0 \\\\tilde{\\\\kappa}>0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each figure shows four panels in a single row, for one cell at a single N N . From left to right: empirical mean of W ~ t i j \\\\tilde{W}_{t}^{ij} under the deterministic initial condition W ~ 0 i j = W ¯ 0 i j \\\\tilde{W}_{0}^{ij}=\\\\bar{W}_{0}^{ij} ; empirical variance under the same initial condition; empirical mean under the Bernoulli initial condition W ~ 0 i j = A 0 i j ∼ Ber ( W ¯ 0 i j ) \\\\tilde{W}_{0}^{ij}=A_{0}^{ij}\\\\sim\\\\mathrm{Ber}(\\\\bar{W}_{0}^{ij}) ; empirical variance under the Bernoulli initial condition. Theoretical curves are overlaid: Ideal Cont. is the closed form of Lemma 1 ; Ideal Disc. is the closed form of Lemma 2 ; Ideal Bern. adds the Corollary 1 term to Ideal Disc. . Since β 0 i j = 0 \\\\beta_{0}^{ij}=0 for the SBM at even N N , Ideal Cont. and Ideal Disc. coincide.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intra-community cell ( W ¯ 0 i j = 0.7 \\\\bar{W}_{0}^{ij}=0.7 )\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inter-community cell ( W ¯ 0 i j = 0.2 \\\\bar{W}_{0}^{ij}=0.2 )\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"G.2 Scalability experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The scalability experiment evaluates whether DiPhon can generate graphs at node sizes outside the training distribution. To this end, we generate all datasets ourselves using standard graph generation tools such as networkx . For each graph family, we build a training set of 300 graphs, a validation set of 200 graphs, and multiple test sets of 100 graphs each. The test sets are generated across node sizes, increasing from the training regime up to the maximum size in steps of 20 nodes. We consider the following three synthetic graph families:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stochastic block model (SBM). The training graphs have node sizes ranging from 40 to 80 and are generated from a two-community SBM with within-community connection probability 0.4 0.4 and across-community connection probability 0.005 0.005 . This family is the most naturally aligned with the graphon setting, since it admits a clear piecewise-constant graphon representation. We do not consider graphs smaller than 40 nodes because, at that scale, it becomes less robust to assess whether a graph displays the expected SBM structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Preferential attachment (PA). The training graphs have node sizes ranging from 40 to 80. Although PA graphs do not admit a standard graphon representation directly, their line graphs do, which makes them a meaningful intermediate case. As with SBMs, we do not consider graphs smaller than 40 nodes because, for very small graphs, the characteristic structural signature of preferential attachment is less reliably assessed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tree. The training graphs have node sizes ranging from 20 to 80. Trees do not admit an associated graphon in the classical sense, so they provide a useful test of performance clearly outside the assumptions under which the theory is derived. In this case, smaller graphs can still be included because it is structurally easy to verify whether a graph is a tree.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The minimum graph size differs across datasets according to the corresponding accuracy metric. For SBM and PA graphs, both training and evaluation start at 40 40 nodes, since correctness is assessed through statistical tests that are less stable at very small graph sizes. For trees, we start at 20 20 nodes, because acyclicity is a deterministic property that can be reliably checked even at smaller sizes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For DiPhon, we use a constant-parameter Jacobi diffusion. This is the simplest implementation of the proposed forward process: it acts pointwise on edge probabilities, remains bounded in [ 0 , 1 ] [0,1] , and preserves the first-moment matching property of the graphon-level dynamics. We use this parametrization to isolate the effect of the proposed diffusion mechanism without adding extra modeling complexity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All methods were trained on the corresponding training datasets and carefully tuned before evaluating out-of-distribution size generalization. For each baseline, we performed an exhaustive grid search over the diffusion hyperparameters and selected configurations that achieved strong performance within the node-size regime observed during training. This ensures that failures at larger graph sizes reflect limitations in size extrapolation rather than poor in-distribution fitting. DiPhon was tuned under the same protocol, using only graphs whose sizes belonged to the training distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each test node size, we generated 100 100 graphs and report the corresponding accuracy. Thus, every point in Figure 3 is computed from 100 100 generated samples. A generated graph is counted as correct when it satisfies the defining structural property of the corresponding graph family. For trees, we use forest accuracy: a sample is considered correct if all its connected components are trees, so that nearly correct acyclic samples are not penalized only for missing a small number of edges. For SBMs and PAs we use statistical tests to assess whether the models are adequate, with details that can be found in the code repository.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For DiPhon, the reverse-time dynamics were simulated using one of three candidate numerical solvers. The solver was selected using only the training-size validation regime and then kept fixed for larger test sizes. We used Heun’s method for SBMs, where preserving block-level edge statistics is the main requirement. For PA graphs and trees, we used the Milstein integrator, which provided more stable extrapolation of structural constraints.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Graphs at different resolutions. We also provide qualitative visualizations of generated samples for the three datasets, comparing different graph sizes and baseline methods. These visualizations illustrate how the generated structures evolve as the target node size increases, and provide an additional qualitative assessment of size extrapolation. Consistently with the quantitative results, DiPhon produces the most faithful samples across datasets and node-size regimes, preserving the characteristic structure of SBMs, PA graphs, and trees more reliably than the competing methods. Some interesting aspects of the visualizations is that PA generates GruM obtains good performance metrics, but overestimates the number of edges with respect to the original data distribution. Also, it can be observed that DiPhon generates better trees for large node sizes, but sometimes it generates small disconnected trees.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work connects two lines of research that have so far been treated separately: diffusion-based generative models on graphs and the theory of graphons as the limit object of dense graph sequences. To the best of our knowledge, no previous diffusion model on graphs has placed the forward dynamics on graphon space. Prior works that use graphons in a generative setting either estimate the graphon itself [ 56 , 55 , 5 ] or summarize it through motifs that serve as a prior for flow matching on graphs [ 53 ] . The choice of the Jacobi process addresses a modelling issue that Gaussian SDEs leave open [ 23 , 22 ] , since it keeps the trajectory inside the edge-probability range by construction. The (approximate) moment matching ensures that, at the discrete level, we mimic the probability distributions of the continuous process, which is itself scale-free. The convergence in homomorphism density to the Erdős–Rényi reference provides a structural analogue of the role played by the Gaussian in score-matching approaches. Beyond the theoretical analysis, the scalability experiments show that a model trained on small graphs transfers its sampling distribution to substantially larger graphs without retraining.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Regarding the main limitations of the ideas presented in this paper, first of all, the theoretical analysis is developed in the dense regime, since this is the regime in which the graphon is the canonical limit object of a sequence of growing graphs. This restriction belongs to the underlying mathematical object and not to the model. Empirically, the scalability experiments include sparse graph families as well, and the model transfers stably to large sizes in that regime, which suggests that the empirical scope of the approach exceeds that of its formal justification. A second limitation is that the equivalence between the two discretization routes, diffuse-then-discretize and discretize-then-diffuse, is established at the level of the first two moments. The second moment is controlled through a closed-form discrepancy that vanishes in the stationary regime, but the full distance between the marginal distributions is not bounded. For generative sampling it is plausible that this distance matters more than the first two moments, and characterizing it lies outside the scope of this work. Finally, the denoising loss requires simulating a forward trajectory per sample at every training step, since the transition of the renormalized Jacobi process does not admit a closed-form density that would allow direct sampling at arbitrary times. This introduces a training overhead with respect to discrete and Gaussian-based approaches whose transition is tractable. We also note that this work focuses on the generation of graphs with binary edges. From a practical standpoint, however, the framework can be naturally extended in several directions: i) node features can be modeled using the same class of Jacobi dynamics; ii) multiple edge categories can be handled through a stick-breaking construction [ 4 ] ; and iii) weighted graphs can be accommodated by applying bounded continuous diffusions to edge weights. These extensions are outside the scope of the present work and should be understood as modeling possibilities rather than fundamental limitations. However, the theoretical guarantees developed in this paper are specific to the binary-edge graphon setting and do not directly carry over to these variants.\"\n }\n]","isConversational":false}}},{"rowIdx":481,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07209"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Continual Learning With Participation Privacy: An Auditable Buffering-Aggregation Recipe"},"authors":{"kind":"list like","value":["T-H. Hubert Chan","Elaine Shi","Mengshi Zhao","Mingxun Zhou"],"string":"[\n \"T-H. Hubert Chan\",\n \"Elaine Shi\",\n \"Mengshi Zhao\",\n \"Mingxun Zhou\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07209v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Modern federated and streaming learning systems release intermediate model snapshots throughout training, so an adversary may observe the entire trajectory ( w ( t ) ) t ≥ 0 (w^{(t)})_{t\\geq 0} rather than only a final model. A canonical instance is streaming stochastic gradient descent (SGD): at step t t , a user contributes a private loss function f t f_{t} and the learner updates using a gradient step based on the current model [ TS13 , KMS+21 ] . Beyond protecting the data within each f t f_{t} , some deployments also require participation privacy : a user may wish to ensure plausible deniability of whether they participated at all. This motivates single-edit (edit-distance) neighboring streams, where two user-update streams are neighbors if one can be obtained from the other by inserting or deleting a single user event."},{"type":"para","text":"Continual release also makes the learning process intrinsically interactive: the released model w ( t ) w^{(t)} influences what happens next, so the update stream can be adaptively generated in response to prior outputs. While classical continual-release primitives such as tree-based private prefix sums [ DNP+10a , CSS11 ] were originally analyzed under statically chosen (Hamming-neighbor) streams, recent work formalizes differential privacy against such adaptive interaction [ JRS+23 ] . Our goal is to make this adaptive viewpoint compatible with the single-edit participation model above."},{"type":"para","text":"The difficulty is that Hamming-style neighboring streams are not the right abstraction for participation privacy on a stream. Most continual DP analyses change the value at a single time index t 0 t_{0} while keeping all other positions aligned [ DWO06 ] . In contrast, a single insertion/deletion shifts the alignment of all subsequent positions, so an edit-neighbor pair can be far from Hamming-neighboring. This also undermines deterministic batching: partitioning the stream into fixed contiguous blocks can cause one edit to shift many later batch boundaries, so standard Hamming-neighbor continual DP guarantees do not directly transfer."},{"type":"para","text":"To address this mismatch, we use a simple modular recipe: randomized buffering and an adaptive-safety certification for downstream continual primitives. We apply a randomized buffering wrapper [ CCM+22 , ZSC+23 ] that introduces random delay and releases user updates in bins of controlled size (e.g., in [ U , 2 U ] [U,2U] ); here U U is a privacy-implied systems cost , calibrated by the target ( ε , δ ) (\\varepsilon,\\delta) and inducing delay. We then invoke standard continual DP primitives (e.g., tree/prefix-sum) to privately aggregate the resulting vector updates, and prove a certification theorem showing when a non-adaptive Hamming-neighbor privacy proof remains valid under feedback. The certification requires fresh per-round randomness together with a stable-context condition: common adaptive updates may change the released statistic, but they must not change the privacy cost of the unique Hamming discrepancy. 1 1 1 An earlier version [ CSZ+26 ] stated the certification condition using independent decomposability alone. Here we use the slightly stronger stable-context condition, formalized in Appendix E as quantitative decomposability. This refinement makes the statement technically precise and is satisfied by the certified cores of the continual-release primitives used in our pipeline."},{"type":"para","text":"Contributions."},{"type":"para","text":"Edit-neighbor continual learning under feedback. We formalize participation privacy via single-edit neighboring user-update streams in a continual learning setting with intermediate releases."},{"type":"para","text":"Modular pipeline with an explicit privacy to latency link. We combine randomized buffering with standard continual DP primitives to obtain trajectory-level privacy for edit-neighbor streams; the target ( ε , δ ) (\\varepsilon,\\delta) determines the buffering level U U , making induced delay explicit."},{"type":"para","text":"Certification theorem for adaptive safety of continual DP primitives. We give a checkable certification condition under which a non-adaptive Hamming-neighbor privacy analysis carries over to the feedback/adaptive setting. The condition combines fresh per-round randomness with a stable-context requirement, and certifies safe reuse of canonical primitives such as tree/prefix-sum mechanisms."},{"type":"para","text":"Results and organization. Our end-to-end guarantee is trajectory-level ( ε , δ ) (\\varepsilon,\\delta) -DP for single-edit neighboring participation streams under feedback, obtained by composing randomized buffering with a continual-release DP aggregator (Theorem 3.1 ); the privacy target explicitly induces a buffering level U ( ε , δ ) U(\\varepsilon,\\delta) and hence delay/staleness. A second ingredient is a reusable certification theorem showing that, for continual primitives satisfying our fresh-randomness and stable-context conditions, a standard non-adaptive privacy analysis lifts to the adaptive/feedback setting (Theorem 4.3 ). Section 2 formalizes the interactive model, edit adjacency, and metrics; Section 3 gives a high-level view of the pipeline and states the main theorem; Section 4 develops the two auditable ingredients (randomized buffering and certification) and combines them via modular composition; Section 5 instantiates the recipe with continual DP-SGD (tree/prefix-sum aggregation). Technical proofs are deferred to the appendix. The relevant experiments are given in [ CSZ+26 ] ."},{"type":"heading","level":2,"text":"1.1 Related Work"},{"type":"para","text":"Continual-Release DP. Canonical continual-release primitives for streaming aggregates include tree-based private prefix sums [ DNP+10a , CSS11 ] and closely related lower-triangular (matrix-style) mechanisms [ DMR+22 ] . These mechanisms are typically analyzed for Hamming-neighboring update streams, where only one time index differs. In contrast, our participation model uses single-edit (insertion/deletion) neighboring streams, where alignment shifts after the edit; this gap is mild for static datasets [ BEB+24 ] but is fundamental for streams."},{"type":"para","text":"Sliding-window variants of private stream release have also been studied for window aggregate queries and graph streams [ CXG+13 , UUA21 ] . These works target recent-window utility, whereas our setting focuses on single-edit participation privacy, where one insertion/deletion shifts all subsequent stream positions and requires an edit-to-Hamming shielding interface."},{"type":"para","text":"Adaptive Streams and Feedback. Continual learning is inherently interactive: released outputs can influence future updates. Recent work formalizes differential privacy against such adaptive interaction (e.g., left-or-right style games and verification viewpoints) [ DMR+22 , JRS+23 ] . Our certification result is complementary: it gives a checkable structural condition under which a standard non-adaptive Hamming-DP analysis of a continual primitive lifts directly to the adaptive setting. The condition is stronger than fresh prefix-causal randomness alone: it also requires that common adaptive context does not change the one-round privacy cost of the unique Hamming discrepancy."},{"type":"para","text":"Privacy-Induced Delayed Reactions. [ CLN+24 ] give a conceptually related delayed-reaction phenomenon for continual observation and online threshold queries. Their Appendix B delays threshold-type outputs on a directly observed online bit stream. Our delay has a different technical role: 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} waits for future arrivals to form bins in [ U , 2 U ] [U,2U] so that a single insertion/deletion, which appears only in the neighboring-stream analysis, becomes a bounded Hamming-style perturbation while the full transcript, including timing and ⊥ \\bot outputs, remains protected."},{"type":"para","text":"Randomized Buffering, RSC, and Obliviousness Lineage. Randomized buffering has appeared as infrastructure in oblivious data structures and related privacy modularity work [ CCM+22 , ZSC+23 ] . A closely related random-partition idea also appears in the Reorder-Slice-Compute (RSC) paradigm of [ CLN+23 ] , where random slicing is used to synchronize neighboring executions and avoid a one-edit domino effect. Our setting differs in that the continual streaming transcript exposes release timing itself: the adversary observes whether each step emits a real block or the dummy symbol ⊥ \\bot . Thus, in our use, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} is an interactive interface that converts single-edit participation streams into a sparse, Hamming-style per-bin update stream while also providing explicit backlog/delay guarantees calibrated by ( ε , δ ) (\\varepsilon,\\delta) ."},{"type":"para","text":"Composition and Concurrency. Beyond classical ( ε , δ ) (\\varepsilon,\\delta) -DP [ DWO06 ] , alternative formalisms (e.g., Rényi divergence and tradeoff-based views) can streamline composition reasoning [ MIR17 , DRS22 , VZ23 , ZZC+24 ] , recovering familiar advanced composition bounds [ DRV10b , KOV15 ] . Concurrent composition in interactive settings has also been studied, including adaptively chosen privacy parameters [ HST+23 , HSV26 ] . These works typically assume each mechanism’s neighboring relation is defined on a single underlying (static or dynamic) database; in our pipeline, we additionally need to reason about neighbor-preserving transformations between components (edit-to-Hamming shielding), which motivates our modular composition viewpoint. Our modular composition theorem can be viewed as combining the refinement-pair reduction of NPDP to standard DP in [ ZZC+24 ] with the interactive/concurrent composition framework of [ HSV26 ] ; for completeness and to match our power-function accounting, we give a self-contained proof in Appendix C ."},{"type":"heading","level":1,"text":"2 Setting and Preliminaries"},{"type":"para","text":"We now fix the interactive streaming-learning model underlying continual release: the adversary observes both the released model trajectory and the release timing, and may influence future user events. We also define edit-style (participation) adjacency and the utility–latency metrics used throughout. With these conventions in place, Section 3 summarizes our buffering–aggregation pipeline and main guarantee."},{"type":"heading","level":4,"text":"Streaming Learning Loop and Transcript."},{"type":"para","text":"At each discrete step t = 1 , 2 , … t=1,2,\\ldots , an adaptive environment produces a user event f t f_{t} (e.g., a private loss or update). The server maintains a model w ( t ) ∈ ℝ d w^{(t)}\\in\\mathbb{R}^{d} and may release intermediate snapshots. To unify per-step and buffered releases, we include an observable timing bit v t ∈ { 0 , 1 } v_{t}\\in\\{0,1\\} : if v t = 1 v_{t}=1 the server incorporates available (buffered) events to form an update g t g_{t} and publishes the new snapshot; if v t = 0 v_{t}=0 it holds w ( t ) = w ( t − 1 ) w^{(t)}=w^{(t-1)} . The adversary’s view up to horizon T T is the transcript 𝗍𝗋 ≤ T := ( ( v t , w ( t ) ) ) t = 0 T , \\mathsf{tr}_{\\leq T}\\;:=\\;\\bigl((v_{t},w^{(t)})\\bigr)_{t=0}^{T}, and our privacy notion is defined with respect to this joint transcript (including timing). All privacy guarantees are required to hold for every horizon T T , and thus apply to the full trajectory."},{"type":"heading","level":4,"text":"Feedback/Adaptive Interaction."},{"type":"para","text":"The environment (and thus the event stream) may depend on the past transcript: formally, f t f_{t} may be chosen as an arbitrary (possibly randomized) function of 𝗍𝗋 < t \\mathsf{tr}_{<t} . All guarantees quantify over such adaptive interaction. When we refer to a non-adaptive privacy analysis for a continual primitive, we mean its input stream is fixed in advance (independent of past releases)."},{"type":"heading","level":4,"text":"Participation Privacy via Single-Edit Adjacency."},{"type":"para","text":"Let ℳ \\mathcal{M} denote the full interactive mechanism (e.g., buffering → \\rightarrow aggregation → \\rightarrow releases) mapping an event stream F = ( f 1 , f 2 , … ) F=(f_{1},f_{2},\\ldots) to a transcript. We use single-edit adjacency: F ∼ edit F ′ F\\sim_{\\mathrm{edit}}F^{\\prime} if F ′ F^{\\prime} is obtained from F F by inserting or deleting exactly one event, shifting subsequent indices; equivalently, for some t 0 t_{0} , either (i) f t ′ = f t f^{\\prime}_{t}=f_{t} for t < t 0 t<t_{0} and f t ′ = f t + 1 f^{\\prime}_{t}=f_{t+1} for all t ≥ t 0 t\\geq t_{0} (deletion), or (ii) the reverse (insertion). The unit of privacy is one participation event; other units (e.g., a session spanning multiple events) reduce via grouping/composition (formal composition tools appear in Appendix A )."},{"type":"heading","level":4,"text":"Differential Privacy (Under Adaptive Interaction)."},{"type":"para","text":"We say ℳ \\mathcal{M} is ( ε , δ ) (\\varepsilon,\\delta) -DP w.r.t. ∼ edit \\sim_{\\mathrm{edit}} if for all F ∼ edit F ′ F\\sim_{\\mathrm{edit}}F^{\\prime} and all measurable transcript-sets S S ,"},{"type":"para","text":"where the probability is over ℳ \\mathcal{M} ’s internal randomness; the quantification over event-generation strategies is captured by the adaptive-interaction model above. Our appendix sometimes uses the equivalent tradeoff-/power-function viewpoint to streamline interactive composition; the main text states guarantees in ( ε , δ ) (\\varepsilon,\\delta) -DP; see Appendix A ."},{"type":"heading","level":4,"text":"Metrics."},{"type":"para","text":"Utility: test accuracy / loss of the final released model by time T T (e.g., w ( T ) w^{(T)} or the last release by T T ). Systems: backlog Q t Q_{t} (pending, not-yet-incorporated events after time t t ) and inclusion delay D ( i ) := min { t ≥ i : f i is incorporated by time t } − i D(i):=\\min\\{t\\geq i:\\ f_{i}\\text{ is incorporated by time }t\\}-i , measured in event arrivals."},{"type":"heading","level":4,"text":"Extension: From One Edit to k k Edits."},{"type":"para","text":"If neighbors may differ by up to k k insertions/deletions, standard group-privacy/composition yields a k k -dependent degradation of ( ε , δ ) (\\varepsilon,\\delta) ; in our pipeline this correspondingly increases the required privacy budgets and thus the induced buffering/latency. Full statements are deferred to the appendix."},{"type":"heading","level":1,"text":"3 Overview"},{"type":"para","text":"This section summarizes our buffering–aggregation pipeline (Fig. 1 ) and the resulting end-to-end guarantee: continual-release ( ε , δ ) (\\varepsilon,\\delta) -DP for single-edit participation streams under feedback, where the adversary observes the released trajectory (and timing). The required buffering level U ( ε , δ ) U(\\varepsilon,\\delta) is privacy-implied and explicitly governs delay."},{"type":"para","text":"We next give the recipe and state the end-to-end guarantee."},{"type":"heading","level":4,"text":"Recipe (buffering → \\rightarrow aggregation → \\rightarrow releases)."},{"type":"para","text":"Given user-event stream ( f t ) (f_{t}) , at each step t ≥ 1 t\\geq 1 , buffering outputs B t B_{t} with | B t | ∈ [ U , 2 U ] |B_{t}|\\in[U,2U] or ⊥ \\bot . Let τ ( 1 ) < τ ( 2 ) < ⋯ \\tau(1)<\\tau(2)<\\cdots be the (random) steps with B τ ( k ) ≠ ⊥ B_{\\tau(k)}\\neq\\bot ."},{"type":"para","text":"Buffer. Run randomized buffering with U ( ε , δ ) U(\\varepsilon,\\delta) to obtain ( B t ) t ≥ 1 (B_{t})_{t\\geq 1} ."},{"type":"para","text":"Update. For each k ≥ 1 k\\geq 1 , form a clipped per-bin update g k := 1 | B τ ( k ) | ∑ f ∈ B τ ( k ) Clip G ( ∇ f ( w ( τ ( k ) − 1 ) ) ) g_{k}:=\\frac{1}{|B_{\\tau(k)}|}\\sum_{f\\in B_{\\tau(k)}}\\textsf{Clip}_{G}\\Bigl(\\!\\nabla f\\!\\bigl(w^{(\\tau(k)-1)}\\bigr)\\Bigr) , where Clip G ( x ) := x ⋅ min { 1 , G ‖ x ‖ 2 } \\textsf{Clip}_{G}(x)~:=~x\\cdot\\min\\!\\left\\{1,\\ \\frac{G}{\\|x\\|_{2}}\\right\\} ."},{"type":"para","text":"Aggregate + release. Feed ( g k ) k ≥ 1 (g_{k})_{k\\geq 1} to a certified continual-release DP primitive and update only at bin times: output w ( τ ( k ) ) w^{(\\tau(k))} after processing g k g_{k} , and hold w ( t ) = w ( τ ( k ) ) w^{(t)}=w^{(\\tau(k))} for τ ( k ) ≤ t < τ ( k + 1 ) \\tau(k)\\leq t<\\tau(k{+}1) (feedback allowed)."},{"type":"para","text":"Theorem 3.1 states the resulting end-to-end privacy."},{"type":"para","text":"Run the above recipe with privacy target ( ε , δ ) (\\varepsilon,\\delta) , allocating ( ε b , δ b ) (\\varepsilon_{b},\\delta_{b}) to buffering and ( ε a , δ a ) (\\varepsilon_{a},\\delta_{a}) to aggregation. Set U = Θ ( 1 ε b log 1 δ b ) U=\\Theta\\!\\Bigl(\\frac{1}{\\varepsilon_{b}}\\log\\frac{1}{\\delta_{b}}\\Bigr) , and plug in any continual-release DP primitive that is adaptively ( ε a , δ a ) (\\varepsilon_{a},\\delta_{a}) -DP for Hamming-neighboring update streams. Then the released model trajectory ( w ( t ) ) t ≥ 0 (w^{(t)})_{t\\geq 0} is ( ε , δ ) (\\varepsilon,\\delta) -DP with respect to single-edit neighboring event streams, even when future events are generated adaptively from past releases (feedback)."},{"type":"para","text":"Proof idea. Randomized buffering turns a single insertion/deletion on the event stream into a bounded Hamming-style perturbation on the emitted update stream; we then apply a certified continual-release DP primitive whose non-adaptive analysis remains valid under feedback, and conclude via modular composition (Appendix C )."},{"type":"heading","level":4,"text":"Certification preview (adaptive safety of continual primitives)."},{"type":"para","text":"We use a reusable certification for feedback: writing x t x_{t} for the per-round input and x 1 : t x_{1:t} for its prefix, a continual-release primitive is independently decomposable if at each release time t t it samples fresh independent randomness R t R_{t} and outputs a deterministic function of ( x 1 : t , R t ) (x_{1:t},R_{t}) . For such primitives, standard non-adaptive ( ε , δ ) (\\varepsilon,\\delta) -DP analyses remain valid under feedback. This covers tree/prefix-sum (Appendix F ) and more general lower-triangular matrix mechanisms (Section 4.2 )."},{"type":"heading","level":4,"text":"SGD instantiation and what we measure."},{"type":"para","text":"We instantiate the pipeline with streaming SGD: each emitted bin B τ ( k ) B_{\\tau(k)} triggers one clipped mini-batch gradient update, privatized by a continual-release DP aggregator (tree/prefix-sum in our experiments). The target ( ε , δ ) (\\varepsilon,\\delta) fixes U ( ε , δ ) U(\\varepsilon,\\delta) , yielding random batch sizes in [ U , 2 U ] [U,2U] and a backlog bound of order O ( 1 ε b log t ( log t + log 1 δ b ) ) O\\!\\bigl(\\tfrac{1}{\\varepsilon_{b}}\\log t\\,(\\log t+\\log\\tfrac{1}{\\delta_{b}})\\bigr) (Lemma 4.2 ). We report accuracy versus ( ε , δ ) (\\varepsilon,\\delta) together with empirical delay distributions implied by U ( ε , δ ) U(\\varepsilon,\\delta) . The theoretical convergence of various SGD variants under DP noise has already been extensively analyzed; we refer the reader to standard works [ JKT12 , SCS13 , TS13 , KMS+21 , DMR+22 ] ."},{"type":"heading","level":4,"text":"Deferred details."},{"type":"para","text":"Sections 4.1 and 4.2 formalize buffering and certification; the end-to-end proof of Theorem 3.1 (via modular composition) and additional instantiations/experiments appear in the appendix."},{"type":"heading","level":1,"text":"4 Auditable Ingredients: Randomized Buffering and Adaptive-Safe Continual-Release DP"},{"type":"para","text":"This section isolates the two ingredients that make the pipeline in Fig. 1 auditable. First, we treat randomized buffering ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} ) as an interface: it converts a single-edit event stream into a sparse per-bin update stream that admits a standard Hamming-neighbor DP analysis, while providing an explicit backlog guarantee. Second, we give a reusable certification theorem for adaptive interaction (output-dependent updates): for independently decomposable (prefix-causal) continual-release mechanisms, a non-adaptive DP proof remains valid even when updates are chosen adaptively as a function of past releases."},{"type":"heading","level":4,"text":"Separation of Roles."},{"type":"para","text":"Randomized buffering is not needed for the certification theorem itself: Theorem 4.3 is a reusable adaptivity lift for independently decomposable continual mechanisms. In our pipeline, buffering serves a different role: it shields the downstream primitive from single-edit adjacency, where one insertion or deletion can shift many later batch boundaries. Thus, our modular proof separates an upstream edit-to-Hamming interface from a downstream adaptive-safety certificate. We do not claim that randomized buffering is the only possible route; alternatives would require either another edit-shielding interface or a direct native proof under edit adjacency."},{"type":"heading","level":2,"text":"4.1 Randomized Buffering Wrapper"},{"type":"heading","level":4,"text":"RandBin Interface."},{"type":"para","text":"We use 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} as a lightweight wrapper that turns an incoming event stream into well-sized bins, while keeping the buffering backlog under control and enabling our edit-to-Hamming reduction downstream. The input is a stream of user events f 1 , f 2 , … f_{1},f_{2},\\ldots (losses, gradients, or updates). At each step t ≥ 1 t\\geq 1 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} outputs a symbol B t B_{t} : either a concrete bin B t B_{t} containing some pending events among those arrived by time t t , or ⊥ \\bot (no emission). Let τ ( 1 ) < τ ( 2 ) < ⋯ \\tau(1)<\\tau(2)<\\cdots be the (random) emission times with B τ ( k ) ≠ ⊥ B_{\\tau(k)}\\neq\\bot . When B t = ⊥ B_{t}=\\bot , nothing is sent to the downstream aggregator and the released model remains unchanged."},{"type":"para","text":"We rely on three interface properties:"},{"type":"para","text":"Bin size. Whenever B t ≠ ⊥ B_{t}\\neq\\bot , the bin size is controlled: | B t | ∈ [ U , 2 U ] |B_{t}|\\in[U,2U] ."},{"type":"para","text":"Backlog control. The number of pending (unbinned) events Q t Q_{t} remains small for all t t (formal bound below), yielding explicit delay guarantees."},{"type":"para","text":"Edit-to-Hamming wrapper: neighbor-preserving (NP) DP. A single insertion/deletion in the input stream induces only a bounded Hamming-style perturbation in the emitted-bin (and hence per-bin update) stream, in the sense needed to compose with a Hamming-DP continual primitive. The details are deferred to Appendix D ."},{"type":"heading","level":4,"text":"Input-Identifiability Convention."},{"type":"para","text":"Lemma 4.1 uses the standard convention that events are identifiable, e.g., by unique IDs or metadata tags, so that after the first mismatch in an edit-neighbor pair the simulation can determine which stream contains the extra event. Anonymous streams with duplicate values require an extra disambiguation assumption; see Appendix D ."},{"type":"heading","level":4,"text":"Systems Metrics: Backlog and Delay."},{"type":"para","text":"The buffering level U ( ε b , δ b ) U(\\varepsilon_{b},\\delta_{b}) controls how aggressively 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} batches the stream and thus the latency regime seen by learning. We track the backlog Q t Q_{t} , the number of pending (unbinned) events after time t t . For an event f i f_{i} , its inclusion delay D ( i ) D(i) is the number of subsequent arrivals until it is placed in an emitted bin. In particular, if Q t ≤ B ( t ) Q_{t}\\leq B(t) for all t ∈ [ i , T ] t\\in[i,T] , then D ( i ) ≤ B ( T ) D(i)\\leq B(T) (and sharper bounds follow from the trajectory of ( Q t ) (Q_{t}) )."},{"type":"heading","level":4,"text":"Interface Guarantees for Modular Composition."},{"type":"para","text":"We will use two interface properties of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} ; proofs and constants appear in Appendix D ."},{"type":"para","text":"For any single-edit neighboring input streams, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} admits a neighbor-preserving paired simulation (with refinement) such that the coupled emitted-bin/update streams differ in a bounded Hamming-style way at the interface required for modular composition with a Hamming-DP continual primitive."},{"type":"para","text":"Intuition: randomized boundaries prevent one insertion/deletion from cascading into many downstream bin shifts; the paired simulation formalizes this stability under interaction."},{"type":"para","text":"With U = Θ ( ε b − 1 log ( 1 / δ b ) ) U=\\Theta(\\varepsilon_{b}^{-1}\\log(1/\\delta_{b})) , for every time t t the backlog satisfies Q t = O ( 1 ε b log t ⋅ ( log t + log 1 δ b ) ) . Q_{t}\\;=\\;O\\!\\Bigl(\\frac{1}{\\varepsilon_{b}}\\log t\\cdot\\Bigl(\\log t+\\log\\frac{1}{\\delta_{b}}\\Bigr)\\Bigr)."},{"type":"para","text":"This bound is obtained by implementing 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} ’s private emission scheduler via the streaming DP prefix-sum mechanism (Theorem F.1 ) and translating its additive-error guarantee into a worst-case queue bound. As a consequence, inclusion delay is controlled at the same order (up to constants) for events arriving by time t t ."},{"type":"para","text":"Intuition: the noisy scheduling forces sufficiently frequent emissions to prevent sustained queue growth, while keeping bin sizes within [ U , 2 U ] [U,2U] ."},{"type":"heading","level":4,"text":"On the Horizon Dependence."},{"type":"para","text":"The log t \\log t dependence in Lemma 4.2 is not merely an artifact of loose algebra. In our implementation, the emission scheduler is a continual private prefix-sum mechanism, and its additive error is translated directly into backlog. On a unit-rate stream, if M t M_{t} is the cumulative number of emitted items by time t t , then Q t = t − M t Q_{t}=t-M_{t} . Thus, within this scheduler family, improving the backlog bound would amount to improving the corresponding continual private counting error. Known continual-counting lower bounds suggest that logarithmic horizon dependence is intrinsic to this proof route [ DNP+10a , CLN+24 ] , although we do not claim a minimax lower bound for all possible NPDP wrappers."},{"type":"heading","level":4,"text":"Scope of the Approximate-DP Regime."},{"type":"para","text":"The bounded-delay guarantee in Lemma 4.2 is tied to the approximate-DP implementation of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} : the proof uses bounded-support buffering noise and an always-valid prefix-sum error bound. For RandBin-style shift masking, pure DP would require unbounded-support noise, and therefore cannot give the same probability-one finite-delay guarantee."},{"type":"heading","level":4,"text":"Positioning."},{"type":"para","text":"Randomized buffering is often used in oblivious data structures to hide access patterns [ CCM+22 , ZSC+23 ] . Our use is different in emphasis: 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} serves as an infrastructure interface that (i) converts single-edit participation streams into a sparse, Hamming-style stream of per-bin updates and (ii) exposes explicit backlog guarantees. Once Lemmas 4.1 – 4.2 hold, the downstream component becomes plug-and-play: any continual primitive that passes our certification in Section 4.2 can be safely reused. Full pseudocode, the NPDP security game (paired simulation + refinement), and sharper tail bounds are deferred to Appendix D ."},{"type":"heading","level":2,"text":"4.2 Certifying Feedback-Safe Continual-Release Primitives"},{"type":"heading","level":4,"text":"Why Certification?"},{"type":"para","text":"Our pipeline treats the continual-release DP primitive as a plug-in component. To make this modularity auditable , we need a certification criterion showing when a standard non-adaptive Hamming-neighbor DP analysis continues to hold when per-round updates are chosen adaptively from past releases. Fresh independent randomness is necessary for this transfer, but not sufficient by itself; Appendix E.1 gives a two-round counterexample. The criterion below therefore combines fresh randomness with a quantitative decomposability condition that prevents adaptive common context from changing the one-round leakage profile."},{"type":"heading","level":4,"text":"Certified Decomposability."},{"type":"para","text":"Write x t x_{t} for the input update at round t t and x 1 : t := ( x 1 , … , x t ) x_{1:t}:=(x_{1},\\dots,x_{t}) . The certification condition has two parts. First, the mechanism must use fresh independent randomness at each round: the release at time t t is generated as"},{"type":"para","text":"where R t R_{t} is independent of the randomness used at other rounds and ℳ t \\mathcal{M}_{t} is deterministic given ( R t , x 1 : t ) (R_{t},x_{1:t}) ."},{"type":"para","text":"Second, the privacy cost of a one-round release must be stable under common context. Informally, if two input histories differ in one update, then the effect of this discrepancy on the round- t t release is determined only by the part of the round- t t aggregate that contains the changed update. Common updates before or after the discrepancy may change the released statistic, but they are shared by both neighboring histories and therefore should not change the privacy cost of the discrepancy itself."},{"type":"para","text":"The formal appendix condition, called quantitative decomposability, rules out this behavior by requiring the one-round privacy profile to depend only on the distance between the relevant aggregates after common aggregation context is removed."},{"type":"para","text":"Importantly, after producing the output y 1 : t y_{1:t} at time t t , the mechanism may apply arbitrary post-processing to ( y 1 , … , y t ) (y_{1},\\dots,y_{t}) and output any derived value at time t t ; by closure under post-processing, this does not weaken the privacy guarantee proved for the underlying releases."},{"type":"para","text":"Checklist 1 (Certification Conditions). A continual-release primitive ℳ \\mathcal{M} passes certification if its core release process satisfies the following conditions: 1. Prefix dependence: the core release at time t t depends only on the input prefix x 1 : t x_{1:t} , not on future inputs. 2. Fresh randomness: the core release at time t t uses randomness R t R_{t} that is independent of the randomness used in all other rounds. 3. Deterministic release map: given ( R t , x 1 : t ) (R_{t},x_{1:t}) , the core release y t = ℳ t ( R t , x 1 : t ) y_{t}=\\mathcal{M}_{t}(R_{t},x_{1:t}) is deterministic. 4. Stable one-round privacy cost: if two prefixes differ in one update, then common aggregation context in the two prefixes does not change the privacy cost of that discrepancy. The appendix formalizes this condition as quantitative decomposability in Definition E.3 . 5. Post-processing only afterward: after the certified core releases are produced, the mechanism may output any function of the released transcript; this is post-processing and does not weaken the privacy guarantee. Consequence: any finite-horizon non-adaptive ( ε , δ ) (\\varepsilon,\\delta) -DP proof for Hamming-neighboring update streams lifts to adaptive interaction."},{"type":"heading","level":4,"text":"Certification Theorem."},{"type":"para","text":"The next theorem formalizes the consequence of Checklist 1 and serves as an audit tool for reusing continual DP primitives inside interactive learning loops. The formal appendix statement is given in terms of power functions; the version below states the resulting ( ε , δ ) (\\varepsilon,\\delta) -DP guarantee."},{"type":"para","text":"Consider Hamming-style neighboring update streams, where two equal-length streams differ in at most one position. Let ℳ \\mathcal{M} be a continual primitive satisfying Checklist 1. If ℳ \\mathcal{M} is non-adaptively ( ε , δ ) (\\varepsilon,\\delta) -DP for Hamming-neighboring update streams for every finite horizon, then ℳ \\mathcal{M} is also ( ε , δ ) (\\varepsilon,\\delta) -DP against adaptive interaction for every finite horizon, where each x t x_{t} may be chosen as an arbitrary function of past releases."