We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L_1 -norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). As immediate consequences, we obtain the following results.

The self-attention mechanism, at the heart of the transformer model, is able to effectively model pairwise interactions between tokens. However, numerous recent works have shown that it is unable to perform basic tasks involving detecting triples of correlated tokens, or compositional tasks where multiple input tokens need to be referenced to generate a result. Some higher-dimensional alternatives to self-attention have been proposed to address this, including higher-order attention (Sanford et al., 2023) and Strassen attention (Kozachinskiy et al., 2025), which can perform some of these polyadic tasks in exchange for slower, superquadratic running times. In this work, we define a vast class of generalizations of self-attention, which we call poly-attention mechanisms. Our mechanisms can incorporate arbitrary higher-order (tensor) computations as well as arbitrary relationship structures between the input tokens, and they include the aforementioned alternatives as special cases. We then systematically study their computational complexity and representational strength, including giving new algorithms and matching complexity-theoretic lower bounds on the time complexity of computing the attention matrix exactly as well as approximately, and tightly determining which polyadic tasks they can each perform. Our results give interesting tradeoffs between different desiderata for these mechanisms, including a tight relationship between how expressive a mechanism is, and how large the coefficients in the model may be so that the mechanism can be approximated in almost-linear time. Notably, we give a new attention mechanism which can be computed exactly in quadratic time, and which can perform function composition for any fixed number of functions. Prior mechanisms, even for just composing two functions, could only be computed in superquadratic time, and our new lower bounds show that faster algorithms for them are not possible.

The Stabbing Planes proof system [8] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations mod2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [25] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.

Differential privacy (DP) is the de facto notion of privacy both in theory and in practice. However, despite its popularity, DP imposes strict requirements which guard against strong worst-case scenarios. For example, it guards against seemingly unrealistic scenarios where an attacker has full information about all but one point in the data set, and still nothing can be learned about the remaining point. While preventing such a strong attack is desirable, many works have explored whether average-case relaxations of DP are easier to satisfy [17, 28, 5, 22]. In this work, we are motivated by the question of whether alternate, weaker notions of privacy are possible: can a weakened privacy notion still guarantee some basic level of privacy, and on the other hand, achieve privacy more efficiently and/or for a substantially broader set of tasks? Our main result shows the answer is no: even in the statistical setting, any reasonable measure of privacy satisfying nontrivial composition is equivalent to DP. To prove this, we identify a core set of four axioms or desiderata: pre-processing invariance, prohibition of blatant non-privacy, strong composition, and linear scalability. Our main theorem shows that any privacy measure satisfying our axioms is equivalent to DP, up to polynomial factors in sample complexity. We complement this result by showing our axioms are minimal: removing any one of our axioms enables ill-behaved measures of privacy. 2012 ACM Subject Classification Theory of computation -> Theory of database privacy and security; Theory of computation -> Machine learning theory

We give a poly(s,1/ε)-query algorithm for testing whether an unknown and arbitrary function f: {0,1}^n →{0,1} is an s-term DNF, in the challenging relative-error framework for Boolean function property testing that was recently introduced and studied in a number of works [CDH+25b, CPPS25a, CPPS25b, CDH+25a]. This gives the first example of a rich and natural class of functions which may depend on a super-constant number of variables and yet is efficiently testable in the relative-error model with constant query complexity. A crucial new ingredient enabling our approach is a novel decomposition of any s-term DNF formula into “local clusters” of terms. Our results demonstrate that this new decomposition can be usefully exploited for algorithms even when the s-term DNF is not explicitly given; we believe that this decomposition may have applications in other contexts.

