A Reduction from Delayed to Immediate Feedback for Online Convex Optimization with Improved Guarantees

We develop a reduction-based framework for online learning with delayed feedback that recovers and improves upon existing results for both first-order and bandit convex optimization. Our approach introduces a continuous-time model under which regret decomposes into a delay-independent learning term and a delay-induced drift term, yielding a delay-adaptive reduction that converts any algorithm for online linear optimization into one that handles round-dependent delays. For bandit convex optimization, we significantly improve existing regret bounds, with delay-dependent terms matching state-of-the-art first-order rates. For first-order feedback, we recover state-of-the-art regret bounds via a simpler, unified analysis. Quantitatively, for bandit convex optimization we obtain $O(\sqrt{d_{\text{tot}}} + T^{\frac{3}{4}}\sqrt{k})$ regret, improving the delay-dependent term from $O(\min\{\sqrt{T d_{\text{max}}},(Td_{\text{tot}})^{\frac{1}{3}}\})$ in previous work to $O(\sqrt{d_{\text{tot}}})$. Here, $k$, $T$, $d_{\text{max}}$, and $d_{\text{tot}}$ denote the dimension, time horizon, maximum delay, and total delay, respectively. Under strong convexity, we achieve $O(\min\{σ_{\text{max}} \ln T, \sqrt{d_{\text{tot}}}\} + (T^2\ln T)^{\frac{1}{3}} {k}^{\frac{2}{3}})$, improving the delay-dependent term from $O(d_{\text{max}} \ln T)$ in previous work to $O(\min\{σ_{\text{max}} \ln T, \sqrt{d_{\text{tot}}}\})$, where $σ_{\text{max}}$ denotes the maximum number of outstanding observations and may be considerably smaller than $d_{\text{max}}$.

On the Hardness of Learning Regular Expressions

Despite the theoretical significance and wide practical use of regular expressions, the computational complexity of learning them has been largely unexplored. We study the computational hardness of improperly learning regular expressions in the PAC model and with membership queries. We show that PAC learning is hard even under the uniform distribution on the hypercube, and also prove hardness of distribution-free learning with membership queries. Furthermore, if regular expressions are extended with complement or intersection, we establish hardness of learning with membership queries even under the uniform distribution. We emphasize that these results do not follow from existing hardness results for learning DFAs or NFAs, since the descriptive complexity of regular languages can differ exponentially between DFAs, NFAs, and regular expressions.

Learning-Augmented Algorithms for Boolean Satisfiability

Learning-augmented algorithms are a prominent recent development in beyond worst-case analysis. In this framework, a problem instance is provided with a prediction (``advice'') from a machine-learning oracle, which provides partial information about an optimal solution, and the goal is to design algorithms that leverage this advice to improve worst-case performance. We study the classic Boolean satisfiability (SAT) decision and optimization problems within this framework using two forms of advice. ``Subset advice" provides a random $\epsilon$ fraction of the variables from an optimal assignment, whereas ``label advice" provides noisy predictions for all variables in an optimal assignment. For the decision problem $k$-SAT, by using the subset advice we accelerate the exponential running time of the PPSZ family of algorithms due to Paturi, Pudlak, Saks and Zane, which currently represent the state of the art in the worst case. We accelerate the running time by a multiplicative factor of $2^{-c}$ in the base of the exponent, where $c$ is a function of $\epsilon$ and $k$. For the optimization problem, we show how to incorporate subset advice in a black-box fashion with any $\alpha$-approximation algorithm, improving the approximation ratio to $\alpha + (1 - \alpha)\epsilon$. Specifically, we achieve approximations of $0.94 + \Omega(\epsilon)$ for MAX-$2$-SAT, $7/8 + \Omega(\epsilon)$ for MAX-$3$-SAT, and $0.79 + \Omega(\epsilon)$ for MAX-SAT. Moreover, for label advice, we obtain near-optimal approximation for instances with large average degree, thereby generalizing recent results on MAX-CUT and MAX-$2$-LIN.

