What Is Important About the No Free Lunch Theorems?

Abstract

The No Free Lunch theorems prove that under a uniform distribution over induction problems (search problems or learning problems), all induction algorithms perform equally. As I discuss in this chapter, the importance of the theorems arises by using them to analyze scenarios involving nonuniform distributions, and to compare different algorithms, without any assumption about the distribution over problems at all. In particular, the theorems prove that anti-cross-validation (choosing among a set of candidate algorithms based on which has worst out-of-sample behavior) performs as well as cross-validation, unless one makes an assumption—which has never been formalized—about how the distribution over induction problems, on the one hand, is related to the set of algorithms one is choosing among using (anti-)cross validation, on the other. In addition, they establish strong caveats concerning the significance of the many results in the literature that establish the strength of a particular algorithm without assuming a particular distribution. They also motivate a “dictionary” between supervised learning and improving blackbox optimization, which allows one to “translate” techniques from supervised learning into the domain of blackbox optimization, thereby strengthening blackbox optimization algorithms. In addition to these topics, I also briefly discuss their implications for philosophy of science.

Appendix

1.1 A.1 NFL and Inner Product Formulas for Search

To begin, expand the performance probability distribution:

$$\displaystyle \begin{aligned} \begin{array}{rcl} P(\phi \mid {\mathcal{A}}, m) & =&\displaystyle \sum_{d^m_Y} P(d^m_Y \mid {\mathcal{A}}, m) P(\phi \mid d^m_Y, {\mathcal{A}}, m) \\ & =&\displaystyle \sum_{d^m_Y} P(d^m_Y \mid {\mathcal{A}}, m) \delta(\phi, \Phi(d^Y_m)), {} \end{array} \end{aligned} $$

(5)

where the delta function equals 1 if its two arguments are equal, and 0 otherwise. The choice of search algorithm affects performance only through the term $P(d^m_Y \mid {\mathcal {A}}, m)$. In turn, this probability of $d^m_Y$ under ${\mathcal {A}}$ is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} P(d^m_Y \mid {\mathcal{A}}, m) & =&\displaystyle \sum_{f} P(d_Y^m \mid f, m, {\mathcal{A}}) P(f \mid m, {\mathcal{A}}) \\ & =&\displaystyle \sum_{f} P(d_Y^m \mid f, m, {\mathcal{A}}) P(f). \end{array} \end{aligned} $$

(6)

Plugging in gives

$$\displaystyle \begin{aligned} P(\phi \mid {\mathcal{A}}, m) &= \sum_{f} P(f) D(f; d^m_Y, {\mathcal{A}}, m), {} \end{aligned} $$

(7)

where

$$\displaystyle \begin{aligned} D(f; d^m_, {\mathcal{A}}, m) &:= \sum_{d^m_Y} P(d^m_Y \mid f, {\mathcal{A}}, m) \delta(\phi, \Phi(d^Y_m)). \end{aligned} $$

(8)

So for any fixed ϕ, $P(\phi \mid {\mathcal {A}}, m)$ is an inner product of two real-valued vectors each indexed by f: $D(f; d^m_Y, {\mathcal {A}}, m)$ and P(f). Note that all the details of how the search algorithm operates are embodied in the first of those vectors. In contrast, the second one is completely independent of the search algorithm.

This notation also allows us to state the NFL for search theorem formally. Let B be any subset of the set of all objective functions, Y ^X. Then Eq. (7) allows us to express the expected performance for functions inside B in terms of expected performance outside of B:

$$\displaystyle \begin{aligned} \sum_{f \in B} {\mathbb{E}}(\Phi \mid f, m, {\mathcal{A}}) &= constant - \sum_{f \in Y^X \setminus B} {\mathbb{E}}(\Phi \mid f, m, {\mathcal{A}}), {} \end{aligned} $$

(9)

where the constant on the right-hand side depends on the performance measure Φ(.) but is independent of both ${\mathcal {A}}$ and B [3]. Expressed differently, Eq. (9) says that $\sum _f E(\Phi \mid f, m, {\mathcal {A}})$ is independent of ${\mathcal {A}}$. This is the core of the NFL for search, as elaborated in the next section.

