Bounded Full-Information Algorithms

• Bounded full-information algorithms are methods that compress or simulate complete information under resource constraints like bounded gains, limited memory, or recall.
• They leverage diverse techniques—from finite-state transducers in online path learning to protocol complexes in shared-memory systems—to achieve near-optimal performance.
• These constructions yield practical benefits, such as tighter regret bounds in online learning and efficient simulations in dynamic pricing and distributed protocols.

A bounded full-information algorithm denotes, across several technical literatures, an algorithmic construction that either operates in a full-information regime under explicit bounds or reproduces the effect of an unbounded full-information process using bounded resources. In online non-additive path learning, the full-information algorithm is Context-dependent Component Hedge (CDCH), whose analysis relies on transition gains bounded by $B$ after reduction to an additive-gain automaton (Cortes et al., 2018). In asynchronous distributed computing, a bounded full-information algorithm is an iterated shared-memory protocol that simulates the unbounded full-information protocol when each shared-memory entry carries only $b$ bits (Toyos-Marfurt et al., 16 Sep 2025). Related bounded variants of full-information learning also arise in bounded-recall online learning (Schneider et al., 2022), general memory-bounded learning under a known underlying distribution (Moshkovitz et al., 2017), and resource-constrained dynamic pricing with known demand, where the full-information algorithm is the Boundary Attracted Re-solve Method (Ao et al., 24 Jan 2025).

1. Terminological scope and recurring structure

The phrase appears in distinct but structurally related settings. In each case, the idealized object is a full-information process: observation of all transition outputs, revelation of the entire reward vector, writing of the entire current view, or knowledge of the demand model. The bound is then imposed on gain magnitude, shared-memory capacity, recall length, working memory, or feasible resource consumption.

Domain	Full-information object	Bounding mechanism
Online path learning	target $y_t$ and output $\mathrm{out}_t(e)$ of every transition $e\in E$	bounded transition gains $B$
Iterated shared memory	unbounded full-information protocol $FI_P$	$b$-bit shared-memory entries
Online learning with bounded recall	reward vector $r_t \in [0,1]^d$ revealed each round	dependence on last $M$ rewards only
Bounded-memory learning	known underlying distribution over $b$0	bounded memory and candidate-set representation
Dynamic pricing with knapsack	known linear demand $b$1	resource capacities $b$2

Taken together, these works suggest a recurring design pattern. One either compresses full information into a lower-dimensional or bounded representation, or one simulates the combinatorial effect of full information indirectly. The technical tools differ sharply—finite-state transducers, protocol complexes, restart-based averaging, separability arguments, and re-solving with boundary attraction—but the underlying question is similar: how much of the power of full information survives after the relevant bound is imposed?

A persistent source of confusion is terminological. In some papers, “bounded” refers to a bounded gain parameter or bounded regret analysis; in others, it refers to bounded memory, bounded recall, or bounded shared-memory entries. The phrase therefore does not designate a single formal model. It designates a family of constructions in which full-information behavior is retained, approximated, or simulated under an explicit resource constraint.

2. CDCH and bounded gains in online non-additive path learning

In “Online Non-Additive Path Learning under Full and Partial Information” (Cortes et al., 2018), the learning problem is defined over a fixed acyclic expert automaton $b$3 whose accepting paths correspond to experts. At round $b$4, the learner selects an accepting path $b$5, the path induces an output sequence $b$6, the adversary provides a target sequence $b$7, and gain is measured through count-based pattern overlap: $b$8

$b$9

This objective is non-additive in the original automaton, so standard additive path-learning updates do not apply directly.

The full-information regime is defined by observing both the target $y_t$0 and the output $y_t$1 of every transition $y_t$2. The algorithm proposed for this setting is Context-dependent Component Hedge (CDCH). Its central construction is an intermediate context-dependent automaton

$y_t$3

obtained by compiling context-dependent rewrite rules into a deterministic finite-state transducer $y_t$4, composing it with the expert automaton, and projecting away the input labels. For every accepting path $y_t$5 in $y_t$6, there exists a unique accepting path $y_t$7 in $y_t$8. The transformed automaton preserves path identity, has a fixed topology over time, has size independent of the output alphabet size $y_t$9, and converts the original count-based gain into an additive path gain: $\mathrm{out}_t(e)$0

Once this reduction is available, CDCH maintains a weight vector $\mathrm{out}_t(e)$1 over transitions of $\mathrm{out}_t(e)$2 in the unit-flow polytope $\mathrm{out}_t(e)$3. The multiplicative update is

$\mathrm{out}_t(e)$4

followed by relative entropy projection back to $\mathrm{out}_t(e)$5: $\mathrm{out}_t(e)$6 Prediction is performed by decomposing $\mathrm{out}_t(e)$7 into a convex combination of path distributions over accepting paths in $\mathrm{out}_t(e)$8, then sampling one path.

