From 1d1bfe130f51a07a32e3d4c6735f287980513400 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Wed, 19 Sep 2018 15:46:42 +0200 Subject: [PATCH] update on gradient descent --- .../gaussian/autocorrelation.eps | 41947 ++++------- .../gaussian/autocorrelation.png | Bin 20894 -> 21158 bytes .../gaussian/blocking.eps | 1150 +- .../gaussian/blocking.png | Bin 41759 -> 39532 bytes .../gaussian/data.eps | 62198 +++++++--------- .../gaussian/data.png | Bin 31922 -> 32346 bytes .../gaussian/hist.eps | 572 +- .../gaussian/hist.png | Bin 28235 -> 27884 bytes doc/src/Splines/Splines.do.txt | 110 + 9 files changed, 42291 insertions(+), 63686 deletions(-) diff --git a/doc/Programs/ResamplingAnalysisScripts/gaussian/autocorrelation.eps b/doc/Programs/ResamplingAnalysisScripts/gaussian/autocorrelation.eps index 71aaeb3ba..ba93a33cd 100644 --- a/doc/Programs/ResamplingAnalysisScripts/gaussian/autocorrelation.eps +++ b/doc/Programs/ResamplingAnalysisScripts/gaussian/autocorrelation.eps @@ -1,7 +1,7 @@ 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z3$yeV_Pz~8C_f1+MHJF^L4!RZSOT|?Uhn`(L|qA^8N4*66!N#6g~4cLRaFNAWhpq< z5-&!93kO17H}2*#_PTdgUIZbg5=#{xM`TvQj*dhEcJ|0|KC!$+% W!)GF`9B+doDJ!VSzdvnq@BaV;dTJy9 diff --git a/doc/src/Splines/Splines.do.txt b/doc/src/Splines/Splines.do.txt index 198e3fa48..e881fd8fc 100644 --- a/doc/src/Splines/Splines.do.txt +++ b/doc/src/Splines/Splines.do.txt @@ -6,6 +6,116 @@ DATE: today !split ===== Optimization, the central part of any Machine Learning algortithm ===== +Almost every problem in machine learning and data science starts with +a dataset $X$, a model $g(\theta)$, which is a function of the +parameters $\theta$ and a cost function $C(X, g(\theta))$ that allows +us to judge how well the model $g(\theta)$ explains the observations +$X$. The model is fit by finding the values of $\theta$ that minimize +the cost function. Ideally we would be able to solve for $\theta$ +analytically, however this is not possible in general and we must use +some approximative/numerical method to compute the minimum. + +The method of steepest descent +The basic idea of gradient descent is that a function $F(\mathbf{x})$, $ \mathbf{x} \equiv (x_1,\cdots,x_n)$, decreases fastest if one goes from $\bf {x}$ in the direction of the negative gradient $-\nabla F(\mathbf{x})$. + +It can be shown that if $$ +\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ \gamma_k > 0 +$$ for $\gamma_k$ small enough, then $F(\mathbf{x}_{k+1}) \leq F(\mathbf{x}_k)$. This means that for a sufficiently small $\gamma_k$ we are always moving towards smaller function values, i.e a minimum. + +This observation is the basis of the method of steepest descent, which is also referred to as just gradient descent (GD). One starts with an initial guess $\mathbf{x}_0$ for a minimum of $F$ and compute new approximations according to + +$$ +\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. +$$ +The parameter $\gamma_k$ is often referred to as the step length or the learning rate in the context of ML. + +Ideally the sequence $\{ \mathbf{x}_k \}_{k=0}$ converges to a global minimum of the function $F$. In general we do not know if we are in a global or local minimum. In the special case when $F$ is a convex function, all local minima are also global minima, so in this case gradient descent can converge to the global solution. The advantage of this scheme is that it is conceptually simple and straightforward to implement. However the method in this form has some severe limitations: + +In machine learing we are often faced with non-convex high dimensional cost functions with many local minimum. Since GD is deterministic we will get stuck in a local minimum, if the method converges, unless we have a very good intial guess. This also implies that the scheme is sensitive to the chosen initial condition. + +Note that gradient is a function of $\mathbf{x} = (x_1,\cdots,x_n)$ which makes it expensive to compute numerically. + +GD is sensitive to the choice of learning rate $\gamma_k$. This is due to the fact that we are only guaranteed that $F(\mathbf{x}_{k+1}) \leq F(\mathbf{x}_k)$ for sufficiently small $\gamma_k$. The problem is to determine an optimal learning rate. If the learning rate is chosen to small the method will take a long to converge and if it is to large we can experience erratic behavior. + +Many of these shortcomings can be alleviated by introducing randomness. One such method is that of Stochastic Gradient Descent (SGD). + + +Stochastic Gradient Descent +Stochastic gradient descent (SGD) and variants thereof address some of the shortcomings of the Gradient descent method discussed above. + +The underlying idea of SGD comes from the observation that the cost function, which we want to minimize, can almost always be written as a sum over $n$ datapoints $\{\mathbf{x}_i\}_{i=1}^n$, $$C(\mathbf{\theta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, \mathbf{\theta}). $$ This in turn means that the gradient can be computed as a sum over $i$-gradients $$\nabla_\theta C(\mathbf{\theta}) = \sum_i^n \nabla_\theta c_i(\mathbf{x}_i, \mathbf{\theta}). $$ + +Now, stochasticity/randomness is introduced by only taking the gradient on a subset of the data called minibatches. +If there are $n$ datapoints and the size of each minibatch is $M$, there will be $n/M$ minibatches. We denote these minibatches by $B_k$ where $k=1,\cdots,n/M$. + +As an example, suppose we have $10$ datapoints $( \mathbf{x}_1, \cdots, \mathbf{x}_{10} )$ and we choose to have $M=5$ minibathces, then each minibatch contains two datapoints. In particular we have $B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = (\mathbf{x}_9,\mathbf{x}_{10})$. Note that if you choose $M=1$ you have only a single batch with all datapoints and on the other extreme, you may choose $M=n$ resulting in a minibatch for each datapoint, i.e $B_k = \mathbf{x}_k$. + +The idea is now to approximate the gradient by replacing the sum over all datapoints with a sum over the datapoints in one the minibatches picked at random in each gradient descent step $$\nabla_\theta C(\mathbf{\theta}) = \sum_{i=1}^n \nabla_\theta c_i(\mathbf{x}_i, \mathbf{\theta}) \rightarrow \sum_{i \in B_k}^n \nabla_\theta c_i(\mathbf{x}_i, \mathbf{\theta}). $$ + +Thus a gradient descent step now looks like $$ \theta_{j+1} = \theta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\theta c_i(\mathbf{x}_i, \mathbf{\theta}) $$ where $k$ is picked at random with equal probability from $[1,n/M]$. An iteration over the number of minibathces (n/M) is commonly referred to as an epoch. Thus it is typical to choose a number of epochs and for each epoch iterate over the number of minibatches, as exemplified in the code below. + + +!bc pycod +import numpy as np + +n = 100 #100 datapoints +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +n_epochs = 10 #number of epochs + +j = 0 +for epoch in range(1,n_epochs+1): + for i in range(m): + k = np.random.randint(m) #Pick the k-th minibatch at random + #Compute the gradient using the data in minibatch Bk + #Compute new suggestion for theta + j += 1 +!ec + +Taking the gradient only on a subset of the data has two important benefits. First, it introduces randomness which decreases the chance that our opmization scheme gets stuck in a local minima. Second, if the size of the minibatches are small relative to the number of datapoints ($M < n$), the computation of the gradient is much cheaper since we sum over the datapoints in the k-th minibatch and not all $n$ datapoints. + +A natural question is when do we stop the search for a new minimum? One possibility is to compute the full gradient after a given number of epochs and check if the norm of the gradient is smaller than some threshold and stop if true. However, the condition that the gradient is zero is valid also for local minima, so this would only tell us that we are close to a local/global minimum. However, we could also evaluate the cost function at this point, store the result and continue the search. If the test kicks in at a later stage we can compare the values of the cost function and keep the $\theta$ that gave the lowest value. + +Another approach is to let the step length $\gamma_j$ depend on the number of epochs in such a way that it becomes very small after a reasonable time such that we do not move at all. + +As an example, let $e = 0,1,2,3,\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \cdot m + i$ where $m$ is the number of minibatches and $i=0,\cdots,m-1$. Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in "time" t. + +In this way we can fix the number of epochs, compute $\theta$ and evaluate the cost function at the end. Repeating the computation will give a different result since the scheme is random by design. Then we pick the final $\theta$ that gives the lowest value of the cost function. + +!bc pycod +import numpy as np + +def step_length(t,t0,t1): + return t0/(t+t1) + +n = 100 #100 datapoints +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +n_epochs = 500 #number of epochs +t0 = 1.0 +t1 = 10 + +gamma_j = t0/t1 +j = 0 +for epoch in range(1,n_epochs+1): + for i in range(m): + k = np.random.randint(m) #Pick the k-th minibatch at random + #Compute the gradient using the data in minibatch Bk + #Compute new suggestion for theta + t = epoch*m+i + gamma_j = step_length(t,t0,t1) + j += 1 + +print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) +!ec + + +We will look at a class of methods for computing minima of functions known as gradient descent and its generalizations. This note is meant as a quick introduction/crash course on gradient methods and mathematical details are kept brief. + + + + + We will explain the general approach for minimizing a multi-variate function and set the terminology in this section.