(The convergence proof of SGD for nonconvex problems indeed assumes three times differentiable cost functions.) As a typical case in machine learning, we use SGD in a setting where it is not guaranteed to converge in theory, but behaves well in practice, as shown in our experiments. Remark on SGD convergence and sampling strategy. We adopt the feature-sign algorithm for efficiently solving the exact solution to the sparse coding problem. The most computationally intensive step in Algorithm 2.1 is solving the sparse coding (step 3). The proof of the above equations is given in the Appendix. (2.12) β S C ⁎ = 0, β S ⁎ = ( D S T D S + λ 2 I ) − 1 ∇ A S ,Īnd S are the indices of the nonzero coefficients of A. A middle ground to both is the method considering a mini-batch stochastic gradient descent, where the loss is computed over a small batch of samples, thus preventing weight updates from being too erratic and keeping the process computationally manageable at the same time. This is known as batch gradient descent.Įven though this is theoretically more acceptable, the computational overhead increases significantly for a higher number of samples, which is less than linear returns. However, a gradient corresponding to a combined error over all samples more updates the weight in a much more appropriate direction. Weight updates due to single samples can be erratic. This can be achieved by calculating a combined error on all training samples and finding a resultant gradient, which is a much better option for a globally acceptable weight update. However, in machine learning a statistically sound approach is to consider high number of training samples to reduce uncertainty. This is in concordance with standard optimization protocol. The most straightforward way to implement this is to pass one sample through the network, compute the error, calculate the gradients, and then update the weights. However, unlike the pseudo-inverse approach, the Lyapunov method does not guarantee that the primary task of end-effector positioning will be accomplished with zero error while at the same time exploiting redundancy to satisfy secondary tasks. The gradient descent method is more computationally attractive than the pseudo-inverse approach because J T is much more straightforward to compute than J + (although some of the extensions given in give up some, or most, of this efficiency). The method can be readily extended to incorporate null-space motions for the purpose of satisfying additional subtasks. However, usually A is taken to be a constant matrix (e.g., A = αI, 0 < α ∈ ℝ, in the standard gradient descent algorithm), and historically (at least in the robotics literature) a distinction between the pseudo-inverse methods and the Lyapunov/gradient descent methods has been maintained.Īs described, the Lyapunov method does not exploit manipulator redundancy. Thus the pseudo-inverse methods are actually special cases of the more general class of Lyapunov methods. This is just the pseudo-inverse method if one takes x ˙ = x ˜ = x − x d. If the manipulator Jacobian, J, is full rank (the arm is not at a kinematic singularity), we can take Q = I and A = −1, yielding θ ˙ = − J T ( J J T ) − 1 x ˜ = − J + x ˜. If derivatives are replaced with finite differences and Q = I, this method becomes equivalent to the iterative gradient descent method. A fixed point may be reached with nonzero positioning error θ ˙ = − Q J ( θ ) T A T x ˜ is 0 when either x ˜ = 0 (and thus the actual position is the desired position x d) or when A T x ˜ is in the nullspace of J (θ) T, which can occur when J (θ) is singular, Since asymptotically V ˙ → 0 as t → ∞, we have V ˙ = 0. Other popular gradient descent methods used in deep learning include the adaptive gradient (Adagrad), adaptive moment estimation (Adam), and adaptive learning rate (Adadelta). Its effect can be thought of as a ball rolling down a hill in the weights space and picking up pace to roll up the opposite slope and potentially escape a local minimum. For example, the momentum method (and variant Nesterov momentum ) helps escape from a local minimum by diminishing the fluctuations in weights updates over consecutive iterations. Moreover, training methods have been developed to escape from local minima, making on-line learning obsolete. However, batch training is often preferred as it can be very efficiently implemented with modern computers. Batch learning, by averaging the gradients, removes noise and is more prone to being caught in local minima. With on-line learning the stochastic error surface is noisy which helps escaping such local minima. Ī common issue with gradient based optimization methods is being trapped in a local minimum far from the desired global minimum.
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