Predictive Complexity for Games with Finite Outcome Spaces

Abstract

Predictive complexity is a generalization of Kolmogorov complexity motivated by an on-line prediction scenario. It quantifies the “unpredictability” of a sequence in a particular prediction environment. This chapter surveys key results on predictive complexity for games with finitely many outcomes. The issues of existence, non-existence, uniqueness, and linear inequalities are covered.

Appendix: Enumerating Superloss Processes

In this appendix we will discuss the question of effective enumeration of superloss processes. We reproduce and analyze the construction from [15].

A process $L:\varOmega ^*\rightarrow [0,+\infty ]$ is finitary if the set $\{{\mathbf {x}}\in \varOmega ^*\mid L({\mathbf {x}})<+\infty \}$ is finite. A process L is dyadic if its values are dyadic rationals or $+\infty$.

We call a dyadic finitary superloss process L verifiable if for every ${\mathbf {x}}\in \varOmega$ there is $\gamma \in \varGamma$ such that $L({\mathbf {x}}\omega )-L({\mathbf {x}})>\lambda (\gamma ,\omega )$ for all $\omega \in \varOmega$. The inequality is equivalent to $e^{L({\mathbf {x}})-L({\mathbf {x}}\omega )}<e^{-\lambda (\gamma ,\omega )}$. Since $\lambda$ is continuous, the inequalities will still hold within a small vicinity of $\gamma$. Recall that for computable games we postulated the existence of an effective dense dyadic sequence $\gamma _i$. Thus for a computable game if we are given a finite list $\ell$ of pairs $({\mathbf {x}}_s,r_s)\in \varOmega ^*\times {\mathbb D}$, $s=1,2,\ldots ,S$, such that

$$\begin{aligned} N_\ell ({\mathbf {x}})= {\left\{ \begin{array}{ll} \min \{\mathop {\mathrm{{d}}}\nolimits (r_s)\mid ({\mathbf {x}},r_s) \text{ is } \text{ in } \text{ the } \text{ list }\} &{} \text{ if } {\mathbf {x}}={\mathbf {x}}_s \text{ for } \text{ some }\\ &{}\;\;\;\; s=1,2,\ldots ,S\,\,;\\ +\infty &{} \text{ otherwise } \end{array}\right. } \end{aligned}$$

(8.18)

is a verifiable dyadic finitary superloss process, we will be able to confirm that.

Therefore verifiable dyadic finitary superloss processes can be effectively enumerated. Let $P_i$, $i\in {\mathbb N}$, be an effective enumeration of programs such that each $P_i$ outputs a finite list of pairs $({\mathbf {x}}_s,r_s)\in \varOmega ^*\times {\mathbb D}$, $s=1,2,\ldots ,S_i$ (the program must halt after finitely many steps) defining a verifiable dyadic finitary superloss process $N_i$ as in (8.18) and every verifiable dyadic finitary superloss process is calculated by some $P_i$.

Pick a universal partial computable function M(i, j) on ${\mathbb N}^2$. Universality means that every partial computable function on the integers coincides with some $M(i,\cdot )$. Put $M^*(i,j)=M(i,j)$ if

• 1. the function M is defined on all pairs $(i,j')$ with $j'\le j$,

• 2. all outputs $M(i,j')$ with $j'\le j$ encode lists of pairs $({\mathbf {x}},r)\in \varOmega ^*\times {\mathbb D}$,

• 3. all $N_{M(i,j')}$, where $j'\le j$, are verifiable dyadic finitary superloss processes, and

• 4. $N_{M(i,j')}$ never exceeds $N_{M(i,j'+1)}$ (i.e., for all ${\mathbf {x}}\in \varOmega ^*$ we have $N_{M(i,j')}({\mathbf {x}})\ge N_{M(i,j'+1)}({\mathbf {x}})$), $j'=1,2,\ldots ,j-1$.

and let $M^*(i,j)$ be undefined otherwise.

For every $i\in {\mathbb N}$ define a process $k_i$ by

$$ k_i({\mathbf {x}})=\inf _jN_{M^*(i,j)}\,\,, $$

where the infimum is taken over all j such that $M^*(i,j)$ is defined. Clearly, $k_i$ is an upper semicomputable superloss process. Indeed, if $M^*(i,j)$ is undefined from some j on, then $k_i$ is a finitary superloss process. Otherwise each value $k_i({\mathbf {x}})$ is the limit of a non-increasing sequence of $L_j({\mathbf {x}})=N_{M^*(i,j)}({\mathbf {x}})$. Since each $L_j$ is a superloss process, for every ${\mathbf {x}}$ there is a $\gamma _j\in \varGamma$ such that (8.2) holds for $L_j$ for all $\omega \in \varOmega$. Since $\varGamma$ is compact, there is a converging subsequence of $\gamma _j$ and therefore (8.2) holds in the limit and thus $k_i$ is a superloss process. Since the partial function $f({\mathbf {x}},n)=L_n({\mathbf {x}})$ is uniformly computable, $k_i$ is upper semicomputable.

