Careful with that Scalpel: Improving Gradient Surgery with an EMA

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

Riemannian Diffusion Schrödinger Bridge

Score-based generative models exhibit state of the art performance on density estimation and generative modeling tasks. These models typically assume that the data geometry is flat, yet recent extensions have been developed to synthesize data living on Riemannian manifolds. Existing methods to accelerate sampling of diffusion models are typically not applicable in the Riemannian setting and Riemannian score-based methods have not yet been adapted to the important task of interpolation of datasets. To overcome these issues, we introduce \emph{Riemannian Diffusion Schr\"odinger Bridge}. Our proposed method generalizes Diffusion Schr\"odinger Bridge introduced in \cite{debortoli2021neurips} to the non-Euclidean setting and extends Riemannian score-based models beyond the first time reversal. We validate our proposed method on synthetic data and real Earth and climate data.

Rethinking Initialization of the Sinkhorn Algorithm

Computing an optimal transport (OT) coupling between distributions plays an increasingly important role in machine learning. While OT problems can be solved as linear programs, adding an entropic smoothing term is known to result in solvers that are faster and more robust to outliers, differentiable and easier to parallelize. The Sinkhorn fixed point algorithm is the cornerstone of these approaches, and, as a result, multiple attempts have been made to shorten its runtime using, for instance, annealing, momentum or acceleration. The premise of this paper is that \textit{initialization} of the Sinkhorn algorithm has received comparatively little attention, possibly due to two preconceptions: as the regularized OT problem is convex, it may not be worth crafting a tailored initialization as \textit{any} is guaranteed to work; secondly, because the Sinkhorn algorithm is often differentiated in end-to-end pipelines, data-dependent initializations could potentially bias gradient estimates obtained by unrolling iterations. We challenge this conventional wisdom and show that carefully chosen initializations can result in dramatic speed-ups, and will not bias gradients which are computed with implicit differentiation. We detail how initializations can be recovered from closed-form or approximate OT solutions, using known results in the 1D or Gaussian settings. We show empirically that these initializations can be used off-the-shelf, with little to no tuning, and result in consistent speed-ups for a variety of OT problems.

Riemannian Score-Based Generative Modeling

Score-based generative models (SGMs) are a novel class of generative models demonstrating remarkable empirical performance. One uses a diffusion to add progressively Gaussian noise to the data, while the generative model is a "denoising" process obtained by approximating the time-reversal of this "noising" diffusion. However, current SGMs make the underlying assumption that the data is supported on a Euclidean manifold with flat geometry. This prevents the use of these models for applications in robotics, geoscience or protein modeling which rely on distributions defined on Riemannian manifolds. To overcome this issue, we introduce Riemannian Score-based Generative Models (RSGMs) which extend current SGMs to the setting of compact Riemannian manifolds. We illustrate our approach with earth and climate science data and show how RSGMs can be accelerated by solving a Schr\"odinger bridge problem on manifolds.

Simulating Diffusion Bridges with Score Matching

We consider the problem of simulating diffusion bridges, i.e. diffusion processes that are conditioned to initialize and terminate at two given states. Diffusion bridge simulation has applications in diverse scientific fields and plays a crucial role for statistical inference of discretely-observed diffusions. This is known to be a challenging problem that has received much attention in the last two decades. In this work, we first show that the time-reversed diffusion bridge process can be simulated if one can time-reverse the unconditioned diffusion process. We introduce a variational formulation to learn this time-reversal that relies on a score matching method to circumvent intractability. We then consider another iteration of our proposed methodology to approximate the Doob's $h$-transform defining the diffusion bridge process. As our approach is generally applicable under mild assumptions on the underlying diffusion process, it can easily be used to improve the proposal bridge process within existing methods and frameworks. We discuss algorithmic considerations and extensions, and present some numerical results.

Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling

Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schr\"odinger Bridge problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).

Differentiable Particle Filtering via Entropy-Regularized Optimal Transport

Particle Filtering (PF) methods are an established class of procedures for performing inference in non-linear state-space models. Resampling is a key ingredient of PF, necessary to obtain low variance likelihood and states estimates. However, traditional resampling methods result in PF-based loss functions being non-differentiable with respect to model and PF parameters. In a variational inference context, resampling also yields high variance gradient estimates of the PF-based evidence lower bound. By leveraging optimal transport ideas, we introduce a principled differentiable particle filter and provide convergence results. We demonstrate this novel method on a variety of applications.