Machine Learning { Brett Bernstein Recitation 1: Gradients and Directional Derivatives Intro Question d 1. We are given the data set (x1; y1);:::; (xn; yn) where xi R and yi R. We want to fit a linear function to this data by performing empirical2 risk minimization.2 More precisely, we are using the hypothesis space = f(x) = wT x w Rd and the loss function `(a; y) = (a y)2. Given an initial guessF fw ~ for the empiricalj 2 riskg minimizing parameter vector, how− could we improve our guess? y x Figure 1: Data Set With d = 1 Multivariable Differentiation Differential calculus allows us to convert non-linear problems into local linear problems, to which we can apply the well-developed techniques of linear algebra. Here we will review some of the important concepts needed in the rest of the course. 1 Single Variable Differentiation To gain intuition, we first recall the single variable case. Let f : R R be differentiable. The derivative ! f(x + h) f(x) f 0(x) = lim − h 0 ! h gives us a local linear approximation of f near x. This is seen more clearly in the following form: f(x + h) = f(x) + hf 0(x) + o(h) as h 0, ! where o(h) represents a function g(h) with g(h)=h 0 as h 0. This can be used to show ! ! that if x is a local extremum of f then f 0(x) = 0. Points with f 0(x) = 0 are called critical points. f (x + h, f(x + h)) f(x + h) (f(x)+ hf ′(x)) − (x + h, f(x)+ hf ′(x)) (x, f(x)) Figure 2: 1D Linear Approximation By Derivative Multivariate Differentiation More generally, we will look at functions f : Rn R. In the single-variable case, the derivative was just a number that signified how much! the function increased when we moved in the positive x-direction. In the multivariable case, we have many possible directions we T n can move along from a given point x = (x1; : : : ; xn) R . 2 2 Figure 3: Multiple Possible Directions for f : R2 R ! If we fix a direction u we can compute the directional derivative f 0(x; u) as f(x + hu) f(x) f 0(x; u) = lim − : h 0 ! h This allows us to turn our multidimensional problem into a 1-dimensional computation. For instance, f(x + hu) = f(x) + hf 0(x; u) + o(h); mimicking our earlier 1-d formula. This says that nearby x we can get a good approximation of f(x+hu) using the linear approximation f(x)+hf 0(x; u). In particular, if f 0(x; u) < 0 (such a u is called a descent direction) then for sufficiently small h > 0 we have f(x + hu) < f(x). 3 Figure 4: Directional Derivative as a Slope of a Slice i 1 z }|− { Let ei = (0; 0;:::; 0; 1; 0;:::; 0) be the ith standard basis vector. The directional deriva- tive in the direction ei is called the ith partial derivative and can be written in several ways: @ f(x) = @xi f(x) = @if(x) = f 0(x; ei): @xi We say that f is differentiable at x if f(x + v) f(x) gT v lim − − = 0; v 0 v 2 ! k k for some g Rn (note that the limit for v is taken in Rn). This g is uniquely determined, and is called2 the gradient of f at x denoted by f(x). It can be shown that the gradient is the vector of partial derivatives: r 0 1 @x1 f(x) . f(x) = B . C : r @ A @xn f(x) 4 The kth entry of the gradient (i.e., the kth partial derivative) is the approximate change in f due to a small positive change in xk. Sometimes we will split the variables of f into two parts. For instance, we could write f(x; w) with x Rp and w Rq. It is often useful to take the gradient with respect to some 2 2 of the variables. Here we would write x or w to specify which part: r r 0 1 0 1 @x1 f(x; w) @w1 f(x; w) . xf(x; w) := B . C and wf(x; w) := B . C : r @ A r @ A @xp f(x; w) @wq f(x; w) Analogous to the univariate case, can express the condition for differentiability in terms of a gradient approximation: T f(x + v) = f(x) + f(x) v + o( v 2): r k k The approximation f(x + v) f(x) + f(x)T v gives a tangent plane at the point x as we let v vary. ≈ r Figure 5: Tangent Plane for f : R2 R ! 