Novi Quadrianto

Myles Bartlett, Sara Romiti, Viktoriia Sharmanska et al.

We propose Okapi, a simple, efficient, and general method for robust semi-supervised learning based on online statistical matching. Our method uses a nearest-neighbours-based matching procedure to generate cross-domain views for a consistency loss, while eliminating statistical outliers. In order to perform the online matching in a runtime- and memory-efficient way, we draw upon the self-supervised literature and combine a memory bank with a slow-moving momentum encoder. The consistency loss is applied within the feature space, rather than on the predictive distribution, making the method agnostic to both the modality and the task in question. We experiment on the WILDS 2.0 datasets Sagawa et al., which significantly expands the range of modalities, applications, and shifts available for studying and benchmarking real-world unsupervised adaptation. Contrary to Sagawa et al., we show that it is in fact possible to leverage additional unlabelled data to improve upon empirical risk minimisation (ERM) results with the right method. Our method outperforms the baseline methods in terms of out-of-distribution (OOD) generalisation on the iWildCam (a multi-class classification task) and PovertyMap (a regression task) image datasets as well as the CivilComments (a binary classification task) text dataset. Furthermore, from a qualitative perspective, we show the matches obtained from the learned encoder are strongly semantically related. Code for our paper is publicly available at https://github.com/wearepal/okapi/.

Sara Romiti, Christopher Inskip, Viktoriia Sharmanska et al.

Machine learning classifiers are typically trained to minimise the average error across a dataset. Unfortunately, in practice, this process often exploits spurious correlations caused by subgroup imbalance within the training data, resulting in high average performance but highly variable performance across subgroups. Recent work to address this problem proposes model patching with CAMEL. This previous approach uses generative adversarial networks to perform intra-class inter-subgroup data augmentations, requiring (a) the training of a number of computationally expensive models and (b) sufficient quality of model's synthetic outputs for the given domain. In this work, we propose RealPatch, a framework for simpler, faster, and more data-efficient data augmentation based on statistical matching. Our framework performs model patching by augmenting a dataset with real samples, mitigating the need to train generative models for the target task. We demonstrate the effectiveness of RealPatch on three benchmark datasets, CelebA, Waterbirds and a subset of iWildCam, showing improvements in worst-case subgroup performance and in subgroup performance gap in binary classification. Furthermore, we conduct experiments with the imSitu dataset with 211 classes, a setting where generative model-based patching such as CAMEL is impractical. We show that RealPatch can successfully eliminate dataset leakage while reducing model leakage and maintaining high utility. The code for RealPatch can be found at https://github.com/wearepal/RealPatch.

Gergely Dániel Németh, Miguel Ángel Lozano, Novi Quadrianto et al.

Federated learning (FL) has been proposed as a privacy-preserving approach in distributed machine learning. A federated learning architecture consists of a central server and a number of clients that have access to private, potentially sensitive data. Clients are able to keep their data in their local machines and only share their locally trained model's parameters with a central server that manages the collaborative learning process. FL has delivered promising results in real-life scenarios, such as healthcare, energy, and finance. However, when the number of participating clients is large, the overhead of managing the clients slows down the learning. Thus, client selection has been introduced as a strategy to limit the number of communicating parties at every step of the process. Since the early naïve random selection of clients, several client selection methods have been proposed in the literature. Unfortunately, given that this is an emergent field, there is a lack of a taxonomy of client selection methods, making it hard to compare approaches. In this paper, we propose a taxonomy of client selection in Federated Learning that enables us to shed light on current progress in the field and identify potential areas of future research in this promising area of machine learning.

Ainhize Barrainkua, Paula Gordaliza, Jose A. Lozano et al.

Human lives are increasingly being affected by the outcomes of automated decision-making systems and it is essential for the latter to be, not only accurate, but also fair. The literature of algorithmic fairness has grown considerably over the last decade, where most of the approaches are evaluated under the strong assumption that the train and test samples are independently and identically drawn from the same underlying distribution. However, in practice, dissimilarity between the training and deployment environments exists, which compromises the performance of the decision-making algorithm as well as its fairness guarantees in the deployment data. There is an emergent research line that studies how to preserve fairness guarantees when the data generating processes differ between the source (train) and target (test) domains, which is growing remarkably. With this survey, we aim to provide a wide and unifying overview on the topic. For such purpose, we propose a taxonomy of the existing approaches for fair classification under distribution shift, highlight benchmarking alternatives, point out the relation with other similar research fields and eventually, identify future venues of research.

