ML Interview Q Series: What issues arise when the cost function must handle partially observed labels (e.g., partial annotation or weak supervision), and how could you address them?
📚 Browse the full ML Interview series here.
Hint: Consider estimating missing labels or using expected values in the loss.
Comprehensive Explanation
When labels are missing or only partially observed, the most immediate challenge is that the training objective cannot be straightforwardly computed in standard supervised learning fashion. A typical loss function expects fully known labels for each training instance, but under partial annotation or weak supervision, we might only have access to partial or incomplete signals about the true target.
This partial availability introduces several interconnected problems. First, models can become biased if they only use the small subset of examples that have labels. Second, ignoring unlabeled parts of the data can reduce effective sample size, leading to underfitting and poor generalization. Third, models may overfit to noisy or partially incorrect labels if the weakly supervised signals are themselves unreliable. Finally, the very definition of a "loss" becomes trickier, as we need a procedure for weighting or imputing labels that are unknown or only weakly specified.
Addressing these issues often involves a method to either (1) infer or estimate missing labels, or (2) treat the missing labels as latent variables and incorporate an expectation-based approach in the loss. One popular approach is to rely on an expectation of the loss over the unknown label distribution. Below is a conceptual expression for an expected loss in the presence of missing labels, which can be central to solutions that integrate partial supervision:
Here, f_{theta}(x_i) is the model’s prediction function parameterized by theta in plain text notation. The outer summation runs over the N training instances. For each instance, we take an expectation over the unknown or partially observed labels. In practice, this expectation can be approximated through various techniques:
In some scenarios, you might have partially observed labels in the form of constraints (for example, we might know a label is from a certain set of possible classes, but not which one exactly). In others, you might have multiple noisy labeling functions, all of which can be combined to give a probabilistic estimate of the label. By modeling this partial knowledge as a distribution over possible labels, you then compute the expected loss for each sample.
Strategies include iterative Expectation-Maximization (EM) procedures, pseudo-labeling, or introducing latent variables that represent the unknown labels, and marginalizing them out in the objective function. EM-like methods repeatedly estimate the most probable labels (E-step) given current model parameters, then optimize the parameters to best fit these labels (M-step). Alternatively, weak supervision frameworks, where multiple noisy or partial label sources are combined, seek to estimate the underlying true label distributions and produce label estimates that can be utilized in a standard training pipeline.
From a practical perspective, it is also important to handle the potential biases or errors introduced by these estimation processes. It can help to have validation schemes that can detect if the model is overfitting to mislabeled data, or if certain subsets of partially observed labels degrade performance. Furthermore, domain knowledge can guide how to incorporate constraints or partial information. For instance, if you know that a label must come from a fixed subset of classes, that knowledge can drastically reduce the uncertainty in the missing labels and improve the stability of training.
Potential Follow-up Questions
How can we design a custom cost function to incorporate partial labels in a deep learning framework?
One way is to augment the standard loss function so that it can be computed even when some labels are missing. Instead of a direct label comparison, you can encode the partially known label as a probability distribution across possible classes. Then the cost function involves computing the expectation under that distribution. In a framework like PyTorch or TensorFlow, you might define a custom forward pass that checks if a label is fully known. If not, you compute the expected loss over the set of possible labels. This can involve techniques such as masking or weighting certain terms in the loss so that missing labels are not penalized in the same way as fully observed labels.
Another practical step is to ensure you do not silently ignore examples with missing labels, because that might reduce your effective dataset size. Instead, carefully incorporate them via a partial supervision strategy that can still backpropagate meaningful gradients. This could be done through a mixture of standard cross-entropy for known labels and a special partial-label loss term for unknown labels.
How does an Expectation-Maximization (EM) algorithm help with partial annotation?
In partial annotation scenarios, EM is often used when you can treat the unknown labels as latent variables. During the E-step, you use the current model parameters to infer or “soft-assign” a distribution over these unknown labels. This might be done by running a forward pass with the model to estimate probabilities for each possible class. Then in the M-step, you treat these inferred labels (or distributions) as if they were known, and perform a parameter update that tries to minimize the loss with respect to these newly assigned labels. You alternate between these two steps until convergence.
A major benefit is that EM systematically integrates the uncertainty in missing labels. However, EM can converge to local minima, so initialization and careful hyperparameter tuning are often key. You also need to watch out for overfitting, because if the E-step assigns incorrect labels consistently, the M-step might reinforce that incorrect labeling, leading to a suboptimal solution.
Are there pitfalls or constraints when applying these methods to large-scale data with partial labeling?
One challenge in large-scale settings is the computational overhead. If you are performing an EM-like approach or a complex labeling function that needs iterative refinement, the cost of repeated passes over a massive dataset can be high. Efficient implementations and parallelization strategies become vital.
In addition, partial labels can be extremely noisy. You might have scenarios in which weak supervision signals are contradictory. Careful design of your data pipeline (e.g., weighting certain label sources more than others) can reduce the impact of noise. You also want to maintain a robust validation approach, because you need a way to reliably evaluate whether your approach to handling partial labels is actually improving the model rather than introducing systematic biases.
