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An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of ...
These are called margin-based loss functions. Choosing a margin-based loss function amounts to choosing . Selection of a loss function within this framework impacts the optimal which minimizes the expected risk, see empirical risk minimization.
As defined above, the Huber loss function is strongly convex in a uniform neighborhood of its minimum =; at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points = and =. These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance ...
The green and blue functions both incur zero loss on the given data points. A learned model can be induced to prefer the green function, which may generalize better to more points drawn from the underlying unknown distribution, by adjusting , the weight of the regularization term.
A decision rule is a function:, where upon observing , we choose to take action (). Also define a loss function L : Θ × A → R {\displaystyle L:\Theta \times {\mathcal {A}}\rightarrow \mathbb {R} } , which specifies the loss we would incur by taking action a ∈ A {\displaystyle a\in {\mathcal {A}}} when the true state of nature is θ ∈ Θ ...
For regularized least squares the square loss function is introduced: = = (, ()) = = (()) However, if the functions are from a relatively unconstrained space, such as the set of square-integrable functions on X {\displaystyle X} , this approach may overfit the training data, and lead to poor generalization.
The objective function takes a set of hyperparameters and returns the associated loss. [3] Cross-validation is often used to estimate this generalization performance, and therefore choose the set of values for hyperparameters that maximize it. [4]
While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. A too high learning rate will make the learning jump over minima but a too low learning rate will either take too long to converge or get stuck in an undesirable local minimum.