Galileo Galileo has many self-appointed intellectual heirs these days. Whether it’s a claim that the election has been stolen, that COVID-19 is less fatal than the flu, that climate change or evolution are hoaxes, or that P=NP, we keep hearing from people considering themselves as bold truth-tellers railing against conventional wisdom. We are encouraged to “teach the debate” and that if only paid attention, we will see that their Tweet, declaration, or arxiv paper contains an irrefutable proof of their assertions.

In the words of Ted Cruz, “They brand you a heretic. Today, the global warming alarmists are the equivalent of the flat-Earthers. It used to be [that] it is accepted scientific wisdom the Earth is flat, and this heretic named Galileo was branded a denier”.

Of course by Galileo’s time it was well known that the earth was spherical, and Magellan circumnavigated the earth more than 40 years before Galileo was born. But putting aside Cruz’s confusion of flat earth and geocentrism, the story of heliocentric theory is not one of an outsider railing against the scientific mainstream. Galileo himself was a chaired professor of mathematics at the University of Padua, and later philosopher and mathematician to the grand duke of Tuscany. He was very much part of the scientific establishment of his time. Moreover, though Galileo did provide important evidence for heliocentrism, he was not the only one doing so. Kepler found a heliocentric model with elliptical orbits that actually made correct predictions, and, though it took a decade or so, Kepler’s book eventually became the standard textbook for astronomy.

My point in this post is not to rehash the history of heliocentrism or Galileo but rather to call out a misconception which, to use Sean Carrol’s phrasing, amounts to valorization of puzzle-solving over wisdom.

People valorize a certain puzzle-solving kind of intelligence. And solving puzzles is important. But the ultimate goal isn't to be clever, it's to be correct. For that, knowing what information to pay attention to and what ideas to take seriously is more relevant.

— Sean Carroll (@seanmcarroll) November 22, 2020

It is tempting to think that an argument, regardless whether it comes from an expert or a random person on Twitter, can be presented in a self-contained way and judged on its merits. However, this is not how things work in any interesting setting. Even in the case of a purported P vs NP proof, there is background knowledge on computational complexity without which it would be hard to spot holes in the argument. This is doubly so for any claim involving empirical facts, whether it’s about elections, infections, climate etc. It is not possible to evaluate such claims without context, and to get this context you need to turn to the experts that have studied the topic.

I have written before in defense of expertise (see also here) but Carroll puts it very well. Another way to say it is that the operational interpretation of the common refrain

Extraordinary claims require extraordinary evidence

is

Treat claims conforming to conventional wisdom with charity, and claims disputing it with skepticism.

(There is a question of how to define “conventional wisdom” but interestingly there is usually agreement in practice by both sides. Most “deniers” of various sorts are proud of going against conventional wisdom, but don’t acknowledge that this means they are more likely to be wrong.)

As an example, even if someone has expertise in analytic number theory, and so presumably has plenty of so-called “puzzle-solving intelligence”, that doesn’t mean that they can evaluate a statistical claim on election fraud and their analysis should be considered evidence (apparently at this point the number theorist himself agrees). We can try to read and debunk what they wrote, or we can assume that if there was evidence for large-scale fraud, then the president of the United States and his well-funded campaign would have managed to find actual statisticians and experts on election to make the case.

There can be debate if Trump’s attempt to overthrow the election should be considered as dangerous or merely absurd, but the constant attacks on the very notions of truth, science, and expertise are causing far-reaching harm.

(H/T: Scott Aaronson, who makes a similar point around the election conspiracies.)

I am teaching deep learing this week in Harvard’s CS 182 (Artificial Intelligence) course. As I’m preparing the back-propagation lecture, Preetum Nakkiran told me about Andrej Karpathy’s awesome micrograd package which implements automatic differentiation for scalar variables in very few lines of code.

