Improving SGD convergence by tracing multiple promising directions and estimating distance to minimum
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Deep neural networks are usually trained with stochastic gradient descent (SGD), which optimizes θ∈R^D parameters to minimize objective function using very rough approximations of gradient, averaging to the real gradient. To improve its convergence, there is used some state representing the current situation, like momentum being local average of gradients - gradient calculated in successive step updates the state, which directly governs the update of θ parameters. This high dimensional optimization might have more than one direction worth tracing, also the used methods do not try to estimate distance to the minimum for adapting the step size. Modelling Hessian would try to trace all the directions, also estimating distance to minimum, however, it is seen as too costly. We can reduce this representation to make it more practical: try to model some d≪ D of its most promising (eigen)directions, using d which provides the best tradeoff. There will be discussed using state as parametrisation with approximated Hessian: H≈∑_i=1^d λ_i v_i v_i^T, preferably using only positive eigenvalues, trying to model the lowest valley nearby. In every step we can use the calculated gradient to update such local model, then shift the θ parameters toward closest minimum of such approximation of local behavior of objective function. Comparing to the use of single momentum vector, it would remember and update multiple (d) promising directions, and adapt step size using estimated distance to the minimum.
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