Weintroduce asimple butgeneral online learning frameworkinwhich alearner plays against an adversary in a vector-valued game that changes every round. Even though the learner'sobjectiveis not convex-concave(and so the minimax theorem does not apply), we giveasimple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret.

Thesetting isarepeated game between the Player and Nature, where at each stage both pick actions based on the contexts.

In this work, we give a general approach for improving regret bounds in online submodular maximization by exploiting"first-order" regret boundsfor online linearoptimization.

Conformal inference [Vovk et al., 2005] provides a general procedure for constructing prediction sets with guaranteed coverage, under the assumption that the data are exchangeable. This assumption, however, is often too restrictive: it typically fails in sequential or time-dependent settings such as time series forecasting, where the distribution of observations may shift over time. To address this issue, Gibbs and Cand` es [2021] introduced Adaptive Conformal Inference (ACI), which extends Conformal Prediction (CP) to adversarial environments. ACI adapts to distribution shifts by updating prediction intervals in response to observed outcomes, ensuring that the empirical coverage converges to the desired level. While effective in maintaining coverage, ACI and its extensions generally lack efficiency guarantees-for instance, there is no control over the average length of prediction intervals in adversarial regimes. In this work, we study sequential conformal inference as a repeated two-player finite game and invoke Blackwell's theory of approachability to characterize feasible objectives. Building on this perspective, we design a calibration-based algorithm that ensures asymptotic validity while achieving asymptotic efficiency under mild assumptions. Our approach relies on the notion of opportunistic approachability [Bernstein et al., 2014], which allows the learner to exploit potential restrictions in the opponent's play. We argue that such assumptions better fit the typical use cases of ACI-such as distributional drift or regime switching-than the fully adversarial setting.

We introduce a simple but general online learning framework in which a learner plays against an adversary in a vector-valued game that changes every round. Even though the learner's objective is not convex-concave (and so the minimax theorem does not apply), we give a simple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret.

Player and Nature, where at each stage both pick actions based on the contexts.

We study the problem of Active Simple Hypothesis Testing (ASHT) whe re an agent is faced with the problem of choosing between m different simple hypotheses after observing T samples. At the end of T samples, it has to output one of the m hypothesis. The distinguishing difference from the usual hypothes is testing scenario is the ability to choose one of K actions and observe the corresponding sample for that action. Th is ability to control the samples in this way makes the problem more interesting and difficult compared to the usual hypothesis testing with no control over the sample generation. The performance of the agent is meas ured in terms of the error probability its decision incurs. The above theoretical problem is a model for many practica l scenarios-A cosmetic drug trial often involve a testing period where the outcome of interest is to identify the best product after the trial period, choosing a channel from a set of channels before commencing communications, placeme nt of sensors in certain set of positions so as to minimize signal error. Any situation which require a period of testing b efore committing to a final decision with only certain fixed budget of samples (that is an inability to request additio nal samples) can be modeled effectively using ASHT and its more general version - Fixed Budget Best Arm Identific ation (FB-BAI). We intend to study the ASHT problem in the large deviation setting with the quantity of interest being the minimax error exponent over the hypotheses, that is, the worst case er ror exponent over the hypotheses.

We revisit Blackwell's celebrated approachability problem which considers a repeated vector-valued game between a player and an adversary. Motivated by settings in which the action set of the player or adversary (or both) is difficult to optimize over, for instance when it corresponds to the set of all possible solutions to some NP-Hard optimization problem, we ask what can the player guarantee \textit{efficiently}, when only having access to these sets via approximation algorithms with ratios $\alpha_{\mX} \geq 1$ and $ 1 \geq \alpha_{\mY} > 0$, respectively. Assuming the player has monotone preferences, in the sense that he does not prefer a vector-valued loss $\ell_1$ over $\ell_2$ if $\ell_2 \leq \ell_1$, we establish that given a Blackwell instance with an approachable target set $S$, the downward closure of the appropriately-scaled set $\alpha_{\mX}\alpha_{\mY}^{-1}S$ is \textit{efficiently} approachable with optimal rate. In case only the player's or adversary's set is equipped with an approximation algorithm, we give simpler and more efficient algorithms.

Abernethy et al. (2011) showed that Blackwell approachability and no-regret learning are equivalent, in the sense that any algorithm that solves a specific Blackwell approachability instance can be converted to a sublinear regret algorithm for a specific no-regret learning instance, and vice versa. In this paper, we study a more fine-grained form of such reductions, and ask when this translation between problems preserves not only a sublinear rate of convergence, but also preserves the optimal rate of convergence. That is, in which cases does it suffice to find the optimal regret bound for a no-regret learning instance in order to find the optimal rate of convergence for a corresponding approachability instance? We show that the reduction of Abernethy et al. (2011) does not preserve rates: their reduction may reduce a $d$-dimensional approachability instance $I_1$ with optimal convergence rate $R_1$ to a no-regret learning instance $I_2$ with optimal regret-per-round of $R_2$, with $R_{2}/R_{1}$ arbitrarily large (in particular, it is possible that $R_1 = 0$ and $R_{2} > 0$). On the other hand, we show that it is possible to tightly reduce any approachability instance to an instance of a generalized form of regret minimization we call improper $\phi$-regret minimization (a variant of the $\phi$-regret minimization of Gordon et al. (2008) where the transformation functions may map actions outside of the action set). Finally, we characterize when linear transformations suffice to reduce improper $\phi$-regret minimization problems to standard classes of regret minimization problems in a rate preserving manner. We prove that some improper $\phi$-regret minimization instances cannot be reduced to either subclass of instance in this way, suggesting that approachability can capture some problems that cannot be phrased in the language of online learning.

Blackwell's approachability is a very general sequential decision framework where a Decision Maker obtains vector-valued outcomes, and aims at the convergence of the average outcome to a given "target" set. Blackwell gave a sufficient condition for the decision maker having a strategy guaranteeing such a convergence against an adversarial environment, as well as what we now call the Blackwell's algorithm, which then ensures convergence. Blackwell's approachability has since been applied to numerous problems, in online learning and game theory, in particular. We extend this framework by allowing the outcome function and the dot product to be time-dependent. We establish a general guarantee for the natural extension to this framework of Blackwell's algorithm. In the case where the target set is an orthant, we present a family of time-dependent dot products which yields different convergence speeds for each coordinate of the average outcome. We apply this framework to the Big Match (one of the most important toy examples of stochastic games) where an $\epsilon$-uniformly optimal strategy for Player I is given by Blackwell's algorithm in a well-chosen auxiliary approachability problem.