Dravyansh Sharma

Maria-Florina Balcan, Avrim Blum, Kiriaki Fragkia et al. · cmu

Large language models with chain-of-thought generation have demonstrated great potential for producing complex mathematical proofs. However, their reasoning can often go astray, leading to increasing interest in formal and learned verifiers. A major challenge in learning verifiers, especially when their output will be used by the prover, is that this feedback loop may produce substantial distribution shift. Motivated by this challenge, we propose an online learning framework for learning chain-of-thought verifiers that, given a problem and a sequence of reasoning steps, check the correctness of the solution. Highlighting the asymmetric role of soundness (failure in catching errors in a proof) and completeness (flagging correct proofs as wrong) mistakes of the verifier, we introduce novel extensions of the Littlestone dimension which tightly characterize the mistake bounds for learning a verifier in the realizable setting. We provide optimal algorithms for finding the Pareto-frontier (the smallest total number of mistakes given a budget of soundness mistakes) as well as minimizing a linear combination of asymmetric costs. We further show how our learned verifiers can be used to boost the accuracy of a collection of weak provers, and enable generation of proofs beyond what they were trained on. With the mild assumption that one of the provers can generate the next reasoning step correctly with some minimal probability, we show how to learn a strong prover with small error and abstention rates.

Maria-Florina Balcan, Avrim Blum, Steve Hanneke et al.

Data poisoning attacks, in which an adversary corrupts a training set with the goal of inducing specific desired mistakes, have raised substantial concern: even just the possibility of such an attack can make a user no longer trust the results of a learning system. In this work, we show how to achieve strong robustness guarantees in the face of such attacks across multiple axes. We provide robustly-reliable predictions, in which the predicted label is guaranteed to be correct so long as the adversary has not exceeded a given corruption budget, even in the presence of instance targeted attacks, where the adversary knows the test example in advance and aims to cause a specific failure on that example. Our guarantees are substantially stronger than those in prior approaches, which were only able to provide certificates that the prediction of the learning algorithm does not change, as opposed to certifying that the prediction is correct, as we are able to achieve in our work. Remarkably, we provide a complete characterization of learnability in this setting, in particular, nearly-tight matching upper and lower bounds on the region that can be certified, as well as efficient algorithms for computing this region given an ERM oracle. Moreover, for the case of linear separators over logconcave distributions, we provide efficient truly polynomial time algorithms (i.e., non-oracle algorithms) for such robustly-reliable predictions. We also extend these results to the active setting where the algorithm adaptively asks for labels of specific informative examples, and the difficulty is that the adversary might even be adaptive to this interaction, as well as to the agnostic learning setting where there is no perfect classifier even over the uncorrupted data.

Maria-Florina Balcan, Mikhail Khodak, Dravyansh Sharma et al.

An important unresolved challenge in the theory of regularization is to set the regularization coefficients of popular techniques like the ElasticNet with general provable guarantees. We consider the problem of tuning the regularization parameters of Ridge regression, LASSO, and the ElasticNet across multiple problem instances, a setting that encompasses both cross-validation and multi-task hyperparameter optimization. We obtain a novel structural result for the ElasticNet which characterizes the loss as a function of the tuning parameters as a piecewise-rational function with algebraic boundaries. We use this to bound the structural complexity of the regularized loss functions and show generalization guarantees for tuning the ElasticNet regression coefficients in the statistical setting. We also consider the more challenging online learning setting, where we show vanishing average expected regret relative to the optimal parameter pair. We further extend our results to tuning classification algorithms obtained by thresholding regression fits regularized by Ridge, LASSO, or ElasticNet. Our results are the first general learning-theoretic guarantees for this important class of problems that avoid strong assumptions on the data distribution. Furthermore, our guarantees hold for both validation and popular information criterion objectives.

Maria-Florina Balcan, Christopher Seiler, Dravyansh Sharma

Data-driven algorithm design is a promising, learning-based approach for beyond worst-case analysis of algorithms with tunable parameters. An important open problem is the design of computationally efficient data-driven algorithms for combinatorial algorithm families with multiple parameters. As one fixes the problem instance and varies the parameters, the "dual" loss function typically has a piecewise-decomposable structure, i.e. is well-behaved except at certain sharp transition boundaries. In this work we initiate the study of techniques to develop efficient ERM learning algorithms for data-driven algorithm design by enumerating the pieces of the sum dual loss functions for a collection of problem instances. The running time of our approach scales with the actual number of pieces that appear as opposed to worst case upper bounds on the number of pieces. Our approach involves two novel ingredients -- an output-sensitive algorithm for enumerating polytopes induced by a set of hyperplanes using tools from computational geometry, and an execution graph which compactly represents all the states the algorithm could attain for all possible parameter values. We illustrate our techniques by giving algorithms for pricing problems, linkage-based clustering and dynamic-programming based sequence alignment.

