Speculative decoding is an effective method for lossless acceleration of large language models during inference. It uses a fast model to draft a block of tokens which are then verified in parallel by the target model, and provides a guarantee that the output is distributed identically to a sample from the target model. In prior works, draft verification is performed independently token-by-token. Surprisingly, we show that this approach is not optimal. We propose *Block Verification*, a simple draft verification algorithm that verifies the entire block jointly and provides additional wall-clock speedup. We prove that the proposed mechanism is optimal in the expected number of tokens produced each iteration and specifically is never worse than the standard token-level verification. Empirically, block verification provides modest but consistent wall-clock speedups over the standard token verification algorithm of 5\%-8\% in a range of tasks and datasets. Given that block verification does not increase code complexity, maintains the strong lossless guarantee of the standard speculative decoding verification algorithm, cannot deteriorate performance, and, in fact, consistently improves it, it can be used as a good default in speculative decoding implementations.

Language model alignment is a critical step in training modern generative language models. Alignment targets to improve win rate of a sample from the aligned model against the base model. Today, we are increasingly using inference-time algorithms (e.g., Best-of-$N$ , controlled decoding, tree search) to decode from language models rather than standard sampling. We show that this train/test mismatch makes standard RLHF framework sub-optimal in view of such inference-time methods. To this end, we propose a framework for inference-aware alignment (InfAlign), which aims to optimize *inference-time win rate* of the aligned policy against the base model. We prove that for any inference-time decoding procedure, the optimal aligned policy is the solution to the standard RLHF problem with a *transformation* of the reward. This motivates us to provide the calibrate-and-transform RL (InfAlign-CTRL) algorithm to solve this problem, which involves a reward calibration step and a KL-regularized reward maximization step with a transformation of the calibrated reward. For best-of-$N$ sampling and best-of-$N$ jailbreaking, we propose specific transformations offering up to 3-8% improvement on inference-time win rates. Finally, we also show that our proposed reward calibration method is a strong baseline for optimizing standard win rate.

In the private set union problem each user owns a bag of at most $k$ items (from some large universe of items), and we are interested in computing the union of the items in the bags of all of the users. This is trivial without privacy, but a differentially private algorithm must be careful about reporting items contained in only a small number of bags. We consider differentially private algorithms that always report a subset of the union, and define the utility of an algorithm to be the expected size of the subset that it reports. Because the achievable utility varies significantly with the dataset, we introduce the *utility ratio*, which normalizes utility by a dataset-specific upper bound and characterizes a mechanism by its lowest normalized utility across all datasets. We then develop algorithms with guaranteed utility ratios and complement them with bounds on the best possible utility ratio. Prior work has shown that a single algorithm can be simultaneously optimal for all datasets when $k=1$, but we show that instance-optimal algorithms do not exist when $k>1$, and characterize how performance degrades as $k$ grows. At the same time, we design a private algorithm that achieves the maximum possible utility, regardless of $k$, when the item histogram matches a prior prediction (for instance, from a previous data release) and degrades gracefully with the $L_\infty$ distance between the prediction and the actual histogram when the prediction is imperfect.

Training machine learning models with differential privacy (DP) has received increasing interest in recent years. One of the most popular algorithms for training differentially private models is differentially private stochastic gradient descent (DPSGD) and its variants, where at each step gradients are clipped and combined with some noise. Given the increasing usage of DPSGD, we ask the question: is DPSGD alone sufficient to find a good minimizer for every dataset under privacy constraints? As a first step towards answering this question, we show that even for the simple case of linear classification, unlike non-private optimization, (private) feature preprocessing is vital for differentially private optimization. In detail, we first show theoretically that there exists an example where without feature preprocessing, DPSGD incurs a privacy error proportional to the maximum norm of features over all samples. We then propose an algorithm called *DPSGD-F*, which combines DPSGD with feature preprocessing and prove that for classification tasks, it incurs a privacy error proportional to the diameter of the features $\max_{x, x' \in D} \|x - x'\|_2$. We then demonstrate the practicality of our algorithm on image classification benchmarks.