How do you evaluate performance when the training data are partially labeled?
It is usually necessary to have a cleanly labeled hold-out set, even if it is smaller, so that you can evaluate the final model performance without the confounding effect of missing labels. This fully annotated test or validation set becomes your gold standard for tuning hyperparameters and checking for overfitting. If no fully labeled subset is available, cross-validation techniques become more difficult to set up. In such extreme cases, you might rely on indirect measures of performance (like training stability, agreement across multiple noisy sources, or external tasks that use the trained model’s features).
Another possibility is active learning, where you iteratively ask for ground truth labels on a carefully chosen subset of the unlabeled data. This can help refine your model’s estimates of missing labels and give you a small but accurate validation set for monitoring performance.
Below are additional follow-up questions
What happens if the missing or partial labels are systematically biased toward certain classes?
Systematic bias in missing or partial labels can distort model training. If specific classes or categories are more likely to be missing or inaccurately labeled, the model may become overconfident in the classes that are more frequently observed. In real-world scenarios, this can arise from sampling methods, data collection protocols, or labeling processes that inadvertently emphasize certain outcomes. When a class is chronically underrepresented or missing in your partially labeled dataset, the model may have insufficient exposure and fail to generalize.
A deeper issue appears when this bias manifests differently across subpopulations. For instance, if certain demographic groups are more frequently missing labels, your model’s performance may degrade or become discriminatory in that subgroup. Addressing this requires careful data analysis or building domain knowledge to detect skew in label distribution. Techniques such as reweighting specific classes, collecting additional labels for those underrepresented portions, or incorporating domain constraints can mitigate these imbalances.
How do you handle the possibility that weakly supervised signals themselves could have different degrees of reliability?
When using multiple weak or partial supervision sources, each signal could have a distinct reliability or noise level. If you treat all sources equally, the model might overweight highly noisy signals. Conversely, ignoring some weaker signals may discard potentially valuable information. A real-world example is when you have programmatically generated labels that differ in accuracy depending on the data domain.
To address this, you can explicitly model the reliability of each source. One approach is to introduce a learnable parameter or confidence score for each supervision channel. During training, if certain signals consistently match high-confidence predictions, you could dynamically increase their contribution. But if a source is found to be highly contradictory, you can decrease its influence. In practice, you might have an extra “calibration” step that looks at how each source correlates with a small set of ground-truth labels, thereby estimating how trustworthy the source is. Potential pitfalls include overfitting these reliability estimates and ignoring rare but informative signals. Validation and cross-checking are crucial to ensure that reliability modeling is stable.
Can transfer learning methods amplify errors from partially labeled data?
Transfer learning typically involves pretraining a model on a large corpus of data, often with more complete labels or more abundant self-supervisory signals, and then fine-tuning on a target dataset that may contain partial labels. If the partial labels in the target domain are heavily biased or noisy, the fine-tuning phase can degrade or “catastrophically forget” the robust features learned during pretraining. This degradation is especially pronounced when the target task is quite different from the pretraining task.
In addition, if the partial labels do not cover the entire label space effectively, the pretrained model may overfit the subset of labels it sees, ignoring other relevant features. The best practice is to carefully select the layers or parameters you want to fine-tune so you do not over-adapt the model to untrustworthy labels. Monitoring training metrics over time can help detect if the model starts diverging from the robust representations gained during pretraining. Another strategy is to freeze large portions of the pretrained network and only learn a small number of parameters on top, thus reducing your reliance on partially labeled data.
How does partial labeling impact confidence calibration in model outputs?
Confidence calibration measures how well predicted probabilities match actual likelihoods (i.e., if a model predicts class A with probability 0.8, it should be correct about 80% of the time). With partial labels, the model might learn incorrectly calibrated confidence estimates. For instance, if some labels are unknown or replaced with a distributional guess, the model can end up systematically underestimating or overestimating the confidence for certain classes.
A major risk arises when partial annotations lead to artificially inflated certainty in certain regions of the feature space. The model may have encountered examples marked with partial labels that strongly suggest a single class, even when, in reality, multiple classes are plausible. Such scenarios can arise if partial labels come from “strong but incomplete” heuristics. To handle this, you might apply post-hoc calibration techniques like temperature scaling or isotonic regression, specifically validating them on a small set of fully verified labels. Another strategy is to incorporate uncertainty in the loss function from the outset by encouraging more dispersed predictions when the label is uncertain. Yet, overdoing it can degrade performance on the examples that do have accurate labels.
How does partial annotation work in structured prediction tasks like sequence labeling or object detection?
In structured prediction tasks, the model often needs to predict multiple outputs simultaneously (e.g., the sequence of tokens in named entity recognition or bounding boxes in object detection). Partial labels in these contexts are more intricate because they could apply to only some parts of the structure. For instance, in sequence labeling, some token labels might be unknown or only weakly indicated, while others are known. Similarly, for object detection, some bounding boxes might be fully annotated, while others are missing or only roughly localized.
