Self-attention is essential to Transformer architectures, yet how information is embedded in the self-attention matrices and how different objective functions impact this process remains unclear. We present a mathematical framework to analyze self-attention matrices by deriving the structures governing their weight updates. Using this framework, we demonstrate that bidirectional training induces symmetry in the weight matrices, while autoregressive training results in directionality and column dominance. Our theoretical findings are validated across multiple Transformer models — including ModernBERT, GPT, LLaMA3, and Mistral — and input modalities like text, vision, and audio. Finally, we apply these insights by showing that symmetric initialization improves the performance of encoder-only models on language tasks. This mathematical analysis offers a novel theoretical perspective on how information is embedded through self-attention, thereby improving the interpretability of Transformer models.

Identifying unfamiliar inputs, also known as out-of-distribution (OOD) detection, is a crucial property of any decision making process. A simple and empirically validated technique is based on deep ensembles where the variance of predictions over different neural networks acts as a substitute for input uncertainty. Nevertheless, a theoretical understanding of the inductive biases leading to the performance of deep ensemble's uncertainty estimation is missing. To improve our description of their behavior, we study deep ensembles with large layer widths operating in simplified linear training regimes, in which the functions trained with gradient descent can be described by the neural tangent kernel. We identify two sources of noise, each inducing a distinct inductive bias in the predictive variance at initialization. We further show theoretically and empirically that both noise sources affect the predictive variance of non-linear deep ensembles in toy models and realistic settings after training. Finally, we propose practical ways to eliminate part of these noise sources leading to significant changes and improved OOD detection in trained deep ensembles.

The success of deep learning sparked interest in whether the brain learns by using similar techniques for assigning credit to each synaptic weight for its contribution to the network output. However, the majority of current attempts at biologically-plausible learning methods are either non-local in time, require highly specific connectivity motifs, or have no clear link to any known mathematical optimization method. Here, we introduce Deep Feedback Control (DFC), a new learning method that uses a feedback controller to drive a deep neural network to match a desired output target and whose control signal can be used for credit assignment. The resulting learning rule is fully local in space and time and approximates Gauss-Newton optimization for a wide range of feedback connectivity patterns. To further underline its biological plausibility, we relate DFC to a multi-compartment model of cortical pyramidal neurons with a local voltage-dependent synaptic plasticity rule, consistent with recent theories of dendritic processing. By combining dynamical system theory with mathematical optimization theory, we provide a strong theoretical foundation for DFC that we corroborate with detailed results on toy experiments and standard computer-vision benchmarks.

Since embedded systems design involves stringent design constraints, designing a system for reliability requires optimization under tolerable overhead constraints. This paper presents a novel reliability-driven compilation scheme for software program reliability optimization under tolerable overhead constraints. Our scheme exploits program-level error masking and propagation properties to perform reliability-driven prioritization of instructions and selective protection during compilation. To enable this, we develop statistical models for estimating error masking and propagation probabilities. Our scheme provides significant improvement in reliability efficiency (avg. 30%-60%) compared to state-of-the-art program-level protection schemes.

State-of-the-art reliability optimizing schemes deploy spatial or temporal redundancy for the complete functionality. This introduces significant performance/area overhead which is often prohibitive within the stringent design constraints of embedded systems. This paper presents a novel scheme for selective software reliability optimization constraint under user-provided tolerable performance overhead constraint. To enable this scheme, statistical models for quantifying software resilience and error masking properties at function and instruction level are proposed. These models leverage a whole new range of reliability optimization. Given a tolerable performance overhead, our scheme selectively protects the reliability-wise most important instructions based on their masking probability, vulnerability, and redundancy overhead. Compared to state-of-the-art [7], our scheme provides a 4.84X improved reliability at 50% tolerable performance overhead constraint.