Let$\boldsymbol{p}$denote a reference generative language model. Let$\boldsymbol{r}$denote a reward model that returns a scalar to capture the degree at which a draw from$\boldsymbol{p}$is preferred. The goal of language model alignment is to alter$\boldsymbol{p}$to a new distribution$\phi$that results in a higher expected reward while keeping$\phi$close to$\boldsymbol{p}$. A popular alignment method is the KL-constrained reinforcement learning$(\boldsymbol{RL})$, which chooses a distribution$\Phi_\Delta$that maximizes$E_{\phi_{\Delta}}\boldsymbol{r}(\boldsymbol{y})$subject to a relative entropy constraint$D_{\mathrm{K}\mathrm{L}}(\phi_{\Delta}\Vert \boldsymbol{p})\leq\Delta$. Another simple alignment method is best-of-N, where$N$samples are drawn from$\boldsymbol{p}$and one with highest reward is selected. In this paper, we offer a closed-form characterization of the optimal KL-constrained RL solution. We then demonstrate that any alignment method that achieves a comparable trade-off between KL divergence and expected reward must approximate the optimal KL-constrained RL solution in terms of relative entropy. To analyze the properties of alignment methods, we introduce two simplifying assumptions: we let the language model be memoryless, and the reward model be linear. Although these assumptions may not reflect complex real-world scenarios, they enable a precise characterization of the asymptotic (in the sequence length) behavior of the best-of-N and the KL-constrained RL methods, in terms of information-theoretic quantities.1

We study the sample complexity of causal structure learning on a two-variable system with observational and experimental data. Specifically, for two variables $X$ and $Y$, we consider the classical scenario where either $X$ causes $Y$, $Y$ causes $X$, or there is an unmeasured confounder between $X$ and $Y$. Let $m_1$ be the number of observational samples of $(X,Y)$, and let $m_2$ be the number of interventional samples where either $X$ or $Y$ has been subject to an external intervention. We show that if $X$ and $Y$ are over a finite domain of size $k$ and are significantly correlated, the minimum $m_2$ needed is sublinear in $k$. Moreover, as $m_1$ grows, the minimum $m_2$ needed to identify the causal structure decreases. In fact, we can give a tight characterization of the tradeoff between $m_1$ and $m_2$ when $m_1 = O(k)$ or is sufficiently large. We build upon techniques for closeness testing when $m_1$ is small (e.g., sublinear in $k$), and for non-parametric density estimation when $m_2$ is large. Our hardness results are based on carefully constructing causal models whose marginal and interventional distributions form hard instances of canonical results on property testing.

We study discrete distribution estimation under user-level local differential privacy (LDP). In user-level $\varepsilon$-LDP, each user has a $m\ge1$ samples and the privacy of all $m$ samples must be preserved simultaneously. We resolve the following dilemma: While on the one hand having more samples per user should provide more information about the underlying distribution, on the other hand, guaranteeing privacy of all $m$ samples should make estimation task more difficult. We obtain tight bounds for this problem under almost all parameter regimes. Perhaps surprisingly, we show that in suitable parameter regimes, having $m$ samples per user is equivalent to having $m$ times more users, each with only one sample. Our results demonstrate interesting phase transitions for $m$ and the privacy parameter $\varepsilon$ in the estimation risk. Finally, connecting with recent results on shuffled DP, we show that combined with random shuffling, our algorithm leads to optimal error guarantees (up to logarithmic factors) under the central model of user-level DP in certain parameter regimes. We provide several simulations to verify our theoretical findings.

We study the problem of differentially private (DP) stochastic convex optimization (SCO) under the notion of user-level differential privacy. In this problem, there are $n$ users, each contributing $m>1$ samples to the input dataset of the private SCO algorithm, and the notion of indistinguishability embedded in DP is w.r.t. replacing the entire local dataset of any given user. Under smoothness conditions of the loss, we establish the optimal rates for user-level DP-SCO in both the central and local models of DP. In particular, we show, roughly, that the optimal rate is $\frac{1}{\sqrt{nm}}+\frac{\sqrt{d}}{\varepsilon n \sqrt{m}}$ in the central setting and is $\frac{\sqrt{d}}{\varepsilon \sqrt{nm}}$ in the local setting, where $d$ is the dimensionality of the problem and $\varepsilon$ is the privacy parameter. Our algorithms combine new user-level DP mean estimation techniques with carefully designed first-order stochastic optimization methods. For the central DP setting, our optimal rate improves over the rate attained for the same setting in Levy et al. (2021) by $\sqrt{d}$ factor. One of the main ingredients that enabled such an improvement is a novel application of the generalization properties of DP in the context of multi-pass stochastic gradient methods.

