Sham M. Kakade
Publications
From Interface to Inference: Eliciting Any-Order Inference from Any-Order Models
Many discrete reasoning tasks, such as code generation, are inherently non-causal: programmers move between high-level structure and local details, a process we call any-order inference. For autoregressive language models, which lack a native any-order interface, non-causal abilities such as infilling and next-edit prediction require hand-designed mechanisms. Can we instead design models that natively support any-order inference? Masked diffusion models have recently emerged as compelling candidates, as their any-order training objective naturally offers an any-order prediction interface. This interface, however, does not automatically yield any-order inference. We demonstrate that this interface-inference gap stems from positional uncertainty: fixed-canvas, token-level models may know what semantic component should appear without knowing where to place it. In light of this, we propose two complementary approaches: (1) Insertion-based masked diffusion, building on FlexMDM (Kim et al, 2025), relaxes fixed-position commitments via insertions, enabling generation across non-contiguous regions. (2) Latent-space masked diffusion shifts prediction to coarser semantic segments, enabling search over latent generation orders. Empirically, we train a 7B FlexMDM for Python coding and a 125M LatentMDM for GSM8K and show that both approaches induce distinct any-order inference behaviors and improve downstream performance. We release our codebase at https://github.com/SeunggeunKimkr/genuine-any-order.
Compute Efficiency and Serial Runtime Tradeoffs for Stochastic Momentum Methods
Stochastic momentum methods such as heavy ball (HB), Nesterov momentum, and variants of Accelerated SGD (ASGD) [Kidambi et al., 2018] are widely used in modern training, but their stochastic benefits depend on two distinct quantities: serial runtime, the number of iterations needed to reach a target accuracy, and compute efficiency (CE), the inverse total gradient-query or FLOP cost. Larger batches reduce serial runtime without hurting CE only when the contraction gap grows linearly with batch size. We study stochastic HB and ASGD for consistent linear regression with Gaussian covariates and prove finite-dimensional, discrete-time lower bounds on their batch-size tradeoffs. Our first result shows that HB does not improve the CE frontier over SGD for arbitrary spectra; rather, it preserves SGD-level CE over a larger batch-size window, allowing larger batches to reduce serial runtime until HB reaches its deterministic accelerated scale. This window can be a factor $\sqrtκ$ larger than the SGD critical batch size. For ASGD, the picture is more spectrum-dependent: for rapidly decaying power-law spectra, ASGD improves small-batch CE over HB/SGD, but as batch size grows it trades this CE advantage for improved serial runtime. Synthetic linear-regression experiments verify these qualitative regimes, including near-overlap of ASGD and HB for slowly decaying spectra and the predicted CE--serial tradeoff for rapidly decaying spectra.
Peer-Predictive Self-Training for Language Model Reasoning
Mechanisms for continued self-improvement of language models without external supervision remain an open challenge. We propose Peer-Predictive Self-Training (PST), a label-free fine-tuning framework in which multiple language models improve collaboratively by leveraging a cross-model aggregated response as an internal training signal. Given a prompt question, the models generate responses sequentially; the final aggregated answer, often more reliable than individual responses in practice, serves as an internal target for learning. We measure how informative each intermediate response is about the aggregate using pointwise mutual information (PMI), and use this signal to scale self-training updates. Responses already aligned with the aggregate are updated less, while less informative or misaligned responses are updated more. On mathematical reasoning benchmarks (SimulEq, Math500, and MultiArith), PST improves exact-match accuracy by 2.2 to 4.3 percentage points across Gemma-2-2B, LLaMA-3.2-1B, and Qwen-2.5-1.5B, and reduces the average generator-verifier gap (GV-Gap) by 26 to 40 percent, while requiring no external supervision or teacher-student hierarchy and relying solely on cross-model interactions. These results suggest that cross-model generations and peer-predictive feedback can serve as an effective approach for self-supervised training.
Evaluating Relational Reasoning in LLMs with REL
Relational reasoning is the ability to infer relations that jointly bind multiple entities, attributes, or variables. This ability is central to scientific reasoning, but existing evaluations of relational reasoning in large language models often focus on structured inputs such as tables, graphs, or synthetic tasks, and do not isolate the difficulty introduced by higher-arity relational binding. We study this problem through the lens of Relational Complexity (RC), which we define as the minimum number of independent entities or operands that must be simultaneously bound to apply a relation. RC provides a principled way to vary reasoning difficulty while controlling for confounders such as input size, vocabulary, and representational choices. Building on RC, we introduce REL, a generative benchmark framework spanning algebra, chemistry, and biology that varies RC within each domain. Across frontier LLMs, performance degrades consistently and monotonically as RC increases, even when the total number of entities is held fixed. This failure mode persists with increased test-time compute and in-context learning, suggesting a limitation tied to the arity of the required relational binding rather than to insufficient inference steps or lack of exposure to examples. Our results identify a regime of higher-arity reasoning in which current models struggle, and motivate re-examining benchmarks through the lens of relational complexity.