We introduce a new family of propositional proof systems, denoted , for an arbitrary TFNP search problem R. Informally, a refutation of a CNF formula F in is given by a polynomial-time reduction from the false-clause search problem Search_F to R, combined with an Extended Frege proof that the reduction is correct. These are motivated in two ways: 1. They are the propositional translations of witnessing theorems in bounded arithmetic, by which proofs of ∀ Σ^b_1 formulas ϕ in a theory T imply algorithms solving the search problem for ϕ in a TFNP class corresponding to T. 2. They are a white-box analogue of the characterizations of proof systems using decision tree reductions to black-box TFNP problems. We consider the proof system , where Iter is a complete problem for PLS. We prove that is polynomially equivalent to the sequent calculus G_1, and also to the implicit Resolution proof system [EF, Resolution]. Hence G_1 and [EF, Resolution] are equivalent, which is the first characterization of an implicit proof system by a classical proof system beyond the work of Wang. We also consider for general TFNP relations R. We observe that if EF can prove that a search problem R is in FP, then is polynomially equivalent to EF. This contrasts to our above result, which shows that Extended-Frege provable reductions to Iter, a problem widely believed not to be in FP, yields a proof system (G_1) that is believed to be stronger than Extended Frege. Finally, we show that for any proof system P which is sufficiently strong, there is a polynomial-time computable search problem R_P ∈ FP such that is polynomially equivalent to P. Letting P = [EF, Resolution] and combining our two results shows that is polynomially equivalent to .

We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with log(N)-depth list decoding (RNC^1) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for (polylog(N)-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.

We study the relative-error property testing model for Boolean functions that was recently introduced in the work of [10]. In relative-error testing, the testing algorithm gets uniform random satisfying assignments as well as black-box queries to f, and it must accept f with high probability whenever f has the property that is being tested and reject any f that is relative-error far from having the property. Here the relative-error distance from f to a function g is measured with respect to vertical bar f(-1)(1)vertical bar rather than with respect to the entire domain size 2(n) as in the Hamming distance measure that is used in the standard model; thus, unlike the standard model, relative-error testing allows us to study the testability of sparse Boolean functions that have few satisfying assignments. It was shown in [10] that relative-error testing is at least as difficult as standard-model property testing, but for many natural and important Boolean function classes the precise relationship between the two notions is unknown. In this paper we consider the well-studied and fundamental properties of being a conjunction and being a decision list. In the relative-error setting, we give an efficient one-sided error tester for conjunctions with running time and query complexity O(1/epsilon). Secondly, we give a two-sided relative-error (O) over tilde (1/epsilon) tester for decision lists, matching the query complexity of the state-of-the-art algorithm in the standard model [4, 19].

This paper considers the junta testing problem in a recently introduced "relative error" variant of the standard Boolean function property testing model. In relative-error testing we measure the distance from f to g, where f, g : {0, 1}(n) -> {0, 1}, by the ratio of vertical bar f(-1)(1) Delta g(-1)(1)vertical bar (the number of inputs on which f and g disagree) to vertical bar f(-1)(1)vertical bar (the number of satisfying assignments of f), and we give the testing algorithm both black-box access to f and also access to independent uniform samples from f(-1)(1). Chen et al. (2025a) observed that the class of k-juntas is poly(2(k), 1/epsilon)-query testable in the relativeerror model, and asked whether poly(k, 1/epsilon) queries is achievable. We answer this question affirmatively by giving a (O) over tilde (k/epsilon)-query algorithm, matching the optimal complexity achieved in the less challenging standard model. Moreover, as our main result, we show that any subclass of k-juntas that is closed under permuting variables is relative-error testable with a similar complexity. This gives highly efficient relative-error testing algorithms for a number of well-studied function classes, including size-k decision trees, size-k branching programs, and size-k Boolean formulas.