We propose a new definition of instance optimality for differentially private estimation algorithms. Our definition requires an optimal algorithm to compete, simultaneously for every dataset $D$, with the best private benchmark algorithm that (a) knows $D$ in advance and (b) is evaluated by its worst-case performance on large subsets of $D$. That is, the benchmark algorithm need not perform well when potentially extreme points are added to $D$; it only has to handle the removal of a small number of real data points that already exist. This makes our benchmark significantly stronger than those proposed in prior work. We nevertheless show, for real-valued datasets, how to construct private algorithms that achieve our notion of instance optimality when estimating a broad class of dataset properties, including means, quantiles, and $\ell_p$-norm minimizers. For means in particular, we provide a detailed analysis and show that our algorithm simultaneously matches or exceeds the asymptotic performance of existing algorithms under a range of distributional assumptions.

Motivated by real-life deployments of multi-round federated analytics with secure aggregation, we investigate the fundamental communication-accuracy tradeoffs of the heavy hitter discovery and approximate (open-domain) histogram problems under a linear sketching constraint. We propose efficient algorithms based on local subsampling and invertible bloom look-up tables (IBLTs). We also show that our algorithms are information-theoretically optimal for a broad class of interactive schemes. The results show that the linear sketching constraint does increase the communication cost for both tasks by introducing an extra linear dependence on the number of users in a round. Moreover, our results also establish a separation between the communication cost for heavy hitter discovery and approximate histogram in the multi-round setting. The dependence on the number of rounds $R$ is at most logarithmic for heavy hitter discovery whereas that of approximate histogram is $\Theta(\sqrt{R})$. We also empirically demonstrate our findings.

We study the problem of discrete distribution estimation in KL divergence and provide concentration bounds for the Laplace estimator. We show that the deviation from mean scales as $\sqrt k /n$ when n≥k, improving upon the best prior result of k/n. We also establish a matching lower bound that shows that our bounds are tight up to polylogarithmic factors.

We consider distributed parameter estimation using interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a unified framework enabling us to derive a variety of (tight) minimax lower bounds for different parametric families of distributions, both continuous and discrete, under any $\ell_p$ loss. Our lower bound framework is versatile and yields “plug-and-play” bounds that are widely applicable to a large range of estimation problems, and, for the prototypical case of the Gaussian family, circumvents limitations of previous techniques. In particular, our approach recovers bounds obtained using data processing inequalities and Cramér–Rao bounds, two other alternative approaches for proving lower bounds in our setting of interest. Further, for the families considered, we complement our lower bounds with matching upper bounds.

Autoregressive sampling from large language models has led to state-of-the-art results in several natural language tasks. However, autoregressive sampling generates tokens one at a time making it slow, and even prohibitive in certain tasks. One way to speed up sampling is *speculative decoding*: use a small model to sample a *draft* (block or sequence of tokens), and then score all tokens in the draft by the large language model in parallel. A subset of the tokens in the draft are accepted (and the rest rejected) based on a statistical method to guarantee that the final output follows the distribution of the large model. In this work, we provide a principled understanding of speculative decoding through the lens of optimal transport (OT) with *membership cost*. This framework can be viewed as an extension of the well-known *maximal-coupling* problem. This new formulation enables us to generalize the speculative decoding method to allow for a set of $k$ candidates at the token-level, which leads to an improved optimal membership cost. We show that the optimal draft selection algorithm (transport plan) can be computed via linear programming, whose best-known runtime is exponential in $k$. We then propose a valid draft selection algorithm whose acceptance probability is $(1-1/e)$-optimal multiplicatively. Moreover, it can be computed in time almost linear with size of domain of a single token. Using this new draft selection algorithm, we develop a new autoregressive sampling algorithm called *SpecTr*, which provides speedup in decoding while ensuring that there is no quality degradation in the decoded output. We experimentally demonstrate that for state-of-the-art large language models, the proposed approach achieves a wall clock speedup of 2.13X, a further 1.37X speedup over speculative decoding on standard benchmarks.