},{"type":"heading","level":4,"text":"Proof Sketch."},{"type":"para","text":"(Full proof in Appendix E ) The proof works in the paired-simulation game and proceeds by induction on the horizon. If the first input pair already contains the unique Hamming discrepancy, then all later paired inputs must be equal componentwise. Checklist 1 ensures that these later common inputs only add common context and do not change the one-round privacy cost of the discrepancy; the per-round privacy losses then tensorize exactly. If the first input pair is equal, we condition on the common first release, fix the first input as common context, and apply the induction hypothesis to the shifted mechanism on the remaining rounds. Fresh independent randomness is used to separate the per-round contributions, while the stable-context condition prevents adaptive common inputs from amplifying the leakage of the unique discrepancy."},{"type":"heading","level":4,"text":"Mini-Audit 1: Tree / Prefix-Sum."},{"type":"para","text":"For the tree mechanism, the certified core is the stream of newly revealed noisy tree-node values. Each such node value depends only on the updates in its interval and on its own fresh noise. If two update streams differ in one coordinate, then only the tree nodes whose intervals contain that coordinate see different sums; all other contributions are common to the two executions. Thus the one-round privacy cost is determined only by the changed update’s contribution to the affected node, not by the surrounding common updates. The published prefix sum at time t t is a deterministic function of the noisy node values revealed so far, hence post-processing. Therefore the usual non-adaptive Hamming-neighbor DP analysis for tree prefix sums carries over to adaptive update choices by Theorem 4.3 ."},{"type":"heading","level":4,"text":"Mini-Audit 2: Lower-Triangular (Prefix) Linear Mechanisms."},{"type":"para","text":"Many continual primitives can be written as a certified core followed by linear post-processing. Concretely, suppose the core releases noisy statistics"},{"type":"para","text":"where the noises η t \\eta_{t} are fresh and independent across t t . Each core release depends only on the input prefix x 1 : t x_{1:t} and its own fresh noise. If two update streams differ in one coordinate, then the two linear statistics differ only through that coordinate’s contribution; all common updates contribute equally to both executions. Thus the one-round privacy cost is stable under common context."},{"type":"para","text":"Any published output of the form y t = ∑ s ≤ t A t , s z s y_{t}=\\sum_{s\\leq t}A_{t,s}\\,z_{s} is a deterministic function of ( z 1 , … , z t ) (z_{1},\\dots,z_{t}) and therefore post-processing. Hence any non-adaptive Hamming-neighbor DP bound for the core stream ( z t ) (z_{t}) remains valid under adaptive interaction, and the same holds for the released stream ( y t ) (y_{t}) by Theorem 4.3 . For details, see Appendix F.1 ."},{"type":"heading","level":4,"text":"Scope Boundary."},{"type":"para","text":"Theorem 4.3 is a sufficient certification condition, not a characterization of adaptive privacy. Fresh randomness alone is not enough; Appendix E.1 gives a two-round separation. The missing ingredient is the stable-context condition in Checklist 1, which prevents adaptive common inputs from changing the leakage profile of an earlier Hamming discrepancy. Conversely, failure of Checklist 1 does not imply that adaptive privacy is false; it means that a separate interaction-aware proof is needed, as for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} in Appendix D ."},{"type":"heading","level":2,"text":"4.3 Putting the Ingredients Together"},{"type":"para","text":"We prove end-to-end privacy by a modular composition argument that composes two interactive mechanisms in series. Our general composition theorem (Appendix C ) applies to any upstream wrapper that is ( ε b , δ b ) (\\varepsilon_{b},\\delta_{b}) -NPDP and any downstream continual primitive that is ( ε a , δ a ) (\\varepsilon_{a},\\delta_{a}) -DP for Hamming-neighboring update streams and is feedback-safe by certification. The pipeline in Fig. 1 is one instantiation, with 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} as the wrapper and a continual DP aggregator as the downstream mechanism."},{"type":"para","text":"Section 4.1 supplies the wrapper side: 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} implements an edit-to-Hamming reduction, i.e., a single insertion/deletion in the event stream induces only a bounded Hamming-style change in the induced (per-bin) update stream (Lemma 4.1 ). Section 4.2 supplies the downstream side: for primitives that pass Checklist 1, a standard non-adaptive DP proof for Hamming-neighboring inputs remains valid when updates are chosen adaptively from past releases (Theorem 4.3 )."},{"type":"heading","level":4,"text":"Modular Composition (Informal)."},{"type":"para","text":"View buffering as producing (i) a visible transcript (e.g., whether a bin is emitted) and (ii) a hidden interface stream of per-bin updates. The downstream continual primitive consumes this interface stream and releases the trajectory observed by the adversary."},{"type":"para","text":"The end-to-end guarantee follows from two conditions:"},{"type":"para","text":"Wrapper condition (NPDP). The wrapper is ( ε b , δ b ) (\\varepsilon_{b},\\delta_{b}) -NPDP for single-edit neighboring event streams, and the induced interface streams can be aligned to be Hamming-neighboring (Lemma 4.1 )."},{"type":"para","text":"Downstream condition (Hamming-DP + certification). The downstream primitive is ( ε a , δ a ) (\\varepsilon_{a},\\delta_{a}) -DP for Hamming-neighboring update streams on its own input, and passes Checklist 1, hence remains DP under adaptive interaction (Theorem 4.3 )."},{"type":"para","text":"Conclusion. Under these conditions, the composed pipeline is ( ε b + ε a , δ b + δ a ) (\\varepsilon_{b}{+}\\varepsilon_{a},\\;\\delta_{b}{+}\\delta_{a}) -DP with respect to single-edit neighboring event streams, even under adaptive interaction. This is the only composition step used to derive Theorem 3.1 ; the formal NPDP definition, alignment/refinement machinery, and the general modular composition theorem are deferred to Appendix C ."},{"type":"heading","level":4,"text":"Takeaway."},{"type":"para","text":"Sections 4.1 – 4.3 justify treating the pipeline as plug-and-play: once the wrapper and certification conditions hold, we can focus on the DP-SGD instantiation and evaluation."},{"type":"heading","level":1,"text":"5 Instantiation: Continual DP-SGD Pipeline"},{"type":"para","text":"This section instantiates the buffering–aggregation recipe (Section 4 ) with streaming SGD (Algorithm 2 ). Each arriving user event f t f_{t} is buffered by 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} . When a bin is emitted, we form a clipped mini-batch gradient and pass a (stepsize-scaled) update to a continual-release DP aggregator. In our experiments, the aggregator 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\mathsf{PrivStreamSum} is either (i) the standard tree-based prefix-sum mechanism [ CSS10 ] , or (ii) the lower-triangular matrix factorization prefix-sum mechanism [ FHU23 ] . Their formal DP and error/consistency guarantees are summarized in Appendix F . As in Fig. 1 , the adversary observes the released models and the release timing (i.e., whether B t = ⊥ B_{t}=\\bot at each step), while bin contents and internal states are hidden."},{"type":"heading","level":4,"text":"Fixed vs. Tuned Parameters."},{"type":"para","text":"The privacy target ( ε , δ ) (\\varepsilon,\\delta) and its allocation into ( ε b , δ b ) (\\varepsilon_{b},\\delta_{b}) and ( ε a , δ a ) (\\varepsilon_{a},\\delta_{a}) determine (i) the buffering level U ( ε b , δ b ) U(\\varepsilon_{b},\\delta_{b}) and (ii) the aggregation budget for 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\mathsf{PrivStreamSum} . Given a learning task, the remaining choices follow standard DP-SGD practice: the clipping norm G G and stepsize schedule ( η k ) (\\eta_{k}) are tuned for utility. The release schedule is induced by 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} : bins are emitted at random times, and satisfy | B t | ∈ [ U , 2 U ] |B_{t}|\\in[U,2U] whenever B t ≠ ⊥ B_{t}\\neq\\bot . The aggregator 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\mathsf{PrivStreamSum} is used as a plug-in, subject to: (i) ( ε a , δ a ) (\\varepsilon_{a},\\delta_{a}) -DP for Hamming-neighboring update streams on its own input, and (ii) passing Checklist 1 so its non-adaptive DP analysis remains valid under adaptive interaction (Section 4.2 )."},{"type":"heading","level":4,"text":"Privacy-Budget Split."},{"type":"para","text":"We use a symmetric 50 / 50 50/50 split between buffering and aggregation as a simple default. Since privacy composes as ( ε , δ ) = ( ε b + ε a , δ b + δ a ) (\\varepsilon,\\delta)=(\\varepsilon_{b}+\\varepsilon_{a},\\delta_{b}+\\delta_{a}) and latency scales as U = Θ ( ε b − 1 log ( 1 / δ b ) ) U=\\Theta(\\varepsilon_{b}^{-1}\\log(1/\\delta_{b})) , any fixed split ε b = c ε \\varepsilon_{b}=c\\varepsilon , ε a = ( 1 − c ) ε \\varepsilon_{a}=(1-c)\\varepsilon with c ∈ ( 0 , 1 ) c\\in(0,1) changes the asymptotic guarantees only by constant factors; assigning constant fractions of δ \\delta changes log ( 1 / δ b ) \\log(1/\\delta_{b}) only by an additive O ( 1 ) O(1) term. The best finite-sample split can be chosen empirically by sweeping c c and measuring the latency–accuracy tradeoff."},{"type":"heading","level":4,"text":"Mini-Batch View."},{"type":"para","text":"When a bin is emitted, the gradient estimate uses the standard mini-batch average 1 | B t | ∑ f ∈ B t ∇ f ( w ) \\frac{1}{|B_{t}|}\\sum_{f\\in B_{t}}\\nabla f(w) . Since 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} enforces | B t | ∈ [ U , 2 U ] |B_{t}|\\in[U,2U] , the batch size is controlled and varies by at most a factor of 2 2 ."},{"type":"heading","level":4,"text":"Implementation Notes and Measured Latency."},{"type":"para","text":"The server maintains (a) the current released model w ( t ) w^{(t)} , (b) the FIFO buffer state of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} , and (c) the internal state of 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\mathsf{PrivStreamSum} (e.g., the tree nodes). The model is updated only on emission times and is held fixed otherwise, yielding a piecewise-constant released trajectory. To quantify privacy-implied latency, we log the backlog Q t Q_{t} (pending events after time t t ). For an event f i f_{i} , its inclusion delay is D ( i ) := min { t ≥ i : f i ∈ B t } − i D(i):=\\min\\{t\\geq i:\\;f_{i}\\in B_{t}\\}-i , which can be computed directly from the same FIFO queue evolution underlying ( Q t ) (Q_{t}) ; we report summary statistics and empirical CDFs in the experiments."},{"type":"heading","level":4,"text":"Other Optimizers."},{"type":"para","text":"The same template applies to streaming optimizers whose per-bin updates can be encoded as a clipped, sensitivity-bounded stream for the certified continual-release aggregator: replace g k = Clip ( ⋅ ) g_{k}=\\textsf{Clip}(\\cdot) and u k u_{k} by the desired update encoding, while keeping 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} as the edit-to-Hamming wrapper and 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\mathsf{PrivStreamSum} as the aggregation primitive. Variants with additional state, such as momentum or adaptive preconditioning, require a separate audit of Theorem 4.3 ’s conditions; falling outside the theorem does not by itself imply failure of adaptive privacy."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"We studied continual learning under adaptive interaction when privacy is defined over edit-style (single-insertion/deletion) user streams, where naive continual-release analyses do not directly apply. Our main recipe composes a randomized buffering wrapper 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} , which reduces single-edit neighbors to a Hamming-style per-bin update interface with explicit backlog guarantees, with a certified continual-release DP primitive (tree-based prefix sums in our experiments), and then invokes modular composition to obtain end-to-end ( ε , δ ) (\\varepsilon,\\delta) -DP."},{"type":"para","text":"We formalize the setting as outlined in the introduction. We first describe the concept of an abstract interactive mechanism, from which we will derive other forms of interactive mechanisms later."},{"type":"heading","level":4,"text":"Instantiation: Mapping the Abstract Interaction to Our Transcript."},{"type":"para","text":"The abstract interaction formalism above subsumes the continual-learning setting of Section 2 . Concretely, we instantiate the mechanism’s input messages as the user-side stream x t := f t x_{t}:=f_{t} (or, when modeling a subcomponent, x t x_{t} may represent an emitted bin B t B_{t} or a per-bin update g t g_{t} ), and we instantiate the mechanism’s outputs as the released observations y t := ( v t , w ( t ) ) y_{t}:=(v_{t},w^{(t)}) , so that the induced transcript is 𝗍𝗋 ≤ T = ( ( v t , w ( t ) ) ) t = 0 T \\mathsf{tr}_{\\leq T}=((v_{t},w^{(t)}))_{t=0}^{T} . The adversary’s view function ν 𝒜 \\nu_{{\\mathcal{A}}} reveals exactly the observable portion of each release—the timing bit and model snapshot—matching the trajectory-and-timing threat model in Section 2 ."},{"type":"para","text":"Bounded Termination. In addition to finite ℛ \\mathcal{R} and 𝒰 \\mathcal{U} , we consider mechanisms that always terminate in a bounded number of steps. Specifically, for each mechanism ℳ \\mathcal{M} , there exists T ℳ ∈ ℤ T_{\\mathcal{M}}\\in\\mathbb{Z} such that ℳ \\mathcal{M} always terminates in at most T ℳ T_{\\mathcal{M}} steps. This is without loss of generality, because any privacy notion defined for mechanisms with bounded termination can be extended naturally to unbounded mechanisms. We simply require that, for any T ∈ ℤ + T\\in\\mathbb{Z}_{+} , the truncated mechanism obtained by running the (unbounded) mechanism for T T steps will satisfy the privacy notion defined for bounded mechanisms. This matches the “for every horizon T T ” convention in Section 2 ."},{"type":"para","text":"Mechanism View. The view observed by a mechanism ℳ \\mathcal{M} up to time step t t consists of its generated random seed r r and its received input messages x [ 1 . . t ] x[1..t] up to time t t ."},{"type":"para","text":"Since ℳ \\mathcal{M} can recover its own output sequence from its random seed and input sequence, it suffices to include the latter two in its view."},{"type":"para","text":"Similarity Between Mechanisms. To formalize privacy notions later, we need a way to quantify the similarity between two mechanisms. Since we consider randomized mechanisms, we will use power functions to compare how different two distributions are. It is known that power functions are general enough to capture all divergences satisfying the data processing inequality. Recall that in this work, we focus on finite sample spaces."},{"type":"para","text":"A divergence measure is a function 𝖣 \\mathsf{D} that takes two distributions P P and Q Q on the same space such that 𝖣 ( P ∥ Q ) ≥ 0 \\mathsf{D}(P\\|Q)\\geq 0 , where equality holds iff the distributions P = Q P=Q are identical."},{"type":"para","text":"A divergence 𝖣 \\mathsf{D} satisfies the data processing inequality if, for any stochastic transformation (or channel) T T that maps the original space to another space, the following inequality holds:"},{"type":"para","text":"where T ( P ) T(P) and T ( Q ) T(Q) are the resulting distributions after applying the transformation T T to P P and Q Q , respectively."},{"type":"para","text":"Suppose P P and Q Q are distributions on the same finite sample space Ω \\Omega , i.e., P P and Q Q are vectors in ℝ ≥ 0 Ω \\mathbb{R}^{\\Omega}_{\\geq 0} whose coordinates sum to 1. The power function 𝖯𝗈𝗐 ( P ∥ Q ) : [ 0 , 1 ] → [ 0 , 1 ] \\mathsf{Pow}(P\\|Q):[0,1]\\rightarrow[0,1] can be defined in terms of the fractional knapsack problem ."},{"type":"para","text":"Given a collection Ω \\Omega of items, suppose ω ∈ Ω \\omega\\in\\Omega has weight P ( ω ) P(\\omega) and value Q ( ω ) Q(\\omega) . Then, given α ∈ [ 0 , 1 ] \\alpha\\in[0,1] , 𝖯𝗈𝗐 ( P ∥ Q ) ( α ) \\mathsf{Pow}(P\\|Q)(\\alpha) is the maximum value attained with weight capacity constraint α \\alpha , where items may be taken fractionally."},{"type":"para","text":"Intuition. When two distributions P P and Q Q are the same, it is clear that given any capacity constraint α \\alpha , the maximum reward is also α \\alpha ; this means the power function is exactly the identity function. However, if the two distributions are very different, this means that there are items whose reward-to-weight ratios are large; hence, in this case, the power function can initially grow faster than the identity function."},{"type":"para","text":"Partial Order on Power Functions. Pointwise comparison naturally induces a partial order on power functions. We denote f 1 ≤ f 2 f_{1}\\leq f_{2} if for all α ∈ [ 0 , 1 ] \\alpha\\in[0,1] , f 1 ( α ) ≤ f 2 ( α ) f_{1}(\\alpha)\\leq f_{2}(\\alpha) , where a larger f 2 f_{2} indicates that the two distributions are more different."},{"type":"para","text":"For any two distributions P P and Q Q on the same sample space, 𝖯𝗈𝗐 ( P ∥ Q ) ≥ 𝖨𝖽 \\mathsf{Pow}(P\\|Q)\\geq\\mathsf{Id} , where 𝖨𝖽 : [ 0 , 1 ] → [ 0 , 1 ] \\mathsf{Id}:[0,1]\\rightarrow[0,1] is the identity function, and equality holds iff the distributions P = Q P=Q are identical."},{"type":"para","text":"Moreover, power functions satisfy the data processing inequality , i.e., for any stochastic transformation (or channel) T T that maps the original space to another space, the following inequality holds:"},{"type":"para","text":"Differential Privacy in Terms of Power Function. Given ε ≥ 0 \\varepsilon\\geq 0 and 0 ≤ δ ≤ 1 0\\leq\\delta\\leq 1 , the power function 𝖣𝖯 ε , δ : [ 0 , 1 ] → [ 0 , 1 ] \\mathsf{DP}_{\\varepsilon,\\delta}:[0,1]\\rightarrow[0,1] can be described as follows (see Figure 2 ):"},{"type":"para","text":"In the x y xy -plane, starting at ( 0 , δ ) (0,\\delta) , the function increases linearly with slope e ε e^{\\varepsilon} until it touches the line y = 1 − x y=1-x at ( 1 − δ 1 + e ε , e ε + δ 1 + e ε ) (\\frac{1-\\delta}{1+e^{\\varepsilon}},\\frac{e^{\\varepsilon}+\\delta}{1+e^{\\varepsilon}}) ."},{"type":"para","text":"After touching the line y = 1 − x y=1-x , the slope changes to e − ε e^{-\\varepsilon} , until the line segment reaches ( 1 − δ , 1 ) (1-\\delta,1) , where the curve remains horizontally till the end ( 1 , 1 ) (1,1) is reached."},{"type":"para","text":"In other words,"},{"type":"para","text":"𝖣𝖯 ε , δ ( α ) = { δ + e ε ⋅ α , for 0 ≤ α ≤ 1 − δ 1 + e ε ; 1 − e − ε ( 1 − δ ) + e − ε α , for 1 − δ 1 + e ε < α ≤ 1 − δ ; 1 , for 1 − δ < α ≤ 1 . \\mathsf{DP}_{\\varepsilon,\\delta}(\\alpha)=\\begin{cases}\\delta+e^{\\varepsilon}\\cdot\\alpha,&\\text{for }0\\leq\\alpha\\leq\\frac{1-\\delta}{1+e^{\\varepsilon}};\\\\ 1-e^{-\\varepsilon}(1-\\delta)+e^{-\\varepsilon}\\alpha,&\\text{for }\\frac{1-\\delta}{1+e^{\\varepsilon}}<\\alpha\\leq 1-\\delta;\\\\ 1,&\\text{for }1-\\delta<\\alpha\\leq 1.\\end{cases}"},{"type":"para","text":"Below are some facts on power functions that we will use, which are usually equivalently stated in the literature in terms of the tradeoff function 𝖳 ( P ∥ Q ) ( α ) := 1 − 𝖯𝗈𝗐 ( P ∥ Q ) ( α ) \\mathsf{T}(P\\|Q)(\\alpha):=1-\\mathsf{Pow}(P\\|Q)(\\alpha) ."},{"type":"para","text":"Given two distributions P P and Q Q on the same sample space Ω \\Omega , 𝖯𝗈𝗐 ( P ∥ Q ) ≤ 𝖣𝖯 ε , δ \\mathsf{Pow}(P\\|Q)\\leq\\mathsf{DP}_{\\varepsilon,\\delta} is equivalent to the statement that for all S ⊆ Ω S\\subseteq\\Omega , P ( S ) ≤ e ε ⋅ Q ( S ) + δ P(S)\\leq e^{\\varepsilon}\\cdot Q(S)+\\delta and Q ( S ) ≤ e ε ⋅ P ( S ) + δ Q(S)\\leq e^{\\varepsilon}\\cdot P(S)+\\delta ."},{"type":"para","text":"( Triangle Inequality for Power Functions )[Theorem 2.14 in [ DRS22 ] ] Suppose X X , Y Y , Z Z are distributions on the same sample space such that 𝖯𝗈𝗐 ( X ∥ Y ) ≤ g 1 \\mathsf{Pow}(X\\|Y)\\leq g_{1} and 𝖯𝗈𝗐 ( Y ∥ Z ) ≤ g 2 \\mathsf{Pow}(Y\\|Z)\\leq g_{2} . Then, 𝖯𝗈𝗐 ( X ∥ Z ) ≤ g 2 ∘ g 1 \\mathsf{Pow}(X\\|Z)\\leq g_{2}\\circ g_{1} , where the composition is defined as ( g 2 ∘ g 1 ) ( x ) := g 2 ( g 1 ( x ) ) (g_{2}\\circ g_{1})(x):=g_{2}(g_{1}(x)) . In other words, 𝖯𝗈𝗐 ( X ∥ Z ) ≤ 𝖯𝗈𝗐 ( Y ∥ Z ) ∘ 𝖯𝗈𝗐 ( X ∥ Y ) \\mathsf{Pow}(X\\|Z)\\leq\\mathsf{Pow}(Y\\|Z)\\circ\\mathsf{Pow}(X\\|Y) ."},{"type":"para","text":"Example. Consider g = 𝖣𝖯 ε , δ g=\\mathsf{DP}_{\\varepsilon,\\delta} . Then, g ∘ g ≤ 𝖣𝖯 2 ε , δ ( 1 + e ε ) g\\circ g\\leq\\mathsf{DP}_{2\\varepsilon,\\delta(1+e^{\\varepsilon})} . As we shall see, this is relevant to the scenario of multi-hop neighbors – aka group DP in the literature."},{"type":"para","text":"Given a collection 𝒮 \\mathcal{S} of power functions, its supremum sup ( 𝒮 ) = sup f ∈ 𝒮 f \\sup(\\mathcal{S})=\\sup_{f\\in\\mathcal{S}}f is the least power function f ^ \\widehat{f} such that for all 𝔣 ∈ 𝒮 \\mathfrak{f}\\in\\mathcal{S} , f ≤ f ^ f\\leq\\widehat{f} ."},{"type":"para","text":"Formally, for each α ∈ [ 0 , 1 ] \\alpha\\in[0,1] , f ^ ( α ) := \\widehat{f}(\\alpha):="},{"type":"para","text":"Suppose f f and g g are power functions that are attained by the corresponding distributions in 𝖯𝗈𝗐 ( P 1 ∥ P 2 ) = f \\mathsf{Pow}(P_{1}\\|P_{2})=f and g = 𝖯𝗈𝗐 ( Q 1 ∥ Q 2 ) g=\\mathsf{Pow}(Q_{1}\\|Q_{2}) . Then, their corresponding tensor product f ⊗ g = 𝖯𝗈𝗐 ( P 1 × Q 1 ∥ P 2 × Q 2 ) f\\otimes g=\\mathsf{Pow}(P_{1}\\times Q_{1}\\|P_{2}\\times Q_{2}) is defined in terms of the corresponding product distributions."},{"type":"para","text":"Suppose Λ \\Lambda is an indexing set and f f is a power function such that for each λ ∈ Λ \\lambda\\in\\Lambda , a pair ( P ( λ ) , Q ( λ ) ) (P(\\lambda),Q(\\lambda)) of distributions satisfy 𝖯𝗈𝗐 ( P ( λ ) ∥ Q ( λ ) ) ≤ f \\mathsf{Pow}(P(\\lambda)\\|Q(\\lambda))\\leq f ."},{"type":"para","text":"Then, for any distribution L L on Λ \\Lambda , 𝖯𝗈𝗐 ( P ( L ) ∥ Q ( L ) ) ≤ f \\mathsf{Pow}(P(L)\\|Q(L))\\leq f ."},{"type":"para","text":"Suppose P P and P ′ P^{\\prime} are distributions on the sample space Ω \\Omega . For each ω ∈ Ω \\omega\\in\\Omega , Q ( ω ) Q(\\omega) and Q ′ ( ω ) Q^{\\prime}(\\omega) are distributions on sample space Ω ′ \\Omega^{\\prime} . If 𝖯𝗈𝗐 ( P ∥ P ′ ) ≤ f \\mathsf{Pow}(P\\|P^{\\prime})\\leq f and 𝖯𝗈𝗐 ( Q ( ω ) ∥ Q ′ ( ω ) ) ≤ g \\mathsf{Pow}(Q(\\omega)\\|Q^{\\prime}(\\omega))\\leq g for all ω ∈ Ω \\omega\\in\\Omega , then 𝖯𝗈𝗐 ( ( P , Q ( P ) ) ∥ ( P ′ , Q ′ ( P ′ ) ) ) ≤ f ⊗ g . \\mathsf{Pow}((P,Q(P))\\|(P^{\\prime},Q^{\\prime}(P^{\\prime})))\\leq f\\otimes g."},{"type":"para","text":"Distinguishing between Interactive Mechanisms. Suppose we wish to compare the behavior of two mechanisms ℳ 0 \\mathcal{M}_{0} and ℳ 1 \\mathcal{M}_{1} , to which we only have oracle accesses. An adversary 𝒜 : ℛ × 𝒰 ∗ → 𝒰 \\mathcal{A}:\\mathcal{R}\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} is also an interactive mechanism. The adversary has some view function ν 𝒜 : 𝒰 → 𝒰 \\nu_{\\mathcal{A}}:\\mathcal{U}\\rightarrow\\mathcal{U} that specifies what information it can gather from the output of the mechanism. However, later when we instantiate from the abstract mechanism, we will use the structure of the message space to clarify what the adversary can observe."},{"type":"para","text":"We next define the interaction 𝒜 ↔ ℳ {\\mathcal{A}}\\leftrightarrow\\mathcal{M} between the two mechanisms. The adversary 𝒜 {\\mathcal{A}} repeatedly sends messages to ℳ \\mathcal{M} . At each time step, the adversary determines the next message to be sent to ℳ \\mathcal{M} , based on the history of prior interactions. This process repeats iteratively, until the mechanism ℳ \\mathcal{M} outputs 𝗁𝖺𝗅𝗍 \\mathsf{halt} in some time step. Formally, it is captured by the following description."},{"type":"para","text":"Suppose the interaction 𝒜 ↔ ℳ {\\mathcal{A}}\\leftrightarrow\\mathcal{M} terminates at time step t t (which may be a random quantity). Then, the adversary view is defined as:"},{"type":"para","text":"𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ ) := ( r 𝒜 ; v [ 0 . . t ] ) \\mathsf{view}({\\mathcal{A}}\\leftrightarrow\\mathcal{M}):=(r_{\\mathcal{A}};v[0..t]) ."},{"type":"para","text":"Terminology (“non-adaptive”). In this paper, when we say “non-adaptive” for a continual mechanism’s privacy analysis, we mean the mechanism’s input stream is fixed in advance (equivalently, the environment/adversary that provides inputs does not condition on past outputs)."},{"type":"para","text":"An adversary transition function 𝒜 : ℛ × 𝒰 ∗ → 𝒰 \\mathcal{A}:\\mathcal{R}\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} is non-adaptive, if there exists a transition function of the form 𝒜 ^ : ℛ × ℤ → 𝒰 \\widehat{\\mathcal{A}}:\\mathcal{R}\\times\\mathbb{Z}\\rightarrow\\mathcal{U} such that for any length- t t sequence v [ 0 . . t − 1 ] v[0..t-1] , 𝒜 ( r ; v [ 0 . . t − 1 ] ) = 𝒜 ^ ( r ; t ) \\mathcal{A}(r;v[0..t-1])=\\widehat{\\mathcal{A}}(r;t) ."},{"type":"para","text":"We can next define a distance notion between two mechanisms based on power functions. Intuitively, it quantifies the difficulty of distinguishing between two mechanisms."},{"type":"para","text":"Given two interactive mechanisms ℳ 0 \\mathcal{M}_{0} and ℳ 1 \\mathcal{M}_{1} , we overload the notation and denote 𝖯𝗈𝗐 ( ℳ 0 ∥ ℳ 1 ) ≤ f \\mathsf{Pow}\\left(\\mathcal{M}_{0}\\|\\mathcal{M}_{1}\\right)\\leq f , for some power function f f , if for all adversaries 𝒜 {\\mathcal{A}} , 𝖯𝗈𝗐 ( 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ 0 ) ∥ 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ 1 ) ) ≤ f . \\mathsf{Pow}\\left(\\mathsf{view}({\\mathcal{A}}\\leftrightarrow\\mathcal{M}_{0})\\|\\mathsf{view}({\\mathcal{A}}\\leftrightarrow\\mathcal{M}_{1})\\right)\\leq f."},{"type":"para","text":"If we have the weaker requirement that the inequality holds for just non-adaptive adversaries, we denote: 𝖯𝗈𝗐 NA ( ℳ 0 ∥ ℳ 1 ) ≤ f \\mathsf{Pow}^{{\\textsc{NA}}}\\left(\\mathcal{M}_{0}\\|\\mathcal{M}_{1}\\right)\\leq f ."},{"type":"para","text":"Simulation of Two Close Interactive Mechanisms with Close Random Seeds. The following result from [ VZ23 ] states that if two mechanisms are close in the sense of Definition A.13 , then it is possible to simulate either one of them exactly with the same mechanism transition function that initially takes a random seed generated from one of the two appropriate distributions. Because the proof uses induction and Cartesian products of sample spaces, only the special case of finite sample and message spaces and mechanisms with bounded termination is considered. It is plausible that the result could be generalized to unbounded spaces, but that would likely require the Axiom of Choice, which often falls outside the scope of the computer science community."},{"type":"para","text":"This is a very useful lemma in proving privacy composition results, because instead of interacting with two potentially different mechanisms in every time step, we can view the adversary as interacting with the same mechanism transition function but with initial random seeds drawn from two different distributions. One useful analogy is that after a sensitive database is anonymized with appropriate noise, then it can be queried unlimited number of times without privacy degradation because of data post-processing."},{"type":"para","text":"Suppose ℳ 0 \\mathcal{M}_{0} and ℳ 1 : ℛ × 𝒰 ∗ → 𝒰 \\mathcal{M}_{1}:\\mathcal{R}\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} are two interactive mechanisms with bounded termination, and f f is a power function such that 𝖯𝗈𝗐 ( ℳ 0 ∥ ℳ 1 ) ≤ f \\mathsf{Pow}(\\mathcal{M}_{0}\\|\\mathcal{M}_{1})\\leq f in the sense of Definition A.13 ."},{"type":"para","text":"Then, there exist two distributions N 0 N_{0} and N 1 N_{1} on some sample space Ω \\Omega and a transition function 𝒫 : Ω × 𝒰 ∗ → 𝒰 \\mathcal{P}:\\Omega\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} such that 𝖯𝗈𝗐 ( N 0 ∥ N 1 ) ≤ f \\mathsf{Pow}(N_{0}\\|N_{1})\\leq f , and, for all adversaries 𝒜 {\\mathcal{A}} , the following holds for both b ∈ { 0 , 1 } b\\in\\{0,1\\} : 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ b ) ≡ 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ 𝒫 ( N b ) ) , \\mathsf{view}({\\mathcal{A}}\\leftrightarrow\\mathcal{M}_{b})\\equiv\\mathsf{view}({\\mathcal{A}}\\leftrightarrow\\mathcal{P}(N_{b})),"},{"type":"para","text":"where 𝒫 ( N ) \\mathcal{P}({N}) is the interactive mechanism that uses the specified transition function 𝒫 \\mathcal{P} with a random seed drawn from the distribution N N ."},{"type":"heading","level":4,"text":"Interactive Mechanism with Hidden Output."},{"type":"para","text":"We next consider a special form of interactive mechanism with a transition function of the form ℳ : ℛ × 𝒳 ∗ → 𝒱 × 𝒴 \\mathcal{M}:\\mathcal{R}\\times\\mathcal{X}^{*}\\to\\mathcal{V}\\times\\mathcal{Y} . Here, we use 𝒳 \\mathcal{X} to denote the input space. In each time step, the mechanism produces ( v , y ) ∈ 𝒱 × 𝒴 (v,y)\\in\\mathcal{V}\\times\\mathcal{Y} , where the adversary can observe v v from the view space 𝒱 \\mathcal{V} and the output y y from the output space 𝒴 \\mathcal{Y} is potentially hidden from the adversary, which may be passed to another interactive mechanism (that could however possibly leak information about y y ). If an adversary only interacts with ℳ \\mathcal{M} , then it can be described by a transition function with the form 𝒜 : ℛ × 𝒱 ∗ → 𝒳 {\\mathcal{A}}:\\mathcal{R}\\times\\mathcal{V}^{*}\\to\\mathcal{X} ."},{"type":"para","text":"Neighboring Input Sequences. Similar to the conventional differential privacy, we need a neighboring notion for input sequences such that intuitively the behaviors of a mechanism on two neighboring input sequences should be similar."},{"type":"para","text":"We assume there is a neighboring relation ∼ 𝒳 ∗ \\sim_{\\mathcal{X}^{*}} (or just ∼ \\sim for succinctness) that satisfies the property that if two sequences x [ 1 . . t ] ∼ x ′ [ 1 . . t ′ ] x[1..t]\\sim x^{\\prime}[1..t^{\\prime}] are neighboring, then the following holds:"},{"type":"para","text":"They have the same length, i.e., t = t ′ t=t^{\\prime} ."},{"type":"para","text":"All their corresponding prefixes are neighboring, i.e., for all 1 ≤ τ < t 1\\leq\\tau<t , x [ 1 . . τ ] ∼ x ′ [ 1 . . τ ] x[1..\\tau]\\sim x^{\\prime}[1..\\tau] ."},{"type":"para","text":"Privacy for Interactive Mechanisms. Since the inputs for an interactive mechanism in later time steps can be influenced by its earlier behavior, one cannot formally define its privacy just based on neighboring input sequences. In the literature, adaptive differential privacy [ JRS+23 ] is based on a construction that we call paired simulation ."},{"type":"para","text":"Suppose an interactive mechanism is given by the transition function ℳ : ℛ × 𝒳 ∗ → 𝒱 × 𝒴 \\mathcal{M}:\\mathcal{R}\\times\\mathcal{X}^{*}\\to\\mathcal{V}\\times\\mathcal{Y} and the random seed distribution R ℳ R_{\\mathcal{M}} , where a neighboring relation is defined on 𝒳 ∗ \\mathcal{X}^{*} ."},{"type":"para","text":"Then, the paired simulation consists of a canonical pair of interactive mechanisms ( ℳ 0 𝗉𝖺𝗂𝗋 , ℳ 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{M}^{\\mathsf{pair}}_{0},\\mathcal{M}^{\\mathsf{pair}}_{1}) such that for each b ∈ { 0 , 1 } b\\in\\{0,1\\} , the transition function has the form ℳ b 𝗉𝖺𝗂𝗋 : ℛ × ( 𝒳 2 ) ∗ → 𝒱 × 𝒴 \\mathcal{M}^{\\mathsf{pair}}_{b}:\\mathcal{R}\\times(\\mathcal{X}^{2})^{*}\\to\\mathcal{V}\\times\\mathcal{Y} . Moreover, in the paired simulation, an adversary with transition function 𝒜 : ℛ × 𝒱 ∗ → 𝒳 2 \\mathcal{A}:\\mathcal{R}\\times\\mathcal{V}^{*}\\to\\mathcal{X}^{2} interacts with ℳ b 𝗉𝖺𝗂𝗋 \\mathcal{M}^{\\mathsf{pair}}_{b} as follows."},{"type":"para","text":"Given an interactive mechanism ℳ \\mathcal{M} , suppose ( ℳ 0 𝗉𝖺𝗂𝗋 , ℳ 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{M}^{\\mathsf{pair}}_{0},\\mathcal{M}^{\\mathsf{pair}}_{1}) is the paired simulation as in Definition B.2 . For a power function f f , the mechanism ℳ \\mathcal{M} is adaptively f f -differentially private (DP) if"},{"type":"para","text":"𝖯𝗈𝗐 ( ℳ 0 𝗉𝖺𝗂𝗋 ∥ ℳ 1 𝗉𝖺𝗂𝗋 ) ≤ f \\mathsf{Pow}(\\mathcal{M}^{\\mathsf{pair}}_{0}\\|\\mathcal{M}^{\\mathsf{pair}}_{1})\\leq f , for all adaptive adversaries in the sense of Definition A.13 . Similarly, non-adaptively f f -DP is defined by 𝖯𝗈𝗐 NA ( ℳ 0 𝗉𝖺𝗂𝗋 ∥ ℳ 1 𝗉𝖺𝗂𝗋 ) ≤ f \\mathsf{Pow}^{\\textsc{NA}}(\\mathcal{M}^{\\mathsf{pair}}_{0}\\|\\mathcal{M}^{\\mathsf{pair}}_{1})\\leq f ."},{"type":"para","text":"Multi-Hop Neighboring Notion. In the literature, group DP or multi-hop neighbors can be generalized to sequences readily."},{"type":"para","text":"For k = 1 k=1 , 1 1 -hop neighboring is the same as that in Assumption B.1 . For k ≥ 2 k\\geq 2 , two sequences x x and x ′ x^{\\prime} are k k -hop neighboring if there exist x 1 , … , x k − 1 x_{1},\\ldots,x_{k-1} such that for 0 ≤ i < k 0\\leq i<k , x i x_{i} and x i + 1 x_{i+1} are 1 1 -hop neighboring, where x 0 = x x_{0}=x and x k = x ′ x_{k}=x^{\\prime} ."},{"type":"para","text":"We also generalize Definition B.2 to multi-hop neighboring simulation."},{"type":"para","text":"Generalizing the description in Definition B.2 , in each time step t t , the adversary generates a tuple ( x t ( 0 ) , x t ( 1 ) , … , x t ( k ) ) (x^{(0)}_{t},x^{(1)}_{t},\\ldots,x^{(k)}_{t}) such that for each 0 ≤ i < k 0\\leq i<k , x ( i ) [ 1 . . t ] x^{(i)}[1..t] and x ( i + 1 ) [ 1 . . t ] x^{(i+1)}[1..t] are neighboring sequences."},{"type":"para","text":"Moreover, for b ∈ [ 0 . . k ] b\\in[0..k] , ℳ b 𝗁𝗈𝗉 \\mathcal{M}^{\\mathsf{hop}}_{b} is a simulation of ℳ \\mathcal{M} that takes the b b -th component x t ( b ) x^{(b)}_{t} at each step t t ."},{"type":"para","text":"Suppose ℳ \\mathcal{M} is adaptively f f -DP as in Definition B.3 . Then, in a k k -hop neighboring simulation as in Definition B.5 , we have: 𝖯𝗈𝗐 ( ℳ 0 𝗁𝗈𝗉 ∥ ℳ k 𝗁𝗈𝗉 ) ≤ f ∘ k \\mathsf{Pow}(\\mathcal{M}^{\\mathsf{hop}}_{0}\\|\\mathcal{M}^{\\mathsf{hop}}_{k})\\leq f^{\\circ k} ,"},{"type":"para","text":"where f ∘ 1 = f f^{\\circ 1}=f and f ∘ ( i + 1 ) = f ∘ i ∘ f f^{\\circ(i+1)}=f^{\\circ i}\\circ f ."},{"type":"para","text":"This follows from Definition B.3 and Fact A.6 . ∎"},{"type":"para","text":"Modular Composition of Interactive Mechanisms. As mentioned above, the hidden output of one mechanism may be fed as the input of another mechanism. To this end, we define a specific form of composition that we call modular composition . The adversary only directly supplies an input to the “head” mechanism of the list, where each intermediate mechanism receives its input from the (hidden) output from the previous mechanism."},{"type":"para","text":"Suppose two interactive mechanisms are given by the transition functions ℳ : ℛ 1 × 𝒳 ∗ → 𝒱 × 𝒴 \\mathcal{M}:\\mathcal{R}_{1}\\times\\mathcal{X}^{*}\\rightarrow\\mathcal{V}\\times\\mathcal{Y} and 𝒩 : ℛ 2 × 𝒴 ∗ → 𝒲 × 𝒵 \\mathcal{N}:\\mathcal{R}_{2}\\times\\mathcal{Y}^{*}\\to\\mathcal{W}\\times\\mathcal{Z} , with random seed distributions R ℳ R_{\\mathcal{M}} and R 𝒩 R_{\\mathcal{N}} , respectively. Then, an adversary with transition function 𝒜 : ℛ × ( 𝒱 × 𝒲 ) ∗ → 𝒳 \\mathcal{A}:\\mathcal{R}\\times(\\mathcal{V}\\times\\mathcal{W})^{*}\\to\\mathcal{X} interacts with the modular composition ℳ → 𝒩 \\mathcal{M}\\to\\mathcal{N} as above."},{"type":"para","text":"Neighbor-Preserving Paired Simulation. To consider composition of differentially oblivious mechanisms [ CCM+22 ] , the framework neighbor-preserving differential obliviousness (NPDO) [ ZSC+23 , ZZC+24 ] has been introduced to formalize the condition that if neighboring inputs are submitted to a mechanism, then the corresponding outputs would somehow be neighboring for a subsequent mechanism. To generalize this to interactive mechanisms, we will need to introduce neighbor-preserving paired simulation. In addition to neighboring input sequences in 𝒳 ∗ \\mathcal{X}^{*} for mechanism 𝒩 \\mathcal{N} , we assume that there is a similar notion of neighboring sequences in 𝒴 ∗ \\mathcal{Y}^{*} (satisfying Assumption B.1 ) for mechanism 𝒩 \\mathcal{N} ."},{"type":"para","text":"Given an interactive mechanism with transition function ℳ : ℛ × 𝒳 ∗ → 𝒱 × 𝒴 \\mathcal{M}:\\mathcal{R}\\times\\mathcal{X}^{*}\\rightarrow\\mathcal{V}\\times\\mathcal{Y} and random seed distribution R ℳ R_{\\mathcal{M}} , a neighbor-preserving paired simulation (which might not be unique) is a pair ( ℳ 0 𝗇𝗉𝗉 , ℳ 1 𝗇𝗉𝗉 ) (\\mathcal{M}^{\\mathsf{npp}}_{0},\\mathcal{M}^{\\mathsf{npp}}_{1}) of interactive mechanisms such that the following holds."},{"type":"para","text":"There exists possibly another collection ℛ ′ \\mathcal{R}^{\\prime} of random seeds and an appropriate distribution R ℳ ′ R^{\\prime}_{\\mathcal{M}} such that for each b ∈ { 0 , 1 } b\\in\\{0,1\\} , the corresponding mechanism has a transition function of the form ℳ b 𝗇𝗉𝗉 : ℛ × ℛ ′ × ( 𝒳 2 ) ∗ → 𝒱 × 𝒴 2 \\mathcal{M}^{\\mathsf{npp}}_{b}:\\mathcal{R}\\times\\mathcal{R}^{\\prime}\\times(\\mathcal{X}^{2})^{*}\\to\\mathcal{V}\\times\\mathcal{Y}^{2} , where the random seed distribution on ℛ \\mathcal{R} still follows R ℳ R_{\\mathcal{M}} , and the random seed from ℛ ′ \\mathcal{R}^{\\prime} is sampled from R ℳ ′ R^{\\prime}_{\\mathcal{M}} ."},{"type":"para","text":"In other words, in addition to a random seed in ℛ \\mathcal{R} , an extra random seed in ℛ ′ \\mathcal{R}^{\\prime} may be taken. Just like paired simulation in Definition B.2 , a sequence of input pairs are also taken. However, the transition function returns a tuple in 𝒱 × 𝒴 2 \\mathcal{V}\\times\\mathcal{Y}^{2} , i.e., in addition to v ∈ 𝒱 v\\in\\mathcal{V} , a pair ( y 0 , y 1 ) ∈ 𝒴 2 (y_{0},y_{1})\\in\\mathcal{Y}^{2} is returned."},{"type":"para","text":"For each b ∈ { 0 , 1 } b\\in\\{0,1\\} , r ∈ ℛ r\\in\\mathcal{R} , r ′ ∈ ℛ ′ r^{\\prime}\\in\\mathcal{R}^{\\prime} , x ( 0 ) [ 1 . . t ] x^{(0)}[1..t] and x ( 1 ) [ 1 . . t ] ∈ 𝒳 ∗ x^{(1)}[1..t]\\in\\mathcal{X}^{*} ,"},{"type":"para","text":"if x ( 0 ) [ 1 . . t ] x^{(0)}[1..t] and x ( 1 ) [ 1 . . t ] x^{(1)}[1..t] are not neighboring sequences, then ℳ b 𝗇𝗉𝗉 ( r , r ′ ; x ( 0 ) [ 1 . . t ] , x ( 1 ) [ 1 . . t ] ) = ⊥ \\mathcal{M}^{\\mathsf{npp}}_{b}(r,r^{\\prime};x^{(0)}[1..t],x^{(1)}[1..t])=\\bot , and the whole simulation terminates at time step t t ."},{"type":"para","text":"Otherwise, the pair ( y ( 0 ) [ 1 . . t ] , y ( 1 ) [ 1 . . t ] ) (y^{(0)}[1..t],y^{(1)}[1..t]) of output sequences produced in the first t t time steps are neighboring in 𝒴 ∗ \\mathcal{Y}^{*} ."},{"type":"para","text":"If for some ( v , y 0 , y 1 ) ∈ 𝒱 × 𝒴 2 (v,y_{0},y_{1})\\in\\mathcal{V}\\times\\mathcal{Y}^{2} , ℳ b 𝗇𝗉𝗉 ( r , r ′ ; x ( 0 ) [ 1 . . t ] , x ( 1 ) [ 1 . . t ] ) = ( v , y 0 , y 1 ) \\mathcal{M}^{\\mathsf{npp}}_{b}(r,r^{\\prime};x^{(0)}[1..t],x^{(1)}[1..t])=(v,y_{0},y_{1}) , then ℳ ( r ; x ( b ) [ 1 . . t ] ) = ( v , y b ) \\mathcal{M}(r;x^{(b)}[1..t])=(v,y_{b}) ."},{"type":"para","text":"In other words, we require consistency between ℳ b 𝗇𝗉𝗉 \\mathcal{M}^{\\mathsf{npp}}_{b} and ℳ b 𝗉𝖺𝗂𝗋 \\mathcal{M}^{\\mathsf{pair}}_{b} (from Definition B.2 ). Specifically, the projection of ℳ b 𝗇𝗉𝗉 \\mathcal{M}^{\\mathsf{npp}}_{b} ’s output onto ℳ b 𝗉𝖺𝗂𝗋 \\mathcal{M}^{\\mathsf{pair}}_{b} ’s range must match the behavior of ℳ b 𝗉𝖺𝗂𝗋 \\mathcal{M}^{\\mathsf{pair}}_{b} ."},{"type":"para","text":"However, unlike paired simulation ℳ b 𝗉𝖺𝗂𝗋 \\mathcal{M}^{\\mathsf{pair}}_{b} whose behavior is totally determined by the given ℳ \\mathcal{M} , in the tuple ( v , y 0 , y 1 ) (v,y_{0},y_{1}) returned above by ℳ b 𝗇𝗉𝗉 \\mathcal{M}^{\\mathsf{npp}}_{b} , the value y b ¯ y_{\\overline{b}} (where b ≠ b ¯ b\\neq\\overline{b} ) can depend on all the input parameters in a manner that is not specified by ℳ \\mathcal{M} ."},{"type":"para","text":"Augmented Adversary View. When an adversary interacts with ℳ b 𝗇𝗉𝗉 \\mathcal{M}^{\\mathsf{npp}}_{b} , the adversarial view consists of its randomness and the sequence in ( 𝒱 × 𝒴 2 ) ∗ (\\mathcal{V}\\times\\mathcal{Y}^{2})^{*} produced until termination."},{"type":"para","text":"Given a power function f f , an interactive mechanism ℳ \\mathcal{M} is said to be adaptively f f -neighbor-preserving differentially private (NPDP), if there exists a neighbor-preserving paired simulation ( ℳ 0 𝗇𝗉𝗉 , ℳ 1 𝗇𝗉𝗉 ) (\\mathcal{M}^{\\mathsf{npp}}_{0},\\mathcal{M}^{\\mathsf{npp}}_{1}) such that 𝖯𝗈𝗐 ( ℳ 0 𝗇𝗉𝗉 ∥ ℳ 1 𝗇𝗉𝗉 ) ≤ f \\mathsf{Pow}(\\mathcal{M}^{\\mathsf{npp}}_{0}\\|\\mathcal{M}^{\\mathsf{npp}}_{1})\\leq f , in the sense of Definition A.13 ."},{"type":"para","text":"Observe that if one wishes to prove that a mechanism is NPDP as in Definition B.9 , there is potentially some freedom to construct an appropriate NPP simulation."},{"type":"para","text":"On the contrary, for non-interactive mechanisms (that only run for one time step), the recent result on universally closest distribution refinements [ CX25 ] implies that a “best possible” NPP simulation may be used."},{"type":"para","text":"Unfortunately, this does not seem to generalize to interactive mechanisms. The high level reason is that there may be no closest distribution refinements on two random sequences of length t + 1 t+1 such that its prefix projection would give a closest distribution refinement for the corresponding prefixes of length t t . In other words, if one constructs the “best possible” NPP up to time step t t , it may not result in the best NPP if the process enters step t + 1 t+1 ."},{"type":"para","text":"We next describe a modular composition theorem for interactive mechanisms. At a high level, the theorem states that if the first stage is neighbor-preserving (NPDP) with privacy profile f f , and the second stage is (adaptive) DP with profile g g on the intermediate stream, then the end-to-end interactive pipeline is (adaptive) DP with profile f ⊗ g f\\otimes g ."},{"type":"heading","level":4,"text":"How this theorem relates to existing composition frameworks."},{"type":"para","text":"Our theorem can be viewed as a direct consequence of two existing ideas, once NPDP is interpreted through paired simulation ."},{"type":"para","text":"First, it is shown [ ZZC+24 ] that neighbor-preserving notions (including differential obliviousness/NPDO) admit an equivalent characterization in terms of a paired simulation that outputs a refinement pair . Informally, in the standard secret-bit distinguishing game, an adversary provides at each time step a neighboring pair ( x t ( 0 ) , x t ( 1 ) ) (x_{t}^{(0)},x_{t}^{(1)}) , and a secret bit b ∈ { 0 , 1 } b\\in\\{0,1\\} selects which component is the “real” input. A mechanism is DP if the adversary’s views under b = 0 b=0 and b = 1 b=1 are close. The refinement-pair view resolves the usual obstacle in modular composition: even if the real world uses x ( b ) x^{(b)} and produces a real intermediate output y ( b ) y^{(b)} , one also needs a synthetic neighbor y ¯ ( 1 − b ) \\overline{y}^{(1-b)} so that the next stage can be analyzed in a paired simulation. The NPDP guarantee precisely asserts the existence of such a paired simulator that, for each b b , produces a pair ( y ( 0 ) , y ( 1 ) ) (y^{(0)},y^{(1)}) whose b b -marginal matches the real execution, and such that the two resulting pair-distributions (for b = 0 b=0 vs. b = 1 b=1 ) are close."