SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Remark 4. Unlike prior works such as [76, 8], we do not need a pre-specified bound on the gradient of the losses. Consequently Eq.(1) holds for the smallest value of the Lipschitzness coefficient  $G$ , leading to tight regret bounds. Further, the projection step in Line 2 of Algorithm 2 is done only to safeguard our theory against pathological scenarios with unbounded Lipschitz constant for losses. In our experiments, we do not perform such projections.
We next show that the performance guarantee in Theorem 2 is optimal (modulo factors of  $\log T$ ) in a minimax sense.
Theorem 5. Let  $V_{T} \leq 64T$ . There exists a loss function, a domain  $\mathcal{D}$  (in Assumption 2), and a choice of adversarial strategy for generating the data such that for any algorithm, we have  $\sum_{t=1}^{T} E([L_{t}(\hat{q}_{t})] - L_{t}(q_{t})) = \Omega\left(\max\{T^{2/3}V_{T}^{1/3}, \sqrt{T}\}\right)$ , where  $\hat{q}_{t} \in \mathcal{D}$  is the weight estimated by the algorithm and  $q_{t} \in \mathcal{D}$  is the label marginal at round  $t$  chosen by the adversary. Here the expectation is taken with respect to the randomness in the algorithm and the adversary.
4 Supervised Online Label Shift
In this section, we focus on the SOLS problem where the labels are revealed to the learner after it makes decisions. Framework 3 summarizes our setup. Let  $f_{t}^{*} \coloneqq \mathrm{argmin}_{f \in \mathcal{H}} L_{t}(f)$  be the population minimiser. We aim to control the dynamic regret against the best sequence of hypotheses in hindsight:
$$
R _ {\text {d y n a m i c}} ^ {\mathcal {H}} (T) =: \sum_ {t = 1} ^ {T} L _ {t} \left(f _ {t}\right) - L _ {t} \left(f _ {t} ^ {*}\right). \tag {3}
$$
If the SOLS problem is convex, it reduces to OCO [37, 54] and existing works provide  $\tilde{O}(T^{2/3}V_T^{1/3})$  dynamic regret guarantees [82]. However, in practice, since loss functions are seldom convex with respect to model parameters in modern machine learning, the performance bounds of OCO algorithms cease to hold true. In our work, we extend the generalization guarantees of ERM from statistical learning theory [11] to the SOLS problem. All proofs of next sub-section are deferred to App. E.
4.1 Proposed algorithms and performance guarantees
We start by providing a simple initial algorithm whose computational complexity and flexibility will be improved later. Note that due to the label shift assumption, for any  $j, t \in [T]$ , we have  $E_{(x,y) \sim Q_t}[\ell(f(x), y)] = E_{(x,y) \sim Q_j}\left[\frac{q_t(y)}{q_j(y)}\ell(f(x), y)\right]$ . Here we assume that the true label marginals  $q_t(y) > 0$  for all  $t \in [T]$  and all  $y \in [K]$ . Based on this, we propose a simple weighted ERM approach (Algorithm 4) where we use an online regression oracle to estimate the label marginals from the (noisy) empirical label marginals computed with observed labeled data. With weighted
ERM and plug-in estimates of importance weights, we can obtain our classifier  $f_{t}$ . One can expect that by adequately choosing the online regression oracle ALG, the risk of the hypothesis  $f_{t}$  computed will be close to that of  $f_{t}^{*}$ . Here the degree of closeness will also depend on the number of data points seen thus far. Consequently, Algorithm 4 controls the dynamic regret (Eq.(3)) in a graceful manner. We have the following performance guarantee:
Theorem 6. Suppose the true label marginal satisfies  $\min_{t,k} q_t(k) \geq \mu > 0$ . Choose the online regression oracle in Algorithm 4 as FLH-FTL (App. F) or Aligator from Baby et al. [7] with its predictions clipped such that  $\hat{q}_t[k] \geq \mu$ . Then with probability at least  $1 - \delta$ , Algorithm 4 produces hypotheses with  $R_{dynamic}^{\mathcal{H}} = \tilde{O}\left(T^{2/3} V_T^{1/3} + \sqrt{T \log(|\mathcal{H}| / \delta)}\right)$ , where  $V_T = \sum_{t=2}^{T} \| q_t - q_{t-1} \|$ . Further, this result is attained without any prior knowledge of the variation budget  $V_T$ .
The above rate contains the sum of two terms. The second term is the familiar rate seen in the supervised statistical learning theory literature under iid data [11]. The first term reflects the price we pay for adapting to distributional drift in the label margins. While we prove this result for finite hypothesis sets, the extension to infinite sets is direct by standard covering net arguments [70].
