arXiv subjects

Richard Nock

Publications and source records attributed to Richard Nock.

At least 37 records · Page 2Linked to original sources

arXiv2022

Proper-Composite Loss Functions in Arbitrary Dimensions

The study of a machine learning problem is in many ways is difficult to separate from the study of the loss function being used. One avenue of inquiry has been to look at these loss functions in terms of their properties as scoring rules via the proper-composite representation, in which predictions are mapped to probability distributions which are then scored via a scoring rule. However, recent research so far has primarily been concerned with analysing the (typically) finite-dimensional conditional risk problem on the output space, leaving aside the larger total risk minimisation. We generalise a number of these results to an infinite dimensional setting and in doing so we are able to exploit the familial resemblance of density and conditional density estimation to provide a simple characterisation of the canonical link.

cs.LG↗

arXiv2022

What killed the Convex Booster ?

A landmark negative result of Long and Servedio established a worst-case spectacular failure of a supervised learning trio (loss, algorithm, model) otherwise praised for its high precision machinery. Hundreds of papers followed up on the two suspected culprits: the loss (for being convex) and/or the algorithm (for fitting a classical boosting blueprint). Here, we call to the half-century+ founding theory of losses for class probability estimation (properness), an extension of Long and Servedio's results and a new general boosting algorithm to demonstrate that the real culprit in their specific context was in fact the (linear) model class. We advocate for a more general stanpoint on the problem as we argue that the source of the negative result lies in the dark side of a pervasive -- and otherwise prized -- aspect of ML: \textit{parameterisation}.

cs.LG↗

arXiv2022

New Tricks for Estimating Gradients of Expectations

We introduce a family of pairwise stochastic gradient estimators for gradients of expectations, which are related to the log-derivative trick, but involve pairwise interactions between samples. The simplest example of our new estimator, dubbed the fundamental trick estimator, is shown to arise from either a) introducing and approximating an integral representation based on the fundamental theorem of calculus, or b) applying the reparameterisation trick to an implicit parameterisation under infinitesimal perturbation of the parameters. From the former perspective we generalise to a reproducing kernel Hilbert space representation, giving rise to a locality parameter in the pairwise interactions mentioned above, yielding our representer trick estimator. The resulting estimators are unbiased and shown to offer an independent component of useful information in comparison with the log-derivative estimator. We provide a further novel theoretical analysis which further characterises the variance reduction afforded by the new techniques. Promising analytical and numerical examples confirm the theory and intuitions behind the new estimators.

cs.LG↗

arXiv2022

Manifold Learning Benefits GANs

In this paper, we improve Generative Adversarial Networks by incorporating a manifold learning step into the discriminator. We consider locality-constrained linear and subspace-based manifolds, and locality-constrained non-linear manifolds. In our design, the manifold learning and coding steps are intertwined with layers of the discriminator, with the goal of attracting intermediate feature representations onto manifolds. We adaptively balance the discrepancy between feature representations and their manifold view, which is a trade-off between denoising on the manifold and refining the manifold. We find that locality-constrained non-linear manifolds outperform linear manifolds due to their non-uniform density and smoothness. We also substantially outperform state-of-the-art baselines.

cs.CV↗

arXiv2022

Generative Trees: Adversarial and Copycat

While Generative Adversarial Networks (GANs) achieve spectacular results on unstructured data like images, there is still a gap on tabular data, data for which state of the art supervised learning still favours to a large extent decision tree (DT)-based models. This paper proposes a new path forward for the generation of tabular data, exploiting decades-old understanding of the supervised task's best components for DT induction, from losses (properness), models (tree-based) to algorithms (boosting). The \textit{properness} condition on the supervised loss -- which postulates the optimality of Bayes rule -- leads us to a variational GAN-style loss formulation which is \textit{tight} when discriminators meet a calibration property trivially satisfied by DTs, and, under common assumptions about the supervised loss, yields "one loss to train against them all" for the generator: the $χ^2$. We then introduce tree-based generative models, \textit{generative trees} (GTs), meant to mirror on the generative side the good properties of DTs for classifying tabular data, with a boosting-compliant \textit{adversarial} training algorithm for GTs. We also introduce \textit{copycat training}, in which the generator copies at run time the underlying tree (graph) of the discriminator DT and completes it for the hardest discriminative task, with boosting compliant convergence. We test our algorithms on tasks including fake/real distinction, training from fake data and missing data imputation. Each one of these tasks displays that GTs can provide comparatively simple -- and interpretable -- contenders to sophisticated state of the art methods for data generation (using neural network models) or missing data imputation (relying on multiple imputation by chained equations with complex tree-based modeling).

