Optimal Actions

A model first sees an image from one physical measurement experiment, such as how far a block coasted, and must answer a question about a new trial, such as whether the block will pass a target after a fixed push. The initial experiment may provide enough information to answer, or the model may need another measurement, such as the object's mass, friction, restitution, or spring stiffness. We study whether vision language models can decide when to answer immediately and, when more evidence is needed, which experiment to perform. Current physical reasoning benchmarks usually evaluate only the final answer, so they do not directly measure this decision-making ability. We introduce a controlled evaluation where each problem provides one measurement image and four possible physical worlds created by combining two possible masses and two possible values of another relevant property. The model must either stop and answer or select the cheapest additional experiment that can resolve the question. We construct matched problem pairs where changing either the observed measurement or the question changes the optimal action. Since all possible worlds and experiment costs are known, we can explicitly determine the optimal choice. Across six open models and 144 physical parameter sets, direct responses repeat the same action for 95.1% to 100% of image pairs even when the correct action changes. Brief reasoning improves action switching, but the best model makes both decisions correctly for only 5.9% of image pairs. Additional analysis reveals failures in measurement interpretation, physical reasoning, and response formatting. By evaluating evidence selection separately from final answers, our benchmark reveals limitations in physical reasoning that conventional answer accuracy can overlook.

We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0,1]$, and a unique optimal arm. For each suboptimal arm $i$, let $Δ_i=μ_*-μ_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}Δ_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}<Δ_i\le2^{-r}$, and let $\mathrm{Ent}(I)=\sum_{r:p_r>0} p_r\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-δ$ on every Gaussian instance, the optimal expected number of samples on a given instance, averaged over all permutations of the arm labels, is within absolute constant factors of $H(\log(1/δ)+\mathrm{Ent}(I))$. Moreover, there is an algorithm, independent of the instance, whose expected number of samples is bounded by a constant multiple of this quantity plus $g^{-2}\log\log(e^e/g)$, where $g=\min_{i\ne *}Δ_i$ is the gap to the closest competitor.

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of $\log T$ between the regret upper bound $O(d \log T)$ and the lower bound $Ω(d)$ is unresolved (here $T$ is the total number of rounds and $d$ is the dimension). When the action set is M-convex, the regret is known to be bounded by $O(d \log d)$, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake in the case where the correct action is uniformly separated from the other candidates, and applies it to online gradient descent, the online Newton step, and MetaGrad. The number of mistakes is then bounded, for all three, by a quantity independent of $T$; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes $O(d^2)$ when the forward problem is an integer linear program, that is, the factor $\log T$ is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.

The Independent Chip Model (ICM) converts tournament chips into reference prize equity, and policies are routinely constructed against those values. Because ICM reads only stack sizes, it omits action order, blind obligations, and seat rotation, and it does not price the elimination pressure a big stack puts on the short stacks it can bust. Those omissions can alter the successor-state contrasts that determine a move. We introduce Strategic-Continuation Optimization (SCO), a policy-construction method that enumerates current-hand outcomes, maps them to successor states, prices those states with continuation values computed from the finite tournament model, and optimizes and freezes the resulting current-hand policy. The fixed-ICM comparison policy changes one thing only: the same optimizer solves the same game with successor states priced by analytic ICM, so the two policies differ only through that pricing. We evaluate the resulting policies in a three-player jam/fold tournament with a $1M prize pool. Relative to the frozen strategic-continuation benchmark, analytic ICM has $9{,}036 mean absolute value error across all 2,838 state--seat entries. That value error rewrites the ranges it prices: measured against each decision point's own fixed-ICM jam range, SCO moves the jam frequency by an average of 14.08%. To price those different moves, we compare all 946 states and three policy owners while changing only the focal policy and holding both opponents and the continuation evaluator fixed. The policy produced by SCO earns $214.33 more prize equity per hand on average and is favored in 2,433 of 2,838 matched units. The ordering survives replacing the solver-built opponent with two LLMs and with a family of non-modeling threshold players. This value-to-policy-to-cost chain shows directly when ICM becomes an inadequate objective for tournament strategy construction.