I couldn’t resist using this to show how simple back-propagation and stochastic gradient descents are. To make sure we leave nothing “under the hood” we will not import anything from the package but rather only copy paste the few things we need. I hope that the text below is generally accessible to anyone familiar with partial derivatives. See this colab notebook for all the code in this tutorial. In particular, aside from libraries for plotting and copy pasting a few dozen lines from Karpathy this code uses absolutely no libraries (no numpy, no pytorch, etc..) and can train (slowly..) neural networks using stochastic gradient descent. (This notebook builds the code more incrementally.)

Automatic differentiation is a mechanism that allows you to write a Python functions such as

def f(x,y): return (x+y)+x**3

and enables one to automatically obtain the partial derivatives and . Numerically we could do this by choosing some small value and computing both and .
However, if we generalize this approach to variables, we get an algorithm that requires roughly evaluations of . Back-propagation enables computing all of the partial derivatives at only constant overhead over the cost of a single evaluation of .

Back propagation and the chain rule

Back-propagation is a direct implication of the multi-variate chain rule. Let’s illustrate this for the case of two variables. Suppose that and are differentiable functions, and define

.

That is, we have the following situation:

where is the value

Then the chain rule states that

You can take this on faith, but it also has a simple proof. To see the intuition, note that for small , and . For small , . Hence, if we ignore terms with powers of delta two or higher,

Meaning that which is what we needed to show.

The chain rule generalizes naturally to the case that is a function of more variables than . Generally, if the value is obtained by first computing some intermediate values from and then computing in some arbitrary way from , then .

As a corollary, if you already managed to compute the values , and you kept track of the way that were obtained from , then you can compute .

This suggests a simple recursive algorithm by which you compute the derivative of the final value with respect to an intermediate value in the computation using recursive calls to compute the values for all the values that were directly computed from . Back propagation is this algorithm.

Implementing automatic differentiation using back propagation in Python

We now describe how to do this in Python, following Karpathy’s code. The basic class we use is Value. Every member of Value is a container that holds:

1. The actual scalar (i.e., floating point) value that holds. We call this data.
2. The gradient of with respect to some future unknown value that will use it. We call this grad and it is initialized to zero.
3. Pointers to all the values that were used in the computation of . We call this _prev
4. The method that adds (using the current value of and other values) the contribution of to the gradient of all its previous values to their gradients. We call this function _backward. Specifically, at the time we call _backward we assume that u.grad already contains where is the final value we are interested in. For every value that was used to compute , we add to v.grad the quantity . For the latter quantity we need to keep track of how was computed from .
5. If we call the method backwards (without an underscore) on a variable then this will compute the derivative of with respect to for all values that were used in the computation of . We do this by applying _backward to and then recursively (just like in DFS) going over the “children” (values used to compute ), calling _backward on each one and keeping track the ones we visited just like the Depth First Search (DFS) algorithm.

Let’s now describe this in code. We start off with a simple version that only supports addition and multiplication. The constructor for the class is the following:

class Value:
    """ stores a single scalar value and its gradient """

    def __init__(self, data, _children=()):
        self.data = data
        self.grad = 0
        self._backward = lambda: None
        self._prev = set(_children)

which fairly directly matches the description above. This constructor creates a value not using prior ones, which is why the _backward function is empty.
However, we can also create values by adding or multiplying prior ones, by adding the following methods:

  def __add__(self, other):
        other = other if isinstance(other, Value) else Value(other)
        out = Value(self.data + other.data, (self, other))

        def _backward():
            self.grad += out.grad
            other.grad += out.grad
        out._backward = _backward

        return out

    def __mul__(self, other):
        other = other if isinstance(other, Value) else Value(other)
        out = Value(self.data * other.data, (self, other))

        def _backward():
            self.grad += other.data * out.grad
            other.grad += self.data * out.grad
        out._backward = _backward

        return out

That is, if we create by adding the values and , then the _backward function of works by adding w.grad to both u.grad and v.grad.
If we is obtain by multiplying and then we add w.grad v.data to u.grad and similarly add w.grad u.data to v.grad.