The regret guarantee is given in terms of the length $\mathrm{out}_t(e)$9 of the longest path in $e\in E$0, the number $e\in E$1 of transitions in $e\in E$2, and the upper bound $e\in E$3 on transition gains: $e\in E$4 The paper notes that this is more favorable than earlier full-information methods based on $e\in E$5 losses and is tighter than the prior RRWM-based approach by a factor of $e\in E$6. A common misconception is that full information alone makes the problem additive; the construction of $e\in E$7 shows that additivity is achieved only after an explicit automata-theoretic transformation.

3. Simulating the unbounded full-information protocol with bounded entries

“Space-Time Trade-off in Bounded Iterated Memory” (Toyos-Marfurt et al., 16 Sep 2025) studies a different use of the phrase. Here the reference object is the unbounded full-information protocol in asynchronous shared memory, where in every round each process shares its complete knowledge of the system. The question is how many extra rounds are required to implement that protocol when each shared-memory entry has only $e\in E$8 bits.

The model is a family of iterated shared-memory algorithms $e\in E$9, organized into layers $B$0, with one shared register per process in each layer. In round $B$1, process $B$2 writes an encoding of its current local state to $B$3, reads according to a write-read pattern $B$4, and updates its state: $B$5 If the encoding and state-update functions are the identity, the protocol is the full-information protocol $B$6. Under bounded entries, the number of non-$B$7 values available per entry is bounded by

$B$8

The paper frames simulation topologically, via the protocol complex $B$9. The goal is not bit-for-bit reproduction of the unbounded protocol, but generation of a protocol complex isomorphic to that of $FI_P$0. This distinction is crucial: the bounded protocol may encode states differently, provided the resulting simplicial complex matches the combinatorial structure of the unbounded full-information evolution.

The central necessary condition is stated through vertex distinguishability. If

$FI_P$1

then

$FI_P$2

In other words, every face of the input complex must become distinguishable at some round. This is the combinatorial mechanism behind the space-time trade-off.

The main asymptotic lower bound states that for $FI_P$3, simulating $FI_P$4 iterations of the unbounded full-information protocol requires

$FI_P$5

rounds. For iterated collect, the bound is tight: $FI_P$6 For snapshot-based models, the paper gives

$FI_P$7

and a construction with

$FI_P$8

The constructive component is the Greedy Star algorithm, which builds a sequence of encoding functions by selecting stars of vertices and assigning unique encodings inside each selected star so that those stars become distinguishable. The paper concludes that the algorithm is asymptotically optimal for iterated collect and within a linear factor $FI_P$9 of optimal for snapshot-based models. A notable special case is that for two processes, the round complexity is $b$0, achievable with 2-bit registers.

4. Bounded recall as a bounded full-information learner

In “Online Learning with Bounded Recall” (Schneider et al., 2022), the underlying environment is full-information online learning: on each round $b$1, the learner chooses a distribution $b$2, the adversary reveals a reward vector $b$3, and performance is measured by per-round regret

$b$4

The bound is on recall. An algorithm is $b$5-bounded-recall if

$b$6

with $b$7 for $b$8. If the rule does not depend on $b$9, the algorithm is stationary.

The paper establishes a sharp lower bound: for every $r_t \in [0,1]^d$0, every $r_t \in [0,1]^d$1-bounded-recall algorithm, and every $r_t \in [0,1]^d$2, there exists a distribution over length-$r_t \in [0,1]^d$3 instances such that

$r_t \in [0,1]^d$4

It also gives a simple nonstationary baseline, PeriodicRestart, which partitions time into blocks of length $r_t \in [0,1]^d$5 and restarts a standard low-regret algorithm in each block. If the base algorithm $r_t \in [0,1]^d$6 satisfies $r_t \in [0,1]^d$7, then

$r_t \in [0,1]^d$8

The main negative result concerns the natural idea of running Hedge, FTRL, or a related mean-based learner on the last $r_t \in [0,1]^d$9 rounds only. For any $M$0-mean-based algorithm and any $M$1, there exists a length-$M$2 instance with two actions such that

$M$3

Thus the sliding-window mean-based approach can suffer constant regret per round. This directly contradicts the common intuition that truncating history in a standard full-information algorithm preserves no-regret behavior.