},{"type":"para","text":"Second, a general framework for adaptive privacy of continual mechanisms [ HSV26 ] is developed to prove concurrent composition theorems under interactive adversaries. In particular, once a mechanism is expressed in the standard secret-bit game (and the controller only interacts with downstream submechanisms through their privatized outputs), privacy of the overall system follows from standard interactive composition and post-processing. Under the paired-simulation perspective, our two-stage pipeline ℳ → 𝒩 \\mathcal{M}\\rightarrow\\mathcal{N} fits this template: the NPDP stage can be viewed as a mechanism that, on secret bit b b , outputs a paired intermediate stream ( y t ( 0 ) , y t ( 1 ) ) (y^{(0)}_{t},y^{(1)}_{t}) (together with visible outputs v t v_{t} ), and the DP stage 𝒩 \\mathcal{N} is then run on y ( b ) y^{(b)} . Thus, one may reduce modular composition to ordinary interactive composition: compose the lifted NPDP mechanism (as DP-on-the-bit with profile f f ) with the second-stage DP mechanism (profile g g ), and finally apply post-processing to remove the extra synthetic transcript."},{"type":"para","text":"While this reduction gives a useful conceptual guide and could be formalized by instantiating the conditions of [ HSV26 ] together with the refinement-pair characterization of [ ZZC+24 ] , we include below a self-contained proof tailored to our setting and our power-function accounting. In particular, a more general concurrent setting (multiple submechanisms with interleaving calls) is studied in [ HSV26 ] , whereas we only need the serial modular composition used by our buffering–aggregation pipeline. Providing a direct proof also makes explicit how the privacy profile tensor f ⊗ g f\\otimes g arises in our notation, and it cleanly interfaces with the certification arguments in other sections."},{"type":"para","text":"Suppose an interactive mechanism with transition function ℳ : ℛ 1 × 𝒳 ∗ → 𝒱 × 𝒴 \\mathcal{M}:\\mathcal{R}_{1}\\times\\mathcal{X}^{*}\\to\\mathcal{V}\\times\\mathcal{Y} is adaptively f f -NPDP (as in Definition B.9 ). Suppose another interactive mechanism 𝒩 : ℛ 2 × 𝒴 ∗ → 𝒲 × 𝒵 \\mathcal{N}:\\mathcal{R}_{2}\\times\\mathcal{Y}^{*}\\to\\mathcal{W}\\times\\mathcal{Z} is adaptively g g -DP (as in Definition B.3 ). Then, the modular composition (as in Definition B.7 ) ℳ → 𝒩 : ℛ 1 × ℛ 2 × 𝒳 ∗ → 𝒱 × 𝒲 × 𝒵 \\mathcal{M}\\rightarrow\\mathcal{N}:\\mathcal{R}_{1}\\times\\mathcal{R}_{2}\\times\\mathcal{X}^{*}\\to\\mathcal{V}\\times\\mathcal{W}\\times\\mathcal{Z} is adaptively f ⊗ g f\\otimes g -DP."},{"type":"para","text":"We denote the resulting interactive mechanism from the modular composition by ℒ := ℳ → 𝒩 \\mathcal{L}:=\\mathcal{M}\\rightarrow\\mathcal{N} . In view of Definition B.3 , we consider the paired simulation ( ℒ 0 𝗉𝖺𝗂𝗋 , ℒ 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{L}^{\\mathsf{pair}}_{0},\\mathcal{L}^{\\mathsf{pair}}_{1}) as described in Definition B.2 . Specifically, our goal is to show that for any adversary 𝒜 \\mathcal{A} , the two distributions 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℒ 0 𝗉𝖺𝗂𝗋 ) \\mathsf{view}(\\mathcal{A}\\leftrightarrow\\mathcal{L}^{\\mathsf{pair}}_{0}) and 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℒ 1 𝗉𝖺𝗂𝗋 ) \\mathsf{view}(\\mathcal{A}\\leftrightarrow\\mathcal{L}^{\\mathsf{pair}}_{1}) are “close” in the sense quantified by the power function f ⊗ g f\\otimes g ."},{"type":"para","text":"Our proof strategy is to construct an augmented pair ( ℒ ^ 0 , ℒ ^ 1 ) (\\widehat{\\mathcal{L}}_{0},\\widehat{\\mathcal{L}}_{1}) of interactive mechanisms and an augmented adversary 𝒜 ^ \\widehat{\\mathcal{A}} whose views have the same distributions as those from ( ℒ 0 𝗉𝖺𝗂𝗋 , ℒ 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{L}^{\\mathsf{pair}}_{0},\\mathcal{L}^{\\mathsf{pair}}_{1}) and 𝒜 \\mathcal{A} ."},{"type":"para","text":"We next use Fact A.14 to show that instead of considering the pair ( ℒ 0 𝗉𝖺𝗂𝗋 , ℒ 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{L}^{\\mathsf{pair}}_{0},\\mathcal{L}^{\\mathsf{pair}}_{1}) with the same random seed distribution, we can equivalently simulate the pair with some common transition function, but using two different random seed distributions. By the data processing inequality, it suffices to analyze the alternative simulation and the power function of the two random seed distributions. Because ℳ \\mathcal{M} is f f -NPDP, let ( ℳ 0 𝗇𝗉𝗉 , ℳ 1 𝗇𝗉𝗉 ) (\\mathcal{M}^{\\mathsf{npp}}_{0},\\mathcal{M}^{\\mathsf{npp}}_{1}) be the corresponding NPP simulation as guaranteed by Definition B.9 such that 𝖯𝗈𝗐 ( ℳ 0 𝗇𝗉𝗉 ∥ ℳ 1 𝗇𝗉𝗉 ) ≤ f \\mathsf{Pow}(\\mathcal{M}^{\\mathsf{npp}}_{0}\\|\\mathcal{M}^{\\mathsf{npp}}_{1})\\leq f . By Fact A.14 , there exist some interactive mechanism 𝒫 \\mathcal{P} and two random seed distributions F 0 F_{0} and F 1 F_{1} such that 𝖯𝗈𝗐 ( F 0 ∥ F 1 ) ≤ f \\mathsf{Pow}(F_{0}\\|F_{1})\\leq f and for each b ∈ { 0 , 1 } b\\in\\{0,1\\} , 𝒫 ( F b ) \\mathcal{P}(F_{b}) is an exact simulation of ℳ b 𝗇𝗉𝗉 \\mathcal{M}^{\\mathsf{npp}}_{b} (with the appropriate random seed distribution R ℳ × R ℳ ′ R_{\\mathcal{M}}\\times R^{\\prime}_{\\mathcal{M}} ) for any adversary. Specifically, for each b b , ℳ b 𝗇𝗉𝗉 ( R ℳ × R ℳ ′ ) \\mathcal{M}^{\\mathsf{npp}}_{b}(R_{\\mathcal{M}}\\times R^{\\prime}_{\\mathcal{M}}) and 𝒫 ( F b ) \\mathcal{P}(F_{b}) are equivalent."},{"type":"para","text":"Similarly, since 𝒩 \\mathcal{N} is g g -DP, the paired simulation ( 𝒩 0 𝗉𝖺𝗂𝗋 , 𝒩 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{N}^{\\mathsf{pair}}_{0},\\mathcal{N}^{\\mathsf{pair}}_{1}) satisfies 𝖯𝗈𝗐 ( 𝒩 0 𝗉𝖺𝗂𝗋 ∥ 𝒩 1 𝗉𝖺𝗂𝗋 ) ≤ g \\mathsf{Pow}(\\mathcal{N}^{\\mathsf{pair}}_{0}\\|\\mathcal{N}^{\\mathsf{pair}}_{1})\\leq g . Again, by Fact A.14 , there exist some interactive mechanism 𝒬 \\mathcal{Q} and two random seed distributions G 0 G_{0} and G 1 G_{1} such that 𝖯𝗈𝗐 ( G 0 ∥ G 1 ) ≤ g \\mathsf{Pow}(G_{0}\\|G_{1})\\leq g and for each b ∈ { 0 , 1 } b\\in\\{0,1\\} , 𝒬 ( G b ) \\mathcal{Q}(G_{b}) is an exact simulation of 𝒩 b 𝗉𝖺𝗂𝗋 \\mathcal{N}^{\\mathsf{pair}}_{b} (with the appropriate random seed distribution R 𝒩 R_{\\mathcal{N}} ) for any adversary. Specifically, for each b b , 𝒩 b 𝗉𝖺𝗂𝗋 ( R 𝒩 ) \\mathcal{N}^{\\mathsf{pair}}_{b}(R_{\\mathcal{N}}) and 𝒬 ( G b ) \\mathcal{Q}(G_{b}) are equivalent."},{"type":"para","text":"Given an adversary 𝒜 \\mathcal{A} that interacts with ℒ b 𝗉𝖺𝗂𝗋 \\mathcal{L}^{\\mathsf{pair}}_{b} for b ∈ { 0 , 1 } b\\in\\{0,1\\} , we describe how an augmented adversary 𝒜 ^ \\widehat{\\mathcal{A}} interacts with ℒ ^ b \\widehat{\\mathcal{L}}_{b} in the above colored box. Because we follow exactly the same structure as in Definition B.7 , after comparing the two descriptions line-by-line, it follows that 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℒ b 𝗉𝖺𝗂𝗋 ) \\mathsf{view}(\\mathcal{A}\\leftrightarrow\\mathcal{L}^{\\mathsf{pair}}_{b}) has the same distribution as 𝗏𝗂𝖾𝗐 ( 𝒜 ^ ↔ ℒ ^ b ) \\mathsf{view}(\\widehat{\\mathcal{A}}\\leftrightarrow\\widehat{\\mathcal{L}}_{b}) ."},{"type":"para","text":"As above, we use Fact A.14 , for each b ∈ { 0 , 1 } b\\in\\{0,1\\} , we can equivalently replace ℳ b 𝗇𝗉𝗉 ( R ℳ , R ℳ ′ ) \\mathcal{M}^{\\mathsf{npp}}_{b}(R_{\\mathcal{M}},R^{\\prime}_{\\mathcal{M}}) with 𝒫 ( F b ) \\mathcal{P}(F_{b}) , and 𝒩 b 𝗉𝖺𝗂𝗋 ( R 𝒩 ) \\mathcal{N}^{\\mathsf{pair}}_{b}(R_{\\mathcal{N}}) with 𝒬 ( G b ) \\mathcal{Q}(G_{b}) in the above description."},{"type":"para","text":"Hence, the first inequality below follows from the data processing inequality:"},{"type":"para","text":"where the second inequality follows from the data processing inequality and the exact simulations, and the last inequality follows from Fact A.10 . ∎"},{"type":"para","text":"In the literature, neighboring streams are typically defined in the Hamming sense. Specifically, given a collection 𝒪 \\mathcal{O} of objects, an (infinite) input stream is x : ℤ ≥ 1 → 𝒪 x:\\mathbb{Z}_{\\geq 1}\\rightarrow\\mathcal{O} . In our applications, each object in 𝒪 \\mathcal{O} captures the characteristics of a user."},{"type":"para","text":"Hamming-Style Neighboring Two streams x ∼ H x ′ x\\sim_{H}x^{\\prime} are neighboring in the Hamming sense if there exists some t ∈ ℤ t\\in\\mathbb{Z} such that σ ( τ ) = σ ′ ( τ ) \\sigma(\\tau)=\\sigma^{\\prime}(\\tau) for all τ ≠ t \\tau\\neq t ."},{"type":"para","text":"Edit-Style Neighboring. We are interested in the neighboring notion when an element may be inserted or deleted from a stream."},{"type":"para","text":"Given a stream x x and t ∈ ℤ t\\in\\mathbb{Z} , we use 𝖣𝖾𝗅𝖾𝗍𝖾 ( x ; t ) \\mathsf{Delete}(x;t) to denote the stream x ′ x^{\\prime} such that for all τ < t \\tau<t , x ′ [ τ ] = x [ τ ] x^{\\prime}[\\tau]=x[\\tau] and for all τ ≥ t \\tau\\geq t , x ′ [ τ ] = x [ τ + 1 ] x^{\\prime}[\\tau]=x[\\tau+1] ; in other words, the element at time t t is deleted in stream x x to produce x ′ x^{\\prime} ."},{"type":"para","text":"Two streams x ∼ 𝖤 x ′ x\\sim_{\\mathsf{E}}x^{\\prime} are neighboring in the edit sense if one stream can be obtained from the other by such a deletion operation."},{"type":"para","text":"Two finite sequences of the same length are neighboring (in the edit sense) if they are prefixes of such neighboring streams."},{"type":"heading","level":2,"text":"D.1 Motivating Example: Private Prefix-Sum for Edit-Style Neighboring Streams"},{"type":"para","text":"The input for the prefix-sum problem is a stream σ : ℤ > 0 → ℤ \\sigma:\\mathbb{Z}_{>0}\\to\\mathbb{Z} of integers, and the (accurate) output is a vector c c of prefix-sums, i.e., for t ≥ 1 t\\geq 1 , c [ t ] = ∑ i = 1 t σ [ i ] c[t]=\\sum_{i=1}^{t}\\sigma[i] . The well-known private prefix-sum mechanism [ DNP+10a , CSS11 ] based on the binary tree structure is designed for neighboring streams in the Hamming sense. At the step t 0 t_{0} at which the two streams differ, the difference must be bounded, for instance | σ ( t 0 ) − σ ′ ( t 0 ) | ≤ 1 |\\sigma(t_{0})-\\sigma^{\\prime}(t_{0})|\\leq 1 ."},{"type":"para","text":"Hamming-style neighboring captures the scenario when a user wants some protection on its data contents. However, whether a user has participated in the process may be regarded as sensitive information. This is captured by a neighboring notion in the edit sense. However, deleting one element from a stream can change the element for every step. For instance, if we delete the first element from the stream σ = ( 1 , 0 , 1 , 0 , … ) \\sigma=(1,0,1,0,\\ldots) , the resulting stream σ ′ = ( 0 , 1 , 0 , 1 , … ) \\sigma^{\\prime}=(0,1,0,1,\\ldots) will have every position flipped! This is as far away from being Hamming-neighboring as one can get, and indeed the DP mechanism based on binary trees is not designed to provide good privacy guarantees for these two streams. Specifically, for all odd time steps t t , the adversary can observe the noisy σ [ t ] + N t \\sigma[t]+N_{t} , where N t N_{t} is independently generated noise centered at 0. Therefore, by averaging over these numbers for odd time steps, the adversary can distinguish between σ \\sigma and σ ′ \\sigma^{\\prime} almost for certain."},{"type":"para","text":"Instead of re-designing the DP prefix-sum mechanism to cater for edit-style neighboring streams, our goal is to design a pre-processing mechanism 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} that somehow transforms edit-style neighboring streams into Hamming-style neighboring streams. Then, using the modular composition framework, we pass the processed stream to the tree-based DP prefix-sum mechanism."},{"type":"para","text":"Random Buffer to the Rescue. Edit-style neighboring notion has been considered in the context of oblivious data structures [ CCM+22 , ZSC+23 ] , and the high-level idea is to use a buffering technique to introduce random delays into the streams in the hope of eventually aligning the elements from the two edit-style neighboring streams. We give a simplified view of the procedure that processes a stream σ \\sigma of elements. It maintains a buffer queue that is initially empty. At each step t t , the element σ [ t ] \\sigma[t] enters the end of the queue. The procedure then makes two random decisions: (i) whether to remove any element from the head of the queue; (ii) if yes, then remove a random number of elements from the head of the queue and return their sum in the output stream."},{"type":"para","text":"Consider applying this random procedure to two edit-style neighboring streams, where we use alphabets to label bits at different position:"},{"type":"para","text":"σ = [ a , b , c , d , e , f ] = [ 1 , 0 , 1 , 0 , 1 , 0 ] \\sigma=[a,b,c,d,e,f]=[1,0,1,0,1,0] and σ ′ = [ b , c , d , e , f , g ] = [ 0 , 1 , 0 , 1 , 0 , 1 ] \\sigma^{\\prime}=[b,c,d,e,f,g]=[0,1,0,1,0,1] ."},{"type":"para","text":"Below are possible outcomes of the random batching process for each stream, where we sum up the bits in each non-empty batch."},{"type":"para","text":"Note that at the end of step 6, the queue contains { f } \\{f\\} for the case of σ \\sigma and contains { f , g } \\{f,g\\} for the case of σ ′ \\sigma^{\\prime} . In this particular lucky scenario, we see that the 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} procedure has converted edit-style neighboring streams to Hamming-style neighboring streams. However, observe that these two outcomes from the two scenarios do not occur with exactly the same probability, but similar probabilities if the 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} procedure is designed carefully. Moreover, for easy illustration, we give an oversimplified description of Hamming-style neighboring streams. We next give the formal and technical details."},{"type":"heading","level":2,"text":"D.2 Technical Description of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin}"},{"type":"para","text":"We use the following Hamming-style neighboring notion on bin sequences, which are suitable for private mechanisms that are designed expecting Hamming-style neighboring streams."},{"type":"para","text":"We use ℬ \\mathcal{B} to denote the collection of concrete bins, where a concrete bin contains an array of objects with some specified maximum array length. In our usage, a concrete bin is non-empty and is typically opaque in the sense that the adversary cannot observe its contents."},{"type":"para","text":"A bin sequence of length T T is denoted by B : [ 1 . . T ] → ℬ ∪ { ⊥ } B:[1..T]\\rightarrow\\mathcal{B}\\cup\\{\\bot\\} , where ⊥ \\bot means that no concrete bin is returned in that time step. Adjacent concrete bins with indices i < j i<j in a sequence B B mean that for i < t < j i<t<j , B [ t ] = ⊥ B[t]=\\bot ."},{"type":"para","text":"Two bin sequences B ∼ 𝖡 B ′ B\\sim_{\\mathsf{B}}B^{\\prime} of the same length T T are neighboring if the following holds."},{"type":"para","text":"For all t ∈ [ 1 . . T ] t\\in[1..T] , B [ t ] = ⊥ B[t]=\\bot iff B ′ [ t ] = ⊥ B^{\\prime}[t]=\\bot ."},{"type":"para","text":"Except for at most 2 indices t t in [ 1 . . T ] [1..T] , B [ t ] = B ′ [ t ] B[t]=B^{\\prime}[t] ."},{"type":"para","text":"If B B and B ′ B^{\\prime} differ in exactly one index, the corresponding index refers to the last concrete bin in both sequences. Moreover, the two bin contents must have the following form."},{"type":"para","text":"There exist elements w w and u u and (possibly empty) element arrays σ 1 \\sigma_{1} and σ 2 \\sigma_{2} such that the two differing bins from B B and B ′ B^{\\prime} contain the arrays:"},{"type":"para","text":"[ σ 1 , w , σ 2 ] [\\sigma_{1},w,\\sigma_{2}] and [ σ 1 , σ 2 , u ] [\\sigma_{1},\\sigma_{2},u] ."},{"type":"para","text":"If B B and B ′ B^{\\prime} differ in exactly two indices i < j i<j , then the two indices correspond to adjacent concrete bins in both sequences. Moreover, there exist elements w w and u u and element arrays σ 1 \\sigma_{1} , σ 2 \\sigma_{2} and σ 3 \\sigma_{3} such that two adjacent concrete bins from each bin sequence are illustrated as follows:"},{"type":"para","text":"[ σ 1 , w , σ 2 ] , ⊥ , … , ⊥ , [ u , σ 3 ] [\\sigma_{1},w,\\sigma_{2}],\\bot,\\ldots,\\bot,[u,\\sigma_{3}]"},{"type":"para","text":"[ σ 1 , σ 2 , u ] , ⊥ , … , ⊥ , [ σ 3 ] [\\sigma_{1},\\sigma_{2},u],\\bot,\\ldots,\\bot,[\\sigma_{3}]"},{"type":"para","text":"Observe that if each of two neighboring bin sequences are concatenated, two resulting item sequences will have lengths differ by at most 1, in which case deleting one element from the longer item sequence results in the shorter item sequence."},{"type":"para","text":"We need the following primitives to describe the 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} mechanism."},{"type":"para","text":"Let α > 1 \\alpha>1 . The symmetric geometric distribution Geom ( α ) \\text{Geom}(\\alpha) takes integer values such that the probability mass function at k k is α − 1 α + 1 ⋅ α − | k | . \\frac{\\alpha-1}{\\alpha+1}\\cdot\\alpha^{-|k|}."},{"type":"para","text":"It is known that using Geom ( e ε ) \\text{Geom}(e^{\\varepsilon}) as additive noise can mask two integers differing by 1 with ( ε , 0 ) (\\varepsilon,0) -DP. However, if we want the noise to have bounded support, then we truncate the noise and get only ( ε , δ ) (\\varepsilon,\\delta) -DP."},{"type":"para","text":"Let ε > 0 \\varepsilon>0 , δ ∈ ( 0 , 1 ) \\delta\\in(0,1) . Let U U be the smallest even positive integer such that Pr | Geom ( e ε ) | ≥ U 2 ≤ δ \\Pr{|\\text{Geom}(e^{\\varepsilon})|\\geq\\frac{U}{2}}\\leq\\delta , where U = O ( 1 ε log 1 δ ) . U=O(\\frac{1}{\\varepsilon}\\log\\frac{1}{\\delta}). The truncated geometric distribution G ( ε , δ ) G(\\varepsilon,\\delta) has support [ − U 2 , U 2 ] [-\\frac{U}{2},\\frac{U}{2}] , and is obtained by first sampling r r from Geom ( e ε ) \\text{Geom}(e^{\\varepsilon}) and then truncating r r within [ − U 2 , U 2 ] [-\\frac{U}{2},\\frac{U}{2}] , i.e., return min ( max ( − U 2 , r ) , U 2 ) \\min(\\max(-\\frac{U}{2},r),\\frac{U}{2}) ."},{"type":"para","text":"Differentially Private Streaming Prefix Sum with Always Error Bounds. We use the ( ε , δ ) (\\varepsilon,\\delta) -DP interactive mechanism from Theorem F.1 , denoted as 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\mathsf{StreamSum}_{\\varepsilon,\\delta} . Observe that 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\mathsf{StreamSum}_{\\varepsilon,\\delta} is not necessarily private against adaptive adversaries. Even though we will prove that 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} is adaptively NPDP, we do not need the adaptive property from 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\mathsf{StreamSum}_{\\varepsilon,\\delta} , but rather that the additive error from all time steps t t be bounded by some function E ( t ) E(t) with probability 1."},{"type":"para","text":"Mechanism Intuition. In Algorithm 3 , we reinterpret 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} (which was introduced in [ CCM+22 , ZSC+23 ] ) as an interactive mechanism. On a high level, the mechanism maintains an internal buffer that is a subarray. In each time step t t , the mechanism takes an element x [ t ] x[t] from the input stream and appends it to the end of the buffer. The mechanism decides according to some rules if a new bin should be created from a prefix of the buffer contents. If yes, a prefix from the buffer is removed to form a new bin that is returned in this step; otherwise, nothing is returned in this step and we denote the output by ⊥ \\bot ."},{"type":"para","text":"At the end of every step t t , with probability 1, the concatenation of the concrete bins that have been output so far is a prefix of the input stream with length at least t − O ( 1 ε ⋅ log t ⋅ ( log t + log 1 δ ) ) t-O(\\frac{1}{\\varepsilon}\\cdot\\log t\\cdot(\\log t+\\log\\frac{1}{\\delta})) ."},{"type":"para","text":"Observe that because the additive error of 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\mathsf{StreamSum}_{\\varepsilon,\\delta} holds with probability 1, the 𝖡𝗎𝖿 \\mathsf{Buf} will never run into underflow in line 15 . Actually, to ensure no underflow, it suffices to have t < S [ 𝖨𝗇𝖽𝖾𝗑 ] + E [ 𝖨𝗇𝖽𝖾𝗑 ] t<S[\\mathsf{Index}]+E[\\mathsf{Index}] in line 12 ; we shall see that the extra term 2 U 2U is crucial to argue that 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} is private against adaptive adversaries."},{"type":"para","text":"Note that at the end of time step t t , we have the invariant that t < S [ 𝖨𝗇𝖽𝖾𝗑 ] + E [ 𝖨𝗇𝖽𝖾𝗑 ] + 2 U t<S[\\mathsf{Index}]+E[\\mathsf{Index}]+2U ; this holds because S [ 𝖨𝗇𝖽𝖾𝗑 + 1 ] − S [ 𝖨𝗇𝖽𝖾𝗑 ] ≥ U ≥ 1 S[\\mathsf{Index}+1]-S[\\mathsf{Index}]\\geq U\\geq 1 and E [ 𝖨𝗇𝖽𝖾𝗑 + 1 ] ≥ E [ 𝖨𝗇𝖽𝖾𝗑 ] E[\\mathsf{Index}+1]\\geq E[\\mathsf{Index}] ."},{"type":"para","text":"Moreover, at this moment, the number of elements in the queue 𝖡𝗎𝖿 \\mathsf{Buf} is"},{"type":"para","text":"using the crude bound 𝖨𝗇𝖽𝖾𝗑 ≤ t \\mathsf{Index}\\leq t . ∎"},{"type":"heading","level":2,"text":"D.3 Adaptive Privacy Analysis"},{"type":"para","text":"Privacy of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} . Observe that in each step, the view of the adversary just consists of one bit of information: whether a concrete bin or ⊥ \\bot is returned. Moreover, the distribution of the view is always the same, regardless of what the input stream is. However, as noted in [ ZSC+23 ] , when a mechanism has hidden outputs that will be passed to another mechanism, one needs to consider the joint distribution of the view and the output, which is why NPDO was defined, which we have extended to interactive mechanisms in Definition B.9 ."},{"type":"para","text":"Construction of Neighbor-Preserving Paired (NPP) Simulation for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} . The proofs in [ ZSC+23 ] already contain an implicit description of how an NPP simulation for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} is constructed, but we will make it explicit here. Recall that the adversary will generate two input sequences ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) in an NPP simulation in Definition B.8 . For b ∈ { 0 , 1 } b\\in\\{0,1\\} , the mechanism ℳ b 𝗇𝗉𝗉 \\mathcal{M}^{\\mathsf{npp}}_{b} will do two things in each step t t ."},{"type":"para","text":"Simulate the behavior of the original mechanism ℳ \\mathcal{M} on the sequence x ( b ) [ 1 . . t ] x^{(b)}[1..t] to produce the corresponding view v [ t ] v[t] and output y ( b ) [ t ] y^{(b)}[t] ."},{"type":"para","text":"Generate another output y ( b ¯ ) [ t ] y^{(\\overline{b})}[t] such that the output sequences y ( b ) [ 1 . . t ] y^{(b)}[1..t] and y ( b ¯ ) [ 1 . . t ] y^{(\\overline{b})}[1..t] are neighboring."},{"type":"para","text":"The counter-intuitive point is that y ( b ¯ ) [ 1 . . t ] y^{(\\overline{b})}[1..t] is not necessarily the simulation of the original ℳ \\mathcal{M} on the other input sequence x ( b ¯ ) x^{(\\overline{b})} . In fact, it is possible that y ( b ¯ ) [ 1 . . t ] y^{(\\overline{b})}[1..t] may not even be in the support of ℳ \\mathcal{M} on x ( b ¯ ) x^{(\\overline{b})} ."},{"type":"para","text":"In general, extra randomness other than that used in the original ℳ \\mathcal{M} may be needed to generate y ( b ¯ ) [ t ] y^{(\\overline{b})}[t] . However, for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} , no extra randomness is needed."},{"type":"para","text":"Technical Assumption on Input Pair. Note that in Definition B.8 , the adversary needs to ensure that at every time step t t , the pair ( x ( 0 ) [ 1 . . t ] , x ( 1 ) [ 1 . . t ] ) (x^{(0)}[1..t],x^{(1)}[1..t]) of input sequences generated so far are neighboring (in the edit sense)."},{"type":"para","text":"For instance, for t = 4 t=4 , the pair ( [ 0 , 1 , 0 , 1 ] , [ 1 , 0 , 1 , 0 ] ) ([0,1,0,1],[1,0,1,0]) of sequences are neighboring in the edit sense, but the issue is that we do not know which stream has an element deleted. If next input pair at t = 5 t=5 is ( 0 , a ) (0,a) , then we know that in the two streams ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) , the first element from x ( 0 ) x^{(0)} is deleted to produce x ( 1 ) x^{(1)} ; on the other hand, if the next input pair at t = 5 t=5 is ( a , 1 ) (a,1) , then the opposite is true."},{"type":"para","text":"However, in the construction of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 b 𝗇𝗉𝗉 \\mathsf{RandBin}^{\\mathsf{npp}}_{b} , we would need to know whether an element is deleted from x ( 0 ) x^{(0)} or x ( 1 ) x^{(1)} . Observe that this issue will not arise if a stream contains only distinct elements. In practice, this is a valid assumption because an item is typically tagged with unique metadata."},{"type":"para","text":"Hence, in the proof, we assume that if the adversary first generates x ( 0 ) [ t ] ≠ x ( 1 ) [ t ] x^{(0)}[t]\\neq x^{(1)}[t] at step t t , then when we receive a pair at the beginning of step t + 1 t+1 , we will know from which stream an element is deleted. This will not be too late, because the additive error for the prefix-sum S [ 𝖨𝗇𝖽𝖾𝗑 ] S[\\mathsf{Index}] is at most E [ 𝖨𝗇𝖽𝖾𝗑 ] E[\\mathsf{Index}] , we can ensure that the 𝖡𝗎𝖿 \\mathsf{Buf} queue will never be empty, i.e., an element arriving at step t t will never be returned in a concrete bin in the same step t t ."},{"type":"para","text":"We focus on the description of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 \\mathsf{RandBin}^{\\mathsf{npp}}_{0} (because the behavior of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 \\mathsf{RandBin}^{\\mathsf{npp}}_{1} is symmetric); recall that the adversary generates a pair of time sequences ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) ."},{"type":"para","text":"The first component simulates 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} from Algorithm 3 on the sequence x ( 0 ) x^{(0)} ; in every step t t , it will output y ( 0 ) [ t ] y^{(0)}[t] that is either a concrete bin or ⊥ \\bot ."},{"type":"para","text":"We describe the second component more carefully, which is how y ( 1 ) [ t ] y^{(1)}[t] is constructed in each step t t . Note that since the two output sequences must be neighboring as in Definition D.2 , we will make sure y ( 0 ) [ t ] = ⊥ y^{(0)}[t]=\\bot iff y ( 1 ) [ t ] = ⊥ y^{(1)}[t]=\\bot . Moreover, we need to ensure that at most two adjacent concrete bins are different in a specific way. We consider two cases."},{"type":"para","text":"Suppose an element x ( 0 ) [ t ] = w x^{(0)}[t]=w is deleted from x ( 0 ) x^{(0)} to produce the stream x ( 1 ) x^{(1)} . In the output sequence y ( 0 ) y^{(0)} , we focus on two adjacent concrete bins that contains the deleted item w w and the next concrete bin:"},{"type":"para","text":"[ σ 1 , w , σ 2 ] , ⊥ , … , ⊥ , [ u , σ 3 ] [\\sigma_{1},w,\\sigma_{2}],\\bot,\\ldots,\\bot,[u,\\sigma_{3}] ,"},{"type":"para","text":"where σ 1 \\sigma_{1} , σ 2 \\sigma_{2} and σ 3 \\sigma_{3} are (potentially empty) subarrays."},{"type":"para","text":"Then, in the output sequence y ( 1 ) y^{(1)} , those two concrete bins are modified to:"},{"type":"para","text":"[ σ 1 , σ 2 , u ] , ⊥ , … , ⊥ , [ σ 3 ] [\\sigma_{1},\\sigma_{2},u],\\bot,\\ldots,\\bot,[\\sigma_{3}] ."},{"type":"para","text":"Note that the bin y ( 0 ) [ τ ] = [ σ 1 , w , σ 2 ] y^{(0)}[\\tau]=[\\sigma_{1},w,\\sigma_{2}] is returned in the output sequence y ( 0 ) y^{(0)} in some step τ > t \\tau>t ; observe that at the end of step τ \\tau , in the remaining 𝖡𝗎𝖿 \\mathsf{Buf} queue, u u should be the first element. Moreover, u u should have already appeared in the input pair together with the last element of σ \\sigma . Hence, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 \\mathsf{RandBin}^{\\mathsf{npp}}_{0} is able to produce y ( 1 ) [ τ ] = [ σ 1 , σ 2 , u ] y^{(1)}[\\tau]=[\\sigma_{1},\\sigma_{2},u] ."},{"type":"para","text":"Moreover, note that if [ u , σ 3 ] [u,\\sigma_{3}] contains exactly U U elements, then [ σ 3 ] [\\sigma_{3}] will not be in the support of simulating the actual 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} ."},{"type":"para","text":"Suppose an element x ( 1 ) [ t ] = w x^{(1)}[t]=w is deleted from x ( 1 ) x^{(1)} to produce the stream x ( 0 ) x^{(0)} . Again, we focus on two adjacent concrete bins from the output sequence y ( 0 ) y^{(0)} :"},{"type":"para","text":"[ σ 1 , σ 2 , u ] , ⊥ , … , ⊥ , [ σ 3 ] [\\sigma_{1},\\sigma_{2},u],\\bot,\\ldots,\\bot,[\\sigma_{3}] ,"},{"type":"para","text":"where [ σ 1 , σ 2 , u ] [\\sigma_{1},\\sigma_{2},u] is the bin that contains the element x ( 0 ) [ t ] x^{(0)}[t] , which is also the first element in the subarray [ σ 2 , u ] [\\sigma_{2},u] . Note that in the case that σ 2 \\sigma_{2} is empty, then u = x ( 0 ) [ t ] u=x^{(0)}[t] ."},{"type":"para","text":"Then, in the output sequence y ( 1 ) y^{(1)} , those two concrete bins are modified to:"},{"type":"para","text":"[ σ 1 , w , σ 2 ] , ⊥ , … , ⊥ , [ u , σ 3 ] [\\sigma_{1},w,\\sigma_{2}],\\bot,\\ldots,\\bot,[u,\\sigma_{3}]"},{"type":"para","text":"Note that if [ σ 3 ] [\\sigma_{3}] contains 2 U 2U elements, then [ u , σ 3 ] [u,\\sigma_{3}] will not appear in the support of simulating the actual 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} ."},{"type":"para","text":"From construction, the two output bin sequences y ( 0 ) y^{(0)} and y ( 1 ) y^{(1)} are neighboring as in Definition D.2 ."},{"type":"para","text":"With the formal terminology in place, we can formally restate the result in [ ZSC+23 ] ."},{"type":"para","text":"The interactive mechanism in Algorithm 3 is non-adaptively ( ε , δ ) (\\varepsilon,\\delta) -NPDP. Specifically, in terms of power function, the NPP simulation ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\mathsf{RandBin}^{\\mathsf{npp}}_{0},\\mathsf{RandBin}^{\\mathsf{npp}}_{1}) in Definition D.6 satisfies: 𝖯𝗈𝗐 NA ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 ∥ 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) ≤ 𝖣𝖯 ε , δ \\mathsf{Pow}^{{\\textsc{NA}}}(\\mathsf{RandBin}^{\\mathsf{npp}}_{0}\\|\\mathsf{RandBin}^{\\mathsf{npp}}_{1})\\leq\\mathsf{DP}_{\\varepsilon,\\delta} ."},{"type":"para","text":"Suppose for some power function f f , the interactive mechanism 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} in Algorithm 3 is non-adaptively f f -NPDP (with respect to input neighboring notion in Definition D.1 and output neighboring notion in Definition D.2 ). Then, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} is also adaptively f f -NPDP; specifically, the NPP simulation ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\mathsf{RandBin}^{\\mathsf{npp}}_{0},\\mathsf{RandBin}^{\\mathsf{npp}}_{1}) in Definition D.6 satisfies:"},{"type":"para","text":"𝖯𝗈𝗐 ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 ∥ 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) ≤ f \\mathsf{Pow}(\\mathsf{RandBin}^{\\mathsf{npp}}_{0}\\|\\mathsf{RandBin}^{\\mathsf{npp}}_{1})\\leq f ."},{"type":"para","text":"By joint convexity in Fact A.9 , we assume a deterministic adversary 𝒜 {\\mathcal{A}} that interacts with ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\mathsf{RandBin}^{\\mathsf{npp}}_{0},\\mathsf{RandBin}^{\\mathsf{npp}}_{1}) . Moreover, without loss of generality, to simplify the notation, we assume that there is a canonical sequence x ( t ) := t x(t):=t ."},{"type":"para","text":"Then, for some initial steps t t , the default action of the adversary is to generate a pair ( x t ( 0 ) , x t ( 1 ) ) (x^{(0)}_{t},x^{(1)}_{t}) satisfying x t ( 0 ) = x t ( 1 ) = t x^{(0)}_{t}=x^{(1)}_{t}=t until some critical step t c t_{c} in which the adversary decides (based on its observation) to deviate from the default behavior. It can either:"},{"type":"para","text":"(i) delete from x ( 0 ) x^{(0)} , i.e., for t ≥ t c t\\geq t_{c} , x t ( 0 ) = t + 1 x^{(0)}_{t}=t+1 and x t ( 1 ) = t x^{(1)}_{t}=t , or"},{"type":"para","text":"(ii) delete from x ( 1 ) x^{(1)} , i.e., for t ≥ t c t\\geq t_{c} , x t ( 0 ) = t x^{(0)}_{t}=t and x t ( 1 ) = t + 1 x^{(1)}_{t}=t+1 ."},{"type":"para","text":"Observe that before step t c t_{c} , the two input sequences x ( 0 ) x^{(0)} and x ( 1 ) x^{(1)} are identical. Hence, the single decision made by the adversary is at which time step t c t_{c} it wants to deviate from the default behavior, and it does not really matter whether it deletes from x ( 0 ) x^{(0)} or x ( 1 ) x^{(1)} ."},{"type":"para","text":"Augmented View. We assume a more powerful adversary that can observe, in addition to its normal view (which consists of the contents of both bins y ( 0 ) [ t ] y^{(0)}[t] and y ( 1 ) [ t ] ) y^{(1)}[t]) ), the noisy prefix-sums S [ ⋅ ] S[\\cdot] once they are created. We note that the proof in [ ZSC+23 ] also allows the (non-adaptive) adversary to have this augmented view. However, it is important when the adversary learns about these noisy prefix-sums."},{"type":"para","text":"Default Distribution. Suppose the canonical sequence x x is used in both of ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) . Then, the behaviors of both interactive mechanisms ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\mathsf{RandBin}^{\\mathsf{npp}}_{0},\\mathsf{RandBin}^{\\mathsf{npp}}_{1}) are identical. We use Ω \\Omega to denote the space of views, and 𝒫 \\mathcal{P} to denote the corresponding distribution of views. Each ω ∈ Ω \\omega\\in\\Omega consists of the view and output of every time step t ≥ 1 t\\geq 1 ."},{"type":"para","text":"Decision of Adversary 𝒜 {\\mathcal{A}} . Because we have assumed that 𝒜 {\\mathcal{A}} is deterministic, for each ω ∈ Ω \\omega\\in\\Omega , either (i) 𝒜 {\\mathcal{A}} will never deviate from the default behavior, or (ii) 𝒜 {\\mathcal{A}} will deviate at the beginning of some step t t , in which case the resulting view will be different from ω \\omega starting from step t t onwards. We use Ω t ⊆ Ω \\Omega_{t}\\subseteq\\Omega to denote the collection of view in which the adversary deviates at the beginning of step t t ."},{"type":"para","text":"Consider some ω ∈ Ω t \\omega\\in\\Omega_{t} . As explained above, this means that the adversary decides to deviate at the beginning of step t t , after observing the information in ω \\omega up to step t − 1 t-1 . This means it will generate a pair ( x ( 0 ) [ t ] , x ( 1 ) [ t ] ) (x^{(0)}[t],x^{(1)}[t]) of different inputs, and as a result the distributions of outputs for ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\mathsf{RandBin}^{\\mathsf{npp}}_{0},\\mathsf{RandBin}^{\\mathsf{npp}}_{1}) from step t t onwards may be different. For such ω ∈ Ω t \\omega\\in\\Omega_{t} , we use g ω := 𝖯𝗈𝗐 ( V ω ( 0 ) ∥ V ω ( 1 ) ) g_{\\omega}:=\\mathsf{Pow}(V^{(0)}_{\\omega}\\|V^{(1)}_{\\omega}) to denote the power function between the corresponding two view distributions V ω ( 0 ) V^{(0)}_{\\omega} and V ω ( 1 ) V^{(1)}_{\\omega} starting from step t t ."},{"type":"para","text":"Key Insight. We next show that for ω ∈ Ω t \\omega\\in\\Omega_{t} , the decision of 𝒜 {\\mathcal{A}} to deviate the beginning of step t t is equivalent to a randomized non-adaptive adversary that interacts with ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\mathsf{RandBin}^{\\mathsf{npp}}_{0},\\mathsf{RandBin}^{\\mathsf{npp}}_{1}) . Suppose i i is the index of concrete bins such that S [ i − 1 ] + E [ i − 1 ] + 2 U < t ≤ S [ i ] + E [ i ] + 2 U S[i-1]+E[i-1]+2U<t\\leq S[i]+E[i]+2U . First, analyze what information the adversary has gathered till the beginning of step t t :"},{"type":"para","text":"At the end of step S [ i − 1 ] + E [ i − 1 ] + 2 U S[i-1]+E[i-1]+2U , the adversary learns the exact count C [ i − 1 ] C[i-1] (and also C [ i − 2 ] , … , C [ 1 ] C[i-2],\\ldots,C[1] ), because so far the two output sequences are identical. Moreover, it also learns the noisy prefix-sum S [ i ] S[i] that is an estimate of ∑ j = 1 i C [ j ] \\sum_{j=1}^{i}C[j] ."},{"type":"para","text":"At the beginning of step t t , it knows the exact number of elements in the 𝖡𝗎𝖿 \\mathsf{Buf} queue, which is larger than 2 U 2U . Hence, the input pair of elements generated at time t t will definitely not go into the concrete bin with index i i . This is why we need the term 2 U 2U in line 12 ."},{"type":"para","text":"The crucial observation is that at this moment, the number C [ i + 1 ] C[i+1] of elements in the ( i + 1 ) (i+1) -st concrete bin has not been sampled yet."},{"type":"para","text":"Hence, if the adversary 𝒜 {\\mathcal{A}} decides to deviate at the beginning of step t t , it will know that the element at step t t will be the N ω := ( t − ∑ j = 1 i C [ j ] ) N_{\\omega}:=(t-\\sum_{j=1}^{i}C[j]) -th element in the 𝖡𝗎𝖿 \\mathsf{Buf} queue."},{"type":"para","text":"The adversary does not know C [ i ] C[i] , but based on the knowledge of S [ i ] , S [ i − 1 ] , … , S [ 1 ] S[i],S[i-1],\\ldots,S[1] and C [ i − 1 ] , … , C [ 1 ] C[i-1],\\ldots,C[1] , it knows the conditional distribution of C [ i ] C[i] , and hence, can derive the distribution of N ω N_{\\omega} ."},{"type":"para","text":"Conclusion. For ω ∈ Ω t \\omega\\in\\Omega_{t} , the adversary 𝒜 {\\mathcal{A}} deciding to deviate at the beginning of step t t will be equivalent to another instance where a randomized non-adaptive adversary decides upfront to deviate at some step according to the distribution N ω N_{\\omega} . From the hypothesis, we conclude that the power function satisfies g ω = 𝖯𝗈𝗐 ( V ω ( 0 ) ∥ V ω ( 1 ) ) ≤ f g_{\\omega}=\\mathsf{Pow}(V^{(0)}_{\\omega}\\|V^{(1)}_{\\omega})\\leq f ."},{"type":"para","text":"Then, the whole proof concludes because:"},{"type":"para","text":"where the inequality follows from the composition rule of power functions in Fact A.10 ."},{"type":"para","text":"∎"},{"type":"para","text":"Fact D.7 and Lemma D.8 immediately give the following corollary."},{"type":"para","text":"The interactive mechanism 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\mathsf{RandBin} in Algorithm 3 is adaptively ( ε , δ ) (\\varepsilon,\\delta) -NPDP with input neighboring notion ∼ 𝖤 \\sim_{\\mathsf{E}} and output neighboring notion ∼ 𝖡 \\sim_{\\mathsf{B}} ."},{"type":"para","text":"This section proves the adaptive-safety certification used in the modular pipeline. The goal is to identify structural conditions under which an ordinary non-adaptive privacy analysis of a continual mechanism already implies privacy against adaptive adversaries. The result applies to Hamming-style neighboring input streams and is stated in the power-function language used throughout the appendix."},{"type":"para","text":"The first condition is a fresh-randomness, prefix-causality condition. At round t t , the mechanism may depend on the entire input prefix x [ 1 . . t ] x[1..t] , but the only randomness used for the round- t t output is a fresh seed ω t \\omega_{t} independent of all other rounds. This rules out hidden cross-round correlations introduced by reused noise, while still allowing the release at time t t to be an arbitrary deterministic function of the input prefix."},{"type":"para","text":"An (abstract) interactive mechanism with transition function ℳ : ℛ × 𝒰 ∗ → 𝒰 \\mathcal{M}:\\mathcal{R}\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} is independently decomposable if the following properties hold."},{"type":"para","text":"Each random seed r ∈ ℛ r\\in\\mathcal{R} takes the form r = ( ω 1 , ω 2 , … ) ∈ Ω ℕ r=(\\omega_{1},\\omega_{2},\\ldots)\\in\\Omega^{\\mathbb{N}} . Moreover, the mechanism samples ω t \\omega_{t} independently from some distribution R t R_{t} in time step t t ."},{"type":"para","text":"For each t ≥ 1 t\\geq 1 , there exists a deterministic function ℳ t \\mathcal{M}_{t} such that the output y t ← ℳ t ( ω t ; x [ 1 . . t ] ) ∈ 𝒰 y_{t}\\leftarrow\\mathcal{M}_{t}(\\omega_{t};x[1..t])\\in\\mathcal{U} depends only on x [ 1 . . t ] x[1..t] and the randomness ω t \\omega_{t} sampled in step t t ."},{"type":"heading","level":4,"text":"Quantitatively Decomposable Mechanisms."},{"type":"para","text":"Independent decomposability alone records where the randomness enters, but does not quantify how much a change in the input prefix can affect a one-round release. We therefore add a quantitative condition. The condition says that, for each round t t , the power function between the two possible round- t t output laws is determined exactly by the distance between two aggregates of selected input coordinates. The invariant-pair condition below ensures that common aggregation context can be ignored: only the part of the aggregate containing the Hamming discrepancy contributes to the distance."},{"type":"para","text":"Let d : 𝒰 × 𝒰 → ℝ ≥ 0 d:\\mathcal{U}\\times\\mathcal{U}\\rightarrow\\mathbb{R}_{\\geq 0} be a metric, and let ( 𝒰 , ⊕ , 0 ⊕ ) (\\mathcal{U},\\oplus,0_{\\oplus}) be a monoid. We say that ( d , ⊕ ) (d,\\oplus) is an invariant pair (under common aggregation context) if for all a , b , u , u ′ ∈ 𝒰 a,b,u,u^{\\prime}\\in\\mathcal{U} ,"},{"type":"para","text":"Notation. For an index set I = { τ 1 < τ 2 < ⋯ < τ k } ⊆ [ 1 . . t ] I=\\{\\tau_{1}<\\tau_{2}<\\cdots<\\tau_{k}\\}\\subseteq[1..t] and an input history x [ 1 . . t ] ∈ 𝒰 t x[1..t]\\in\\mathcal{U}^{t} , define"},{"type":"para","text":"For I = ∅ I=\\emptyset , we use the convention ⨁ τ ∈ I x [ τ ] := 0 ⊕ \\bigoplus_{\\tau\\in I}x[\\tau]:=0_{\\oplus} . Since ⊕ \\oplus is associative and the elements are aggregated in increasing index order, this expression is well-defined without requiring commutativity."},{"type":"para","text":"An interactive mechanism ℳ : ℛ × 𝒰 ∗ → 𝒰 \\mathcal{M}:\\mathcal{R}\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} is quantitatively decomposable with respect to an invariant pair ( d , ⊕ ) (d,\\oplus) in the sense of Definition E.2 if it is independently decomposable in the sense of Definition E.1 , and there exist a sequence of index sets { I t } t ≥ 1 \\{I_{t}\\}_{t\\geq 1} with I t ⊆ [ 1 . . t ] I_{t}\\subseteq[1..t] and, for every t ≥ 1 t\\geq 1 and θ ≥ 0 \\theta\\geq 0 , a power function g t ( θ ) : [ 0 , 1 ] → [ 0 , 1 ] g_{t}(\\theta):[0,1]\\rightarrow[0,1] such that the following hold."},{"type":"para","text":"For every t ≥ 1 t\\geq 1 , g t ( θ ) ≤ g t ( θ ′ ) g_{t}(\\theta)\\leq g_{t}(\\theta^{\\prime}) whenever 0 ≤ θ ≤ θ ′ 0\\leq\\theta\\leq\\theta^{\\prime} ."},{"type":"para","text":"For every t ≥ 1 t\\geq 1 and every pair of input histories x [ 1 . . t ] , x ′ [ 1 . . t ] ∈ 𝒰 t x[1..t],x^{\\prime}[1..t]\\in\\mathcal{U}^{t} ,"},{"type":"para","text":"Here, ℳ t ( R t ; x [ 1 . . t ] ) \\mathcal{M}_{t}(R_{t};x[1..t]) denotes the distribution of ℳ t ( ω t ; x [ 1 . . t ] ) \\mathcal{M}_{t}(\\omega_{t};x[1..t]) when ω t \\omega_{t} is sampled from R t R_{t} ."},{"type":"para","text":"We can now state the certification theorem. The only-if direction is immediate, since non-adaptive adversaries are a special case of adaptive adversaries. The content is the converse: for quantitatively decomposable mechanisms, adaptivity does not increase the worst-case power-function profile. Thus any finite-horizon non-adaptive f f -DP proof automatically lifts to the paired-simulation adaptive game."