Polynomial-Time Minimax Prediction for Stationary Renewal Processes

Prediction aggregation aims to combine information from multiple predictors into a more informative one. We study this question in the setting of calibrated predictors, where each prediction must equal the conditional expectation of the quantity being predicted given the predictor's signal. Given several calibrated input predictors and the feature distribution, but not the underlying Bayes probabilities, we ask when one can construct refined calibrated predictors that preserve the information in the original predictors and cannot be further refined using the available information. We formulate calibrated predictors as signaling schemes and define refinement through feature-independent garblings: a predictor refines another if its signal can simulate the other's signal. Constructibility is characterized through observable linear information: each signal corresponds to a vector over the feature space, and a new signal is constructible exactly when its vector lies in the linear span of the input signal vectors. Under this formulation, we establish a sharp algorithmic picture. For deterministic output predictors, bilateral refinement admits a polynomial-time algorithm based on a bipartite graph between the two input signal partitions, while refinement with an arbitrary number of input predictors is $\mathsf{NP}$-hard. In contrast, when randomized output predictors are allowed, we give a polynomial-time algorithm for any number of input predictors by decomposing constructible signal vectors into extreme rays of the associated polyhedral cone.

A memory that reproduces every current prediction may still have forgotten how to learn from the next observation. We characterize this distinction for exact cumulative learning and attention. For smooth convex residual losses with unrestricted real features and labels, a continuous recurrent state of fixed dimension exists exactly when the loss is polynomial. With ordinary, unweighted observations, degree $D$ requires precisely $r=\binom{d+D}{D}-1$ coordinates; even among histories with one fixed current predictor, exactly $r-d$ coordinates can remain necessary for continuation. Nonpolynomial losses have unbounded requirements on such equal-prediction histories. Bounded scalar logistic examples retain this obstruction after fixing any finite derivative jet. For ridge regression, the hidden evidence is the design matrix: the exact state costs $dp+d(d+1)/2$, versus $dp$ for the predictor. On a well-conditioned zero-prediction family, optimal $B$-bit compression has average one-update error of order $d^{-1}2^{-2B/s_d}$, $s_d=d(d+1)/2$, even with arbitrary randomized codes. Kernel attention gives a second exact closure classification. The results identify what future updates reveal, rather than proposing a new regression algorithm or asserting trained-model gains.

Lempel-Ziv Plurality (LZP) is an algorithm of Kosoy that predicts the next symbol of a string by using its Lempel-Ziv factorization, and makes at most $O(z\log n)$ mistakes on every string of length $n$ with $z$ LZ77 phrases. We give efficient online implementations of LZP and of a new variant of it. Our first implementation maintains a run-length encoded Burrows-Wheeler transform (RLBWT) of the reversed history. It implements LZP, uses $O(z\log z\log^2 n)$ bits, and has worst-case update and prediction times $O(\log z+\log\log n)$ and $O(σ(\log z+\log\log n))$, respectively, where $σ$ is the size of the alphabet. This is an exponential improvement over Kosoy's implementation, whose update time is $O(z\log^7 n)$. We then introduce the Lempel-Ziv Majority (LZM) predictor, which has the same mistake bound and update time, and predicts in $O(1)$ time independently of $σ$. These two predictors accept new symbols at any time, without knowing the alphabet in advance. For an alphabet known in advance, a different data structure gives $O(σ(1+\log z/\log\log n))$ time per symbol, which is constant when $σ=O(1)$ and $z=\mathrm{polylog}(n)$.