PLS: Prediction with Limited Selectivity

• PLS is an online prediction model where forecasters can only pick prediction windows at prescribed times, restricting flexibility compared to fully adaptive methods.
• It employs a block vector representation and the concept of approximate uniformity to quantify how the geometry of allowed prediction times influences predictive accuracy.
• The model achieves optimal error bounds using randomized halving strategies, and it motivates further research to refine complexity measures beyond approximate uniformity.

Searching arXiv for the primary paper and closely related selective-prediction context. Prediction with Limited Selectivity (PLS) is an online prediction model in which a forecaster may choose the prediction window only at prescribed times rather than arbitrarily throughout the horizon. In the formulation introduced in "Online Prediction with Limited Selectivity" (Liu et al., 13 Aug 2025), the total horizon is $n$, the admissible prediction-start times form a prespecified set $T \subseteq \{0,1,\dots,n-1\}$, and Nature secretly chooses a bounded sequence $x_1,\dots,x_n \in [0,1]$. The forecaster observes the sequence online and, at exactly one time $t \in T$, must choose a window length $w \in \{1,\dots,n-t\}$ and produce a prediction $\hat\mu$ for the future average $\mu=\frac1w\sum_{i=1}^w x_{t+i}$. The model isolates the effect of limiting the forecaster’s selectivity: many selective-prediction guarantees rely on the ability to predict at any time, whereas PLS asks how prediction quality degrades when this freedom is restricted.

1. Formal model and prediction protocol

An instance of PLS is the pair $I=(n,T)$, where $n$ is a positive integer and $T$ is the set of allowed stopping times. The online protocol is one-shot. For $T \subseteq \{0,1,\dots,n-1\}$0, the forecaster observes $T \subseteq \{0,1,\dots,n-1\}$1 in order. At any time $T \subseteq \{0,1,\dots,n-1\}$2—including possibly $T \subseteq \{0,1,\dots,n-1\}$3—having seen $T \subseteq \{0,1,\dots,n-1\}$4, the forecaster must choose a window length $T \subseteq \{0,1,\dots,n-1\}$5 and make a single prediction $T \subseteq \{0,1,\dots,n-1\}$6 for the true future average

$T \subseteq \{0,1,\dots,n-1\}$7

After this prediction, the game ends. The forecaster must guarantee to make exactly one prediction by time $T \subseteq \{0,1,\dots,n-1\}$8, so that $T \subseteq \{0,1,\dots,n-1\}$9 remains feasible (Liu et al., 13 Aug 2025).

The loss is squared error:

$x_1,\dots,x_n \in [0,1]$0

where the expectation is over the internal randomization of the forecasting algorithm $x_1,\dots,x_n \in [0,1]$1. This definition places PLS in the adversarial online-prediction tradition: the sequence is bounded but otherwise unrestricted, and the performance criterion does not assume stochastic generation, expert advice, or a reference distribution.

A useful structural representation is obtained by writing $x_1,\dots,x_n \in [0,1]$2, setting $x_1,\dots,x_n \in [0,1]$3 and $x_1,\dots,x_n \in [0,1]$4, and defining block lengths

$x_1,\dots,x_n \in [0,1]$5

Often one ignores the final trivial block by assuming $x_1,\dots,x_n \in [0,1]$6 and requiring a prediction by $x_1,\dots,x_n \in [0,1]$7, so that $x_1,\dots,x_n \in [0,1]$8 and the instance is encoded by the block vector $x_1,\dots,x_n \in [0,1]$9 with $t \in T$0. This block representation converts the combinatorics of allowed prediction times into a sequence of contiguous intervals.

2. Performance objectives and instance structure

The primary worst-case objective is the optimal squared error on a fixed instance. For any possibly randomized forecaster $t \in T$1 on instance $t \in T$2,

$t \in T$3

and the optimal achievable error is

$t \in T$4

This is an instance-by-instance minimax quantity: it asks how well any online strategy can perform against the worst bounded sequence compatible with the admissible prediction times (Liu et al., 13 Aug 2025).

