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Optimal Ordered Problem Solver (OOPS)

Our recent OOPS [54,52] is a simple, general, theoretically sound, time-optimal way of searching for a universal behavior or program that solves each problem in a sequence of computational problems, continually organizing and managing and reusing earlier acquired knowledge. For example, the $n$-th problem may be to compute the $n$-th event from previous events (prediction), or to find a faster way through a maze than the one found during the search for a solution to the $n-1$-th problem (optimization).

Let us first introduce the important concept of bias-optimality, which is a pragmatic definition of time-optimality, as opposed to the asymptotic optimality of both LSEARCH and HSEARCH, which may be viewed as academic exercises demonstrating that the $O()$ notation can sometimes be practically irrelevant despite its wide use in theoretical computer science. Unlike asymptotic optimality, bias-optimality does not ignore huge constant slowdowns:

\begin{definition}[{\sc Bias-Optimal Searchers}]
Given is a problem class $\cal ...
...\leq P(p \mid r)~T_{max}/n$.
It is {\em bias-optimal} if $n=1$.
This definition makes intuitive sense: the most probable candidates should get the lion's share of the total search time, in a way that precisely reflects the initial bias. Now we are ready to provide a general overview of the basic ingredients of OOPS [54,52]:

Primitives. We start with an initial set of user-defined primitive behaviors. Primitives may be assembler-like instructions or time-consuming software, such as, say, theorem provers, or matrix operators for neural network-like parallel architectures, or trajectory generators for robot simulations, or state update procedures for multiagent systems, etc. Each primitive is represented by a token. It is essential that those primitives whose runtimes are not known in advance can be interrupted at any time.

Task-specific prefix codes. Complex behaviors are represented by token sequences or programs. To solve a given task represented by task-specific program inputs, OOPS tries to sequentially compose an appropriate complex behavior from primitive ones, always obeying the rules of a given user-defined initial programming language. Programs are grown incrementally, token by token; their beginnings or prefixes are immediately executed while being created; this may modify some task-specific internal state or memory, and may transfer control back to previously selected tokens (e.g., loops). To add a new token to some program prefix, we first have to wait until the execution of the prefix so far explicitly requests such a prolongation, by setting an appropriate signal in the internal state. Prefixes that cease to request any further tokens are called self-delimiting programs or simply programs (programs are their own prefixes). Binary self-delimiting programs were studied by [30] and [9] in the context of Turing machines [64] and the theory of Kolmogorov complexity and algorithmic probability [59,28]. OOPS, however, uses a more practical, not necessarily binary framework.

The program construction procedure above yields task-specific prefix codes on program space: with any given task, programs that halt because they have found a solution or encountered some error cannot request any more tokens. Given the current task-specific inputs, no program can be the prefix of another one. On a different task, however, the same program may continue to request additional tokens. This is important for our novel approach--incrementally growing self-delimiting programs are unnecessary for the asymptotic optimality properties of LSEARCH and HSEARCH, but essential for OOPS.

Access to previous solutions. Let $p^n$ denote a found prefix solving the first $n$ tasks. The search for $p^{n+1}$ may greatly profit from the information conveyed by (or the knowledge embodied by) $p^1, p^2, \ldots, p^n$ which are stored or frozen in special nonmodifiable memory shared by all tasks, such that they are accessible to $p^{n+1}$ (this is another difference to nonincremental LSEARCH and HSEARCH). For example, $p^{n+1}$ might execute a token sequence that calls $p^{n-3}$ as a subprogram, or that copies $p^{n-17}$ into some internal modifiable task-specific memory, then modifies the copy a bit, then applies the slightly edited copy to the current task. In fact, since the number of frozen programs may grow to a large value, much of the knowledge embodied by $p^j$ may be about how to access and edit and use older $p^i$ ($i<j$).

Bias. The searcher's initial bias is embodied by initial, user-defined, task dependent probability distributions on the finite or infinite search space of possible program prefixes. In the simplest case we start with a maximum entropy distribution on the tokens, and define prefix probabilities as the products of the probabilities of their tokens. But prefix continuation probabilities may also depend on previous tokens in context sensitive fashion.

Self-computed suffix probabilities. In fact, we permit that any executed prefix assigns a task-dependent, self-computed probability distribution to its own possible continuations. This distribution is encoded and manipulated in task-specific internal memory. So unlike with ALS [57] we do not use a prewired learning scheme to update the probability distribution. Instead we leave such updates to prefixes whose online execution modifies the probabilities of their suffixes. By, say, invoking previously frozen code that redefines the probability distribution on future prefix continuations, the currently tested prefix may completely reshape the most likely paths through the search space of its own continuations, based on experience ignored by nonincremental LSEARCH and HSEARCH. This may introduce significant problem class-specific knowledge derived from solutions to earlier tasks.

