Vinod Raman

Steve Hanneke, Shay Moran, Vinod Raman et al.

We study multiclass classification in the agnostic adversarial online learning setting. As our main result, we prove that any multiclass concept class is agnostically learnable if and only if its Littlestone dimension is finite. This solves an open problem studied by Daniely, Sabato, Ben-David, and Shalev-Shwartz (2011,2015) who handled the case when the number of classes (or labels) is bounded. We also prove a separation between online learnability and online uniform convergence by exhibiting an easy-to-learn class whose sequential Rademacher complexity is unbounded. Our learning algorithm uses the multiplicative weights algorithm, with a set of experts defined by executions of the Standard Optimal Algorithm on subsequences of size Littlestone dimension. We argue that the best expert has regret at most Littlestone dimension relative to the best concept in the class. This differs from the well-known covering technique of Ben-David, Pál, and Shalev-Shwartz (2009) for binary classification, where the best expert has regret zero.

Ananth Raman, Vinod Raman, Unique Subedi et al.

We study online multiclass classification under bandit feedback. We extend the results of Daniely and Helbertal [2013] by showing that the finiteness of the Bandit Littlestone dimension is necessary and sufficient for bandit online learnability even when the label space is unbounded. Moreover, we show that, unlike the full-information setting, sequential uniform convergence is necessary but not sufficient for bandit online learnability. Our result complements the recent work by Hanneke, Moran, Raman, Subedi, and Tewari [2023] who show that the Littlestone dimension characterizes online multiclass learnability in the full-information setting even when the label space is unbounded.

Vinod Raman, Unique Subedi, Ambuj Tewari

We study a variant of online multiclass classification where the learner predicts a single label but receives a \textit{set of labels} as feedback. In this model, the learner is penalized for not outputting a label contained in the revealed set. We show that unlike online multiclass learning with single-label feedback, deterministic and randomized online learnability are \textit{not equivalent} even in the realizable setting with set-valued feedback. Accordingly, we give two new combinatorial dimensions, named the Set Littlestone and Measure Shattering dimension, that tightly characterize deterministic and randomized online learnability respectively in the realizable setting. In addition, we show that the Measure Shattering dimension characterizes online learnability in the agnostic setting and tightly quantifies the minimax regret. Finally, we use our results to establish bounds on the minimax regret for three practical learning settings: online multilabel ranking, online multilabel classification, and real-valued prediction with interval-valued response.

Vinod Raman, Unique Subedi, Ananth Raman et al.

In online binary classification under \emph{apple tasting} feedback, the learner only observes the true label if it predicts ``1". First studied by \cite{helmbold2000apple}, we revisit this classical partial-feedback setting and study online learnability from a combinatorial perspective. We show that the Littlestone dimension continues to provide a tight quantitative characterization of apple tasting in the agnostic setting, closing an open question posed by \cite{helmbold2000apple}. In addition, we give a new combinatorial parameter, called the Effective width, that tightly quantifies the minimax expected mistakes in the realizable setting. As a corollary, we use the Effective width to establish a \emph{trichotomy} of the minimax expected number of mistakes in the realizable setting. In particular, we show that in the realizable setting, the expected number of mistakes of any learner, under apple tasting feedback, can be $Θ(1), Θ(\sqrt{T})$, or $Θ(T)$. This is in contrast to the full-information realizable setting where only $Θ(1)$ and $Θ(T)$ are possible.

Vinod Raman, Unique Subedi, Ambuj Tewari

Recently, Montasser et al. [2019] showed that finite VC dimension is not sufficient for proper adversarially robust PAC learning. In light of this hardness, there is a growing effort to study what type of relaxations to the adversarially robust PAC learning setup can enable proper learnability. In this work, we initiate the study of proper learning under relaxations of the worst-case robust loss. We give a family of robust loss relaxations under which VC classes are properly PAC learnable with sample complexity close to what one would require in the standard PAC learning setup. On the other hand, we show that for an existing and natural relaxation of the worst-case robust loss, finite VC dimension is not sufficient for proper learning. Lastly, we give new generalization guarantees for the adversarially robust empirical risk minimizer.

Vinod Raman, Ambuj Tewari

Boosting is a fundamental approach in machine learning that enjoys both strong theoretical and practical guarantees. At a high-level, boosting algorithms cleverly aggregate weak learners to generate predictions with arbitrarily high accuracy. In this way, boosting algorithms convert weak learners into strong ones. Recently, Brukhim et al. extended boosting to the online agnostic binary classification setting. A key ingredient in their approach is a clean and simple reduction to online convex optimization, one that efficiently converts an arbitrary online convex optimizer to an agnostic online booster. In this work, we extend this reduction to multiclass problems and give the first boosting algorithm for online agnostic mutliclass classification. Our reduction also enables the construction of algorithms for statistical agnostic, online realizable, and statistical realizable multiclass boosting.

