A Theoretical Analysis of Joint Weight–Bias Hessian-Based Initialization Strategies in Deep Learning: Impact on Model Convergence, Curvature, and Accuracy
Keywords:
neural network initialization, Hessian, bias initialization, curvature, variance preservation, second-order optimization, loss landscape, joint parameterization.Abstract
Conventional neural-network initialization is largely motivated by controlling activation and gradient moments across layers. Xavier/Glorot and He/Kaiming provide effective variance-based rules, but variance preservation alone does not specify how initialization relates to local optimization geometry. Curvature-aware analyses provide a complementary second-order view, while related work has examined scaling, bias behavior, initialization geometry, and mathematically constrained initialization. This paper develops a theoretical Joint Weight–Bias Hessian Initialization Framework in which the weights and biases of a layer are represented by one parameter vector and analyzed through the complete block Hessian. A variance-allocation coordinate r controls how a fixed target pre-activation variance is divided between weights and bias, and a normalized curvature–variance criterion provides a common objective for studying that allocation. The scalar linear-quadratic model is solved exactly, including the joint Hessian, its spectrum, and the variance-constrained family. An important boundary result is established: under standardized scalar assumptions, J(r)=0 for every feasible r, so the scalar model cannot identify a unique allocation. This is a limitation of the simplified model rather than evidence against the framework. The paper therefore presents a mathematically explicit, testable framework and a controlled future protocol for nonlinear MLP and CNN validation, without claiming empirical accuracy or convergence improvements.
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