We study what can be recovered about the transition probabilities of a Markov decision process from optimal actions alone. This is closely related to the inverse problem considered by Letcher et al., who ask when the dynamics can be recovered from numerical $Q$-values. Here the numerical values themselves are not observed; only the optimal actions are known, for every reward in a given class. For state-action rewards $r(s,a)$, knowing the optimal actions for every reward also tells us how much better one action is than another when each is followed by the same fixed policy. This is still not enough to determine the transition probabilities uniquely. We prove that two kernels give the same optimal actions for every reward exactly when $$Q_{s,a} = \Bigl(P_{s,a}+\tfrac1γe_s^{\mathsf T}(L-I)\Bigr)L^{-1}$$ for one invertible matrix $L$ satisfying $L\mathbf 1=\mathbf 1$. Near a kernel with strictly positive entries, there is an $n(n-1)$-dimensional family of different kernels with this property. The result is unchanged if we consider only rewards having a unique optimal action at every state. We then compare this with rewards of the forms $r(s)$ and $r(s,a,s')$. Rewards that depend on the next state can usually recover the transition kernel itself: every row at a state with at least two actions is determined, and we describe exactly when a row at a state with one action can remain hidden. State rewards reveal less: two kernels give the same optimal actions exactly when every deterministic policy is optimal for the same set of rewards. The results show how the form of the reward affects what can be learned about the dynamics from optimal actions alone.

We consider a stochastic multi-objective bandit problem where, at each round, the agent selects a slate of $k$ arms and observes their $d$-dimensional reward vectors under semi-bandit feedback. We do not aim at identifying a single optimal arm; instead, we consider the problem of maintaining a small set of actions that jointly approximate the Pareto frontier. We formalize this objective through the dominated hypervolume induced by the selected subset of arms, and define an $α$-approximate hypervolume regret with respect to the best size-$k$ subset achievable in hindsight, where $α= 1 - 1/e$ reflects the approximation guarantee of greedy maximization for monotone submodular functions. To address this problem, we introduce \textit{THV-UCB}, an optimistic algorithm that selects arms greedily based on optimistic estimates of their marginal hypervolume contributions. We establish a gap-free regret bound $\tilde{O}(d\sqrt{nkT})$ that holds on every instance, together with a gap-dependent bound $\tilde{O}(nk^{2.5}/Δ_{\min})$ that becomes polylogarithmic in $T$ once the arms are sufficiently well separated. Our results provide theoretical support for using small subsets to approximate Pareto fronts in various multi-objective applications.

Modern reasoning models depend on reasoning data, today sourced from human annotations or distilled from stronger LLMs. However, a rich and largely untapped source of supervision lies in expert systems (e.g., game engines, classical planners, theorem provers), which routinely produce near-optimal actions across diverse domains. But these experts are silent: they commit to an action without writing down the chain of thought (CoT) behind it. Recovering that CoT as natural-language reasoning would distill expert knowledge into a student that generalizes beyond the demonstrated actions. We treat it as a latent variable and study how to recover it from the action alone. Our approach, LeAct (Learning to reason from Actions), optimizes this latent variable: the student samples candidate CoTs for each expert action, and we retain those that measurably improve its own probability of recovering the action. Across imperfect-information games at multiple scales and a simulated robotics benchmark, LeAct reaches the solver's numerical floor on small enumerable games. At larger scale, it is $5\times$ closer to the solver than the strongest expert-iteration baseline. At Flop Hold'em ($\sim 10^9$ infosets), LeAct wins head-to-head by $+60$ mbb/g, and on the robotics probe it is the only training recipe that improves on direct imitation. We present a principled framework and the result: expert systems become a categorically new source of reasoning teachers for foundation models.