In order to show that this construction allows us to enumerate all superloss processes, we need to prove that every superloss process is the limit of a uniformly computable non-increasing sequence of verifiable dyadic finitary superloss processes. We will formulate a sufficient condition for that.

Consider a game $\mathfrak G=\langle \varOmega ,\varGamma ,\lambda \rangle$ with $\varOmega =\left\{ \omega ^{(0)},\omega ^{(1)},\ldots ,\omega ^{(M-1)}\right\}$. Consider the partial function $H:[0,+\infty ]^{M}\rightarrow {\mathbb R}$ defined by

$$ H(x_0,x_1,\ldots ,x_{M-1})=\max \{h\ge 0\mid \text{ there } \text{ is } \gamma \in \varGamma \text{ such } \text{ that } \\ \qquad \qquad \quad \qquad \qquad \qquad \quad x_m-h\ge \lambda \left( \gamma ,\omega ^{(m)}\right) , m=0,1,\ldots ,M-1\} $$

(if the set is empty, the function is undefined). Note that the maximum is achieved because $\varGamma$ is compact and $\lambda$ is continuous. Let us call $\mathfrak G$ a game with effective minorization if H is computable where it is defined (here we assume that $+\infty$ is given to us as a special symbol).

Lemma 8.6

Let a computable binary game $\langle \mathbb {B},[0, 1],\lambda \rangle$ have a monotonic loss function so that $\lambda (\gamma ,0)$ is non-decreasing and $\lambda (\gamma ,1)$ is non-increasing. Then the function H is computable where it is defined.

This lemma implies that the binary square-loss, absolute-loss, and logarithmic games are games with effective minorization.

Proof

The system of inequalities

$$\begin{aligned} x_0-h&\ge \lambda (\gamma ,0)\,\,;\\ x_1-h&\ge \lambda (\gamma ,1) \end{aligned}$$

is equivalent to

$$\begin{aligned} e^h&\le e^{x_0}e^{-\lambda (\gamma ,0)}\,\,;\\ e^h&\le e^{x_1}e^{-\lambda (\gamma ,1)}\,\,. \end{aligned}$$

The maximum h is achieved where the monotonic graphs of the functions on the right-hand side intersect.    $\square$

Lemma 8.7

If $\mathfrak G$ is a computable game with effective minorization, then every upper semicomputable superloss process L is the infimum of a non-increasing effective sequence of verifiable finitary superloss processes.

Proof

We say that a process $L_1$ majorizes a process $L_2$ if $L_1({\mathbf {x}})\ge L_2({\mathbf {x}})$ for all ${\mathbf {x}}\in \varOmega ^*$. A set of pairs $({\mathbf {x}}_s,r_s)\in \varOmega ^*\times {\mathbb D}$, $s=1,2,\ldots ,S$, majorizes a process $L_2$ if $\mathop {\mathrm{{d}}}\nolimits (r_s)\ge L_2({\mathbf {x}}_s)$ for all $s=1,2,\ldots ,S$.

Lemma 8.8

For a finite set of pairs $({\mathbf {x}}_s,r_s)\in \varOmega ^*\times {\mathbb D}$, $s=1,2,\ldots ,S$, that majorize some superloss process there is a maximum finitary superloss process N that the set majorizes, i.e., there is a finitary superloss process N majorized by the set of pairs and majorizing every other superloss process majorized by the set of pairs.

Proof

Let $n=\max _s|{\mathbf {x}}_s|$ be the maximum length of a sequence in the set. If $|{\mathbf {x}}|>n$ we let $N({\mathbf {x}})=+\infty$. If $|{\mathbf {x}}|=n$ we let $N({\mathbf {x}})$ to be the minimum of $\mathop {\mathrm{{d}}}\nolimits (r_s)$ such that $({\mathbf {x}},r_s)$ is in the set or $+\infty$ if there are none. Clearly, for every superloss process $L'$ majorized by the set of pairs, we have $N({\mathbf {x}})\ge L'({\mathbf {x}})$ so far.

For sequences ${\mathbf {x}}$ of smaller length we define $N({\mathbf {x}})$ by induction from larger lengths to smaller by setting $N({\mathbf {x}})$ to be the minimum of $\mathop {\mathrm{{d}}}\nolimits (r_s)$ such that the pair $({\mathbf {x}},r_s)$ is in the set and $H\left( N\left( {\mathbf {x}}\omega ^{(0)}\right) ,N\left( {\mathbf {x}}\omega ^{(1)}\right) ,\ldots , N\left( {\mathbf {x}}\omega ^{(M-1)}\right) \right)$. It is easy to see that if for some superloss process $L'$ we have $N\left( {\mathbf {x}}\omega ^{(m)}\right) \ge L'\left( {\mathbf {x}}\omega ^{(m)}\right)$ for all $m=0,1,\ldots ,M-1$, then $H\ge L'({\mathbf {x}})$. Lemma 8.8 is proved.   $\square$