Vinod Raman, Unique Subedi, Ambuj Tewari

We consider the problem of learning linear operators under squared loss between two infinite-dimensional Hilbert spaces in the online setting. We show that the class of linear operators with uniformly bounded $p$-Schatten norm is online learnable for any $p \in [1, \infty)$. On the other hand, we prove an impossibility result by showing that the class of uniformly bounded linear operators with respect to the operator norm is \textit{not} online learnable. Moreover, we show a separation between sequential uniform convergence and online learnability by identifying a class of bounded linear operators that is online learnable but uniform convergence does not hold. Finally, we prove that the impossibility result and the separation between uniform convergence and learnability also hold in the batch setting.

Vinod Raman, Unique Subedi, Ambuj Tewari

Multilabel ranking is a central task in machine learning. However, the most fundamental question of learnability in a multilabel ranking setting with relevance-score feedback remains unanswered. In this work, we characterize the learnability of multilabel ranking problems in both batch and online settings for a large family of ranking losses. Along the way, we give two equivalence classes of ranking losses based on learnability that capture most, if not all, losses used in practice.

Vinod Raman, Unique Subedi, Ambuj Tewari

We consider the problem of learning multioutput function classes in the batch and online settings. In both settings, we show that a multioutput function class is learnable if and only if each single-output restriction of the function class is learnable. This provides a complete characterization of the learnability of multilabel classification and multioutput regression in both batch and online settings. As an extension, we also consider multilabel learnability in the bandit feedback setting and show a similar characterization as in the full-feedback setting.

Vinod Raman, Unique Subedi, Ambuj Tewari

We study the online learnability of hypothesis classes with respect to arbitrary, but bounded loss functions. No characterization of online learnability is known at this level of generality. We give a new scale-sensitive combinatorial dimension, named the sequential minimax dimension, and show that it gives a tight quantitative characterization of online learnability. In addition, we show that the sequential minimax dimension subsumes most existing combinatorial dimensions in online learning theory.

Jixuan Leng, Si Si, Hsiang-Fu Yu et al.

Preference optimization is widely used to align Large Language Models (LLMs) with preference feedback. However, most existing methods train on a single positive-negative pair per prompt, discarding additional supervision available in preference datasets that typically contain multiple candidate responses. Motivated by this limitation, recent work explores group-wise preference optimization, which jointly contrasts multiple responses for the same prompt, but its empirical behavior and scalability remain underexplored due to the memory overhead of group-coupled objectives. In this work, we introduce a memory-efficient group-wise preference optimization algorithm that preserves gradients while decoupling samples during backpropagation, substantially reducing peak memory usage, which enables scalable training with larger group sizes. Across both offline and online alignment settings, we show that leveraging multiple responses consistently outperforms single-pair training. Furthermore, incorporating a negative log-likelihood (NLL) term on positive responses is critical for both performance gains and training stability.

Travis Dick, Matthew Joseph, Vinod Raman

We study several problems in differentially private domain discovery, where each user holds a subset of items from a shared but unknown domain, and the goal is to output an informative subset of items. For set union, we show that the simple baseline Weighted Gaussian Mechanism (WGM) has a near-optimal $\ell_1$ missing mass guarantee on Zipfian data as well as a distribution-free $\ell_\infty$ missing mass guarantee. We then apply the WGM as a domain-discovery precursor for existing known-domain algorithms for private top-$k$ and $k$-hitting set and obtain new utility guarantees for their unknown domain variants. Finally, experiments demonstrate that all of our WGM-based methods are competitive with or outperform existing baselines for all three problems.

Alex Bie, Travis Dick, Alex Kulesza et al.

Modern AI systems have been successfully deployed to win medals at international math competitions, assist with research workflows, and prove novel technical lemmas. However, despite their progress at advanced levels of mathematics, they remain stubbornly bad at basic arithmetic, consistently failing on the simple task of adding two numbers. We present a systematic investigation of this phenomenon. We demonstrate empirically that all frontier models suffer significantly degraded accuracy for integer addition as the number of digits increases. Furthermore, we show that most errors made by these models are highly interpretable and can be attributed to either operand misalignment or a failure to correctly carry; these two error classes explain 87.9%, 62.9%, and 92.4% of Claude Opus 4.1, GPT-5, and Gemini 2.5 Pro errors, respectively. Finally, we show that misalignment errors are frequently related to tokenization, and that carrying errors appear largely as independent random failures.