In this work we observe a tight connection between three topics: NC0 cryptography, NC0 range avoidance, and static data structure lower bounds. Using this connection, we leverage techniques from the cryptanalysis of NC0 PRGs to prove state-of-the-art results in the latter two subjects. Our main result is an improvement to the best known static data structure lower bounds, breaking a barrier which has stood for several decades. Prior to our work, the best known lower bound for any explicit problem with M inputs and N queries was S >= N-1/t (logM)(1-1/t) for any setting of the word length w (where S = space and t = time) [1]. We prove, for the same class of explicit problems considered in [1], a quadratically stronger space lower bound of the form S >= (Omega) over tilde ( N-2/t center dot (logM)(1-2/t) center dot 2(-O(w))) for all even t > 0. Second, for the restricted class of nonadaptive bit probe data structures, we improve on this lower bound polynomially: for all odd constants t > 1 we give an explicit problem with N queries and M <= N-O(1) inputs and prove a lower bound S >= Omega(N-2/t + epsilon(t)) for some constant epsilon(t) > 0 depending only on t. Our results build off of an exciting body of work on refuting semi-random CSPs (e.g., [2]-[4]). We then utilize our explicit cell probe lower bounds to obtain the best known unconditional algorithms for NC0 range avoidance: we can solve any instance with stretch n (sic) m in polynomial time once m >> n(t/2) when t is even; with the aid of an NP oracle we can solve any instance with m > n(t/2-epsilon) when t is odd for some constant epsilon > 0. Finally, using our main correspondence we establish some barrier results for obtaining significant improvements to our cell probe lower bounds: (i) near-optimal space lower bounds for an explicit problem with t = 4, w = 1 implies EXPNP not subset of NC1; (ii) under the widelybelieved assumption that polynomial-stretch NC0 PRGs exist, there is no natural proof of a lower bound of the form S >= N (Omega(1)) when t = omega(1), w = 1.

Continual learning seeks to enable machine learning systems to solve an increasing corpus of tasks sequentially. A critical challenge for continual learning is forgetting, where the performance on previously learned tasks decreases as new tasks are introduced. One of the commonly used techniques to mitigate forgetting, sample replay, has been shown empirically to reduce forgetting by retaining some examples from old tasks and including them in new training episodes. In this work, we provide a theoretical analysis of sample replay in an over-parameterized continual linear regression setting, where each task is given by a linear subspace and with enough replay samples, one would be able to eliminate forgetting. Our analysis focuses on sample replay and highlights the role of the replayed samples and the relationship between task subspaces. Surprisingly, we find that, even in a noiseless setting, forgetting can be non-monotonic with respect to the number of replay samples. We present tasks where replay can be harmful with respect to worst-case settings, and also in distributional settings where replay of randomly selected samples increases forgetting in expectation. We also give empirical evidence that harmful replay is not limited to training with linear models by showing similar behavior for a neural networks equipped with SGD. Through experiments on a commonly used benchmark, we provide additional evidence that, even in seemingly benign scenarios, performance of the replay heavily depends on the choice of replay samples and the relationship between tasks.

The recent explosion in the capabilities of large language models has led to a wave of interest in how best to prompt a model to perform a given task. While it may be tempting to simply choose a prompt based on average performance on a validation set, this can lead to a deployment where unexpectedly poor responses are generated, especially for the worst-off users. To mitigate this prospect, we propose Prompt Risk Control, a lightweight framework for selecting a prompt based on rigorous upper bounds on families of informative risk measures. We offer methods for producing bounds on a diverse set of metrics, including quantities that measure worst-case responses and disparities in generation quality across the population of users. In addition, we extend the underlying statistical bounding techniques to accommodate the possibility of distribution shifts in deployment. Experiments on applications such as open-ended chat, medical question summarization, and code generation highlight how such a framework can foster responsible deployment by reducing the risk of the worst outcomes.

