As the problem of minimizing functionals on the Wasserstein space encompasses many applications in machine learning, different optimization algorithms on $\mathbb{R}^d$ have received their counterpart analog on the Wasserstein space. We focus here on lifting two explicit algorithms: mirror descent and preconditioned gradient descent. These algorithms have been introduced to better capture the geometry of the function to minimize and are provably convergent under appropriate (namely relative) smoothness and convexity conditions. Adapting these notions to the Wasserstein space, we prove guarantees of convergence of some Wasserstein-gradient-based discrete-time schemes for new pairings of objective functionals and regularizers. The difficulty here is to carefully select along which curves the functionals should be smooth and convex. We illustrate the advantages of adapting the geometry induced by the regularizer on ill-conditioned optimization tasks, and showcase the improvement of choosing different discrepancies and geometries in a computational biology task of aligning single-cells.

Automatically designing fast and space-efficient digital circuits is challenging because circuits are discrete, must exactly implement the desired logic, and are costly to simulate. We address these challenges with CircuitVAE, a search algorithm that embeds computation graphs in a continuous space and optimizes a learned surrogate of physical simulation by gradient descent. By carefully controlling overfitting of the simulation surrogate and ensuring diverse exploration, our algorithm is highly sample-efficient, yet gracefully scales to large problem instances and high sample budgets. We test CircuitVAE by designing binary adders across a large range of sizes, IO timing constraints, and sample budgets. Our method excels at designing large circuits, where other algorithms struggle: compared to reinforcement learning and genetic algorithms, CircuitVAE typically finds 64-bit adders which are smaller and faster using less than half the sample budget. We also find CircuitVAE can design state-of-the-art adders in a real-world chip, demonstrating that our method can outperform commercial tools in a realistic setting.

Machine learning models often incorporate vast amounts of data, raising significant privacy concerns. Machine unlearning, the ability to remove the influence of specific data points from a trained model, addresses these concerns. This paper explores practical methods for implementing machine unlearning, focusing on a first-epoch gradient-ascent approach. Key findings include: 1. Single vs. Multi-Epoch Unlearning: First-epoch gradient unlearning is more effective than multi-epoch gradients. 2. Layer-Based Unlearning: The embedding layer in GPT-2 is crucial for effective unlearning. Gradients from the output layers (11 and 12) have no impact. Efficient unlearning can be achieved using only the embedding layer, halving space complexity. 3. Influence Functions & Scoring: Techniques like Hessian Vector Product and the dot product of activations and tensors are used for quantifying unlearning. 4. Gradient Ascent Considerations: Calibration is necessary to avoid overexposing the model to specific data points during unlearning, which could prematurely terminate the process. 5. Fuzzy Matching vs. Iterative Unlearning: Fuzzy matching techniques shift the model to a new optimum, while iterative unlearning provides a more complete modality. Our empirical evaluation confirms that first-epoch gradient ascent for machine unlearning is more effective than whole-model gradient ascent. These results highlight the potential of machine unlearning for enhancing data privacy and compliance with regulations such as GDPR and CCPA. The study underscores the importance of formal methods to comprehensively evaluate the unlearning process.

By leveraging the kernel trick in the output space, kernel-induced losses provide a principled way to define structured output prediction tasks for a wide variety of output modalities. In particular, they have been successfully used in the context of surrogate non-parametric regression, where the kernel trick is typically exploited in the input space as well. However, when inputs are images or texts, more expressive models such as deep neural networks seem more suited than non-parametric methods. In this work, we tackle the question of how to train neural networks to solve structured output prediction tasks, while still benefiting from the versatility and relevance of kernel-induced losses. We design a novel family of deep neural architectures, whose last layer predicts in a data-dependent finite-dimensional subspace of the infinite-dimensional output feature space deriving from the kernel-induced loss. This subspace is chosen as the span of the eigenfunctions of a randomly-approximated version of the empirical kernel covariance operator. Interestingly, this approach unlocks the use of gradient descent algorithms (and consequently of any neural architecture) for structured prediction. Experiments on synthetic tasks as well as real-world supervised graph prediction problems show the relevance of our method.

In this paper, we examine the long-run distribution of stochastic gradient descent (SGD) in general, non-convex problems. Specifically, we seek to understand which regions of the problem's state space are more likely to be visited by SGD, and by how much. Using an approach based on the theory of large deviations and randomly perturbed dynamical systems, we show that the long-run distribution of SGD resembles the Boltzmann-Gibbs distribution of equilibrium thermodynamics with temperature equal to the method's step-size and energy levels determined by the problem's objective and the statistics of the noise. In particular, we show that, in the long run, (a) the problem's critical region is visited exponentially more often than any non-critical region; (b) the iterates of SGD are exponentially concentrated around the problem's minimum energy state (which does not always coincide with the global minimum of the objective); (c) all other connected components of critical points are visited with frequency that is exponentially proportional to their energy level; and, finally (d) any component of local maximizers or saddle points is "dominated" by a component of local minimizers which is visited exponentially more often.