Maria-Florina Balcan, Steve Hanneke, Rattana Pukdee et al.

The problem of designing learners that provide guarantees that their predictions are provably correct is of increasing importance in machine learning. However, learning theoretic guarantees have only been considered in very specific settings. In this work, we consider the design and analysis of reliable learners in challenging test-time environments as encountered in modern machine learning problems: namely `adversarial' test-time attacks (in several variations) and `natural' distribution shifts. In this work, we provide a reliable learner with provably optimal guarantees in such settings. We discuss computationally feasible implementations of the learner and further show that our algorithm achieves strong positive performance guarantees on several natural examples: for example, linear separators under log-concave distributions or smooth boundary classifiers under smooth probability distributions.

Maria-Florina Balcan, Anh Tuan Nguyen, Dravyansh Sharma

Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm's hyperparameters using problem instances drawn from the problem distribution of the target application domain. This can be achieved by maximizing empirical utilities that measure the algorithms' performance as a function of their hyperparameters, using problem instances. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piecewise discontinuous structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the \textit{Pfaffian GJ framework}, an extension of the classical \textit{GJ framework}, that is capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a \textit{refined piecewise structure}, which automatically translates to learning guarantees using our proposed framework.

Maria-Florina Balcan, Matteo Pozzi, Dravyansh Sharma

Overcoming the impact of selfish behavior of rational players in multiagent systems is a fundamental problem in game theory. Without any intervention from a central agent, strategic users take actions in order to maximize their personal utility, which can lead to extremely inefficient overall system performance, often indicated by a high Price of Anarchy. Recent work (Lin et al. 2021) investigated and formalized yet another undesirable behavior of rational agents, that of avoiding freely available information about the game for selfish reasons, leading to worse social outcomes. A central planner can significantly mitigate these issues by injecting a subsidy to reduce certain costs associated with the system and obtain net gains in the system performance. Crucially, the planner needs to determine how to allocate this subsidy effectively. We formally show that designing subsidies that perfectly optimize the social good, in terms of minimizing the Price of Anarchy or preventing the information avoidance behavior, is computationally hard under standard complexity theoretic assumptions. On the positive side, we show that we can learn provably good values of subsidy in repeated games coming from the same domain. This data-driven subsidy design approach avoids solving computationally hard problems for unseen games by learning over polynomially many games. We also show that optimal subsidy can be learned with no-regret given an online sequence of games, under mild assumptions on the cost matrix. Our study focuses on two distinct games: a Bayesian extension of the well-studied fair cost-sharing game, and a component maintenance game with engineering applications.

Avrim Blum, Marten Garicano, Kavya Ravichandran et al.

The improving multi-armed bandits problem is a formal model for allocating effort under uncertainty, motivated by scenarios such as investing research effort into new technologies, performing clinical trials, and hyperparameter selection from learning curves. Each pull of an arm provides reward that increases monotonically with diminishing returns. A growing line of work has designed algorithms for improving bandits, albeit with somewhat pessimistic worst-case guarantees. Indeed, strong lower bounds of $Ω(k)$ and $Ω(\sqrt{k})$ multiplicative approximation factors are known for both deterministic and randomized algorithms (respectively) relative to the optimal arm, where $k$ is the number of bandit arms. In this work, we propose two new parameterized families of bandit algorithms and bound the sample complexity of learning the near-optimal algorithm from each family using offline data. The first family we define includes the optimal randomized algorithm from prior work. We show that an appropriately chosen algorithm from this family can achieve stronger guarantees, with optimal dependence on $k$, when the arm reward curves satisfy additional properties related to the strength of concavity. Our second family contains algorithms that both guarantee best-arm identification on well-behaved instances and revert to worst case guarantees on poorly-behaved instances. Taking a statistical learning perspective on the bandit rewards optimization problem, we achieve stronger data-dependent guarantees without the need for actually verifying whether the assumptions are satisfied.