The backward function is obtained by setting the gradient of the current value to and then running _backwards on all other values in reverse topological order:

def backward(self, visited= None): # slightly shorter code to fit in the blog 
    if visited is None:
        visited= set([self])
        self.grad = 1
    self._backward()
    for child in self._prev:
        if not child in visited:
            visited.add(child)
            child.backward(visited)

For example, if we run the following code

a = Value(5)
print(a.grad)
def f(x): return (x+2)**2 + x**3
f(a).backward()
print(a.grad)

then the values printed will be 0 and 89 since the derivative of equals .

In the notebook you can see that we implement also the power function, and have some “convenience methods” (division etc..).

Linear regression using back propagation and stochastic gradient descent

In stochastic gradient descent we are given some data and want to find an hypothesis that minimizes the empirical loss where is a loss function mapping two labels to a real number. If we let be the -th term of this sum, then, identifying with the parameters (i.e., real numbers) that specify it, stochastic gradient descent is the following algorithm:

1. Set to be a random vector. Set to be some small number (e.g., )
2. For (where is the number of epochs):

• For : (in random order)
  • Let

If is specified by the parameters is the vector . This is exactly the vector we can obtain using back propagation.

For example, if we want a linear model, we can use as our parameters and the function will be . We can generate random points X,Y as follows:

Now we can define a linear model as follows:

class Linear:
  def __init__(self):
    self.a,self.b = Value(random.random()),Value(random.random())
  def __call__(self,x): return self.a*x+self.b

  def zero_grad(self):
    self.a.grad, self.b.grad = 0,0

And train it directly by using SGD:

η = 0.03, epochs = 20
for t in range(epochs):
  for x,y in zip(X,Y):
    model.zero_grad()
    loss = (model(x)-y)**2
    loss.backward()
    model.a , model.b = (model.a - η*model.a.grad  , model.b - η*model.b.grad)

Which as you can see works very well:

From linear classifiers to Neural Networks.

The above was somewhat of an “overkill” for linear models, but the beautify of automatic differentiation is that we can easily use more complex computation.

We can follow Karpathy’s demo and us the same approach to train a neural network.

We will use a neural network that takes two inputs and has two hidden layers of width 16. A neuron that takes input will apply the ReLU function () to where are its weight parameters. (It’s easy to add support for relu for our Value class. Also we won’t have a bias term in this example.)

The code for this Neural Network is as follows: (when Value() is called without a parameter the value is random number in )

def Neuron(weights,inputs, relu =True):
  # Evaluate neuron with given weights on given inputs
  v =  sum(weights[i]*x for i,x in enumerate(inputs))
  return v.relu() if relu else v


class Net:
  # Depth 3 fully connected neural net with one two inputs and output
  def __init__(self,  N=16):
    self.layer_1 = [[Value(),Value()] for i in range(N)]
    self.layer_2 = [ [Value() for j in range(N)] for i in range(N)]
    self.output =  [ Value() for i in range(N)]
    self.parameters = [v for L in [self.layer_1,self.layer_2,[self.output]] for w in L for v in w]


  def __call__(self,x):
    layer_1_vals = [Neuron(w,x) for w in self.layer_1]
    layer_2_vals = [Neuron(w,layer_1_vals) for w in self.layer_2]
    return Neuron(self.output,layer_2_vals,relu=False) 
    # the last output does not have the ReLU on top

  def zero_grad(self):
    for p in self.parameters:
      p.grad=0

We can train it in the same way as above.
We will follow Karpathy and train it to classify the following points:

The training code is very similar, with the following differences:

• Instead of the square loss, we use the function which is if . This makes sense since our data labels will be and we say we classify correctly if we get the same sign. We get zero loss if we classify correctly all samples with a margin of at least .
• Instead of stochastic gradient descent we will do standard gradient descent, using all the datapoints before taking a gradient step. The optimal for neural networks is actually often something in the middle – batch gradient descent where we take a batch of samples and perform the gradient over them.