The model also supports average-case analysis when the stopping-time set itself is random. If $t \in T$5 is sampled according to some distribution—for example, by including each $t \in T$6 independently with probability $t \in T$7—one studies high-probability bounds on $t \in T$8. This separates two sources of difficulty: the adversarial uncertainty in the sequence values and the geometric constraints imposed by the random availability of prediction times.

The central structural insight is that the hardness of an instance is governed not merely by $t \in T$9 but by how the allowed times partition the horizon. Long irregular gaps can sharply constrain feasible window choices. Conversely, when many consecutive blocks have comparable lengths, the instance behaves more like the fully selective setting. This motivates an instance-dependent complexity measure based on the block vector rather than on the raw count of stopping times alone.

3. Approximate uniformity as the complexity measure

The paper introduces approximate uniformity for a block vector $w \in \{1,\dots,n-t\}$0:

$w \in \{1,\dots,n-t\}$1

Its stated intuition is: “How many roughly-equal-sized blocks can one merge to form a long segment?” (Liu et al., 13 Aug 2025)

This quantity is instance-dependent. It is large when the instance contains a long run of consecutive blocks whose sizes are all within a constant factor of one another, and small when the horizon is dominated by highly nonuniform gaps. In the fully selective case, where $w \in \{1,\dots,n-t\}$2, one has $w \in \{1,\dots,n-t\}$3, so the classical $w \in \{1,\dots,n-t\}$4 guarantee is recovered. Approximate uniformity therefore interpolates between unrestricted selectivity and heavily constrained timing.

The complexity measure is operational rather than purely descriptive. It is built to enter both upper and lower bounds on $w \in \{1,\dots,n-t\}$5. The paper’s analysis shows that approximate uniformity captures a substantial part of the hardness of the problem, although not all of it. This suggests that the geometry of allowed stopping times can be compressed into a single scalar parameter for many purposes, but not without some loss of resolution.

4. Optimal worst-case error bounds

The main upper bound states that on every PLS instance with block vector $w \in \{1,\dots,n-t\}$6, there exists a randomized forecaster $w \in \{1,\dots,n-t\}$7 such that

$w \in \{1,\dots,n-t\}$8

for a universal constant $w \in \{1,\dots,n-t\}$9; all logarithms are taken base $\hat\mu$0 for concreteness (Liu et al., 13 Aug 2025). The proof proceeds in two stages. First, when all block lengths are within a constant factor of one another so that $\hat\mu$1, one can generalize the Drucker–QV random-halving algorithm: a scale $\hat\mu$2 is selected randomly, a fair coin determines whether to predict across the central $\hat\mu$3 blocks or recurse on one half, and the predictor uses the mean of the previous $\hat\mu$4 blocks to forecast the next $\hat\mu$5. This yields $\hat\mu$6 error on approximately uniform instances. Second, for a general block vector, a greedy merging argument produces a merge $\hat\mu$7 with $\hat\mu$8 uniform blocks; since $\hat\mu$9 can only decrease when blocks merge, the uniform-case guarantee transfers.

The complementary lower bound states that for every PLS instance $\mu=\frac1w\sum_{i=1}^w x_{t+i}$0 and every forecaster $\mu=\frac1w\sum_{i=1}^w x_{t+i}$1,

$\mu=\frac1w\sum_{i=1}^w x_{t+i}$2

Two constructions underlie this result. The $\mu=\frac1w\sum_{i=1}^w x_{t+i}$3 term comes from a random sequence that is constant on each block, with independent Bernoulli$\mu=\frac1w\sum_{i=1}^w x_{t+i}$4 block values; a block-overlap lemma shows that any chosen prediction window must overlap some single block by at least $\mu=\frac1w\sum_{i=1}^w x_{t+i}$5, forcing conditional variance at least $\mu=\frac1w\sum_{i=1}^w x_{t+i}$6. The $\mu=\frac1w\sum_{i=1}^w x_{t+i}$7 term comes from coupling the block means along a carefully built ternary tree whose internal edges inject independent noise; for any admissible prediction time and window length, the conditional variance of the future average remains $\mu=\frac1w\sum_{i=1}^w x_{t+i}$8.