We study high-dimensional sparse estimation under three natural constraints: communication constraints, local privacy constraints, and linear measurements (compressive sensing). Without sparsity assumptions, it has been established that interactivity cannot improve the minimax rates of estimation under these information constraints. The question of whether interactivity helps with natural inference tasks has been a topic of active research. We settle this question in the affirmative for the prototypical problems of high-dimensional sparse mean estimation and compressive sensing, by demonstrating a gap between interactive and noninteractive protocols. We further establish that the gap increases when we have more structured sparsity: for \emph{block sparsity} this gap can be as large as \emph{polynomial} in the dimensionality. Thus, the more structured the sparsity is, the greater is the advantage of interaction. Proving the lower bounds requires a careful breaking of a sum of correlated random variables into independent components using Baranyai’s theorem on decomposition of hypergraphs, which might be of independent interest.

We study the problem of distributed mean estimation and optimization under communication constraints. We propose a correlated quantization protocol whose error guarantee depends on the deviation of data points instead of their absolute range. The design doesn’t need any prior knowledge on the concentration property of the dataset, which is required to get such dependence in previous works. We show that applying the proposed protocol as a sub-routine in distributed optimization algorithms leads to better convergence rates. We also prove the optimality of our protocol under mild assumptions. Experimental results show that our proposed algorithm outperforms existing mean estimation protocols on a diverse set of tasks.

We consider the problem of estimating sparse discrete distributions under local differential privacy (LDP) and communication constraints. We characterize the sample complexity for sparse estimation under LDP constraints up to a constant factor, and the sample complexity under communication constraints up to a logarithmic factor. Our upper bounds under LDP are based on the Hadamard Response, a private coin scheme that requires only one bit of communication per user. Under communication constraints we propose public coin schemes based on random hashing functions. Our tight lower bounds are based on recently proposed method of chi squared contractions.

Le Cam’s method, Fano’s inequality, and Assouad’s lemma are three widely used techniques to prove lower bounds for statistical estimation tasks. We propose their analogues under central differential privacy. Our results are simple, easy to apply and we use them to establish sample complexity bounds in several estimation tasks. \\{We} establish the optimal sample complexity of discrete distribution estimation under total variation distance and $\ell_2$ distance. We also provide lower bounds for several other distribution classes, including product distributions and Gaussian mixtures that are tight up to logarithmic factors. The technical component of our paper relates coupling between distributions to the sample complexity of estimation under differential privacy.

We study robust testing and estimation of discrete distributions in the strong contamination model. Our results cover both centralized setting and distributed setting with general local information constraints including communication and LDP constraints. Our technique relates the strength of manipulation attacks to the earth-mover distance using Hamming distance as the metric between messages (samples) from the users. In the centralized setting, we provide optimal error bounds for both learning and testing. Our lower bounds under local information constraints build on the recent lower bound methods in distributed inference. In the communication constrained setting, we develop novel algorithms based on random hashing and an L1-L1 isometry.

We study the role of interactivity in distributed statistical inference under information constraints, e.g., communication constraints and local differential privacy. We focus on the tasks of goodness-of-fit testing and estimation of discrete distributions. From prior work, these tasks are well understood under noninteractive protocols. Extending these approaches directly for interactive protocols is difficult due to correlations that can build due to interactivity; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and testing under local differential privacy and communication constraints. We also provide an example of a natural testing problem where interactivity helps.