Jiaxun Li, Vinod Raman, Ambuj Tewari

We study generation through the lens of statistical learning theory. First, we abstract and formalize the results of Gold [1967], Angluin [1979], Angluin [1980] and Kleinberg and Mullainathan [2024] in terms of a binary hypothesis class defined over an abstract example space. Then, we extend the notion of "generation" from Kleinberg and Mullainathan [2024] to two new settings, we call "uniform" and "non-uniform" generation, and provide a characterization of which hypothesis classes are uniformly and non-uniformly generatable. As is standard in learning theory, our characterizations are in terms of the finiteness of a new combinatorial dimension termed the Closure dimension. By doing so, we are able to compare generatability with predictability (captured via PAC and online learnability) and show that these two properties of hypothesis classes are incompatible -- there are classes that are generatable but not predictable and vice versa. Finally, we extend our results to capture prompted generation and give a complete characterization of which classes are prompt generatable, generalizing some of the work by Kleinberg and Mullainathan [2024].

Ananth Raman, Vinod Raman

We continue to study the learning-theoretic foundations of generation by extending the results from Kleinberg and Mullainathan [2024] and Li et al. [2024] to account for noisy example streams. In the noiseless setting of Kleinberg and Mullainathan [2024] and Li et al. [2024], an adversary picks a hypothesis from a binary hypothesis class and provides a generator with a sequence of its positive examples. The goal of the generator is to eventually output new, unseen positive examples. In the noisy setting, an adversary still picks a hypothesis and a sequence of its positive examples. But, before presenting the stream to the generator, the adversary inserts a finite number of negative examples. Unaware of which examples are noisy, the goal of the generator is to still eventually output new, unseen positive examples. In this paper, we provide necessary and sufficient conditions for when a binary hypothesis class can be noisily generatable. We provide such conditions with respect to various constraints on the number of distinct examples that need to be seen before perfect generation of positive examples. Interestingly, for finite and countable classes we show that generatability is largely unaffected by the presence of a finite number of noisy examples.

Charlotte Peale, Vinod Raman, Omer Reingold

We introduce "representative generation," extending the theoretical framework for generation proposed by Kleinberg et al. (2024) and formalized by Li et al. (2024), to additionally address diversity and bias concerns in generative models. Our notion requires outputs of a generative model to proportionally represent groups of interest from the training data. We characterize representative uniform and non-uniform generation, introducing the "group closure dimension" as a key combinatorial quantity. For representative generation in the limit, we analyze both information-theoretic and computational aspects, demonstrating feasibility for countably infinite hypothesis classes and collections of groups under certain conditions, but proving a negative result for computability using only membership queries. This contrasts with Kleinberg et al.'s (2024) positive results for standard generation in the limit. Our findings provide a rigorous foundation for developing more diverse and representative generative models.

Steve Hanneke, Vinod Raman, Amirreza Shaeiri et al.

We consider the problem of multiclass transductive online learning when the number of labels can be unbounded. Previous works by Ben-David et al. [1997] and Hanneke et al. [2023b] only consider the case of binary and finite label spaces, respectively. The latter work determined that their techniques fail to extend to the case of unbounded label spaces, and they pose the question of characterizing the optimal mistake bound for unbounded label spaces. We answer this question by showing that a new dimension, termed the Level-constrained Littlestone dimension, characterizes online learnability in this setting. Along the way, we show that the trichotomy of possible minimax rates of the expected number of mistakes established by Hanneke et al. [2023b] for finite label spaces in the realizable setting continues to hold even when the label space is unbounded. In particular, if the learner plays for $T \in \mathbb{N}$ rounds, its minimax expected number of mistakes can only grow like $Θ(T)$, $Θ(\log T)$, or $Θ(1)$. To prove this result, we give another combinatorial dimension, termed the Level-constrained Branching dimension, and show that its finiteness characterizes constant minimax expected mistake-bounds. The trichotomy is then determined by a combination of the Level-constrained Littlestone and Branching dimensions. Quantitatively, our upper bounds improve upon existing multiclass upper bounds in Hanneke et al. [2023b] by removing the dependence on the label set size. In doing so, we explicitly construct learning algorithms that can handle extremely large or unbounded label spaces. A key and novel component of our algorithm is a new notion of shattering that exploits the sequential nature of transductive online learning. Finally, we complete our results by proving expected regret bounds in the agnostic setting, extending the result of Hanneke et al. [2023b].

