Bilevel optimization has emerged as a technique for addressing a wide range of machine learning problems that involve an outer objective implicitly determined by the minimizer of an inner problem. In this paper, we investigate the generalization properties for kernel bilevel optimization problems where the inner objective is optimized over a Reproducing Kernel Hilbert Space. This setting enables rich function approximation while providing a foundation for rigorous theoretical analysis. In this context, we establish novel generalization error bounds for the bilevel problem under finite-sample approximation. Our approach adopts a functional perspective, inspired by (Petrulionyte et al., 2024), and leverages tools from empirical process theory and maximal inequalities for degenerate $U$-processes to derive uniform error bounds. These generalization error estimates allow to characterize the statistical accuracy of gradient-based methods applied to the empirical discretization of the bilevel problem.

In this empirical article, we introduce INNAprop, an optimization algorithm that combines the INNA method with the RMSprop adaptive gradient scaling. After giving geometrical insights, we evaluate INNAprop on CIFAR-10, Food101, and ImageNet with ResNets, VGG, DenseNet, and ViT, and on GPT-2 (OpenWeb-Text) train from scratch and with LoRA fine-tuning (E2E). INNAprop consistently matches or outperforms AdamW both in training speed and accuracy, with minimal hyperparameter tuning in large-scale settings. As deep learning models grow in size, massive computational resources are needed for training, representing significant challenges in terms of financial costs, energy consumption, and processing time (Susnjak et al., 2024; Varoquaux et al., 2024). According to the UN's Environment Programme Training, the Big Tech sector produced between two and three percent of the world's carbon emissions in 2021; some estimations for the year 2023 go beyond 4%, see the latest Stand.earth For instance, training GPT-3 is estimated to require 1,287 megawatt-hours (MWh) of electricity, equivalent to the annual usage of over 100 U.S. households (Anthony et al., 2020; Patterson et al., 2021). Similarly, the financial cost of specialized hardware and cloud computing is extremely high. OpenAI claimed that the training cost for GPT-4 (Achiam et al., 2023) exceeded 100 million dollars.

The differentiation of iterative algorithms has been a subject of research since the 1990s (Gilbert, 1992; Christianson, 1994; Beck, 1994), and was succinctly described as "piggyback differentiation" by Griewank and Faure (2003). This idea has gained renewed interest within the machine learning community, particularly for applications such as hyperparameter optimization (Maclaurin et al., 2015; Franceschi et al., 2017), metalearning (Finn et al., 2017; Rajeswaran et al., 2019), and learning discretization of total variation (Chambolle and Pock, 2021; Bogensperger et al., 2022). When applied to an optimization problem, an important theoretical concern is the convergence of the derivatives of iterates to the derivatives of the solution. Traditional guarantees focus on asymptotic convergence to the solution derivative, as described by the implicit function theorem (Gilbert, 1992; Christianson, 1994; Beck, 1994). This issue has inspired recent works for smooth optimization algorithms (Mehmood and Ochs, 2020, 2022), generic nonsmooth iterations (Bolte et al., 2022), and second-order methods (Bolte et al., 2023).

Motivated by the widespread use of approximate derivatives in machine learning and optimization, we study inexact subgradient methods with non-vanishing additive errors and step sizes. In the nonconvex semialgebraic setting, under boundedness assumptions, we prove that the method provides points that eventually fluctuate close to the critical set at a distance proportional to $\epsilon^\rho$ where $\epsilon$ is the error in subgradient evaluation and $\rho$ relates to the geometry of the problem. In the convex setting, we provide complexity results for the averaged values. We also obtain byproducts of independent interest, such as descent-like lemmas for nonsmooth nonconvex problems and some results on the limit of affine interpolants of differential inclusions.

Differentiating the solution of a machine learning problem is a important task, e.g., in hyperparameters optimization [9], in neural architecture search [26] and when using convex layers [3]. There are two main ways to achieve this goal: automatic differentiation (AD) and implicit differentiation (ID). Automatic differentiation implements the idea of evaluating derivatives through the compositional rules of differential calculus in a user-transparent way. It is a mature concept [23] implemented in several machine learning frameworks [31, 16, 1]. However, the time and memory complexity incurred may become prohibitive as soon as the computational graph becomes bigger, a typical example being unrolling iterative optimization algorithms such as gradient descent [5]. The alternative, implicit differentiation, is not always accessible: it does not solely relies on the compositional rules of differential calculus and usually requires solving a linear system. The user needs to implement custom rules in an automatic differentiation framework (as done, for example, in [4]) or use dedicated libraries such as [11, 3, 10] implementing these rules for given models. Provided that the implementation is carefully done, this is most of the time the gold standard for the task of differentiating problem solutions.

Using the notion of conservative gradient, we provide a simple model to estimate the computational costs of the backward and forward modes of algorithmic differentiation for a wide class of nonsmooth programs. The overhead complexity of the backward mode turns out to be independent of the dimension when using programs with locally Lipschitz semi-algebraic or definable elementary functions. This considerably extends Baur-Strassen's smooth cheap gradient principle. We illustrate our results by establishing fast backpropagation results of conservative gradients through feedforward neural networks with standard activation and loss functions. Nonsmooth backpropagation's cheapness contrasts with concurrent forward approaches, which have, to this day, dimensional-dependent worst-case overhead estimates. We provide further results suggesting the superiority of backward propagation of conservative gradients. Indeed, we relate the complexity of computing a large number of directional derivatives to that of matrix multiplication, and we show that finding two subgradients in the Clarke subdifferential of a function is an NP-hard problem.

Minibatch decomposition methods for empirical risk minimization are commonly analysed in a stochastic approximation setting, also known as sampling with replacement. On the other hands modern implementations of such techniques are incremental: they rely on sampling without replacement, for which available analysis are much scarcer. We provide convergence guaranties for the latter variant by analysing a versatile incremental gradient scheme. For this scheme, we consider constant, decreasing or adaptive step sizes. In the smooth setting we obtain explicit complexity estimates in terms of epoch counter. In the nonsmooth setting we prove that the sequence is attracted by solutions of optimality conditions of the problem.

We show that the derivatives of the Sinkhorn-Knopp algorithm, or iterative proportional fitting procedure, converge towards the derivatives of the entropic regularization of the optimal transport problem with a locally uniform linear convergence rate.

In theory, the choice of ReLU'(0) in [0, 1] for a neural network has a negligible influence both on backpropagation and training. Yet, in the real world, 32 bits default precision combined with the size of deep learning problems makes it a hyperparameter of training methods. We investigate the importance of the value of ReLU'(0) for several precision levels (16, 32, 64 bits), on various networks (fully connected, VGG, ResNet) and datasets (MNIST, CIFAR10, SVHN). We observe considerable variations of backpropagation outputs which occur around half of the time in 32 bits precision. The effect disappears with double precision, while it is systematic at 16 bits. For vanilla SGD training, the choice ReLU'(0) = 0 seems to be the most efficient. We also evidence that reconditioning approaches as batch-norm or ADAM tend to buffer the influence of ReLU'(0)'s value. Overall, the message we want to convey is that algorithmic differentiation of nonsmooth problems potentially hides parameters that could be tuned advantageously.

Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning. In this work we articulate the relationships between differentiation of programs as implemented in practice and differentiation of nonsmooth functions. To this end we provide a simple class of functions, a nonsmooth calculus, and show how they apply to stochastic approximation methods. We also evidence the issue of artificial critical points created by algorithmic differentiation and show how usual methods avoid these points with probability one.