Vision-language models (VLMs) can describe a scene, but can they act well within one? We study whether VLMs can make sound strategic decisions, using soccer as an objective testbed with quantifiably-valued actions. We introduce SportD, a dataset and evaluation consisting of 1421 decision scenarios across professional men's and women's soccer games, where a VLM must decide what action to take next. Models on average select the optimal action around 27% of the time, less often than the professional players, and capture markedly less of the value at stake. Furthermore, they exhibit a clear preference for safer actions, favoring lower-variance, lower-value choices that also make less physical progress toward goal. Frontier VLMs are better at estimating whether an action will succeed, placing the highest-success-probability action among their top choices in 72-85% of cases. Yet VLMs systematically conflate likelihood with value, assigning higher value to actions that are more likely to succeed ($ρ=+0.30$ to $+0.52$), despite no such relationship in the ground truth ($ρ=-0.08$). Modifying the deliberation instructions to encourage risk-taking brings the frontier models closer to the players' skill levels. SportD opens a new direction for rigorously evaluating physical strategic decision-making in VLMs, showing that careful decomposition of their choices can reveal the mechanisms underlying systematic biases such as risk aversion.

We consider Bayesian bandit models and prove that Thompson sampling makes at most twice the expected number of mistakes (selections of a suboptimal arm) as any other policy. Our analysis applies as long as the latent arm processes are independent and each arm evolves only when played. For stochastic bandits with best arm defined via mean reward, this confirms a conjecture of Guha and Munagala from 2014, where the factor $2$ is already best possible. The result holds under any nonincreasing sequence of round weights, including fixed horizon and geometric discounting.

Autonomous driving requires long-horizon closedloop decision making in dynamic traffic environments. Latent world models offer an effective framework for this problem by enabling imagination-based decision making in compact latent spaces. However, multi-source observations contain controlirrelevant redundancy, whereas reliable driving decisions rely on risk-relevant relations, future dynamics, and continuous action adjustments. This mismatch makes observation reconstruction and absolute action modeling suboptimal for learning decisionrelevant latent dynamics. We propose LIDAR-AD, a decoderfree Latent-Interaction Dreamer with Action-Residual Chains for autonomous driving. LIDAR-AD replaces observation reconstruction with redundancy-reduced latent alignment, encouraging compact representations of risk-relevant relations in multi-source driving inputs. It further models vehicle control as residual action updates and uses residual-action sequence contrastive learning to align multi-step residual-driven rollouts with future latent states. A deterministic analysis shows that the latent-tanh residual parameterization preserves interior action reachability while representing smooth long-horizon control as compact local updates. Together, these designs improve risk-aware state abstraction, continuous-control modeling, and long-horizon dynamics prediction. Extensive experiments across diverse simulated driving scenarios demonstrate that LIDAR-AD consistently outperforms world-model baselines, achieving the highest reward and the best success rate among learning-based methods. Evaluations on nuPlan-derived log-reconstructed scenarios further demonstrate the transferability of LIDAR-AD under real-world traffic layouts.

AlphaZero has demonstrated that a neural-guided Monte Carlo Tree Search can achieve superhuman performance, but strong play does not necessarily imply perfect play. We study this gap in two oracle-evaluable domains with contrasting structure: Connect Four, a solved partisan game with exact game-theoretic values, and Chomp, an impartial game whose optimal play is governed by Grundy-number structure. Under a unified self-play $+$ MCTS pipeline, we compare vanilla AlphaZero, a multi-frame variant (limited to Chomp), and an AlphaZero Auxiliary Loss (AZAL) that adds oracle-derived policy supervision. We find that vanilla AlphaZero achieves strong play across both domains but cannot preserve the exact trajectories required for optimal play: in Connect Four, it fails to maintain the optimal line of play, while in Chomp, it fails to consistently restore the $g=0$ invariant. On rectangular Chomp boards, multi-frame inputs alone do not remove this gap. Nevertheless, AZAL substantially improves oracle consistency across multi-seeded full-game traces and sampled-state evaluations. On Chomp, AZAL reaches perfect full-game oracle consistency on 10x11 and high but not complete consistency on 9x10; on Connect Four, AZAL improves oracle-match rate and delays the first oracle mistake, but does not reach perfect play.

We present FootsiesGym, an open-source environment for learning in a non-trivial two-player, zero-sum, imperfect-information game. Built on HiFight's minimalist 2D fighting game Footsies, it isolates the cyclic, non-transitive strategic interactions of fighting game neutral play while remaining simple enough for efficient analysis. We provide a vectorized simulator that enables high-throughput training on standard hardware, making the environment accessible and reproducible. We describe the design of the environment, benchmark several reinforcement learning algorithms, and discuss open research directions it enables. The code is available at https://github.com/como-research/FootsiesGym.

