# Shark Conventions for Derivatives¶

## Background¶

Gradient-based optimization is the most common reason for computing derivatives in Shark. Gradient-based optimizers operate on objective functions, however, these will often delegate the job further on to models, kernels, and loss or more general cost functions. Thus, all these types of objects have the capability to compute derivatives foreseen in their respective interfaces. However, getting the best performance when evaluating derivatives is not always straightforward. Since Shark aims for maximal speed, the library enforces a very specific evaluation scheme for derivative computations. The design rationale will be explained in the following.

## An Example: The Derivative of the Error¶

Let’s consider a simple example, namely the derivative of a squared error term

$E = \frac 1 N \sum_{i=1}^N L(f_w(x_i), t_i)$

w.r.t. a parameter $$w_i$$ of a parametric family $$f_w(x)$$ of models, evaluated with the squared loss $$L(y, t) = (y-t)^2$$:

\begin{align}\begin{aligned}\frac{\partial E}{\partial w_k} &= \frac 1 N \sum_{i=1}^N L(f_w(x_i), t_i)\\ &= \frac 1 N \sum_{i=1}^N \frac{\partial L}{\partial y}(f_w(x_i), t_i) \frac{\partial f_w}{\partial w_k}(x_i)\\ &= \sum_{i=1}^N \frac 2 N (f_w(x_i) - t_i) \frac{\partial f_w}{\partial w}(x_i)\end{aligned}\end{align}

The derivative involves the chain rule for the combination of model and loss. The term can be understood as a weighted sum of the partial model derivatives, where the weights are the loss derivatives. Note how these weights do not require the model derivatives, but they do depend on the model’s output values. This is the general situation when chaining computations:

$\frac{\partial f \circ g(x)}{\partial w_k} = f'(g(x)) g'(x)$

The value $$g(x)$$ is needed as the point in which the derivative $$f'$$ is to be evaluated, and it is used rather independent of the derivative $$g'(x)$$.

In a typical error function the overall derivative is a weighted sum of model derivatives, evaluated in different points. The weights require only the model evaluations in these points, not their derivatives. This hints at the following order of evaluation:

• evaluate the model values $$y_i = f_w(x_i)$$,
• evaluate the loss derivatives $$\frac{\partial L}{\partial y}(y_i, t_i)$$,
• evaluate the model derivatives $$\frac{\partial f}{\partial w_i}(x_i)$$ and compute their weighted sum.

## Two-stage Derivative Computation¶

The order of computation as laid out above is a necessity for the efficient evaluation of derivatives of chains. We lift this necessity to a principle: first evaluate, then differentiate. In other words, always call eval on an object before calling evalDerivative or similar functions. Otherwise the results of the derivative are undefined. This holds for models and kernels. Objective functions and losses can compute both at once since they can be interpreted as the ends of the chain of computation - the loss is required to evaluate and return the full derivative while the objective function returns the final summed result.

In simple situations the order of evaluation is not crucial. However, in general the requirement to evaluate before computing derivatives is not restrictive at all. So even if there is no natural order of calls, the order is dictated by the Shark interface design.

The rationale behind this design is that there are often strong synergies between the computation of the value and its derivative. More often than not, the derivative is a cheap byproduct once the value has been computed. Thus, efficient evaluation of both the value and the derivative requires either the computation of both at the same time, or the storage of intermediate results. The first way is not viable in case of chained computations. Therefore Shark has decided to take the second route, and to store intermediate values for derivative computations in the object. This state is written by the evaluation method and read by the derivative computations.