Ainhize Barrainkua, Paula Gordaliza, Jose A. Lozano et al.

Several recent works encourage the use of a Bayesian framework when assessing performance and fairness metrics of a classification algorithm in a supervised setting. We propose the Uncertainty Matters (UM) framework that generalizes a Beta-Binomial approach to derive the posterior distribution of any criteria combination, allowing stable performance assessment in a bias-aware setting.We suggest modeling the confusion matrix of each demographic group using a Multinomial distribution updated through a Bayesian procedure. We extend UM to be applicable under the popular K-fold cross-validation procedure. Experiments highlight the benefits of UM over classical evaluation frameworks regarding informativeness and stability.

Thomas Kehrenberg, Myles Bartlett, Viktoriia Sharmanska et al.

When trained on diverse labeled data, machine learning models have proven themselves to be a powerful tool in all facets of society. However, due to budget limitations, deliberate or non-deliberate censorship, and other problems during data collection and curation, the labeled training set might exhibit a systematic shortage of data for certain groups. We investigate a scenario in which the absence of certain data is linked to the second level of a two-level hierarchy in the data. Inspired by the idea of protected groups from algorithmic fairness, we refer to the partitions carved by this second level as "subgroups"; we refer to combinations of subgroups and classes, or leaves of the hierarchy, as "sources". To characterize the problem, we introduce the concept of classes with incomplete subgroup support. The representational bias in the training set can give rise to spurious correlations between the classes and the subgroups which render standard classification models ungeneralizable to unseen sources. To overcome this bias, we make use of an additional, diverse but unlabeled dataset, called the "deployment set", to learn a representation that is invariant to subgroup. This is done by adversarially matching the support of the training and deployment sets in representation space. In order to learn the desired invariance, it is paramount that the sets of samples observed by the discriminator are balanced by class; this is easily achieved for the training set, but requires using semi-supervised clustering for the deployment set. We demonstrate the effectiveness of our method with experiments on several datasets and variants of the problem.

Gergely Dániel Németh, Miguel Ángel Lozano, Novi Quadrianto et al.

Federated Learning (FL) has been proposed as a privacy-preserving solution for distributed machine learning, particularly in heterogeneous FL settings where clients have varying computational capabilities and thus train models with different complexities compared to the server's model. However, FL is not without vulnerabilities: recent studies have shown that it is susceptible to membership inference attacks (MIA), which can compromise the privacy of client data. In this paper, we examine the intersection of these two aspects, heterogeneous FL and its privacy vulnerabilities, by focusing on the role of client model integration, the process through which the server integrates parameters from clients' smaller models into its larger model. To better understand this process, we first propose a taxonomy that categorizes existing heterogeneous FL methods and enables the design of seven novel heterogeneous FL model integration strategies. Using CIFAR-10, CIFAR-100, and FEMNIST vision datasets, we evaluate the privacy and accuracy trade-offs of these approaches under three types of MIAs. Our findings reveal significant differences in privacy leakage and performance depending on the integration method. Notably, introducing randomness in the model integration process enhances client privacy while maintaining competitive accuracy for both the clients and the server. This work provides quantitative light on the privacy-accuracy implications client model integration in heterogeneous FL settings, paving the way towards more secure and efficient FL systems.

Ainhize Barrainkua, Santiago Mazuelas, Novi Quadrianto et al.

As automated classification systems become increasingly prevalent, concerns have emerged over their potential to reinforce and amplify existing societal biases. In the light of this issue, many methods have been proposed to enhance the fairness guarantees of classifiers. Most of the existing interventions assume access to group information for all instances, a requirement rarely met in practice. Fairness without access to demographic information has often been approached through robust optimization techniques,which target worst-case outcomes over a set of plausible distributions known as the uncertainty set. However, their effectiveness is strongly influenced by the chosen uncertainty set. In fact, existing approaches often overemphasize outliers or overly pessimistic scenarios, compromising both overall performance and fairness. To overcome these limitations, we introduce SPECTRE, a minimax-fair method that adjusts the spectrum of a simple Fourier feature mapping and constrains the extent to which the worst-case distribution can deviate from the empirical distribution. We perform extensive experiments on the American Community Survey datasets involving 20 states. The safeness of SPECTRE comes as it provides the highest average values on fairness guarantees together with the smallest interquartile range in comparison to state-of-the-art approaches, even compared to those with access to demographic group information. In addition, we provide a theoretical analysis that derives computable bounds on the worst-case error for both individual groups and the overall population, as well as characterizes the worst-case distributions responsible for these extremal performances