In a pair of recent breakthroughs [1], [2] it was shown that the classes S-2(E), ZPE(NP), and Sigma(E)(2) require exponential circuit complexity, giving the first unconditional improvements to a classical result of Kannan [3]. These results were obtained by designing a surprising new algorithm for the total search problem Range Avoidance: given a circuit C : {0, 1}(n) -> {0, 1}(n+1), find an n+1-bit string outside its range. Range Avoidance is a member of the class TF Sigma(P)(2) of total search problems in the second level of the polynomial hierarchy, analogous to its better-known counterpart TFNP in the first level. TF Sigma(P)(2) was only recently introduced in [4] and its structure is not well understood. We investigate here the extent to which algorithms of the kind in [1], [2] can be applied to other search problems in this class, and prove a variety of results both positive and negative. On the positive side we show that Li's Range Avoidance algorithm [2] can be improved to give a reduction from Range Avoidance to a natural total search problem we call the Linear Ordering Principle or "LOP": given a circuit <: {0, 1}(n)x{0, 1}(n) -> {0, 1} purportedly defining a total order on {0, 1}(n), find either a witness that < is not a total order or else a minimal element in the ordering. The problem LOP is quite interesting in its own right, as it defines a natural syntactic subclass "L-2(P)" of S-2(P) which nonetheless maintains most of the interesting properties of S-2(P); in particular we show that L-2(P) contains MA and that its exponential analogue L-2(E) requires 2(n)/n size circuits. Both of these are consequences of our reduction from Range Avoidance to LOP. On the negative side we prove that the algorithms developed in [1], [2] cannot be extended to Strong Range Avoidance, a problem considered in the same paper which first introduced Range Avoidance [4]. In this problem we are given a circuit C : {0, 1}(n)\{0(n)} -> {0, 1}(n), and once again seek a point outside its range. We give a separation in the decision tree (oracle) model showing that this problem cannot be solved in FP parallel to Sigma P2, which in particular rules out all of the new kinds of algorithms considered in [1], [2]. This black box separation is derived from a novel depth 3 AC(0) circuit lower bound for a total search problem, which we believe is of independent interest from the perspective of circuit complexity: we show that unlike previous depth 3 lower bounds, ours cannot be proven by reduction from a decision problem, and thus requires new techniques specifically tailored to total search problems. Proving lower bounds of this kind was recently proposed by Vyas and Williams in the context of the original (Weak) Avoid problem [5].

A reliable deep learning system should be able to accurately express its confidence with respect to its predictions, a quality known as calibration. One of the most effective ways to produce reliable confidence estimates with a pre-trained model is by applying a post-hoc recalibration method. Popular recalibration methods like temperature scaling are typically fit on a small amount of data and work in the model's output space, as opposed to the more expressive feature embedding space, and thus usually have only one or a handful of parameters. However, the target distribution to which they are applied is often complex and difficult to fit well with such a function. To this end we propose selective recalibration, where a selection model learns to reject some user-chosen proportion of the data in order to allow the recalibrator to focus on regions of the input space that can be well-captured by such a model. We provide theoretical analysis to motivate our algorithm, and test our method through comprehensive experiments on difficult medical imaging and zero-shot classification tasks. Our results show that selective recalibration consistently leads to significantly lower calibration error than a wide range of selection and recalibration baselines.

The complexity class PPP contains all total search problems many-one reducible to the Pigeon problem, where we are given a succinct encoding of a function mapping n+1 pigeons to n holes, and must output two pigeons that collide in a hole. PPP is one of the “original five” syntactically-defined subclasses of TFNP, and has been extensively studied due to the strong connections between its defining problem — the pigeonhole principle — and problems in cryptography, extremal combinatorics, proof complexity, and other fields. However, despite its importance, PPP appears to be less robust than the other important TFNP subclasses. In particular, unlike all other major TFNP subclasses, it was conjectured by Buss and Johnson that PPP is not closed under Turing reductions, and they called for a black-box separation in order to provide evidence for this conjecture. The question of whether PPP contains its Turing closure was further highlighted by Daskalakis in his recent IMU Abacus Medal Lecture. In this work we prove that PPP is indeed not Turing-closed in the black-box setting, affirmatively resolving the above conjecture and providing strong evidence that PPP is not Turing-closed. In fact, we are able to separate PPP from its non-adaptive Turing closure, in which all calls to the Pigeon oracle must be made in parallel. This differentiates PPP from all other important TFNP subclasses, and especially from its closely-related subclass PWPP — defined by reducibility to the weak pigeonhole principle — which is known to be non-adaptively Turing-closed. Our proof requires developing new tools for PPP lower bounds, and creates new connections between PPP and the theory of pseudoexpectation operators used for Sherali-Adams and Sum-of-Squares lower bounds. In particular, we introduce a new type of pseudoexpectation operator that is precisely tailored for lower bounds against black-box PPP, which may be of independent interest.

