However, this method can be a very simple and effective solution to allow a movement even in regions where the gradient is close to zero. The variance should be carefully chosen (for example, it could decay exponentially during the epochs). Now we’re going to discuss some common methods to improve the performance of a vanilla Stochastic Gradient Descent algorithm.Ī very simple approach to the problem of plateaus is adding a small noisy term (Gaussian noise) to the gradient. In this case, the point is maximum considering an orthogonal projection and a minimum for another one. If the Hessian matrix has both positive and negative eigenvalues (and the gradient is null), the Hessian is said to be indefinite and the point is called saddle point.
If the Hessian matrix is (m x m) where m is the number of parameters, the second condition is equivalent to say that all m eigenvalues must be non-negative (or that all principal minors composed with the first n rows and n columns must be non-negative).įrom a probabilistic viewpoint, P(H positive semi-def.) → 0 when m → ∞, therefore local minima are rare in deep models (this doesn’t mean that local minima are impossible, but their relative weight is lower and lower in deeper models).
When the system is relatively shallow, it’s easier to find local minima where the training process can stop, while in deeper models, the probability of a local minimum becomes smaller and, instead, saddle points become more and more likely. Unfortunately, the reality is a little bit different, in particular in deep models, where the number of parameters is in the order of ten or one hundred million. In a standard optimization problem, without particular requirements, the algorithm converges in a limited number of iterations. In this context, we assume that Stochastic Gradient Descent operates on batch sizes equal or greater than 1. In some books, the expression “Stochastic Gradient Descent” refers to an algorithm which operates on a batch size equal to 1, while “Mini-batch Gradient Descent” is adopted when the batch size is greater than 1. In this post, we’re going to analyze how it works and the most important variations that can speed up the convergence in deep models.įirst of all, it’s necessary to standardize the naming. However, the vanilla algorithm has many limitations, in particular when the system is ill-conditioned and could never find the global minimum. Stochastic Gradient Descent (SGD) is a very powerful technique, currently employed to optimize all deep learning models. This article was written by Giuseppe Bonaccorso.
#Stochastic gradient descent update#
The update step of the algorithm is the same where is replaced by. In this version, instead of simply choosing one index at random, a subset is chosen at random. These component functions are chosen at random at each step. There is a variant of stochastic gradient descent called mini-batch, where instead of using only one component function, several components functions are used. Where is the expectation taken over dthe distribution of (which is chosen randomly, remember). Correctnessįurthermore, is an unbiased estimate for, indeed: Which explains why is computationaly cheaper than. The gradient of a sum is the sum of the gradients, so: This algorithm works because is an unbiased estimate of that is much simpler to compute. ) and that SGD still converges after only a few hundreds of iterations (i.e. In machine learning, it often happens that we have hundreds of thousands, or millions of training examples (i.e. In practice SGD is worth a shot when we have a very large number of training examples. Is it really faster than gradient descent?įor iterations, and training examples, gradient descent computes gradients while stochastic gradient descent computes only gradients. While the update rule for stochastic gradient descent is: The update rule for gradient descent at step t is: randint ( 0, len ( f ) - 1 ) # Update step x = x - step_size * gradient ( of = f, at = x ) return x Comparison with gradient descent """ x = x_0 for step in range ( max_steps ): # Choose the component function at random n = random. Def stochastic_gradient_descent ( f, x_0, max_steps, step_size ): """
0 kommentar(er)
