the model into the center of the loss basin to get a lower test loss. However, weight averaging does not necessarily work in all cases. For example, the network obtained after weight averaging gets a larger loss under a large learning rate with a large around step (e.g.,  $lr = 0.1$ ,  $k = 10000$ ). In this case, the model is located on a peak between different basins rather than in the center of a basin. Moreover, in the case of around step  $k = 1$ , the weight averaging also does not achieve better performance. Nevertheless, such extreme cases do not prevent weight averaging from being a useful tool to speed up the training process. The center of the basin in loss landscape, which requires multi-step gradient descent to reach, can be reached by only one weight averaging step. At other learning rates and around steps  $k$ , the models after weight averaging all result in a lower loss than the individual model. Such phenomena encourage us to explore more methods to find more optimal solutions in the loss landscape in the future.
B Steady-state and Convergence Analysis of Lookaround
We use quadratic functions to analyze the steady-state and convergence analysis of Lookaround. First, we present the proof of Proposition 2.
Proposition 2 (Steady-state risk). Let  $0 < \gamma < 1 / L$  be the learning rate satisfying  $L = \max_{i}a_{i}$ . One can obtain that in the noisy quadratic setup, the variances of SGD, Lookahead [49] and Lookaround converge to the following fixed matrix:
$$
V _ {S G D} ^ {*} = \frac {\gamma^ {2} \mathbf {A} ^ {2} \Sigma^ {2}}{\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2}}, \tag {7}
$$
$$
V _ {\text {L o o k a h e a d}} ^ {*} = \underbrace {\frac {\alpha^ {2} \left(\mathbf {I} - \left(\mathbf {I} - \gamma \mathbf {A}\right) ^ {2 k}\right)}{\alpha^ {2} \left(\mathbf {I} - \left(\mathbf {I} - \gamma \mathbf {A}\right) ^ {2 k}\right) + 2 \alpha (1 - \alpha) \left(\mathbf {I} - \left(\mathbf {I} - \gamma \mathbf {A}\right) ^ {k}\right)}} _ {\leqslant \mathbf {I}, i f \alpha \in (0, 1)} V _ {S G D} ^ {*}, \tag {8}
$$
$$
V _ {L o o k a r o u n d} ^ {*} = \underbrace {\frac {\alpha^ {2} (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}) + 2 \alpha (1 - \alpha) (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {k})}{\alpha^ {2} (d \mathbf {I} - (d - 1) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})}} _ {\leqslant \mathbf {I}, i f d \geqslant 3 a n d \alpha \in [ 1 / 2 , 1)} V _ {L o o k a h e a d} ^ {*}. \tag {9}
$$
Figure 7: The test loss landscape of ResNet50 on CIFAR100. The diamond block (V-network) represents the pretrained model trained with random vertical flipping. Then we can use random horizontal flipping and RandAugment to finetune V-network to get H-network and R-network.
From Wu et al. [47], we can obtain the following dynamics of SGD with learning rate  $\gamma$ :
$$
\mathbb {E} \left[ \theta^ {(t + 1)} \right] = (\mathbf {I} - \gamma \mathbf {A}) \mathbb {E} \left[ \theta^ {(t)} \right],
$$
$$
\mathbb {V} \left[ \theta^ {(t + 1)} \right] = (\mathbf {I} - \gamma \mathbf {A}) ^ {2} \mathbb {V} \left[ \theta^ {(t)} \right] + \gamma^ {2} \mathbf {A} ^ {2} \Sigma .
$$
Lemma 1. The expectation and variance of lookaround have the following iterates:
$$
\mathbb {E} \left[ \phi^ {(t + 1)} \right] = \left(\mathbf {I} - \gamma \mathbf {A}\right) ^ {k} \mathbb {E} \left[ \phi^ {(t)} \right], \tag {10}
$$
$$
\mathbb {V} \left[ \phi^ {(t + 1)} \right] = \frac {d - 1}{d} \left(\mathbf {I} - \gamma \mathbf {A}\right) ^ {2 k} \mathbb {V} \left[ \phi^ {(t)} \right] + \frac {\gamma^ {2} \mathbf {A} ^ {2} \Sigma \left(\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}\right)}{d \left(\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2}\right)}. \tag {11}
$$
$\phi_t$  yields  $\phi_{t+1}$  by performing an around step and an average step.
Proof. The expected iterate follows from SGD:
$$
\begin{array}{l} \mathbb {E} \left[ \phi^ {t + 1} \right] = \mathbb {E} \left[ \frac {1}{d} \sum_ {i} \theta_ {t, i, k} \right] = \sum_ {i} \frac {1}{d} \mathbb {E} \left[ \theta_ {t, i, k} \right] \\ = \sum_ {i} \frac {1}{d} (\mathbf {I} - \gamma \mathbf {A}) ^ {k} \mathbb {E} [ \theta_ {t, i, 0} ] = (\mathbf {I} - \gamma \mathbf {A}) ^ {k} \mathbb {E} [ \phi^ {t} ]. \\ \end{array}
$$
For the variance of  $\phi^{t + 1}$ , we can break it down into two parts as follows:
$$
\mathbb {V} [ \phi^ {t + 1} ] = \mathbb {V} [ \frac {1}{d} \sum_ {i} \theta_ {t, i, k} ] = \sum_ {i} ^ {d} \frac {1}{d ^ {2}} \mathbb {V} [ \theta_ {t, i, k} ] + \sum_ {i \neq j, 1 \leqslant i, j \leqslant d} \frac {1}{d ^ {2}} \operatorname {c o v} (\theta_ {t, i, k}, \theta_ {t, j, k}).