Prediction markets aggregate dispersed beliefs into prices that act as probabilistic forecasts of uncertain events. Classical theory establishes a clean equivalence between forecasting accuracy and trading profit, but only for the specific automated market maker (AMM) design. However, the largest exchanges today are based on central limit order books in which informed forecasters routinely lose money while uninformed strategies can profit on simple heuristics. We resolve this discrepancy by establishing a formal equivalence between predictive accuracy and profitability. For any strictly proper scoring rule $S$, we exhibit a "proper" betting strategy that depends only on the forecaster's prediction $\mathbf{p}$ and the market price $\mathbf{q}$, and earns positive expected profit whenever $\mathbf{p}$ outperforms $\mathbf{q}$ under $S$ and the market has sufficient liquidity. Moreover, this proper betting is essentially the only strategy with such robust profitability guarantee. The proof rests on a decomposition of expected profit that strictly generalizes the classical AMM guarantee and also explains how strategies can profit without an accuracy edge. Empirically, across thousands of forecasts by AI models, proper betting is the only strategy that reliably converts accuracy into profit, and we further identify systematic forecasting personas and show how the optimal proper strategy varies across them. A month-long live deployment on Kalshi achieves $+80.33\%$ return on investment with a Sharpe ratio of $3.35$.

Deploying multimodal foundation models as closed-loop policies increasingly requires conditioning actions on observations that are no longer visible. However, existing benchmarks either expose the full state, conflate hidden-state reconstruction with other agent skills, or test recall only after an episode has ended. We introduce RNG-Bench (Reconstructive Non-Markov Games), a benchmark suite designed to isolate a base model's ability to reconstruct past observations and act on them during multi-step interaction. RNG-Bench includes two complementary games: Matching Pairs, where card identities briefly revealed at specific locations must later be recalled, and 3D Maze, where egocentric views must be integrated into a spatial map. Both games are evaluated under a unified harness with three controlled difficulty axes: grid size, visual pattern, and observation modality. The benchmark further introduces a head-to-head duel protocol to control for instance-level variance and a Memory Gap metric that disentangles forgetting from poor action selection. The hardest configurations require contexts of roughly 128K tokens and 350 image inputs per episode, and remain far from saturated by frontier MLLMs. Memory Gap analysis shows that most residual errors stem from forgetting earlier observations rather than from suboptimal decision making. Finally, fine-tuning Qwen3.5-9B on optimal-policy rollouts and filtered model demonstrations improves performance on RNG-Bench and transfers to existing benchmarks without degrading general multimodal capability.

We propose Ambient Diffusion Policy, a simple and principled method for imitation learning from suboptimal data in robotics. High-quality, task-specific robot data is expensive and time-consuming to collect, while suboptimal datasets with lower-quality or out-of-distribution demonstrations are abundant. Existing methods that co-train on both data sources in robotics often fail to separate the meaningful and the harmful features in the suboptimal samples. In contrast, our method extracts only the useful features by introducing a new axis to co-training in robotics: noise-dependent data usage. Ambient Diffusion Policy restricts the contribution of suboptimal data during training to only the high and low diffusion times. To rigorously justify our approach, we first observe that robot action data exhibits a spectral power law. This induces two important properties on the optimal Diffusion Policy that we exploit: a global-to-local hierarchy and locality. We theoretically formalize this discussion using a simplified model. Our experiments validate Ambient Diffusion Policy on four types of suboptimal action data (noisy trajectories, sim-to-real gap, task mismatch, and large-scale data mixtures) across six tasks. The results show that it effectively learns from arbitrary sources of suboptimal data. Notably, it outperforms existing co-training baselines by up to 33% when scaled to Open X-Embodiment - a large dataset with heterogeneous data quality and unstructured distribution shifts. Overall, Ambient Diffusion Policy increases the utility of suboptimal demonstrations and expands the set of usable data sources in robotics.