},{"type":"para","text":"Consider the Hamming-style neighboring relation on equal-length input sequences in 𝒰 \\mathcal{U} , where two sequences are neighboring if they differ in at most one position. Let f f be a power function. Suppose ℳ : ℛ × 𝒰 ∗ → 𝒰 \\mathcal{M}:\\mathcal{R}\\times\\mathcal{U}^{*}\\rightarrow\\mathcal{U} is quantitatively decomposable in the sense of Definition E.3 . If ℳ \\mathcal{M} is non-adaptively f f -DP in the paired-simulation sense of Definition B.3 , for every finite horizon, then ℳ \\mathcal{M} is adaptively f f -DP in the same sense, for every finite horizon."},{"type":"para","text":"We prove the following finite-horizon statement by induction on T T : for every power function f f , if ℳ \\mathcal{M} is non-adaptively f f -DP up to horizon T T , then ℳ \\mathcal{M} is adaptively f f -DP up to horizon T T . This implies the theorem because the theorem assumes the non-adaptive guarantee for every finite horizon."},{"type":"para","text":"We first reduce to deterministic adversaries. A randomized adversary can be viewed as first sampling its random seed and then running a deterministic adversary. For every fixed seed, the deterministic-adversary case gives the desired power-function bound. Taking the mixture over the same seed distribution under the two secret bits preserves the bound by the joint convexity property in Fact A.9 . Hence it suffices to consider deterministic adaptive adversaries."},{"type":"para","text":"We also assume, without loss of generality, that the adversary observes the entire output of ℳ \\mathcal{M} in each round. If the real adversary observes only a view of the output, the real view is obtained from the full-output transcript by post-processing, and the desired bound follows from data processing. If the mechanism has an initial output before the adversary sends any input, this initial output has the same distribution under the two secret bits; it may be included as an additional common first observation and does not affect the privacy bound. Thus we count rounds from the first adversarial input."},{"type":"para","text":"Let 𝒜 {\\mathcal{A}} be a deterministic adaptive adversary in the paired simulation ( ℳ 0 𝗉𝖺𝗂𝗋 , ℳ 1 𝗉𝖺𝗂𝗋 ) (\\mathcal{M}^{\\mathsf{pair}}_{0},\\mathcal{M}^{\\mathsf{pair}}_{1}) . We pad terminated executions with a fixed halt symbol, so all transcripts have length T T . This padding is post-processing and does not change the privacy claim. We must prove"},{"type":"para","text":"The base case T = 1 T=1 is immediate. With only one input round, the adversary has no previous mechanism output on which to adapt, so the adaptive and non-adaptive paired simulations coincide."},{"type":"para","text":"Assume the statement holds for horizon T − 1 T-1 , and consider horizon T > 1 T>1 . Let"},{"type":"para","text":"be the first pair of inputs produced by 𝒜 {\\mathcal{A}} . Since 𝒜 {\\mathcal{A}} is deterministic, this pair is fixed. Let"},{"type":"para","text":"be the first-round output distribution under secret bit b b ."},{"type":"para","text":"We split into two cases."},{"type":"para","text":"Case 1: the first round is the challenge round. Assume x ( 0 ) [ 1 ] ≠ x ( 1 ) [ 1 ] x^{(0)}[1]\\neq x^{(1)}[1] , and write"},{"type":"para","text":"Because the paired simulation only allows Hamming-neighboring prefixes, once the first inputs differ, every later input pair produced by the adversary must have equal components. That is, on every non-terminated transcript prefix and for every t ≥ 2 t\\geq 2 , the adversary must output a pair of the form ( x t , x t ) (x_{t},x_{t}) . We use the finite-horizon convention that terminated executions are padded with a fixed halt symbol; the padded rounds have identical outputs under the two secret bits and hence contribute the identity power function."},{"type":"para","text":"For each round t t , define"},{"type":"para","text":"We first show that, at every round t t and every transcript prefix that can arise before round t t , the conditional next-output distributions under the two secret bits have power function exactly h t h_{t} ."},{"type":"para","text":"Fix such a transcript prefix. Since the adversary is deterministic, conditioning on this common prefix fixes all input pairs generated so far. As argued above, the two input histories entering round t t differ only in the first coordinate. If 1 ∉ I t 1\\notin I_{t} , then the two aggregates"},{"type":"para","text":"are identical. Hence their distance is zero. Applying Definition E.3 to identical histories gives g t ( 0 ) = 𝖨𝖽 g_{t}(0)=\\mathsf{Id} , and therefore the one-round power function is exactly 𝖨𝖽 \\mathsf{Id} ."},{"type":"para","text":"If 1 ∈ I t 1\\in I_{t} , then the two aggregates have the form"},{"type":"para","text":"for the same common suffix aggregate c c over the remaining indices in I t I_{t} . By the invariant-pair property,"},{"type":"para","text":"Hence Definition E.3 gives the exact one-round identity"},{"type":"para","text":"Thus the one-round power function is exactly h t h_{t} , and h t h_{t} depends only on t t , not on the transcript prefix."},{"type":"para","text":"We now justify that these exact one-round identities tensorize to an exact identity for the whole adaptive transcript in this case. Let P b P_{b} be the distribution of the full length- T T transcript under secret bit b b . For a transcript prefix s = y [ 1 . . t − 1 ] s=y[1..t-1] , let K t , s b K^{b}_{t,s} be the conditional distribution of y t y_{t} given the prefix s s under bit b b . By independent decomposability, K t , s b K^{b}_{t,s} is generated using the fresh seed R t R_{t} , which is independent of the previous seeds. From the preceding paragraph,"},{"type":"para","text":"for every prefix s s that can arise."},{"type":"para","text":"We use the following finite-experiment characterization of power functions. For two finite distributions P P and Q Q , the power function 𝖯𝗈𝗐 ( P ∥ Q ) \\mathsf{Pow}(P\\|Q) is determined by the decreasing rearrangement of the likelihood ratio Q ( ω ) / P ( ω ) Q(\\omega)/P(\\omega) under ω ∼ P \\omega\\sim P ; equivalently, two pairs of finite distributions with the same power function have the same likelihood ratio distribution under their first distribution, up to the usual splitting of atoms. Moreover, tensor product of power functions corresponds to taking the product of independent likelihood ratios."},{"type":"para","text":"Apply this characterization to the conditional kernels. For each prefix s s , the likelihood ratio"},{"type":"para","text":"has, under y t ∼ K t , s 0 y_{t}\\sim K^{0}_{t,s} , the likelihood-ratio distribution determined by h t h_{t} . This distribution is the same for every prefix s s . Since R t R_{t} is fresh and independent of the previous seeds, the conditional likelihood-ratio increment at round t t is independent of the past after conditioning on the prefix, and its distribution is determined only by h t h_{t} ."},{"type":"para","text":"For a full transcript y [ 1 . . T ] y[1..T] , the likelihood ratio between P 1 P_{1} and P 0 P_{0} factors as"},{"type":"para","text":"Under P 0 P_{0} , the factors in this product have the same joint distribution as independent likelihood-ratio variables whose one-step power functions are h 1 , … , h T h_{1},\\ldots,h_{T} . Therefore the power function of the full adaptive transcript pair is exactly the tensor product:"},{"type":"para","text":"where the second equality uses 𝖨𝖽 ⊗ q = q \\mathsf{Id}\\otimes q=q for every power function q q ."},{"type":"para","text":"Finally, consider any non-adaptive adversary that submits the fixed first pair x ( 0 ) [ 1 ] , x ( 1 ) [ 1 ] x^{(0)}[1],x^{(1)}[1] and then submits identical inputs in every later round. The later common inputs may be chosen arbitrarily. By the invariant pair property, they only contribute common aggregation context and therefore do not change the one-step power labels h t h_{t} . For this non-adaptive input pair, the same tensor product is the power function of the transcript pair:"},{"type":"para","text":"Since ℳ \\mathcal{M} is non-adaptively f f -DP up to horizon T T , this tensor product is at most f f . Hence"},{"type":"para","text":"in Case 1."},{"type":"para","text":"Case 2: the first round is not the challenge round. Assume x ( 0 ) [ 1 ] = x ( 1 ) [ 1 ] x^{(0)}[1]=x^{(1)}[1] . Denote this common value by c c . Then U ( 0 ) = U ( 1 ) U^{(0)}=U^{(1)} as distributions, and hence"},{"type":"para","text":"We now define a shifted mechanism for the remaining T − 1 T-1 rounds after the common first input c c has been fixed. Let ℳ ( c ) \\mathcal{M}^{(c)} be the mechanism whose round- s s output, for 1 ≤ s ≤ T − 1 1\\leq s\\leq T-1 , is"},{"type":"para","text":"This mechanism uses the independent seeds R 2 , … , R T R_{2},\\ldots,R_{T} , so it is independently decomposable. It is also quantitatively decomposable. Indeed, for round s s define"},{"type":"para","text":"If 1 ∉ I s + 1 1\\notin I_{s+1} , then the aggregate used by ℳ s + 1 \\mathcal{M}_{s+1} is exactly the aggregate over I s ( c ) I^{(c)}_{s} in the suffix history. If 1 ∈ I s + 1 1\\in I_{s+1} , then the aggregate has the common left context c c , and the invariant-pair property removes this common context. Therefore, for every two suffix histories z [ 1 . . s ] z[1..s] and z ′ [ 1 . . s ] z^{\\prime}[1..s] ,"},{"type":"para","text":"Thus ℳ ( c ) \\mathcal{M}^{(c)} satisfies the same structural assumptions as ℳ \\mathcal{M} , with the time indices shifted by one."},{"type":"para","text":"Let Λ \\Lambda be the set of all Hamming-neighboring pairs of suffix histories of length T − 1 T-1 . For"},{"type":"para","text":"let V λ ( b ) V_{\\lambda}^{(b)} be the distribution of the length- ( T − 1 ) (T-1) transcript generated by ℳ ( c ) \\mathcal{M}^{(c)} on the fixed suffix input z ( b ) [ 1 . . T − 1 ] z^{(b)}[1..T-1] . Define"},{"type":"para","text":"By definition of the supremum of power functions, g g is a power function."},{"type":"para","text":"We claim that ℳ ( c ) \\mathcal{M}^{(c)} is non-adaptively g g -DP up to horizon T − 1 T-1 . For deterministic non-adaptive suffix adversaries this is immediate from the definition of g g . Randomized non-adaptive suffix adversaries are mixtures of deterministic ones, and the same bound follows from Fact A.9 ."},{"type":"para","text":"Moreover, g ≤ f g\\leq f . To see this, fix any λ ∈ Λ \\lambda\\in\\Lambda and consider the non-adaptive length- T T adversary for ℳ \\mathcal{M} that submits the common first input c c in both branches and then submits the suffix pair λ \\lambda . The first-round output has the same distribution under the two secret bits, and the remaining transcript has law V λ ( b ) V_{\\lambda}^{(b)} under bit b b . Hence the power function of this full non-adaptive transcript pair is"},{"type":"para","text":"The non-adaptive f f -DP assumption for ℳ \\mathcal{M} gives g λ ≤ f g_{\\lambda}\\leq f for every λ ∈ Λ \\lambda\\in\\Lambda . Taking the supremum over λ \\lambda gives g ≤ f g\\leq f ."},{"type":"para","text":"Now condition on a possible first-round output u u in the support of U ( 0 ) = U ( 1 ) U^{(0)}=U^{(1)} . After observing u u , the original deterministic adaptive adversary 𝒜 {\\mathcal{A}} induces a deterministic adaptive suffix adversary 𝒜 u {\\mathcal{A}}_{u} for the shifted mechanism ℳ ( c ) \\mathcal{M}^{(c)} : it continues exactly as 𝒜 {\\mathcal{A}} would continue after the first transcript symbol u u . Let"},{"type":"para","text":"be the distribution of the remaining transcript under secret bit b b , and write"},{"type":"para","text":"Since ℳ ( c ) \\mathcal{M}^{(c)} is quantitatively decomposable and non-adaptively g g -DP up to horizon T − 1 T-1 , the induction hypothesis gives"},{"type":"para","text":"for every such first-round output u u ."},{"type":"para","text":"Finally, the full transcript under secret bit b b has the same distribution as"},{"type":"para","text":"where U ∼ U ( 0 ) = U ( 1 ) U\\sim U^{(0)}=U^{(1)} . The first coordinate has identical distribution under the two secret bits, and, conditional on U = u U=u , the continuation distributions have power function at most g g . Applying Fact A.10 with first-stage profile 𝖨𝖽 \\mathsf{Id} and second-stage profile g g yields"},{"type":"para","text":"This proves Case 2."},{"type":"para","text":"The two cases complete the induction step. Hence the finite-horizon adaptive f f -DP guarantee holds for every deterministic adaptive adversary. As argued at the beginning, joint convexity then extends the same guarantee to randomized adaptive adversaries. Therefore ℳ \\mathcal{M} is adaptively f f -DP for every finite horizon. ∎"},{"type":"para","text":"Definition E.3 states quantitative decomposability using a scalar distance parameter. This is sufficient for the theorem as used in the main text. Some standard vector-valued mechanisms, however, have an exact one-round power profile that depends on the full difference vector rather than only on its norm. For example, for coordinate-wise Laplace noise on ℝ k \\mathbb{R}^{k} , the exact profile of u + η u+\\eta versus u ′ + η u^{\\prime}+\\eta may depend on the vector u − u ′ u-u^{\\prime} ."},{"type":"para","text":"The theorem and proof extend verbatim to this setting by replacing the scalar distance d ( u , u ′ ) d(u,u^{\\prime}) with an invariant difference map Δ ( u , u ′ ) \\Delta(u,u^{\\prime}) taking values in some parameter space, such as Δ ( u , u ′ ) = u − u ′ ∈ ℝ k \\Delta(u,u^{\\prime})=u-u^{\\prime}\\in\\mathbb{R}^{k} , and replacing g t ( θ ) g_{t}(\\theta) by a family g t ( v ) g_{t}(v) indexed by this difference parameter. The only property needed in the proof is invariance under common aggregation context:"},{"type":"para","text":"Under this vector-indexed variant, the binary-tree prefix-sum mechanism with coordinate-wise Laplace noise is covered exactly for vector updates in ℝ k \\mathbb{R}^{k} . In the main text, we state the simpler scalar version because the usual ( ε , δ ) (\\varepsilon,\\delta) consequences can also be obtained by upper-bounding the vector-indexed profile in terms of a norm, such as ‖ u − u ′ ‖ 1 \\|u-u^{\\prime}\\|_{1} for coordinate-wise Laplace noise."},{"type":"heading","level":2,"text":"E.1 Why Independent Decomposability Alone Is Insufficient"},{"type":"para","text":"The certification theorem in Section E requires quantitative decomposability, not merely independent decomposability. The following example shows why the stronger condition is needed. Even when each round uses fresh independent randomness and the round- t t output is a deterministic function of the input prefix and the fresh seed, a non-adaptive approximate-DP guarantee need not lift to adaptive adversaries. The obstruction is that the one-round leakage can depend on the adaptive common context, rather than only on the location and magnitude of the Hamming discrepancy."},{"type":"heading","level":4,"text":"Counterexample with Fresh Independent Randomness."},{"type":"para","text":"Consider a two-round mechanism with input and output alphabets { 0 , 1 } \\{0,1\\} . Write x 1 = b x_{1}=b and x 2 = q x_{2}=q . Using independent fresh randomness in the two rounds, the mechanism outputs Y 1 , Y 2 ∈ { 0 , 1 } Y_{1},Y_{2}\\in\\{0,1\\} with"},{"type":"para","text":"and"},{"type":"para","text":"Since each output is a deterministic function of the current prefix and a fresh independent seed, the mechanism is independently decomposable in the sense of Definition E.1 ."},{"type":"para","text":"For every fixed input stream ( b , q ) (b,q) , the transcript distribution is the product distribution of Y 1 Y_{1} and Y 2 Y_{2} . Since two-sided ( 0 , δ ) (0,\\delta) -DP is equivalent to total variation distance at most δ \\delta , it suffices to check the four Hamming-neighboring pairs:"},{"type":"para","text":"Hence the mechanism is non-adaptively ( 0 , 0.09 ) (0,0.09) -DP."},{"type":"para","text":"Now consider the deterministic adaptive adversary that chooses the challenge pair x 1 ( 0 ) = 0 x^{(0)}_{1}=0 and x 1 ( 1 ) = 1 x^{(1)}_{1}=1 at round 1 1 , and, after observing Y 1 = y Y_{1}=y , submits the common second input x 2 ( 0 ) = x 2 ( 1 ) = y x^{(0)}_{2}=x^{(1)}_{2}=y . This is a valid adaptive Hamming-neighbor adversary: the two submitted streams differ only at coordinate 1 1 , but their common second input depends on the first release."},{"type":"para","text":"The resulting transcript distributions are:"},{"type":"para","text":"For S = { ( 0 , 1 ) , ( 1 , 0 ) } S=\\{(0,1),(1,0)\\} , we have"},{"type":"para","text":"Therefore,"},{"type":"para","text":"so the mechanism is not adaptively ( 0 , 0.09 ) (0,0.09) -DP."},{"type":"para","text":"This does not contradict Theorem E.4 , because the theorem assumes quantitative decomposability. The example shows that fresh independent randomness alone is not enough: the adaptive adversary can choose a common later input that changes how the earlier Hamming discrepancy is revealed. Quantitative decomposability rules out precisely this behavior by forcing the one-round power function to depend only on the distance between the relevant aggregates, with common aggregation context removed by the invariant-pair property."},{"type":"para","text":"There have been numerous works on differentially private prefix-sum algorithms since the binary tree mechanism [ CSS10 , CSS11 ] was proposed. We state the properties of the variant that are needed for our applications and briefly outline how they can be achieved from existing works."},{"type":"para","text":"Problem Setting. For each time step t ≥ 1 t\\geq 1 , an input stream has some integer x t ∈ 𝒳 = [ a . . b ] x_{t}\\in\\mathcal{X}=[a..b] from some known range that is passed to the interactive mechanism 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\mathsf{StreamSum} that returns a value S t S_{t} that is supposed to be an estimation of the sum ∑ τ = 1 t x τ \\sum_{\\tau=1}^{t}x_{\\tau} . The additive error is | S t − ∑ τ = 1 t x τ | |S_{t}-\\sum_{\\tau=1}^{t}x_{\\tau}| . The adversary observes S t S_{t} ."},{"type":"para","text":"Neighboring Notion. Two input streams are neighboring if they differ in at most one time step, and the corresponding two values differ by 1."},{"type":"para","text":"As aforementioned, a mechanism with unbounded termination satisfies ( ε , δ ) (\\varepsilon,\\delta) -DP if for every T > 0 T>0 , the truncated mechanism run for T T steps is ( ε , δ ) (\\varepsilon,\\delta) -DP."},{"type":"para","text":"For any ε > 0 , δ ∈ ( 0 , 1 ) \\varepsilon>0,\\delta\\in(0,1) and an interval [ a . . b ] ⊆ ℤ [a..b]\\subseteq\\mathbb{Z} of integers, there exists an ( ε , δ ) (\\varepsilon,\\delta) -differentially private interactive mechanism 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\mathsf{StreamSum}_{\\varepsilon,\\delta} , such that given a stream { x t } \\{x_{t}\\} of integers in [ a . . b ] [a..b] , the algorithm outputs a number S t S_{t} in each time step t t such that the following hold."},{"type":"para","text":"Bounded Error: With probability 1, at each time step t t , the output has additive error at most E ( t ) := O ( 1 ε ⋅ log t ⋅ ( log t + log 1 δ ) ) E(t):=O\\left(\\frac{1}{\\varepsilon}\\cdot\\log t\\cdot(\\log t+\\log\\frac{1}{\\delta})\\right) ."},{"type":"para","text":"Consistency: With probability 1, for each time step t t , the increment S t − S t − 1 ∈ [ a . . b ] S_{t}-S_{t-1}\\in[a..b] is in the above range."},{"type":"para","text":"As in [ CCM+22 ] , we start with a pure ε \\varepsilon -DP prefix sum algorithm that has high probability guarantee on the additive error."},{"type":"para","text":"For any ε > 0 \\varepsilon>0 , there exists an ( ε , 0 ) (\\varepsilon,0) -differentially private interactive mechanism 𝖧𝗒𝖻𝗋𝗂𝖽𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε \\mathsf{HybridStreamSum}_{\\varepsilon} that takes an input stream { x t } \\{x_{t}\\} of integers and output S t S_{t} at each time step t t with the following properties."},{"type":"para","text":"Bounded Error: Fix any 0 < δ < 1 0<\\delta<1 . With probability 1 − δ 1-\\delta , for every time step t t , the output at time t t has an additive error of at most O ( 1 ε log t ( log t + log 1 δ ) ) O(\\frac{1}{\\varepsilon}\\log t(\\log t+\\log\\frac{1}{\\delta})) ."},{"type":"para","text":"Structure: In each time step t t , the mechanism actually allows the adversary to observe noisy counts of the form ∑ i ∈ I x i \\sum_{i\\in I}x_{i} masked with independent noise, where the index interval has the form I = [ c . . t ] I=[c..t] . The output S t S_{t} can be constructed deterministically from the noisy counts released so far."},{"type":"para","text":"Fix ε \\varepsilon and δ \\delta . Starting from the mechanism 𝖧𝗒𝖻𝗋𝗂𝖽𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε \\mathsf{HybridStreamSum}_{\\varepsilon} in Fact F.2 , we modify it step by step to achieve the desired properties."},{"type":"para","text":"Always Error Bound. If the output S t S_{t} is too large, i.e., S t > ∑ i = 1 t x i + E ( t ) S_{t}>\\sum_{i=1}^{t}x_{i}+E(t) , return ∑ i = 1 t x i + E ( t ) \\sum_{i=1}^{t}x_{i}+E(t) ; if the output is too small, i.e., S t < ∑ i = 1 t x i − E ( t ) S_{t}<\\sum_{i=1}^{t}x_{i}-E(t) , return ∑ i = 1 t x i − E ( t ) \\sum_{i=1}^{t}x_{i}-E(t) ."},{"type":"para","text":"The argument in Theorem 3.2 of [ CCM+22 ] shows this can turn a high probability additive error bound into an additive error bound with probability 1, at the cost of getting ( ε , δ ) (\\varepsilon,\\delta) -DP."},{"type":"para","text":"Consistency. The argument in Lemma 5.2 of [ CSS11 ] shows that a further truncation operation can achieve consistency without increasing the additive error bound."},{"type":"para","text":"∎"},{"type":"para","text":"The interactive mechanism 𝖧𝗒𝖻𝗋𝗂𝖽𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε \\mathsf{HybridStreamSum}_{\\varepsilon} in Fact F.2 is adaptively ( ε , 0 ) (\\varepsilon,0) -DP, and can be made to be consistent."},{"type":"para","text":"The structure property in Fact F.2 implies that noisy counts obey the independent decomposability property in Definition E.1 . Adaptability follows from Theorem E.4 ."},{"type":"para","text":"Consistency can be achieved, because the modification can be done deterministically from the released noisy counts only. ∎"},{"type":"para","text":"It is not clear whether the “Always Error Bound” can be achieved under the independent decomposability framework, because the truncation needs to refer back to the private inputs, in addition to the released noisy counts."},{"type":"heading","level":2,"text":"F.1 Prefix Sums via Lower-Triangular Matrix Factorizations"},{"type":"para","text":"We briefly restate the prefix-sum workload in the language of the matrix mechanism . Fix a horizon T T and write the stream as a vector x ∈ ℝ T x\\in\\mathbb{R}^{T} . The prefix sums are the linear workload"},{"type":"para","text":"i.e., ( M 𝖼𝗈𝗎𝗇𝗍 x ) t = ∑ i = 1 t x i (M_{\\mathsf{count}}x)_{t}=\\sum_{i=1}^{t}x_{i} . We use the event-level neighboring notion: x ∼ x ′ x\\sim x^{\\prime} iff x − x ′ = ± Δ e i x-x^{\\prime}=\\pm\\Delta e_{i} for some coordinate i ∈ [ T ] i\\in[T] , where Δ > 0 \\Delta>0 is the sensitivity unit (in the above we take Δ = 1 \\Delta=1 )."},{"type":"heading","level":4,"text":"Privacy Notions: Renyi DP (RDP) and zCDP."},{"type":"para","text":"Let P , Q P,Q be distributions with densities p , q p,q w.r.t. a common base measure. For α > 1 \\alpha>1 , the (order- α \\alpha ) Renyi divergence is"},{"type":"para","text":"An interactive mechanism ℳ \\mathcal{M} is ( α , ε ) (\\alpha,\\varepsilon) -RDP if for all neighboring x ∼ x ′ x\\sim x^{\\prime} ,"},{"type":"para","text":"It is ρ \\rho -zCDP if for all α > 1 \\alpha>1 and all neighboring x ∼ x ′ x\\sim x^{\\prime} ,"},{"type":"para","text":"Equivalently, ρ \\rho -zCDP means ( α , ρ α ) (\\alpha,\\rho\\alpha) -RDP holds simultaneously for every α > 1 \\alpha>1 ."},{"type":"heading","level":4,"text":"Converting RDP to ( ε , δ ) (\\varepsilon,\\delta) -DP."},{"type":"para","text":"If ℳ \\mathcal{M} satisfies ( α , ε α ) (\\alpha,\\varepsilon_{\\alpha}) -RDP for some α > 1 \\alpha>1 , then for every δ ∈ ( 0 , 1 ) \\delta\\in(0,1) it satisfies ( ε , δ ) (\\varepsilon,\\delta) -DP with"},{"type":"para","text":"One may optimize over α > 1 \\alpha>1 to get the tightest ( ε , δ ) (\\varepsilon,\\delta) bound:"},{"type":"heading","level":4,"text":"Lower-Triangular Factorization Mechanism."},{"type":"para","text":"Let M 𝖼𝗈𝗎𝗇𝗍 = L R M_{\\mathsf{count}}=LR be a (possibly rectangular) factorization where R ∈ ℝ m × T R\\in\\mathbb{R}^{m\\times T} and L ∈ ℝ T × m L\\in\\mathbb{R}^{T\\times m} are lower-triangular in time (i.e., causal: the t t -th output depends only on x [ 1 . . t ] x[1..t] through R R and L L ). Define the column-sensitivity of R R by"},{"type":"para","text":"Consider the Gaussian matrix mechanism"},{"type":"para","text":"Since L ( ⋅ ) L(\\cdot) is post-processing, privacy is governed entirely by the Gaussian release of R x Rx ."},{"type":"para","text":"For neighboring x ∼ x ′ x\\sim x^{\\prime} with x − x ′ = ± Δ e i x-x^{\\prime}=\\pm\\Delta e_{i} , we have ‖ R ( x − x ′ ) ‖ 2 ≤ Δ s ( R ) \\|R(x-x^{\\prime})\\|_{2}\\leq\\Delta\\,s(R) . Hence the mechanism ℳ L , R , σ \\mathcal{M}_{L,R,\\sigma} satisfies:"},{"type":"para","text":"zCDP: ρ \\rho -zCDP with"},{"type":"para","text":"RDP: for every order α > 1 \\alpha>1 , ( α , ε α ) (\\alpha,\\varepsilon_{\\alpha}) -RDP with"},{"type":"heading","level":4,"text":"Binary Tree Mechanism as a Factorization."},{"type":"para","text":"Assume first T = 2 h T=2^{h} for an integer h ≥ 0 h\\geq 0 (otherwise pad to the next power of two, which only changes constants by replacing h h with ⌈ log 2 T ⌉ \\lceil\\log_{2}T\\rceil ). Index the nodes of the complete binary tree over [ T ] [T] by dyadic intervals; let q ∈ ℝ m q\\in\\mathbb{R}^{m} be the vector of all dyadic-interval sums, so that q = R 𝗍𝗋𝖾𝖾 x q=R_{\\mathsf{tree}}x where R 𝗍𝗋𝖾𝖾 R_{\\mathsf{tree}} is a 0 / 1 0/1 incidence matrix (leaf i i contributes to exactly its h + 1 h+1 ancestors). Each prefix [ 1 . . t ] [1..t] has a canonical dyadic partition of size at most h + 1 h+1 , so there is a reconstruction matrix L 𝗍𝗋𝖾𝖾 L_{\\mathsf{tree}} with M 𝖼𝗈𝗎𝗇𝗍 = L 𝗍𝗋𝖾𝖾 R 𝗍𝗋𝖾𝖾 M_{\\mathsf{count}}=L_{\\mathsf{tree}}R_{\\mathsf{tree}} ."},{"type":"para","text":"For T = 2 h T=2^{h} , every column of R 𝗍𝗋𝖾𝖾 R_{\\mathsf{tree}} has exactly h + 1 h+1 ones, hence"},{"type":"para","text":"Therefore, the Gaussian tree mechanism L 𝗍𝗋𝖾𝖾 ( R 𝗍𝗋𝖾𝖾 x + z ) L_{\\mathsf{tree}}(R_{\\mathsf{tree}}x+z) with z ∼ 𝒩 ( 0 , σ 2 I m ) z\\sim\\mathcal{N}(0,\\sigma^{2}I_{m}) is ρ 𝗍𝗋𝖾𝖾 \\rho_{\\mathsf{tree}} -zCDP with"},{"type":"para","text":"and for every α > 1 \\alpha>1 it is ( α , ε α , 𝗍𝗋𝖾𝖾 ) (\\alpha,\\varepsilon_{\\alpha,\\mathsf{tree}}) -RDP with"},{"type":"para","text":"For general T T , the same holds with h := ⌈ log 2 T ⌉ h:=\\lceil\\log_{2}T\\rceil ."},{"type":"heading","level":4,"text":"A Smooth Lower-Triangular Factorization [ FHU23 ] ."},{"type":"para","text":"Fichtenberger et al. give an explicit Toeplitz lower-triangular factorization M 𝖼𝗈𝗎𝗇𝗍 = L R M_{\\mathsf{count}}=LR with L = R L=R defined by a scalar sequence f ( ⋅ ) f(\\cdot) :"},{"type":"para","text":"and 0 otherwise. Let γ \\gamma be the Euler–Mascheroni constant and define"},{"type":"para","text":"Let L , R L,R be the above factorization. Then M 𝖼𝗈𝗎𝗇𝗍 = L R M_{\\mathsf{count}}=LR and"},{"type":"para","text":"Consequently, the mechanism L ( R x + z ) L(Rx+z) with z ∼ 𝒩 ( 0 , σ 2 I T ) z\\sim\\mathcal{N}(0,\\sigma^{2}I_{T}) is ρ 𝖥𝖧𝖴 \\rho_{\\mathsf{FHU}} -zCDP with"},{"type":"para","text":"and for every α > 1 \\alpha>1 it is ( α , ε α , 𝖥𝖧𝖴 ) (\\alpha,\\varepsilon_{\\alpha,\\mathsf{FHU}}) -RDP with"},{"type":"heading","level":4,"text":"Takeaway."},{"type":"para","text":"Both mechanisms fit the same template L ( R x + z ) L(Rx+z) , but yield different (exact) privacy constants through the column ℓ 2 \\ell_{2} sensitivity s ( R ) s(R) : the binary tree gives s ( R ) 2 = Θ ( log 2 T ) s(R)^{2}=\\Theta(\\log_{2}T) , while the smooth Toeplitz factorization satisfies s ( R ) 2 ≤ 1 − 1 − γ π + ln T π + 2 T s(R)^{2}\\leq 1-\\frac{1-\\gamma}{\\pi}+\\frac{\\ln T}{\\pi}+\\frac{2}{T} ."},{"type":"para","text":"As the most relevant related works have already been covered in the introduction, we will elaborate further on some related aspects."},{"type":"para","text":"Choices of Privacy Measure. The central idea in differential privacy [ DWO06 ] is that if V 0 V_{0} and V 1 V_{1} are two distributions of outputs produced by a mechanism from two neighboring inputs, then those two distributions should be close. The classical ( ε , δ ) (\\varepsilon,\\delta) -DP notion uses two parameters to quantify closeness, where a smaller value in each parameter means that the two distributions are closer. However, since distributions are inherently complex objects, some information on the two distributions will be inevitably lost when they are compared using just two parameters. When the same Gaussian noise is used to mask two different vectors, researchers have discovered that the Rényi divergence [ MIR17 ] can capture the variance of the Gaussian distribution perfectly, and hence, can quantify the closeness of two such distributions (with the same variance but different means)."},{"type":"para","text":"In general, any useful way to quantify privacy guarantees must satisfy the property that if the output satisfies certain privacy requirement, then any further processing of the output cannot violate that specific requirement. This can be formally formulated by requiring that the divergence – used for measuring how different two distributions are – must satisfy the data processing inequality ; it is worth noting that some common distance notion such as the ℓ 2 \\ell_{2} -norm does not satisfy this property."},{"type":"para","text":"Instead of just using a few parameters to capture the closeness of two distributions, tradeoff functions [ DRS22 ] have been proposed to define differential privacy, because a tradeoff function can capture all the essential information about how two distributions differ in the sense that any divergence satisfying the data processing inequality can be recovered from the tradeoff function. Indeed, tradeoff functions offer a powerful tool to describe the composition of private mechanisms."},{"type":"para","text":"However, one notational inconvenience is that a larger tradeoff (measured by pointwise comparison) means that the two distributions are closer, which has the opposite interpretation from other divergence parameters such as ε \\varepsilon and δ \\delta . In fact, an ad hoc concept of generalized probability distance has been defined in [ VZ23 ] to reverse the direction of the inequalities such that it will be consistent with the notion of distance. On the other hand, a simpler way to achieve this notation consistency is to replace a tradeoff function with its complement that is known as a power function , which naturally preserves all the equivalent mathematical properties. As we shall see in Definition A.3 , a power function also has an intuitive description using the fractional knapsack problem."},{"type":"para","text":"As illustrated in [ VZ23 , ZZC+24 ] , if one uses such a powerful tool to define any new notion of differential privacy, then a single composition theorem (such as our Theorem C.1 ) will be sufficient to recover any composition result from the classical notion to the new notion of privacy. Therefore, it would not be necessary to reconstruct individual advanced composition theorems [ DRV10b , KOV15 ] ."},{"type":"para","text":"Hamming vs Edit Neighboring Notions. In [ BEB+24 ] , two neighboring notions are considered for static databases:"},{"type":"para","text":"Hamming-style. Two databases have the same number of elements, and they differ in at most one element."},{"type":"para","text":"Edit-style. One element from one database is deleted to form the other database."},{"type":"para","text":"They considered fixed-size mini-batches, which may be sampled in two ways: with or without replacement. Since they considered sampling from static databases, the difference between Hamming- vs edit-style neighboring static databases would not have such a stark contrast as streams, as deleting the first element of a stream can cause it to change in every position."},{"type":"para","text":"Other Recent Works on Concurrent Composition. A more general notion of concurrent composition is considered in [ HST+23 ] , where the privacy parameters of mechanisms can be adaptively chosen. However, as in [ VZ23 ] , each interactive mechanism is associated with a single static database on which its neighboring relation is defined; the adversary may interact adaptively with the mechanism, but neighboring inputs differ only in this underlying database, not in a dynamic stream of updates."},{"type":"para","text":"Concurrent composition for mechanisms with adaptively chosen privacy parameters are also considered in [ HSV26 ] , but for neighboring dynamic databases. They also give a formulation based on adaptive DP [ DMR+22 ] , expressed via a left-or-right style distinguishing game with a verification function, which is essentially the same as our paired simulation in Definition B.2 ."},{"type":"para","text":"Note that both works consider composition where each mechanism has a single neighboring relation on its dataset (static or dynamic). In order to capture modular composition in which neighboring notions are defined separately on both the input and the output of a mechanism—and to reason about neighbor-preserving transformations that change the neighboring structure—we need a more refined notion of neighbor-preserving paired simulation, given in Definition B.8 ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Modern federated and streaming learning systems release intermediate model snapshots throughout training, so an adversary may observe the entire trajectory ( w ( t ) ) t ≥ 0 (w^{(t)})_{t\\\\geq 0} rather than only a final model. A canonical instance is streaming stochastic gradient descent (SGD): at step t t , a user contributes a private loss function f t f_{t} and the learner updates using a gradient step based on the current model [ TS13 , KMS+21 ] . Beyond protecting the data within each f t f_{t} , some deployments also require participation privacy : a user may wish to ensure plausible deniability of whether they participated at all. This motivates single-edit (edit-distance) neighboring streams, where two user-update streams are neighbors if one can be obtained from the other by inserting or deleting a single user event.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Continual release also makes the learning process intrinsically interactive: the released model w ( t ) w^{(t)} influences what happens next, so the update stream can be adaptively generated in response to prior outputs. While classical continual-release primitives such as tree-based private prefix sums [ DNP+10a , CSS11 ] were originally analyzed under statically chosen (Hamming-neighbor) streams, recent work formalizes differential privacy against such adaptive interaction [ JRS+23 ] . Our goal is to make this adaptive viewpoint compatible with the single-edit participation model above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The difficulty is that Hamming-style neighboring streams are not the right abstraction for participation privacy on a stream. Most continual DP analyses change the value at a single time index t 0 t_{0} while keeping all other positions aligned [ DWO06 ] . In contrast, a single insertion/deletion shifts the alignment of all subsequent positions, so an edit-neighbor pair can be far from Hamming-neighboring. This also undermines deterministic batching: partitioning the stream into fixed contiguous blocks can cause one edit to shift many later batch boundaries, so standard Hamming-neighbor continual DP guarantees do not directly transfer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address this mismatch, we use a simple modular recipe: randomized buffering and an adaptive-safety certification for downstream continual primitives. We apply a randomized buffering wrapper [ CCM+22 , ZSC+23 ] that introduces random delay and releases user updates in bins of controlled size (e.g., in [ U , 2 U ] [U,2U] ); here U U is a privacy-implied systems cost , calibrated by the target ( ε , δ ) (\\\\varepsilon,\\\\delta) and inducing delay. We then invoke standard continual DP primitives (e.g., tree/prefix-sum) to privately aggregate the resulting vector updates, and prove a certification theorem showing when a non-adaptive Hamming-neighbor privacy proof remains valid under feedback. The certification requires fresh per-round randomness together with a stable-context condition: common adaptive updates may change the released statistic, but they must not change the privacy cost of the unique Hamming discrepancy. 1 1 1 An earlier version [ CSZ+26 ] stated the certification condition using independent decomposability alone. Here we use the slightly stronger stable-context condition, formalized in Appendix E as quantitative decomposability. This refinement makes the statement technically precise and is satisfied by the certified cores of the continual-release primitives used in our pipeline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Edit-neighbor continual learning under feedback. We formalize participation privacy via single-edit neighboring user-update streams in a continual learning setting with intermediate releases.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Modular pipeline with an explicit privacy to latency link. We combine randomized buffering with standard continual DP primitives to obtain trajectory-level privacy for edit-neighbor streams; the target ( ε , δ ) (\\\\varepsilon,\\\\delta) determines the buffering level U U , making induced delay explicit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Certification theorem for adaptive safety of continual DP primitives. We give a checkable certification condition under which a non-adaptive Hamming-neighbor privacy analysis carries over to the feedback/adaptive setting. The condition combines fresh per-round randomness with a stable-context requirement, and certifies safe reuse of canonical primitives such as tree/prefix-sum mechanisms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results and organization. Our end-to-end guarantee is trajectory-level ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP for single-edit neighboring participation streams under feedback, obtained by composing randomized buffering with a continual-release DP aggregator (Theorem 3.1 ); the privacy target explicitly induces a buffering level U ( ε , δ ) U(\\\\varepsilon,\\\\delta) and hence delay/staleness. A second ingredient is a reusable certification theorem showing that, for continual primitives satisfying our fresh-randomness and stable-context conditions, a standard non-adaptive privacy analysis lifts to the adaptive/feedback setting (Theorem 4.3 ). Section 2 formalizes the interactive model, edit adjacency, and metrics; Section 3 gives a high-level view of the pipeline and states the main theorem; Section 4 develops the two auditable ingredients (randomized buffering and certification) and combines them via modular composition; Section 5 instantiates the recipe with continual DP-SGD (tree/prefix-sum aggregation). Technical proofs are deferred to the appendix. The relevant experiments are given in [ CSZ+26 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Continual-Release DP. Canonical continual-release primitives for streaming aggregates include tree-based private prefix sums [ DNP+10a , CSS11 ] and closely related lower-triangular (matrix-style) mechanisms [ DMR+22 ] . These mechanisms are typically analyzed for Hamming-neighboring update streams, where only one time index differs. In contrast, our participation model uses single-edit (insertion/deletion) neighboring streams, where alignment shifts after the edit; this gap is mild for static datasets [ BEB+24 ] but is fundamental for streams.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sliding-window variants of private stream release have also been studied for window aggregate queries and graph streams [ CXG+13 , UUA21 ] . These works target recent-window utility, whereas our setting focuses on single-edit participation privacy, where one insertion/deletion shifts all subsequent stream positions and requires an edit-to-Hamming shielding interface.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Adaptive Streams and Feedback. Continual learning is inherently interactive: released outputs can influence future updates. Recent work formalizes differential privacy against such adaptive interaction (e.g., left-or-right style games and verification viewpoints) [ DMR+22 , JRS+23 ] . Our certification result is complementary: it gives a checkable structural condition under which a standard non-adaptive Hamming-DP analysis of a continual primitive lifts directly to the adaptive setting. The condition is stronger than fresh prefix-causal randomness alone: it also requires that common adaptive context does not change the one-round privacy cost of the unique Hamming discrepancy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Privacy-Induced Delayed Reactions. [ CLN+24 ] give a conceptually related delayed-reaction phenomenon for continual observation and online threshold queries. Their Appendix B delays threshold-type outputs on a directly observed online bit stream. Our delay has a different technical role: 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} waits for future arrivals to form bins in [ U , 2 U ] [U,2U] so that a single insertion/deletion, which appears only in the neighboring-stream analysis, becomes a bounded Hamming-style perturbation while the full transcript, including timing and ⊥ \\\\bot outputs, remains protected.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Randomized Buffering, RSC, and Obliviousness Lineage. Randomized buffering has appeared as infrastructure in oblivious data structures and related privacy modularity work [ CCM+22 , ZSC+23 ] . A closely related random-partition idea also appears in the Reorder-Slice-Compute (RSC) paradigm of [ CLN+23 ] , where random slicing is used to synchronize neighboring executions and avoid a one-edit domino effect. Our setting differs in that the continual streaming transcript exposes release timing itself: the adversary observes whether each step emits a real block or the dummy symbol ⊥ \\\\bot . Thus, in our use, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} is an interactive interface that converts single-edit participation streams into a sparse, Hamming-style per-bin update stream while also providing explicit backlog/delay guarantees calibrated by ( ε , δ ) (\\\\varepsilon,\\\\delta) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Composition and Concurrency. Beyond classical ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP [ DWO06 ] , alternative formalisms (e.g., Rényi divergence and tradeoff-based views) can streamline composition reasoning [ MIR17 , DRS22 , VZ23 , ZZC+24 ] , recovering familiar advanced composition bounds [ DRV10b , KOV15 ] . Concurrent composition in interactive settings has also been studied, including adaptively chosen privacy parameters [ HST+23 , HSV26 ] . These works typically assume each mechanism’s neighboring relation is defined on a single underlying (static or dynamic) database; in our pipeline, we additionally need to reason about neighbor-preserving transformations between components (edit-to-Hamming shielding), which motivates our modular composition viewpoint. Our modular composition theorem can be viewed as combining the refinement-pair reduction of NPDP to standard DP in [ ZZC+24 ] with the interactive/concurrent composition framework of [ HSV26 ] ; for completeness and to match our power-function accounting, we give a self-contained proof in Appendix C .