Comparing BFGS and OGR for Second-Order Optimization

Across standard test functions and ablations with/without line search, OGR variants match or outperform BFGS in final objective and step efficiency, with particular gains in nonconvex landscapes where saddle handling matters. Exact Hessians (via AD) are used only as an oracle baseline to evaluate estimation quality, not to form steps. II. Online Gradient Regression (OGR) Online Gradient Regression (OGR) is a second-order optimization framework that accelerates stochastic gradient descent (SGD) by online least-squares regression of noisy gradients to infer local curvature and the distance to a stationary point [3]. The central assumption is that, in a small neighborhood, the objective F (θ) is well-approximated by a quadratic model, so the gradient varies approximately linearly with the parameters. OGR maintains exponentially weighted statistics of recent (θ t, g t) pairs and updates a local model each iteration at negligible extra cost compared to computing the gradient itself [2], [3]. A. Direct multivariate approach In given time T, based on recent gradients g t R d and positions θ t R d for t < T, we would like to locally approximate behavior with 2nd order polynomial using parametrization: f (θ) = h + 1 2 (θ p) T H(θ p) f = H(θ p) for Hessian H R d d and p R d position of saddle or extremum. For local behavior we will work on averages with weights w t further decreasing exponentially, defining averages: v null t