The positive result is AverageRestart. For each $M$4, one runs the base algorithm $M$5 on the suffix $M$6, obtaining $M$7, and plays

$M$8

The rule is stationary because it depends only on the last $M$9 rewards. If the base algorithm satisfies $b$00, then

$b$01

With Hedge as the base learner, this yields

$b$02

matching the lower bound up to constants. The paper further shows that ordering matters: any bounded-recall algorithm that plays a symmetric function of the past $b$03 losses must incur constant regret per round on some instance.

5. General memory-bounded learning under known distribution

“A General Memory-Bounded Learning Algorithm” (Moshkovitz et al., 2017) studies a realizable online labeled-example model in which the learner receives examples $b$04, the target $b$05 belongs to a hypothesis class $b$06, and the underlying distribution over $b$07 is known. The paper phrases the main text in terms of the uniform distribution over $b$08. The learning objective is not to store examples explicitly, but to store only a few important bits that give sufficient information.

The algorithmic state is a candidate set $b$09, initially $b$10. The central combinatorial condition is separability. Writing the class as a bipartite graph with hypotheses on one side and examples on the other, and using the density

$b$11

the graph is $b$12-separable if any $b$13 that is not $b$14-tight admits $b$15 and disjoint $b$16 such that

$b$17

and

$b$18

The paper explicitly describes separability as the anti-mixing notion and notes that the negation of mixing and anti-mixing are similar but not identical.

Two subroutines drive the algorithm. Is-close(h,\epsilon,k) tests whether a hypothesis $b$19 is close to the target $b$20 by drawing labeled examples and checking disagreement frequency. Estimate(S,\tau,k) estimates $b$21 by sampling labeled examples and returning the ratio of positive labels among examples with $b$22. The guarantees are concentration bounds for closeness testing and density estimation.

The main loop has two cases. If $b$23 is $b$24-tight, an oracle returns a witness hypothesis $b$25 such that $b$26, the learner runs Is-close, and either halts with $b$27 or deletes the entire $b$28-ball: $b$29 If $b$30 is not tight, the oracle returns $b$31 with

$b$32

and $b$33. The learner estimates $b$34 and deletes either $b$35 or $b$36 according to which side is inconsistent with the estimate. The paper’s theorem states that for any hypothesis class $b$37 that is $b$38-separable, Algorithm 1 is a bounded-memory learner for $b$39. It is also shown to be a statistical query algorithm and therefore robust to classification noise.

6. Boundary attraction in full-information dynamic pricing

“Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices” (Ao et al., 24 Jan 2025) treats the full-information setting of dynamic pricing with knapsack. The seller has $b$40 products, $b$41 resources, horizon $b$42, initial capacities $b$43, consumption matrix $b$44, remaining inventory $b$45, and chooses prices $b$46. Demand follows the linear model

$b$47

with realized demand

$b$48

revenue

$b$49

and inventory dynamics

$b$50

The full-information analysis assumes the linear demand function is known, noise is independent across time and sub-Gaussian with parameter $b$51, and $b$52, which implies $b$53 is invertible.

The fluid benchmark solves

$b$54

with optimal value $b$55. Regret is

$b$56

The full-information algorithm is the Boundary Attracted Re-solve Method. At time $b$57, it resolves the fluid problem using remaining inventory $b$58 and remaining horizon $b$59, producing $b$60. It then modifies the target demand by rounding small components down to zero: $b$61 The posted price $b$62 is chosen so that $b$63.

This boundary attraction replaces standard re-solving by “resolve + boundary attraction.” The purpose is to handle near-zero demand coordinates without invoking the non-degeneracy condition often required in earlier logarithmic-regret analyses. The proof uses a hybrid policy decomposition,

$b$64

and analyzes three cases: all demand components large, all components small, and the mixed case. The per-step regret is controlled by a term proportional to $b$65, so summing over $b$66 yields logarithmic growth. The theorem states that if $b$67, then

$b$68

A recurrent misconception is that logarithmic regret here depends on a stable optimal basis. The stated result avoids that requirement by using the boundary-attraction threshold together with the quadratic structure induced by $b$69.