Source-linked AI summary
Sequence prediction under a lying oracle
Puspabeethi Samanta, Nikhil Karamchandani, Jayakrishnan Nair
TL;DR
The paper studies sequential probability assignment when prediction costs reflect comparative queries to an oracle that may lie. It proposes separate algorithms for stochastic and adversarial environments and proves logarithmic regret upper bounds in both settings.
Problem
Sequential prediction lacks a loss function capturing the extra query complexity caused by unreliable oracle responses in twenty-questions-style identification.
Method
The paper analyzes a lie-motivated perturbation of log loss, using add-beta estimation for stochastic outcomes and EWOO for adversarial outcomes.
Results
O(mk log n) adversarial regret is achieved by EWOO, while add-beta achieves logarithmic stochastic regret scaling linearly with alphabet size m and permitted lies k.
Takeaways & Limitations
The lie-motivated loss retains log-loss-like logarithmic regret behavior across stochastic and adversarial sequential prediction settings.
Takeaways & Limitations
The paper identifies proving regret lower bounds and studying other loss functions for different noise or lying models as future research.
Abstract
from arXiv · showhide
We consider the problem of sequential prediction of an $m$-ary sequence, where at each epoch, (i) the environment selects an outcome from an $m$-ary alphabet, (ii) the learner selects a probability distribution over the same alphabet (unaware of the outcome generated by the environment), and finally, (iii) the learner incurs a cost that depends on the probability assigned to the outcome. The cost function we consider captures the complexity of predicting the outcome generated by the environment, in a scenario where the aforementioned prediction is performed via comparative queries to a lying oracle. We consider both stochastic and adversarial environments, propose algorithms for both settings, and establish logarithmic upper bounds on their regret.
I. INTRODUCTION … C. Worst-case regret
The paper studies sequential prediction with a perturbed logarithmic loss motivated by comparative queries to an oracle that may lie. It analyzes estimator performance under i.i.d. outcomes and worst-case regret under arbitrary sequences, including numerical experiments.
- I. INTRODUCTION: Allowing up to k lies increases query complexity by an overhead of order k log log(1/µ(x)) beyond log(1/µ(x)).This extends the noiseless twenty questions interpretation, where query complexity scales as log(1/µ(x)).
- I. INTRODUCTION: The perturbed loss combines classical logarithmic loss with an additive complexity term modeling unreliable oracle responses.The parameter k represents allowed lies, while c ≥1 is a constant.
- C. Worst-case regret: The perturbed loss is exp-concave, enabling the use of the Exponentially Weighted Online Optimizer to obtain adversarial regret guarantees.EWOO is considered for the arbitrary-sequence setting.
- I. INTRODUCTION: Numerical experiments examine how regret depends on the horizon n, the lie parameter k, and the choice of estimator.These experiments complement the theoretical analysis of the proposed loss and algorithms.
- A. General Model: Sequential prediction assigns a probability mass function over an m-ary alphabet before observing each outcome and then incurs a loss depending on that prediction.The learner uses only prior outcomes when forming its distribution.
- B. Stochastic regret: In the stochastic setting, outcomes are i.i.d. from an unknown fixed distribution p, and performance is measured by cumulative expected regret relative to predicting p.The paper analyzes the add-β family of estimators under the perturbed loss.
- C. Worst-case regret: In the adversarial setting, the outcome sequence is arbitrary, and regret compares cumulative learner loss with that of the best fixed distribution chosen in hindsight.The worst-case regret is obtained by taking a supremum over all sequences in Y^n.
D. Loss Function · III. STOCHASTIC SETTING
The paper studies a loss motivated by guessing with lies, generalizing log loss through a parameter k. In the stochastic setting, the add-β estimator achieves logarithmic cumulative regret, with dependence on support size and k matching the stated interpretation.
- D. Loss Function: The loss function is motivated by the problem of guessing with lies.
- D. Loss Function: The loss is defined as l(p, y) := f(p(y)) using a logarithmic expression involving p(y) and k.
- D. Loss Function: For k = 0, the loss matches the widely studied log loss function.Here, k ≥ 0 and c ≥ 1 are constants; k is an integer and c is real.
- III. STOCHASTIC SETTING: The stochastic analysis uses the add-β estimator, equivalent to a Bayesian posterior under a symmetric Dirichlet prior.It is also the convex combination of a uniform prior and the empirical estimate.
- III. STOCHASTIC SETTING: Theorem 1 considers an m-ary sequence whose symbols are generated i.i.d. from a fixed distribution p and establishes a logarithmic cumulative-regret bound.
- III. STOCHASTIC SETTING: The regret scales logarithmically with the time horizon n and linearly with the support size m.This behavior is identical to the regret behavior for log loss, corresponding to k = 0.
- III. STOCHASTIC SETTING: The regret grows linearly with k, which represents the number of permitted lies in the query model.