cs.LG↗

arXiv2022

Being Properly Improper

Properness for supervised losses stipulates that the loss function shapes the learning algorithm towards the true posterior of the data generating distribution. Unfortunately, data in modern machine learning can be corrupted or twisted in many ways. Hence, optimizing a proper loss function on twisted data could perilously lead the learning algorithm towards the twisted posterior, rather than to the desired clean posterior. Many papers cope with specific twists (e.g., label/feature/adversarial noise), but there is a growing need for a unified and actionable understanding atop properness. Our chief theoretical contribution is a generalization of the properness framework with a notion called twist-properness, which delineates loss functions with the ability to "untwist" the twisted posterior into the clean posterior. Notably, we show that a nontrivial extension of a loss function called $α$-loss, which was first introduced in information theory, is twist-proper. We study the twist-proper $α$-loss under a novel boosting algorithm, called PILBoost, and provide formal and experimental results for this algorithm. Our overarching practical conclusion is that the twist-proper $α$-loss outperforms the proper $\log$-loss on several variants of twisted data.

cs.LG↗

arXiv2021

The hyperbolic Voronoi diagram in arbitrary dimension

We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power diagram, requiring however algebraic arithmetic. By considering the lesser-known Beltrami hemisphere model of hyperbolic geometry, we overcome the arithmetic limitations of Klein construction. Finally, we characterize the bisectors and geodesics in the other Poincar\' e upper half-space, the Poincaré ball, and the Lorentz hyperboloid models, and discusses on degenerate cases for which the dual hyperbolic Delaunay complex is not a triangulation.

cs.CG↗

arXiv2021

On $w$-mixtures: Finite convex combinations of prescribed component distributions

We consider the space of $w$-mixtures which is defined as the set of finite statistical mixtures sharing the same prescribed component distributions closed under convex combinations. The information geometry induced by the Bregman generator set to the Shannon negentropy on this space yields a dually flat space called the mixture family manifold. We show how the Kullback-Leibler (KL) divergence can be recovered from the corresponding Bregman divergence for the negentropy generator: That is, the KL divergence between two $w$-mixtures amounts to a Bregman Divergence (BD) induced by the Shannon negentropy generator. Thus the KL divergence between two Gaussian Mixture Models (GMMs) sharing the same Gaussian components is equivalent to a Bregman divergence. This KL-BD equivalence on a mixture family manifold implies that we can perform optimal KL-averaging aggregation of $w$-mixtures without information loss. More generally, we prove that the statistical skew Jensen-Shannon divergence between $w$-mixtures is equivalent to a skew Jensen divergence between their corresponding parameters. Finally, we state several properties, divergence identities, and inequalities relating to $w$-mixtures.

cs.LG↗

arXiv2021

Advances and Open Problems in Federated Learning

Federated learning (FL) is a machine learning setting where many clients (e.g. mobile devices or whole organizations) collaboratively train a model under the orchestration of a central server (e.g. service provider), while keeping the training data decentralized. FL embodies the principles of focused data collection and minimization, and can mitigate many of the systemic privacy risks and costs resulting from traditional, centralized machine learning and data science approaches. Motivated by the explosive growth in FL research, this paper discusses recent advances and presents an extensive collection of open problems and challenges.

cs.LG↗

arXiv2020

All your loss are belong to Bayes

Loss functions are a cornerstone of machine learning and the starting point of most algorithms. Statistics and Bayesian decision theory have contributed, via properness, to elicit over the past decades a wide set of admissible losses in supervised learning, to which most popular choices belong (logistic, square, Matsushita, etc.). Rather than making a potentially biased ad hoc choice of the loss, there has recently been a boost in efforts to fit the loss to the domain at hand while training the model itself. The key approaches fit a canonical link, a function which monotonically relates the closed unit interval to R and can provide a proper loss via integration. In this paper, we rely on a broader view of proper composite losses and a recent construct from information geometry, source functions, whose fitting alleviates constraints faced by canonical links. We introduce a trick on squared Gaussian Processes to obtain a random process whose paths are compliant source functions with many desirable properties in the context of link estimation. Experimental results demonstrate substantial improvements over the state of the art.

