Given a training set, a loss function, and a neural network architecture, it is often taken for granted that optimal network parameters exist, and a common practice is to apply available optimization algorithms to search for them. In this work, we show that the existence of an optimal solution is not always guaranteed, especially in the context of sparse ReLU neural networks.In particular, we first show that optimization problems involving deep networks with certain sparsity patterns do not always have optimal parameters, and that optimization algorithms may then diverge. Via a new topological relation between sparse ReLU neural networks and their linear counterparts, we derive --using existing tools from real algebraic geometry-- an algorithm to verify that a given sparsity pattern suffers from this issue. Then, the existence of a global optimum is proved for every concrete optimization problem involving a shallow sparse ReLU neural network of output dimension one. Overall, the analysis is based on the investigation of two topological properties of the space of functions implementable as sparse ReLU neural networks: a best approximation property, and a closedness property, both in the uniform norm. This is studied both for (finite) domains corresponding to practical training on finite training sets, and for more general domains such as the unit cube. This allows us to provide conditions for the guaranteed existence of an optimum given a sparsity pattern. The results apply not only to several sparsity patterns proposed in recent works on network pruning/sparsification, but also to classical dense neural networks, including architectures not covered by existing results.

First-order optimization methods tend to inherently favor certain solutions over others when minimizing a given training objective with multiple local optima. This phenomenon, known as \emph{implicit bias}, plays a critical role in understanding the generalization capabilities of optimization algorithms. Recent research has revealed that gradient-descent-based methods exhibit an implicit bias for the $\ell_2$-maximal margin classifier in the context of separable binary classification. In contrast, generic optimization methods, such as mirror descent and steepest descent, have been shown to converge to maximal margin classifiers defined by alternative geometries. However, while gradient-descent-based algorithms demonstrate fast implicit bias rates, the implicit bias rates of generic optimization methods have been relatively slow. To address this limitation, in this paper, we present a series of state-of-the-art implicit bias rates for mirror descent and steepest descent algorithms. Our primary technique involves transforming a generic optimization algorithm into an online learning dynamic that solves a regularized bilinear game, providing a unified framework for analyzing the implicit bias of various optimization methods. The accelerated rates are derived leveraging the regret bounds of online learning algorithms within this game framework.

We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations.

For instance, the US Federal Reserve publishes synthetic market stress scenarios given by the constrained time series for financial institutions to assess their performance in hypothetical recessions.Existing approaches for generating constrained time series usually penalize training loss to enforce constraints, and reject non-conforming samples. However, these approaches would require re-training if we change constraints, and rejection sampling can be computationally expensive, or impractical for complex constraints.In this paper, we propose a novel set of methods to tackle the constrained time series generation problem and provide efficient sampling while ensuring the realism of generated time series. In particular, we frame the problem using a constrained optimization framework and then we propose a set of generative methods including'GuidedDiffTime', a guided diffusion model. We empirically evaluate our work on several datasets for financial and energy data, where incorporating constraints is critical. We show that our approaches outperform existing work both qualitatively and quantitatively, and that'GuidedDiffTime' does not require re-training for new constraints, resulting in a significant carbon footprint reduction, up to 92% w.r.t.

Optimal auction design is a fundamental problem in algorithmic game theory. This problem is notoriously difficult already in very simple settings. Recent work in differentiable economics showed that neural networks can efficiently learn known optimal auction mechanisms and discover interesting new ones. In an attempt to theoretically justify their empirical success, we focus on one of the first such networks, RochetNet, and a generalized version for affine maximizer auctions. We prove that they satisfy mode connectivity, i.e., locally optimal solutions are connected by a simple, piecewise linear path such that every solution on the path is almost as good as one of the two local optima. Mode connectivity has been recently investigated as an intriguing empirical and theoretically justifiable property of neural networks used for prediction problems. Our results give the first such analysis in the context of differentiable economics, where neural networks are used directly for solving non-convex optimization problems.

Generative models have significantly influenced both vision and language domains, ushering in innovative multimodal applications. Although these achievements have motivated exploration in scientific and engineering fields, challenges emerge, particularly in constrained settings with limited data where precision is crucial. Traditional engineering optimization methods rooted in physics often surpass generative models in these contexts. To address these challenges, we introduce Diffusion Optimization Models (DOM) and Trajectory Alignment (TA), a learning framework that demonstrates the efficacy of aligning the sampling trajectory of diffusion models with the trajectory derived from physics-based iterative optimization methods. This alignment ensures that the sampling process remains grounded in the underlying physical principles.

Coreset is a small set that provides a data summary for a large dataset, such that training solely on the small set achieves competitive performance compared with a large dataset. In rehearsal-based continual learning, the coreset is typically used in the memory replay buffer to stand for representative samples in previous tasks, and the coreset selection procedure is typically formulated as a bilevel problem. However, the typical bilevel formulation for coreset selection explicitly performs optimization over discrete decision variables with greedy search, which is computationally expensive. Several works consider other formulations to address this issue, but they ignore the nested nature of bilevel optimization problems and may not solve the bilevel coreset selection problem accurately. To address these issues, we propose a new bilevel formulation, where the inner problem tries to find a model which minimizes the expected training error sampled from a given probability distribution, and the outer problem aims to learn the probability distribution with approximately $K$ (coreset size) nonzero entries such that learned model in the inner problem minimizes the training error over the whole data. To ensure the learned probability has approximately $K$ nonzero entries, we introduce a novel regularizer based on the smoothed top-$K$ loss in the upper problem.

The strength of modern generative models lies in their ability to be controlled through prompts. Hard prompts comprise interpretable words and tokens, and are typically hand-crafted by humans. Soft prompts, on the other hand, consist of continuous feature vectors. These can be discovered using powerful optimization methods, but they cannot be easily edited, re-used across models, or plugged into a text-based interface. We describe an easy-to-use approach to automatically optimize hard text prompts through efficient gradient-based optimization. Our approach can be readily applied to text-to-image and text-only applications alike. This method allows API users to easily generate, discover, and mix and match image concepts without prior knowledge of how to prompt the model. Furthermore, using our method, we can bypass token-level content filters imposed by Midjourney by optimizing through the open-sourced text encoder.

We study the convergence behavior of the celebrated temporal-difference (TD) learning algorithm. By looking at the algorithm through the lens of optimization, we first argue that TD can be viewed as an iterative optimization algorithm where the function to be minimized changes per iteration. By carefully investigating the divergence displayed by TD on a classical counter example, we identify two forces that determine the convergent or divergent behavior of the algorithm. We next formalize our discovery in the linear TD setting with quadratic loss and prove that convergence of TD hinges on the interplay between these two forces. We extend this optimization perspective to prove convergence of TD in a much broader setting than just linear approximation and squared loss. Our results provide a theoretical explanation for the successful application of TD in reinforcement learning.

Linear scalarization, i.e., combining all loss functions by a weighted sum, has been the default choice in the literature of multi-task learning (MTL) since its inception. In recent years, there is a surge of interest in developing Specialized Multi-Task Optimizers (SMTOs) that treat MTL as a multi-objective optimization problem. However, it remains open whether there is a fundamental advantage of SMTOs over scalarization. In fact, heated debates exist in the community comparing these two types of algorithms, mostly from an empirical perspective. To approach the above question, in this paper, we revisit scalarization from a theoretical perspective.