We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only 𝒪(log(n)) workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time. First, we design a polynomial time algorithm for directed s-t connectivity using n / 2^{Θ(√{log n})} catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [James Cook and Edward Pyne, 2026]. Furthermore, using only 𝒪(log(n)) random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms. Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fréchet Distance, again using n / 2^{Θ(√{log n})} catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].

Motivated by the goal of showing stronger structural results about the complexity of learning, we study the learnability of strong concept classes beyond P/poly , such as PSPACE/poly and E/poly . We show the following: (Unconditional Lower Bounds for Learning) Building on Klivans et al. (2013), we prove unconditionally that BPE/poly cannot be weakly learned in polynomial time over the uniform distribution, even with membership and equivalence queries. (Robustness of Learning) For the concept classes EXP/poly and PSPACE/poly , we unconditionally show that worst-case and average-case learning are equivalent, that PAC -learnability and learnability over the uniform distribution are equivalent, and that membership queries do not help in either case. (Reducing Succinct Search to Decision for Learning) For the decision problems R Kt and R KS capturing the complexity of learning EXP/poly and PSPACE/poly , respectively, we show a succinct search to decision reduction: for each of these problems, the problem is in BPP iff there is a probabilistic polynomial-time algorithm computing circuits encoding proofs for positive instances of the problem. This is shown via a more general result giving succinct search to decision results for PSPACE, EXP and NEXP , which might be of independent interest. (Implausibility of Oblivious Strongly Black-Box Reductions showing NP -hardness of learning NP/poly ) We define a natural notion of hardness of learning with respect to oblivious strongly blackbox reductions. We show that learning PSPACE/poly is PSPACE hard with respect to oblivious strongly black-box reductions. On the other hand, if learning NP/poly is NP -hard with respect to oblivious strongly black-box reductions, the polynomial hierarchy collapses.

We continue the study of doubly-efficient proof systems for verifying agnostic PAC learning, for which we obtain the following results. We construct an interactive protocol for learning the t largest Fourier characters of a given function f ∶ {0,1}n → {0,1} up to an arbitrarily small error, wherein the verifier uses poly(t) random examples. This improves upon the Interactive Goldreich-Levin protocol of Goldwasser, Rothblum, Shafer, and Yehudayoff (ITCS 2021) whose sample complexity is poly(t,n). For agnostically learning the class AC0[2] under the uniform distribution, we build on the work of Carmosino, Impagliazzo, Kabanets, and Kolokolova (APPROX/RANDOM 2017) and design an interactive protocol, where given a function f ∶ {0,1}n → {0,1}, the verifier learns the closest hypothesis up to polylog(n) multiplicative factor, using quasi-polynomially many random examples. In contrast, this class has been notoriously resistant even for constructing realisable learners (without a prover) using random examples. For agnostically learning k-juntas under the uniform distribution, we obtain an interactive protocol, where the verifier uses O(2k) random examples to a given function f ∶ {0,1}n → {0,1}. Crucially, the sample complexity of the verifier is independent of n. We also show that if we do not insist on doubly-efficient proof systems, then the model becomes trivial. Specifically, we show a protocol for an arbitrary class C of Boolean functions in the distribution-free setting, where the verifier uses O(1) labeled examples to learn f.