1.2 A.2 NFL for Search When We Average over P(f)’s

To derive the NFL theorem that applies when we vary over P(f)’s, first recall our simplifying assumption that both X and Y  are finite (as they will be when doing search on any digital computer). Due to this, any P(f) is a finite-dimensional real-valued vector living on a simplex Ω. Let π refer to a generic element of Ω. So ∫_Ω dπ P(f∣π) is the average probability of any one particular f, if one uniformly averages over all distributions on f’s. By symmetry, this integral must be a constant, independent of f. In addition, as mentioned above, Eq. (9) tells us that $\sum _{f \in B} {\mathbb {E}}(\Phi \mid f, m, {\mathcal {A}})$ is independent of ${\mathcal {A}}$. Therefore for any two search algorithms ${\mathcal {A}}$ and ${\mathcal {B}}$,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{A}}) & =&\displaystyle \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{B}}) , \\ \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{A}}) \bigg[\int_\Omega d\pi \; P(f \mid \pi)\bigg] & =&\displaystyle \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{B}}) \bigg[\int_\Omega d\pi \; P(f \mid \pi)\bigg] , \\ \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{A}}) \bigg[\int_\Omega d\pi \; \pi(f)\bigg] & =&\displaystyle \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{B}}) \bigg[\int_\Omega d\pi \; \pi(f)\bigg], \\ \int_\Omega d\pi \; \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{A}}) \pi(f) & =&\displaystyle \int_\Omega d\pi \; \sum_f {\mathbb{E}}(\Phi \mid f, m, {\mathcal{B}}) \pi(f), \end{array} \end{aligned} $$

(10)

that is,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \int_\Omega d\pi \; {\mathbb{E}}_\pi(\Phi \mid m, {\mathcal{A}}) & =&\displaystyle \int_\Omega d\pi \; {\mathbb{E}}_\pi(\Phi \mid m, {\mathcal{B}}). {} \end{array} \end{aligned} $$

(11)

We can re-express this result as the statement that $\int _\Omega d\pi \; {\mathbb {E}}_\pi (\Phi \mid m, {\mathcal {A}})$ is independent of ${\mathcal {A}}$.

Next, let Π be any subset of Ω. Then our result that $\int _\Omega d\pi \; {\mathbb {E}}_\pi (\Phi \mid m, {\mathcal {A}})$ is independent of ${\mathcal {A}}$ implies

$$\displaystyle \begin{aligned} \begin{array}{rcl} \int_{\pi \in \Pi} d\pi \; {\mathbb{E}}_\pi(\Phi \mid m, {\mathcal{A}}) & =&\displaystyle constant - \int_{\pi \in \Omega \setminus \Pi} d\pi \; {\mathbb{E}}_\pi(\Phi \mid m, {\mathcal{A}}), {} \end{array} \end{aligned} $$

(12)

where the constant depends on Φ but is independent of both ${\mathcal {A}}$ and Π. So if any search algorithm performs particularly well for one set of P(f)’s, Π, it must perform correspondingly poorly on all other P(f)’s. This is the NFL theorem for search when P(f)’s vary.

1.3 A.3 NFL and Inner Product Formulas for Supervised Learning

To state the supervised learning inner product and NFL theorems requires introducing some more notation. Conventionally, these theorems are presented in the version where both the learning algorithm and the target function are stochastic. (In contrast, the restrictions for search—presented above—conventionally involve a deterministic search algorithm and deterministic objective function.) This makes the statement of the restrictions for supervised learning intrinsically more complicated.

Let X be a finite input space, Y  a finite output space, and say we have a target distribution f(y _f ∈ Y ∣x ∈ X), along with a training set $d = (d^m_X, d^m_Y)$ of m pairs $\{(d^m_X(i) \in X, d^m_Y(i) \in Y)\}$, which is stochastically generated according to a distribution P(d∣f) (conventionally called a likelihood, or “data-generation process”). Assume that based on d we have a hypothesis distribution h(y _h ∈ Y ∣x ∈ X). (The creation of h from d—specified in toto by the distribution P(h∣d)—is conventionally called the learning algorithm.) In addition, let L(y _h, y _f) be a loss function taking $Y \times Y \rightarrow {\mathbb {R}}$. Finally, let C(f, h, d) be an off-training set cost function,²

$$\displaystyle \begin{aligned} \begin{array}{rcl} C(f, h, d) \propto \sum_{y_f \in Y, y_h \in Y} \sum_{q \in X \setminus d^m_X} P(q) L(y_f, y_h) f(y_f \mid q) h(y_h \mid q), \end{array} \end{aligned} $$

(13)

where P(q) is some probability distribution over X assigning nonzero measure to $X \setminus d^m_X$.

All aspects of any supervised learning scenario—including the prior, the learning algorithm, the data likelihood function, etc.—are given by the joint distribution P(f, h, d, c) (where c is the value of the cost function) and its marginals. In particular, in [9] it is proven that the probability of a particular cost value c is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} P(c \mid d ) & =&\displaystyle \int df dh \; P(h \mid d)P(f \mid d)M_{c,d}(f, h) {} \end{array} \end{aligned} $$

(14)

for a matrix M _c,d that is symmetric in its arguments so long as the loss function is. P(f∣d) ∝ P(d∣f)P(f) is the posterior probability that the real world has produced a target f for you to try to learn, given that you only know d. It has nothing to do with your learning algorithm. In contrast, P(h∣d) is the specification of your learning algorithm. It has nothing to do with the distribution of targets f in the real world. So Eq. (14) tells us that as long as the loss function is symmetric, how “aligned” you (the learning algorithm) are with the real world (the posterior) determines how well you will generalize.

This supervised learning inner product formula results in a set of NFL theorems for supervised learning, once one imposes some additional conditions on the loss function. See [9] for details.