Human-in-the-loop reinforcement learning (HIL-RL) improves sample efficiency in real-robot manipulation through online human intervention. However, successful trajectories may include suboptimal actions that deviate from the desired task-execution path and force human intervention. Existing HIL-RL methods typically apply the consistent credit assignment principle to all transitions, uniformly propagating discounted terminal rewards through suboptimal segments, ignoring the actual contribution of each transition to task success. This overestimates Q-values for critic learning and indirectly misguides actor updates toward suboptimal behavior patterns. To this end, we propose PACT, a Preference-calibrated Actor-Critic Training framework that leverages the implicit preference signals induced by intervention to perform credit reassignment on identified suboptimal segments while directly guiding policy training for unbiased critic-actor learning. Specifically, we first design a progress model that learns from human demonstration and identifies suboptimal segments for credit correction. Then, from the human action and resampled policy action at the intervention state, we build preference pairs to define a counterfactual advantage that penalizes Bellman targets of the identified suboptimal segment, enabling directional credit calibration. Moreover, we directly align the policy with human corrective actions in the bounded mean space, providing an additional signal beyond critic-guided updates. Across five real-robot manipulation tasks, PACT improves the average success rate by 24.5% and achieves 1.3 times faster convergence, thereby improving both RL sample efficiency and performance. Code is available at https://anonymous.4open.science/r/HILRL-A1X-BC05.

Incentive design studies how a central authority can influence strategic agents through payments, subsidies, or taxes, so that individual objectives align with collective welfare. This paper introduces a No-Regret Adaptive Incentive Design (RAID) framework for nonlinear games with continuous action spaces and private agent costs. In this framework, the authority (planner) designs incentives that regulate the Nash equilibrium toward a socially optimal action profile, while simultaneously learning agents' unknown preferences from repeated strategic responses. We formulate the RAID problem and construct a least-squares estimator whose strong consistency requires only diminishing excitation. Leveraging this weak excitation requirement, we propose a switching incentive policy that alternates between probing (exploration) and estimate-based (exploitation) incentives. The resulting policy achieves an $O(t^{-0.5})$ parameter estimation rate and accumulates $O(t^{0.5}\log t)$ squared social-cost regret, almost surely. We further extend the framework to an endogenous-noise response model, where standard least-squares estimation is biased due to an error-in-variables correlation between the noise and agent responses. We utilize a repeated-sampling estimator and corresponding switching policy that retain the same almost-sure convergence and regret rates. Numerical experiments validate the effectiveness and predicted convergence rates of the method.

Gaussian process-based Bayesian optimization (BO) is a popular approach for expensive black-box optimization, but its performance often degrades on complex multimodal or high-dimensional problems. Trust region-based BO mitigates this issue by focusing on local regions, and recent studies suggest that selecting an effective region can be formulated as a multi-armed bandit problem. We propose a trajectory-aware framework that integrates best-arm identification (BAI) with trust region-based BO to efficiently solve multimodal optimization problems. Our method extrapolates the optimization trajectories of multiple locally initialized optimizers to predict their final performance and progressively eliminates suboptimal candidates via BAI. We theoretically show that the proposed BAI-guided BO converges faster to the global optimum than conventional BO under mild assumptions, and demonstrate its effectiveness through extensive experiments on synthetic and real-world benchmarks.

Driving a system from one state to another through targeted interventions is a fundamental challenge in science, yet most predictive models offer limited mechanistic insight and no principled framework for decision-making. Here we present COAST (Causally Optimal Actions for State Transitions), a causal-intelligence approach for the in-silico design of constrained interventions that induce user-defined state transitions. Given data characterizing source and target states, COAST learns context-specific causal graphs and structural causal models, attributes observed distributional shifts to mechanism-level causal drivers, and introduces a novel constraint-aware multi-objective optimization formulation that balances transition efficacy, intervention complexity, and target-state stability. The approach is modular and domain-agnostic, integrating feature selection, causal discovery, causal modeling, and intervention identification and evaluation through interchangeable components. Across synthetic benchmarks and real biological datasets, COAST recovers key causal drivers and identifies robust single- and multi-target intervention strategies that achieve desired state transitions, accompanied by transparent mechanistic rationales to guide experimental validation.