Hardness magnification reduces major complexity separations (such as EXP ⊈ NC 1 ) to proving lower bounds for some natural problem Q against weak circuit models. Several recent works [ 11 , 13 , 14 , 40 , 42 , 43 , 46 ] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than Q , while Q itself is susceptible to lower bounds, but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: – Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [ 51 ] ? – Can we adapt known lower-bound techniques to establish the desired lower bound for Q ? We establish that some instantiations of hardness magnification overcome the natural proofs barrier in the following sense: slightly superlinear-size circuit lower bounds for certain versions of the minimum circuit-size problem imply the non-existence of natural proofs. As the non-existence of natural proofs implies the non-existence of efficient learning algorithms, we show that certain magnification theorems not only imply strong worst-case circuit lower bounds but also rule out the existence of efficient learning algorithms. Hardness magnification might sidestep natural proofs, but we identify a source of difficulty when trying to adapt existing lower-bound techniques to prove strong lower bounds via magnification. This is captured by a locality barrier : existing magnification theorems unconditionally show that the problems Q considered above admit highly efficient circuits extended with small fan-in oracle gates, while lower-bound techniques against weak circuit models quite often easily extend to circuits containing such oracles. This explains why direct adaptations of certain lower bounds are unlikely to yield strong complexity separations via hardness magnification.

Recent results have shown that algorithms for learning the optimal commitment in a Stackelberg game are susceptible to manipulation by the follower. These learning algorithms operate by querying the best responses of the follower, who consequently can deceive the algorithm by using fake best responses, typically by responding according to fake payoffs that are different from the actual ones. For this strategic behavior to be successful, the main challenge faced by the follower is to pinpoint the fake payoffs that would make the learning algorithm output a commitment that benefits them the most. While this problem has been considered before, the related literature has only focused on a simple setting where the follower can only choose from a finite set of payoff matrices, thus leaving the general version of the problem unanswered. In this paper, we fill this gap by showing that it is always possible for the follower to efficiently compute (near-)optimal fake payoffs, for various scenarios of learning interaction between the leader and the follower. Our results also establish an interesting connection between the follower’s deception and the leader’s maximin utility: through deception, the follower can induce almost any (fake) Stackelberg equilibrium if and only if the leader obtains at least their maximin utility in this equilibrium.

Hardness magnification reduces major complexity separations (such as EXP ⊈ NC^1) to proving lower bounds for some natural problem Q against weak circuit models. Several recent works [Igor Carboni Oliveira and Rahul Santhanam, 2018; Dylan M. McKay et al., 2019; Lijie Chen and Roei Tell, 2019; Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019; Igor Carboni Oliveira, 2019; Lijie Chen et al., 2019] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than Q, while Q itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: - Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [Alexander A. Razborov and Steven Rudich, 1997]? - Can we adapt known lower bound techniques to establish the desired lower bound for Q? We establish that some instantiations of hardness magnification overcome the natural proofs barrier in the following sense: slightly superlinear-size circuit lower bounds for certain versions of the minimum circuit size problem MCSP imply the non-existence of natural proofs. As a corollary of our result, we show that certain magnification theorems not only imply strong worst-case circuit lower bounds but also rule out the existence of efficient learning algorithms. Hardness magnification might sidestep natural proofs, but we identify a source of difficulty when trying to adapt existing lower bound techniques to prove strong lower bounds via magnification. This is captured by a locality barrier: existing magnification theorems unconditionally show that the problems Q considered above admit highly efficient circuits extended with small fan-in oracle gates, while lower bound techniques against weak circuit models quite often easily extend to circuits containing such oracles. This explains why direct adaptations of certain lower bounds are unlikely to yield strong complexity separations via hardness magnification.

Recent results in the ML community have revealed that learning algorithms used to compute the optimal strategy for the leader to commit to in a Stackelberg game, are susceptible to manipulation by the follower. Such a learning algorithm operates by querying the best responses or the payoffs of the follower, who consequently can deceive the algorithm by responding as if their payoffs were much different than what they actually are. For this strategic behavior to be successful, the main challenge faced by the follower is to pinpoint the payoffs that would make the learning algorithm compute a commitment so that best responding to it maximizes the follower's utility, according to the true payoffs. While this problem has been considered before, the related literature only focused on the simplified scenario in which the payoff space is finite, thus leaving the general version of the problem unanswered. In this paper, we fill this gap by showing that it is always possible for the follower to efficiently compute (near-)optimal payoffs for various scenarios of learning interaction between the leader and the follower.

We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p.