Dravyansh Sharma, Maxwell Jones

Computational efficiency is a major bottleneck in using classic graph-based approaches for semi-supervised learning on datasets with a large number of unlabeled examples. Known techniques to improve efficiency typically involve an approximation of the graph regularization objective, but suffer two major drawbacks - first the graph is assumed to be known or constructed with heuristic hyperparameter values, second they do not provide a principled approximation guarantee for learning over the full unlabeled dataset. Building on recent work on learning graphs for semi-supervised learning from multiple datasets for problems from the same domain, and leveraging techniques for fast approximations for solving linear systems in the graph Laplacian matrix, we propose algorithms that overcome both the above limitations. We show a formal separation in the learning-theoretic complexity of sparse and dense graph families. We further show how to approximately learn the best graphs from the sparse families efficiently using the conjugate gradient method. Our approach can also be used to learn the graph efficiently online with sub-linear regret, under mild smoothness assumptions. Our online learning results are stated generally, and may be useful for approximate and efficient parameter tuning in other problems. We implement our approach and demonstrate significant ($\sim$10-100x) speedups over prior work on semi-supervised learning with learned graphs on benchmark datasets.

Maria-Florina Balcan, Anh Tuan Nguyen, Dravyansh Sharma

Modern machine learning algorithms, especially deep learning based techniques, typically involve careful hyperparameter tuning to achieve the best performance. Despite the surge of intense interest in practical techniques like Bayesian optimization and random search based approaches to automating this laborious and compute intensive task, the fundamental learning theoretic complexity of tuning hyperparameters for deep neural networks is poorly understood. Inspired by this glaring gap, we initiate the formal study of hyperparameter tuning complexity in deep learning through a recently introduced data driven setting. We assume that we have a series of deep learning tasks, and we have to tune hyperparameters to do well on average over the distribution of tasks. A major difficulty is that the utility function as a function of the hyperparameter is very volatile and furthermore, it is given implicitly by an optimization problem over the model parameters. To tackle this challenge, we introduce a new technique to characterize the discontinuities and oscillations of the utility function on any fixed problem instance as we vary the hyperparameter; our analysis relies on subtle concepts including tools from differential/algebraic geometry and constrained optimization. This can be used to show that the learning theoretic complexity of the corresponding family of utility functions is bounded. We instantiate our results and provide sample complexity bounds for concrete applications tuning a hyperparameter that interpolates neural activation functions and setting the kernel parameter in graph neural networks.

Dravyansh Sharma, Arun Sai Suggala

Classic algorithms for stochastic bandits typically use hyperparameters that govern their critical properties such as the trade-off between exploration and exploitation. Tuning these hyperparameters is a problem of great practical significance. However, this is a challenging problem and in certain cases is information theoretically impossible. To address this challenge, we consider a practically relevant transfer learning setting where one has access to offline data collected from several bandit problems (tasks) coming from an unknown distribution over the tasks. Our aim is to use this offline data to set the hyperparameters for a new task drawn from the unknown distribution. We provide bounds on the inter-task (number of tasks) and intra-task (number of arm pulls for each task) sample complexity for learning near-optimal hyperparameters on unseen tasks drawn from the distribution. Our results apply to several classic algorithms, including tuning the exploration parameters in UCB and LinUCB and the noise parameter in GP-UCB. Our experiments indicate the significance and effectiveness of the transfer of hyperparameters from offline problems in online learning with stochastic bandit feedback.

Maria-Florina Balcan, Dravyansh Sharma

Decision trees and their ensembles are popular in machine learning as easy-to-understand models. Several techniques have been proposed in the literature for learning tree-based classifiers, with different techniques working well for data from different domains. In this work, we develop approaches to design tree-based learning algorithms given repeated access to data from the same domain. We study multiple formulations covering different aspects and popular techniques for learning decision tree based approaches. We propose novel parameterized classes of node splitting criteria in top-down algorithms, which interpolate between popularly used entropy and Gini impurity based criteria, and provide theoretical bounds on the number of samples needed to learn the splitting function appropriate for the data at hand. We also study the sample complexity of tuning prior parameters in Bayesian decision tree learning, and extend our results to decision tree regression. We further consider the problem of tuning hyperparameters in pruning the decision tree for classical pruning algorithms including min-cost complexity pruning. In addition, our techniques can be used to optimize the explainability versus accuracy trade-off when using decision trees. We extend our results to tuning popular tree-based ensembles, including random forests and gradient-boosted trees. We demonstrate the significance of our approach on real world datasets by learning data-specific decision trees which are simultaneously more accurate and interpretable.

Maria-Florina Balcan, Avrim Blum, Zhiyuan Li et al.

Chain-of-Thought reasoning has emerged as a powerful approach for solving complex mathematical and logical problems. However, it can often veer off track through incorrect or unsubstantiated inferences. Formal mathematical reasoning, which can be checked with a formal verifier, is one approach to addressing this issue. However, currently LLMs are simply not good enough to solve complex problems in a formal way, and even just formalizing an informal problem statement can be challenging. Motivated by this fact, in this work we consider the problem of learning reliable verifiers for natural language Chain-of-Thought reasoning. That is, given a problem statement and step-by-step solution in natural language, the aim of the verifier is to output [Yes] if the reasoning steps in the solution are all valid, and [No] otherwise. In this work we give a formal PAC-learning framework for studying this problem. We propose and analyze several natural verification goals, at different levels of strength, in this framework. We provide sample complexity upper-bounds for learning verifiers satisfying these goals, as well as lower-bound and impossibility results for learning other natural verification objectives without additional assumptions.