Leonidas Gee, Andrea Zugarini, Novi Quadrianto

To reduce the inference cost of large language models, model compression is increasingly used to create smaller scalable models. However, little is known about their robustness to minority subgroups defined by the labels and attributes of a dataset. In this paper, we investigate the effects of 18 different compression methods and settings on the subgroup robustness of BERT language models. We show that worst-group performance does not depend on model size alone, but also on the compression method used. Additionally, we find that model compression does not always worsen the performance on minority subgroups. Altogether, our analysis serves to further research into the subgroup robustness of model compression.

Thomas Kehrenberg, Javier Sanguino, Jose A. Lozano et al.

The field of performative prediction had its beginnings in 2020 with the seminal paper "Performative Prediction" by Perdomo et al., which established a novel machine learning setup where the deployment of a predictive model causes a distribution shift in the environment, which in turn causes a mismatch between the distribution expected by the predictive model and the real distribution. This shift is defined by a so-called distribution map. In the half-decade since, a literature has emerged which has, among other things, introduced new solution concepts to the original setup, extended the setup, offered new theoretical analyses, and examined the intersection of performative prediction and other established fields. In this survey, we first lay out the performative prediction setting and explain the different optimization targets: performative stability and performative optimality. We introduce a new way of classifying different performative prediction settings, based on how much information is available about the distribution map. We survey existing implementations of distribution maps and existing methods to address the problem of performative prediction, while examining different ways to categorize them. Finally, we point out known and previously unknown connections that can be drawn to other fields, in the hopes of stimulating future research.

Ainhize Barrainkua, Giovanni De Toni, Jose Antonio Lozano et al.

Machine learning based predictions are increasingly used in sensitive decision-making applications that directly affect our lives. This has led to extensive research into ensuring the fairness of classifiers. Beyond just fair classification, emerging legislation now mandates that when a classifier delivers a negative decision, it must also offer actionable steps an individual can take to reverse that outcome. This concept is known as algorithmic recourse. Nevertheless, many researchers have expressed concerns about the fairness guarantees within the recourse process itself. In this work, we provide a holistic theoretical characterization of unfairness in algorithmic recourse, formally linking fairness guarantees in recourse and classification, and highlighting limitations of the standard equal cost paradigm. We then introduce a novel fairness framework based on social burden, along with a practical algorithm (MISOB), broadly applicable under real-world conditions. Empirical results on real-world datasets show that MISOB reduces the social burden across all groups without compromising overall classifier accuracy.

Javier Sanguino, Thomas Kehrenberg, Jose A. Lozano et al.

Performative Prediction addresses scenarios where deploying a model induces a distribution shift in the input data, such as individuals modifying their features and reapplying for a bank loan after rejection. Literature has had a theoretical perspective giving mathematical guarantees for convergence (either to the stable or optimal point). We believe that visualization of the loss landscape can complement this theoretical advances with practical insights. Therefore, (1) we introduce a simple decoupled risk visualization method inspired in the two-step process that performative prediction is. Our approach visualizes the risk landscape with respect to two parameter vectors: model parameters and data parameters. We use this method to propose new properties of the interest points, to examine how existing algorithms traverse the risk landscape and perform under more realistic conditions, including strategic classification with non-linear models. (2) Building on this decoupled risk visualization, we introduce a novel setting - extended Performative Prediction - which captures scenarios where the distribution reacts to a model different from the decision-making one, reflecting the reality that agents often lack full access to the deployed model.

Gergely D. Németh, Eros Fanì, Yeat Jeng Ng et al.