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Setting and Preliminaries\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now fix the interactive streaming-learning model underlying continual release: the adversary observes both the released model trajectory and the release timing, and may influence future user events. We also define edit-style (participation) adjacency and the utility–latency metrics used throughout. With these conventions in place, Section 3 summarizes our buffering–aggregation pipeline and main guarantee.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Streaming Learning Loop and Transcript.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At each discrete step t = 1 , 2 , … t=1,2,\\\\ldots , an adaptive environment produces a user event f t f_{t} (e.g., a private loss or update). The server maintains a model w ( t ) ∈ ℝ d w^{(t)}\\\\in\\\\mathbb{R}^{d} and may release intermediate snapshots. To unify per-step and buffered releases, we include an observable timing bit v t ∈ { 0 , 1 } v_{t}\\\\in\\\\{0,1\\\\} : if v t = 1 v_{t}=1 the server incorporates available (buffered) events to form an update g t g_{t} and publishes the new snapshot; if v t = 0 v_{t}=0 it holds w ( t ) = w ( t − 1 ) w^{(t)}=w^{(t-1)} . The adversary’s view up to horizon T T is the transcript 𝗍𝗋 ≤ T := ( ( v t , w ( t ) ) ) t = 0 T , \\\\mathsf{tr}_{\\\\leq T}\\\\;:=\\\\;\\\\bigl((v_{t},w^{(t)})\\\\bigr)_{t=0}^{T}, and our privacy notion is defined with respect to this joint transcript (including timing). All privacy guarantees are required to hold for every horizon T T , and thus apply to the full trajectory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Feedback/Adaptive Interaction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The environment (and thus the event stream) may depend on the past transcript: formally, f t f_{t} may be chosen as an arbitrary (possibly randomized) function of 𝗍𝗋 < t \\\\mathsf{tr}_{<t} . All guarantees quantify over such adaptive interaction. When we refer to a non-adaptive privacy analysis for a continual primitive, we mean its input stream is fixed in advance (independent of past releases).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Participation Privacy via Single-Edit Adjacency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ℳ \\\\mathcal{M} denote the full interactive mechanism (e.g., buffering → \\\\rightarrow aggregation → \\\\rightarrow releases) mapping an event stream F = ( f 1 , f 2 , … ) F=(f_{1},f_{2},\\\\ldots) to a transcript. We use single-edit adjacency: F ∼ edit F ′ F\\\\sim_{\\\\mathrm{edit}}F^{\\\\prime} if F ′ F^{\\\\prime} is obtained from F F by inserting or deleting exactly one event, shifting subsequent indices; equivalently, for some t 0 t_{0} , either (i) f t ′ = f t f^{\\\\prime}_{t}=f_{t} for t < t 0 t<t_{0} and f t ′ = f t + 1 f^{\\\\prime}_{t}=f_{t+1} for all t ≥ t 0 t\\\\geq t_{0} (deletion), or (ii) the reverse (insertion). The unit of privacy is one participation event; other units (e.g., a session spanning multiple events) reduce via grouping/composition (formal composition tools appear in Appendix A ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Differential Privacy (Under Adaptive Interaction).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We say ℳ \\\\mathcal{M} is ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP w.r.t. ∼ edit \\\\sim_{\\\\mathrm{edit}} if for all F ∼ edit F ′ F\\\\sim_{\\\\mathrm{edit}}F^{\\\\prime} and all measurable transcript-sets S S ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the probability is over ℳ \\\\mathcal{M} ’s internal randomness; the quantification over event-generation strategies is captured by the adaptive-interaction model above. Our appendix sometimes uses the equivalent tradeoff-/power-function viewpoint to streamline interactive composition; the main text states guarantees in ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP; see Appendix A .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Metrics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Utility: test accuracy / loss of the final released model by time T T (e.g., w ( T ) w^{(T)} or the last release by T T ). Systems: backlog Q t Q_{t} (pending, not-yet-incorporated events after time t t ) and inclusion delay D ( i ) := min { t ≥ i : f i is incorporated by time t } − i D(i):=\\\\min\\\\{t\\\\geq i:\\\\ f_{i}\\\\text{ is incorporated by time }t\\\\}-i , measured in event arrivals.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Extension: From One Edit to k k Edits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If neighbors may differ by up to k k insertions/deletions, standard group-privacy/composition yields a k k -dependent degradation of ( ε , δ ) (\\\\varepsilon,\\\\delta) ; in our pipeline this correspondingly increases the required privacy budgets and thus the induced buffering/latency. Full statements are deferred to the appendix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Overview\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section summarizes our buffering–aggregation pipeline (Fig. 1 ) and the resulting end-to-end guarantee: continual-release ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP for single-edit participation streams under feedback, where the adversary observes the released trajectory (and timing). The required buffering level U ( ε , δ ) U(\\\\varepsilon,\\\\delta) is privacy-implied and explicitly governs delay.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next give the recipe and state the end-to-end guarantee.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Recipe (buffering → \\\\rightarrow aggregation → \\\\rightarrow releases).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given user-event stream ( f t ) (f_{t}) , at each step t ≥ 1 t\\\\geq 1 , buffering outputs B t B_{t} with | B t | ∈ [ U , 2 U ] |B_{t}|\\\\in[U,2U] or ⊥ \\\\bot . Let τ ( 1 ) < τ ( 2 ) < ⋯ \\\\tau(1)<\\\\tau(2)<\\\\cdots be the (random) steps with B τ ( k ) ≠ ⊥ B_{\\\\tau(k)}\\\\neq\\\\bot .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Buffer. Run randomized buffering with U ( ε , δ ) U(\\\\varepsilon,\\\\delta) to obtain ( B t ) t ≥ 1 (B_{t})_{t\\\\geq 1} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Update. For each k ≥ 1 k\\\\geq 1 , form a clipped per-bin update g k := 1 | B τ ( k ) | ∑ f ∈ B τ ( k ) Clip G ( ∇ f ( w ( τ ( k ) − 1 ) ) ) g_{k}:=\\\\frac{1}{|B_{\\\\tau(k)}|}\\\\sum_{f\\\\in B_{\\\\tau(k)}}\\\\textsf{Clip}_{G}\\\\Bigl(\\\\!\\\\nabla f\\\\!\\\\bigl(w^{(\\\\tau(k)-1)}\\\\bigr)\\\\Bigr) , where Clip G ( x ) := x ⋅ min { 1 , G ‖ x ‖ 2 } \\\\textsf{Clip}_{G}(x)~:=~x\\\\cdot\\\\min\\\\!\\\\left\\\\{1,\\\\ \\\\frac{G}{\\\\|x\\\\|_{2}}\\\\right\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Aggregate + release. Feed ( g k ) k ≥ 1 (g_{k})_{k\\\\geq 1} to a certified continual-release DP primitive and update only at bin times: output w ( τ ( k ) ) w^{(\\\\tau(k))} after processing g k g_{k} , and hold w ( t ) = w ( τ ( k ) ) w^{(t)}=w^{(\\\\tau(k))} for τ ( k ) ≤ t < τ ( k + 1 ) \\\\tau(k)\\\\leq t<\\\\tau(k{+}1) (feedback allowed).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorem 3.1 states the resulting end-to-end privacy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Run the above recipe with privacy target ( ε , δ ) (\\\\varepsilon,\\\\delta) , allocating ( ε b , δ b ) (\\\\varepsilon_{b},\\\\delta_{b}) to buffering and ( ε a , δ a ) (\\\\varepsilon_{a},\\\\delta_{a}) to aggregation. Set U = Θ ( 1 ε b log 1 δ b ) U=\\\\Theta\\\\!\\\\Bigl(\\\\frac{1}{\\\\varepsilon_{b}}\\\\log\\\\frac{1}{\\\\delta_{b}}\\\\Bigr) , and plug in any continual-release DP primitive that is adaptively ( ε a , δ a ) (\\\\varepsilon_{a},\\\\delta_{a}) -DP for Hamming-neighboring update streams. Then the released model trajectory ( w ( t ) ) t ≥ 0 (w^{(t)})_{t\\\\geq 0} is ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP with respect to single-edit neighboring event streams, even when future events are generated adaptively from past releases (feedback).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proof idea. Randomized buffering turns a single insertion/deletion on the event stream into a bounded Hamming-style perturbation on the emitted update stream; we then apply a certified continual-release DP primitive whose non-adaptive analysis remains valid under feedback, and conclude via modular composition (Appendix C ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Certification preview (adaptive safety of continual primitives).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use a reusable certification for feedback: writing x t x_{t} for the per-round input and x 1 : t x_{1:t} for its prefix, a continual-release primitive is independently decomposable if at each release time t t it samples fresh independent randomness R t R_{t} and outputs a deterministic function of ( x 1 : t , R t ) (x_{1:t},R_{t}) . For such primitives, standard non-adaptive ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP analyses remain valid under feedback. This covers tree/prefix-sum (Appendix F ) and more general lower-triangular matrix mechanisms (Section 4.2 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"SGD instantiation and what we measure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We instantiate the pipeline with streaming SGD: each emitted bin B τ ( k ) B_{\\\\tau(k)} triggers one clipped mini-batch gradient update, privatized by a continual-release DP aggregator (tree/prefix-sum in our experiments). The target ( ε , δ ) (\\\\varepsilon,\\\\delta) fixes U ( ε , δ ) U(\\\\varepsilon,\\\\delta) , yielding random batch sizes in [ U , 2 U ] [U,2U] and a backlog bound of order O ( 1 ε b log t ( log t + log 1 δ b ) ) O\\\\!\\\\bigl(\\\\tfrac{1}{\\\\varepsilon_{b}}\\\\log t\\\\,(\\\\log t+\\\\log\\\\tfrac{1}{\\\\delta_{b}})\\\\bigr) (Lemma 4.2 ). We report accuracy versus ( ε , δ ) (\\\\varepsilon,\\\\delta) together with empirical delay distributions implied by U ( ε , δ ) U(\\\\varepsilon,\\\\delta) . The theoretical convergence of various SGD variants under DP noise has already been extensively analyzed; we refer the reader to standard works [ JKT12 , SCS13 , TS13 , KMS+21 , DMR+22 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Deferred details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sections 4.1 and 4.2 formalize buffering and certification; the end-to-end proof of Theorem 3.1 (via modular composition) and additional instantiations/experiments appear in the appendix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Auditable Ingredients: Randomized Buffering and Adaptive-Safe Continual-Release DP\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section isolates the two ingredients that make the pipeline in Fig. 1 auditable. First, we treat randomized buffering ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} ) as an interface: it converts a single-edit event stream into a sparse per-bin update stream that admits a standard Hamming-neighbor DP analysis, while providing an explicit backlog guarantee. Second, we give a reusable certification theorem for adaptive interaction (output-dependent updates): for independently decomposable (prefix-causal) continual-release mechanisms, a non-adaptive DP proof remains valid even when updates are chosen adaptively as a function of past releases.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Separation of Roles.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Randomized buffering is not needed for the certification theorem itself: Theorem 4.3 is a reusable adaptivity lift for independently decomposable continual mechanisms. In our pipeline, buffering serves a different role: it shields the downstream primitive from single-edit adjacency, where one insertion or deletion can shift many later batch boundaries. Thus, our modular proof separates an upstream edit-to-Hamming interface from a downstream adaptive-safety certificate. We do not claim that randomized buffering is the only possible route; alternatives would require either another edit-shielding interface or a direct native proof under edit adjacency.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Randomized Buffering Wrapper\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"RandBin Interface.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} as a lightweight wrapper that turns an incoming event stream into well-sized bins, while keeping the buffering backlog under control and enabling our edit-to-Hamming reduction downstream. The input is a stream of user events f 1 , f 2 , … f_{1},f_{2},\\\\ldots (losses, gradients, or updates). At each step t ≥ 1 t\\\\geq 1 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} outputs a symbol B t B_{t} : either a concrete bin B t B_{t} containing some pending events among those arrived by time t t , or ⊥ \\\\bot (no emission). Let τ ( 1 ) < τ ( 2 ) < ⋯ \\\\tau(1)<\\\\tau(2)<\\\\cdots be the (random) emission times with B τ ( k ) ≠ ⊥ B_{\\\\tau(k)}\\\\neq\\\\bot . When B t = ⊥ B_{t}=\\\\bot , nothing is sent to the downstream aggregator and the released model remains unchanged.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We rely on three interface properties:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bin size. Whenever B t ≠ ⊥ B_{t}\\\\neq\\\\bot , the bin size is controlled: | B t | ∈ [ U , 2 U ] |B_{t}|\\\\in[U,2U] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Backlog control. The number of pending (unbinned) events Q t Q_{t} remains small for all t t (formal bound below), yielding explicit delay guarantees.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Edit-to-Hamming wrapper: neighbor-preserving (NP) DP. A single insertion/deletion in the input stream induces only a bounded Hamming-style perturbation in the emitted-bin (and hence per-bin update) stream, in the sense needed to compose with a Hamming-DP continual primitive. The details are deferred to Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Input-Identifiability Convention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Lemma 4.1 uses the standard convention that events are identifiable, e.g., by unique IDs or metadata tags, so that after the first mismatch in an edit-neighbor pair the simulation can determine which stream contains the extra event. Anonymous streams with duplicate values require an extra disambiguation assumption; see Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Systems Metrics: Backlog and Delay.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The buffering level U ( ε b , δ b ) U(\\\\varepsilon_{b},\\\\delta_{b}) controls how aggressively 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} batches the stream and thus the latency regime seen by learning. We track the backlog Q t Q_{t} , the number of pending (unbinned) events after time t t . For an event f i f_{i} , its inclusion delay D ( i ) D(i) is the number of subsequent arrivals until it is placed in an emitted bin. In particular, if Q t ≤ B ( t ) Q_{t}\\\\leq B(t) for all t ∈ [ i , T ] t\\\\in[i,T] , then D ( i ) ≤ B ( T ) D(i)\\\\leq B(T) (and sharper bounds follow from the trajectory of ( Q t ) (Q_{t}) ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Interface Guarantees for Modular Composition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We will use two interface properties of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} ; proofs and constants appear in Appendix D .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any single-edit neighboring input streams, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} admits a neighbor-preserving paired simulation (with refinement) such that the coupled emitted-bin/update streams differ in a bounded Hamming-style way at the interface required for modular composition with a Hamming-DP continual primitive.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intuition: randomized boundaries prevent one insertion/deletion from cascading into many downstream bin shifts; the paired simulation formalizes this stability under interaction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With U = Θ ( ε b − 1 log ( 1 / δ b ) ) U=\\\\Theta(\\\\varepsilon_{b}^{-1}\\\\log(1/\\\\delta_{b})) , for every time t t the backlog satisfies Q t = O ( 1 ε b log t ⋅ ( log t + log 1 δ b ) ) . Q_{t}\\\\;=\\\\;O\\\\!\\\\Bigl(\\\\frac{1}{\\\\varepsilon_{b}}\\\\log t\\\\cdot\\\\Bigl(\\\\log t+\\\\log\\\\frac{1}{\\\\delta_{b}}\\\\Bigr)\\\\Bigr).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This bound is obtained by implementing 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} ’s private emission scheduler via the streaming DP prefix-sum mechanism (Theorem F.1 ) and translating its additive-error guarantee into a worst-case queue bound. As a consequence, inclusion delay is controlled at the same order (up to constants) for events arriving by time t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intuition: the noisy scheduling forces sufficiently frequent emissions to prevent sustained queue growth, while keeping bin sizes within [ U , 2 U ] [U,2U] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"On the Horizon Dependence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The log t \\\\log t dependence in Lemma 4.2 is not merely an artifact of loose algebra. In our implementation, the emission scheduler is a continual private prefix-sum mechanism, and its additive error is translated directly into backlog. On a unit-rate stream, if M t M_{t} is the cumulative number of emitted items by time t t , then Q t = t − M t Q_{t}=t-M_{t} . Thus, within this scheduler family, improving the backlog bound would amount to improving the corresponding continual private counting error. Known continual-counting lower bounds suggest that logarithmic horizon dependence is intrinsic to this proof route [ DNP+10a , CLN+24 ] , although we do not claim a minimax lower bound for all possible NPDP wrappers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Scope of the Approximate-DP Regime.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The bounded-delay guarantee in Lemma 4.2 is tied to the approximate-DP implementation of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} : the proof uses bounded-support buffering noise and an always-valid prefix-sum error bound. For RandBin-style shift masking, pure DP would require unbounded-support noise, and therefore cannot give the same probability-one finite-delay guarantee.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Positioning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Randomized buffering is often used in oblivious data structures to hide access patterns [ CCM+22 , ZSC+23 ] . Our use is different in emphasis: 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} serves as an infrastructure interface that (i) converts single-edit participation streams into a sparse, Hamming-style stream of per-bin updates and (ii) exposes explicit backlog guarantees. Once Lemmas 4.1 – 4.2 hold, the downstream component becomes plug-and-play: any continual primitive that passes our certification in Section 4.2 can be safely reused. Full pseudocode, the NPDP security game (paired simulation + refinement), and sharper tail bounds are deferred to Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Certifying Feedback-Safe Continual-Release Primitives\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why Certification?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our pipeline treats the continual-release DP primitive as a plug-in component. To make this modularity auditable , we need a certification criterion showing when a standard non-adaptive Hamming-neighbor DP analysis continues to hold when per-round updates are chosen adaptively from past releases. Fresh independent randomness is necessary for this transfer, but not sufficient by itself; Appendix E.1 gives a two-round counterexample. The criterion below therefore combines fresh randomness with a quantitative decomposability condition that prevents adaptive common context from changing the one-round leakage profile.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Certified Decomposability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Write x t x_{t} for the input update at round t t and x 1 : t := ( x 1 , … , x t ) x_{1:t}:=(x_{1},\\\\dots,x_{t}) . The certification condition has two parts. First, the mechanism must use fresh independent randomness at each round: the release at time t t is generated as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where R t R_{t} is independent of the randomness used at other rounds and ℳ t \\\\mathcal{M}_{t} is deterministic given ( R t , x 1 : t ) (R_{t},x_{1:t}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, the privacy cost of a one-round release must be stable under common context. Informally, if two input histories differ in one update, then the effect of this discrepancy on the round- t t release is determined only by the part of the round- t t aggregate that contains the changed update. Common updates before or after the discrepancy may change the released statistic, but they are shared by both neighboring histories and therefore should not change the privacy cost of the discrepancy itself.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The formal appendix condition, called quantitative decomposability, rules out this behavior by requiring the one-round privacy profile to depend only on the distance between the relevant aggregates after common aggregation context is removed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Importantly, after producing the output y 1 : t y_{1:t} at time t t , the mechanism may apply arbitrary post-processing to ( y 1 , … , y t ) (y_{1},\\\\dots,y_{t}) and output any derived value at time t t ; by closure under post-processing, this does not weaken the privacy guarantee proved for the underlying releases.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Checklist 1 (Certification Conditions). A continual-release primitive ℳ \\\\mathcal{M} passes certification if its core release process satisfies the following conditions: 1. Prefix dependence: the core release at time t t depends only on the input prefix x 1 : t x_{1:t} , not on future inputs. 2. Fresh randomness: the core release at time t t uses randomness R t R_{t} that is independent of the randomness used in all other rounds. 3. Deterministic release map: given ( R t , x 1 : t ) (R_{t},x_{1:t}) , the core release y t = ℳ t ( R t , x 1 : t ) y_{t}=\\\\mathcal{M}_{t}(R_{t},x_{1:t}) is deterministic. 4. Stable one-round privacy cost: if two prefixes differ in one update, then common aggregation context in the two prefixes does not change the privacy cost of that discrepancy. The appendix formalizes this condition as quantitative decomposability in Definition E.3 . 5. Post-processing only afterward: after the certified core releases are produced, the mechanism may output any function of the released transcript; this is post-processing and does not weaken the privacy guarantee. Consequence: any finite-horizon non-adaptive ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP proof for Hamming-neighboring update streams lifts to adaptive interaction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Certification Theorem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The next theorem formalizes the consequence of Checklist 1 and serves as an audit tool for reusing continual DP primitives inside interactive learning loops. The formal appendix statement is given in terms of power functions; the version below states the resulting ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP guarantee.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider Hamming-style neighboring update streams, where two equal-length streams differ in at most one position. Let ℳ \\\\mathcal{M} be a continual primitive satisfying Checklist 1. If ℳ \\\\mathcal{M} is non-adaptively ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP for Hamming-neighboring update streams for every finite horizon, then ℳ \\\\mathcal{M} is also ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP against adaptive interaction for every finite horizon, where each x t x_{t} may be chosen as an arbitrary function of past releases.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Proof Sketch.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Full proof in Appendix E ) The proof works in the paired-simulation game and proceeds by induction on the horizon. If the first input pair already contains the unique Hamming discrepancy, then all later paired inputs must be equal componentwise. Checklist 1 ensures that these later common inputs only add common context and do not change the one-round privacy cost of the discrepancy; the per-round privacy losses then tensorize exactly. If the first input pair is equal, we condition on the common first release, fix the first input as common context, and apply the induction hypothesis to the shifted mechanism on the remaining rounds. Fresh independent randomness is used to separate the per-round contributions, while the stable-context condition prevents adaptive common inputs from amplifying the leakage of the unique discrepancy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Mini-Audit 1: Tree / Prefix-Sum.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the tree mechanism, the certified core is the stream of newly revealed noisy tree-node values. Each such node value depends only on the updates in its interval and on its own fresh noise. If two update streams differ in one coordinate, then only the tree nodes whose intervals contain that coordinate see different sums; all other contributions are common to the two executions. Thus the one-round privacy cost is determined only by the changed update’s contribution to the affected node, not by the surrounding common updates. The published prefix sum at time t t is a deterministic function of the noisy node values revealed so far, hence post-processing. Therefore the usual non-adaptive Hamming-neighbor DP analysis for tree prefix sums carries over to adaptive update choices by Theorem 4.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Mini-Audit 2: Lower-Triangular (Prefix) Linear Mechanisms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Many continual primitives can be written as a certified core followed by linear post-processing. Concretely, suppose the core releases noisy statistics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the noises η t \\\\eta_{t} are fresh and independent across t t . Each core release depends only on the input prefix x 1 : t x_{1:t} and its own fresh noise. If two update streams differ in one coordinate, then the two linear statistics differ only through that coordinate’s contribution; all common updates contribute equally to both executions. Thus the one-round privacy cost is stable under common context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Any published output of the form y t = ∑ s ≤ t A t , s z s y_{t}=\\\\sum_{s\\\\leq t}A_{t,s}\\\\,z_{s} is a deterministic function of ( z 1 , … , z t ) (z_{1},\\\\dots,z_{t}) and therefore post-processing. Hence any non-adaptive Hamming-neighbor DP bound for the core stream ( z t ) (z_{t}) remains valid under adaptive interaction, and the same holds for the released stream ( y t ) (y_{t}) by Theorem 4.3 . For details, see Appendix F.1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Scope Boundary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorem 4.3 is a sufficient certification condition, not a characterization of adaptive privacy. Fresh randomness alone is not enough; Appendix E.1 gives a two-round separation. The missing ingredient is the stable-context condition in Checklist 1, which prevents adaptive common inputs from changing the leakage profile of an earlier Hamming discrepancy. Conversely, failure of Checklist 1 does not imply that adaptive privacy is false; it means that a separate interaction-aware proof is needed, as for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} in Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Putting the Ingredients Together\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We prove end-to-end privacy by a modular composition argument that composes two interactive mechanisms in series. Our general composition theorem (Appendix C ) applies to any upstream wrapper that is ( ε b , δ b ) (\\\\varepsilon_{b},\\\\delta_{b}) -NPDP and any downstream continual primitive that is ( ε a , δ a ) (\\\\varepsilon_{a},\\\\delta_{a}) -DP for Hamming-neighboring update streams and is feedback-safe by certification. The pipeline in Fig. 1 is one instantiation, with 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} as the wrapper and a continual DP aggregator as the downstream mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 4.1 supplies the wrapper side: 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} implements an edit-to-Hamming reduction, i.e., a single insertion/deletion in the event stream induces only a bounded Hamming-style change in the induced (per-bin) update stream (Lemma 4.1 ). Section 4.2 supplies the downstream side: for primitives that pass Checklist 1, a standard non-adaptive DP proof for Hamming-neighboring inputs remains valid when updates are chosen adaptively from past releases (Theorem 4.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Modular Composition (Informal).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"View buffering as producing (i) a visible transcript (e.g., whether a bin is emitted) and (ii) a hidden interface stream of per-bin updates. The downstream continual primitive consumes this interface stream and releases the trajectory observed by the adversary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The end-to-end guarantee follows from two conditions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Wrapper condition (NPDP). The wrapper is ( ε b , δ b ) (\\\\varepsilon_{b},\\\\delta_{b}) -NPDP for single-edit neighboring event streams, and the induced interface streams can be aligned to be Hamming-neighboring (Lemma 4.1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Downstream condition (Hamming-DP + certification). The downstream primitive is ( ε a , δ a ) (\\\\varepsilon_{a},\\\\delta_{a}) -DP for Hamming-neighboring update streams on its own input, and passes Checklist 1, hence remains DP under adaptive interaction (Theorem 4.3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conclusion. Under these conditions, the composed pipeline is ( ε b + ε a , δ b + δ a ) (\\\\varepsilon_{b}{+}\\\\varepsilon_{a},\\\\;\\\\delta_{b}{+}\\\\delta_{a}) -DP with respect to single-edit neighboring event streams, even under adaptive interaction. This is the only composition step used to derive Theorem 3.1 ; the formal NPDP definition, alignment/refinement machinery, and the general modular composition theorem are deferred to Appendix C .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Takeaway.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sections 4.1 – 4.3 justify treating the pipeline as plug-and-play: once the wrapper and certification conditions hold, we can focus on the DP-SGD instantiation and evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Instantiation: Continual DP-SGD Pipeline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section instantiates the buffering–aggregation recipe (Section 4 ) with streaming SGD (Algorithm 2 ). Each arriving user event f t f_{t} is buffered by 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} . When a bin is emitted, we form a clipped mini-batch gradient and pass a (stepsize-scaled) update to a continual-release DP aggregator. In our experiments, the aggregator 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\\\mathsf{PrivStreamSum} is either (i) the standard tree-based prefix-sum mechanism [ CSS10 ] , or (ii) the lower-triangular matrix factorization prefix-sum mechanism [ FHU23 ] . Their formal DP and error/consistency guarantees are summarized in Appendix F . As in Fig. 1 , the adversary observes the released models and the release timing (i.e., whether B t = ⊥ B_{t}=\\\\bot at each step), while bin contents and internal states are hidden.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Fixed vs. Tuned Parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The privacy target ( ε , δ ) (\\\\varepsilon,\\\\delta) and its allocation into ( ε b , δ b ) (\\\\varepsilon_{b},\\\\delta_{b}) and ( ε a , δ a ) (\\\\varepsilon_{a},\\\\delta_{a}) determine (i) the buffering level U ( ε b , δ b ) U(\\\\varepsilon_{b},\\\\delta_{b}) and (ii) the aggregation budget for 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\\\mathsf{PrivStreamSum} . Given a learning task, the remaining choices follow standard DP-SGD practice: the clipping norm G G and stepsize schedule ( η k ) (\\\\eta_{k}) are tuned for utility. The release schedule is induced by 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} : bins are emitted at random times, and satisfy | B t | ∈ [ U , 2 U ] |B_{t}|\\\\in[U,2U] whenever B t ≠ ⊥ B_{t}\\\\neq\\\\bot . The aggregator 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\\\mathsf{PrivStreamSum} is used as a plug-in, subject to: (i) ( ε a , δ a ) (\\\\varepsilon_{a},\\\\delta_{a}) -DP for Hamming-neighboring update streams on its own input, and (ii) passing Checklist 1 so its non-adaptive DP analysis remains valid under adaptive interaction (Section 4.2 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Privacy-Budget Split.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use a symmetric 50 / 50 50/50 split between buffering and aggregation as a simple default. Since privacy composes as ( ε , δ ) = ( ε b + ε a , δ b + δ a ) (\\\\varepsilon,\\\\delta)=(\\\\varepsilon_{b}+\\\\varepsilon_{a},\\\\delta_{b}+\\\\delta_{a}) and latency scales as U = Θ ( ε b − 1 log ( 1 / δ b ) ) U=\\\\Theta(\\\\varepsilon_{b}^{-1}\\\\log(1/\\\\delta_{b})) , any fixed split ε b = c ε \\\\varepsilon_{b}=c\\\\varepsilon , ε a = ( 1 − c ) ε \\\\varepsilon_{a}=(1-c)\\\\varepsilon with c ∈ ( 0 , 1 ) c\\\\in(0,1) changes the asymptotic guarantees only by constant factors; assigning constant fractions of δ \\\\delta changes log ( 1 / δ b ) \\\\log(1/\\\\delta_{b}) only by an additive O ( 1 ) O(1) term. The best finite-sample split can be chosen empirically by sweeping c c and measuring the latency–accuracy tradeoff.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Mini-Batch View.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When a bin is emitted, the gradient estimate uses the standard mini-batch average 1 | B t | ∑ f ∈ B t ∇ f ( w ) \\\\frac{1}{|B_{t}|}\\\\sum_{f\\\\in B_{t}}\\\\nabla f(w) . Since 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} enforces | B t | ∈ [ U , 2 U ] |B_{t}|\\\\in[U,2U] , the batch size is controlled and varies by at most a factor of 2 2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Implementation Notes and Measured Latency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The server maintains (a) the current released model w ( t ) w^{(t)} , (b) the FIFO buffer state of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} , and (c) the internal state of 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\\\mathsf{PrivStreamSum} (e.g., the tree nodes). The model is updated only on emission times and is held fixed otherwise, yielding a piecewise-constant released trajectory. To quantify privacy-implied latency, we log the backlog Q t Q_{t} (pending events after time t t ). For an event f i f_{i} , its inclusion delay is D ( i ) := min { t ≥ i : f i ∈ B t } − i D(i):=\\\\min\\\\{t\\\\geq i:\\\\;f_{i}\\\\in B_{t}\\\\}-i , which can be computed directly from the same FIFO queue evolution underlying ( Q t ) (Q_{t}) ; we report summary statistics and empirical CDFs in the experiments.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Other Optimizers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The same template applies to streaming optimizers whose per-bin updates can be encoded as a clipped, sensitivity-bounded stream for the certified continual-release aggregator: replace g k = Clip ( ⋅ ) g_{k}=\\\\textsf{Clip}(\\\\cdot) and u k u_{k} by the desired update encoding, while keeping 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} as the edit-to-Hamming wrapper and 𝖯𝗋𝗂𝗏𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\\\mathsf{PrivStreamSum} as the aggregation primitive. Variants with additional state, such as momentum or adaptive preconditioning, require a separate audit of Theorem 4.3 ’s conditions; falling outside the theorem does not by itself imply failure of adaptive privacy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We studied continual learning under adaptive interaction when privacy is defined over edit-style (single-insertion/deletion) user streams, where naive continual-release analyses do not directly apply. Our main recipe composes a randomized buffering wrapper 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} , which reduces single-edit neighbors to a Hamming-style per-bin update interface with explicit backlog guarantees, with a certified continual-release DP primitive (tree-based prefix sums in our experiments), and then invokes modular composition to obtain end-to-end ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formalize the setting as outlined in the introduction. We first describe the concept of an abstract interactive mechanism, from which we will derive other forms of interactive mechanisms later.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Instantiation: Mapping the Abstract Interaction to Our Transcript.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The abstract interaction formalism above subsumes the continual-learning setting of Section 2 . Concretely, we instantiate the mechanism’s input messages as the user-side stream x t := f t x_{t}:=f_{t} (or, when modeling a subcomponent, x t x_{t} may represent an emitted bin B t B_{t} or a per-bin update g t g_{t} ), and we instantiate the mechanism’s outputs as the released observations y t := ( v t , w ( t ) ) y_{t}:=(v_{t},w^{(t)}) , so that the induced transcript is 𝗍𝗋 ≤ T = ( ( v t , w ( t ) ) ) t = 0 T \\\\mathsf{tr}_{\\\\leq T}=((v_{t},w^{(t)}))_{t=0}^{T} . The adversary’s view function ν 𝒜 \\\\nu_{{\\\\mathcal{A}}} reveals exactly the observable portion of each release—the timing bit and model snapshot—matching the trajectory-and-timing threat model in Section 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bounded Termination. In addition to finite ℛ \\\\mathcal{R} and 𝒰 \\\\mathcal{U} , we consider mechanisms that always terminate in a bounded number of steps. Specifically, for each mechanism ℳ \\\\mathcal{M} , there exists T ℳ ∈ ℤ T_{\\\\mathcal{M}}\\\\in\\\\mathbb{Z} such that ℳ \\\\mathcal{M} always terminates in at most T ℳ T_{\\\\mathcal{M}} steps. This is without loss of generality, because any privacy notion defined for mechanisms with bounded termination can be extended naturally to unbounded mechanisms. We simply require that, for any T ∈ ℤ + T\\\\in\\\\mathbb{Z}_{+} , the truncated mechanism obtained by running the (unbounded) mechanism for T T steps will satisfy the privacy notion defined for bounded mechanisms. This matches the “for every horizon T T ” convention in Section 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mechanism View. The view observed by a mechanism ℳ \\\\mathcal{M} up to time step t t consists of its generated random seed r r and its received input messages x [ 1 . . t ] x[1..t] up to time t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since ℳ \\\\mathcal{M} can recover its own output sequence from its random seed and input sequence, it suffices to include the latter two in its view.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similarity Between Mechanisms. To formalize privacy notions later, we need a way to quantify the similarity between two mechanisms. Since we consider randomized mechanisms, we will use power functions to compare how different two distributions are. It is known that power functions are general enough to capture all divergences satisfying the data processing inequality. Recall that in this work, we focus on finite sample spaces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A divergence measure is a function 𝖣 \\\\mathsf{D} that takes two distributions P P and Q Q on the same space such that 𝖣 ( P ∥ Q ) ≥ 0 \\\\mathsf{D}(P\\\\|Q)\\\\geq 0 , where equality holds iff the distributions P = Q P=Q are identical.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A divergence 𝖣 \\\\mathsf{D} satisfies the data processing inequality if, for any stochastic transformation (or channel) T T that maps the original space to another space, the following inequality holds:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where T ( P ) T(P) and T ( Q ) T(Q) are the resulting distributions after applying the transformation T T to P P and Q Q , respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose P P and Q Q are distributions on the same finite sample space Ω \\\\Omega , i.e., P P and Q Q are vectors in ℝ ≥ 0 Ω \\\\mathbb{R}^{\\\\Omega}_{\\\\geq 0} whose coordinates sum to 1. The power function 𝖯𝗈𝗐 ( P ∥ Q ) : [ 0 , 1 ] → [ 0 , 1 ] \\\\mathsf{Pow}(P\\\\|Q):[0,1]\\\\rightarrow[0,1] can be defined in terms of the fractional knapsack problem .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a collection Ω \\\\Omega of items, suppose ω ∈ Ω \\\\omega\\\\in\\\\Omega has weight P ( ω ) P(\\\\omega) and value Q ( ω ) Q(\\\\omega) . Then, given α ∈ [ 0 , 1 ] \\\\alpha\\\\in[0,1] , 𝖯𝗈𝗐 ( P ∥ Q ) ( α ) \\\\mathsf{Pow}(P\\\\|Q)(\\\\alpha) is the maximum value attained with weight capacity constraint α \\\\alpha , where items may be taken fractionally.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intuition. When two distributions P P and Q Q are the same, it is clear that given any capacity constraint α \\\\alpha , the maximum reward is also α \\\\alpha ; this means the power function is exactly the identity function. However, if the two distributions are very different, this means that there are items whose reward-to-weight ratios are large; hence, in this case, the power function can initially grow faster than the identity function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Partial Order on Power Functions. Pointwise comparison naturally induces a partial order on power functions. We denote f 1 ≤ f 2 f_{1}\\\\leq f_{2} if for all α ∈ [ 0 , 1 ] \\\\alpha\\\\in[0,1] , f 1 ( α ) ≤ f 2 ( α ) f_{1}(\\\\alpha)\\\\leq f_{2}(\\\\alpha) , where a larger f 2 f_{2} indicates that the two distributions are more different.