Idan Attias, Avrim Blum, Keziah Naggita et al.

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

Ally Yalei Du, Eric Huang, Dravyansh Sharma

Graph-based semi-supervised learning is a powerful paradigm in machine learning for modeling and exploiting the underlying graph structure that captures the relationship between labeled and unlabeled data. A large number of classical as well as modern deep learning based algorithms have been proposed for this problem, often having tunable hyperparameters. We initiate a formal study of tuning algorithm hyperparameters from parameterized algorithm families for this problem. We obtain novel $O(\log n)$ pseudo-dimension upper bounds for hyperparameter selection in three classical label propagation-based algorithm families, where $n$ is the number of nodes, implying bounds on the amount of data needed for learning provably good parameters. We further provide matching $Ω(\log n)$ pseudo-dimension lower bounds, thus asymptotically characterizing the learning-theoretic complexity of the parameter tuning problem. We extend our study to selecting architectural hyperparameters in modern graph neural networks. We bound the Rademacher complexity for tuning the self-loop weighting in recently proposed Simplified Graph Convolution (SGC) networks. We further propose a tunable architecture that interpolates graph convolutional neural networks (GCN) and graph attention networks (GAT) in every layer, and provide Rademacher complexity bounds for tuning the interpolation coefficient.

Maria-Florina Balcan, Saumya Goyal, Dravyansh Sharma

Modern regression problems often involve high-dimensional data and a careful tuning of the regularization hyperparameters is crucial to avoid overly complex models that may overfit the training data while guaranteeing desirable properties like effective variable selection. We study the recently introduced direction of tuning regularization hyperparameters in linear regression across multiple related tasks. We obtain distribution-dependent bounds on the generalization error for the validation loss when tuning the L1 and L2 coefficients, including ridge, lasso and the elastic net. In contrast, prior work develops bounds that apply uniformly to all distributions, but such bounds necessarily degrade with feature dimension, d. While these bounds are shown to be tight for worst-case distributions, our bounds improve with the "niceness" of the data distribution. Concretely, we show that under additional assumptions that instances within each task are i.i.d. draws from broad well-studied classes of distributions including sub-Gaussians, our generalization bounds do not get worse with increasing d, and are much sharper than prior work for very large d. We also extend our results to a generalization of ridge regression, where we achieve tighter bounds that take into account an estimate of the mean of the ground truth distribution.

Dravyansh Sharma

Gradient-based iterative optimization methods are the workhorse of modern machine learning. They crucially rely on careful tuning of parameters like learning rate and momentum. However, one typically sets them using heuristic approaches without formal near-optimality guarantees. Recent work by Gupta and Roughgarden studies how to learn a good step-size in gradient descent. However, like most of the literature with theoretical guarantees for gradient-based optimization, their results rely on strong assumptions on the function class including convexity and smoothness which do not hold in typical applications. In this work, we develop novel analytical tools for provably tuning hyperparameters in gradient-based algorithms that apply to non-convex and non-smooth functions. We obtain matching sample complexity bounds for learning the step-size in gradient descent shown for smooth, convex functions in prior work (up to logarithmic factors) but for a much broader class of functions. Our analysis applies to gradient descent on neural networks with commonly used activation functions (including ReLU, sigmoid and tanh). We extend our framework to tuning multiple hyperparameters, including tuning the learning rate schedule, simultaneously tuning momentum and step-size, and pre-training the initialization vector. Our approach can be used to bound the sample complexity for minimizing both the validation loss as well as the number of gradient descent iterations.

Dravyansh Sharma, Alec Sun

Machine learning is now ubiquitous in societal decision-making, for example in evaluating job candidates or loan applications, and it is increasingly important to take into account how classified agents will react to the learning algorithms. The majority of recent literature on strategic classification has focused on reducing and countering deceptive behaviors by the classified agents, but recent work of Attias et al. identifies surprising properties of learnability when the agents genuinely improve in order to attain the desirable classification, such as smaller generalization error than standard PAC-learning. In this paper we characterize so-called learnability with improvements across multiple new axes. We introduce an asymmetric variant of minimally consistent concept classes and use it to provide an exact characterization of proper learning with improvements in the realizable setting. While prior work studies learnability only under general, arbitrary agent improvement regions, we give positive results for more natural Euclidean ball improvement sets. In particular, we characterize improper learning under a mild generative assumption on the data distribution. We further show how to learn in more challenging settings, achieving lower generalization error under well-studied bounded noise models and obtaining mistake bounds in realizable and agnostic online learning. We resolve open questions posed by Attias et al. for both proper and improper learning.