Federated Learning (FL) enables decentralized training of machine learning models on distributed data while preserving privacy. However, in real-world FL settings, client data is often non-identically distributed and imbalanced, resulting in statistical data heterogeneity which impacts the generalization capabilities of the server's model across clients, slows convergence and reduces performance. In this paper, we address this challenge by proposing first a characterization of statistical data heterogeneity by means of 6 metrics of global and client attribute imbalance, class imbalance, and spurious correlations. Next, we create and share 7 computer vision datasets for binary and multiclass image classification tasks in Federated Learning that cover a broad range of statistical data heterogeneity and hence simulate real-world situations. Finally, we propose FEDDIVERSE, a novel client selection algorithm in FL which is designed to manage and leverage data heterogeneity across clients by promoting collaboration between clients with complementary data distributions. Experiments on the seven proposed FL datasets demonstrate FEDDIVERSE's effectiveness in enhancing the performance and robustness of a variety of FL methods while having low communication and computational overhead.

Leonidas Gee, Wing Yan Li, Viktoriia Sharmanska et al.

The cost of deploying vision transformers increasingly represents a barrier to wider industrial adoption. Existing compression techniques require additional end-to-end fine-tuning or incur a significant drawback to runtime, making them ill-suited for online (real-time) inference, where a prediction is made on any new input as it comes in. We introduce the $\textbf{Visual Word Tokenizer}$ (VWT), a training-free method for reducing power costs while retaining performance and runtime. The VWT groups visual subwords (image patches) that are frequently used into visual words while infrequent ones remain intact. To do so, $\textit{intra}$-image or $\textit{inter}$-image statistics are leveraged to identify similar visual concepts for sequence compression. Experimentally, we demonstrate a reduction in wattage of up to 25% with only a 20% increase in runtime at most. Comparative approaches of 8-bit quantization and token merging achieve a lower or similar power efficiency but exact a higher toll on runtime (up to 100% or more). Our results indicate that VWTs are well-suited for efficient online inference with a marginal compromise on performance.

Ainhize Barrainkua, Paula Gordaliza, Jose A. Lozano et al.

This paper introduces a novel approach to bolster algorithmic fairness in scenarios where sensitive information is only partially known. In particular, we propose to leverage instances with uncertain identity with regards to the sensitive attribute to train a conventional machine learning classifier. The enhanced fairness observed in the final predictions of this classifier highlights the promising potential of prioritizing ambiguity (i.e., non-normativity) as a means to improve fairness guarantees in real-world classification tasks.

Thomas Kehrenberg, Myles Bartlett, Oliver Thomas et al.

We propose to learn invariant representations, in the data domain, to achieve interpretability in algorithmic fairness. Invariance implies a selectivity for high level, relevant correlations w.r.t. class label annotations, and a robustness to irrelevant correlations with protected characteristics such as race or gender. We introduce a non-trivial setup in which the training set exhibits a strong bias such that class label annotations are irrelevant and spurious correlations cannot be distinguished. To address this problem, we introduce an adversarially trained model with a null-sampling procedure to produce invariant representations in the data domain. To enable disentanglement, a partially-labelled representative set is used. By placing the representations into the data domain, the changes made by the model are easily examinable by human auditors. We show the effectiveness of our method on both image and tabular datasets: Coloured MNIST, the CelebA and the Adult dataset.

Viktoriia Sharmanska, Lisa Anne Hendricks, Trevor Darrell et al.

Computer vision algorithms, e.g. for face recognition, favour groups of individuals that are better represented in the training data. This happens because of the generalization that classifiers have to make. It is simpler to fit the majority groups as this fit is more important to overall error. We propose to create a balanced training dataset, consisting of the original dataset plus new data points in which the group memberships are intervened, minorities become majorities and vice versa. We show that current generative adversarial networks are a powerful tool for learning these data points, called contrastive examples. We experiment with the equalized odds bias measure on tabular data as well as image data (CelebA and Diversity in Faces datasets). Contrastive examples allow us to expose correlations between group membership and other seemingly neutral features. Whenever a causal graph is available, we can put those contrastive examples in the perspective of counterfactuals.

Bradley Butcher, Vincent S. Huang, Jeremy Reffin et al.

In solving real-world problems like changing healthcare-seeking behaviors, designing interventions to improve downstream outcomes requires an understanding of the causal links within the system. Causal Bayesian Networks (BN) have been proposed as one such powerful method. In real-world applications, however, confidence in the results of BNs are often moderate at best. This is due in part to the inability to validate against some ground truth, as the DAG is not available. This is especially problematic if the learned DAG conflicts with pre-existing domain doctrine. At the policy level, one must justify insights generated by such analysis, preferably accompanying them with uncertainty estimation. Here we propose a causal extension to the datasheet concept proposed by Gebru et al (2018) to include approximate BN performance expectations for any given dataset. To generate the results for a prototype Causal Datasheet, we constructed over 30,000 synthetic datasets with properties mirroring characteristics of real data. We then recorded the results given by state-of-the-art structure learning algorithms. These results were used to populate the Causal Datasheet, and recommendations were automatically generated dependent on expected performance. As a proof of concept, we used our Causal Datasheet Generation Tool (CDG-T) to assign expected performance expectations to a maternal health survey we conducted in Uttar Pradesh, India.