Vinod Raman, Ambuj Tewari

We study online classification when the learner has access to predictions about future examples. We design an online learner whose expected regret is never worse than the worst-case regret, gracefully improves with the quality of the predictions, and can be significantly better than the worst-case regret when the predictions of future examples are accurate. As a corollary, we show that if the learner is always guaranteed to observe data where future examples are easily predictable, then online learning can be as easy as transductive online learning. Our results complement recent work in online algorithms with predictions and smoothed online classification, which go beyond a worse-case analysis by using machine-learned predictions and distributional assumptions respectively.

Vinod Raman, Hilal Asi, Satyen Kale

Recent advances in test-time alignment methods, such as Best-of-N sampling, offer a simple and effective way to steer language models (LMs) toward preferred behaviors using reward models (RM). However, these approaches can be computationally expensive, especially when applied uniformly across prompts without accounting for differences in alignment difficulty. In this work, we propose a prompt-adaptive strategy for Best-of-N alignment that allocates inference-time compute more efficiently. Motivated by latency concerns, we develop a two-stage algorithm: an initial exploratory phase estimates the reward distribution for each prompt using a small exploration budget, and a second stage adaptively allocates the remaining budget using these estimates. Our method is simple, practical, and compatible with any LM-RM combination. Empirical results on prompts from the AlpacaEval, HH-RLHF, and PKU-SafeRLHF datasets for 12 LM/RM pairs and 50 different batches of prompts show that our adaptive strategy outperforms the uniform allocation with the same inference budget. Moreover, we show that our adaptive strategy remains competitive against uniform allocations with 20 percent larger inference budgets and improves in performance as the batch size grows.

Vinod Raman, Unique Subedi, Ambuj Tewari

We study the problem of learning to predict the next state of a dynamical system when the underlying evolution function is unknown. Unlike previous work, we place no parametric assumptions on the dynamical system, and study the problem from a learning theory perspective. We define new combinatorial measures and dimensions and show that they quantify the optimal mistake and regret bounds in the realizable and agnostic settings respectively. By doing so, we find that in the realizable setting, the total number of mistakes can grow according to \emph{any} increasing function of the time horizon $T$. In contrast, we show that in the agnostic setting under the commonly studied notion of Markovian regret, the only possible rates are $Θ(T)$ and $\tildeΘ(\sqrt{T})$.

Vinod Raman, Shenghao Xie, Samson Zhou

Motivated by the predictable nature of real-life in data streams, we study online regression when the learner has access to predictions about future examples. In the extreme case, called transductive online learning, the sequence of examples is revealed to the learner before the game begins. For this setting, we fully characterize the minimax expected regret in terms of the fat-shattering dimension, establishing a separation between transductive online regression and (adversarial) online regression. Then, we generalize this setting by allowing for noisy or \emph{imperfect} predictions about future examples. Using our results for the transductive online setting, we develop an online learner whose minimax expected regret matches the worst-case regret, improves smoothly with prediction quality, and significantly outperforms the worst-case regret when future example predictions are precise, achieving performance similar to the transductive online learner. This enables learnability for previously unlearnable classes under predictable examples, aligning with the broader learning-augmented model paradigm.

Yusuf Kalayci, Vinod Raman, Shaddin Dughmi

Large language model (LLM) generation often requires balancing output quality against inference cost, especially when using multiple generations. We introduce a new framework for inference-time optimization based on the classical Pandora's Box problem. Viewing each generation as opening a costly "box" with random reward, we develop algorithms that decide when to stop generating without knowing the underlying reward distribution. Our first contribution is a UCB-style Pandora's Box algorithm, which achieves performance that is provably close to Weitzman's algorithm, the optimal strategy when the distribution is known. We further adapt this method to practical LLM settings by addressing reward scaling across prompts via a Bradley-Terry inspired transformation. This leads to an adaptive inference-time optimization method that normalizes rewards and learns stopping thresholds on the fly. Experiments on the AlpacaFarm and HH-RLHF datasets, using multiple LLM-reward model pairs, show that our adaptive strategy can obtain the same performance as non-adaptive Best-of-N sampling while requiring 15-35 percent fewer generations on average. Our results establish a principled bridge between optimal stopping theory and inference-time scaling, providing both theoretical performance bounds and practical efficiency gains for LLM deployment.