IV. ADVERSARIAL SETTING
For arbitrary outcome sequences, the paper applies EWOO after establishing exponential concavity of the loss, obtaining logarithmic worst-case regret. The adversarial regret scales as O(mk log n), similarly to the stochastic add-β estimator.
- EWOO method: EWOO addresses arbitrary outcome sequences by producing probability assignments as weighted mixtures over the simplex, with weights based on cumulative loss in hindsight.The weights are proportional to an exponential function of cumulative loss.
- Exp-concavity: The loss f(p(y_t)) is convex and α-exponentially concave for a suitable α.The simpler result states that f(x) is a** log_2 e-exp-concave.
- Worst-case regret: a* log_2 n + 2 log_2 e is the EWOO estimator’s logarithmic worst-case cumulative regret bound.Here, a* is the constant specified in equation (9).
- Worst-case regret: O(mk log n) is the adversarial regret scaling for EWOO, using the lower bound a* ≥ (2 ln 2 − 1)/k.The paper notes this scaling is similar to the stochastic regret of the add-β estimator.
V. SIMULATION EXPERIMENTS
The section presents numerical evaluations intended to support the paper’s theoretical results.
- Numerical evaluations are conducted to support the theoretical results.
A. Implementations in Stochastic setting … VII. PROOFS
The implementations validate logarithmic regret in the stochastic setting, with regret increasing in the number of lies and the optimal smoothing parameter depending on k and the horizon. In the adversarial setting, EWOO is evaluated numerically, while the paper concludes with regret upper bounds for add-β and EWOO and identifies lower bounds and other loss functions as future directions.
- A. Implementations in Stochastic setting: Stochastic experiments use a Bernoulli source with m = 2, n = 1000, and regret averaged over 500 i.i.d. sequences to compare add-β estimators.The estimators use β = 1/2 or 1, corresponding to Jeffreys’ prior and the uniform prior, respectively.
- A. Implementations in Stochastic setting: Regret scales logarithmically with the number of rounds for k = 0, 1, 2 under the Bernoulli(.25) source, and increases as k increases.Here, k is the number of lies in the query model.
- B. Implementations in Adversarial setting: Adversarial experiments evaluate EWOO on binary sequences 0^n−11 with n varying from 1 to 500, computing the best fixed distribution in hindsight numerically.The simplex integral in EWOO is approximated by discretization.
- VI. CONCLUDING REMARKS: The paper studies a perturbation of log loss motivated by “twenty questions with a liar” in both stochastic and adversarial settings.It provides regret upper bounds for add-β and EWOO, respectively.
- VII. PROOFS: The concluding discussion identifies proving regret lower bounds and considering other loss functions as future research directions.These directions follow the study of sequential probability assignment under the liar-motivated loss.
A. Proof of Theorem 1
The proof constructs the add-beta estimator, expresses expected cumulative regret, and bounds its components using indicator events and binomial counts.
- The proof uses the add-beta estimator p_t specified in equation (8).
- Expected cumulative regret is expressed as the quantity to be bounded.
- The argument introduces indicator events and separately bounds components A and B.
- The proof uses the binomial distribution of N_j,t−1 and treats the case N_j,t−1 = 0.Specifically, N_j,t−1 ∼ Bin(t − 1, p(j)).
1) Upper bound on A:
The upper-bound analysis shows that the contribution involving A is uniformly bounded in n because exponential decay dominates logarithmic growth. The section also introduces a proposition bounding an auxiliary expectation and completes the proof of Theorem 1.
- 1) Upper bound on A:: Exponential decay of (1 −pmin)t−1 dominates the remaining logarithmic factors, making the relevant series converge.The resulting summation is bounded by a constant independent of n.
- 1) Upper bound on A:: Proposition 1 provides a bound for the expectation of D for every t and j.The proof is deferred to the end of the section before the bound is substituted into the main argument.
- 1) Upper bound on A:: The proof combines the bounds on A and D to establish Theorem 1.The argument uses exponential convergence and a standard harmonic bound before concluding.
B. Proof of Theorem 2
The proof establishes exp-concavity of the loss by rewriting it through an adversarial binary vector and analyzing a logarithmic function’s concavity on (0,1). It then derives simpler sufficient conditions on the exp-concavity parameter via explicit bounds on a*.
- Concavity analysis: Exp-concavity is proved by studying the first and second derivatives of an auxiliary function g(x) and imposing its concavity over x∈(0,1).The proof uses positivity of the denominator to convert the concavity requirement into an inequality.
- Parameter condition: For every a≤a*, the function f(x) is a/log^2 e-exp-concave, because the defining requirement is satisfied throughout that parameter range.Here a* belongs to (0,1].
- Simplified bound: A simpler sufficient threshold a** is introduced with a**<a*, and f(x) remains exp-concave for all a≤a**.The proof establishes a*≥a** and then proceeds toward an explicit lower bound on a**.