Remark 7. Theorem 6 requires that the estimates of the label marginals to be clipped from below by  $\mu$ . This is done only to facilitate theoretical guarantees by enforcing that the importance weights used in Eq.(2) do not become unbounded. However, note that only the labels we actually observe enters the objective in Eq.(2). In particular, if a label has very low probability of getting sampled at a round, then it is unlikely that it enters the objective. Due to this reason, in our experiments, we haven't used the clipping operation (see Section 5 and Appendix  $F$  for more details).
The proof of the theorem uses concentration arguments to establish that the risk of the hypothesis  $f_{t}$  is close to the risk of the optimal  $f_{t}^{*}$ . However, unlike the standard offline supervised setting with iid data, for any fixed hypothesis, the terms in the summation of Eq.(2) are correlated through the estimates of the online regression oracle. We handle it by introducing uncorrelated surrogate random variables and bounding the associated discrepancy. Next, we show (near) minimax optimality of the guarantee in Theorem 6.
Theorem 8. Let  $V_{T} \leq T / 8$ . There exists a choice of hypothesis class, loss function, and adversarial strategy of generating the data such that  $R_{\text{dynamic}}^{\mathcal{H}} = \Omega \left( T^{2/3} V_{T}^{1/3} + \sqrt{T \log(|\mathcal{H}|)} \right)$ , where the expectation is taken with respect to randomness in the algorithm and adversary.
Remark 9. Though the rates in Theorems 5 and 8 are similar, we note that the corresponding regret definitions are different. Hence the minimax rates are not directly comparable between the supervised and unsupervised settings.
Even though Algorithm 4 has attractive performance guarantees, it requires retraining with weighted ERM at every round. This can be computationally expensive. To alleviate this issue, we design a new online change point detection algorithm (Algorithm 5 in App. D) that can adaptively discover time intervals where the label margins change slow enough. We show that the new online change point detection algorithm can be used to significantly reduce the number of retraining steps without sacrificing statistical efficiency (up to constants). Due to space constraints, we defer the exact details to App. D. We remark that our change point detection algorithm is applicable to general online regression problems and hence can be of independent interest to online learning community.
Remark 10. Algorithm 5 helps to reduce the run-time complexity. However, both Algorithms 4 and 5 have the drawback of storing all data points accumulated over the online rounds. This is reminiscent to FTL / FTRL type algorithms from online learning. We leave the task of deriving theoretical guarantees with reduced storage complexity under non-convex losses as an important future direction.
5 Experiments2
5.1 UOLS Setup and Results
Setup Following the dataset setup of Bai et al. [8], we conducted experiments on synthetic and common benchmark data such as MNIST [50], CIFAR-10 [49], Fashion [77], EuroSAT [40], Arxiv [15], and SHL [31, 71]. For each dataset, the original data is split into labeled data available
$^{2}$ Code is publicly available at https://github.com/Anon-djiwh/OnlineLabelShift.
(b)
Figure 2: Results on the UOLS problem. (a) and (b): Ablation on CIFAR10 with monotone shift over sizes of holdout data used to update model parameters and compute confusion matrix, with amount of training data held fixed. FLH-FTL (ours) outperforms all other alternatives throughout in classification error and mean square error in label marginal estimation. Unlike the alternatives, the performance of FLH-FTL (ours) is unaffected by the decrease in amount of holdout data. (c): CIFAR10 results with monotone shift using varying amount of training data, with the remaining labeled data used as holdout (total number of samples fixed to  $50\mathrm{k}$ ). The performance of FLH-FTL is minimally impacted by the reduction in the quantity of holdout data, thus yielding the greatest advantage from utilizing a larger volume of training data.
Methods	Synthetic	MNIST	CIFAR	EuroSAT	Fashion	ArXiv
Ber	Sin	Ber	Sin	Ber	Sin	Ber	Sin	Ber	Sin	Ber	Sin
Base	8.6±0.2	8.2±0.3	4.9±0.4	3.9±0.0	16±0	16±0	13±0	13±0	15±0	15±0	23±1	19±0
OFC	6.4±0.6	5.5±0.2	4.4±0.5	3.2±0.3	12±1	11±0	11±1	10±1	7.9±0.1	7.1±0.1	20±2	15±0
Oracle	3.7±0.8	3.9±0.2	2.5±0.5	1.5±0.1	5.4±0.5	5.8±0.1	3.9±0.3	4.1±0.1	3.7±0.2	3.6±0.1	7.7±1.0	5.1±0.1
FTH	6.5±0.6	5.7±0.3	4.5±0.6	3.3±0.2	11±0	11±0	10±0	9.6±0.0	8.5±0.3	6.9±0.4	20±1	14±0
FTFWH	6.6±0.5	5.7±0.3	4.5±0.6	3.3±0.2	11±1	11±0	9.8±0.4	9.6±0.1	8.2±0.6	6.9±0.4	20±1	14±0
ROGD	7.9±0.3	7.2±0.6	6.2±2.8	4.4±1.5	16±3	13±0	14±1	13±1	10±1	8.2±0.7	23±2	17±1
UOGD	8.1±0.6	7.5±0.6	5.4±0.6	4.0±0.0	14±0	14±1	10±1	9.8±0.7	11±2	11±2	21±1	17±1
ATLAS	8.0±1.0	7.5±0.6	5.2±0.6	3.7±0.2	13±0	13±1	10±1	9.9±0.7	12±2	12±2	21±1	16±0
FLH-FTL (ours)	5.4±0.7	5.4±0.4	4.4±0.7	3.3±0.2	10±0	11±0	9.2±0.4	9.6±0.1	7.7±0.4	7.0±0.0	19±1	14±0
Synthetic	MNIST	CIFAR	EuroSAT	Fashion	ArXiv
Ber	Sin	Ber	Sin	Ber	Sin	Ber	Sin	Ber	Sin	Ber	Sin
FTH	0.19±0.01	0.10±0.00	0.27±0.00	0.14±0.00	0.27±0.01	0.14±0.00	0.27±0.00	0.14±0.00	0.29±0.01	0.14±0.01	0.29±0.01	0.15±0.00
FTFWH	0.19±0.02	0.09±0.00	0.26±0.02	0.13±0.00	0.25±0.02	0.13±0.00	0.25±0.01	0.13±0.00	0.25±0.04	0.14±0.01	0.27±0.02	0.15±0.00
ROGD	0.29±0.03	0.24±0.01	0.41±0.08	0.37±0.06	0.39±0.04	0.30±0.05	0.43±0.04	0.35±0.03	0.37±0.02	0.30±0.01	0.34±0.03	0.28±0.01
FLH-FTL (ours)	0.10±0.01	0.08±0.00	0.15±0.01	0.12±0.00	0.17±0.01	0.13±0.00	0.16±0.01	0.13±0.00	0.18±0.02	0.14±0.01	0.23±0.01	0.15±0.00
Table 1: Results for UOLS problems under sinusoidal (Sin) and Bernoulli (Ber) shifts. Top: Classification Error. Bottom: Mean-squared error in estimating label marginal. For both, lower is better. Across all datasets, we observe that FLH-FTL (ours) often improves over best alternatives.
during offline training and validation, and the unlabeled data that we observe during online learning. We experiment with varying sizes of holdout offline data which is used to obtain the confusion matrix and update the model parameters to adapt to OLS to probe the sample efficiency of all the methods. In contrast to previous works [8, 76], we have chosen to use a smaller amount of holdout offline data for our main experiments. We made this decision because the standard practice for deployment involves training and validating models on training and holdout splits, respectively (e.g., with k-fold cross-validation). Then, the final model is deployed by training on all available data (i.e., the union of train and holdout) with the identified hyperparameters. However, to employ UOLS techniques in practice, practitioners must hold out data that was not seen during training to update the model during online adaptation. Therefore, methods that are efficient with respect to the amount of offline holdout data required might be preferable.
Base	Oracle	ROGD	FTH	FTFWH	FLH-FTL (ours)
CI Err	18±1	6.3±1.3	19±3	14±2	14±2	13±2
MSE	NA	0.0±0.0	0.3±0.0	0.3±0.0	0.3±0.0	0.2±0.0
Table 2: Results with a Random Forest classifier on MNIST dataset. Note that methods that update model parameters are not applicable here. FLH-FTL outperforms existing alternatives for both accuracy and label marginal estimation.
CT (base)	CT-RS (ours) w FTH	CT-RS (ours) w FLH-FTL	w-ERM (oracle)
CI Err	20.0±0.5	18.38±0.4	17.12±0.8	16.32±0.7
MSE	NA	0.18±0.01	0.12±0.01	NA
Table 3: Results on SOLS setup on CIFAR10 SOLS with Bernoulli shift. CT with RS improves over the base model (CT) and achieves competitive performance with respect to weighted ERM oracle. MNIST results are similar (see App. F).
