Nong Minh Hieu

Nong Minh Hieu

We establish the finite-sample concentration rate for the Median-of-Incomplete-U-Statistics (MIU), an efficient robust estimator for the expectation of symmetric kernels.

Nong Minh Hieu, Antoine Ledent

Contrastive Representation Learning (CRL) has achieved strong empirical success in multiple machine learning disciplines, yet its theoretical sample complexity remains poorly understood. Existing analyses usually assume that input tuples are identically and independently distributed, an assumption violated in most practical settings where contrastive tuples are constructed from a finite pool of labeled data, inducing dependencies among tuples. While one recent work analyzed this learning setting using U-Statistics to estimate the population risk, the techniques used therein require the risk of each class to concentrate uniformly, making excess risk bounds scale in the order of $ρ_{\min}^{-{1}/{2}}$ where $ρ_{\min}$ denotes the probability of the rarest class. Such a dependency can be overly pessimistic in the extreme multiclass settings where there are many tail classes which contribute minimally to the overall population risk. Our contributions are two-fold. Firstly, we improve upon the previous work and prove a bound with a sample complexity of the same order as the number of classes $R$, regardless of the distribution over classes. Furthermore, we formulate a different estimator that captures the concentration of the risk \textit{across classes}, enabling sharper bounds in extreme multi-class learning scenarios, especially where class distributions are long-tailed. Under mild assumptions on the class distributions, the resulting sample complexity is $\mathcal{O}(k)$ where $k$ is the number of samples per tuple.

Nong Minh Hieu, Antoine Ledent

Contrastive Representation Learning (CRL) has achieved impressive success in various domains in recent years. Nevertheless, the theoretical understanding of the generalization behavior of CRL has remained limited. Moreover, to the best of our knowledge, the current literature only analyzes generalization bounds under the assumption that the data tuples used for contrastive learning are independently and identically distributed. However, in practice, we are often limited to a fixed pool of reusable labeled data points, making it inevitable to recycle data across tuples to create sufficiently large datasets. Therefore, the tuple-wise independence condition imposed by previous works is invalidated. In this paper, we provide a generalization analysis for the CRL framework under non-$i.i.d.$ settings that adheres to practice more realistically. Drawing inspiration from the literature on U-statistics, we derive generalization bounds which indicate that the required number of samples in each class scales as the logarithm of the covering number of the class of learnable feature representations associated to that class. Next, we apply our main results to derive excess risk bounds for common function classes such as linear maps and neural networks.

Nong Minh Hieu, Antoine Ledent, Yunwen Lei et al.

In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles. On the one hand, we derive a parameter-counting bound that scales with the overall size of the neural networks. On the other hand, we provide a norm-based bound that scales with the norms of neural networks' weight matrices. Ignoring logarithmic factors, the bounds are independent of $k$, the size of the tuples provided for contrastive learning. To the best of our knowledge, this property is only shared by one other work, which employed a different proof strategy and suffers from very strong exponential dependence on the depth of the network which is due to a use of the peeling technique. Our results circumvent this by leveraging powerful results on covering numbers with respect to uniform norms over samples. In addition, we utilize loss augmentation techniques to further reduce the dependency on matrix norms and the implicit dependence on network depth. In fact, our techniques allow us to produce many bounds for the contrastive learning setting with similar architectural dependencies as in the study of the sample complexity of ordinary loss functions, thereby bridging the gap between the learning theories of contrastive learning and DNNs.

Antoine Ledent, Mun Chong Soo, Nong Minh Hieu

We study a matrix completion problem where both the ground truth $R$ matrix and the unknown sampling distribution $P$ over observed entries are low-rank matrices, and \textit{share a common subspace}. We assume that a large amount $M$ of \textit{unlabeled} data drawn from the sampling distribution $P$ is available, together with a small amount $N$ of labeled data drawn from the same distribution and noisy estimates of the corresponding ground truth entries. This setting is inspired by recommender systems scenarios where the unlabeled data corresponds to `implicit feedback' (consisting in interactions such as purchase, click, etc. ) and the labeled data corresponds to the `explicit feedback', consisting of interactions where the user has given an explicit rating to the item. Leveraging powerful results from the theory of low-rank subspace recovery, together with classic generalization bounds for matrix completion models, we show error bounds consisting of a sum of two error terms scaling as $\widetilde{O}\left(\sqrt{\frac{nd}{M}}\right)$ and $\widetilde{O}\left(\sqrt{\frac{dr}{N}}\right)$ respectively, where $d$ is the rank of $P$ and $r$ is the rank of $M$. In synthetic experiments, we confirm that the true generalization error naturally splits into independent error terms corresponding to the estimations of $P$ and and the ground truth matrix $\ground$ respectively. In real-life experiments on Douban and MovieLens with most explicit ratings removed, we demonstrate that the method can outperform baselines relying only on the explicit ratings, demonstrating that our assumptions provide a valid toy theoretical setting to study the interaction between explicit and implicit feedbacks in recommender systems.