In this paper, we study the role of the critic in actor--critic for entropy-regularized, finite, discounted environments. We establish that, when the critic is exact, using the latter as a baseline is a variance-reduction method in a strong sense. In this case, actor--critic with stochastic gradients matches the sample complexity of deterministic policy gradient, reaching an $ε$-optimal regularized value with $\tilde{O}(\log(1/ε))$ samples. In practice, the critic is learned alongside the actor: the variance of the actor update is then influenced by the critic's variance and bias. Specifically, when the critic has a sufficiently small error, the variance reduction and rapid convergence are preserved. This suggests to learn the critic first, keeping it up to date after each actor update, underscoring the crucial role of accurate critic estimation in actor--critic methods.

Rearranging densely packed tabletop objects is challenging when parallel-gripper picks are infeasible without sufficient clearance around an object. This work studies the problem characteristics for practically motivated settings with uniformly sized blocks placed at planar tabletop grid locations. Since purely prehensile removal can become infeasible, a directional knock primitive is therefore introduced and the optimal knock-pick variant of the problem is formulated. The work proposes a series of abstractions wherein minimal constraining gadgets are covered to identify the necessary knocks. Utilizing a maximum-weight perfect matching on a graphical abstraction yields efficient polynomial-time computation of the optimal plan that minimizes the number of actions. Experiments are reported for increasing grid sizes in synthetic settings as well as in IsaacSim. The theoretical observations provide a promising stepping stone towards rigorously building efficient manipulation strategies that interleave prehensile and non-prehensile actions.

This paper addresses the problem of learning to sparsify stochastic linear bandits, where a decision-maker sequentially selects actions from a high-dimensional space subject to a sparsity constraint on the number of nonzero elements in the action vector. The key challenge lies in minimizing cumulative regret while tackling the potential NP-hardness of finding optimal sparse actions due to the inherent combinatorial structure of the problem. We propose an adaptively phased exploration and exploitation algorithmic framework, utilizing ordinary least squares for parameter learning and specialized subroutines for sparse action selection. When the action set is a Euclidean ball, optimal sparse actions can be efficiently computed, enabling us to establish a $\tilde{\mathcal{O}}(d\sqrt{T})$ regret, where $d$ is the dimension of the action vector and $T$ is the time horizon length. For general convex and compact action sets where finding optimal sparse actions is intractable, we employ a greedy subroutine. For general strongly convex action sets, we derive a $\tilde{\mathcal{O}}(d \sqrt{T})$ $α$-regret; for general compact sets lacking strong convexity, we establish a $\tilde{\mathcal{O}}(d T^{2/3})$ $α$-regret, where $α$ pertains to the approximation ratio of the greedy algorithm. Finally, we validate the performance of our algorithms using extensive experiments including an application to recommendation system.

Poker is an imperfect information game that has served as a long-standing benchmark for decision-making under uncertainty. To maximize utility beyond the Nash equilibrium, an agent can deviate from Nash-equilibrium policies to exploit suboptimal play. We introduce AlphaExploitem, which extends the competitive RL poker agent AlphaHoldem by using a hierarchical transformer encoder that enables reasoning over previously played hands and modifying the training procedure with the inclusion of a diverse pool of exploitable opponents to facilitate learning to exploit. We train and evaluate AlphaExploitem on two standard benchmarks for imperfect-information games. Empirically, AlphaExploitem successfully exploits weak play by both in- and out-of-distribution opponents, without losing performance against NE opponents.

Recent developments in digital platforms have highlighted the prevalence of open systems, where agents can arrive and depart over time. While bandit learning in open systems has recently received initial attention, existing work imposes structural assumptions that are frequently violated in practice. A learning paradigm for general open systems creates fresh challenges: newly arriving agents induce endogenous non-stationarity; agent patterns determine how quickly information accumulates; and new agents make regret scale further with the time horizon. To this end, we formulate a unified open-system bandit problem with general dynamics, including heterogeneous rewards and general agent patterns. We introduce new concepts to capture the inherent complexities: the \emph{pre-training degree} of new agents quantifies how much information an agent carries upon entry, \emph{stability} measures the impact of new agents on the system, and \emph{global dynamic regret} compares the cumulative expected reward of all active agents with that of the varying optimal arms. We develop certified global-UCB learning methodologies with provable guarantees. Our regret bounds reveal that entry uncertainty enters linearly via the pre-training degree, while in stable regimes, regret is governed by the time needed to identify a persistent optimal arm, as well as by the agent patterns. We further show that these dependencies are tight via lower bounds in hard instances.