The resulting code is the following:

for t in range(epochs):
  loss = sum([(1+ -y*model(x)).relu() for (x,y) in zip(X,Y)])/len(X)
  model.zero_grad()
  loss.backward()
  for p in model.parameters:
    p.data -= η*p.grad

If we use this, we get a decent approximation for this training set (see image below). As Karpathy shows, by adjusting the learning rate and using regularization, one can in fact get 100% accuracy.

Update 11/30: Thanks to Gollamudi Tarakaram for pointing out a typo in a previous version.

One of the fascinating lines of research in recent years has been a convergence between the statistical physics and theoretical computer science points of view on optimization problems.
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This blog post is mainly a note to myself (i.e., I’m the “dummy” 😃), trying to work out some basic facts in some of this line of work. it was inspired by this excellent talk of Eliran Subag, itself part of a great Simons institute workshop which I am still planning to watch the talks of. I am posting this in case it’s useful for others, but this is quite rough, missing many references, and I imagine I have both math mistakes as well as inaccuracies in how I refer to the literature – would be grateful for comments!

In computer science, optimization is the task of finding an assignment that minimizes some function . In statistical physics we think of as the states of particles, and as the energy of this state. Finding the minimum assignment corresponds to finding the ground state, and another computational problem is sampling from the Gibbs distribution where the probability of is proportional to for some .

Two prototypical examples of such problems are:

1. Random 3SAT – in this case and is the number of clauses violated by the assignment for a random formula.
2. Sherrington-Kirpatrick model – in this case and where are independent normal variables with variance for and variance for . (Another way to say it is that is the matrix where ‘s entries are chosen i.i.d from .)

The physics and CS intuition is that these two problems have very different computational properties. For random 3SAT (of the appropriate density), it is believed that the set of solutions is “shattered” in the sense that it is partitioned to exponentially many clusters, separated from one another by large distance. It is conjectured that in this setting the problem will be computationally hard. Similarly from the statistical physics point of view, it is conjectured that if we were to start with the uniform distribution (i.e., a “hot” system) and “lower the temperature” (increase ) at a rate that is not exponentially slow then we will get “stuck” at a “metastable” state. This is analogous how when we heat up sand and then cool it quickly then rather than returning to its original state, the sand will get stuck at the metastable state of glass.

In contrast for the Sherrington-Kirpatrick (SK) model, the geometry is more subtle, but interestingly this enables better algorithms. The SK model is extermely widely studied, with hundreds of papers, and was the inspiration for the simulated annealing algorithm. If memory serves me right, Sherrington and Kirpatrick made the wrong conjecture on the energy of the ground state, and then Parisi came up in 1979 with a wonderful and hugely influential way to compute this value. Parisi’s calculation was heuristic, but about 30 years later, first Talagrand and later Panchenko proved rigorously many of Parisi’s conjectures. (See this survey of Panchenko.)

Recently Montanari gave a polynomial time algorithm to find a state with energy that is arbitrarily close to the ground state’s. The algorithm relies on Parisi’s framework and in particular on the fact that the solution space has a property known as “full replica symmetry breaking (RSB)” / “ultrametricity”. Parisi’s derivations (and hence also Montanari’s analysis) are highly elaborate and I admit that I have not yet been able to fully follow it. The nice thing is that (as we’ll see) it is possible to describe at least some of the algorithmic results without going into this theory. In the end of the post I will discuss a bit some of the relation to this theory, which is the underlying inspiration for Subag’s results described here.

Note: These papers and this blog post deal with the search problem of finding a solution that minimizes the objective. The refutation problem of certifying that this minimum is at least for some has often been studied. The computational complexity of these problems need not be identical. In particular there are cases where the search problem has an efficient algorithm achieving value but the best refutation algorithm can only certify that the value is at most for .