Hilal Asi, Vinod Raman, Kunal Talwar · apple-ml

We design new differentially private algorithms for the problems of adversarial bandits and bandits with expert advice. For adversarial bandits, we give a simple and efficient conversion of any non-private bandit algorithm to a private bandit algorithm. Instantiating our conversion with existing non-private bandit algorithms gives a regret upper bound of $O\left(\frac{\sqrt{KT}}{\sqrtε}\right)$, improving upon the existing upper bound $O\left(\frac{\sqrt{KT \log(KT)}}ε\right)$ for all $ε\leq 1$. In particular, our algorithms allow for sublinear expected regret even when $ε\leq \frac{1}{\sqrt{T}}$, establishing the first known separation between central and local differential privacy for this problem. For bandits with expert advice, we give the first differentially private algorithms, with expected regret $O\left(\frac{\sqrt{NT}}{\sqrtε}\right), O\left(\frac{\sqrt{KT\log(N)}\log(KT)}ε\right)$, and $\tilde{O}\left(\frac{N^{1/6}K^{1/2}T^{2/3}\log(NT)}{ε^{1/3}} + \frac{N^{1/2}\log(NT)}ε\right)$, where $K$ and $N$ are the number of actions and experts respectively. These rates allow us to get sublinear regret for different combinations of small and large $K, N$ and $ε.$

Seamus Somerstep, Vinod Raman, Unique Subedi et al.

Using the bit string generation problem as a case study, we theoretically compare two standard methods for adapting large language models to new tasks. The first, referred to as supervised fine-tuning, involves training a new next token predictor on good generations. The second method, Best-of-N, trains a reward model to select good responses from a collection generated by an unaltered base model. If the learning setting is realizable, we find that supervised fine-tuning outperforms BoN through a better dependence on the response length in its rate of convergence. If realizability fails, then depending on the failure mode, BoN can enjoy a better rate of convergence in either n or a rate of convergence with better dependence on the response length.

Aadirupa Saha, Vinod Raman, Hilal Asi

We design differentially private algorithms for the problem of prediction with expert advice under dynamic regret, also known as tracking the best expert. Our work addresses three natural types of adversaries, stochastic with shifting distributions, oblivious, and adaptive, and designs algorithms with sub-linear regret for all three cases. In particular, under a shifting stochastic adversary where the distribution may shift $S$ times, we provide an $ε$-differentially private algorithm whose expected dynamic regret is at most $O\left( \sqrt{S T \log (NT)} + \frac{S \log (NT)}ε\right)$, where $T$ and $N$ are the epsilon horizon and number of experts, respectively. For oblivious adversaries, we give a reduction from dynamic regret minimization to static regret minimization, resulting in an upper bound of $O\left(\sqrt{S T \log(NT)} + \frac{S T^{1/3}\log(T/δ) \log(NT)}{ε^{2/3}}\right)$ on the expected dynamic regret, where $S$ now denotes the allowable number of switches of the best expert. Finally, similar to static regret, we establish a fundamental separation between oblivious and adaptive adversaries for the dynamic setting: while our algorithms show that sub-linear regret is achievable for oblivious adversaries in the high-privacy regime $ε\le \sqrt{S/T}$, we show that any $(ε, δ)$-differentially private algorithm must suffer linear dynamic regret under adaptive adversaries for $ε\le \sqrt{S/T}$. Finally, to complement this lower bound, we give an $ε$-differentially private algorithm that attains sub-linear dynamic regret under adaptive adversaries whenever $ε\gg \sqrt{S/T}$.

Vinod Raman, Unique Subedi, Ambuj Tewari

We study online classification under smoothed adversaries. In this setting, at each time point, the adversary draws an example from a distribution that has a bounded density with respect to a fixed base measure, which is known apriori to the learner. For binary classification and scalar-valued regression, previous works \citep{haghtalab2020smoothed, block2022smoothed} have shown that smoothed online learning is as easy as learning in the iid batch setting under PAC model. However, we show that smoothed online classification can be harder than the iid batch classification when the label space is unbounded. In particular, we construct a hypothesis class that is learnable in the iid batch setting under the PAC model but is not learnable under the smoothed online model. Finally, we identify a condition that ensures that the PAC learnability of a hypothesis class is sufficient for its smoothed online learnability.

Vinod Raman, Daniel T. Zhang, Young Hun Jung et al.

We present online boosting algorithms for multilabel ranking with top-k feedback, where the learner only receives information about the top k items from the ranking it provides. We propose a novel surrogate loss function and unbiased estimator, allowing weak learners to update themselves with limited information. Using these techniques we adapt full information multilabel ranking algorithms (Jung and Tewari, 2018) to the top-k feedback setting and provide theoretical performance bounds which closely match the bounds of their full information counterparts, with the cost of increased sample complexity. These theoretical results are further substantiated by our experiments, which show a small gap in performance between the algorithms for the top-k feedback setting and that for the full information setting across various datasets.