The Elements of Differentiable Programming

Artificial intelligence has recently experienced remarkable advances, fueled by large models, vast datasets, accelerated hardware, and, last but not least, the transformative power of differentiable programming. This new programming paradigm enables end-to-end differentiation of complex computer programs (including those with control flows and data structures), making gradient-based optimization of program parameters possible. As an emerging paradigm, differentiable programming builds upon several areas of computer science and applied mathematics, including automatic differentiation, graphical models, optimization and statistics. This book presents a comprehensive review of the fundamental concepts useful for differentiable programming. We adopt two main perspectives, that of optimization and that of probability, with clear analogies between the two. Differentiable programming is not merely the differentiation of programs, but also the thoughtful design of programs intended for differentiation. By making programs differentiable, we inherently introduce probability distributions over their execution, providing a means to quantify the uncertainty associated with program outputs.

On the Interplay Between Stepsize Tuning and Progressive Sharpening

Recent empirical work has revealed an intriguing property of deep learning models by which the sharpness (largest eigenvalue of the Hessian) increases throughout optimization until it stabilizes around a critical value at which the optimizer operates at the edge of stability, given a fixed stepsize (Cohen et al, 2022). We investigate empirically how the sharpness evolves when using stepsize-tuners, the Armijo linesearch and Polyak stepsizes, that adapt the stepsize along the iterations to local quantities such as, implicitly, the sharpness itself. We find that the surprisingly poor performance of a classical Armijo linesearch may be well explained by its tendency to ever-increase the sharpness of the objective in the full or large batch regimes. On the other hand, we observe that Polyak stepsizes operate generally at the edge of stability or even slightly beyond, while outperforming its Armijo and constant stepsizes counterparts. We conclude with an analysis that suggests unlocking stepsize tuners requires an understanding of the joint dynamics of the step size and the sharpness.

Distributionally Robust Optimization with Bias and Variance Reduction

We consider the distributionally robust optimization (DRO) problem with spectral risk-based uncertainty set and $f$-divergence penalty. This formulation includes common risk-sensitive learning objectives such as regularized condition value-at-risk (CVaR) and average top-$k$ loss. We present Prospect, a stochastic gradient-based algorithm that only requires tuning a single learning rate hyperparameter, and prove that it enjoys linear convergence for smooth regularized losses. This contrasts with previous algorithms that either require tuning multiple hyperparameters or potentially fail to converge due to biased gradient estimates or inadequate regularization. Empirically, we show that Prospect can converge 2-3$\times$ faster than baselines such as stochastic gradient and stochastic saddle-point methods on distribution shift and fairness benchmarks spanning tabular, vision, and language domains.

Dual Gauss-Newton Directions for Deep Learning

Inspired by Gauss-Newton-like methods, we study the benefit of leveraging the structure of deep learning objectives, namely, the composition of a convex loss function and of a nonlinear network, in order to derive better direction oracles than stochastic gradients, based on the idea of partial linearization. In a departure from previous works, we propose to compute such direction oracles via their dual formulation, leading to both computational benefits and new insights. We demonstrate that the resulting oracles define descent directions that can be used as a drop-in replacement for stochastic gradients, in existing optimization algorithms. We empirically study the advantage of using the dual formulation as well as the computational trade-offs involved in the computation of such oracles.

Modified Gauss-Newton Algorithms under Noise

Gauss-Newton methods and their stochastic version have been widely used in machine learning and signal processing. Their nonsmooth counterparts, modified Gauss-Newton or prox-linear algorithms, can lead to contrasting outcomes when compared to gradient descent in large-scale statistical settings. We explore the contrasting performance of these two classes of algorithms in theory on a stylized statistical example, and experimentally on learning problems including structured prediction. In theory, we delineate the regime where the quadratic convergence of the modified Gauss-Newton method is active under statistical noise. In the experiments, we underline the versatility of stochastic (sub)-gradient descent to minimize nonsmooth composite objectives.

Stochastic Optimization for Spectral Risk Measures

Spectral risk objectives - also called $L$-risks - allow for learning systems to interpolate between optimizing average-case performance (as in empirical risk minimization) and worst-case performance on a task. We develop stochastic algorithms to optimize these quantities by characterizing their subdifferential and addressing challenges such as biasedness of subgradient estimates and non-smoothness of the objective. We show theoretically and experimentally that out-of-the-box approaches such as stochastic subgradient and dual averaging are hindered by bias and that our approach outperforms them.

Iterative Linear Quadratic Optimization for Nonlinear Control: Differentiable Programming Algorithmic Templates

We present the implementation of nonlinear control algorithms based on linear and quadratic approximations of the objective from a functional viewpoint. We present a gradient descent, a Gauss-Newton method, a Newton method, differential dynamic programming approaches with linear quadratic or quadratic approximations, various line-search strategies, and regularized variants of these algorithms. We derive the computational complexities of all algorithms in a differentiable programming framework and present sufficient optimality conditions. We compare the algorithms on several benchmarks, such as autonomous car racing using a bicycle model of a car. The algorithms are coded in a differentiable programming language in a publicly available package.

Target Propagation via Regularized Inversion

Target Propagation (TP) algorithms compute targets instead of gradients along neural networks and propagate them backward in a way that is similar yet different than gradient back-propagation (BP). The idea was first presented as a perturbative alternative to back-propagation that may achieve greater accuracy in gradient evaluation when training multi-layer neural networks (LeCun et al., 1989). However, TP has remained more of a template algorithm with many variations than a well-identified algorithm. Revisiting insights of LeCun et al., (1989) and more recently of Lee et al. (2015), we present a simple version of target propagation based on regularized inversion of network layers, easily implementable in a differentiable programming framework. We compare its computational complexity to the one of BP and delineate the regimes in which TP can be attractive compared to BP. We show how our TP can be used to train recurrent neural networks with long sequences on various sequence modeling problems. The experimental results underscore the importance of regularization in TP in practice.

Differentiable Programming à la Moreau

The notion of a Moreau envelope is central to the analysis of first-order optimization algorithms for machine learning. Yet, it has not been developed and extended to be applied to a deep network and, more broadly, to a machine learning system with a differentiable programming implementation. We define a compositional calculus adapted to Moreau envelopes and show how to integrate it within differentiable programming. The proposed framework casts in a mathematical optimization framework several variants of gradient back-propagation related to the idea of the propagation of virtual targets.