5 If f is differentiable, we can use the gradient to compute an arbitrary directional deriva- tive: T f 0(x; u) = f(x) u: r From this expression we can prove that (assuming f(x) = 0) r 6 f(x) f(x) arg max f 0(x; u) = r and arg min f 0(x; u) = r : u 2=1 f(x) 2 u 2=1 − f(x) 2 k k kr k k k kr k In words, we say that the gradient points in the direction of steepest ascent, and the negative gradient points in the direction of steepest descent. As in the 1-dimensional case, if f : Rn R is differentiable and x is a local extremum of f then we must have f(x) = 0. Points x!with f(x) = 0 are called critical points. As we will see later in the course,r if a function is differentiabler and convex, then a point is critical if and only if it is a global minimum. Figure 6: Critical Points of f : R2 R ! Computing Gradients A simple method to compute the gradient of a function is to compute each partial derivative separately. For example, if f : R3 R is given by ! x1+2x2+3x3 f(x1; x2; x3) = log(1 + e ) 6 then we can directly compute ex1+2x2+3x3 2ex1+2x2+3x3 3ex1+2x2+3x3 @x f(x1; x2; x3) = ;@x f(x1; x2; x3) = ;@x f(x1; x2; x3) = 1 1 + ex1+2x2+3x3 2 1 + ex1+2x2+3x3 3 1 + ex1+2x2+3x3 and obtain 0 1 ex1+2x2+3x3 B x1+2x2+3x3 C B 1 + e C B 2ex1+2x2+3x3 C f(x1; x2; x3) = B C : r B x1+2x2+3x3 C B 1 + e C @ 3ex1+2x2+3x3 A 1 + ex1+2x2+3x3 Alternatively, we could let w = (1; 2; 3)T and write T f(x) = log(1 + ew x): Then we can apply a version of the chain rule which says that if g : R R and h : Rn R are differentiable then ! ! g(h(x)) = g0(h(x)) h(x): r r Applying the chain rule twice (for log and exp) we obtain 1 T f(x) = ew xw; r 1 + ewT x T where we use the fact that x(w x) = w. This last expression is more concise, and is more amenable to vectorized computationr in many languages. Another useful technique is to compute a general directional derivative and then infer the gradient from the computation. For example, let f : Rn R be given by ! f(x) = Ax y 2 = (Ax y)T (Ax y) = xT AT Ax 2yT Ax + yT y; k − k2 − − − m n m T for some A R × and y R . Assuming f is differentiable (so that f 0(x; v) = f(x) v) we have 2 2 r f(x + tv) = (x + tv)T AT A(x + tv) 2yT A(x + tv) + yT y − = xT AT Ax + t2vT AT Av + 2txT AT Av 2yT Ax 2tyT Av + yT y − − = f(x) + t(2xT AT A 2yT A)v + t2vT AT Av: − Thus we have f(x + tv) f(x) − = (2xT AT A 2yT A)v + tvT AT Av: t − Taking the limit as t 0 shows ! T T T T T T T T T f 0(x; v) = (2x A A 2y A)v = f(x) = (2x A A 2y A) = 2A Ax 2A y: − ) r − − 7 Assume the columns of the data matrix A have been centered (by subtracting their respective means). We can interpret f(x) = 2AT (Ax y) as (up to scaling) the covariance between the features and the residual.r − Using the above calculation we can determine the critical points of f. Let's assume here that A has full column rank. Then AT A is invertible, and the unique critical point is T 1 T x = (A A)− A y. As we will see later in the course, this is a global minimum since f is convex (the Hessian of f satisfies 2f(x) = 2AT A 0). r (?) Proving Differentiability With a little extra work we can make the previous technique give a proof of differentiability. Using the computation above, we can rewrite f(x + v) as f(x) plus terms depending on v: f(x + v) = f(x) + (2xT AT A 2yT A)v + vT AT Av: − Note that T T 2 2 2 v A Av Av 2 A 2 v 2 2 = k k k k k k = A 2 v 2 0; v 2 v 2 ≤ v 2 k k k k ! k k k kj k k as v 2 0. (This section is starred since we used the matrix norm A 2 here.) This shows f(xk+kv)! above has the form k k T f(x + v) = f(x) + f(x) v + o( v 2): r k k This proves that f is differentiable and that f(x) = 2AT Ax 2AT y: r − Another method we could have used to establish differentiability is to observe that the partial derivatives are all continuous.