William L. Brown

Arpit Agarwal, William Brown

When an Agent visits a platform recommending a menu of content to select from, their choice of item depends not only on fixed preferences, but also on their prior engagements with the platform. The Recommender's primary objective is typically to encourage content consumption which optimizes some reward, such as ad revenue, but they often also aim to ensure that a wide variety of content is consumed by the Agent over time. We formalize this problem as an adversarial bandit task. At each step, the Recommender presents a menu of $k$ (out of $n$) items to the Agent, who selects one item in the menu according to their unknown preference model, which maps their history of past items to relative selection probabilities. The Recommender then observes the Agent's chosen item and receives bandit feedback of the item's reward. In addition to optimizing reward from selected items, the Recommender must also ensure that the total distribution of chosen items has sufficiently high entropy. We define a class of preference models which are locally learnable, i.e. behavior over the entire domain can be estimated by only observing behavior in a small region; this includes models representable by bounded-degree polynomials as well as functions with a sparse Fourier basis. For this class, we give an algorithm for the Recommender which obtains $\tilde{O}(T^{3/4})$ regret against all item distributions satisfying two conditions: they are sufficiently diversified, and they are instantaneously realizable at any history by some distribution over menus. We show that these conditions are closely connected: all sufficiently high-entropy distributions are instantaneously realizable at any item history. We also give a set of negative results justifying our assumptions, in the form of a runtime lower bound for non-local learning and linear regret lower bounds for alternate benchmarks.

Arpit Agarwal, William Brown

We consider a bandit recommendations problem in which an agent's preferences (representing selection probabilities over recommended items) evolve as a function of past selections, according to an unknown $\textit{preference model}$. In each round, we show a menu of $k$ items (out of $n$ total) to the agent, who then chooses a single item, and we aim to minimize regret with respect to some $\textit{target set}$ (a subset of the item simplex) for adversarial losses over the agent's choices. Extending the setting from Agarwal and Brown (2022), where uniform-memory agents were considered, here we allow for non-uniform memory in which a discount factor is applied to the agent's memory vector at each subsequent round. In the "long-term memory" regime (when the effective memory horizon scales with $T$ sublinearly), we show that efficient sublinear regret is obtainable with respect to the set of $\textit{everywhere instantaneously realizable distributions}$ (the "EIRD set", as formulated in prior work) for any $\textit{smooth}$ preference model. Further, for preferences which are bounded above and below by linear functions of memory weight (we call these "scale-bounded" preferences) we give an algorithm which obtains efficient sublinear regret with respect to nearly the $\textit{entire}$ item simplex. We show an NP-hardness result for expanding to targets beyond EIRD in general. In the "short-term memory" regime (when the memory horizon is constant), we show that scale-bounded preferences again enable efficient sublinear regret for nearly the entire simplex even without smoothness if losses do not change too frequently, yet we show an information-theoretic barrier for competing against the EIRD set under arbitrary smooth preference models even when losses are constant.

William Brown, Jon Schneider, Kiran Vodrahalli

We consider a number of questions related to tradeoffs between reward and regret in repeated gameplay between two agents. To facilitate this, we introduce a notion of $\textit{generalized equilibrium}$ which allows for asymmetric regret constraints, and yields polytopes of feasible values for each agent and pair of regret constraints, where we show that any such equilibrium is reachable by a pair of algorithms which maintain their regret guarantees against arbitrary opponents. As a central example, we highlight the case one agent is no-swap and the other's regret is unconstrained. We show that this captures an extension of $\textit{Stackelberg}$ equilibria with a matching optimal value, and that there exists a wide class of games where a player can significantly increase their utility by deviating from a no-swap-regret algorithm against a no-swap learner (in fact, almost any game without pure Nash equilibria is of this form). Additionally, we make use of generalized equilibria to consider tradeoffs in terms of the opponent's algorithm choice. We give a tight characterization for the maximal reward obtainable against $\textit{some}$ no-regret learner, yet we also show a class of games in which this is bounded away from the value obtainable against the class of common "mean-based" no-regret algorithms. Finally, we consider the question of learning reward-optimal strategies via repeated play with a no-regret agent when the game is initially unknown. Again we show tradeoffs depending on the opponent's learning algorithm: the Stackelberg strategy is learnable in exponential time with any no-regret agent (and in polynomial time with any no-$\textit{adaptive}$-regret agent) for any game where it is learnable via queries, and there are games where it is learnable in polynomial time against any no-swap-regret agent but requires exponential time against a mean-based no-regret agent.