Two searches. Essentially, OOPS provides equal resources for two near-bias-optimal searches (Def. 1) that run in parallel until $p^{n+1}$ is discovered and stored in non-modifiable memory. The first is exhaustive; it systematically tests all possible prefixes on all tasks up to $n+1$. Alternative prefixes are tested on all current tasks in parallel while still growing; once a task is solved, we remove it from the current set; prefixes that fail on a single task are discarded. The second search is much more focused; it only searches for prefixes that start with $p^n$, and only tests them on task $n+1$, which is safe, because we already know that such prefixes solve all tasks up to $n$.

Bias-optimal backtracking. HSEARCH and LSEARCH assume potentially infinite storage. Hence they may largely ignore questions of storage management. In any practical system, however, we have to efficiently reuse limited storage. Therefore, in both searches of OOPS, alternative prefix continuations are evaluated by a novel, practical, token-oriented backtracking procedure that can deal with several tasks in parallel, given some code bias in the form of previously found code. The procedure always ensures near-bias-optimality (Def. 1): no candidate behavior gets more time than it deserves, given the probabilistic bias. Essentially we conduct a depth-first search in program space, where the branches of the search tree are program prefixes, and backtracking (partial resets of partially solved task sets and modifications of internal states and continuation probabilities) is triggered once the sum of the runtimes of the current prefix on all current tasks exceeds the prefix probability multiplied by the total search time so far.

In case of unknown, infinite task sequences we can typically never know whether we already have found an optimal solver for all tasks in the sequence. But once we unwittingly do find one, at most half of the total future run time will be wasted on searching for alternatives. Given the initial bias and subsequent bias shifts due to $p^1, p^2, \ldots, $ no other bias-optimal searcher can expect to solve the $n+1$-th task set substantially faster than OOPS. A by-product of this optimality property is that it gives us a natural and precise measure of bias and bias shifts, conceptually related to Solomonoff's conceptual jump size of [61,62].

Since there is no fundamental difference between domain-specific problem-solving programs and programs that manipulate probability distributions and thus essentially rewrite the search procedure itself, we collapse both learning and metalearning in the same time-optimal framework.

An example initial language. For an illustrative application, we wrote an interpreter for a stack-based universal programming language inspired by FORTH [35], with initial primitives for defining and calling recursive functions, iterative loops, arithmetic operations, and domain-specific behavior. Optimal metasearching for better search algorithms is enabled through the inclusion of bias-shifting instructions that can modify the conditional probabilities of future search options in currently running program prefixes.

Experiments. Using the assembler-like language mentioned above, we first teach OOPS something about recursion, by training it to construct samples of the simple context free language $\{ 1^k2^k \}$ ($k$ 1's followed by $k$ 2's), for $k$ up to 30 (in fact, the system discovers a universal solver for all $k$). This takes roughly 0.3 days on a standard personal computer (PC). Thereafter, within a few additional days, OOPS demonstrates incremental knowledge transfer: it exploits aspects of its previously discovered universal $1^k2^k$-solver, by rewriting its search procedure such that it more readily discovers a universal solver for all $k$ disk Towers of Hanoi problems--in the experiments it solves all instances up to $k=30$ (solution size $2^k-1$), but it would also work for $k>30$. Previous, less general reinforcement learners and nonlearning AI planners tend to fail for much smaller instances.

Future research may focus on devising particularly compact, particularly reasonable sets of initial codes with particularly broad practical applicability. It may turn out that the most useful initial languages are not traditional programming languages similar to the FORTH-like one, but instead based on a handful of primitive instructions for massively parallel cellular automata [65,67,74,72], or on a few nonlinear operations on matrix-like data structures such as those used in recurrent neural network research [69,44,4]. For example, we could use the principles of OOPS to create a non-gradient-based, near-bias-optimal variant of Hochreiter's successful recurrent network metalearner [19]. It should also be of interest to study probabilistic Speed Prior-based OOPS variants [55] and to devise applications of OOPS-like methods as components of universal reinforcement learners (see below). In ongoing work, we are applying OOPS to the problem of optimal trajectory planning for robotics in a realistic physics simulation. This involves the interesting trade-off between comparatively fast program-composing primitives or ``thinking primitives'' and time-consuming ``action primitives'', such as stretch-arm-until-touch-sensor-input.

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Next: OOPS-Based Reinforcement Learning Up: The New AI: General Previous: Optimal Universal Search Algorithms
Juergen Schmidhuber 2003-02-04