arXiv subjects

Toniann Pitassi

Publications and source records attributed to Toniann Pitassi.

At least 19 recordsLinked to original sources

arXiv2026

Provable Reductions in TFNP

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 $\forall \Sigma^b_1$ formulas $\phi$ in a theory $T$ imply algorithms solving the search problem for $\phi$ 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 \in $ FP such that is polynomially equivalent to $P$. Letting $P =$ [EF, Resolution] and combining our two results shows that is polynomially equivalent to .

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arXiv2026

Poly-attention: a general scheme for higher-order self-attention

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 and Strassen attention, 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 trade-offs 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.

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arXiv2026

Every Bit Counts: A Theoretical Study of Precision-Expressivity Tradeoffs in Quantized Transformers

Quantization reduces the numerical precision of Transformer computations and is widely used to accelerate inference, yet its effect on expressivity remains poorly characterized. We demonstrate a fine-grained theoretical tradeoff between expressivity and precision: For every p we exhibit a function {\Gamma}, inspired by the equality function, and prove that a one-layer softmax Transformer can compute {\Gamma}, with p bits of precision, but not with p-1 bits of precision. This result concretely explains the widely observed phenomenon of empirical loss of expressivity when quantization is used. Practically, it suggests that tasks requiring equality-like comparisons (exact match, membership, etc.) are especially sensitive to quantization. Dropping even one bit can cross a threshold where the model cannot represent the needed comparison reliably. Thus, it paves the way for developing heuristics that will help practitioners choose how much quantization is possible: the precision should be chosen as a function of the length of equality to be checked for the specific task. Our proofs combine explicit finite-precision Transformer constructions with communication-complexity lower bounds, yielding a tight "one-bit" threshold.

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arXiv2026

High Rate Efficient Local List Decoding from HDX

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 ($\mathrm{polylog(N)}$-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.

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arXiv2026

DNF formulas are efficiently testable with relative error

We give a poly$(s,1/\epsilon)$-query algorithm for testing whether an unknown and arbitrary function $f: \{0,1\}^n \to \{0,1\}$ 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.

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arXiv2025

Differential privacy from axioms

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 [HWR13,WLF16,BF16,LWX23]. 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.

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arXiv2025

Fully-Fluctuating Participation in Sleepy Consensus

Proof-of-work allows Bitcoin to boast security amidst arbitrary fluctuations in participation of miners throughout time, so long as, at any point in time, a majority of hash power is honest. In recent years, however, the pendulum has shifted in favor of proof-of-stake-based consensus protocols. There, the sleepy model is the most prominent model for handling fluctuating participation of nodes. However, to date, no protocol in the sleepy model rivals Bitcoin in its robustness to drastic fluctuations in participation levels, with state-of-the-art protocols making various restrictive assumptions. In this work, we present a new adversary model, called external adversary. Intuitively, in our model, corrupt nodes do not divulge information about their secret keys. In this model, we show that protocols in the sleepy model can meaningfully claim to remain secure against fully fluctuating participation, without compromising efficiency or corruption resilience. Our adversary model is quite natural, and arguably naturally captures the process via which malicious behavior arises in protocols, as opposed to traditional worst-case modeling. On top of which, the model is also theoretically appealing, circumventing a barrier established in a recent work of Malkhi, Momose, and Ren.

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arXiv2025

Replay Can Provably Increase Forgetting

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.

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arXiv2025

Testing Juntas and Junta Subclasses with Relative Error

This papers 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 $f,g: \{0,1\}^n \to \{0,1\}$, by the ratio of $|f^{-1}(1) \triangle g^{-1}(1)|$ (the number of inputs on which $f$ and $g$ disagree) to $|f^{-1}(1)|$ (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. (SODA 2025) observed that the class of $k$-juntas is $\text{poly}(2^k,1/\epsilon)$-query testable in the relative-error model, and asked whether $\text{poly}(k,1/\epsilon)$ queries is achievable. We answer this question affirmatively by giving a $\tilde{O}(k/\epsilon)$-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.

cs.CC↗

arXiv2025

Relative-error testing of conjunctions and decision lists

We study the relative-error property testing model for Boolean functions that was recently introduced in the work of Chen et al. (SODA 2025). 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 $|f^{-1}(1)|$ 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 Chen et al. (SODA 2025) 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 $\tilde{O}$$(1/\epsilon)$ tester for decision lists, matching the query complexity of the state-of-the-art algorithm in the standard model Bshouty (RANDOM 2020) and Diakonikolas et al. (FOCS 2007).

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arXiv2024

Improving Predictor Reliability with Selective Recalibration

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 \textit{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.

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arXiv2023

Prompt Risk Control: A Rigorous Framework for Responsible Deployment of Large Language Models

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.

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arXiv2023

Distribution-Free Statistical Dispersion Control for Societal Applications

Explicit finite-sample statistical guarantees on model performance are an important ingredient in responsible machine learning. Previous work has focused mainly on bounding either the expected loss of a predictor or the probability that an individual prediction will incur a loss value in a specified range. However, for many high-stakes applications, it is crucial to understand and control the dispersion of a loss distribution, or the extent to which different members of a population experience unequal effects of algorithmic decisions. We initiate the study of distribution-free control of statistical dispersion measures with societal implications and propose a simple yet flexible framework that allows us to handle a much richer class of statistical functionals beyond previous work. Our methods are verified through experiments in toxic comment detection, medical imaging, and film recommendation.

cs.LG↗

arXiv2023

An improved protocol for ExactlyN with more than 3 players

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.

cs.CC↗

arXiv2023

Optimal Non-Adaptive Cell Probe Dictionaries and Hashing

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.

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arXiv2023

On the algebraic proof complexity of Tensor Isomorphism

The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA '17). Our main results are an $\Omega(n)$ lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices, or deriving $BA=I$ from $AB=I$. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication $AB=I \rightarrow BA=I$. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.

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arXiv2023

Stability is Stable: Connections between Replicability, Privacy, and Adaptive Generalization

The notion of replicable algorithms was introduced in Impagliazzo et al. [STOC '22] to describe randomized algorithms that are stable under the resampling of their inputs. More precisely, a replicable algorithm gives the same output with high probability when its randomness is fixed and it is run on a new i.i.d. sample drawn from the same distribution. Using replicable algorithms for data analysis can facilitate the verification of published results by ensuring that the results of an analysis will be the same with high probability, even when that analysis is performed on a new data set. In this work, we establish new connections and separations between replicability and standard notions of algorithmic stability. In particular, we give sample-efficient algorithmic reductions between perfect generalization, approximate differential privacy, and replicability for a broad class of statistical problems. Conversely, we show any such equivalence must break down computationally: there exist statistical problems that are easy under differential privacy, but that cannot be solved replicably without breaking public-key cryptography. Furthermore, these results are tight: our reductions are statistically optimal, and we show that any computational separation between DP and replicability must imply the existence of one-way functions. Our statistical reductions give a new algorithmic framework for translating between notions of stability, which we instantiate to answer several open questions in replicability and privacy. This includes giving sample-efficient replicable algorithms for various PAC learning, distribution estimation, and distribution testing problems, algorithmic amplification of $\delta$ in approximate DP, conversions from item-level to user-level privacy, and the existence of private agnostic-to-realizable learning reductions under structured distributions.

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arXiv2022

Quantile Risk Control: A Flexible Framework for Bounding the Probability of High-Loss Predictions

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.

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