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any two distributions P P and Q Q on the same sample space, 𝖯𝗈𝗐 ( P ∥ Q ) ≥ 𝖨𝖽 \\\\mathsf{Pow}(P\\\\|Q)\\\\geq\\\\mathsf{Id} , where 𝖨𝖽 : [ 0 , 1 ] → [ 0 , 1 ] \\\\mathsf{Id}:[0,1]\\\\rightarrow[0,1] is the identity function, and equality holds iff the distributions P = Q P=Q are identical.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, power functions satisfy the data processing inequality , i.e., for any stochastic transformation (or channel) T T that maps the original space to another space, the following inequality holds:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Differential Privacy in Terms of Power Function. Given ε ≥ 0 \\\\varepsilon\\\\geq 0 and 0 ≤ δ ≤ 1 0\\\\leq\\\\delta\\\\leq 1 , the power function 𝖣𝖯 ε , δ : [ 0 , 1 ] → [ 0 , 1 ] \\\\mathsf{DP}_{\\\\varepsilon,\\\\delta}:[0,1]\\\\rightarrow[0,1] can be described as follows (see Figure 2 ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the x y xy -plane, starting at ( 0 , δ ) (0,\\\\delta) , the function increases linearly with slope e ε e^{\\\\varepsilon} until it touches the line y = 1 − x y=1-x at ( 1 − δ 1 + e ε , e ε + δ 1 + e ε ) (\\\\frac{1-\\\\delta}{1+e^{\\\\varepsilon}},\\\\frac{e^{\\\\varepsilon}+\\\\delta}{1+e^{\\\\varepsilon}}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After touching the line y = 1 − x y=1-x , the slope changes to e − ε e^{-\\\\varepsilon} , until the line segment reaches ( 1 − δ , 1 ) (1-\\\\delta,1) , where the curve remains horizontally till the end ( 1 , 1 ) (1,1) is reached.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In other words,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝖣𝖯 ε , δ ( α ) = { δ + e ε ⋅ α , for 0 ≤ α ≤ 1 − δ 1 + e ε ; 1 − e − ε ( 1 − δ ) + e − ε α , for 1 − δ 1 + e ε < α ≤ 1 − δ ; 1 , for 1 − δ < α ≤ 1 . \\\\mathsf{DP}_{\\\\varepsilon,\\\\delta}(\\\\alpha)=\\\\begin{cases}\\\\delta+e^{\\\\varepsilon}\\\\cdot\\\\alpha,&\\\\text{for }0\\\\leq\\\\alpha\\\\leq\\\\frac{1-\\\\delta}{1+e^{\\\\varepsilon}};\\\\\\\\ 1-e^{-\\\\varepsilon}(1-\\\\delta)+e^{-\\\\varepsilon}\\\\alpha,&\\\\text{for }\\\\frac{1-\\\\delta}{1+e^{\\\\varepsilon}}<\\\\alpha\\\\leq 1-\\\\delta;\\\\\\\\ 1,&\\\\text{for }1-\\\\delta<\\\\alpha\\\\leq 1.\\\\end{cases}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Below are some facts on power functions that we will use, which are usually equivalently stated in the literature in terms of the tradeoff function 𝖳 ( P ∥ Q ) ( α ) := 1 − 𝖯𝗈𝗐 ( P ∥ Q ) ( α ) \\\\mathsf{T}(P\\\\|Q)(\\\\alpha):=1-\\\\mathsf{Pow}(P\\\\|Q)(\\\\alpha) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given two distributions P P and Q Q on the same sample space Ω \\\\Omega , 𝖯𝗈𝗐 ( P ∥ Q ) ≤ 𝖣𝖯 ε , δ \\\\mathsf{Pow}(P\\\\|Q)\\\\leq\\\\mathsf{DP}_{\\\\varepsilon,\\\\delta} is equivalent to the statement that for all S ⊆ Ω S\\\\subseteq\\\\Omega , P ( S ) ≤ e ε ⋅ Q ( S ) + δ P(S)\\\\leq e^{\\\\varepsilon}\\\\cdot Q(S)+\\\\delta and Q ( S ) ≤ e ε ⋅ P ( S ) + δ Q(S)\\\\leq e^{\\\\varepsilon}\\\\cdot P(S)+\\\\delta .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"( Triangle Inequality for Power Functions )[Theorem 2.14 in [ DRS22 ] ] Suppose X X , Y Y , Z Z are distributions on the same sample space such that 𝖯𝗈𝗐 ( X ∥ Y ) ≤ g 1 \\\\mathsf{Pow}(X\\\\|Y)\\\\leq g_{1} and 𝖯𝗈𝗐 ( Y ∥ Z ) ≤ g 2 \\\\mathsf{Pow}(Y\\\\|Z)\\\\leq g_{2} . Then, 𝖯𝗈𝗐 ( X ∥ Z ) ≤ g 2 ∘ g 1 \\\\mathsf{Pow}(X\\\\|Z)\\\\leq g_{2}\\\\circ g_{1} , where the composition is defined as ( g 2 ∘ g 1 ) ( x ) := g 2 ( g 1 ( x ) ) (g_{2}\\\\circ g_{1})(x):=g_{2}(g_{1}(x)) . In other words, 𝖯𝗈𝗐 ( X ∥ Z ) ≤ 𝖯𝗈𝗐 ( Y ∥ Z ) ∘ 𝖯𝗈𝗐 ( X ∥ Y ) \\\\mathsf{Pow}(X\\\\|Z)\\\\leq\\\\mathsf{Pow}(Y\\\\|Z)\\\\circ\\\\mathsf{Pow}(X\\\\|Y) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Example. Consider g = 𝖣𝖯 ε , δ g=\\\\mathsf{DP}_{\\\\varepsilon,\\\\delta} . Then, g ∘ g ≤ 𝖣𝖯 2 ε , δ ( 1 + e ε ) g\\\\circ g\\\\leq\\\\mathsf{DP}_{2\\\\varepsilon,\\\\delta(1+e^{\\\\varepsilon})} . As we shall see, this is relevant to the scenario of multi-hop neighbors – aka group DP in the literature.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a collection 𝒮 \\\\mathcal{S} of power functions, its supremum sup ( 𝒮 ) = sup f ∈ 𝒮 f \\\\sup(\\\\mathcal{S})=\\\\sup_{f\\\\in\\\\mathcal{S}}f is the least power function f ^ \\\\widehat{f} such that for all 𝔣 ∈ 𝒮 \\\\mathfrak{f}\\\\in\\\\mathcal{S} , f ≤ f ^ f\\\\leq\\\\widehat{f} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Formally, for each α ∈ [ 0 , 1 ] \\\\alpha\\\\in[0,1] , f ^ ( α ) := \\\\widehat{f}(\\\\alpha):=\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose f f and g g are power functions that are attained by the corresponding distributions in 𝖯𝗈𝗐 ( P 1 ∥ P 2 ) = f \\\\mathsf{Pow}(P_{1}\\\\|P_{2})=f and g = 𝖯𝗈𝗐 ( Q 1 ∥ Q 2 ) g=\\\\mathsf{Pow}(Q_{1}\\\\|Q_{2}) . Then, their corresponding tensor product f ⊗ g = 𝖯𝗈𝗐 ( P 1 × Q 1 ∥ P 2 × Q 2 ) f\\\\otimes g=\\\\mathsf{Pow}(P_{1}\\\\times Q_{1}\\\\|P_{2}\\\\times Q_{2}) is defined in terms of the corresponding product distributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose Λ \\\\Lambda is an indexing set and f f is a power function such that for each λ ∈ Λ \\\\lambda\\\\in\\\\Lambda , a pair ( P ( λ ) , Q ( λ ) ) (P(\\\\lambda),Q(\\\\lambda)) of distributions satisfy 𝖯𝗈𝗐 ( P ( λ ) ∥ Q ( λ ) ) ≤ f \\\\mathsf{Pow}(P(\\\\lambda)\\\\|Q(\\\\lambda))\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, for any distribution L L on Λ \\\\Lambda , 𝖯𝗈𝗐 ( P ( L ) ∥ Q ( L ) ) ≤ f \\\\mathsf{Pow}(P(L)\\\\|Q(L))\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose P P and P ′ P^{\\\\prime} are distributions on the sample space Ω \\\\Omega . For each ω ∈ Ω \\\\omega\\\\in\\\\Omega , Q ( ω ) Q(\\\\omega) and Q ′ ( ω ) Q^{\\\\prime}(\\\\omega) are distributions on sample space Ω ′ \\\\Omega^{\\\\prime} . If 𝖯𝗈𝗐 ( P ∥ P ′ ) ≤ f \\\\mathsf{Pow}(P\\\\|P^{\\\\prime})\\\\leq f and 𝖯𝗈𝗐 ( Q ( ω ) ∥ Q ′ ( ω ) ) ≤ g \\\\mathsf{Pow}(Q(\\\\omega)\\\\|Q^{\\\\prime}(\\\\omega))\\\\leq g for all ω ∈ Ω \\\\omega\\\\in\\\\Omega , then 𝖯𝗈𝗐 ( ( P , Q ( P ) ) ∥ ( P ′ , Q ′ ( P ′ ) ) ) ≤ f ⊗ g . \\\\mathsf{Pow}((P,Q(P))\\\\|(P^{\\\\prime},Q^{\\\\prime}(P^{\\\\prime})))\\\\leq f\\\\otimes g.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Distinguishing between Interactive Mechanisms. Suppose we wish to compare the behavior of two mechanisms ℳ 0 \\\\mathcal{M}_{0} and ℳ 1 \\\\mathcal{M}_{1} , to which we only have oracle accesses. An adversary 𝒜 : ℛ × 𝒰 ∗ → 𝒰 \\\\mathcal{A}:\\\\mathcal{R}\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} is also an interactive mechanism. The adversary has some view function ν 𝒜 : 𝒰 → 𝒰 \\\\nu_{\\\\mathcal{A}}:\\\\mathcal{U}\\\\rightarrow\\\\mathcal{U} that specifies what information it can gather from the output of the mechanism. However, later when we instantiate from the abstract mechanism, we will use the structure of the message space to clarify what the adversary can observe.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next define the interaction 𝒜 ↔ ℳ {\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{M} between the two mechanisms. The adversary 𝒜 {\\\\mathcal{A}} repeatedly sends messages to ℳ \\\\mathcal{M} . At each time step, the adversary determines the next message to be sent to ℳ \\\\mathcal{M} , based on the history of prior interactions. This process repeats iteratively, until the mechanism ℳ \\\\mathcal{M} outputs 𝗁𝖺𝗅𝗍 \\\\mathsf{halt} in some time step. Formally, it is captured by the following description.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose the interaction 𝒜 ↔ ℳ {\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{M} terminates at time step t t (which may be a random quantity). Then, the adversary view is defined as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ ) := ( r 𝒜 ; v [ 0 . . t ] ) \\\\mathsf{view}({\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{M}):=(r_{\\\\mathcal{A}};v[0..t]) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Terminology (“non-adaptive”). In this paper, when we say “non-adaptive” for a continual mechanism’s privacy analysis, we mean the mechanism’s input stream is fixed in advance (equivalently, the environment/adversary that provides inputs does not condition on past outputs).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An adversary transition function 𝒜 : ℛ × 𝒰 ∗ → 𝒰 \\\\mathcal{A}:\\\\mathcal{R}\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} is non-adaptive, if there exists a transition function of the form 𝒜 ^ : ℛ × ℤ → 𝒰 \\\\widehat{\\\\mathcal{A}}:\\\\mathcal{R}\\\\times\\\\mathbb{Z}\\\\rightarrow\\\\mathcal{U} such that for any length- t t sequence v [ 0 . . t − 1 ] v[0..t-1] , 𝒜 ( r ; v [ 0 . . t − 1 ] ) = 𝒜 ^ ( r ; t ) \\\\mathcal{A}(r;v[0..t-1])=\\\\widehat{\\\\mathcal{A}}(r;t) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can next define a distance notion between two mechanisms based on power functions. Intuitively, it quantifies the difficulty of distinguishing between two mechanisms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given two interactive mechanisms ℳ 0 \\\\mathcal{M}_{0} and ℳ 1 \\\\mathcal{M}_{1} , we overload the notation and denote 𝖯𝗈𝗐 ( ℳ 0 ∥ ℳ 1 ) ≤ f \\\\mathsf{Pow}\\\\left(\\\\mathcal{M}_{0}\\\\|\\\\mathcal{M}_{1}\\\\right)\\\\leq f , for some power function f f , if for all adversaries 𝒜 {\\\\mathcal{A}} , 𝖯𝗈𝗐 ( 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ 0 ) ∥ 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ 1 ) ) ≤ f . \\\\mathsf{Pow}\\\\left(\\\\mathsf{view}({\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{M}_{0})\\\\|\\\\mathsf{view}({\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{M}_{1})\\\\right)\\\\leq f.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If we have the weaker requirement that the inequality holds for just non-adaptive adversaries, we denote: 𝖯𝗈𝗐 NA ( ℳ 0 ∥ ℳ 1 ) ≤ f \\\\mathsf{Pow}^{{\\\\textsc{NA}}}\\\\left(\\\\mathcal{M}_{0}\\\\|\\\\mathcal{M}_{1}\\\\right)\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Simulation of Two Close Interactive Mechanisms with Close Random Seeds. The following result from [ VZ23 ] states that if two mechanisms are close in the sense of Definition A.13 , then it is possible to simulate either one of them exactly with the same mechanism transition function that initially takes a random seed generated from one of the two appropriate distributions. Because the proof uses induction and Cartesian products of sample spaces, only the special case of finite sample and message spaces and mechanisms with bounded termination is considered. It is plausible that the result could be generalized to unbounded spaces, but that would likely require the Axiom of Choice, which often falls outside the scope of the computer science community.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is a very useful lemma in proving privacy composition results, because instead of interacting with two potentially different mechanisms in every time step, we can view the adversary as interacting with the same mechanism transition function but with initial random seeds drawn from two different distributions. One useful analogy is that after a sensitive database is anonymized with appropriate noise, then it can be queried unlimited number of times without privacy degradation because of data post-processing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose ℳ 0 \\\\mathcal{M}_{0} and ℳ 1 : ℛ × 𝒰 ∗ → 𝒰 \\\\mathcal{M}_{1}:\\\\mathcal{R}\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} are two interactive mechanisms with bounded termination, and f f is a power function such that 𝖯𝗈𝗐 ( ℳ 0 ∥ ℳ 1 ) ≤ f \\\\mathsf{Pow}(\\\\mathcal{M}_{0}\\\\|\\\\mathcal{M}_{1})\\\\leq f in the sense of Definition A.13 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, there exist two distributions N 0 N_{0} and N 1 N_{1} on some sample space Ω \\\\Omega and a transition function 𝒫 : Ω × 𝒰 ∗ → 𝒰 \\\\mathcal{P}:\\\\Omega\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} such that 𝖯𝗈𝗐 ( N 0 ∥ N 1 ) ≤ f \\\\mathsf{Pow}(N_{0}\\\\|N_{1})\\\\leq f , and, for all adversaries 𝒜 {\\\\mathcal{A}} , the following holds for both b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} : 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℳ b ) ≡ 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ 𝒫 ( N b ) ) , \\\\mathsf{view}({\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{M}_{b})\\\\equiv\\\\mathsf{view}({\\\\mathcal{A}}\\\\leftrightarrow\\\\mathcal{P}(N_{b})),\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒫 ( N ) \\\\mathcal{P}({N}) is the interactive mechanism that uses the specified transition function 𝒫 \\\\mathcal{P} with a random seed drawn from the distribution N N .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Interactive Mechanism with Hidden Output.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next consider a special form of interactive mechanism with a transition function of the form ℳ : ℛ × 𝒳 ∗ → 𝒱 × 𝒴 \\\\mathcal{M}:\\\\mathcal{R}\\\\times\\\\mathcal{X}^{*}\\\\to\\\\mathcal{V}\\\\times\\\\mathcal{Y} . Here, we use 𝒳 \\\\mathcal{X} to denote the input space. In each time step, the mechanism produces ( v , y ) ∈ 𝒱 × 𝒴 (v,y)\\\\in\\\\mathcal{V}\\\\times\\\\mathcal{Y} , where the adversary can observe v v from the view space 𝒱 \\\\mathcal{V} and the output y y from the output space 𝒴 \\\\mathcal{Y} is potentially hidden from the adversary, which may be passed to another interactive mechanism (that could however possibly leak information about y y ). If an adversary only interacts with ℳ \\\\mathcal{M} , then it can be described by a transition function with the form 𝒜 : ℛ × 𝒱 ∗ → 𝒳 {\\\\mathcal{A}}:\\\\mathcal{R}\\\\times\\\\mathcal{V}^{*}\\\\to\\\\mathcal{X} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neighboring Input Sequences. Similar to the conventional differential privacy, we need a neighboring notion for input sequences such that intuitively the behaviors of a mechanism on two neighboring input sequences should be similar.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We assume there is a neighboring relation ∼ 𝒳 ∗ \\\\sim_{\\\\mathcal{X}^{*}} (or just ∼ \\\\sim for succinctness) that satisfies the property that if two sequences x [ 1 . . t ] ∼ x ′ [ 1 . . t ′ ] x[1..t]\\\\sim x^{\\\\prime}[1..t^{\\\\prime}] are neighboring, then the following holds:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"They have the same length, i.e., t = t ′ t=t^{\\\\prime} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All their corresponding prefixes are neighboring, i.e., for all 1 ≤ τ < t 1\\\\leq\\\\tau<t , x [ 1 . . τ ] ∼ x ′ [ 1 . . τ ] x[1..\\\\tau]\\\\sim x^{\\\\prime}[1..\\\\tau] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Privacy for Interactive Mechanisms. Since the inputs for an interactive mechanism in later time steps can be influenced by its earlier behavior, one cannot formally define its privacy just based on neighboring input sequences. In the literature, adaptive differential privacy [ JRS+23 ] is based on a construction that we call paired simulation .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose an interactive mechanism is given by the transition function ℳ : ℛ × 𝒳 ∗ → 𝒱 × 𝒴 \\\\mathcal{M}:\\\\mathcal{R}\\\\times\\\\mathcal{X}^{*}\\\\to\\\\mathcal{V}\\\\times\\\\mathcal{Y} and the random seed distribution R ℳ R_{\\\\mathcal{M}} , where a neighboring relation is defined on 𝒳 ∗ \\\\mathcal{X}^{*} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, the paired simulation consists of a canonical pair of interactive mechanisms ( ℳ 0 𝗉𝖺𝗂𝗋 , ℳ 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{M}^{\\\\mathsf{pair}}_{0},\\\\mathcal{M}^{\\\\mathsf{pair}}_{1}) such that for each b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , the transition function has the form ℳ b 𝗉𝖺𝗂𝗋 : ℛ × ( 𝒳 2 ) ∗ → 𝒱 × 𝒴 \\\\mathcal{M}^{\\\\mathsf{pair}}_{b}:\\\\mathcal{R}\\\\times(\\\\mathcal{X}^{2})^{*}\\\\to\\\\mathcal{V}\\\\times\\\\mathcal{Y} . Moreover, in the paired simulation, an adversary with transition function 𝒜 : ℛ × 𝒱 ∗ → 𝒳 2 \\\\mathcal{A}:\\\\mathcal{R}\\\\times\\\\mathcal{V}^{*}\\\\to\\\\mathcal{X}^{2} interacts with ℳ b 𝗉𝖺𝗂𝗋 \\\\mathcal{M}^{\\\\mathsf{pair}}_{b} as follows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given an interactive mechanism ℳ \\\\mathcal{M} , suppose ( ℳ 0 𝗉𝖺𝗂𝗋 , ℳ 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{M}^{\\\\mathsf{pair}}_{0},\\\\mathcal{M}^{\\\\mathsf{pair}}_{1}) is the paired simulation as in Definition B.2 . For a power function f f , the mechanism ℳ \\\\mathcal{M} is adaptively f f -differentially private (DP) if\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝖯𝗈𝗐 ( ℳ 0 𝗉𝖺𝗂𝗋 ∥ ℳ 1 𝗉𝖺𝗂𝗋 ) ≤ f \\\\mathsf{Pow}(\\\\mathcal{M}^{\\\\mathsf{pair}}_{0}\\\\|\\\\mathcal{M}^{\\\\mathsf{pair}}_{1})\\\\leq f , for all adaptive adversaries in the sense of Definition A.13 . Similarly, non-adaptively f f -DP is defined by 𝖯𝗈𝗐 NA ( ℳ 0 𝗉𝖺𝗂𝗋 ∥ ℳ 1 𝗉𝖺𝗂𝗋 ) ≤ f \\\\mathsf{Pow}^{\\\\textsc{NA}}(\\\\mathcal{M}^{\\\\mathsf{pair}}_{0}\\\\|\\\\mathcal{M}^{\\\\mathsf{pair}}_{1})\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Multi-Hop Neighboring Notion. In the literature, group DP or multi-hop neighbors can be generalized to sequences readily.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For k = 1 k=1 , 1 1 -hop neighboring is the same as that in Assumption B.1 . For k ≥ 2 k\\\\geq 2 , two sequences x x and x ′ x^{\\\\prime} are k k -hop neighboring if there exist x 1 , … , x k − 1 x_{1},\\\\ldots,x_{k-1} such that for 0 ≤ i < k 0\\\\leq i<k , x i x_{i} and x i + 1 x_{i+1} are 1 1 -hop neighboring, where x 0 = x x_{0}=x and x k = x ′ x_{k}=x^{\\\\prime} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also generalize Definition B.2 to multi-hop neighboring simulation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Generalizing the description in Definition B.2 , in each time step t t , the adversary generates a tuple ( x t ( 0 ) , x t ( 1 ) , … , x t ( k ) ) (x^{(0)}_{t},x^{(1)}_{t},\\\\ldots,x^{(k)}_{t}) such that for each 0 ≤ i < k 0\\\\leq i<k , x ( i ) [ 1 . . t ] x^{(i)}[1..t] and x ( i + 1 ) [ 1 . . t ] x^{(i+1)}[1..t] are neighboring sequences.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, for b ∈ [ 0 . . k ] b\\\\in[0..k] , ℳ b 𝗁𝗈𝗉 \\\\mathcal{M}^{\\\\mathsf{hop}}_{b} is a simulation of ℳ \\\\mathcal{M} that takes the b b -th component x t ( b ) x^{(b)}_{t} at each step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose ℳ \\\\mathcal{M} is adaptively f f -DP as in Definition B.3 . Then, in a k k -hop neighboring simulation as in Definition B.5 , we have: 𝖯𝗈𝗐 ( ℳ 0 𝗁𝗈𝗉 ∥ ℳ k 𝗁𝗈𝗉 ) ≤ f ∘ k \\\\mathsf{Pow}(\\\\mathcal{M}^{\\\\mathsf{hop}}_{0}\\\\|\\\\mathcal{M}^{\\\\mathsf{hop}}_{k})\\\\leq f^{\\\\circ k} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where f ∘ 1 = f f^{\\\\circ 1}=f and f ∘ ( i + 1 ) = f ∘ i ∘ f f^{\\\\circ(i+1)}=f^{\\\\circ i}\\\\circ f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This follows from Definition B.3 and Fact A.6 . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Modular Composition of Interactive Mechanisms. As mentioned above, the hidden output of one mechanism may be fed as the input of another mechanism. To this end, we define a specific form of composition that we call modular composition . The adversary only directly supplies an input to the “head” mechanism of the list, where each intermediate mechanism receives its input from the (hidden) output from the previous mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose two interactive mechanisms are given by the transition functions ℳ : ℛ 1 × 𝒳 ∗ → 𝒱 × 𝒴 \\\\mathcal{M}:\\\\mathcal{R}_{1}\\\\times\\\\mathcal{X}^{*}\\\\rightarrow\\\\mathcal{V}\\\\times\\\\mathcal{Y} and 𝒩 : ℛ 2 × 𝒴 ∗ → 𝒲 × 𝒵 \\\\mathcal{N}:\\\\mathcal{R}_{2}\\\\times\\\\mathcal{Y}^{*}\\\\to\\\\mathcal{W}\\\\times\\\\mathcal{Z} , with random seed distributions R ℳ R_{\\\\mathcal{M}} and R 𝒩 R_{\\\\mathcal{N}} , respectively. Then, an adversary with transition function 𝒜 : ℛ × ( 𝒱 × 𝒲 ) ∗ → 𝒳 \\\\mathcal{A}:\\\\mathcal{R}\\\\times(\\\\mathcal{V}\\\\times\\\\mathcal{W})^{*}\\\\to\\\\mathcal{X} interacts with the modular composition ℳ → 𝒩 \\\\mathcal{M}\\\\to\\\\mathcal{N} as above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neighbor-Preserving Paired Simulation. To consider composition of differentially oblivious mechanisms [ CCM+22 ] , the framework neighbor-preserving differential obliviousness (NPDO) [ ZSC+23 , ZZC+24 ] has been introduced to formalize the condition that if neighboring inputs are submitted to a mechanism, then the corresponding outputs would somehow be neighboring for a subsequent mechanism. To generalize this to interactive mechanisms, we will need to introduce neighbor-preserving paired simulation. In addition to neighboring input sequences in 𝒳 ∗ \\\\mathcal{X}^{*} for mechanism 𝒩 \\\\mathcal{N} , we assume that there is a similar notion of neighboring sequences in 𝒴 ∗ \\\\mathcal{Y}^{*} (satisfying Assumption B.1 ) for mechanism 𝒩 \\\\mathcal{N} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given an interactive mechanism with transition function ℳ : ℛ × 𝒳 ∗ → 𝒱 × 𝒴 \\\\mathcal{M}:\\\\mathcal{R}\\\\times\\\\mathcal{X}^{*}\\\\rightarrow\\\\mathcal{V}\\\\times\\\\mathcal{Y} and random seed distribution R ℳ R_{\\\\mathcal{M}} , a neighbor-preserving paired simulation (which might not be unique) is a pair ( ℳ 0 𝗇𝗉𝗉 , ℳ 1 𝗇𝗉𝗉 ) (\\\\mathcal{M}^{\\\\mathsf{npp}}_{0},\\\\mathcal{M}^{\\\\mathsf{npp}}_{1}) of interactive mechanisms such that the following holds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There exists possibly another collection ℛ ′ \\\\mathcal{R}^{\\\\prime} of random seeds and an appropriate distribution R ℳ ′ R^{\\\\prime}_{\\\\mathcal{M}} such that for each b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , the corresponding mechanism has a transition function of the form ℳ b 𝗇𝗉𝗉 : ℛ × ℛ ′ × ( 𝒳 2 ) ∗ → 𝒱 × 𝒴 2 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b}:\\\\mathcal{R}\\\\times\\\\mathcal{R}^{\\\\prime}\\\\times(\\\\mathcal{X}^{2})^{*}\\\\to\\\\mathcal{V}\\\\times\\\\mathcal{Y}^{2} , where the random seed distribution on ℛ \\\\mathcal{R} still follows R ℳ R_{\\\\mathcal{M}} , and the random seed from ℛ ′ \\\\mathcal{R}^{\\\\prime} is sampled from R ℳ ′ R^{\\\\prime}_{\\\\mathcal{M}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In other words, in addition to a random seed in ℛ \\\\mathcal{R} , an extra random seed in ℛ ′ \\\\mathcal{R}^{\\\\prime} may be taken. Just like paired simulation in Definition B.2 , a sequence of input pairs are also taken. However, the transition function returns a tuple in 𝒱 × 𝒴 2 \\\\mathcal{V}\\\\times\\\\mathcal{Y}^{2} , i.e., in addition to v ∈ 𝒱 v\\\\in\\\\mathcal{V} , a pair ( y 0 , y 1 ) ∈ 𝒴 2 (y_{0},y_{1})\\\\in\\\\mathcal{Y}^{2} is returned.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , r ∈ ℛ r\\\\in\\\\mathcal{R} , r ′ ∈ ℛ ′ r^{\\\\prime}\\\\in\\\\mathcal{R}^{\\\\prime} , x ( 0 ) [ 1 . . t ] x^{(0)}[1..t] and x ( 1 ) [ 1 . . t ] ∈ 𝒳 ∗ x^{(1)}[1..t]\\\\in\\\\mathcal{X}^{*} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"if x ( 0 ) [ 1 . . t ] x^{(0)}[1..t] and x ( 1 ) [ 1 . . t ] x^{(1)}[1..t] are not neighboring sequences, then ℳ b 𝗇𝗉𝗉 ( r , r ′ ; x ( 0 ) [ 1 . . t ] , x ( 1 ) [ 1 . . t ] ) = ⊥ \\\\mathcal{M}^{\\\\mathsf{npp}}_{b}(r,r^{\\\\prime};x^{(0)}[1..t],x^{(1)}[1..t])=\\\\bot , and the whole simulation terminates at time step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Otherwise, the pair ( y ( 0 ) [ 1 . . t ] , y ( 1 ) [ 1 . . t ] ) (y^{(0)}[1..t],y^{(1)}[1..t]) of output sequences produced in the first t t time steps are neighboring in 𝒴 ∗ \\\\mathcal{Y}^{*} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If for some ( v , y 0 , y 1 ) ∈ 𝒱 × 𝒴 2 (v,y_{0},y_{1})\\\\in\\\\mathcal{V}\\\\times\\\\mathcal{Y}^{2} , ℳ b 𝗇𝗉𝗉 ( r , r ′ ; x ( 0 ) [ 1 . . t ] , x ( 1 ) [ 1 . . t ] ) = ( v , y 0 , y 1 ) \\\\mathcal{M}^{\\\\mathsf{npp}}_{b}(r,r^{\\\\prime};x^{(0)}[1..t],x^{(1)}[1..t])=(v,y_{0},y_{1}) , then ℳ ( r ; x ( b ) [ 1 . . t ] ) = ( v , y b ) \\\\mathcal{M}(r;x^{(b)}[1..t])=(v,y_{b}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In other words, we require consistency between ℳ b 𝗇𝗉𝗉 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b} and ℳ b 𝗉𝖺𝗂𝗋 \\\\mathcal{M}^{\\\\mathsf{pair}}_{b} (from Definition B.2 ). Specifically, the projection of ℳ b 𝗇𝗉𝗉 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b} ’s output onto ℳ b 𝗉𝖺𝗂𝗋 \\\\mathcal{M}^{\\\\mathsf{pair}}_{b} ’s range must match the behavior of ℳ b 𝗉𝖺𝗂𝗋 \\\\mathcal{M}^{\\\\mathsf{pair}}_{b} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, unlike paired simulation ℳ b 𝗉𝖺𝗂𝗋 \\\\mathcal{M}^{\\\\mathsf{pair}}_{b} whose behavior is totally determined by the given ℳ \\\\mathcal{M} , in the tuple ( v , y 0 , y 1 ) (v,y_{0},y_{1}) returned above by ℳ b 𝗇𝗉𝗉 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b} , the value y b ¯ y_{\\\\overline{b}} (where b ≠ b ¯ b\\\\neq\\\\overline{b} ) can depend on all the input parameters in a manner that is not specified by ℳ \\\\mathcal{M} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Augmented Adversary View. When an adversary interacts with ℳ b 𝗇𝗉𝗉 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b} , the adversarial view consists of its randomness and the sequence in ( 𝒱 × 𝒴 2 ) ∗ (\\\\mathcal{V}\\\\times\\\\mathcal{Y}^{2})^{*} produced until termination.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a power function f f , an interactive mechanism ℳ \\\\mathcal{M} is said to be adaptively f f -neighbor-preserving differentially private (NPDP), if there exists a neighbor-preserving paired simulation ( ℳ 0 𝗇𝗉𝗉 , ℳ 1 𝗇𝗉𝗉 ) (\\\\mathcal{M}^{\\\\mathsf{npp}}_{0},\\\\mathcal{M}^{\\\\mathsf{npp}}_{1}) such that 𝖯𝗈𝗐 ( ℳ 0 𝗇𝗉𝗉 ∥ ℳ 1 𝗇𝗉𝗉 ) ≤ f \\\\mathsf{Pow}(\\\\mathcal{M}^{\\\\mathsf{npp}}_{0}\\\\|\\\\mathcal{M}^{\\\\mathsf{npp}}_{1})\\\\leq f , in the sense of Definition A.13 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that if one wishes to prove that a mechanism is NPDP as in Definition B.9 , there is potentially some freedom to construct an appropriate NPP simulation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the contrary, for non-interactive mechanisms (that only run for one time step), the recent result on universally closest distribution refinements [ CX25 ] implies that a “best possible” NPP simulation may be used.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unfortunately, this does not seem to generalize to interactive mechanisms. The high level reason is that there may be no closest distribution refinements on two random sequences of length t + 1 t+1 such that its prefix projection would give a closest distribution refinement for the corresponding prefixes of length t t . In other words, if one constructs the “best possible” NPP up to time step t t , it may not result in the best NPP if the process enters step t + 1 t+1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next describe a modular composition theorem for interactive mechanisms. At a high level, the theorem states that if the first stage is neighbor-preserving (NPDP) with privacy profile f f , and the second stage is (adaptive) DP with profile g g on the intermediate stream, then the end-to-end interactive pipeline is (adaptive) DP with profile f ⊗ g f\\\\otimes g .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"How this theorem relates to existing composition frameworks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our theorem can be viewed as a direct consequence of two existing ideas, once NPDP is interpreted through paired simulation .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, it is shown [ ZZC+24 ] that neighbor-preserving notions (including differential obliviousness/NPDO) admit an equivalent characterization in terms of a paired simulation that outputs a refinement pair . Informally, in the standard secret-bit distinguishing game, an adversary provides at each time step a neighboring pair ( x t ( 0 ) , x t ( 1 ) ) (x_{t}^{(0)},x_{t}^{(1)}) , and a secret bit b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} selects which component is the “real” input. A mechanism is DP if the adversary’s views under b = 0 b=0 and b = 1 b=1 are close. The refinement-pair view resolves the usual obstacle in modular composition: even if the real world uses x ( b ) x^{(b)} and produces a real intermediate output y ( b ) y^{(b)} , one also needs a synthetic neighbor y ¯ ( 1 − b ) \\\\overline{y}^{(1-b)} so that the next stage can be analyzed in a paired simulation. The NPDP guarantee precisely asserts the existence of such a paired simulator that, for each b b , produces a pair ( y ( 0 ) , y ( 1 ) ) (y^{(0)},y^{(1)}) whose b b -marginal matches the real execution, and such that the two resulting pair-distributions (for b = 0 b=0 vs. b = 1 b=1 ) are close.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, a general framework for adaptive privacy of continual mechanisms [ HSV26 ] is developed to prove concurrent composition theorems under interactive adversaries. In particular, once a mechanism is expressed in the standard secret-bit game (and the controller only interacts with downstream submechanisms through their privatized outputs), privacy of the overall system follows from standard interactive composition and post-processing. Under the paired-simulation perspective, our two-stage pipeline ℳ → 𝒩 \\\\mathcal{M}\\\\rightarrow\\\\mathcal{N} fits this template: the NPDP stage can be viewed as a mechanism that, on secret bit b b , outputs a paired intermediate stream ( y t ( 0 ) , y t ( 1 ) ) (y^{(0)}_{t},y^{(1)}_{t}) (together with visible outputs v t v_{t} ), and the DP stage 𝒩 \\\\mathcal{N} is then run on y ( b ) y^{(b)} . Thus, one may reduce modular composition to ordinary interactive composition: compose the lifted NPDP mechanism (as DP-on-the-bit with profile f f ) with the second-stage DP mechanism (profile g g ), and finally apply post-processing to remove the extra synthetic transcript.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While this reduction gives a useful conceptual guide and could be formalized by instantiating the conditions of [ HSV26 ] together with the refinement-pair characterization of [ ZZC+24 ] , we include below a self-contained proof tailored to our setting and our power-function accounting. In particular, a more general concurrent setting (multiple submechanisms with interleaving calls) is studied in [ HSV26 ] , whereas we only need the serial modular composition used by our buffering–aggregation pipeline. Providing a direct proof also makes explicit how the privacy profile tensor f ⊗ g f\\\\otimes g arises in our notation, and it cleanly interfaces with the certification arguments in other sections.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose an interactive mechanism with transition function ℳ : ℛ 1 × 𝒳 ∗ → 𝒱 × 𝒴 \\\\mathcal{M}:\\\\mathcal{R}_{1}\\\\times\\\\mathcal{X}^{*}\\\\to\\\\mathcal{V}\\\\times\\\\mathcal{Y} is adaptively f f -NPDP (as in Definition B.9 ). Suppose another interactive mechanism 𝒩 : ℛ 2 × 𝒴 ∗ → 𝒲 × 𝒵 \\\\mathcal{N}:\\\\mathcal{R}_{2}\\\\times\\\\mathcal{Y}^{*}\\\\to\\\\mathcal{W}\\\\times\\\\mathcal{Z} is adaptively g g -DP (as in Definition B.3 ). Then, the modular composition (as in Definition B.7 ) ℳ → 𝒩 : ℛ 1 × ℛ 2 × 𝒳 ∗ → 𝒱 × 𝒲 × 𝒵 \\\\mathcal{M}\\\\rightarrow\\\\mathcal{N}:\\\\mathcal{R}_{1}\\\\times\\\\mathcal{R}_{2}\\\\times\\\\mathcal{X}^{*}\\\\to\\\\mathcal{V}\\\\times\\\\mathcal{W}\\\\times\\\\mathcal{Z} is adaptively f ⊗ g f\\\\otimes g -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We denote the resulting interactive mechanism from the modular composition by ℒ := ℳ → 𝒩 \\\\mathcal{L}:=\\\\mathcal{M}\\\\rightarrow\\\\mathcal{N} . In view of Definition B.3 , we consider the paired simulation ( ℒ 0 𝗉𝖺𝗂𝗋 , ℒ 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{L}^{\\\\mathsf{pair}}_{0},\\\\mathcal{L}^{\\\\mathsf{pair}}_{1}) as described in Definition B.2 . Specifically, our goal is to show that for any adversary 𝒜 \\\\mathcal{A} , the two distributions 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℒ 0 𝗉𝖺𝗂𝗋 ) \\\\mathsf{view}(\\\\mathcal{A}\\\\leftrightarrow\\\\mathcal{L}^{\\\\mathsf{pair}}_{0}) and 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℒ 1 𝗉𝖺𝗂𝗋 ) \\\\mathsf{view}(\\\\mathcal{A}\\\\leftrightarrow\\\\mathcal{L}^{\\\\mathsf{pair}}_{1}) are “close” in the sense quantified by the power function f ⊗ g f\\\\otimes g .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our proof strategy is to construct an augmented pair ( ℒ ^ 0 , ℒ ^ 1 ) (\\\\widehat{\\\\mathcal{L}}_{0},\\\\widehat{\\\\mathcal{L}}_{1}) of interactive mechanisms and an augmented adversary 𝒜 ^ \\\\widehat{\\\\mathcal{A}} whose views have the same distributions as those from ( ℒ 0 𝗉𝖺𝗂𝗋 , ℒ 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{L}^{\\\\mathsf{pair}}_{0},\\\\mathcal{L}^{\\\\mathsf{pair}}_{1}) and 𝒜 \\\\mathcal{A} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next use Fact A.14 to show that instead of considering the pair ( ℒ 0 𝗉𝖺𝗂𝗋 , ℒ 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{L}^{\\\\mathsf{pair}}_{0},\\\\mathcal{L}^{\\\\mathsf{pair}}_{1}) with the same random seed distribution, we can equivalently simulate the pair with some common transition function, but using two different random seed distributions. By the data processing inequality, it suffices to analyze the alternative simulation and the power function of the two random seed distributions. Because ℳ \\\\mathcal{M} is f f -NPDP, let ( ℳ 0 𝗇𝗉𝗉 , ℳ 1 𝗇𝗉𝗉 ) (\\\\mathcal{M}^{\\\\mathsf{npp}}_{0},\\\\mathcal{M}^{\\\\mathsf{npp}}_{1}) be the corresponding NPP simulation as guaranteed by Definition B.9 such that 𝖯𝗈𝗐 ( ℳ 0 𝗇𝗉𝗉 ∥ ℳ 1 𝗇𝗉𝗉 ) ≤ f \\\\mathsf{Pow}(\\\\mathcal{M}^{\\\\mathsf{npp}}_{0}\\\\|\\\\mathcal{M}^{\\\\mathsf{npp}}_{1})\\\\leq f . By Fact A.14 , there exist some interactive mechanism 𝒫 \\\\mathcal{P} and two random seed distributions F 0 F_{0} and F 1 F_{1} such that 𝖯𝗈𝗐 ( F 0 ∥ F 1 ) ≤ f \\\\mathsf{Pow}(F_{0}\\\\|F_{1})\\\\leq f and for each b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , 𝒫 ( F b ) \\\\mathcal{P}(F_{b}) is an exact simulation of ℳ b 𝗇𝗉𝗉 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b} (with the appropriate random seed distribution R ℳ × R ℳ ′ R_{\\\\mathcal{M}}\\\\times R^{\\\\prime}_{\\\\mathcal{M}} ) for any adversary. Specifically, for each b b , ℳ b 𝗇𝗉𝗉 ( R ℳ × R ℳ ′ ) \\\\mathcal{M}^{\\\\mathsf{npp}}_{b}(R_{\\\\mathcal{M}}\\\\times R^{\\\\prime}_{\\\\mathcal{M}}) and 𝒫 ( F b ) \\\\mathcal{P}(F_{b}) are equivalent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similarly, since 𝒩 \\\\mathcal{N} is g g -DP, the paired simulation ( 𝒩 0 𝗉𝖺𝗂𝗋 , 𝒩 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{N}^{\\\\mathsf{pair}}_{0},\\\\mathcal{N}^{\\\\mathsf{pair}}_{1}) satisfies 𝖯𝗈𝗐 ( 𝒩 0 𝗉𝖺𝗂𝗋 ∥ 𝒩 1 𝗉𝖺𝗂𝗋 ) ≤ g \\\\mathsf{Pow}(\\\\mathcal{N}^{\\\\mathsf{pair}}_{0}\\\\|\\\\mathcal{N}^{\\\\mathsf{pair}}_{1})\\\\leq g . Again, by Fact A.14 , there exist some interactive mechanism 𝒬 \\\\mathcal{Q} and two random seed distributions G 0 G_{0} and G 1 G_{1} such that 𝖯𝗈𝗐 ( G 0 ∥ G 1 ) ≤ g \\\\mathsf{Pow}(G_{0}\\\\|G_{1})\\\\leq g and for each b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , 𝒬 ( G b ) \\\\mathcal{Q}(G_{b}) is an exact simulation of 𝒩 b 𝗉𝖺𝗂𝗋 \\\\mathcal{N}^{\\\\mathsf{pair}}_{b} (with the appropriate random seed distribution R 𝒩 R_{\\\\mathcal{N}} ) for any adversary. Specifically, for each b b , 𝒩 b 𝗉𝖺𝗂𝗋 ( R 𝒩 ) \\\\mathcal{N}^{\\\\mathsf{pair}}_{b}(R_{\\\\mathcal{N}}) and 𝒬 ( G b ) \\\\mathcal{Q}(G_{b}) are equivalent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given an adversary 𝒜 \\\\mathcal{A} that interacts with ℒ b 𝗉𝖺𝗂𝗋 \\\\mathcal{L}^{\\\\mathsf{pair}}_{b} for b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , we describe how an augmented adversary 𝒜 ^ \\\\widehat{\\\\mathcal{A}} interacts with ℒ ^ b \\\\widehat{\\\\mathcal{L}}_{b} in the above colored box. Because we follow exactly the same structure as in Definition B.7 , after comparing the two descriptions line-by-line, it follows that 𝗏𝗂𝖾𝗐 ( 𝒜 ↔ ℒ b 𝗉𝖺𝗂𝗋 ) \\\\mathsf{view}(\\\\mathcal{A}\\\\leftrightarrow\\\\mathcal{L}^{\\\\mathsf{pair}}_{b}) has the same distribution as 𝗏𝗂𝖾𝗐 ( 𝒜 ^ ↔ ℒ ^ b ) \\\\mathsf{view}(\\\\widehat{\\\\mathcal{A}}\\\\leftrightarrow\\\\widehat{\\\\mathcal{L}}_{b}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As above, we use Fact A.14 , for each b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , we can equivalently replace ℳ b 𝗇𝗉𝗉 ( R ℳ , R ℳ ′ ) \\\\mathcal{M}^{\\\\mathsf{npp}}_{b}(R_{\\\\mathcal{M}},R^{\\\\prime}_{\\\\mathcal{M}}) with 𝒫 ( F b ) \\\\mathcal{P}(F_{b}) , and 𝒩 b 𝗉𝖺𝗂𝗋 ( R 𝒩 ) \\\\mathcal{N}^{\\\\mathsf{pair}}_{b}(R_{\\\\mathcal{N}}) with 𝒬 ( G b ) \\\\mathcal{Q}(G_{b}) in the above description.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence, the first inequality below follows from the data processing inequality:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the second inequality follows from the data processing inequality and the exact simulations, and the last inequality follows from Fact A.10 . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the literature, neighboring streams are typically defined in the Hamming sense. Specifically, given a collection 𝒪 \\\\mathcal{O} of objects, an (infinite) input stream is x : ℤ ≥ 1 → 𝒪 x:\\\\mathbb{Z}_{\\\\geq 1}\\\\rightarrow\\\\mathcal{O} . In our applications, each object in 𝒪 \\\\mathcal{O} captures the characteristics of a user.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hamming-Style Neighboring Two streams x ∼ H x ′ x\\\\sim_{H}x^{\\\\prime} are neighboring in the Hamming sense if there exists some t ∈ ℤ t\\\\in\\\\mathbb{Z} such that σ ( τ ) = σ ′ ( τ ) \\\\sigma(\\\\tau)=\\\\sigma^{\\\\prime}(\\\\tau) for all τ ≠ t \\\\tau\\\\neq t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Edit-Style Neighboring. We are interested in the neighboring notion when an element may be inserted or deleted from a stream.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a stream x x and t ∈ ℤ t\\\\in\\\\mathbb{Z} , we use 𝖣𝖾𝗅𝖾𝗍𝖾 ( x ; t ) \\\\mathsf{Delete}(x;t) to denote the stream x ′ x^{\\\\prime} such that for all τ < t \\\\tau<t , x ′ [ τ ] = x [ τ ] x^{\\\\prime}[\\\\tau]=x[\\\\tau] and for all τ ≥ t \\\\tau\\\\geq t , x ′ [ τ ] = x [ τ + 1 ] x^{\\\\prime}[\\\\tau]=x[\\\\tau+1] ; in other words, the element at time t t is deleted in stream x x to produce x ′ x^{\\\\prime} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two streams x ∼ 𝖤 x ′ x\\\\sim_{\\\\mathsf{E}}x^{\\\\prime} are neighboring in the edit sense if one stream can be obtained from the other by such a deletion operation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two finite sequences of the same length are neighboring (in the edit sense) if they are prefixes of such neighboring streams.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.1 Motivating Example: Private Prefix-Sum for Edit-Style Neighboring Streams\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The input for the prefix-sum problem is a stream σ : ℤ > 0 → ℤ \\\\sigma:\\\\mathbb{Z}_{>0}\\\\to\\\\mathbb{Z} of integers, and the (accurate) output is a vector c c of prefix-sums, i.e., for t ≥ 1 t\\\\geq 1 , c [ t ] = ∑ i = 1 t σ [ i ] c[t]=\\\\sum_{i=1}^{t}\\\\sigma[i] . The well-known private prefix-sum mechanism [ DNP+10a , CSS11 ] based on the binary tree structure is designed for neighboring streams in the Hamming sense. At the step t 0 t_{0} at which the two streams differ, the difference must be bounded, for instance | σ ( t 0 ) − σ ′ ( t 0 ) | ≤ 1 |\\\\sigma(t_{0})-\\\\sigma^{\\\\prime}(t_{0})|\\\\leq 1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hamming-style neighboring captures the scenario when a user wants some protection on its data contents. However, whether a user has participated in the process may be regarded as sensitive information. This is captured by a neighboring notion in the edit sense. However, deleting one element from a stream can change the element for every step. For instance, if we delete the first element from the stream σ = ( 1 , 0 , 1 , 0 , … ) \\\\sigma=(1,0,1,0,\\\\ldots) , the resulting stream σ ′ = ( 0 , 1 , 0 , 1 , … ) \\\\sigma^{\\\\prime}=(0,1,0,1,\\\\ldots) will have every position flipped! This is as far away from being Hamming-neighboring as one can get, and indeed the DP mechanism based on binary trees is not designed to provide good privacy guarantees for these two streams. Specifically, for all odd time steps t t , the adversary can observe the noisy σ [ t ] + N t \\\\sigma[t]+N_{t} , where N t N_{t} is independently generated noise centered at 0. Therefore, by averaging over these numbers for odd time steps, the adversary can distinguish between σ \\\\sigma and σ ′ \\\\sigma^{\\\\prime} almost for certain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instead of re-designing the DP prefix-sum mechanism to cater for edit-style neighboring streams, our goal is to design a pre-processing mechanism 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} that somehow transforms edit-style neighboring streams into Hamming-style neighboring streams. Then, using the modular composition framework, we pass the processed stream to the tree-based DP prefix-sum mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Random Buffer to the Rescue. Edit-style neighboring notion has been considered in the context of oblivious data structures [ CCM+22 , ZSC+23 ] , and the high-level idea is to use a buffering technique to introduce random delays into the streams in the hope of eventually aligning the elements from the two edit-style neighboring streams. We give a simplified view of the procedure that processes a stream σ \\\\sigma of elements. It maintains a buffer queue that is initially empty. At each step t t , the element σ [ t ] \\\\sigma[t] enters the end of the queue. The procedure then makes two random decisions: (i) whether to remove any element from the head of the queue; (ii) if yes, then remove a random number of elements from the head of the queue and return their sum in the output stream.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider applying this random procedure to two edit-style neighboring streams, where we use alphabets to label bits at different position:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"σ = [ a , b , c , d , e , f ] = [ 1 , 0 , 1 , 0 , 1 , 0 ] \\\\sigma=[a,b,c,d,e,f]=[1,0,1,0,1,0] and σ ′ = [ b , c , d , e , f , g ] = [ 0 , 1 , 0 , 1 , 0 , 1 ] \\\\sigma^{\\\\prime}=[b,c,d,e,f,g]=[0,1,0,1,0,1] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Below are possible outcomes of the random batching process for each stream, where we sum up the bits in each non-empty batch.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that at the end of step 6, the queue contains { f } \\\\{f\\\\} for the case of σ \\\\sigma and contains { f , g } \\\\{f,g\\\\} for the case of σ ′ \\\\sigma^{\\\\prime} . In this particular lucky scenario, we see that the 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} procedure has converted edit-style neighboring streams to Hamming-style neighboring streams. However, observe that these two outcomes from the two scenarios do not occur with exactly the same probability, but similar probabilities if the 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} procedure is designed carefully. Moreover, for easy illustration, we give an oversimplified description of Hamming-style neighboring streams. We next give the formal and technical details.