Explicit Density Approximation for Neural Implicit Samplers Using a Bernstein-Based Convex Divergence

Rank-based statistical metrics, such as the invariant statistical loss (ISL), have recently emerged as robust and practically effective tools for training implicit generative models. In this work, we introduce dual-ISL, a novel likelihood-free objective for training implicit generative models that interchanges the roles of the target and model distributions in the ISL framework, yielding a convex optimization problem in the space of model densities. We prove that the resulting rank-based discrepancy $d_K$ is i) continuous under weak convergence and with respect to the $L^1$ norm, and ii) convex in its first argument-properties not shared by classical divergences such as KL or Wasserstein distances. Building on this, we develop a theoretical framework that interprets $d_K$ as an $L^2$-projection of the density ratio $q = p/\tilde p$ onto a Bernstein polynomial basis, from which we derive exact bounds on the truncation error, precise convergence rates, and a closed-form expression for the truncated density approximation. We further extend our analysis to the multivariate setting via random one-dimensional projections, defining a sliced dual-ISL divergence that retains both convexity and continuity. We empirically show that these theoretical advantages translate into practical ones. Specifically, across several benchmarks dual-ISL converges more rapidly, delivers markedly smoother and more stable training, and more effectively prevents mode collapse than classical ISL and other leading implicit generative methods-while also providing an explicit density approximation.

Robust training of implicit generative models for multivariate and heavy-tailed distributions with an invariant statistical loss

Traditional implicit generative models are capable of learning highly complex data distributions. However, their training involves distinguishing real data from synthetically generated data using adversarial discriminators, which can lead to unstable training dynamics and mode dropping issues. In this work, we build on the \textit{invariant statistical loss} (ISL) method introduced in \cite{de2024training}, and extend it to handle heavy-tailed and multivariate data distributions. The data generated by many real-world phenomena can only be properly characterised using heavy-tailed probability distributions, and traditional implicit methods struggle to effectively capture their asymptotic behavior. To address this problem, we introduce a generator trained with ISL, that uses input noise from a generalised Pareto distribution (GPD). We refer to this generative scheme as Pareto-ISL for conciseness. Our experiments demonstrate that Pareto-ISL accurately models the tails of the distributions while still effectively capturing their central characteristics. The original ISL function was conceived for 1D data sets. When the actual data is $n$-dimensional, a straightforward extension of the method was obtained by targeting the $n$ marginal distributions of the data. This approach is computationally infeasible and ineffective in high-dimensional spaces. To overcome this, we extend the 1D approach using random projections and define a new loss function suited for multivariate data, keeping problems tractable by adjusting the number of projections. We assess its performance in multidimensional generative modeling and explore its potential as a pretraining technique for generative adversarial networks (GANs) to prevent mode collapse, reporting promising results and highlighting its robustness across various hyperparameter settings.

Training Implicit Generative Models via an Invariant Statistical Loss

Implicit generative models have the capability to learn arbitrary complex data distributions. On the downside, training requires telling apart real data from artificially-generated ones using adversarial discriminators, leading to unstable training and mode-dropping issues. As reported by Zahee et al. (2017), even in the one-dimensional (1D) case, training a generative adversarial network (GAN) is challenging and often suboptimal. In this work, we develop a discriminator-free method for training one-dimensional (1D) generative implicit models and subsequently expand this method to accommodate multivariate cases. Our loss function is a discrepancy measure between a suitably chosen transformation of the model samples and a uniform distribution; hence, it is invariant with respect to the true distribution of the data. We first formulate our method for 1D random variables, providing an effective solution for approximate reparameterization of arbitrary complex distributions. Then, we consider the temporal setting (both univariate and multivariate), in which we model the conditional distribution of each sample given the history of the process. We demonstrate through numerical simulations that this new method yields promising results, successfully learning true distributions in a variety of scenarios and mitigating some of the well-known problems that state-of-the-art implicit methods present.