cs.LG↗

arXiv2020

Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family

It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $α$-divergences, and Jeffreys' divergences between densities belonging to a same exponential family have generic closed-form formulas relying on the strictly convex and real-analytic cumulant function characterizing the exponential family. In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function and highlight the role of quasi-arithmetic means and their multivariate mean operator extensions. In practice, these cumulant-free formulas are handy when implementing these (dis)similarities using legacy Application Programming Interfaces (APIs) since our method requires only to partially factorize the densities canonically of the considered exponential family.

math.ST↗

arXiv2020

Generalised Lipschitz Regularisation Equals Distributional Robustness

The problem of adversarial examples has highlighted the need for a theory of regularisation that is general enough to apply to exotic function classes, such as universal approximators. In response, we give a very general equality result regarding the relationship between distributional robustness and regularisation, as defined with a transportation cost uncertainty set. The theory allows us to (tightly) certify the robustness properties of a Lipschitz-regularised model with very mild assumptions. As a theoretical application we show a new result explicating the connection between adversarial learning and distributional robustness. We then give new results for how to achieve Lipschitz regularisation of kernel classifiers, which are demonstrated experimentally.

cs.LG↗

arXiv2020

Supervised Learning: No Loss No Cry

Supervised learning requires the specification of a loss function to minimise. While the theory of admissible losses from both a computational and statistical perspective is well-developed, these offer a panoply of different choices. In practice, this choice is typically made in an \emph{ad hoc} manner. In hopes of making this procedure more principled, the problem of \emph{learning the loss function} for a downstream task (e.g., classification) has garnered recent interest. However, works in this area have been generally empirical in nature. In this paper, we revisit the {\sc SLIsotron} algorithm of Kakade et al. (2011) through a novel lens, derive a generalisation based on Bregman divergences, and show how it provides a principled procedure for learning the loss. In detail, we cast {\sc SLIsotron} as learning a loss from a family of composite square losses. By interpreting this through the lens of \emph{proper losses}, we derive a generalisation of {\sc SLIsotron} based on Bregman divergences. The resulting {\sc BregmanTron} algorithm jointly learns the loss along with the classifier. It comes equipped with a simple guarantee of convergence for the loss it learns, and its set of possible outputs comes with a guarantee of agnostic approximability of Bayes rule. Experiments indicate that the {\sc BregmanTron} substantially outperforms the {\sc SLIsotron}, and that the loss it learns can be minimized by other algorithms for different tasks, thereby opening the interesting problem of \textit{loss transfer} between domains.

cs.LG↗

arXiv2020

Boosted and Differentially Private Ensembles of Decision Trees

Boosted ensemble of decision tree (DT) classifiers are extremely popular in international competitions, yet to our knowledge nothing is formally known on how to make them \textit{also} differential private (DP), up to the point that random forests currently reign supreme in the DP stage. Our paper starts with the proof that the privacy vs boosting picture for DT involves a notable and general technical tradeoff: the sensitivity tends to increase with the boosting rate of the loss, for any proper loss. DT induction algorithms being fundamentally iterative, our finding implies non-trivial choices to select or tune the loss to balance noise against utility to split nodes. To address this, we craft a new parametererized proper loss, called the M$α$-loss, which, as we show, allows to finely tune the tradeoff in the complete spectrum of sensitivity vs boosting guarantees. We then introduce \textit{objective calibration} as a method to adaptively tune the tradeoff during DT induction to limit the privacy budget spent while formally being able to keep boosting-compliant convergence on limited-depth nodes with high probability. Extensive experiments on 19 UCI domains reveal that objective calibration is highly competitive, even in the DP-free setting. Our approach tends to very significantly beat random forests, in particular on high DP regimes ($\varepsilon \leq 0.1$) and even with boosted ensembles containing ten times less trees, which could be crucial to keep a key feature of DT models under differential privacy: interpretability.

cs.LG↗

arXiv2019

Integral Privacy for Sampling

Differential privacy is a leading protection setting, focused by design on individual privacy. Many applications, in medical / pharmaceutical domains or social networks, rather posit privacy at a group level, a setting we call integral privacy. We aim for the strongest form of privacy: the group size is in particular not known in advance. We study a problem with related applications in domains cited above that have recently met with substantial recent press: sampling. Keeping correct utility levels in such a strong model of statistical indistinguishability looks difficult to be achieved with the usual differential privacy toolbox because it would typically scale in the worst case the sensitivity by the sample size and so the noise variance by up to its square. We introduce a trick specific to sampling that bypasses the sensitivity analysis. Privacy enforces an information theoretic barrier on approximation, and we show how to reach this barrier with guarantees on the approximation of the target non private density. We do so using a recent approach to non private density estimation relying on the original boosting theory, learning the sufficient statistics of an exponential family with classifiers. Approximation guarantees cover the mode capture problem. In the context of learning, the sampling problem is particularly important: because integral privacy enjoys the same closure under post-processing as differential privacy does, any algorithm using integrally privacy sampled data would result in an output equally integrally private. We also show that this brings fairness guarantees on post-processing that would eventually elude classical differential privacy: any decision process has bounded data-dependent bias when the data is integrally privately sampled. Experimental results against private kernel density estimation and private GANs displays the quality of our results.

stat.ML↗

arXiv2019

Monge blunts Bayes: Hardness Results for Adversarial Training

The last few years have seen a staggering number of empirical studies of the robustness of neural networks in a model of adversarial perturbations of their inputs. Most rely on an adversary which carries out local modifications within prescribed balls. None however has so far questioned the broader picture: how to frame a resource-bounded adversary so that it can be severely detrimental to learning, a non-trivial problem which entails at a minimum the choice of loss and classifiers. We suggest a formal answer for losses that satisfy the minimal statistical requirement of being proper. We pin down a simple sufficient property for any given class of adversaries to be detrimental to learning, involving a central measure of "harmfulness" which generalizes the well-known class of integral probability metrics. A key feature of our result is that it holds for all proper losses, and for a popular subset of these, the optimisation of this central measure appears to be independent of the loss. When classifiers are Lipschitz -- a now popular approach in adversarial training --, this optimisation resorts to optimal transport to make a low-budget compression of class marginals. Toy experiments reveal a finding recently separately observed: training against a sufficiently budgeted adversary of this kind improves generalization.

cs.LG↗

arXiv2019

Adversarial Networks and Autoencoders: The Primal-Dual Relationship and Generalization Bounds

Since the introduction of Generative Adversarial Networks (GANs) and Variational Autoencoders (VAE), the literature on generative modelling has witnessed an overwhelming resurgence. The impressive, yet elusive empirical performance of GANs has lead to the rise of many GAN-VAE hybrids, with the hopes of GAN level performance and additional benefits of VAE, such as an encoder for feature reduction, which is not offered by GANs. Recently, the Wasserstein Autoencoder (WAE) was proposed, achieving performance similar to that of GANs, yet it is still unclear whether the two are fundamentally different or can be further improved into a unified model. In this work, we study the $f$-GAN and WAE models and make two main discoveries. First, we find that the $f$-GAN and WAE objectives partake in a primal-dual relationship and are equivalent under some assumptions, which then allows us to explicate the success of WAE. Second, the equivalence result allows us to, for the first time, prove generalization bounds for Autoencoder models, which is a pertinent problem when it comes to theoretical analyses of generative models. Furthermore, we show that the WAE objective is related to other statistical quantities such as the $f$-divergence and in particular, upper bounded by the Wasserstein distance, which then allows us to tap into existing efficient (regularized) optimal transport solvers. Our findings thus present the first primal-dual relationship between GANs and Autoencoder models, comment on generalization abilities and make a step towards unifying these models.

stat.ML↗

arXiv2018

Hyperparameter Learning for Conditional Kernel Mean Embeddings with Rademacher Complexity Bounds

Conditional kernel mean embeddings are nonparametric models that encode conditional expectations in a reproducing kernel Hilbert space. While they provide a flexible and powerful framework for probabilistic inference, their performance is highly dependent on the choice of kernel and regularization hyperparameters. Nevertheless, current hyperparameter tuning methods predominantly rely on expensive cross validation or heuristics that is not optimized for the inference task. For conditional kernel mean embeddings with categorical targets and arbitrary inputs, we propose a hyperparameter learning framework based on Rademacher complexity bounds to prevent overfitting by balancing data fit against model complexity. Our approach only requires batch updates, allowing scalable kernel hyperparameter tuning without invoking kernel approximations. Experiments demonstrate that our learning framework outperforms competing methods, and can be further extended to incorporate and learn deep neural network weights to improve generalization.

stat.ML↗