Randomized Geometric Algebra Methods for Convex Neural Networks

We introduce randomized algorithms to Clifford's Geometric Algebra, generalizing randomized linear algebra to hypercomplex vector spaces. This novel approach has many implications in machine learning, including training neural networks to global optimality via convex optimization. Additionally, we consider fine-tuning large language model (LLM) embeddings as a key application area, exploring the intersection of geometric algebra and modern AI techniques. In particular, we conduct a comparative analysis of the robustness of transfer learning via embeddings, such as OpenAI GPT models and BERT, using traditional methods versus our novel approach based on convex optimization. We test our convex optimization transfer learning method across a variety of case studies, employing different embeddings (GPT-4 and BERT embeddings) and different text classification datasets (IMDb, Amazon Polarity Dataset, and GLUE) with a range of hyperparameter settings. Our results demonstrate that convex optimization and geometric algebra not only enhances the performance of LLMs but also offers a more stable and reliable method of transfer learning via embeddings.

Stitching Sub-Trajectories with Conditional Diffusion Model for Goal-Conditioned Offline RL

Offline Goal-Conditioned Reinforcement Learning (Offline GCRL) is an important problem in RL that focuses on acquiring diverse goal-oriented skills solely from pre-collected behavior datasets. In this setting, the reward feedback is typically absent except when the goal is achieved, which makes it difficult to learn policies especially from a finite dataset of suboptimal behaviors. In addition, realistic scenarios involve long-horizon planning, which necessitates the extraction of useful skills within sub-trajectories. Recently, the conditional diffusion model has been shown to be a promising approach to generate high-quality long-horizon plans for RL. However, their practicality for the goal-conditioned setting is still limited due to a number of technical assumptions made by the methods. In this paper, we propose SSD (Sub-trajectory Stitching with Diffusion), a model-based offline GCRL method that leverages the conditional diffusion model to address these limitations. In summary, we use the diffusion model that generates future plans conditioned on the target goal and value, with the target value estimated from the goal-relabeled offline dataset. We report state-of-the-art performance in the standard benchmark set of GCRL tasks, and demonstrate the capability to successfully stitch the segments of suboptimal trajectories in the offline data to generate high-quality plans.

Convex Relaxations of ReLU Neural Networks Approximate Global Optima in Polynomial Time

In this paper, we study the optimality gap between two-layer ReLU networks regularized with weight decay and their convex relaxations. We show that when the training data is random, the relative optimality gap between the original problem and its relaxation can be bounded by a factor of $O(\sqrt{\log n})$, where $n$ is the number of training samples. A simple application leads to a tractable polynomial-time algorithm that is guaranteed to solve the original non-convex problem up to a logarithmic factor. Moreover, under mild assumptions, we show that with random initialization on the parameters local gradient methods almost surely converge to a point that has low training loss. Our result is an exponential improvement compared to existing results and sheds new light on understanding why local gradient methods work well.