Saddle-point models arise throughout optimization, optimal transport, robust learning, and control. In many applications, the relevant function f(x,y) is convex in x and concave in y, and preserving this geometry is essential for obtaining tractable min--max formulations and reliable certificates. We introduce a structured separable decomposition that preserves the convex-concave geometry and prove a complete one-dimensional approximation theorem under a mixed Monge-type convexity condition. We then describe practical saddle network architectures that preserve convexity in x and concavity in y by construction. The proposed architectures require only convexity-preserving neural networks, together with simple output transformations enforcing sign and concavity constraints. Finally, we report numerical benchmarks in dimension 1 and 5, showing that the proposed saddle networks achieve high accuracy on smooth, nonsmooth, and high-rank convex--concave test functions.

Hyperbolic spaces have been quite popular in the recent past for representing hierarchically organized data. Further, several classification algorithms for data in these spaces have been proposed in the literature. These algorithms mainly use either hyperplanes or geodesics for decision boundaries in a large margin classifiers setting leading to a non-convex optimization problem. In this paper, we propose a novel large margin classifier based on horospherical decision boundaries that leads to a geodesically convex optimization problem that can be optimized using any Riemannian gradient descent technique guaranteeing a globally optimal solution.

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

We consider the parallel complexity of submodular function minimization (SFM). We provide a pair of methods which obtain two new query versus depth tradeoffs a submodular function defined on subsets of n elements that has integer values between M and M. The first method has depth 2 and query complexity

Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs--namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.

Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution that best explain the observed data. In the context of text generation, MLE is often used to train generative language models, which can then be used to generate new text. However, we argue that MLE is not always necessary and optimal, especially for closed-ended text generation tasks like machine translation. In these tasks, the goal of model is to generate the most appropriate response, which does not necessarily require it to estimate the entire data distribution with MLE. To this end, we propose a novel class of training objectives based on convex functions, which enables text generation models to focus on highly probable outputs without having to estimate the entire data distribution. We investigate the theoretical properties of the optimal predicted distribution when applying convex functions to the loss, demonstrating that convex functions can sharpen the optimal distribution, thereby enabling the model to better capture outputs with high probabilities. Experiments on various text generation tasks and models show the effectiveness of our approach. It enables autoregressive models to bridge the gap between greedy and beam search, and facilitates the learning of non-autoregressive models with a maximum improvement of 9+ BLEU points. Moreover, our approach also exhibits significant impact on large language models (LLMs), substantially enhancing their generative capability on various tasks.

We revisit first-order optimization under local information constraints such as local privacy, gradient quantization, and computational constraints limiting access to a few coordinates of the gradient. In this setting, the optimization algorithm is not allowed to directly access the complete output of the gradient oracle, but only gets limited information about it subject to the local information constraints. We study the role of adaptivity in processing the gradient output to obtain this limited information from it. We consider optimization for both convex and strongly convex functions and obtain tight or nearly tight lower bounds for the convergence rate, when adaptive gradient processing is allowed. Prior work was restricted to convex functions and allowed only nonadaptive processing of gradients. For both of these function classes and for the three information constraints mentioned above, our lower bound implies that adaptive processing of gradients cannot outperform nonadaptive processing in most regimes of interest. We complement these results by exhibiting a natural optimization problem under information constraints for which adaptive processing of gradient strictly outperforms nonadaptive processing.