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.2 Technical Description of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the following Hamming-style neighboring notion on bin sequences, which are suitable for private mechanisms that are designed expecting Hamming-style neighboring streams.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use ℬ \\\\mathcal{B} to denote the collection of concrete bins, where a concrete bin contains an array of objects with some specified maximum array length. In our usage, a concrete bin is non-empty and is typically opaque in the sense that the adversary cannot observe its contents.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A bin sequence of length T T is denoted by B : [ 1 . . T ] → ℬ ∪ { ⊥ } B:[1..T]\\\\rightarrow\\\\mathcal{B}\\\\cup\\\\{\\\\bot\\\\} , where ⊥ \\\\bot means that no concrete bin is returned in that time step. Adjacent concrete bins with indices i < j i<j in a sequence B B mean that for i < t < j i<t<j , B [ t ] = ⊥ B[t]=\\\\bot .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two bin sequences B ∼ 𝖡 B ′ B\\\\sim_{\\\\mathsf{B}}B^{\\\\prime} of the same length T T are neighboring if the following holds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For all t ∈ [ 1 . . T ] t\\\\in[1..T] , B [ t ] = ⊥ B[t]=\\\\bot iff B ′ [ t ] = ⊥ B^{\\\\prime}[t]=\\\\bot .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Except for at most 2 indices t t in [ 1 . . T ] [1..T] , B [ t ] = B ′ [ t ] B[t]=B^{\\\\prime}[t] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If B B and B ′ B^{\\\\prime} differ in exactly one index, the corresponding index refers to the last concrete bin in both sequences. Moreover, the two bin contents must have the following form.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There exist elements w w and u u and (possibly empty) element arrays σ 1 \\\\sigma_{1} and σ 2 \\\\sigma_{2} such that the two differing bins from B B and B ′ B^{\\\\prime} contain the arrays:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , w , σ 2 ] [\\\\sigma_{1},w,\\\\sigma_{2}] and [ σ 1 , σ 2 , u ] [\\\\sigma_{1},\\\\sigma_{2},u] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If B B and B ′ B^{\\\\prime} differ in exactly two indices i < j i<j , then the two indices correspond to adjacent concrete bins in both sequences. Moreover, there exist elements w w and u u and element arrays σ 1 \\\\sigma_{1} , σ 2 \\\\sigma_{2} and σ 3 \\\\sigma_{3} such that two adjacent concrete bins from each bin sequence are illustrated as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , w , σ 2 ] , ⊥ , … , ⊥ , [ u , σ 3 ] [\\\\sigma_{1},w,\\\\sigma_{2}],\\\\bot,\\\\ldots,\\\\bot,[u,\\\\sigma_{3}]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , σ 2 , u ] , ⊥ , … , ⊥ , [ σ 3 ] [\\\\sigma_{1},\\\\sigma_{2},u],\\\\bot,\\\\ldots,\\\\bot,[\\\\sigma_{3}]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that if each of two neighboring bin sequences are concatenated, two resulting item sequences will have lengths differ by at most 1, in which case deleting one element from the longer item sequence results in the shorter item sequence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We need the following primitives to describe the 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let α > 1 \\\\alpha>1 . The symmetric geometric distribution Geom ( α ) \\\\text{Geom}(\\\\alpha) takes integer values such that the probability mass function at k k is α − 1 α + 1 ⋅ α − | k | . \\\\frac{\\\\alpha-1}{\\\\alpha+1}\\\\cdot\\\\alpha^{-|k|}.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is known that using Geom ( e ε ) \\\\text{Geom}(e^{\\\\varepsilon}) as additive noise can mask two integers differing by 1 with ( ε , 0 ) (\\\\varepsilon,0) -DP. However, if we want the noise to have bounded support, then we truncate the noise and get only ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ε > 0 \\\\varepsilon>0 , δ ∈ ( 0 , 1 ) \\\\delta\\\\in(0,1) . Let U U be the smallest even positive integer such that Pr | Geom ( e ε ) | ≥ U 2 ≤ δ \\\\Pr{|\\\\text{Geom}(e^{\\\\varepsilon})|\\\\geq\\\\frac{U}{2}}\\\\leq\\\\delta , where U = O ( 1 ε log 1 δ ) . U=O(\\\\frac{1}{\\\\varepsilon}\\\\log\\\\frac{1}{\\\\delta}). The truncated geometric distribution G ( ε , δ ) G(\\\\varepsilon,\\\\delta) has support [ − U 2 , U 2 ] [-\\\\frac{U}{2},\\\\frac{U}{2}] , and is obtained by first sampling r r from Geom ( e ε ) \\\\text{Geom}(e^{\\\\varepsilon}) and then truncating r r within [ − U 2 , U 2 ] [-\\\\frac{U}{2},\\\\frac{U}{2}] , i.e., return min ( max ( − U 2 , r ) , U 2 ) \\\\min(\\\\max(-\\\\frac{U}{2},r),\\\\frac{U}{2}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Differentially Private Streaming Prefix Sum with Always Error Bounds. We use the ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP interactive mechanism from Theorem F.1 , denoted as 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\\\mathsf{StreamSum}_{\\\\varepsilon,\\\\delta} . Observe that 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\\\mathsf{StreamSum}_{\\\\varepsilon,\\\\delta} is not necessarily private against adaptive adversaries. Even though we will prove that 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} is adaptively NPDP, we do not need the adaptive property from 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\\\mathsf{StreamSum}_{\\\\varepsilon,\\\\delta} , but rather that the additive error from all time steps t t be bounded by some function E ( t ) E(t) with probability 1.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mechanism Intuition. In Algorithm 3 , we reinterpret 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} (which was introduced in [ CCM+22 , ZSC+23 ] ) as an interactive mechanism. On a high level, the mechanism maintains an internal buffer that is a subarray. In each time step t t , the mechanism takes an element x [ t ] x[t] from the input stream and appends it to the end of the buffer. The mechanism decides according to some rules if a new bin should be created from a prefix of the buffer contents. If yes, a prefix from the buffer is removed to form a new bin that is returned in this step; otherwise, nothing is returned in this step and we denote the output by ⊥ \\\\bot .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the end of every step t t , with probability 1, the concatenation of the concrete bins that have been output so far is a prefix of the input stream with length at least t − O ( 1 ε ⋅ log t ⋅ ( log t + log 1 δ ) ) t-O(\\\\frac{1}{\\\\varepsilon}\\\\cdot\\\\log t\\\\cdot(\\\\log t+\\\\log\\\\frac{1}{\\\\delta})) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that because the additive error of 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\\\mathsf{StreamSum}_{\\\\varepsilon,\\\\delta} holds with probability 1, the 𝖡𝗎𝖿 \\\\mathsf{Buf} will never run into underflow in line 15 . Actually, to ensure no underflow, it suffices to have t < S [ 𝖨𝗇𝖽𝖾𝗑 ] + E [ 𝖨𝗇𝖽𝖾𝗑 ] t<S[\\\\mathsf{Index}]+E[\\\\mathsf{Index}] in line 12 ; we shall see that the extra term 2 U 2U is crucial to argue that 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} is private against adaptive adversaries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that at the end of time step t t , we have the invariant that t < S [ 𝖨𝗇𝖽𝖾𝗑 ] + E [ 𝖨𝗇𝖽𝖾𝗑 ] + 2 U t<S[\\\\mathsf{Index}]+E[\\\\mathsf{Index}]+2U ; this holds because S [ 𝖨𝗇𝖽𝖾𝗑 + 1 ] − S [ 𝖨𝗇𝖽𝖾𝗑 ] ≥ U ≥ 1 S[\\\\mathsf{Index}+1]-S[\\\\mathsf{Index}]\\\\geq U\\\\geq 1 and E [ 𝖨𝗇𝖽𝖾𝗑 + 1 ] ≥ E [ 𝖨𝗇𝖽𝖾𝗑 ] E[\\\\mathsf{Index}+1]\\\\geq E[\\\\mathsf{Index}] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, at this moment, the number of elements in the queue 𝖡𝗎𝖿 \\\\mathsf{Buf} is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"using the crude bound 𝖨𝗇𝖽𝖾𝗑 ≤ t \\\\mathsf{Index}\\\\leq t . ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.3 Adaptive Privacy Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Privacy of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} . Observe that in each step, the view of the adversary just consists of one bit of information: whether a concrete bin or ⊥ \\\\bot is returned. Moreover, the distribution of the view is always the same, regardless of what the input stream is. However, as noted in [ ZSC+23 ] , when a mechanism has hidden outputs that will be passed to another mechanism, one needs to consider the joint distribution of the view and the output, which is why NPDO was defined, which we have extended to interactive mechanisms in Definition B.9 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Construction of Neighbor-Preserving Paired (NPP) Simulation for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} . The proofs in [ ZSC+23 ] already contain an implicit description of how an NPP simulation for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} is constructed, but we will make it explicit here. Recall that the adversary will generate two input sequences ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) in an NPP simulation in Definition B.8 . For b ∈ { 0 , 1 } b\\\\in\\\\{0,1\\\\} , the mechanism ℳ b 𝗇𝗉𝗉 \\\\mathcal{M}^{\\\\mathsf{npp}}_{b} will do two things in each step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Simulate the behavior of the original mechanism ℳ \\\\mathcal{M} on the sequence x ( b ) [ 1 . . t ] x^{(b)}[1..t] to produce the corresponding view v [ t ] v[t] and output y ( b ) [ t ] y^{(b)}[t] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Generate another output y ( b ¯ ) [ t ] y^{(\\\\overline{b})}[t] such that the output sequences y ( b ) [ 1 . . t ] y^{(b)}[1..t] and y ( b ¯ ) [ 1 . . t ] y^{(\\\\overline{b})}[1..t] are neighboring.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The counter-intuitive point is that y ( b ¯ ) [ 1 . . t ] y^{(\\\\overline{b})}[1..t] is not necessarily the simulation of the original ℳ \\\\mathcal{M} on the other input sequence x ( b ¯ ) x^{(\\\\overline{b})} . In fact, it is possible that y ( b ¯ ) [ 1 . . t ] y^{(\\\\overline{b})}[1..t] may not even be in the support of ℳ \\\\mathcal{M} on x ( b ¯ ) x^{(\\\\overline{b})} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In general, extra randomness other than that used in the original ℳ \\\\mathcal{M} may be needed to generate y ( b ¯ ) [ t ] y^{(\\\\overline{b})}[t] . However, for 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} , no extra randomness is needed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Technical Assumption on Input Pair. Note that in Definition B.8 , the adversary needs to ensure that at every time step t t , the pair ( x ( 0 ) [ 1 . . t ] , x ( 1 ) [ 1 . . t ] ) (x^{(0)}[1..t],x^{(1)}[1..t]) of input sequences generated so far are neighboring (in the edit sense).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For instance, for t = 4 t=4 , the pair ( [ 0 , 1 , 0 , 1 ] , [ 1 , 0 , 1 , 0 ] ) ([0,1,0,1],[1,0,1,0]) of sequences are neighboring in the edit sense, but the issue is that we do not know which stream has an element deleted. If next input pair at t = 5 t=5 is ( 0 , a ) (0,a) , then we know that in the two streams ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) , the first element from x ( 0 ) x^{(0)} is deleted to produce x ( 1 ) x^{(1)} ; on the other hand, if the next input pair at t = 5 t=5 is ( a , 1 ) (a,1) , then the opposite is true.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, in the construction of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 b 𝗇𝗉𝗉 \\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{b} , we would need to know whether an element is deleted from x ( 0 ) x^{(0)} or x ( 1 ) x^{(1)} . Observe that this issue will not arise if a stream contains only distinct elements. In practice, this is a valid assumption because an item is typically tagged with unique metadata.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence, in the proof, we assume that if the adversary first generates x ( 0 ) [ t ] ≠ x ( 1 ) [ t ] x^{(0)}[t]\\\\neq x^{(1)}[t] at step t t , then when we receive a pair at the beginning of step t + 1 t+1 , we will know from which stream an element is deleted. This will not be too late, because the additive error for the prefix-sum S [ 𝖨𝗇𝖽𝖾𝗑 ] S[\\\\mathsf{Index}] is at most E [ 𝖨𝗇𝖽𝖾𝗑 ] E[\\\\mathsf{Index}] , we can ensure that the 𝖡𝗎𝖿 \\\\mathsf{Buf} queue will never be empty, i.e., an element arriving at step t t will never be returned in a concrete bin in the same step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We focus on the description of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 \\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0} (because the behavior of 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 \\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1} is symmetric); recall that the adversary generates a pair of time sequences ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first component simulates 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} from Algorithm 3 on the sequence x ( 0 ) x^{(0)} ; in every step t t , it will output y ( 0 ) [ t ] y^{(0)}[t] that is either a concrete bin or ⊥ \\\\bot .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We describe the second component more carefully, which is how y ( 1 ) [ t ] y^{(1)}[t] is constructed in each step t t . Note that since the two output sequences must be neighboring as in Definition D.2 , we will make sure y ( 0 ) [ t ] = ⊥ y^{(0)}[t]=\\\\bot iff y ( 1 ) [ t ] = ⊥ y^{(1)}[t]=\\\\bot . Moreover, we need to ensure that at most two adjacent concrete bins are different in a specific way. We consider two cases.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose an element x ( 0 ) [ t ] = w x^{(0)}[t]=w is deleted from x ( 0 ) x^{(0)} to produce the stream x ( 1 ) x^{(1)} . In the output sequence y ( 0 ) y^{(0)} , we focus on two adjacent concrete bins that contains the deleted item w w and the next concrete bin:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , w , σ 2 ] , ⊥ , … , ⊥ , [ u , σ 3 ] [\\\\sigma_{1},w,\\\\sigma_{2}],\\\\bot,\\\\ldots,\\\\bot,[u,\\\\sigma_{3}] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where σ 1 \\\\sigma_{1} , σ 2 \\\\sigma_{2} and σ 3 \\\\sigma_{3} are (potentially empty) subarrays.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, in the output sequence y ( 1 ) y^{(1)} , those two concrete bins are modified to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , σ 2 , u ] , ⊥ , … , ⊥ , [ σ 3 ] [\\\\sigma_{1},\\\\sigma_{2},u],\\\\bot,\\\\ldots,\\\\bot,[\\\\sigma_{3}] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that the bin y ( 0 ) [ τ ] = [ σ 1 , w , σ 2 ] y^{(0)}[\\\\tau]=[\\\\sigma_{1},w,\\\\sigma_{2}] is returned in the output sequence y ( 0 ) y^{(0)} in some step τ > t \\\\tau>t ; observe that at the end of step τ \\\\tau , in the remaining 𝖡𝗎𝖿 \\\\mathsf{Buf} queue, u u should be the first element. Moreover, u u should have already appeared in the input pair together with the last element of σ \\\\sigma . Hence, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 \\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0} is able to produce y ( 1 ) [ τ ] = [ σ 1 , σ 2 , u ] y^{(1)}[\\\\tau]=[\\\\sigma_{1},\\\\sigma_{2},u] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, note that if [ u , σ 3 ] [u,\\\\sigma_{3}] contains exactly U U elements, then [ σ 3 ] [\\\\sigma_{3}] will not be in the support of simulating the actual 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose an element x ( 1 ) [ t ] = w x^{(1)}[t]=w is deleted from x ( 1 ) x^{(1)} to produce the stream x ( 0 ) x^{(0)} . Again, we focus on two adjacent concrete bins from the output sequence y ( 0 ) y^{(0)} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , σ 2 , u ] , ⊥ , … , ⊥ , [ σ 3 ] [\\\\sigma_{1},\\\\sigma_{2},u],\\\\bot,\\\\ldots,\\\\bot,[\\\\sigma_{3}] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where [ σ 1 , σ 2 , u ] [\\\\sigma_{1},\\\\sigma_{2},u] is the bin that contains the element x ( 0 ) [ t ] x^{(0)}[t] , which is also the first element in the subarray [ σ 2 , u ] [\\\\sigma_{2},u] . Note that in the case that σ 2 \\\\sigma_{2} is empty, then u = x ( 0 ) [ t ] u=x^{(0)}[t] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, in the output sequence y ( 1 ) y^{(1)} , those two concrete bins are modified to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"[ σ 1 , w , σ 2 ] , ⊥ , … , ⊥ , [ u , σ 3 ] [\\\\sigma_{1},w,\\\\sigma_{2}],\\\\bot,\\\\ldots,\\\\bot,[u,\\\\sigma_{3}]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that if [ σ 3 ] [\\\\sigma_{3}] contains 2 U 2U elements, then [ u , σ 3 ] [u,\\\\sigma_{3}] will not appear in the support of simulating the actual 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From construction, the two output bin sequences y ( 0 ) y^{(0)} and y ( 1 ) y^{(1)} are neighboring as in Definition D.2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With the formal terminology in place, we can formally restate the result in [ ZSC+23 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interactive mechanism in Algorithm 3 is non-adaptively ( ε , δ ) (\\\\varepsilon,\\\\delta) -NPDP. Specifically, in terms of power function, the NPP simulation ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0},\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1}) in Definition D.6 satisfies: 𝖯𝗈𝗐 NA ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 ∥ 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) ≤ 𝖣𝖯 ε , δ \\\\mathsf{Pow}^{{\\\\textsc{NA}}}(\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0}\\\\|\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1})\\\\leq\\\\mathsf{DP}_{\\\\varepsilon,\\\\delta} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose for some power function f f , the interactive mechanism 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} in Algorithm 3 is non-adaptively f f -NPDP (with respect to input neighboring notion in Definition D.1 and output neighboring notion in Definition D.2 ). Then, 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} is also adaptively f f -NPDP; specifically, the NPP simulation ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0},\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1}) in Definition D.6 satisfies:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝖯𝗈𝗐 ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 ∥ 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) ≤ f \\\\mathsf{Pow}(\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0}\\\\|\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1})\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By joint convexity in Fact A.9 , we assume a deterministic adversary 𝒜 {\\\\mathcal{A}} that interacts with ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0},\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1}) . Moreover, without loss of generality, to simplify the notation, we assume that there is a canonical sequence x ( t ) := t x(t):=t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, for some initial steps t t , the default action of the adversary is to generate a pair ( x t ( 0 ) , x t ( 1 ) ) (x^{(0)}_{t},x^{(1)}_{t}) satisfying x t ( 0 ) = x t ( 1 ) = t x^{(0)}_{t}=x^{(1)}_{t}=t until some critical step t c t_{c} in which the adversary decides (based on its observation) to deviate from the default behavior. It can either:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(i) delete from x ( 0 ) x^{(0)} , i.e., for t ≥ t c t\\\\geq t_{c} , x t ( 0 ) = t + 1 x^{(0)}_{t}=t+1 and x t ( 1 ) = t x^{(1)}_{t}=t , or\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(ii) delete from x ( 1 ) x^{(1)} , i.e., for t ≥ t c t\\\\geq t_{c} , x t ( 0 ) = t x^{(0)}_{t}=t and x t ( 1 ) = t + 1 x^{(1)}_{t}=t+1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that before step t c t_{c} , the two input sequences x ( 0 ) x^{(0)} and x ( 1 ) x^{(1)} are identical. Hence, the single decision made by the adversary is at which time step t c t_{c} it wants to deviate from the default behavior, and it does not really matter whether it deletes from x ( 0 ) x^{(0)} or x ( 1 ) x^{(1)} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Augmented View. We assume a more powerful adversary that can observe, in addition to its normal view (which consists of the contents of both bins y ( 0 ) [ t ] y^{(0)}[t] and y ( 1 ) [ t ] ) y^{(1)}[t]) ), the noisy prefix-sums S [ ⋅ ] S[\\\\cdot] once they are created. We note that the proof in [ ZSC+23 ] also allows the (non-adaptive) adversary to have this augmented view. However, it is important when the adversary learns about these noisy prefix-sums.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Default Distribution. Suppose the canonical sequence x x is used in both of ( x ( 0 ) , x ( 1 ) ) (x^{(0)},x^{(1)}) . Then, the behaviors of both interactive mechanisms ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0},\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1}) are identical. We use Ω \\\\Omega to denote the space of views, and 𝒫 \\\\mathcal{P} to denote the corresponding distribution of views. Each ω ∈ Ω \\\\omega\\\\in\\\\Omega consists of the view and output of every time step t ≥ 1 t\\\\geq 1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Decision of Adversary 𝒜 {\\\\mathcal{A}} . Because we have assumed that 𝒜 {\\\\mathcal{A}} is deterministic, for each ω ∈ Ω \\\\omega\\\\in\\\\Omega , either (i) 𝒜 {\\\\mathcal{A}} will never deviate from the default behavior, or (ii) 𝒜 {\\\\mathcal{A}} will deviate at the beginning of some step t t , in which case the resulting view will be different from ω \\\\omega starting from step t t onwards. We use Ω t ⊆ Ω \\\\Omega_{t}\\\\subseteq\\\\Omega to denote the collection of view in which the adversary deviates at the beginning of step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider some ω ∈ Ω t \\\\omega\\\\in\\\\Omega_{t} . As explained above, this means that the adversary decides to deviate at the beginning of step t t , after observing the information in ω \\\\omega up to step t − 1 t-1 . This means it will generate a pair ( x ( 0 ) [ t ] , x ( 1 ) [ t ] ) (x^{(0)}[t],x^{(1)}[t]) of different inputs, and as a result the distributions of outputs for ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0},\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1}) from step t t onwards may be different. For such ω ∈ Ω t \\\\omega\\\\in\\\\Omega_{t} , we use g ω := 𝖯𝗈𝗐 ( V ω ( 0 ) ∥ V ω ( 1 ) ) g_{\\\\omega}:=\\\\mathsf{Pow}(V^{(0)}_{\\\\omega}\\\\|V^{(1)}_{\\\\omega}) to denote the power function between the corresponding two view distributions V ω ( 0 ) V^{(0)}_{\\\\omega} and V ω ( 1 ) V^{(1)}_{\\\\omega} starting from step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Key Insight. We next show that for ω ∈ Ω t \\\\omega\\\\in\\\\Omega_{t} , the decision of 𝒜 {\\\\mathcal{A}} to deviate the beginning of step t t is equivalent to a randomized non-adaptive adversary that interacts with ( 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 0 𝗇𝗉𝗉 , 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 1 𝗇𝗉𝗉 ) (\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{0},\\\\mathsf{RandBin}^{\\\\mathsf{npp}}_{1}) . Suppose i i is the index of concrete bins such that S [ i − 1 ] + E [ i − 1 ] + 2 U < t ≤ S [ i ] + E [ i ] + 2 U S[i-1]+E[i-1]+2U<t\\\\leq S[i]+E[i]+2U . First, analyze what information the adversary has gathered till the beginning of step t t :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the end of step S [ i − 1 ] + E [ i − 1 ] + 2 U S[i-1]+E[i-1]+2U , the adversary learns the exact count C [ i − 1 ] C[i-1] (and also C [ i − 2 ] , … , C [ 1 ] C[i-2],\\\\ldots,C[1] ), because so far the two output sequences are identical. Moreover, it also learns the noisy prefix-sum S [ i ] S[i] that is an estimate of ∑ j = 1 i C [ j ] \\\\sum_{j=1}^{i}C[j] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the beginning of step t t , it knows the exact number of elements in the 𝖡𝗎𝖿 \\\\mathsf{Buf} queue, which is larger than 2 U 2U . Hence, the input pair of elements generated at time t t will definitely not go into the concrete bin with index i i . This is why we need the term 2 U 2U in line 12 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The crucial observation is that at this moment, the number C [ i + 1 ] C[i+1] of elements in the ( i + 1 ) (i+1) -st concrete bin has not been sampled yet.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence, if the adversary 𝒜 {\\\\mathcal{A}} decides to deviate at the beginning of step t t , it will know that the element at step t t will be the N ω := ( t − ∑ j = 1 i C [ j ] ) N_{\\\\omega}:=(t-\\\\sum_{j=1}^{i}C[j]) -th element in the 𝖡𝗎𝖿 \\\\mathsf{Buf} queue.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The adversary does not know C [ i ] C[i] , but based on the knowledge of S [ i ] , S [ i − 1 ] , … , S [ 1 ] S[i],S[i-1],\\\\ldots,S[1] and C [ i − 1 ] , … , C [ 1 ] C[i-1],\\\\ldots,C[1] , it knows the conditional distribution of C [ i ] C[i] , and hence, can derive the distribution of N ω N_{\\\\omega} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conclusion. For ω ∈ Ω t \\\\omega\\\\in\\\\Omega_{t} , the adversary 𝒜 {\\\\mathcal{A}} deciding to deviate at the beginning of step t t will be equivalent to another instance where a randomized non-adaptive adversary decides upfront to deviate at some step according to the distribution N ω N_{\\\\omega} . From the hypothesis, we conclude that the power function satisfies g ω = 𝖯𝗈𝗐 ( V ω ( 0 ) ∥ V ω ( 1 ) ) ≤ f g_{\\\\omega}=\\\\mathsf{Pow}(V^{(0)}_{\\\\omega}\\\\|V^{(1)}_{\\\\omega})\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, the whole proof concludes because:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the inequality follows from the composition rule of power functions in Fact A.10 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fact D.7 and Lemma D.8 immediately give the following corollary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interactive mechanism 𝖱𝖺𝗇𝖽𝖡𝗂𝗇 \\\\mathsf{RandBin} in Algorithm 3 is adaptively ( ε , δ ) (\\\\varepsilon,\\\\delta) -NPDP with input neighboring notion ∼ 𝖤 \\\\sim_{\\\\mathsf{E}} and output neighboring notion ∼ 𝖡 \\\\sim_{\\\\mathsf{B}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section proves the adaptive-safety certification used in the modular pipeline. The goal is to identify structural conditions under which an ordinary non-adaptive privacy analysis of a continual mechanism already implies privacy against adaptive adversaries. The result applies to Hamming-style neighboring input streams and is stated in the power-function language used throughout the appendix.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first condition is a fresh-randomness, prefix-causality condition. At round t t , the mechanism may depend on the entire input prefix x [ 1 . . t ] x[1..t] , but the only randomness used for the round- t t output is a fresh seed ω t \\\\omega_{t} independent of all other rounds. This rules out hidden cross-round correlations introduced by reused noise, while still allowing the release at time t t to be an arbitrary deterministic function of the input prefix.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An (abstract) interactive mechanism with transition function ℳ : ℛ × 𝒰 ∗ → 𝒰 \\\\mathcal{M}:\\\\mathcal{R}\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} is independently decomposable if the following properties hold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each random seed r ∈ ℛ r\\\\in\\\\mathcal{R} takes the form r = ( ω 1 , ω 2 , … ) ∈ Ω ℕ r=(\\\\omega_{1},\\\\omega_{2},\\\\ldots)\\\\in\\\\Omega^{\\\\mathbb{N}} . Moreover, the mechanism samples ω t \\\\omega_{t} independently from some distribution R t R_{t} in time step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each t ≥ 1 t\\\\geq 1 , there exists a deterministic function ℳ t \\\\mathcal{M}_{t} such that the output y t ← ℳ t ( ω t ; x [ 1 . . t ] ) ∈ 𝒰 y_{t}\\\\leftarrow\\\\mathcal{M}_{t}(\\\\omega_{t};x[1..t])\\\\in\\\\mathcal{U} depends only on x [ 1 . . t ] x[1..t] and the randomness ω t \\\\omega_{t} sampled in step t t .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Quantitatively Decomposable Mechanisms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Independent decomposability alone records where the randomness enters, but does not quantify how much a change in the input prefix can affect a one-round release. We therefore add a quantitative condition. The condition says that, for each round t t , the power function between the two possible round- t t output laws is determined exactly by the distance between two aggregates of selected input coordinates. The invariant-pair condition below ensures that common aggregation context can be ignored: only the part of the aggregate containing the Hamming discrepancy contributes to the distance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let d : 𝒰 × 𝒰 → ℝ ≥ 0 d:\\\\mathcal{U}\\\\times\\\\mathcal{U}\\\\rightarrow\\\\mathbb{R}_{\\\\geq 0} be a metric, and let ( 𝒰 , ⊕ , 0 ⊕ ) (\\\\mathcal{U},\\\\oplus,0_{\\\\oplus}) be a monoid. We say that ( d , ⊕ ) (d,\\\\oplus) is an invariant pair (under common aggregation context) if for all a , b , u , u ′ ∈ 𝒰 a,b,u,u^{\\\\prime}\\\\in\\\\mathcal{U} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notation. For an index set I = { τ 1 < τ 2 < ⋯ < τ k } ⊆ [ 1 . . t ] I=\\\\{\\\\tau_{1}<\\\\tau_{2}<\\\\cdots<\\\\tau_{k}\\\\}\\\\subseteq[1..t] and an input history x [ 1 . . t ] ∈ 𝒰 t x[1..t]\\\\in\\\\mathcal{U}^{t} , define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For I = ∅ I=\\\\emptyset , we use the convention ⨁ τ ∈ I x [ τ ] := 0 ⊕ \\\\bigoplus_{\\\\tau\\\\in I}x[\\\\tau]:=0_{\\\\oplus} . Since ⊕ \\\\oplus is associative and the elements are aggregated in increasing index order, this expression is well-defined without requiring commutativity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An interactive mechanism ℳ : ℛ × 𝒰 ∗ → 𝒰 \\\\mathcal{M}:\\\\mathcal{R}\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} is quantitatively decomposable with respect to an invariant pair ( d , ⊕ ) (d,\\\\oplus) in the sense of Definition E.2 if it is independently decomposable in the sense of Definition E.1 , and there exist a sequence of index sets { I t } t ≥ 1 \\\\{I_{t}\\\\}_{t\\\\geq 1} with I t ⊆ [ 1 . . t ] I_{t}\\\\subseteq[1..t] and, for every t ≥ 1 t\\\\geq 1 and θ ≥ 0 \\\\theta\\\\geq 0 , a power function g t ( θ ) : [ 0 , 1 ] → [ 0 , 1 ] g_{t}(\\\\theta):[0,1]\\\\rightarrow[0,1] such that the following hold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For every t ≥ 1 t\\\\geq 1 , g t ( θ ) ≤ g t ( θ ′ ) g_{t}(\\\\theta)\\\\leq g_{t}(\\\\theta^{\\\\prime}) whenever 0 ≤ θ ≤ θ ′ 0\\\\leq\\\\theta\\\\leq\\\\theta^{\\\\prime} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For every t ≥ 1 t\\\\geq 1 and every pair of input histories x [ 1 . . t ] , x ′ [ 1 . . t ] ∈ 𝒰 t x[1..t],x^{\\\\prime}[1..t]\\\\in\\\\mathcal{U}^{t} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, ℳ t ( R t ; x [ 1 . . t ] ) \\\\mathcal{M}_{t}(R_{t};x[1..t]) denotes the distribution of ℳ t ( ω t ; x [ 1 . . t ] ) \\\\mathcal{M}_{t}(\\\\omega_{t};x[1..t]) when ω t \\\\omega_{t} is sampled from R t R_{t} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can now state the certification theorem. The only-if direction is immediate, since non-adaptive adversaries are a special case of adaptive adversaries. The content is the converse: for quantitatively decomposable mechanisms, adaptivity does not increase the worst-case power-function profile. Thus any finite-horizon non-adaptive f f -DP proof automatically lifts to the paired-simulation adaptive game.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider the Hamming-style neighboring relation on equal-length input sequences in 𝒰 \\\\mathcal{U} , where two sequences are neighboring if they differ in at most one position. Let f f be a power function. Suppose ℳ : ℛ × 𝒰 ∗ → 𝒰 \\\\mathcal{M}:\\\\mathcal{R}\\\\times\\\\mathcal{U}^{*}\\\\rightarrow\\\\mathcal{U} is quantitatively decomposable in the sense of Definition E.3 . If ℳ \\\\mathcal{M} is non-adaptively f f -DP in the paired-simulation sense of Definition B.3 , for every finite horizon, then ℳ \\\\mathcal{M} is adaptively f f -DP in the same sense, for every finite horizon.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We prove the following finite-horizon statement by induction on T T : for every power function f f , if ℳ \\\\mathcal{M} is non-adaptively f f -DP up to horizon T T , then ℳ \\\\mathcal{M} is adaptively f f -DP up to horizon T T . This implies the theorem because the theorem assumes the non-adaptive guarantee for every finite horizon.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first reduce to deterministic adversaries. A randomized adversary can be viewed as first sampling its random seed and then running a deterministic adversary. For every fixed seed, the deterministic-adversary case gives the desired power-function bound. Taking the mixture over the same seed distribution under the two secret bits preserves the bound by the joint convexity property in Fact A.9 . Hence it suffices to consider deterministic adaptive adversaries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also assume, without loss of generality, that the adversary observes the entire output of ℳ \\\\mathcal{M} in each round. If the real adversary observes only a view of the output, the real view is obtained from the full-output transcript by post-processing, and the desired bound follows from data processing. If the mechanism has an initial output before the adversary sends any input, this initial output has the same distribution under the two secret bits; it may be included as an additional common first observation and does not affect the privacy bound. Thus we count rounds from the first adversarial input.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒜 {\\\\mathcal{A}} be a deterministic adaptive adversary in the paired simulation ( ℳ 0 𝗉𝖺𝗂𝗋 , ℳ 1 𝗉𝖺𝗂𝗋 ) (\\\\mathcal{M}^{\\\\mathsf{pair}}_{0},\\\\mathcal{M}^{\\\\mathsf{pair}}_{1}) . We pad terminated executions with a fixed halt symbol, so all transcripts have length T T . This padding is post-processing and does not change the privacy claim. We must prove\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base case T = 1 T=1 is immediate. With only one input round, the adversary has no previous mechanism output on which to adapt, so the adaptive and non-adaptive paired simulations coincide.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the statement holds for horizon T − 1 T-1 , and consider horizon T > 1 T>1 . Let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"be the first pair of inputs produced by 𝒜 {\\\\mathcal{A}} . Since 𝒜 {\\\\mathcal{A}} is deterministic, this pair is fixed. Let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"be the first-round output distribution under secret bit b b .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We split into two cases.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Case 1: the first round is the challenge round. Assume x ( 0 ) [ 1 ] ≠ x ( 1 ) [ 1 ] x^{(0)}[1]\\\\neq x^{(1)}[1] , and write\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because the paired simulation only allows Hamming-neighboring prefixes, once the first inputs differ, every later input pair produced by the adversary must have equal components. That is, on every non-terminated transcript prefix and for every t ≥ 2 t\\\\geq 2 , the adversary must output a pair of the form ( x t , x t ) (x_{t},x_{t}) . We use the finite-horizon convention that terminated executions are padded with a fixed halt symbol; the padded rounds have identical outputs under the two secret bits and hence contribute the identity power function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each round t t , define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first show that, at every round t t and every transcript prefix that can arise before round t t , the conditional next-output distributions under the two secret bits have power function exactly h t h_{t} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix such a transcript prefix. Since the adversary is deterministic, conditioning on this common prefix fixes all input pairs generated so far. As argued above, the two input histories entering round t t differ only in the first coordinate. If 1 ∉ I t 1\\\\notin I_{t} , then the two aggregates\"\n },\n {\n \"type\": \"para\",\n \"text\": \"are identical. Hence their distance is zero. Applying Definition E.3 to identical histories gives g t ( 0 ) = 𝖨𝖽 g_{t}(0)=\\\\mathsf{Id} , and therefore the one-round power function is exactly 𝖨𝖽 \\\\mathsf{Id} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If 1 ∈ I t 1\\\\in I_{t} , then the two aggregates have the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for the same common suffix aggregate c c over the remaining indices in I t I_{t} . By the invariant-pair property,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence Definition E.3 gives the exact one-round identity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus the one-round power function is exactly h t h_{t} , and h t h_{t} depends only on t t , not on the transcript prefix.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now justify that these exact one-round identities tensorize to an exact identity for the whole adaptive transcript in this case. Let P b P_{b} be the distribution of the full length- T T transcript under secret bit b b . For a transcript prefix s = y [ 1 . . t − 1 ] s=y[1..t-1] , let K t , s b K^{b}_{t,s} be the conditional distribution of y t y_{t} given the prefix s s under bit b b . By independent decomposability, K t , s b K^{b}_{t,s} is generated using the fresh seed R t R_{t} , which is independent of the previous seeds. From the preceding paragraph,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for every prefix s s that can arise.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the following finite-experiment characterization of power functions. For two finite distributions P P and Q Q , the power function 𝖯𝗈𝗐 ( P ∥ Q ) \\\\mathsf{Pow}(P\\\\|Q) is determined by the decreasing rearrangement of the likelihood ratio Q ( ω ) / P ( ω ) Q(\\\\omega)/P(\\\\omega) under ω ∼ P \\\\omega\\\\sim P ; equivalently, two pairs of finite distributions with the same power function have the same likelihood ratio distribution under their first distribution, up to the usual splitting of atoms. Moreover, tensor product of power functions corresponds to taking the product of independent likelihood ratios.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Apply this characterization to the conditional kernels. For each prefix s s , the likelihood ratio\"\n },\n {\n \"type\": \"para\",\n \"text\": \"has, under y t ∼ K t , s 0 y_{t}\\\\sim K^{0}_{t,s} , the likelihood-ratio distribution determined by h t h_{t} . This distribution is the same for every prefix s s . Since R t R_{t} is fresh and independent of the previous seeds, the conditional likelihood-ratio increment at round t t is independent of the past after conditioning on the prefix, and its distribution is determined only by h t h_{t} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a full transcript y [ 1 . . T ] y[1..T] , the likelihood ratio between P 1 P_{1} and P 0 P_{0} factors as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under P 0 P_{0} , the factors in this product have the same joint distribution as independent likelihood-ratio variables whose one-step power functions are h 1 , … , h T h_{1},\\\\ldots,h_{T} . Therefore the power function of the full adaptive transcript pair is exactly the tensor product:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the second equality uses 𝖨𝖽 ⊗ q = q \\\\mathsf{Id}\\\\otimes q=q for every power function q q .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, consider any non-adaptive adversary that submits the fixed first pair x ( 0 ) [ 1 ] , x ( 1 ) [ 1 ] x^{(0)}[1],x^{(1)}[1] and then submits identical inputs in every later round. The later common inputs may be chosen arbitrarily. By the invariant pair property, they only contribute common aggregation context and therefore do not change the one-step power labels h t h_{t} . For this non-adaptive input pair, the same tensor product is the power function of the transcript pair:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since ℳ \\\\mathcal{M} is non-adaptively f f -DP up to horizon T T , this tensor product is at most f f . Hence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"in Case 1.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Case 2: the first round is not the challenge round. Assume x ( 0 ) [ 1 ] = x ( 1 ) [ 1 ] x^{(0)}[1]=x^{(1)}[1] . Denote this common value by c c . Then U ( 0 ) = U ( 1 ) U^{(0)}=U^{(1)} as distributions, and hence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now define a shifted mechanism for the remaining T − 1 T-1 rounds after the common first input c c has been fixed. Let ℳ ( c ) \\\\mathcal{M}^{(c)} be the mechanism whose round- s s output, for 1 ≤ s ≤ T − 1 1\\\\leq s\\\\leq T-1 , is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This mechanism uses the independent seeds R 2 , … , R T R_{2},\\\\ldots,R_{T} , so it is independently decomposable. It is also quantitatively decomposable. Indeed, for round s s define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If 1 ∉ I s + 1 1\\\\notin I_{s+1} , then the aggregate used by ℳ s + 1 \\\\mathcal{M}_{s+1} is exactly the aggregate over I s ( c ) I^{(c)}_{s} in the suffix history. If 1 ∈ I s + 1 1\\\\in I_{s+1} , then the aggregate has the common left context c c , and the invariant-pair property removes this common context. Therefore, for every two suffix histories z [ 1 . . s ] z[1..s] and z ′ [ 1 . . s ] z^{\\\\prime}[1..s] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus ℳ ( c ) \\\\mathcal{M}^{(c)} satisfies the same structural assumptions as ℳ \\\\mathcal{M} , with the time indices shifted by one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let Λ \\\\Lambda be the set of all Hamming-neighboring pairs of suffix histories of length T − 1 T-1 . For\"\n },\n {\n \"type\": \"para\",\n \"text\": \"let V λ ( b ) V_{\\\\lambda}^{(b)} be the distribution of the length- ( T − 1 ) (T-1) transcript generated by ℳ ( c ) \\\\mathcal{M}^{(c)} on the fixed suffix input z ( b ) [ 1 . . T − 1 ] z^{(b)}[1..T-1] . Define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By definition of the supremum of power functions, g g is a power function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We claim that ℳ ( c ) \\\\mathcal{M}^{(c)} is non-adaptively g g -DP up to horizon T − 1 T-1 . For deterministic non-adaptive suffix adversaries this is immediate from the definition of g g . Randomized non-adaptive suffix adversaries are mixtures of deterministic ones, and the same bound follows from Fact A.9 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, g ≤ f g\\\\leq f . To see this, fix any λ ∈ Λ \\\\lambda\\\\in\\\\Lambda and consider the non-adaptive length- T T adversary for ℳ \\\\mathcal{M} that submits the common first input c c in both branches and then submits the suffix pair λ \\\\lambda . The first-round output has the same distribution under the two secret bits, and the remaining transcript has law V λ ( b ) V_{\\\\lambda}^{(b)} under bit b b . Hence the power function of this full non-adaptive transcript pair is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The non-adaptive f f -DP assumption for ℳ \\\\mathcal{M} gives g λ ≤ f g_{\\\\lambda}\\\\leq f for every λ ∈ Λ \\\\lambda\\\\in\\\\Lambda . Taking the supremum over λ \\\\lambda gives g ≤ f g\\\\leq f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now condition on a possible first-round output u u in the support of U ( 0 ) = U ( 1 ) U^{(0)}=U^{(1)} . After observing u u , the original deterministic adaptive adversary 𝒜 {\\\\mathcal{A}} induces a deterministic adaptive suffix adversary 𝒜 u {\\\\mathcal{A}}_{u} for the shifted mechanism ℳ ( c ) \\\\mathcal{M}^{(c)} : it continues exactly as 𝒜 {\\\\mathcal{A}} would continue after the first transcript symbol u u . Let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"be the distribution of the remaining transcript under secret bit b b , and write\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since ℳ ( c ) \\\\mathcal{M}^{(c)} is quantitatively decomposable and non-adaptively g g -DP up to horizon T − 1 T-1 , the induction hypothesis gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for every such first-round output u u .