$$
The covariance of the different models can be calculated in the following way:
$$
\begin{array}{l} \operatorname {c o v} \left(\theta_ {t, i, k}, \theta_ {t, j, k}\right) = \mathbb {E} \left[ \theta_ {t, i, k} \theta_ {t, j, k} \right] - \mathbb {E} \left[ \theta_ {t, i, k} \right] \mathbb {E} \left[ \theta_ {t, j, k} \right] \\ = \mathbb {E} \left[ (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}\right) (\phi^ {t}) ^ {2} ] - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k} \mathbb {E} [ \phi^ {t} ] ^ {2} \\ = (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k} \mathbb {V} [ \phi^ {t} ]. \\ \end{array}
$$
After permuting and regrouping again, we can obtain the iterate with respect to the variance.
$$
\begin{array}{l} \mathbb {V} \left[ \phi^ {t + 1} \right] = \sum_ {i \neq j, 1 \leqslant i, j \leqslant d} \frac {1}{d ^ {2}} \operatorname {c o v} \left(\theta_ {t, i, k}, \theta_ {t, j, k}\right) + \sum_ {i} ^ {d} \frac {1}{d ^ {2}} \mathbb {V} \left[ \theta_ {t, i, k} \right] \\ = \frac {1}{d ^ {2}} (d ^ {2} - d) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k} \mathbb {V} [ \phi^ {t} ] + \frac {1}{d} [ \sum_ {i = 0} ^ {k - 1} (\mathbf {I} - \gamma \mathbf {A}) ^ {2 i} \gamma^ {2} \mathbf {A} ^ {2} \Sigma ] \\ = \frac {d - 1}{d} (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k} \mathbb {V} [ \boldsymbol {\phi} ^ {t} ] + \frac {\gamma^ {2} \mathbf {A} ^ {2} \Sigma (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})}{d (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2})}. \\ \end{array}
$$
Remark. From Equation 10, the expectation term for  $\phi$  in Lookaround eventually converges to 0, as does Lookahead and SGD.
From Zhang et al. [49], we have the following analysis about the variance fixed point of lookahead with learning rate  $\gamma$  and weight factor  $\alpha$ , which represents the average weight factor of models with different trajectory points in the Lookahead optimizer, which is generally (0, 1):
$$
V _ {L o o k a h e a d} ^ {*} = \frac {\alpha^ {2} (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})}{\alpha^ {2} (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}) + 2 \alpha (1 - \alpha) (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {k})} V _ {S G D} ^ {*}. \tag {12}
$$
We now derive the fixed point of the variance, proceed with the proof of Proposition 1:
$$
V _ {L o o k a r o u n d} ^ {*} = \frac {\alpha^ {2} (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}) + 2 \alpha (1 - \alpha) (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {k})}{\alpha^ {2} (d \mathbf {I} - (d - 1) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})} V _ {L o o k a h e a d} ^ {*}.
$$
$$
V _ {L o o k a r o u n d} ^ {*} = \frac {d - 1}{d} (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k} V _ {L o o k a r o u n d} ^ {*} + \frac {\gamma^ {2} \mathbf {A} ^ {2} \Sigma (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})}{d (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2})}.
$$
$$
\begin{array}{l} \Rightarrow V _ {L o o k a r o u n d} ^ {*} = \frac {1}{\mathbf {I} - \frac {d - 1}{d} (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}} \frac {\gamma^ {2} \mathbf {A} ^ {2} \Sigma (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})}{d (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2})} \\ = \frac {\gamma^ {2} \mathbf {A} ^ {2} \Sigma [ \mathbf {I} - (\mathbf {I} - \gamma \mathbf {I}) ^ {2 k} ]}{[ d \mathbf {I} - (d - 1) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k} ] [ \mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2} ]} \\ = \frac {\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}}{d \mathbf {I} - (d - 1) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}} V _ {S G D} ^ {*}. \\ \end{array}
$$
According to Equation 12, we can deduce that
$$
\begin{array}{l} V _ {L o o k a r o u n d} ^ {*} = \frac {\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}}{d \mathbf {I} - (d - 1) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}} V _ {S G D} ^ {*} \\ = \frac {\alpha^ {2} (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k}) + 2 \alpha (1 - \alpha) (\mathbf {I} - (\mathbf {I} - \gamma \mathbf {A}) ^ {k})}{\alpha^ {2} (d \mathbf {I} - (d - 1) (\mathbf {I} - \gamma \mathbf {A}) ^ {2 k})} V _ {\text {L o o k a h e a d}} ^ {*} \\ \end{array}
$$
B.1 Comparing the dynamics of Lookahead
We now proceed with the proof for the range of constraints variable  $\alpha$  in Equation 9. When  $d\geqslant 3$  and  $\alpha \in [0.5,1)$ , the Lookaround method can obtain a smaller variance than the Lookahead method:
Proof. Let  $B = (\mathbf{I} - \gamma \mathbf{A})^k$ , due to  $0 < \gamma < 1 / L$ ,  $L = \max_i a_i$ , so we can have  $B \leqslant \mathbf{I}$ . Substituting the matrix  $\mathbf{B}$  into the expressions for the variance fixed point relation of Lookaround and Lookahead: