SlowGuess/NAACL_2022_OCR huggingface dataset:
To mitigate label noise, we use the approach introduced by Ma et al. (2020), where they propose to add noise robustness to any given loss function by normalizing it across all labels. Ma et al. (2020) further improve convergence by presenting a combination loss function with active and passive loss components, to maximize the posterior for the true class and to minimize the posterior for complementary classes, respectively. In our experiments, we use a combination of Normalized Cross Entropy (NCE) Ma et al. (2020) and Reverse Cross Entropy (RCE) Wang et al. (2019) as the active and passive components, weighted in a ratio 1:2 selected using our validation set.
$$
l o s s _ {r o b u s t} = N C E + 2 * R C E \tag {8}
$$
The NCE and RCE functions are listed below.
$$
N C E = \frac {- \sum_ {k = 1} ^ {K} q (k | x) \log p (k | x)}{- \sum_ {j = 1} ^ {K} \sum_ {k = 1} ^ {K} q (y = j | x) \log p (k | x)} \tag{9}
$$
$$
R C E = - \sum_ {k = 1} ^ {K} p (k | x) \log q (k | x)) \tag {10}
$$
where  $K$  is number of label classes,  $q(k|x)$  denotes the gold label distribution and  $p(k|x)$  denotes the posterior probability distribution.
We use the publicly-provided train and test splits for the sst2 and 20news datasets and further derive a validation split consisting of  $20\%$  ( $v = 0.2$ ) of the train split  $(D_{t})$ , with uniform class distribution. We use another  $1\%$  ( $k = 0.01$ ) of  $D_{t}$  as seed model set,  $D_{s}$ . We choose a small value for  $k$  to mimic a real world scenario where a larger volume of data may be un-annotated in an FL setup. The remaining unlabeled dataset  $D_{u}$  ( $79\%$  of the  $(D_{t})$  is further divided among 15 mutually exclusive subsets  $(D_{u}^{n}, n \in [1, 15])$ , which simulates the data for 15 edge clients. While creating the clients' partitions we ensure that all clients have data with a uniform class distribution which enables us to focus on our model performance in an idealized case. We use the DistilBERT Sanh et al. (2019) base model as the classifier for our tasks and follow the standard fine-tuning setup for text classification. To implement the federated learning pipeline we use the publicly-available FedNLP Lin et al. (2021) benchmark and apply the FedAvg McMahan et al. (2017)
Experimental settings	20 news	sst2
Initial model (Ds)	59.14	77.37
Self training (no feedback)	60.79	77.26
Positive feedback (noisy)	62.10	79.79
All feedback (noisy)	65.01	85.17
Positive feedback (noise robust)	62.33	79.85
All feedback (noise robust)	65.13	85.39
Positive feedback (noise free)	70.44	83.80
All feedback (noise free)	75.13	88.58
Full supervision	82.12	89.12
Table 1: Accuracy of noise robust federated self training with user feedback against various baselines for 20news and sst2 datasets;  $*$ : all models using feedback (with and without noise robustness) are statistically significant against the self training baseline (without feedback), at  $p < 0.05$ .
algorithm to aggregate the client model updates at the server end. We train the model on an 8-GPU (Nvidia V100s) machine for up to 50 rounds with early stopping enabled. Within each round, we use a batch size of 8 to train the client models for 5 epochs each.
5.2 Model Evaluation
We compare our model performance against three baselines:
Initial model This is the seed model  $f_{\mathrm{s}}(x)$  trained on just  $D_{s}$  (1% of the training data).
Self-training We train this model using federated learning with pseudo labels, but do not utilize the user feedback. Hence, at each client, we only have the raw pseudo labels  $\rho_{i}$  for each  $x_{i} \in D_{u}^{n}$  to train on. We use this setup as the primary baseline against which to compare the performance of our models trained with user feedback.
Full supervision We train a model in a federated setting using  $D_{u}^{n}$  and the true gold labels at each client. Although in a real-world setting, the clients will not have access to the gold labels, we establish this benchmark to set an upper bound.
We evaluate performance of our proposed strategy of leveraging user feedback in two settings:
Positive feedback At each client, we train the local version of the model using only the samples in  $D_{pos}^{n}$  and corresponding pseudo labels  $\rho_{i}$ , i.e. only the samples which receive positive feedback. Since this baseline is trained using regular cross entropy, it provides a powerful yet computationally less-intensive alternative to training with both types of feedback, which is especially important in edge
devices with low compute power.
All feedback We utilize all the data samples in  $D_{pos}^{n}$  and  $D_{neg}^{n}$  at each client and train using the loss function described in Section 4.3.
In both the positive and all feedback setups, we evaluate models with and without user feedback noise. For the noise-free scenario, we set  $\gamma = 1$  and  $\delta = 1$  while simulating the user feedback. This leads to perfectly accurate feedback, as discussed in §4.2. For the noisy feedback scenario, we use the noise parameters derived from the Mturk study. We obtain the following dataset-specific values of  $\gamma$  and  $\delta$  by averaging the estimates of  $\gamma$  and  $\delta$  across all annotators:  $(\gamma = 0.79, \delta = 0.55)$  for 20news and  $(\gamma = 0.76, \delta = 0.70)$  for sst2.
Table 1 reports the  $\%$  accuracy for each of the experimental setups described above across both datasets. We observe that in both the noisy and noise-free settings, the introduction of positive user feedback shows a marked improvement in performance when compared to the self-training baseline. There is an additional performance gain when we add negative feedback (all feedback baseline), which signifies the importance of learning from complementary labels. As expected, the improvement is substantially larger in the noise free setting, suggesting the need for model robustness to mitigate the effect of noise. Note that for  $sst2$ , performance of the noise free model with all feedback is very close to that of full supervision, thanks to the fact that complementary labels in the case of binary classification provide same information as true labels. On the other hand, using perfect positive and negative feedback in 20news is still sub-optimal compared to full supervision, since a negative label in this dataset is less informative compared to  $sst2$ .
5.3 Noise Robustness
To mitigate the effects of noise, we replace the traditional cross-entropy loss function with the active-passive loss described in §4.4, using the same experimental setups presented earlier (positive only and all-feedback), with  $\gamma$  and  $\delta$  values from the Mturk study. However, as evident in Table 1, the robust loss functions only seem to confer marginal performance improvements in both datasets. This is likely due to the fact that the noise parameters extracted from Mturk belong to a moderate to low noise regime (Section 3.2), providing limited room for gains with noise robustness.
To further investigate this, we explore two ex
Noise level	Loss	Accuracy
Low	lossrobust	73.29
lossreg	74.30
Adversarial	lossrobust	42.26*
lossreg	25.19
treme cases of user feedback noise for the 20news dataset: i) low noise, where we simulate user feedback with  $\gamma \rightarrow 1$ ,  $\delta \rightarrow 1$  for all the clients, which imitates clients providing correct feedback with very high probability, and ii) adversarial noise, with  $\gamma \rightarrow 0$ ,  $\delta \rightarrow 0$  for all the clients, which captures the possible risk of users deliberately providing incorrect feedback with high probability. In Table 2, we compare the performances of the all feedback model trained with and without noise robustness in these two scenarios. As seen in the table, when user noise is high, the noise-robust loss functions show a statistically significant improvement against the noisy model, highlighting the value of adding noise robustness. In the low noise regime, adding robustness seems to cause negligible degradation in accuracy, but within the bounds of statistical error. Given this, we recommend using noise robustness in all applications of this framework unless it is well known before hand that the feedback has very low noise. We defer the task of developing a noise robustness regime that works for all noise levels to future work.
5.4.1 Varying Degrees of Noise
As discussed in §4.4, the level of feedback noise has a substantial impact on model performance. In this section, we further investigate this effect, simulating user feedback across various noise parameters values, spanning  $\gamma, \delta \in \{0.3, 0.5, 0.7\}$ , to capture different points in the  $\gamma - \delta$  space. Table 3 shows our results for each dataset with the noise robust loss function 8. As expected, as  $\gamma \to 0$  and/or  $\delta \to 0$ , model performance decreases on both datasets. At very low values of  $\delta$  and  $\gamma$ , e.g. both  $\leq 0.5$ , training on the extremely noisy user feedback actually decreases model performance below the original seed model. This is not unexpected, since at  $\delta = 0.5$  and  $\gamma = 0.5$ , user feedback is essentially random noise, and at lower values the
Table 2: Performance analysis of noise robust loss functions trained on all feedback in different noise regimes for the 20news dataset;  $*$ : statistically significant against the adversarial model without robustness at  $p < 0.05$ .
γ/δ	0.7	0.5	0.3
0.7	66.69	63.18	60.66
0.5	65.56	59.15	59.73
0.3	60.01	58.94	58.21
(a) 20news dataset; initial model performance: 59.14, performance with all feedback and no noise  $(\gamma = \delta = 1)$ : 75.13.
γ/δ	0.7	0.5	0.3
0.7	83.86	80.89	76.17
0.5	81.99	77.38	75.07
0.3	78.03	74.41	71.99
(b) sst2 dataset; initial model performance: 77.37, performance with all feedback and no noise  $(\gamma = \delta = 1)$ : 88.58.
Table 3: Model performance (accuracy) at varying values of  $\gamma$  and  $\delta$
feedback is adversarial. These results highlight the importance of evaluating the reliability of user feedback before using it to further train an ML system.
5.4.2 Non-identical User Feedback Behavior
In previous sections, we used identical values of the noise parameters  $\gamma$  and  $\delta$  for all clients in the FL training setup. However, as observed in our Mturk study, real users have different feedback behaviors, with scores spread out over the  $\gamma -\delta$  space. In this section, we simulate non-identical user feedback for four potential user behaviors:
1. Low noise users  $(\gamma \rightarrow 1, \delta \rightarrow 1)$
2. Adversarial/high noise users  $(\gamma \rightarrow 0, \delta \rightarrow 0)$
3. Positive users  $(\gamma \rightarrow 1, \delta \rightarrow 0)$  who provide consistently positive feedback, regardless of the correctness of the model prediction; and
4. Negative users  $(\gamma \rightarrow 0, \delta \rightarrow 1)$  who provide consistently negative feedback.
To simulate non-identical user feedback for the clients, we sample the noise parameters from an appropriately skewed  $\beta (a,b)$  distribution. For example, in order to generate  $\delta$  and  $\gamma$  scores for setup four (negative users), each user needs  $\gamma \approx 0$ ,  $\delta \approx 1$ . To generate these parameters, we sample  $\gamma$  from  $\beta (1,10)$  and  $\delta$  from  $\beta (10,1)$ . Proceeding this way, we can simulate all four user behaviors listed above. Finally, we also conduct an experiment closer to the real-world scenario, where we randomly sample 15 workers each for both datasets from our Mturk study and use their estimated values of  $\gamma$  and  $\delta$  to simulate user feedback.
User Behavior	20news	sst2
Low noise	73.67	88.35
Adversarial	55.86	64.85
Always positive	60.99	77.16
Always negative	58.92	74.13
Real world (mturk study)	65.37	85.61
Table 4: Model performance at various user behaviors.
Table 4 shows our results across all five simulations for both datasets when trained with the noise robust loss function 8. As expected, the best model performance is achieved with the low-noise users, followed by the real-world users sampled from our MTurk study. In the three other simulations (adversarial, consistently positive, consistently negative), user feedback is highly noisy and unreliable, and the models show limited improvement over the initial seed model. Note that the performance in the positive feedback scenario is higher than negative feedback, which can be accredited to the fact that the initial seed model's accuracy is greater  $50\%$  for both datasets (Table 1). With  $>50\%$  accuracy, a majority of the pseudo-labels generated using the seed model will match the gold label. Hence, consistently positive feedback introduces less noise and in turn better performance compared to the all negative feedback model.
In this work, we propose a novel framework for federated learning which leverages noisy user feedback on the edge. Our framework eliminates the need for labeling edge data, in turn improving customer privacy since we no longer need to move data off of edge devices for annotation. In order to evaluate our framework, we propose a method to simulate user feedback on publicly-available datasets and a parametric model to simulate user noise in that feedback. Using that method, we conduct experiments on two benchmark text classification datasets and show that models trained with our framework significantly improve over self-training baselines.
Future work includes deploying our framework in a real world FL application and incorporating live user feedback in model training. We can also explore other noise-robustness strategies for low and medium label-noise scenarios. One such strategy would be to incorporate a measure of user reliability into the calculation of each new global model
in FedAVG - e.g. the updated global model parameters could be computed as a weighted average of client models, with the weight capturing some measure of each client's reliability.
7 Ethics Statement
Our Mturk data collection recruited annotators from across the globe without any constraints on user demographics. The annotators were compensated with above minimum wages and no personal information was collected from the annotators.
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Algorithm 1 lists our training loop.
B Estimators for  $\gamma$  and  $\delta$
Let  $X$  be the data set and  $n$  be the total number of data points. For any data point  $i \in [n]$ , let  $p_i$ ,  $t_i$  and  $f_i$  denote the model predicted label, ground truth label, and user feedback respectively. (Note that  $p_i$  and  $t_i$  take values from the set of labels and  $f_i$  takes values from the set {pos, neg, idx} representing feedbacks positive, negative, and 'I don't know').) By definition, we have
$$
\gamma := \Pr (f _ {i} = p o s \mid p _ {i} = t _ {i})
$$
$$
\delta := \Pr (f _ {i} = n e g \mid p _ {i} \neq t _ {i})
$$
Let us also define
$$
\alpha := \Pr (f _ {i} = n e g \mid p _ {i} = t _ {i})
$$
$$
\beta := \Pr (f _ {i} = p o s \mid p _ {i} \neq t _ {i})
$$
Note that the above definitions imply that
$$
1 - \alpha - \gamma = \Pr (f _ {i} = i d k \mid p _ {i} = t _ {i})
$$
$$
1 - \beta - \delta = \Pr (f _ {i} = i d k \mid p _ {i} \neq t _ {i})
$$
Moreover, let  $a$  denote the accuracy of the labels predicted by the model defined as
$$
a := \Pr (p _ {i} = t _ {i})
$$
Define sets  $\{S_j\}_{j\in [6]}$  such that
$$
S _ {1} := \{i \in [ n ]: f _ {i} = p o s a n d p _ {i} = t _ {i} \}
$$
$$
S _ {2} := \{i \in [ n ]: f _ {i} = p o s a n d p _ {i} \neq t _ {i} \}
$$
$$
S _ {3} := \{i \in [ n ]: f _ {i} = \text {n e g a n d} p _ {i} = t _ {i} \}
$$
$$
S _ {4} := \left\{i \in [ n ]: f _ {i} = \text {n e g a n d} p _ {i} \neq t _ {i} \right\}
$$
$$
S _ {5} := \left\{i \in [ n ]: f _ {i} = i d k \text {a n d} p _ {i} = t _ {i} \right\}
$$
$$
S _ {6} := \{i \in [ n ]: f _ {i} = i d k \text {a n d} p _ {i} \neq t _ {i} \}
$$
define  $n_j \coloneqq |S_j|$ . Note that  $\sum_{j \in [6]} n_j = n$ .
Theorem 1. The maximum likelihood estimators for  $\gamma$  and  $\delta$  are  $n_1 / (n_1 + n_3 + n_5)$  and  $n_4 / (n_2 + n_4 + n_6)$  respectively.
Proof. Now for any data point  $i \in [n]$ , we have
$$
\begin{array}{l} \Pr (i \in S _ {1}) = \Pr (f _ {i} = p o s a n d p _ {i} = t _ {i}) \\ = \Pr (f _ {i} = p o s \mid p _ {i} = t _ {i}) \cdot \Pr (p _ {i} = t _ {i}) \\ = \gamma a. \\ \end{array}
$$
By a similar reasoning, we have
$$
\Pr (i \in S _ {2}) = \beta (1 - a)
$$
$$
\Pr (i \in S _ {3}) = \alpha a
$$
$$
\Pr (i \in S _ {4}) = \delta (1 - a)
$$
$$
\Pr (i \in S _ {5}) = (1 - \alpha - \gamma) a
$$
$$
\Pr (i \in S _ {6}) = (1 - \beta - \delta) (1 - a)
$$
Therefore the likelihood function of the model is
$$
\mathcal {L} (\alpha , \beta , \gamma , \delta \mid X) = \frac {n !}{n _ {1} ! \dots n _ {6} !} (\gamma a) ^ {n _ {1}}
$$
$$
(\beta (1 - a)) ^ {n _ {2}} (\alpha a) ^ {n _ {3}}
$$
$$
\left(\delta (1 - a)\right) ^ {n _ {4}} \left((1 - \alpha - \gamma) a\right) ^ {n _ {5}}
$$
$$
\left((1 - \beta - \delta) (1 - a)\right) ^ {n _ {6}}
$$
and consequently the log-likelihood function is
$$
\log \mathcal {L} (\alpha , \beta , \gamma , \delta \mid X) = \log \left(\frac {n !}{n _ {1} ! \dots n _ {6} !}\right) +
$$
$$
n _ {1} \log (\gamma a) + n _ {2} \log (\beta (1 - a)) +
$$
$$
n _ {3} \log (\alpha a) + n _ {4} \log (\delta (1 - a)) +
$$
$$
n _ {5} \log ((1 - \alpha - \gamma) a) +
$$
$$
n _ {6} \log ((1 - \beta - \delta) (1 - a)) \tag {11}
$$
To obtain MLE estimates of parameters  $\alpha, \beta, \gamma, \delta$ , we wish to solve the following optimization problem
$$
\max  _ {(\alpha , \beta , \gamma , \delta) \in [ 0, 1 ] ^ {4}} \log \mathcal {L} (\alpha , \beta , \gamma , \delta \mid X) \tag {12}
$$
By Fermat's theorem, the optimal solution to the above optimization problem lies at either a boundary point or a stationary point.
The boundary points of the set  $[0,1]^4$  are given by the set
$$
B := \left\{\left(\alpha , \beta , \gamma , \delta\right) \in [ 0, 1 ] ^ {4}: \alpha = 0 \text {o r} \alpha = 1 \text {o r} \right.
$$
$$
\beta = 0 \text {o r} \beta = 1 \text {o r}
$$
$$
\gamma = 0 \text {o r} \gamma = 1 \text {o r}
$$
$$
\delta = 0 \text {o r} \delta = 1 \}
$$
The value of the function  $\log \mathcal{L}(\alpha, \beta, \gamma, \delta)$  is negatively unbounded on the set  $B$ .
On the other hand, the stationary points can be determined by setting the gradient to be zero, i.e., by solving the equation
$$
\nabla_ {(\alpha , \beta , \gamma , \delta)} \log \mathcal {L} (\alpha , \beta , \gamma , \delta \mid X) = 0.
$$
Solving the above equation yields the stationary point  $(\alpha^{*},\beta^{*},\gamma^{*},\delta^{*})$  given as