Analysis of a simpler setting

Luckily, there is a similar computational problem, for which the analysis of analogous algorithm, which was discovered by Subag and was the partial inspiration for Montanari’s work, is much simpler. Specifically, we consider the case where is an element of the unit sphere, and is a degree polynomial with random Gaussian coefficients. Specifically, for every vector , we let where for every , is a random tensor of order whose coefficients are all chosen i.i.d in . (We assume that polynomial does not have constant or linear components.)

Depending on , the computational and geometrical properties of this problem can vary considerably. The case that (i.e., only has a non-zero coeffiecent) corresponds to finding the unit vector minimizing for a random matrix , which of course corresponds to the efficiently solveable minimum eigenvector problem. In contrast, the case corresponds to finding a rank one component of a random three-tensor, which is believed to be computationally difficult. The Parisi calculations give a precise condition on the vector such that if holds then the solution space has the “full RSB” property (and hence the problem is computationally easy) and if does not hold then the solution space does not have this property (and potentially the problem is hard).

These calculations also give rise to the following theorem:

Theorem (Chen and Sen, Proposition 2): If holds then in the limit , , where . (That is, is the second derivative of this univariate polynomial)

We will not discuss the proof of this theorem, but rather how, taking it as a black box, it leads to an algorithm for minimizing that achieves a near-optimal value (assuming holds).

It is a nice exercise to show that for every two vectors , . Hence for any unit vector , is a random variable with mean zero and standard deviation . Since (after some coarsening) the number of unit vectors of dimension can be thought of as for some , and we expect the probability of deviating standard deviations to be , the minimum value of should be for some constant . However determining this constant is non trivial and is the result of the Parisi theory.

To get a better sense for the quantity , let’s consider two simple cases:

• If (i.e., for random matrix ) then and , meaning that . This turns out to be the actual minimum value. Indeed in this case . But the matrix ‘s non diagonal entries are distributed like and the diagonal entries like which means that is distributed as times a random matrix from the Gaussian Orthogonal Ensemble (GOE) where for off diagonal entries and for diagonal entries. The minimum eigenvalue of such matrices is known to be with high probability.
  
• If (i.e. for a random -tensor ) then does not hold. Indeed, in this case the value for large . However I believe (though didn’t find the reference) that the actual minimum tends to with .

While the particular form of the property is not important for this post, there are several equivalent ways to state it, see Proposition 1 in Subag’s paper. One of them is that the function (note the negative exponent) is concave on the interval .
It can be shown that this condition cannot be satisfied if , and that for every setting of , if is large enough then it will be satisfied.

The central result of Subag’s paper is the following:

Theorem: For every , there is a polynomial-time algorithm such that on input random chosen according to the distribution above, with high probability such that .

The algorithm itself, and the idea behind the analysis are quite simple. In some sense it’s the second algorithm you would think of (or at least the second algorithm according to some ordering).

The first algorithm one would think of is gradient descent. We start at some initial point , and repeat the transformation for some small (and normalizing the norm). Unfortunately, we will generally run into saddle points when we do so, with the gradient being zero. In fact, for simplicity, below we will always make the pessimistic assumption that we are constantly on a saddle point. (This assumption turns out to be true in the actual algorithm, and if it was not then we can always use gradient descent until we hit a saddle.)

The second algorithm one could think of would be to use the Hessian instead of the gradient. That is, repeat the transformation where is the minimal eigenvector of (i.e., the Hessian matrix such that ). By the Taylor approximation (and since we assume the gradient is zero) the change in the objective will be roughly . (Because we assume the gradient vanishes, it will not make a difference whether we update with or , but we use for consistency with gradient descent.)

The above approach is promising, but we still need some control over the norm. The way that Subag handles this is that he starts with , and at each step takes a step in an orthogonal direction, and so within steps he will get to a unit norm vector. That is, the algorithm is as follows:

Algorithm:

Input: .