While it is commonly observed in practice that pruning networks to a certain level of sparsity can improve the quality of the features, a theoretical explanation of this phenomenon remains elusive. In this work, we investigate this by demonstrating that a broad class of statistical models can be optimally learned using pruned neural networks trained with gradient descent, in high-dimensions. We consider learning both single-index and multi-index models of the form $y = \sigma^*(\boldsymbol{V}^{\top} \boldsymbol{x}) + \epsilon$, where $\sigma^*$ is a degree-$p$ polynomial, and $\boldsymbol{V} \in \mathbbm{R}^{d \times r}$ with $r \ll d$, is the matrix containing relevant model directions. We assume that $\boldsymbol{V}$ satisfies a certain $\ell_q$-sparsity condition for matrices and show that pruning neural networks proportional to the sparsity level of $\boldsymbol{V}$ improves their sample complexity compared to unpruned networks. Furthermore, we establish Correlational Statistical Query (CSQ) lower bounds in this setting, which take the sparsity level of $\boldsymbol{V}$ into account. We show that if the sparsity level of $\boldsymbol{V}$ exceeds a certain threshold, training pruned networks with a gradient descent algorithm achieves the sample complexity suggested by the CSQ lower bound. In the same scenario, however, our results imply that basis-independent methods such as models trained via standard gradient descent initialized with rotationally invariant random weights can provably achieve only suboptimal sample complexity.

A popular method to perform adversarial attacks on neuronal networks is the so-called fast gradient sign method and its iterative variant. In this paper, we interpret this method as an explicit Euler discretization of a differential inclusion, where we also show convergence of the discretization to the associated gradient flow. To do so, we consider the concept of p-curves of maximal slope in the case $p=\infty$. We prove existence of $\infty$-curves of maximum slope and derive an alternative characterization via differential inclusions. Furthermore, we also consider Wasserstein gradient flows for potential energies, where we show that curves in the Wasserstein space can be characterized by a representing measure on the space of curves in the underlying Banach space, which fulfill the differential inclusion. The application of our theory to the finite-dimensional setting is twofold: On the one hand, we show that a whole class of normalized gradient descent methods (in particular signed gradient descent) converge, up to subsequences, to the flow, when sending the step size to zero. On the other hand, in the distributional setting, we show that the inner optimization task of adversarial training objective can be characterized via $\infty$-curves of maximum slope on an appropriate optimal transport space.

The recent emergence of diffusion models has significantly advanced the precision of learnable priors, presenting innovative avenues for addressing inverse problems. Since inverse problems inherently entail maximum a posteriori estimation, previous works have endeavored to integrate diffusion priors into the optimization frameworks. However, prevailing optimization-based inverse algorithms primarily exploit the prior information within the diffusion models while neglecting their denoising capability. To bridge this gap, this work leverages the diffusion process to reframe noisy inverse problems as a two-variable constrained optimization task by introducing an auxiliary optimization variable. By employing gradient truncation, the projection gradient descent method is efficiently utilized to solve the corresponding optimization problem. The proposed algorithm, termed ProjDiff, effectively harnesses the prior information and the denoising capability of a pre-trained diffusion model within the optimization framework. Extensive experiments on the image restoration tasks and source separation and partial generation tasks demonstrate that ProjDiff exhibits superior performance across various linear and nonlinear inverse problems, highlighting its potential for practical applications. Code is available at https://github.com/weigerzan/ProjDiff/.

We study the generalization of two-layer ReLU neural networks in a univariate nonparametric regression problem with noisy labels. This is a problem where kernels (\emph{e.g.} NTK) are provably sub-optimal and benign overfitting does not happen, thus disqualifying existing theory for interpolating (0-loss, global optimal) solutions. We present a new theory of generalization for local minima that gradient descent with a constant learning rate can \emph{stably} converge to. We show that gradient descent with a fixed learning rate $\eta$ can only find local minima that represent smooth functions with a certain weighted \emph{first order total variation} bounded by $1/\eta - 1/2 + \widetilde{O}(\sigma + \sqrt{\mathrm{MSE}})$ where $\sigma$ is the label noise level, $\mathrm{MSE}$ is short for mean squared error against the ground truth, and $\widetilde{O}(\cdot)$ hides a logarithmic factor. Under mild assumptions, we also prove a nearly-optimal MSE bound of $\widetilde{O}(n^{-4/5})$ within the strict interior of the support of the $n$ data points. Our theoretical results are validated by extensive simulation that demonstrates large learning rate training induces sparse linear spline fits. To the best of our knowledge, we are the first to obtain generalization bound via minima stability in the non-interpolation case and the first to show ReLU NNs without regularization can achieve near-optimal rates in nonparametric regression.

The usual way to interpret language models (LMs) is to test their performance on different benchmarks and subsequently infer their internal processes. In this paper, we present an alternative approach, concentrating on the quality of LM processing, with a focus on their language abilities. To this end, we construct 'linguistic task spaces' -- representations of an LM's language conceptualisation -- that shed light on the connections LMs draw between language phenomena. Task spaces are based on the interactions of the learning signals from different linguistic phenomena, which we assess via a method we call 'similarity probing'. To disentangle the learning signals of linguistic phenomena, we further introduce a method called 'fine-tuning via gradient differentials' (FTGD). We apply our methods to language models of three different scales and find that larger models generalise better to overarching general concepts for linguistic tasks, making better use of their shared structure. Further, the distributedness of linguistic processing increases with pre-training through increased parameter sharing between related linguistic tasks. The overall generalisation patterns are mostly stable throughout training and not marked by incisive stages, potentially explaining the lack of successful curriculum strategies for LMs.