We obtain tight minimax rates for the problem of distributed estimation of discrete distributions under communication constraints, where $n$ users observing $m $ samples each can broadcast only $\ell$ bits. Our main result is a tight characterization (up to logarithmic factors) of the error rate as a function of $m$, $\ell$, the domain size, and the number of users under most regimes of interest. While previous work focused on the setting where each user only holds one sample, we show that as $m$ grows the $\ell_1$ error rate gets reduced by a factor of $\sqrt{m}$ for small $m$. However, for large $m$ we observe an interesting phase transition: the dependence of the error rate on the communication constraint $\ell$ changes from $1/\sqrt{2^{\ell}}$ to $1/\sqrt{\ell}$.

We propose and analyze algorithms to solve a range of learning tasks under user-level differential privacy constraints. Rather than guaranteeing only the privacy of individual samples, user-level DP protects a user's entire contribution ($m \ge 1$ samples), providing more stringent but more realistic protection against information leaks. We show that for high-dimensional meanestimation, empirical risk minimization with smooth losses, stochastic convex optimization, and learning hypothesis classes with finite metric entropy, the privacy cost decreases as $O(1/\sqrt{m})$ as users provide more samples. In contrast, when increasing the number of users $n$, the privacy cost decreases at a faster $O(1/n)$ rate. We complement these results with lower bounds showing the minimax optimality of our algorithms for mean estimation and stochastic convex optimization. Our algorithms rely on novel techniques for private mean estimation in arbitrary dimension with error scaling as the concentration radius $\tau$ of the distribution rather than the entire range.

We study goodness-of-fit of discrete distributions in the distributed setting, where samples are divided between multiple users who can only release a limited amount of information about their samples due to various information constraints. Recently, a subset of the authors showed that having access to a common random seed (i.e., shared randomness) leads to a significant reduction in the sample complexity of this problem. In this work, we provide a complete understanding of the interplay between the amount of shared randomness available, the stringency of information constraints, and the sample complexity of the testing problem by characterizing a tight trade-off between these three parameters. We provide a general distributed goodness-of-fit protocol that as a function of the amount of shared randomness interpolates smoothly between the private- and public-coin sample complexities. We complement our upper bound with a general framework to prove lower bounds on the sample complexity of this testing problems under limited shared randomness. Finally, we instantiate our bounds for the two archetypal information constraints of communication and local privacy, and show that our sample complexity bounds are optimal as a function of all the parameters of the problem, including the amount of shared randomness. A key component of our upper bounds is a new primitive of \textit{domain compression}, a tool that allows us to map distributions to a much smaller domain size while preserving their pairwise distances, using a limited amount of randomness.

Local differential privacy (LDP) is a strong notion of privacy that often leads to a significant drop in utility. The original definition of LDP assumes that all the elements in the data domain are equally sensitive. However, in many real-life applications, some elements are more sensitive than others. We propose a context-aware framework for LDP that allows the privacy level to vary across the data domain, enabling system designers to place privacy constraints where they matter without paying the cost where they do not. For binary data domains, we provide a universally optimal privatization scheme and highlight its connections to Warner’s randomized response and Mangat’s improved response. Motivated by geo-location and web search applications, for k-ary data domains, we consider two special cases of context-aware LDP: block-structured LDP and high-low LDP. We study minimax discrete distribution estimation under both cases and provide communication-efficient, sample-optimal schemes, and information-theoretic lower bounds. We show, using worst-case analyses and experiments on Gowalla’s 3.6 million check-ins to 43,750 locations, that context-aware LDP achieves a far better accuracy under the same number of samples.

We study the problem of estimating $k$-ary distributions under $\eps$-local differential privacy. $n$ samples are distributed across users who send privatized versions of their sample to a central server. All previously known sample optimal algorithms require linear (in $k$) communication from each user in the high privacy regime $(\eps=O(1))$, and run in time that grows as $n\cdot k$, which can be prohibitive for large domain size $k$. We propose Hadamard Response (HR), a local privatization scheme that requires no shared randomness and is symmetric with respect to the users. Our scheme has order optimal sample complexity for all $\eps$, a communication of at most $\log k+2$ bits per user, and nearly linear running time of $\tilde{O}(n + k)$. Our encoding and decoding are based on Hadamard matrices and are simple to implement. The statistical performance relies on the coding theoretic aspects of Hadamard matrices, ie, the large Hamming distance between the rows. An efficient implementation of the algorithm using the Fast Walsh-Hadamard transform gives the computational gains. We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), both theoretically, and experimentally. For $k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR.