## Another Example: The derivative of a concatenation of models¶

Let’s assume the function $$f_w$$ of the previous example would in fact not be a single model, but two models where the output of the first model is the input of the second, such that with $$w=(u,v)$$ being the combined parameter vector of both models, we get $$f_w(x)=g_u(h_v(x))$$.Thus the derivatives are:

$\frac{\partial E}{\partial u_k}f_w(x) =\frac{\partial g_u}{\partial u_k}(h_v(x))$

and

$\frac{\partial E}{\partial v_k}f_w(x) = \frac{\partial g_u}{\partial h_v(x)}(h_v(x)) \frac{\partial h_v}{\partial v_k} h_v(x)$

Please remember that the partial derivatives with respect to the arguments of $$g$$ are full jacobi matrices and not single values or vectors. Thus the computation of $$v_k$$ as stated here requires a matrix-vector product for every parameter $$v_k$$, or a matrix-matrix product if the derivative is computed for all $$v_k$$ at once. But putting this into the the equation of the derivative of the error function of the previous example, we get for the derivative with respect to $$v_k$$:

$\begin{split}\frac{\partial E}{\partial v_k} &= \sum_{i=1}^N \frac 2 N (f_w(x_i) - t_i) \frac{\partial f_w}{\partial v_k}(x_i)\\ &= \sum_{i=1}^N \frac 2 N (f_w(x_i) - t_i) \frac{\partial g_u}{\partial h_v(x_i)}(h_v(x_i)) \frac{\partial h_v}{\partial v_k} h_v(x_i)\end{split}$

now adding braces around the derivative of the loss and the partial derivative of $$g_u$$ we see that this term can be computed as matrix-vector product. Thus the whole Term can be computed using 2N matrix-vector products instead of N matrix-matrix and matrix-vector products! This makes in practice a huge difference.

## Weighted Sums of Derivatives¶

In the first example, the derivative of the squared error w.r.t. a model parameter is a weighted sum of derivatives for single data points. The computation of the weights requires the model’s output values for the same data points. Again, this situation is completely general, and thus Shark makes it a principle: derivatives are returned as weighted sums.

A single call to a derivative function may evaluate the derivative in a whole batch or even in a whole data set of different points. However, in the next processing stage these values will typically all enter the same cost function. Thus, the derivative is a weighted sum, with the cost derivatives being the weights.

Now for chaining of the derivatives as in the second example, we can first evaluate the weighted derivative with respects to the inputs of $$g$$, which amounts to computing the aforementioned bracing. After that the resulting vector can be used to calculate the weighted derivative of $$h$$ with respect to it’s parameters. We can further optimize this scheme by computing both derivatives of $$g$$ at the same time using that again in many cases input and parameter derivative can share a lot of computations.

A well known example which uses both optimizations weighted sum calculation and shared computation of derivatives is the back-propagation of error algorithm, which not only allows for a more efficient computation in terms of the complexity of the algorithm, but also allow for a more efficient optimization for RAM throughput, etc. To check that this is in fact the same algorithm, define $$g$$ and $$h$$ as neuron layers of a three layer neural network.

Thus Shark’s derivative interfaces can be understood as a generalization of the same computational trick. The exact weighting scheme applied slightly varies across the different interfaces, e.g., models versus kernels.

## Batching Derivatives and how to derive them¶

As previously mentioned in short, batching is also applied to derivatives. The net effect of batch computing is not as dramatic for the computation time as the application of weighted derivatives, but still quite significant. However, deriving efficient batched computations of weighted derivatives is not straight forward and we are constantly trying to improve the results. In the case of the parameter derivative for example the input is a matrix of values: every row consists of one weight for every output and each row represents one sample. Computing this derivative naiively, the result would be a three tensor which we need to reduce to a single vector by summing over two dimensions. Thus choosing the order in which this reduction is performed - preferently without actually calculating the big tensor itself - can make a huge difference. The key to success in any case is to use matrix notation wherever possible instead of using elementwise derivations as is often done on a sheet of paper. While Vector and Matrix calculus sems unfamiliar at first glance, it immediately answers the questions about which computations can be grouped together and which efficient linear algebra operations can be used.

Todo

TG: Present one simple and one involved use case? Or is this the wrong place?

OK: We need one tutorial for Kernels and Models which explain how these derivatives can actually b calculated. Maybe introduce your nice scalar product syntax which makes the calculation a breese.