60 problems

Consider the modified version of Algorithm $\mathrm{AhdLAMulti}$ in which the dual price for the first batch is computed using only information from customers arriving in that batc…

Let $a_t$ be an independent and identically distributed process, and let $r_i$ follow either a linear regression model with white noise or a weighted random-walk model. Let $\pi_5$…

Order-optimality conjecture. The order optimality of RCA-M should hold when it is used with any index policy that is order optimal for the rested bandit problem.

Optimality conjecture. For $\Theta=\Theta_{M,\,\lambda}$, an optimal proxy function minimizing the ratio $\overline{V}/\alpha$ is the entropy-type function defined in equation (2).

Let $d$ be the dimension, let $\calA$ denote the action space, and let $\mathfrak R_T^\star(d;\calA)$ be the minimax expected regret after $T$ rounds in stochastic bandit convex op…

Unconditional logarithmic rent conjecture. The bound continues to hold when (L2)--(L4) are dropped: when the excitation is endogenous to the designer's policy, agents learn from th…

The one-pass setting requires controlling the smoothness of the empirical risk at each iteration, which can impose substantially stronger dimension-dependent conditions than in bat…

Jain et al.'s anytime optimality conjecture. In the absence of a priori information about $T$, no stepsize sequence can ensure the information-theoretically optimal error rate for…

Conjecture on modified algorithms. Suitably modified variants of bandit algorithms should similarly achieve Pareto-efficient trade-offs as those achieved by texttt{UCB-f}.

Let $b_t$ be an arbitrary ordered vector of base forecasts, and let $\theta_t$ and $\tilde{\theta}_t$ denote the MultiQT iterates in the general setting. Bounded projection distanc…

Conditional-regret conjecture. Behind all concentration results derived by constructing an appropriate non-negative (super)martingale and then applying Ville's inequality, there ex…

Let $\varepsilon>0$ denote the distance of the arrival rate to the capacity region, and suppose that $\varepsilon$ is unknown to the algorithm. Consider algorithms for the single-s…

PUMMA's translation-invariant mistake-bound conjecture. The mistake bound

An experimental procedure is any procedure for comparing recommendation algorithms in an A/B experiment, and preserving the sign means preserving the sign of the comparison between…

Strong-convexity conjecture. Strong convexity should also hold for general $\gamma$-local intrinsically Lipschitz functions.

Variance-offset conjecture. Variance inflation induced by sequential updates offsets the usual variance underestimation associated with KL-based variational approximations.

Let $T$ be the time horizon, let the rounds be partitioned into batches $(\mathcal{T}_j)_{j=1}^n$ of size $Delta_T$, and let $V_{T,j}=\sum_{t\in\mathcal{T}_j}|m_t-m_{t-1}|$ denote…

Time-varying perturbation conjecture. Optimal regret could still be achieved using

Let $f^* \coloneqq \sum_{S \in \mathcal{S}}\alpha_S\chi_S$ be a Boolean function, and let its leap complexity be the minimum value $l$ for which there is an ordering of its coeffic…

Let OFTRL denote optimistic follow-the-regularized-leader algorithms, and consider variants using dynamic step sizes rather than fixed step sizes. Dynamic-step-size conjecture. The…

Constant-strategy nonexistence conjecture. For $n\geq 6$ experts, there is no globally asymptotically optimal adversary strategy that is constant on $\mathbb{S}_n$. The paper repor…

Non-COMB uniqueness conjecture. The non-COMB strategy $(1,0,1,0,0)$ is the only globally asymptotically optimal strategy for $n=5$ experts. The claim is supported by strong numeric…

COMB optimality conjecture. The COMB strategy is globally asymptotically optimal on $\mathbb{S}_n$ only for $n\leq 4$ experts. The paper reports strong numerical evidence against g…

Gravin's COMB strategy conjecture. The COMB strategy is asymptotically optimal as $T\to\infty$ for all $n$. This was proved in the cited source only for $n=2$ and $n=3$ experts. De…

Averaged TV-risk conjecture. The upper bound above may not be tight; it is an open problem whether it can be improved to