Strategic classification studies the problem where self-interested individuals or agents manipulate their response to obtain favorable decision outcomes made by classifiers, typically turning to dishonest actions when they are less costly than genuine efforts. Prior works have demonstrated a fundamental inability to get out of this conundrum by only focusing on the design of a classifier. We note that prior work also heavily focuses on either one-shot settings or repeated interaction with the same classifier. Real-world decision making is often multi-stage, involving a sequence of potentially different classifiers as an agent progresses. This paper introduces a sequential, stochastic, multi-stage model of strategic classification, by capturing how agents adapt their behavior, through improvement actions (enhancing both observable features and true attributes) and gaming actions (enhancing only observable features), over multiple levels of classification with increasing difficulty as well as reward. For each level, we adopt a selective classifier that can abstain from making a prediction at low confidence. Consequently, a positive (resp. negative) outcome leads to promotion (resp. demotion) of the agent to the next higher (resp. lower) level, while abstention keeps the agent at the same level. We fully characterize the agent's optimal instantaneous action under selective classifiers and compare the long-term properties and utility of the agent repeatedly following an optimal myopic policy of either no-improvement (never choose the improvement action) or no-gaming (never choose the gaming action). We further examine design principles over the sequence of classifiers that yield higher long-term utility for the latter policy, thereby effectively incentivizing genuine effort in the long run.

Existing approaches to reward inference typically assume that humans provide demonstrations according to specific behavior models. However, humans often indicate their goals through a wide range of behaviors, from actions that are suboptimal due to poor planning or execution to behaviors intended to communicate goals rather than achieve them. One existing solution for inferring rewards from such behavior $\unicode{x2013}$ provided it is drawn from the same distribution at training and deployment $\unicode{x2013}$ is to construct a dataset of behavior paired with known rewards, and to learn the mapping from behavior to rewards; however, prior methods in this family face notable limitations, such as restrictions to tabular settings. Given such a dataset, we propose instead that supervised learning offers a parsimonious yet powerful solution, which we term Supervised Reward Inference (SRI). Theoretically, we prove that SRI is asymptotically Bayes-optimal under standard assumptions. Empirically, SRI achieves near-ceiling performance on a prior benchmark for reward inference from suboptimal behavior, while on Meta-World robotics tasks, it infers rewards from even arbitrarily suboptimal demonstrations as accurately as those demonstrations allow. Finally, we demonstrate our framework's universality with straightforward generalizations to action- and goal-prediction.

While Nash equilibrium has emerged as the central game-theoretic solution concept, many important games contain several Nash equilibria and we must determine how to select between them in order to create real strategic agents. Several Nash equilibrium refinement concepts have been proposed and studied for sequential imperfect-information games, the most prominent being trembling-hand perfect equilibrium, quasi-perfect equilibrium, and recently one-sided quasi-perfect equilibrium. These concepts are robust to certain arbitrarily small mistakes, and are guaranteed to always exist; however, we argue that neither of these is the correct concept for developing strong agents in sequential games of imperfect information. We define a new equilibrium refinement concept for extensive-form games called observable perfect equilibrium in which the solution is robust over trembles in publicly-observable action probabilities (not necessarily over all action probabilities that may not be observable by opposing players). Observable perfect equilibrium correctly captures the assumption that the opponent is playing as rationally as possible given mistakes that have been observed (while previous solution concepts do not). We prove that observable perfect equilibrium is always guaranteed to exist, and demonstrate that it leads to a different solution than the prior extensive-form refinements in no-limit poker. We expect observable perfect equilibrium to be a useful equilibrium refinement concept for modeling many important imperfect-information games of interest in artificial intelligence.