Context: Require: Learning rate η , momentum parameter α,Initial parameter θ, initial velocity v. while do Stopping criterion not met Sample a minibatch of examples from the training set m {x(1), . . ., x(m ) } Set g = 0 for do i m = 1 to Compute gradient estimate: g g ← + ∇ θL f( (x^( i ); θ, y^( i)) end for Compute velocity update: v ← αv− ηg Apply update: θ ← θ +v end while Explain
Question
Context: Require: Learning rate η , momentum parameter α,Initial parameter θ, initial velocity v. while do Stopping criterion not met Sample a minibatch of examples from the training set m {x(1), . . ., x(m ) } Set g = 0 for do i m = 1 to Compute gradient estimate: g g ← + ∇ θL f( (x^( i ); θ, y^( i)) end for Compute velocity update: v ← αv− ηg Apply update: θ ← θ +v end while Explain
Solution
This is a description of a variant of the gradient descent algorithm, specifically momentum-based gradient descent, which is commonly used in machine learning and deep learning for optimization. Here's a step-by-step explanation:
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Initialization: Set the learning rate (η), momentum parameter (α), initial parameters (θ), and initial velocity (v). The learning rate controls how much we adjust our parameters in response to the estimated error. The momentum parameter helps accelerate gradient vectors in the right directions, thus leading to faster converging. The initial parameters (θ) are the starting point of our optimization algorithm, and the initial velocity (v) is used to store a fraction of the update direction of the past time step.
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Loop Until Convergence: Repeat the following steps until your stopping criteria are met. The stopping criteria could be a certain number of iterations, a minimum improvement in loss, etc.
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Sample a Mini-batch: Randomly select a subset of the data (a mini-batch) from your training set. This is used to estimate the gradient of the loss function.
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Compute Gradient Estimate: For each example in your mini-batch, compute the gradient of the loss function with respect to the parameters (θ). The gradient is a vector that points in the direction of greatest increase of the function. The loss function (L) measures how well the algorithm is doing on the example. The gradient of the loss function tells us how to change our parameters (θ) to improve the performance of our algorithm. We sum up all these gradients to get g.
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Compute Velocity Update: Multiply the previous velocity by the momentum parameter (α) and subtract the product of the learning rate (η) and the gradient (g). This is the new velocity (v).
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Apply Update: Update the parameters (θ) by adding the velocity (v). This step moves the parameters in the direction of the negative gradient to decrease the loss function.
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Repeat: Go back to step 2 and repeat until the stopping criteria are met.
This algorithm is used to find the parameters (θ) that minimize the loss function. The momentum term (αv) helps the algorithm to not get stuck in local minimums and saddle points.
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