Capacity-Constrained Online Learning with Delays: Scheduling Frameworks and Regret Trade-offs

We study online learning with oblivious losses and delays under a novel ``capacity constraint'' that limits how many past rounds can be tracked simultaneously for delayed feedback. Under ``clairvoyance'' (i.e., delay durations are revealed upfront each round) and/or ``preemptibility'' (i.e., we have ability to stop tracking previously chosen round feedback), we establish matching upper and lower bounds (up to logarithmic terms) on achievable regret, characterizing the ``optimal capacity'' needed to match the minimax rates of classical delayed online learning, which implicitly assume unlimited capacity. Our algorithms achieve minimax-optimal regret across all capacity levels, with performance gracefully degrading under suboptimal capacity. For $K$ actions and total delay $D$ over $T$ rounds, under clairvoyance and assuming capacity $C = \Omega(\log(T))$, we achieve regret $\widetilde{\Theta}(\sqrt{TK + DK/C + D\log(K)})$ for bandits and $\widetilde{\Theta}(\sqrt{(D+T)\log(K)})$ for full-information feedback. When replacing clairvoyance with preemptibility, we require a known maximum delay bound $d_{\max}$, adding $\smash{\widetilde{O}(d_{\max})}$ to the regret. For fixed delays $d$ (i.e., $D=Td$), the minimax regret is $\Theta\bigl(\sqrt{TK(1+d/C)+Td\log(K)}\bigr)$ and the optimal capacity is $\Theta(\min\{K/\log(K),d\}\bigr)$ in the bandit setting, while in the full-information setting, the minimax regret is $\Theta\bigl(\sqrt{T(d+1)\log(K)}\bigr)$ and the optimal capacity is $\Theta(1)$. For round-dependent and fixed delays, our upper bounds are achieved using novel scheduling policies, based on Pareto-distributed proxy delays and batching techniques. Crucially, our work unifies delayed bandits, label-efficient learning, and online scheduling frameworks, demonstrating that robust online learning under delayed feedback is possible with surprisingly modest tracking capacity.

PAC Learning with Improvements

One of the most basic lower bounds in machine learning is that in nearly any nontrivial setting, it takes $\textit{at least}$ $1/\epsilon$ samples to learn to error $\epsilon$ (and more, if the classifier being learned is complex). However, suppose that data points are agents who have the ability to improve by a small amount if doing so will allow them to receive a (desired) positive classification. In that case, we may actually be able to achieve $\textit{zero}$ error by just being "close enough". For example, imagine a hiring test used to measure an agent's skill at some job such that for some threshold $\theta$, agents who score above $\theta$ will be successful and those who score below $\theta$ will not (i.e., learning a threshold on the line). Suppose also that by putting in effort, agents can improve their skill level by some small amount $r$. In that case, if we learn an approximation $\hat{\theta}$ of $\theta$ such that $\theta \leq \hat{\theta} \leq \theta + r$ and use it for hiring, we can actually achieve error zero, in the sense that (a) any agent classified as positive is truly qualified, and (b) any agent who truly is qualified can be classified as positive by putting in effort. Thus, the ability for agents to improve has the potential to allow for a goal one could not hope to achieve in standard models, namely zero error. In this paper, we explore this phenomenon more broadly, giving general results and examining under what conditions the ability of agents to improve can allow for a reduction in the sample complexity of learning, or alternatively, can make learning harder. We also examine both theoretically and empirically what kinds of improvement-aware algorithms can take into account agents who have the ability to improve to a limited extent when it is in their interest to do so.

On the Dichotomy Between Privacy and Traceability in $\ell_p$ Stochastic Convex Optimization

In this paper, we investigate the necessity of memorization in stochastic convex optimization (SCO) under $\ell_p$ geometries. Informally, we say a learning algorithm memorizes $m$ samples (or is $m$-traceable) if, by analyzing its output, it is possible to identify at least $m$ of its training samples. Our main results uncover a fundamental tradeoff between traceability and excess risk in SCO. For every $p\in [1,\infty)$, we establish the existence of a risk threshold below which any sample-efficient learner must memorize a \em{constant fraction} of its sample. For $p\in [1,2]$, this threshold coincides with best risk of differentially private (DP) algorithms, i.e., above this threshold, there are algorithms that do not memorize even a single sample. This establishes a sharp dichotomy between privacy and traceability for $p \in [1,2]$. For $p \in (2,\infty)$, this threshold instead gives novel lower bounds for DP learning, partially closing an open problem in this setup. En route of proving these results, we introduce a complexity notion we term \em{trace value} of a problem, which unifies privacy lower bounds and traceability results, and prove a sparse variant of the fingerprinting lemma.

Fixed Point Computation: Beating Brute Force with Smoothed Analysis

We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational inequality, a problem that subsumes fixed point computation and the computation of a Nash Equilibrium. The algorithm's runtime is bounded by $e^{O(n)}/\varepsilon$, under the smoothed-analysis framework. This is the first known algorithm in such a generality whose runtime is faster than $(1/\varepsilon)^{O(n)}$, which is a time that suffices for an exhaustive search. We complement this result with a lower bound of $e^{\Omega(n)}$ on the query complexity for finding an $O(1)$-approximate fixed point on the unit ball, which holds even in the smoothed-analysis model, yet without the assumption that the function is smooth. Existing lower bounds are only known for the hypercube, and adapting them to the ball does not give non-trivial results even for finding $O(1/\sqrt{n})$-approximate fixed points.