Artyom Gadetsky, Kirill Struminsky, Christopher Robinson et al.

Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradient estimates. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. In particular, the control variates allow us to optimize black-box functions over permutations using stochastic gradient descent. To illustrate the approach, we consider a variety of causal structure learning tasks for continuous and discrete data. We show that our method outperforms competitive relaxation-based optimization methods and is also applicable to non-differentiable score functions.

Novi Quadrianto, Viktoriia Sharmanska, Oliver Thomas

Interpretability and fairness are critical in computer vision and machine learning applications, in particular when dealing with human outcomes, e.g. inviting or not inviting for a job interview based on application materials that may include photographs. One promising direction to achieve fairness is by learning data representations that remove the semantics of protected characteristics, and are therefore able to mitigate unfair outcomes. All available models however learn latent embeddings which comes at the cost of being uninterpretable. We propose to cast this problem as data-to-data translation, i.e. learning a mapping from an input domain to a fair target domain, where a fairness definition is being enforced. Here the data domain can be images, or any tabular data representation. This task would be straightforward if we had fair target data available, but this is not the case. To overcome this, we learn a highly unconstrained mapping by exploiting statistics of residuals - the difference between input data and its translated version - and the protected characteristics. When applied to the CelebA dataset of face images with gender attribute as the protected characteristic, our model enforces equality of opportunity by adjusting the eyes and lips regions. Intriguingly, on the same dataset we arrive at similar conclusions when using semantic attribute representations of images for translation. On face images of the recent DiF dataset, with the same gender attribute, our method adjusts nose regions. In the Adult income dataset, also with protected gender attribute, our model achieves equality of opportunity by, among others, obfuscating the wife and husband relationship. Analyzing those systematic changes will allow us to scrutinize the interplay of fairness criterion, chosen protected characteristics, and prediction performance.

Thomas Kehrenberg, Zexun Chen, Novi Quadrianto

The issue of fairness in machine learning models has recently attracted a lot of attention as ensuring it will ensure continued confidence of the general public in the deployment of machine learning systems. We focus on mitigating the harm incurred by a biased machine learning system that offers better outputs (e.g. loans, job interviews) for certain groups than for others. We show that bias in the output can naturally be controlled in probabilistic models by introducing a latent target output. This formulation has several advantages: first, it is a unified framework for several notions of group fairness such as Demographic Parity and Equality of Opportunity; second, it is expressed as a marginalisation instead of a constrained problem; and third, it allows the encoding of our knowledge of what unbiased outputs should be. Practically, the second allows us to avoid unstable constrained optimisation procedures and to reuse off-the-shelf toolboxes. The latter translates to the ability to control the level of fairness by directly varying fairness target rates. In contrast, existing approaches rely on intermediate, arguably unintuitive, control parameters such as covariance thresholds.

Pietro Galliani, Amir Dezfouli, Edwin V. Bonilla et al.

We develop an automated variational inference method for Bayesian structured prediction problems with Gaussian process (GP) priors and linear-chain likelihoods. Our approach does not need to know the details of the structured likelihood model and can scale up to a large number of observations. Furthermore, we show that the required expected likelihood term and its gradients in the variational objective (ELBO) can be estimated efficiently by using expectations over very low-dimensional Gaussian distributions. Optimization of the ELBO is fully parallelizable over sequences and amenable to stochastic optimization, which we use along with control variate techniques and state-of-the-art incremental optimization to make our framework useful in practice. Results on a set of natural language processing tasks show that our method can be as good as (and sometimes better than) hard-coded approaches including SVM-struct and CRFs, and overcomes the scalability limitations of previous inference algorithms based on sampling. Overall, this is a fundamental step to developing automated inference methods for Bayesian structured prediction.