Taken together, these theorems show that, up to a possible quadratic gap in the $\mu=\frac1w\sum_{i=1}^w x_{t+i}$9 term, $I=(n,T)$0 captures the hardness of PLS. The statement is deliberately qualified. The paper further notes that the gap is inherent: there exist families with $I=(n,T)$1 but $I=(n,T)$2. Approximate uniformity is therefore informative but not the perfect “right” measure.

5. Random instances and high-probability behavior

For random stopping-time sets, the paper considers inclusion probabilities $I=(n,T)$3 and calls $I=(n,T)$4 $I=(n,T)$5-monotone if it can be partitioned into $I=(n,T)$6 contiguous pieces, each of which is nondecreasing or nonincreasing. A $I=(n,T)$7-random stopping-time set $I=(n,T)$8 includes each $I=(n,T)$9 independently with probability $n$0. Writing $n$1, the high-probability theorem states that if $n$2 is $n$3-monotone, then with probability at least $n$4 over the draw of $n$5,

$n$6

(Liu et al., 13 Aug 2025).

A corollary gives a matching regime up to constants. If

$n$7

for some $n$8, then with high probability the forecaster from the upper-bound theorem achieves

$n$9

and no algorithm can do better than $T$0. In this regime, the random geometry of the admissible times is sufficiently regular that the effective difficulty is governed by the expected number of stopping opportunities.

These results identify a bridge between worst-case instance complexity and average-case random-instance behavior. They show that the instance-dependent parameter $T$1 is not merely a proof device for adversarial constructions: for structured random models of $T$2, it typically attains the scale needed to match the lower bound, thereby recovering the same logarithmic rate that appears in the fully selective setting.

6. Algorithmic construction, limitations, and open directions

The algorithmic core is the RandomSelect procedure, described as a randomized halving by weighted block lengths that returns an index $T$3 and a window size $T$4. The full algorithm reads up to block $T$5, computes the mean over the last $T$6 blocks seen, and predicts the mean over the next $T$7 blocks. Randomly choosing the scale $T$8 and recursing by halving is what ensures small worst-case squared error (Liu et al., 13 Aug 2025).

The assumptions of the model are explicit. The sequence values satisfy $T$9 and may be adversarial; exactly one prediction is allowed; and the algorithm’s only randomness is internal randomization. The problem is therefore sharply delimited: it is not a repeated forecasting setting, an expert-advice framework, or a stochastic-process model. This narrowness is methodologically useful because it isolates the effect of limited temporal selectivity.

The main limitation identified in the paper is the mismatch between the upper bound $T \subseteq \{0,1,\dots,n-1\}$00 and the lower bound $T \subseteq \{0,1,\dots,n-1\}$01 in the approximate-uniformity term. Section 6 shows that this mismatch cannot be removed simply by refining the proof around $T \subseteq \{0,1,\dots,n-1\}$02, because there are families with $T \subseteq \{0,1,\dots,n-1\}$03 but $T \subseteq \{0,1,\dots,n-1\}$04. A plausible implication is that the true complexity of PLS depends on more delicate combinatorial features than the single summary statistic $T \subseteq \{0,1,\dots,n-1\}$05 captures.

The open directions stated for the model are correspondingly structural. They include refining the complexity measure to capture hardness more precisely, for example by allowing “skipping” of small blocks; extending the framework to more general prediction tasks such as smooth or concatenation-concave functions; studying multi-shot selective prediction; and considering the experts setting with limited times to switch. Another direction is to analyze trade-offs when a forecaster may make multiple predictions at limited cost or under an abstention penalty. These questions position PLS as a foundational variant of selective prediction rather than a closed problem.