One practical approach is to split the structure into two sets of positions: positions with known labels and positions with uncertain labels. You then compute a standard loss on the known positions and an expectation-based or constraint-based loss on the uncertain positions. In object detection, you might treat some bounding boxes as “candidate proposals” generated by a region proposal network and partially supervise them with known bounding boxes or class annotations. A common pitfall is double counting or conflicting supervision if some partial labels overlap or contradict each other. Methods that carefully merge these constraints and handle overlaps—often via specialized matching algorithms—are critical.
In real-world production systems, what monitoring or maintenance challenges emerge from using models trained with partial labels?
Models trained with partial labels can drift in unexpected ways once deployed. Because partial label methods often rely on assumptions about label distributions or consistency, shifts in real data distribution can invalidate those assumptions. For example, if a weak supervision rule was dependent on certain textual cues, and user language changes over time, your model’s performance might degrade abruptly without an obvious cause. Monitoring becomes difficult because you might not have fully labeled data to track accuracy in real time.
A practical solution is to implement data and performance drift detection. You might track the distribution of model predictions or certain features and compare them to historical baselines. If these distributions shift significantly, you trigger re-training or data collection for a more up-to-date partial supervision signal. Another challenge is that partial labels might get “corrected” over time (e.g., new data is fully annotated). This requires a pipeline that periodically re-ingests labeled examples and retrains the model to reduce cumulative bias. However, frequent retraining can be computationally expensive, so you need a schedule or active learning routine to decide the best times to update the model.
How do you prevent the model from “cheating” by overfitting to partial signals that do not generalize?
When partial labels come from heuristic features or proxy variables, there is a risk that the model will memorize those proxies instead of truly understanding the underlying concept. For example, in a text classification scenario, partial labels might be based on specific keywords, and the model simply learns to identify these keywords without capturing the broader context. This leads to a type of overfitting where the model sees strong correlations that work for training but fail in real-world or test scenarios.
A key preventative measure is to hold out some data that does not follow the partial labeling scheme. If that is not possible, simulate or generate new examples that intentionally break the heuristic to see if the model still performs well. Regularization strategies like dropout or data augmentation can also help reduce overreliance on particular signals. Additionally, interpretability tools—like saliency maps or feature importance methods—can help detect whether the model is focusing solely on the partial signal. If you discover it is indeed “cheating,” you can refine your partial supervision to reduce that effect, for instance by adding multiple diverse labeling signals instead of a single heuristic.
How do you integrate partial labels from multiple stages of a pipeline when different stages have varying uncertainty?
In complex machine learning pipelines, partial labels might be introduced at different stages. For instance, an upstream classifier might provide a partial classification, then a downstream system refines or provides additional weak annotations. Each stage has its own uncertainty and might propagate errors downstream. Integrating these partial labels requires a carefully designed architecture that recognizes how errors compound.
One possibility is a hierarchical approach: use the first-stage outputs as features rather than ground-truth labels. Then, incorporate any subsequent partial labels as additional constraints or evidence. You can explicitly model the probability that each stage’s partial annotation is correct and weigh them accordingly. This can be managed via a Bayesian network or factor graph, where each node’s state depends on both the model’s learned parameters and the partial labels from the previous stage. However, building such a graph can be computationally heavy, especially for large-scale data, and requires domain expertise to define appropriate conditional dependencies. The complexity is further increased if the partial labels contradict each other. Continuous monitoring of each stage’s reliability can help mitigate these issues.
How do you handle tasks where the label space is very large but only a small fraction of possible labels are observed for each example?
When the label space is vast (e.g., large vocabulary classification or multi-label settings) and only a few relevant labels are partially known for each instance, the model can become confused about which labels are not observed due to irrelevance versus which labels are genuinely unknown. This is common in recommendation systems or extreme multi-label classification (e.g., tagging a piece of content with relevant tags out of thousands).
A typical remedy is to model the label assignment process. You might treat the known labels as positive signals and assume that all other labels are “latent” rather than outright negative. Then you can use techniques like negative sampling or importance weighting to limit the portion of the label space the model sees as potential negatives. Another strategy involves building hierarchical label structures so that partial supervision at higher levels of the hierarchy can inform lower levels. Real-world pitfalls occur when many labels are correlated, leading to confusion if only a small subset is annotated. Ensuring that popular or frequent labels are more thoroughly verified can help prevent the model from missing obvious tags or over-predicting rarely labeled tags.
How do you validate hyperparameters in the presence of partially observed labels?
Hyperparameter tuning often relies on validation sets where you measure performance (accuracy, F1, etc.) for various configurations. If labels in the validation set are also partially observed, typical metrics may not be reliable. You might inflate or deflate certain metrics if the labels are incomplete or estimated. This can lead to selecting suboptimal hyperparameters.
A practical approach is to isolate a truly fully labeled validation subset, even if it is smaller. This ensures you have a trusted reference for measuring actual performance. If that is not possible, you might rely on consistency-based checks (like internal consistency of predictions across different subsets of data) or use external tasks as a proxy for validation. Some teams set up a multi-stage validation where they refine partial labels in a small hold-out set, turning them into near-complete labels for parameter tuning. The main pitfall is that if your partial labels in the validation set are not representative, or if you only rely on incomplete signals, you risk choosing hyperparameters that overfit to the noise or partial nature of your labeling scheme.