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P\nsubseteq NC 1 ). Karchmer, Raz, and Wigderson [13] suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions f◇g. They showed that the validity of this conjecture would imply that P\nsubseteq NC 1 . Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function f.

A reliable deep learning system should be able to accurately express its confidence with respect to its predictions, a quality known as calibration. One of the most effective ways to produce reliable confidence estimates with a pre-trained model is by applying a post-hoc recalibration method. Popular recalibration methods like temperature scaling are typically fit on a small amount of data and work in the model's output space, as opposed to the more expressive feature embedding space, and thus usually have only one or a handful of parameters. However, the target distribution to which they are applied is often complex and difficult to fit well with such a function. To this end we propose selective recalibration, where a selection model learns to reject some user-chosen proportion of the data in order to allow the recalibrator to focus on regions of the input space that can be well-captured by such a model. We provide theoretical analysis to motivate our algorithm, and test our method through comprehensive experiments on difficult medical imaging and zero-shot classification tasks. Our results show that selective recalibration consistently leads to significantly lower calibration error than a wide range of selection and recalibration baselines.

The ExactlyN problem in the number-on-forehead (NOF) communication setting asks k players, each of whom can see every input but their own, if the k input numbers add up to N. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of randomness in communication complexity with many players. It is also tightly connected to the field of combinatorics: its k-party NOF communication complexity is related to the size of the largest corner-free subset in [N]^k-1. In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for 3 players. As an immediate consequence, this also gave a new construction of larger corner-free subsets in [N]^2. Later that year Green gave a further refinement to their argument. These results represent the first improvements to the highest-order term for k=3 since the famous work of Behrend in 1946. In this paper we give a corresponding improvement to the highest-order term for all k>3, the first since Rankin in 1961. That is, we give a more efficient protocol for ExactlyN as well as larger corner-free sets in higher dimensions. Nearly all previous results in this line of research approached the problem from the combinatorics perspective, implicitly resulting in non-constructive protocols for ExactlyN. Approaching the problem from the communication complexity point of view and constructing explicit protocols for ExactlyN was key to the improvements in the k=3 setting. As a further contribution we provide explicit protocols for ExactlyN for any number of players which serves as a base for our improvement.

We present a simple and provably optimal non-adaptive cell probe data structure for the static dictionary problem. Our data structure supports storing a set of n key-value pairs from [u]x[u] using s words of space and answering key lookup queries in t = O(lg(u/n)/ lg(s/n)) nonadaptive probes. This generalizes a solution to the membership problem (i.e., where no values are associated with keys) due to Buhrman et al. We also present matching lower bounds for the non-adaptive static membership problem in the deterministic setting. Our lower bound implies that both our dictionary algorithm and the preceding membership algorithm are optimal, and in particular that there is an inherent complexity gap in these problems between no adaptivity and one round of adaptivity (with which hashing-based algorithms solve these problems in constant time). Using the ideas underlying our data structure, we also obtain the first implementation of a n-wise independent family of hash functions with optimal evaluation time in the cell probe model.

Rigorous guarantees about the performance of predictive algorithms are necessary in order to ensure their responsible use. Previous work has largely focused on bounding the expected loss of a predictor, but this is not sufficient in many risk-sensitive applications where the distribution of errors is important. In this work, we propose a flexible framework to produce a family of bounds on quantiles of the loss distribution incurred by a predictor. Our method takes advantage of the order statistics of the observed loss values rather than relying on the sample mean alone. We show that a quantile is an informative way of quantifying predictive performance, and that our framework applies to a variety of quantile-based metrics, each targeting important subsets of the data distribution. We analyze the theoretical properties of our proposed method and demonstrate its ability to rigorously control loss quantiles on several real-world datasets.