Let $L({\mathbf {x}})=\inf _{n\in {\mathbb N}}\mathop {\mathrm{{d}}}\nolimits (f({\mathbf {x}},n))$ for some partial computable $f:\varOmega ^*\times {\mathbb N}\rightarrow {\mathbb D}$. We keep generating pairs $({\mathbf {x}},f({\mathbf {x}},n))\in \varOmega ^*\times {\mathbb D}$ and every so often (e.g., after every 1000 computation steps) we define a verifiable dyadic finitary superloss process $L_i$; however sometimes we withhold it. The first process $L_1$ is not withheld.

Let $L_j$ be the latest process that was not withheld. The procedure for producing $L_i$ is as follows. Suppose that we have generated S pairs $({\mathbf {x}}_s,r_s)$. Let n be the largest length of ${\mathbf {x}}_s$. Take a dyadic $\varepsilon =2^{-i-k}$ where $2^k$ is the minimum power of 2 exceeding $2n+3$.

There exists a maximum superloss process $N_i({\mathbf {x}})$ majorized by the set of pairs $({\mathbf {x}}_s,r_s)$ produced so far. Since $\mathfrak G$ is a game with effective minorization, the values of $N_i({\mathbf {x}})$ are computable. We will now approximate it with a verifiable finitary superloss process $L_i$. If $N_i({\mathbf {x}})=+\infty$, we let $L_i({\mathbf {x}})=+\infty$. For each ${\mathbf {x}}$ such that $N_i({\mathbf {x}})<+\infty$ we can find dyadic numbers $d'_{\mathbf {x}}$ and $d''_{\mathbf {x}}$ such that $d''_{\mathbf {x}}-d'_{\mathbf {x}}\le \varepsilon /2$ and $d'_{\mathbf {x}}\le N_i({\mathbf {x}})\le d''_{\mathbf {x}}$. Take $L_i({\mathbf {x}})=d''_{\mathbf {x}}+2\varepsilon (|{\mathbf {x}}|+1)$.

We have $L_i({\mathbf {x}}\omega )-L_i({\mathbf {x}})\ge N_i({\mathbf {x}}\omega )-N_i({\mathbf {x}})+\varepsilon$ provided $N({\mathbf {x}}\omega )$ is finite. Thus $L_i$ is a verifiable finitary superloss process.

Let us compare $L_i$ with the latest process $L_j$ that was not withheld. If $L_i({\mathbf {x}})\le L_j({\mathbf {x}})$ for all ${\mathbf {x}}\in \varOmega ^*$ (note that every $L_k({\mathbf {x}})$ is either $+\infty$ or a dyadic number and we can perform this check in finite time), we output $L_i$; otherwise we withhold it.

We need to show that $L({\mathbf {x}})=\inf _iL_i({\mathbf {x}})$, where the infimum is taken over all i such that $L_i$ is not withheld. First note that $L({\mathbf {x}})=\inf _{i\in {\mathbb N}}N_i({\mathbf {x}})$. Indeed, since $N_i$ is maximal by construction, $L({\mathbf {x}})\le N_i({\mathbf {x}})$. On the other hand $L({\mathbf {x}})$ is the infimum of $\mathop {\mathrm{{d}}}\nolimits (r_s)$ such that $({\mathbf {x}},r_s)$ occurs in the enumeration and for every $({\mathbf {x}},r_s)$ there exists $N_i$ majorized by it. Secondly by construction we have

$$ 2^{-i-k+1}\le 2^{-i-k}2(|{\mathbf {x}}|+1)=2\varepsilon (|{\mathbf {x}}|+1)\le L_i({\mathbf {x}})-N_i({\mathbf {x}})\le \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 2\varepsilon (|{\mathbf {x}}|+1)+\frac{\varepsilon }{2}< \varepsilon (2|{\mathbf {x}}|+3)\le 2^{-i} $$

if $N_i({\mathbf {x}})<+\infty$. Since $L({\mathbf {x}})\le N_i({\mathbf {x}})\le L_i({\mathbf {x}}) \le N_i({\mathbf {x}})+2^{-i}$, we get $L({\mathbf {x}})=\inf _iL_i({\mathbf {x}})$. Finally note that $L_j({\mathbf {x}})\ge N_j({\mathbf {x}})+2^{-j-k+1}\ge L({\mathbf {x}})+2^{-j-k+1}$ for some $k\in {\mathbb N}$, i.e., there is a non-zero gap between $L_j$ and L. Therefore infinitely many $L_i$ will not be withheld.    $\square$

Corollary 8.3

If $\mathfrak G$ is a computable game with effective minorization, then there is an enumeration of upper semicomputable processes w.r.t. $\mathfrak G$.