We consider the problems of distribution estimation, and heavy hitter (frequency) estimation under privacy, and communication constraints. While the constraints have been studied separately, optimal schemes for one are sub-optimal for the other. We propose a sample-optimal $\eps$-locally differentially private (LDP) scheme for distribution estimation, where each user communicates one bit, and requires no public randomness. We also show that Hadamard Response, a recently proposed scheme for $\eps$-LDP distribution estimation is also utility-optimal for heavy hitters estimation. Our final result shows that unlike distribution estimation, without public randomness, any utility-optimal heavy hitter estimation algorithm must require $\Omega(\log n)$ bits of communication per user.

We consider the task of estimating the entropy of $k$-ary distributions from samples in the streaming model, where space is limited. Our main contribution is an algorithm that requires $O\left(\frac{k \log (1/\varepsilon)^2}{\varepsilon^3}\right)$ samples and a constant $O(1)$ memory words of space and outputs a $\pm\varepsilon$ estimate of $H(p)$. Without space limitations, the sample complexity has been established as $S(k,\varepsilon)=\Theta\left(\frac k{\varepsilon\log k}+\frac{\log^2 k}{\varepsilon^2}\right)$, which is sub-linear in the domain size $k$, and the current algorithms that achieve optimal sample complexity also require nearly-linear space in $k$. Our algorithm partitions $[0,1]$ into intervals and estimates the entropy contribution of probability values in each interval. The intervals are designed to trade bias and variance. Distribution property estimation and testing with limited memory is a largely unexplored research area. We hope our work will motivate research in this field.

We develop differentially private methods for estimating various distributional properties. Given a sample from a discrete distribution p, some functional f, and accuracy and privacy parameters alpha and epsilon, the goal is to estimate f(p) up to accuracy alpha, while maintaining epsilon-differential privacy of the sample. We prove almost-tight bounds on the sample size required for this problem for several functionals of interest, including support size, support coverage, and entropy. We show that the cost of privacy is negligible in a variety of settings, both theoretically and experimentally. Our methods are based on a sensitivity analysis of several state-of-the-art methods for estimating these properties with sublinear sample complexities

Given n independent draws from a discrete distribution, what is the probability that the next draw will be a symbol that has not appeared before? We study the problem of estimating this missing mass probability under mean squared error. Our results include the following: 1) Mean squared error (MSE) of Good-Turing estimator is [(0608...)/n]+o([1/n]). 2) Minimax MSE for estimating missing mass of uniform distributions is [(0.570...)/(η)]+o([1/(η)]). We prove that the minimax MSE R* of missing mass estimation satisfies [(0.570...)/n]+o([1/n]) ≤ R* ≤ [(0.608...)/n]+o([1/n]I. The upper bound characterizes the maximum MSE of the celebrated Good-Turing (GT) estimator, and the lower bound characterizes the minimax MSE for estimating the missing mass for the class of uniform distributions.

We study the fundamental problems of identity testing (goodness of fit), and closeness testing (two sample test) of distributions over $k$ elements, under differential privacy. While the problems have a long history in statistics, finite sample bounds for these problems have only been established recently. In this work, we derive upper and lower bounds on the sample complexity of both the problems under $(\varepsilon, \delta)$-differential privacy. We provide optimal sample complexity algorithms for identity testing problem for all parameter ranges, and the first results for closeness testing. Our closeness testing bounds are optimal in the sparse regime where the number of samples is at most $k$. Our upper bounds are obtained by privatizing non-private estimators for these problems. The non-private estimators are chosen to have small sensitivity. We propose a general framework to establish lower bounds on the sample complexity of statistical tasks under differential privacy. We show a bound on differentially private algorithms in terms of a coupling between the two hypothesis classes we aim to test. By constructing carefully chosen priors over the hypothesis classes, and using Le Cam's two point theorem we provide a general mechanism for proving lower bounds. We believe that the framework can be used to obtain strong lower bounds for other statistical tasks under privacy.