Viktoriia Sharmanska, Novi Quadrianto, Christoph H. Lampert

We introduce a learning framework called learning using privileged information (LUPI) to the computer vision field. We focus on the prototypical computer vision problem of teaching computers to recognize objects in images. We want the computers to be able to learn faster at the expense of providing extra information during training time. As additional information about the image data, we look at several scenarios that have been studied in computer vision before: attributes, bounding boxes and image tags. The information is privileged as it is available at training time but not at test time. We explore two maximum-margin techniques that are able to make use of this additional source of information, for binary and multiclass object classification. We interpret these methods as learning easiness and hardness of the objects in the privileged space and then transferring this knowledge to train a better classifier in the original space. We provide a thorough analysis and comparison of information transfer from privileged to the original data spaces for both LUPI methods. Our experiments show that incorporating privileged information can improve the classification accuracy. Finally, we conduct user studies to understand which samples are easy and which are hard for human learning, and explore how this information is related to easy and hard samples when learning a classifier.

Daniel Hernández-Lobato, Viktoriia Sharmanska, Kristian Kersting et al.

The learning with privileged information setting has recently attracted a lot of attention within the machine learning community, as it allows the integration of additional knowledge into the training process of a classifier, even when this comes in the form of a data modality that is not available at test time. Here, we show that privileged information can naturally be treated as noise in the latent function of a Gaussian Process classifier (GPC). That is, in contrast to the standard GPC setting, the latent function is not just a nuisance but a feature: it becomes a natural measure of confidence about the training data by modulating the slope of the GPC sigmoid likelihood function. Extensive experiments on public datasets show that the proposed GPC method using privileged noise, called GPC+, improves over a standard GPC without privileged knowledge, and also over the current state-of-the-art SVM-based method, SVM+. Moreover, we show that advanced neural networks and deep learning methods can be compressed as privileged information.

Novi Quadrianto, Viktoriia Sharmanska, David A. Knowles et al.

We propose a probabilistic model to infer supervised latent variables in the Hamming space from observed data. Our model allows simultaneous inference of the number of binary latent variables, and their values. The latent variables preserve neighbourhood structure of the data in a sense that objects in the same semantic concept have similar latent values, and objects in different concepts have dissimilar latent values. We formulate the supervised infinite latent variable problem based on an intuitive principle of pulling objects together if they are of the same type, and pushing them apart if they are not. We then combine this principle with a flexible Indian Buffet Process prior on the latent variables. We show that the inferred supervised latent variables can be directly used to perform a nearest neighbour search for the purpose of retrieval. We introduce a new application of dynamically extending hash codes, and show how to effectively couple the structure of the hash codes with continuously growing structure of the neighbourhood preserving infinite latent feature space.

Sebastien Bratieres, Novi Quadrianto, Zoubin Ghahramani

We introduce a conceptually novel structured prediction model, GPstruct, which is kernelized, non-parametric and Bayesian, by design. We motivate the model with respect to existing approaches, among others, conditional random fields (CRFs), maximum margin Markov networks (M3N), and structured support vector machines (SVMstruct), which embody only a subset of its properties. We present an inference procedure based on Markov Chain Monte Carlo. The framework can be instantiated for a wide range of structured objects such as linear chains, trees, grids, and other general graphs. As a proof of concept, the model is benchmarked on several natural language processing tasks and a video gesture segmentation task involving a linear chain structure. We show prediction accuracies for GPstruct which are comparable to or exceeding those of CRFs and SVMstruct.

Novi Quadrianto, Chao Chen, Christoph Lampert

When searching for characteristic subpatterns in potentially noisy graph data, it appears self-evident that having multiple observations would be better than having just one. However, it turns out that the inconsistencies introduced when different graph instances have different edge sets pose a serious challenge. In this work we address this challenge for the problem of finding maximum weighted cliques. We introduce the concept of most persistent soft-clique. This is subset of vertices, that 1) is almost fully or at least densely connected, 2) occurs in all or almost all graph instances, and 3) has the maximum weight. We present a measure of clique-ness, that essentially counts the number of edge missing to make a subset of vertices into a clique. With this measure, we show that the problem of finding the most persistent soft-clique problem can be cast either as: a) a max-min two person game optimization problem, or b) a min-min soft margin optimization problem. Both formulations lead to the same solution when using a partial Lagrangian method to solve the optimization problems. By experiments on synthetic data and on real social network data, we show that the proposed method is able to reliably find soft cliques in graph data, even if that is distorted by random noise or unreliable observations.