Goal: Find unit approximately minimizing

1. Initialize
2. For : i. Let be a unit vector such that is orthogonal to and . (Since the bottom eigenspace of has large dimention, we can find a vector that is not only nearly minimal eigenvector but also orthogonal to all prior ones. Also, as mentioned, we assume that .) ii. Set .
3. Output

(The fact that the number of steps is and not is absolutely crucial for the algorithm’s success; without it we would not have been able to use the second order contribution that arise from the Hessian.)

If we define to be the minimum eigenvalue at time , we get that the final objective value achieved by the algorithm satisfies

Now due to rotation invariance, the distribution of at the point is the same as at the point . Using concentration of measure arguments, it can be shown that the minimum eigenvalue of will be close with high probability to the minimum eigenvalue of .
Since we can write

where is the minimum eigenvalue of at the point .
Taking to zero, we get that (approximately) the value of the solution output by the algorithm will satisfy

and hence the result will be completed by showing that

To do this, let’s recall the definition of :

where for every , is a random tensor of order whose coefficients are all chosen i.i.d in .

For simplicity, let’s assume that and hence

(The calculations in the general case are similar)

The entry of equals . The contribution of the component to this term only arises from the terms corresponding to either or and hence for it equals which is distributed like . For , since , the contributioon equals which is distributed like .

The contribution from the component comes (in the case ) from all the terms involving that is, which is distributed like . For the contribution will be from the terms involving , with each yielding a contribution of , and hence the result will be distributed like .

(More generally, for larger , the number of terms for distinct is , each contributing a Gaussian of standard deviation , while for we have terms, each contributing a Gaussian of standard deviation .)

Since the sum of Gaussians is a Gaussian we get that is distributed like a Gaussian with variance for , and twice that for . This means that the minimum eigenvalue of equals times the minimum eigenvalue of a random matrix from the Gaussian Orthogonal Ensemble (i.e. is sampled via , ). As mentioned above, it is known that this minimum eigenvalue is , and in fact by the semi-circle law, for every , the number of eigenvalues of value is , and so we can also pick one that is orthogonal to the previous directions. QED

Full replica symmetry breaking and ultra-metricity

The point of this blog post is that at least in the “mixed spin” case considered by Subag, we can understand what the algorithm does and the value that it achieves without needing to go into the theory of the geometry of the solution space, but let me briefly discuss some of this theory. (As I mentioned, I am still reading through this, and so this part should be read with big grains of salt.)

The key object studied in this line of work is the probability distribution of the dot product for and sampled independently from the Gibbs distribution induced by . (This probability distribution will depend on the number of dimensions , but we consider the case that .)

Intuitively, there are several ways this probability distribution can behave, depending on how the solution space is “clustered”:

• If all solutions are inside a single “cluster”, in the sense that they are all of the form where is the “center” of the cluster and is some random vector, then will be concentrated on the point .
  
• If the solutions are inside a finite number of clusters, with centers , then the support of the distribution will be on the points .
  
• Suppose that the solutions are inside a hierarchy of clusters. That is, suppose we have some rooted tree (e.g., think of a depth full binary tree), and we associate a vector with every vertex of , with the property that is orthogonal to all vectors associated with ‘s ancestors on the tree. Now imagine that the distribution is obtained by taking a random leaf of and outputting for all vertices on the path from the root to . In such a case the dot product of and will be taken over all the common ancestors of . As the dimension and depth of the tree goes to infinity, the distribution over dot product can have continuous support, and it is this setting (specifically when the support is an interval ) that is known as full replica symmetry breaking. Because the dot product is determined by common ancestor, for every three vectors in the support of the distribution or . It is this condition that known as ultra-metricity.

In Subag’s algorithm, as mentioned above, at any given step we could make an update of either or . If we think of all the possible choices for the signs in the of the algorithms, we see that the algorithm does not only produce a single vector but actually such vectors that are arranged in an ultrametric tree just as above. Indeed, this ultrametric structure was the inspiration for the algorithm and is the reason why the algorithm produces the correct result precisely in the full replica symmetry breaking regime.

Acknowledgements: Thanks to Tselil Schramm for helpful comments.