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.
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Acknowledgments
The work has been supported by a Leverhulme Trust research project grant RPG-2013-047 “On-line Self-Tuning Learning Algorithms for Handling Historical Information.”
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Appendix: Enumerating Superloss Processes
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
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
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1.
the function M is defined on all pairs \((i,j')\) with \(j'\le j\),
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2.
all outputs \(M(i,j')\) with \(j'\le j\) encode lists of pairs \(({\mathbf {x}},r)\in \varOmega ^*\times {\mathbb D}\),
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3.
all \(N_{M(i,j')}\), where \(j'\le j\), are verifiable dyadic finitary superloss processes, and
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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
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
(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
is equivalent to
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
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. \(\mathfrak G\).
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Kalnishkan, Y. (2015). Predictive Complexity for Games with Finite Outcome Spaces. In: Vovk, V., Papadopoulos, H., Gammerman, A. (eds) Measures of Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-21852-6_8
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