Posted on Categories:Optimization Theory, 优化理论, 优化理论代写, 数学代写, 机器学习代写, 机器学习代考

# 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|CSC4512 Adaptive Stepsize

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## 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Adaptive Stepsize

No general prescriptions for selecting appropriate learning rate; typically no fixed learning rate appropriate for entire learning period

“Bold driver” heuristic: monitor error after each epoch (sweep through entire training set)

1. If error decreases, increase learning rate: $\epsilon=\epsilon * \rho$
2. If error increases, decrease rate, reset parameters:
$$\epsilon=\epsilon * \sigma ; \quad \mathbf{w}^t=\mathbf{w}^{t-1}$$

Sensible choices for parameters: $\rho=1.1, \quad \sigma=0.5$

Momentum

If the error surface is a long and narrow valley, gradient descent goes quickly down the valley walls, but very slowly along the valley floor

We can alleviate this problem by updating parameters using a combination of the previous update and the gradient update:
$$\Delta w_j^t=\beta \Delta w^{t-1}+(1-\beta)\left(-\epsilon \partial E / \partial w_j\left(\mathbf{w}^t\right)\right)$$

Usually $\beta$ is set quite high, about $0.95$.

This is like giving momentum to the weights

## 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Mini-Batch and Online Optimization

When the dataset is large, computing the exact gradient is expensive

This seems wasteful since the only thing we use the gradient for is to compute a small change in the weights, then throw this out and recompute the gradient all over again

An approximate gradient is useful as long as it points in roughly the same direction as the true gradient

One easy way to do this is to divide the dataset into small batches of examples, compute the gradient using a single batch, make an update, then move to the next batch of examples: mini-batch optimization

In the limit, if each batch contains just one example, then this is the ‘online’ learning, or stochastic gradient descent mentioned in Lecture 2.

These methods are much faster than exact gradient descent, and are very effective when combined with momentum, but care must be taken to ensure convergence

Rather than take a fixed step in the direction of the negative gradient or the momentum-smoothed negative gradient, it is possible to do a search along that direction to find the minimum of the function

Usually the search is a bisection, which bounds the nearest local minimum along the line between any two points such that there is a third point $\mathbf{w}_3$ with $E\left(\mathbf{w}_3\right)<E\left(\mathbf{w}_1\right)$ and $E\left(\mathbf{w}_3\right)<E\left(\mathbf{w}_2\right)$

## 数学代写机器学习中的优化理论代写Optimization for Machine Learning代 考|Adaptive Stepsize

“Bold driver”启发式: 在每个 epoch 之后监控错误（扫描整个训线集）

$$\epsilon=\epsilon * \sigma ; \quad \mathbf{w}^t=\mathbf{w}^{t-1}$$

$$\Delta w_j^t=\beta \Delta w^{t-1}+(1-\beta)\left(-\epsilon \partial E / \partial w_j\left(\mathbf{w}^t\right)\right)$$

## MATLAB代写

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