Maria-Florina Balcan, Mikhail Khodak, Dravyansh Sharma et al.

We analyze the meta-learning of the initialization and step-size of learning algorithms for piecewise-Lipschitz functions, a non-convex setting with applications to both machine learning and algorithms. Starting from recent regret bounds for the exponential forecaster on losses with dispersed discontinuities, we generalize them to be initialization-dependent and then use this result to propose a practical meta-learning procedure that learns both the initialization and the step-size of the algorithm from multiple online learning tasks. Asymptotically, we guarantee that the average regret across tasks scales with a natural notion of task-similarity that measures the amount of overlap between near-optimal regions of different tasks. Finally, we instantiate the method and its guarantee in two important settings: robust meta-learning and multi-task data-driven algorithm design.

Maria-Florina Balcan, Dravyansh Sharma

We consider a novel data driven approach for designing learning algorithms that can effectively learn with only a small number of labeled examples. This is crucial for modern machine learning applications where labels are scarce or expensive to obtain. We focus on graph-based techniques, where the unlabeled examples are connected in a graph under the implicit assumption that similar nodes likely have similar labels. Over the past decades, several elegant graph-based semi-supervised learning algorithms for how to infer the labels of the unlabeled examples given the graph and a few labeled examples have been proposed. However, the problem of how to create the graph (which impacts the practical usefulness of these methods significantly) has been relegated to domain-specific art and heuristics and no general principles have been proposed. In this work we present a novel data driven approach for learning the graph and provide strong formal guarantees in both the distributional and online learning formalizations. We show how to leverage problem instances coming from an underlying problem domain to learn the graph hyperparameters from commonly used parametric families of graphs that perform well on new instances coming from the same domain. We obtain low regret and efficient algorithms in the online setting, and generalization guarantees in the distributional setting. We also show how to combine several very different similarity metrics and learn multiple hyperparameters, providing general techniques to apply to large classes of problems. We expect some of the tools and techniques we develop along the way to be of interest beyond semi-supervised learning, for data driven algorithms for combinatorial problems more generally.

Maria-Florina Balcan, Avrim Blum, Dravyansh Sharma et al.

Despite significant advances, deep networks remain highly susceptible to adversarial attack. One fundamental challenge is that small input perturbations can often produce large movements in the network's final-layer feature space. In this paper, we define an attack model that abstracts this challenge, to help understand its intrinsic properties. In our model, the adversary may move data an arbitrary distance in feature space but only in random low-dimensional subspaces. We prove such adversaries can be quite powerful: defeating any algorithm that must classify any input it is given. However, by allowing the algorithm to abstain on unusual inputs, we show such adversaries can be overcome when classes are reasonably well-separated in feature space. We further provide strong theoretical guarantees for setting algorithm parameters to optimize over accuracy-abstention trade-offs using data-driven methods. Our results provide new robustness guarantees for nearest-neighbor style algorithms, and also have application to contrastive learning, where we empirically demonstrate the ability of such algorithms to obtain high robust accuracy with low abstention rates. Our model is also motivated by strategic classification, where entities being classified aim to manipulate their observable features to produce a preferred classification, and we provide new insights into that area as well.

Maria-Florina Balcan, Travis Dick, Dravyansh Sharma

Optimization in the presence of sharp (non-Lipschitz), unpredictable (w.r.t. time and amount) changes is a challenging and largely unexplored problem of great significance. We consider the class of piecewise Lipschitz functions, which is the most general online setting considered in the literature for the problem, and arises naturally in various combinatorial algorithm selection problems where utility functions can have sharp discontinuities. The usual performance metric of $\mathit{static}$ regret minimizes the gap between the payoff accumulated and that of the best fixed point for the entire duration, and thus fails to capture changing environments. Shifting regret is a useful alternative, which allows for up to $s$ environment shifts. In this work we provide an $O(\sqrt{sdT\log T}+sT^{1-β})$ regret bound for $β$-dispersed functions, where $β$ roughly quantifies the rate at which discontinuities appear in the utility functions in expectation (typically $β\ge1/2$ in problems of practical interest). We also present a lower bound tight up to sub-logarithmic factors. We further obtain improved bounds when selecting from a small pool of experts. We empirically demonstrate a key application of our algorithms to online clustering problems on popular benchmarks.