For all datasets except SHL, we simulate online label shifts with four types of shifts studied in Bai et al. [8]: monotone shift, square shift, sinusoidal shift, and Bernoulli shift. For SHL locomotion, we use the real-world shift occurring over time. For architectures, we use an MLP for Fashion, SHL and MNIST, Resnets [39] for EuroSAT, CINIC, and CIFAR, and DistilBERT [62, 75] based models for arXiv. For alternate approaches, along with a base classifier (which does no adaptation) and oracle classifier (which reweight using the true label margins), we make comparisons with adaptation algorithms proposed in prior works [76, 8]. In particular, we compare with ROGD, FTH, FTFWH from Wu et al. [76] and UOGD, ATLAS from Bai et al. [8]. For brevity, we refer to our method as FLH-FTL (though strictly speaking, our methods are based on FLH from Hazan and Seshadhri [38] with online averages as base learners). We run all the online label shift experiments with the time horizon  $T = 1000$  and at each step 10 samples are revealed. We repeat all experiments with 3 seeds to obtain means and standard deviations of the results. For other methods that perform reweighting correction on softmax predictions, we use the labeled holdout data to calibrate the model with temperature scaling, which tunes one temperature parameter [35]. We provide exact details about the datasets, label shift simulations, models, and prior methods in App. F.
Results Overall, across all datasets, we observe that our method FLH-FTL performs better than alternative approaches in terms of both classification error and mean squared error for estimating the label marginal. Note that methods that directly update the model parameters (i.e., UOGD, ATLAS) do not provide any estimate of the label marginal (Table 1). UOGD and ATLAS also require offline holdout labeled data (i.e., from time step 0) to make online updates to the model parameters. For this purpose, we use the same labeled data that we use to compute the confusion matrix.
As we increase the holdout offline labeled dataset size for updating the model parameters (and to compute the confusion matrix), we observe that classification error and MSE with FLH-FTL stay (relatively) constant whereas the classification errors of other alternatives improve (Fig. 2). This highlights that FLH-FTL can be much more sample efficient with respect to the size of the hold-out offline labeled data. Motivated by this observation, we perform an additional experiment in which we increase the offline training data and observe that we can overall improve the classification accuracy significantly with FLH-FTL (Fig. 2). We present results on SHL dataset with similar findings on semi-synthetic datasets in App. G.5. Finally, we also experiment with a random forest model on the MNIST dataset. Note methods that update model parameters (e.g., UOGD and ATLAS) with OGD are not applicable here. Here, we also observe that we improve over existing applicable alternatives (Table 2).
5.2 SOLS setup and results
Setup For the supervised problem, we experiment with MNIST and CIFAR datasets. We simulate a time horizon of  $T = 200$ . For each dataset, at each step, we observe 50 samples with Bernoulli shift. Motivated by our theoretical results with weighted ERM, we propose a simple baseline which continually trains the model at every step instead of starting ERM from scratch every time. We maintain a pool of all the labeled data received till that time step, and at every step, we randomly sample a batch with uniform label marginal to update the model. Finally, we re-weight the updated softmax outputs with estimated label marginal. We call this method Continual Training via Re-Sampling (CT-RS). Its relation as a close variant of weighted ERM is elaborated in App. F.1. To estimate the label marginal, we try FTH and ours FLH-FTL.
Results On both datasets, we observe that empirical performance with CT-RS improves over the naive continual training baseline. Additionally, CT-RS results are competitive with weighted ERM
while being  $5 - 15\times$  faster in terms of computation cost (we include the exact computational cost in App. F.1). Moreover, as in UOLS setup, we observe that FLH-FTL improves over FTH for both target label marginal estimation and classification.
In this work, we focused on unsupervised and supervised online label shift settings. For both settings, we developed algorithms with minimax optimal dynamic regret. Experimental results on both real and semi-synthetic datasets substantiate that our methods improve over prior works both in terms of accuracy and target label marginal estimation.
In future work, we aim to expand our experiments to more real-world label shift datasets. Our work also motivates future work in exploiting other causal structures (e.g. covariate shift) for online distribution shift problems.
SG acknowledges Amazon Graduate Fellowship and JP Morgan AI Ph.D. Fellowship for their support. ZL acknowledges Amazon AI, Salesforce Research, Facebook, UPMC, Abridge, the PwC Center, the Block Center, the Center for Machine Learning and Health, and the CMU Software Engineering Institute (SEI) via Department of Defense contract FA8702-15-D-0002, for their generous support of ACMI Lab's research on machine learning under distribution shift. DB and YW were partially supported by NSF Award #2007117 and a gift from Google.
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