Sample Compression Scheme Reductions

We present novel reductions from sample compression schemes in multiclass classification, regression, and adversarially robust learning settings to binary sample compression schemes. Assuming we have a compression scheme for binary classes of size $f(d_\mathrm{VC})$, where $d_\mathrm{VC}$ is the VC dimension, then we have the following results: (1) If the binary compression scheme is a majority-vote or a stable compression scheme, then there exists a multiclass compression scheme of size $O(f(d_\mathrm{G}))$, where $d_\mathrm{G}$ is the graph dimension. Moreover, for general binary compression schemes, we obtain a compression of size $O(f(d_\mathrm{G})\log|Y|)$, where $Y$ is the label space. (2) If the binary compression scheme is a majority-vote or a stable compression scheme, then there exists an $\epsilon$-approximate compression scheme for regression over $[0,1]$-valued functions of size $O(f(d_\mathrm{P}))$, where $d_\mathrm{P}$ is the pseudo-dimension. For general binary compression schemes, we obtain a compression of size $O(f(d_\mathrm{P})\log(1/\epsilon))$. These results would have significant implications if the sample compression conjecture, which posits that any binary concept class with a finite VC dimension admits a binary compression scheme of size $O(d_\mathrm{VC})$, is resolved (Littlestone and Warmuth, 1986; Floyd and Warmuth, 1995; Warmuth, 2003). Our results would then extend the proof of the conjecture immediately to other settings. We establish similar results for adversarially robust learning and also provide an example of a concept class that is robustly learnable but has no bounded-size compression scheme, demonstrating that learnability is not equivalent to having a compression scheme independent of the sample size, unlike in binary classification, where compression of size $2^{O(d_\mathrm{VC})}$ is attainable (Moran and Yehudayoff, 2016).

Sequential Probability Assignment with Contexts: Minimax Regret, Contextual Shtarkov Sums, and Contextual Normalized Maximum Likelihood

We study the fundamental problem of sequential probability assignment, also known as online learning with logarithmic loss, with respect to an arbitrary, possibly nonparametric hypothesis class. Our goal is to obtain a complexity measure for the hypothesis class that characterizes the minimax regret and to determine a general, minimax optimal algorithm. Notably, the sequential $\ell_{\infty}$ entropy, extensively studied in the literature (Rakhlin and Sridharan, 2015, Bilodeau et al., 2020, Wu et al., 2023), was shown to not characterize minimax risk in general. Inspired by the seminal work of Shtarkov (1987) and Rakhlin, Sridharan, and Tewari (2010), we introduce a novel complexity measure, the \emph{contextual Shtarkov sum}, corresponding to the Shtarkov sum after projection onto a multiary context tree, and show that the worst case log contextual Shtarkov sum equals the minimax regret. Using the contextual Shtarkov sum, we derive the minimax optimal strategy, dubbed \emph{contextual Normalized Maximum Likelihood} (cNML). Our results hold for sequential experts, beyond binary labels, which are settings rarely considered in prior work. To illustrate the utility of this characterization, we provide a short proof of a new regret upper bound in terms of sequential $\ell_{\infty}$ entropy, unifying and sharpening state-of-the-art bounds by Bilodeau et al. (2020) and Wu et al. (2023).

Causal Bandits: The Pareto Optimal Frontier of Adaptivity, a Reduction to Linear Bandits, and Limitations around Unknown Marginals

In this work, we investigate the problem of adapting to the presence or absence of causal structure in multi-armed bandit problems. In addition to the usual reward signal, we assume the learner has access to additional variables, observed in each round after acting. When these variables $d$-separate the action from the reward, existing work in causal bandits demonstrates that one can achieve strictly better (minimax) rates of regret (Lu et al., 2020). Our goal is to adapt to this favorable "conditionally benign" structure, if it is present in the environment, while simultaneously recovering worst-case minimax regret, if it is not. Notably, the learner has no prior knowledge of whether the favorable structure holds. In this paper, we establish the Pareto optimal frontier of adaptive rates. We prove upper and matching lower bounds on the possible trade-offs in the performance of learning in conditionally benign and arbitrary environments, resolving an open question raised by Bilodeau et al. (2022). Furthermore, we are the first to obtain instance-dependent bounds for causal bandits, by reducing the problem to the linear bandit setting. Finally, we examine the common assumption that the marginal distributions of the post-action contexts are known and show that a nontrivial estimate is necessary for better-than-worst-case minimax rates.