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the full transcript under secret bit b b has the same distribution as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where U ∼ U ( 0 ) = U ( 1 ) U\\\\sim U^{(0)}=U^{(1)} . The first coordinate has identical distribution under the two secret bits, and, conditional on U = u U=u , the continuation distributions have power function at most g g . Applying Fact A.10 with first-stage profile 𝖨𝖽 \\\\mathsf{Id} and second-stage profile g g yields\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This proves Case 2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two cases complete the induction step. Hence the finite-horizon adaptive f f -DP guarantee holds for every deterministic adaptive adversary. As argued at the beginning, joint convexity then extends the same guarantee to randomized adaptive adversaries. Therefore ℳ \\\\mathcal{M} is adaptively f f -DP for every finite horizon. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Definition E.3 states quantitative decomposability using a scalar distance parameter. This is sufficient for the theorem as used in the main text. Some standard vector-valued mechanisms, however, have an exact one-round power profile that depends on the full difference vector rather than only on its norm. For example, for coordinate-wise Laplace noise on ℝ k \\\\mathbb{R}^{k} , the exact profile of u + η u+\\\\eta versus u ′ + η u^{\\\\prime}+\\\\eta may depend on the vector u − u ′ u-u^{\\\\prime} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The theorem and proof extend verbatim to this setting by replacing the scalar distance d ( u , u ′ ) d(u,u^{\\\\prime}) with an invariant difference map Δ ( u , u ′ ) \\\\Delta(u,u^{\\\\prime}) taking values in some parameter space, such as Δ ( u , u ′ ) = u − u ′ ∈ ℝ k \\\\Delta(u,u^{\\\\prime})=u-u^{\\\\prime}\\\\in\\\\mathbb{R}^{k} , and replacing g t ( θ ) g_{t}(\\\\theta) by a family g t ( v ) g_{t}(v) indexed by this difference parameter. The only property needed in the proof is invariance under common aggregation context:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under this vector-indexed variant, the binary-tree prefix-sum mechanism with coordinate-wise Laplace noise is covered exactly for vector updates in ℝ k \\\\mathbb{R}^{k} . In the main text, we state the simpler scalar version because the usual ( ε , δ ) (\\\\varepsilon,\\\\delta) consequences can also be obtained by upper-bounding the vector-indexed profile in terms of a norm, such as ‖ u − u ′ ‖ 1 \\\\|u-u^{\\\\prime}\\\\|_{1} for coordinate-wise Laplace noise.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.1 Why Independent Decomposability Alone Is Insufficient\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The certification theorem in Section E requires quantitative decomposability, not merely independent decomposability. The following example shows why the stronger condition is needed. Even when each round uses fresh independent randomness and the round- t t output is a deterministic function of the input prefix and the fresh seed, a non-adaptive approximate-DP guarantee need not lift to adaptive adversaries. The obstruction is that the one-round leakage can depend on the adaptive common context, rather than only on the location and magnitude of the Hamming discrepancy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Counterexample with Fresh Independent Randomness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider a two-round mechanism with input and output alphabets { 0 , 1 } \\\\{0,1\\\\} . Write x 1 = b x_{1}=b and x 2 = q x_{2}=q . Using independent fresh randomness in the two rounds, the mechanism outputs Y 1 , Y 2 ∈ { 0 , 1 } Y_{1},Y_{2}\\\\in\\\\{0,1\\\\} with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since each output is a deterministic function of the current prefix and a fresh independent seed, the mechanism is independently decomposable in the sense of Definition E.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For every fixed input stream ( b , q ) (b,q) , the transcript distribution is the product distribution of Y 1 Y_{1} and Y 2 Y_{2} . Since two-sided ( 0 , δ ) (0,\\\\delta) -DP is equivalent to total variation distance at most δ \\\\delta , it suffices to check the four Hamming-neighboring pairs:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence the mechanism is non-adaptively ( 0 , 0.09 ) (0,0.09) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now consider the deterministic adaptive adversary that chooses the challenge pair x 1 ( 0 ) = 0 x^{(0)}_{1}=0 and x 1 ( 1 ) = 1 x^{(1)}_{1}=1 at round 1 1 , and, after observing Y 1 = y Y_{1}=y , submits the common second input x 2 ( 0 ) = x 2 ( 1 ) = y x^{(0)}_{2}=x^{(1)}_{2}=y . This is a valid adaptive Hamming-neighbor adversary: the two submitted streams differ only at coordinate 1 1 , but their common second input depends on the first release.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The resulting transcript distributions are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For S = { ( 0 , 1 ) , ( 1 , 0 ) } S=\\\\{(0,1),(1,0)\\\\} , we have\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so the mechanism is not adaptively ( 0 , 0.09 ) (0,0.09) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This does not contradict Theorem E.4 , because the theorem assumes quantitative decomposability. The example shows that fresh independent randomness alone is not enough: the adaptive adversary can choose a common later input that changes how the earlier Hamming discrepancy is revealed. Quantitative decomposability rules out precisely this behavior by forcing the one-round power function to depend only on the distance between the relevant aggregates, with common aggregation context removed by the invariant-pair property.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There have been numerous works on differentially private prefix-sum algorithms since the binary tree mechanism [ CSS10 , CSS11 ] was proposed. We state the properties of the variant that are needed for our applications and briefly outline how they can be achieved from existing works.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Problem Setting. For each time step t ≥ 1 t\\\\geq 1 , an input stream has some integer x t ∈ 𝒳 = [ a . . b ] x_{t}\\\\in\\\\mathcal{X}=[a..b] from some known range that is passed to the interactive mechanism 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 \\\\mathsf{StreamSum} that returns a value S t S_{t} that is supposed to be an estimation of the sum ∑ τ = 1 t x τ \\\\sum_{\\\\tau=1}^{t}x_{\\\\tau} . The additive error is | S t − ∑ τ = 1 t x τ | |S_{t}-\\\\sum_{\\\\tau=1}^{t}x_{\\\\tau}| . The adversary observes S t S_{t} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neighboring Notion. Two input streams are neighboring if they differ in at most one time step, and the corresponding two values differ by 1.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As aforementioned, a mechanism with unbounded termination satisfies ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP if for every T > 0 T>0 , the truncated mechanism run for T T steps is ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any ε > 0 , δ ∈ ( 0 , 1 ) \\\\varepsilon>0,\\\\delta\\\\in(0,1) and an interval [ a . . b ] ⊆ ℤ [a..b]\\\\subseteq\\\\mathbb{Z} of integers, there exists an ( ε , δ ) (\\\\varepsilon,\\\\delta) -differentially private interactive mechanism 𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε , δ \\\\mathsf{StreamSum}_{\\\\varepsilon,\\\\delta} , such that given a stream { x t } \\\\{x_{t}\\\\} of integers in [ a . . b ] [a..b] , the algorithm outputs a number S t S_{t} in each time step t t such that the following hold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bounded Error: With probability 1, at each time step t t , the output has additive error at most E ( t ) := O ( 1 ε ⋅ log t ⋅ ( log t + log 1 δ ) ) E(t):=O\\\\left(\\\\frac{1}{\\\\varepsilon}\\\\cdot\\\\log t\\\\cdot(\\\\log t+\\\\log\\\\frac{1}{\\\\delta})\\\\right) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consistency: With probability 1, for each time step t t , the increment S t − S t − 1 ∈ [ a . . b ] S_{t}-S_{t-1}\\\\in[a..b] is in the above range.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As in [ CCM+22 ] , we start with a pure ε \\\\varepsilon -DP prefix sum algorithm that has high probability guarantee on the additive error.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any ε > 0 \\\\varepsilon>0 , there exists an ( ε , 0 ) (\\\\varepsilon,0) -differentially private interactive mechanism 𝖧𝗒𝖻𝗋𝗂𝖽𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε \\\\mathsf{HybridStreamSum}_{\\\\varepsilon} that takes an input stream { x t } \\\\{x_{t}\\\\} of integers and output S t S_{t} at each time step t t with the following properties.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bounded Error: Fix any 0 < δ < 1 0<\\\\delta<1 . With probability 1 − δ 1-\\\\delta , for every time step t t , the output at time t t has an additive error of at most O ( 1 ε log t ( log t + log 1 δ ) ) O(\\\\frac{1}{\\\\varepsilon}\\\\log t(\\\\log t+\\\\log\\\\frac{1}{\\\\delta})) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Structure: In each time step t t , the mechanism actually allows the adversary to observe noisy counts of the form ∑ i ∈ I x i \\\\sum_{i\\\\in I}x_{i} masked with independent noise, where the index interval has the form I = [ c . . t ] I=[c..t] . The output S t S_{t} can be constructed deterministically from the noisy counts released so far.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix ε \\\\varepsilon and δ \\\\delta . Starting from the mechanism 𝖧𝗒𝖻𝗋𝗂𝖽𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε \\\\mathsf{HybridStreamSum}_{\\\\varepsilon} in Fact F.2 , we modify it step by step to achieve the desired properties.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Always Error Bound. If the output S t S_{t} is too large, i.e., S t > ∑ i = 1 t x i + E ( t ) S_{t}>\\\\sum_{i=1}^{t}x_{i}+E(t) , return ∑ i = 1 t x i + E ( t ) \\\\sum_{i=1}^{t}x_{i}+E(t) ; if the output is too small, i.e., S t < ∑ i = 1 t x i − E ( t ) S_{t}<\\\\sum_{i=1}^{t}x_{i}-E(t) , return ∑ i = 1 t x i − E ( t ) \\\\sum_{i=1}^{t}x_{i}-E(t) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The argument in Theorem 3.2 of [ CCM+22 ] shows this can turn a high probability additive error bound into an additive error bound with probability 1, at the cost of getting ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consistency. The argument in Lemma 5.2 of [ CSS11 ] shows that a further truncation operation can achieve consistency without increasing the additive error bound.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interactive mechanism 𝖧𝗒𝖻𝗋𝗂𝖽𝖲𝗍𝗋𝖾𝖺𝗆𝖲𝗎𝗆 ε \\\\mathsf{HybridStreamSum}_{\\\\varepsilon} in Fact F.2 is adaptively ( ε , 0 ) (\\\\varepsilon,0) -DP, and can be made to be consistent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The structure property in Fact F.2 implies that noisy counts obey the independent decomposability property in Definition E.1 . Adaptability follows from Theorem E.4 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consistency can be achieved, because the modification can be done deterministically from the released noisy counts only. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is not clear whether the “Always Error Bound” can be achieved under the independent decomposability framework, because the truncation needs to refer back to the private inputs, in addition to the released noisy counts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"F.1 Prefix Sums via Lower-Triangular Matrix Factorizations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We briefly restate the prefix-sum workload in the language of the matrix mechanism . Fix a horizon T T and write the stream as a vector x ∈ ℝ T x\\\\in\\\\mathbb{R}^{T} . The prefix sums are the linear workload\"\n },\n {\n \"type\": \"para\",\n \"text\": \"i.e., ( M 𝖼𝗈𝗎𝗇𝗍 x ) t = ∑ i = 1 t x i (M_{\\\\mathsf{count}}x)_{t}=\\\\sum_{i=1}^{t}x_{i} . We use the event-level neighboring notion: x ∼ x ′ x\\\\sim x^{\\\\prime} iff x − x ′ = ± Δ e i x-x^{\\\\prime}=\\\\pm\\\\Delta e_{i} for some coordinate i ∈ [ T ] i\\\\in[T] , where Δ > 0 \\\\Delta>0 is the sensitivity unit (in the above we take Δ = 1 \\\\Delta=1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Privacy Notions: Renyi DP (RDP) and zCDP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let P , Q P,Q be distributions with densities p , q p,q w.r.t. a common base measure. For α > 1 \\\\alpha>1 , the (order- α \\\\alpha ) Renyi divergence is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An interactive mechanism ℳ \\\\mathcal{M} is ( α , ε ) (\\\\alpha,\\\\varepsilon) -RDP if for all neighboring x ∼ x ′ x\\\\sim x^{\\\\prime} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is ρ \\\\rho -zCDP if for all α > 1 \\\\alpha>1 and all neighboring x ∼ x ′ x\\\\sim x^{\\\\prime} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Equivalently, ρ \\\\rho -zCDP means ( α , ρ α ) (\\\\alpha,\\\\rho\\\\alpha) -RDP holds simultaneously for every α > 1 \\\\alpha>1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Converting RDP to ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If ℳ \\\\mathcal{M} satisfies ( α , ε α ) (\\\\alpha,\\\\varepsilon_{\\\\alpha}) -RDP for some α > 1 \\\\alpha>1 , then for every δ ∈ ( 0 , 1 ) \\\\delta\\\\in(0,1) it satisfies ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One may optimize over α > 1 \\\\alpha>1 to get the tightest ( ε , δ ) (\\\\varepsilon,\\\\delta) bound:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Lower-Triangular Factorization Mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let M 𝖼𝗈𝗎𝗇𝗍 = L R M_{\\\\mathsf{count}}=LR be a (possibly rectangular) factorization where R ∈ ℝ m × T R\\\\in\\\\mathbb{R}^{m\\\\times T} and L ∈ ℝ T × m L\\\\in\\\\mathbb{R}^{T\\\\times m} are lower-triangular in time (i.e., causal: the t t -th output depends only on x [ 1 . . t ] x[1..t] through R R and L L ). Define the column-sensitivity of R R by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider the Gaussian matrix mechanism\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since L ( ⋅ ) L(\\\\cdot) is post-processing, privacy is governed entirely by the Gaussian release of R x Rx .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For neighboring x ∼ x ′ x\\\\sim x^{\\\\prime} with x − x ′ = ± Δ e i x-x^{\\\\prime}=\\\\pm\\\\Delta e_{i} , we have ‖ R ( x − x ′ ) ‖ 2 ≤ Δ s ( R ) \\\\|R(x-x^{\\\\prime})\\\\|_{2}\\\\leq\\\\Delta\\\\,s(R) . Hence the mechanism ℳ L , R , σ \\\\mathcal{M}_{L,R,\\\\sigma} satisfies:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"zCDP: ρ \\\\rho -zCDP with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RDP: for every order α > 1 \\\\alpha>1 , ( α , ε α ) (\\\\alpha,\\\\varepsilon_{\\\\alpha}) -RDP with\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Binary Tree Mechanism as a Factorization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume first T = 2 h T=2^{h} for an integer h ≥ 0 h\\\\geq 0 (otherwise pad to the next power of two, which only changes constants by replacing h h with ⌈ log 2 T ⌉ \\\\lceil\\\\log_{2}T\\\\rceil ). Index the nodes of the complete binary tree over [ T ] [T] by dyadic intervals; let q ∈ ℝ m q\\\\in\\\\mathbb{R}^{m} be the vector of all dyadic-interval sums, so that q = R 𝗍𝗋𝖾𝖾 x q=R_{\\\\mathsf{tree}}x where R 𝗍𝗋𝖾𝖾 R_{\\\\mathsf{tree}} is a 0 / 1 0/1 incidence matrix (leaf i i contributes to exactly its h + 1 h+1 ancestors). Each prefix [ 1 . . t ] [1..t] has a canonical dyadic partition of size at most h + 1 h+1 , so there is a reconstruction matrix L 𝗍𝗋𝖾𝖾 L_{\\\\mathsf{tree}} with M 𝖼𝗈𝗎𝗇𝗍 = L 𝗍𝗋𝖾𝖾 R 𝗍𝗋𝖾𝖾 M_{\\\\mathsf{count}}=L_{\\\\mathsf{tree}}R_{\\\\mathsf{tree}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For T = 2 h T=2^{h} , every column of R 𝗍𝗋𝖾𝖾 R_{\\\\mathsf{tree}} has exactly h + 1 h+1 ones, hence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, the Gaussian tree mechanism L 𝗍𝗋𝖾𝖾 ( R 𝗍𝗋𝖾𝖾 x + z ) L_{\\\\mathsf{tree}}(R_{\\\\mathsf{tree}}x+z) with z ∼ 𝒩 ( 0 , σ 2 I m ) z\\\\sim\\\\mathcal{N}(0,\\\\sigma^{2}I_{m}) is ρ 𝗍𝗋𝖾𝖾 \\\\rho_{\\\\mathsf{tree}} -zCDP with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and for every α > 1 \\\\alpha>1 it is ( α , ε α , 𝗍𝗋𝖾𝖾 ) (\\\\alpha,\\\\varepsilon_{\\\\alpha,\\\\mathsf{tree}}) -RDP with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For general T T , the same holds with h := ⌈ log 2 T ⌉ h:=\\\\lceil\\\\log_{2}T\\\\rceil .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A Smooth Lower-Triangular Factorization [ FHU23 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fichtenberger et al. give an explicit Toeplitz lower-triangular factorization M 𝖼𝗈𝗎𝗇𝗍 = L R M_{\\\\mathsf{count}}=LR with L = R L=R defined by a scalar sequence f ( ⋅ ) f(\\\\cdot) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and 0 otherwise. Let γ \\\\gamma be the Euler–Mascheroni constant and define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let L , R L,R be the above factorization. Then M 𝖼𝗈𝗎𝗇𝗍 = L R M_{\\\\mathsf{count}}=LR and\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consequently, the mechanism L ( R x + z ) L(Rx+z) with z ∼ 𝒩 ( 0 , σ 2 I T ) z\\\\sim\\\\mathcal{N}(0,\\\\sigma^{2}I_{T}) is ρ 𝖥𝖧𝖴 \\\\rho_{\\\\mathsf{FHU}} -zCDP with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and for every α > 1 \\\\alpha>1 it is ( α , ε α , 𝖥𝖧𝖴 ) (\\\\alpha,\\\\varepsilon_{\\\\alpha,\\\\mathsf{FHU}}) -RDP with\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Takeaway.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Both mechanisms fit the same template L ( R x + z ) L(Rx+z) , but yield different (exact) privacy constants through the column ℓ 2 \\\\ell_{2} sensitivity s ( R ) s(R) : the binary tree gives s ( R ) 2 = Θ ( log 2 T ) s(R)^{2}=\\\\Theta(\\\\log_{2}T) , while the smooth Toeplitz factorization satisfies s ( R ) 2 ≤ 1 − 1 − γ π + ln T π + 2 T s(R)^{2}\\\\leq 1-\\\\frac{1-\\\\gamma}{\\\\pi}+\\\\frac{\\\\ln T}{\\\\pi}+\\\\frac{2}{T} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As the most relevant related works have already been covered in the introduction, we will elaborate further on some related aspects.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Choices of Privacy Measure. The central idea in differential privacy [ DWO06 ] is that if V 0 V_{0} and V 1 V_{1} are two distributions of outputs produced by a mechanism from two neighboring inputs, then those two distributions should be close. The classical ( ε , δ ) (\\\\varepsilon,\\\\delta) -DP notion uses two parameters to quantify closeness, where a smaller value in each parameter means that the two distributions are closer. However, since distributions are inherently complex objects, some information on the two distributions will be inevitably lost when they are compared using just two parameters. When the same Gaussian noise is used to mask two different vectors, researchers have discovered that the Rényi divergence [ MIR17 ] can capture the variance of the Gaussian distribution perfectly, and hence, can quantify the closeness of two such distributions (with the same variance but different means).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In general, any useful way to quantify privacy guarantees must satisfy the property that if the output satisfies certain privacy requirement, then any further processing of the output cannot violate that specific requirement. This can be formally formulated by requiring that the divergence – used for measuring how different two distributions are – must satisfy the data processing inequality ; it is worth noting that some common distance notion such as the ℓ 2 \\\\ell_{2} -norm does not satisfy this property.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instead of just using a few parameters to capture the closeness of two distributions, tradeoff functions [ DRS22 ] have been proposed to define differential privacy, because a tradeoff function can capture all the essential information about how two distributions differ in the sense that any divergence satisfying the data processing inequality can be recovered from the tradeoff function. Indeed, tradeoff functions offer a powerful tool to describe the composition of private mechanisms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, one notational inconvenience is that a larger tradeoff (measured by pointwise comparison) means that the two distributions are closer, which has the opposite interpretation from other divergence parameters such as ε \\\\varepsilon and δ \\\\delta . In fact, an ad hoc concept of generalized probability distance has been defined in [ VZ23 ] to reverse the direction of the inequalities such that it will be consistent with the notion of distance. On the other hand, a simpler way to achieve this notation consistency is to replace a tradeoff function with its complement that is known as a power function , which naturally preserves all the equivalent mathematical properties. As we shall see in Definition A.3 , a power function also has an intuitive description using the fractional knapsack problem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As illustrated in [ VZ23 , ZZC+24 ] , if one uses such a powerful tool to define any new notion of differential privacy, then a single composition theorem (such as our Theorem C.1 ) will be sufficient to recover any composition result from the classical notion to the new notion of privacy. Therefore, it would not be necessary to reconstruct individual advanced composition theorems [ DRV10b , KOV15 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hamming vs Edit Neighboring Notions. In [ BEB+24 ] , two neighboring notions are considered for static databases:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hamming-style. Two databases have the same number of elements, and they differ in at most one element.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Edit-style. One element from one database is deleted to form the other database.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"They considered fixed-size mini-batches, which may be sampled in two ways: with or without replacement. Since they considered sampling from static databases, the difference between Hamming- vs edit-style neighboring static databases would not have such a stark contrast as streams, as deleting the first element of a stream can cause it to change in every position.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Other Recent Works on Concurrent Composition. A more general notion of concurrent composition is considered in [ HST+23 ] , where the privacy parameters of mechanisms can be adaptively chosen. However, as in [ VZ23 ] , each interactive mechanism is associated with a single static database on which its neighboring relation is defined; the adversary may interact adaptively with the mechanism, but neighboring inputs differ only in this underlying database, not in a dynamic stream of updates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Concurrent composition for mechanisms with adaptively chosen privacy parameters are also considered in [ HSV26 ] , but for neighboring dynamic databases. They also give a formulation based on adaptive DP [ DMR+22 ] , expressed via a left-or-right style distinguishing game with a verification function, which is essentially the same as our paired simulation in Definition B.2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that both works consider composition where each mechanism has a single neighboring relation on its dataset (static or dynamic). In order to capture modular composition in which neighboring notions are defined separately on both the input and the output of a mechanism—and to reason about neighbor-preserving transformations that change the neighboring structure—we need a more refined notion of neighbor-preserving paired simulation, given in Definition B.8 .\"\n }\n]","isConversational":false}}},{"rowIdx":482,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07748"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Selective Left-Shift: Turning Test-Time Compute and Difficulty-based Curation into Training Data for Low-Resource Code Generation"},"authors":{"kind":"list like","value":["Didula Samaraweera","Anjana Supun","Srinath Perera"],"string":"[\n \"Didula Samaraweera\",\n \"Anjana Supun\",\n \"Srinath Perera\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07748v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"Large Language Models (LLMs) have achieved remarkable proficiency in code generation for high-resource programming languages(HRPLs) like Python and Java (Chen et al. , 2021 ) . However, their performance sharply declines when applied to Low-Resource Programming Languages (LRPLs) (Cassano et al. , 2023 ; Joel et al. , 2024 ) . This disparity is not merely academic: languages such as Julia (scientific computing) and Ballerina (cloud-native microservices) serve critical roles across diverse software engineering domains. As the ecosystem of programming languages continues to grow, with new languages emerging to address specialized needs, enabling reliable code generation for LRPLs is an increasingly pressing challenge for the software engineering community, mirroring the historical trajectory of low-resource Natural Language Processing (Ranathunga et al. , 2023 ) ."},{"type":"para","text":"Enhancing Small Language Models (SLMs) for LRPLs presents a trilemma of distinct challenges. First , Supervised Fine-Tuning (SFT) is highly effective for smaller parameter models (Giagnorio et al. , 2025 ) , but it is fundamentally bottlenecked by severe data scarcity. For example, datasets like CodeNet (Puri et al. , 2021 ) contain only 10.8k Julia samples compared to 3.2M for Python. Second , Test-time Scaling (e.g. iterative self-correction, reflection-based refinement) effectively boosts performance (Wang et al. , 2025 ; Ren and others, 2024 ) , but introduces unacceptable latency and computational overhead for real-world deployments. Third , applying Reinforcement Learning (RL) directly to a base model suffers from sparse rewards: e.g., the model generates so many syntax errors that it rarely receives 4 positive feedback to learn from."},{"type":"para","text":"This paper uses on two key hypotheses to improve SLMs for LRPLs."},{"type":"para","text":"Our first hypothesis is that inference-time scaling and supervised fine-tuning are not competing strategies but can be complementary phases of a single pipeline . Rather than deploying computationally expensive iterative refinement during live usage, we can “left-shift” this computation, repurposing it strictly as an offline data curation engine. By utilizing iterative compiler and test-case feedback, where compilation errors or a single failed test case showing the expected versus actual output are fed back to the model for self-correction, we synthesize missing ground-truth data to enable high-quality SFT."},{"type":"para","text":"A central challenge in this pipeline is ensuring the quality of the synthetically generated data: an LLM can produce fluent but semantically incorrect code, and training on such examples would embed faulty priors. We address this through a flexible verification strategy adapted to the available resources. When a ground-truth solution exists in a high-resource language (e.g. Python), we can employ differential testing , executing both the candidate LRPL solution and the reference implementation on shared inputs and comparing their outputs. Differential testing is preferred when available because it catches subtle semantic divergences that fixed IO test suites may miss, such as floating-point precision differences, locale-dependent string formatting, or edge cases in boundary conditions that the original test suite does not explicitly cover. When no reference solution is available, we fall back to input/output test verification , running the candidate against a suite of I/O test cases derived from the problem specification. In practice, the majority of our seed problems originate from competitive programming datasets that provide IO test suites but not reference solutions in the target LRPL, making IO verification the dominant verification path in our experiments. This two-tiered approach is deliberately pragmatic: the verification method is determined entirely by what artifacts are available for a given problem, requiring no additional human annotation."},{"type":"para","text":"Our second hypothesis is that the improvement achieved via Reinforcement Learning is dependent on the difficulty of the problems it gets feedback on . With very easy problems, there is little to learn, and with very hard problems, the model’s answers will all be wrong; thus, the model will not know how to make progress. Problems with the correct difficulty can speed up the learning. For example, Pikus et al. (Pikus et al. , 2025 ) have also observed that hard samples help with GRPO learning. This idea can also be seen as a variation of active learning (Ren et al. (Ren et al. , 2021 ) ), where we actively select the right data to be included in the training set to maximize the model outcome."},{"type":"para","text":"Building on this idea, we create a dataset for Reinforcement Learning with Verifiable Rewards (RLVR) training by using difficulty scores (ELO rating for problems) to curate the data. Subsequently, we apply RLVR via Group Relative Policy Optimization (GRPO) (Shao et al. , 2024 ) , directly adopting the language-agnostic IO-based evaluation framework proposed by Boruch-Gruszecki et al. (Boruch-Gruszecki et al. , 2026 ) . To maximize the learning signal, we employ partial rewards proportional to the fraction of test cases passed and successful compilation, allowing the advantage estimates to capture fine-grained differences between completions."},{"type":"para","text":"As we will see in our results analysis, because the SFT prior constrains the action space away from trivial syntax errors, the RL phase can efficiently navigate the richer landscape of functional logic. Furthermore, to prevent the policy from converging to sub-optimal states, we use zero-advantage masking (Chen et al. , 2025 ) : if no completion within a group achieves the maximum reward, the entire group’s advantage is set to zero, effectively discarding the update. This avoids reinforcing mediocre solutions and ensures that the policy only updates toward genuinely strong completions."},{"type":"para","text":"We evaluate our pipeline on Qwen3-8B (Qwen Team, 2025 ) using two benchmarks: MultiPL-E (Cassano et al. , 2023 ) and the Agnostics LiveCodeBench (Ag-LCB) (Boruch-Gruszecki et al. , 2026 ) . For Julia, our full pipeline improves pass@1 by +24.6 points on MultiPL-E and +30.2 points on Ag-LCB. For Ballerina, where the base model has near-zero prior knowledge, we observe +45.3 points improvement on MultiPL-E and +22.1 points on Ag-LCB over the baseline. Notably, the SFT phase alone achieves on-par results with the current state-of-the-art, demonstrating that left-shifted inference-time compute can match expensive online scaling strategies through a one-time offline investment. Furthermore, validating our second hypothesis, we achieved the above results while using 1 3 \\frac{1}{3} of the data and 1 6 \\frac{1}{6} of the cost."},{"type":"para","text":"We further demonstrate that the pipeline generalizes to Ballerina, a cloud-native programming language for integration and microservices that has near-zero representation in LLM pretraining corpora. Unlike Julia, which benefits from some incidental coverage in training data, Ballerina presents a true cold-start scenario: the base model has effectively no prior exposure to its syntax, type system, or standard library. Despite this, our pipeline bootstraps functional Ballerina code generation by relying solely on compiler feedback and language-agnostic test suites, requiring no human-written Ballerina training examples. This result provides strong evidence that any LRPL can be bootstrapped making the pipeline immediately applicable as new programming languages emerge."},{"type":"para","text":"Our contributions are as follows:"},{"type":"para","text":"We propose a three-phase pipeline, offline inference scaling via iterative compiler and test feedback, SFT, and GRPO with a dataset curated by difficulty, that achieves state-of-the-art results on MultiPL-E and Ag-LCB for Julia and Ballerina using an 8B-parameter model, with much fewer training rounds and lower cost."},{"type":"para","text":"We demonstrate that left-shifting inference-time computation to offline data synthesis resolves the data scarcity bottleneck without human annotation, turning expensive iterative refinement into a one-time investment."},{"type":"para","text":"We show that the pipeline generalizes to an extreme LRPL (Ballerina) with minimal adaptation, establishing a practical recipe for bootstrapping code generation for any LRPL."},{"type":"para","text":"We extend the MultiPL-E benchmark for Ballerina, adding a new language to the evaluation suite and providing a standardized testbed for future research on LRPL code generation."},{"type":"heading","level":1,"text":"2. Related Work"},{"type":"para","text":"Benchmarking LLMs on LRPLs relies on frameworks adapted from high-resource environments, such as MultiPL-E (Cassano et al. , 2023 ) and Agnostic-LiveCodeBench (Jain et al. , 2024 ) . While these benchmarks effectively measure performance, improving that performance requires navigating severe data limitations and choosing optimal learning strategies. We organize prior work along three axes that correspond to the phases of our pipeline."},{"type":"heading","level":2,"text":"2.1. SFT and the Challenge of Data Scarcity"},{"type":"para","text":"Supervised Fine-Tuning remains the most prevalent strategy for adapting models to LRPLs. Cassano et al. (Cassano et al. , 2024 ) introduced MultiPL-T, a pipeline for translating training data from high-resource to low-resource languages, demonstrating significant gains for models up to 8B parameters. Giagnorio et al. (Giagnorio et al. , 2025 ) further showed that smaller models ( ∼ \\sim 1B parameters) trained on specialized, domain-specific data often outperform larger, general-purpose models, as SFT can inversely degrade larger models better suited for Retrieval-Augmented Generation. WizardCoder (Luo et al. , 2024 ) and OpenCoder (Huang et al. , 2024 ) advanced SFT through instruction-tuning and open-source training recipes, though their focus remains on high-resource languages. Joel et al. (Joel et al. , 2024 ) provide a comprehensive survey of LLM-based code generation for low-resource and domain-specific languages."},{"type":"para","text":"The fundamental bottleneck for SFT is data scarcity. Datasets like CodeNet (Puri et al. , 2021 ) provide a foundation, but the available data for LRPLs pales in comparison to high-resource languages. Utilizing organic data requires rigorous curation, including deduplication and strict decontamination to prevent benchmark leakage. To bypass organic data limitations, recent research emphasizes synthetic data pipelines: Magicoder (Wei et al. , 2024 ) leverages OSS-Instruct to generate training data from open-source snippets, while KODCODE (Xu et al. , 2025 ) constructs a diverse, verifiable synthetic dataset. Our work extends this line by introducing compiler/test feedback driven data synthesis, where iterative execution and error correction provides verification that purely generative pipelines lack."},{"type":"heading","level":2,"text":"2.2. Reinforcement Learning and Grounding"},{"type":"para","text":"Beyond SFT, Reinforcement Learning is establishing new state-of-the-art results for code generation. Early approaches used Proximal Policy Optimization (PPO): CodeRL (Le et al. , 2022 ) combined pretrained models with execution-based RL rewards, while Shojaee et al. (Shojaee et al. , 2023 ) demonstrated that execution-based feedback outperforms static heuristics for policy optimization. More recently, Group Relative Policy Optimization (GRPO) (Shao et al. , 2024 ) has emerged as a simpler alternative that eliminates the need for a separate value network. CodeReasoner (Tang et al. , 2025 ) applies GRPO to enhance code reasoning in general-purpose models."},{"type":"para","text":"For LRPLs specifically, recent techniques rely on language agnostic datasets and test-feedback loops. Concurrently, self-play frameworks enable iterative refinement without human annotation (Lin et al. , 2025 ) . However, ungrounded self-play often yields sub-optimal results. To address this problem, Boruch-Gruszecki et al. (Boruch-Gruszecki et al. , 2026 ) introduced Agnostics, a universal RL environment capable of learning to synthesize code in any programming language via execution-based rewards."},{"type":"para","text":"Ma et al. (Ma et al. , 2026 ) demonstrate that interleaving online fine-tuning with RL addresses the hardest questions that reinforcement learning alone cannot solve, highlighting the complementary nature of supervised and RL-based training."},{"type":"para","text":"Our work grounds RL with IO tests similar to Boruch-Gruszecki et al. and, crucially, initializes the RL policy from a syntax-primed SFT model rather than the base model, which dramatically reduces the sparse-reward problem."},{"type":"heading","level":2,"text":"2.3. Test-time Scaling and Its Limitations"},{"type":"para","text":"To compensate for limited training data, the current state-of-the-art often allocates more computational resources during execution, a paradigm known as Test-time Scaling. A prominent family of techniques uses iterative refinement : the model generates a candidate solution, receives feedback (compiler errors, test failures, or self-reflection), and refines its output across multiple rounds. ReflectionCoder (Ren and others, 2024 ) leverages reflection sequences to iteratively correct one-off generations, while Origen (Cui and others, 2024 ) uses code-to-code augmentation with self-reflection for RTL code generation. AgentCoder (Huang et al. , 2023 ) employs a multi-agent framework where a programmer agent, test designer agent, and test executor agent collaborate through iterative testing and optimization. CodeSim (Islam et al. , 2025 ) extends this paradigm with simulation-driven planning and debugging."},{"type":"para","text":"Another family uses selection-based scaling : CodeT (Chen et al. , 2022 ) generates multiple candidate solutions along with tests and selects the best via dual execution agreement. CodeChemist (Wang et al. , 2025 ) applies functional knowledge transfer with test-time scaling specifically for low-resource code generation."},{"type":"para","text":"While these methods form a critical pillar of modern code generation, their high computational cost at inference time makes them impractical for efficient SLM deployment. Our work reframes iterative refinement, the most effective form of inference-time scaling, not as a deployment strategy but as an offline data curation mechanism, amortizing the compute cost across all future queries."},{"type":"heading","level":1,"text":"3. Methodology"},{"type":"heading","level":2,"text":"3.1. Pipeline Overview"},{"type":"para","text":"Our pipeline consists of three sequential phases, each addressing a distinct aspect of the LRPL code generation challenge. Figure 2 provides an architectural overview."},{"type":"para","text":"Given a target low-resource language L L , a set of language independent seed problems 𝒫 = { p 1 , … , p N } \\mathcal{P}=\\{p_{1},\\ldots,p_{N}\\} with corresponding test suites 𝒯 = { t 1 , … , t N } \\mathcal{T}=\\{t_{1},\\ldots,t_{N}\\} , and a pretrained base model M θ M_{\\theta} , our goal is to produce a fine-tuned model M θ ∗ M_{\\theta^{*}} that maximizes pass@1 on held-out problems in L L . The pipeline proceeds as follows: (1) Data Synthesis uses M θ M_{\\theta} with iterative compiler feedback to construct a verified dataset 𝒟 L \\mathcal{D}_{L} ; (2) Syntax-Aware SFT fine-tunes M θ M_{\\theta} on 𝒟 L \\mathcal{D}_{L} to produce M θ sft M_{\\theta_{\\text{sft}}} with strong syntactic priors; (3) RLVR applies GRPO starting from M θ sft M_{\\theta_{\\text{sft}}} to optimize functional correctness, producing the final model M θ ∗ M_{\\theta^{*}} ."},{"type":"heading","level":2,"text":"3.2. Phase 1: Data Synthesis via Offline Inference Scaling"},{"type":"para","text":"To overcome the LRPL data bottleneck, we repurpose expensive inference-time scaling, specifically iterative self-correction with compiler/test-case feedback, strictly as an automated, offline data curation engine."},{"type":"heading","level":3,"text":"3.2.1. Problem Formulation."},{"type":"para","text":"Let 𝒫 = { p 1 , … , p N } \\mathcal{P}=\\{p_{1},\\ldots,p_{N}\\} be a set of language independent algorithmic problems, each paired with a test suite t i ∈ 𝒯 t_{i}\\in\\mathcal{T} consisting of IO pairs. For a target language L L , we seek to construct a supervised dataset:"},{"type":"heading","level":3,"text":"3.2.2. Iterative Compiler/Test Feedback Refinement."},{"type":"para","text":"For each problem p i p_{i} , the base model M θ M_{\\theta} generates an initial candidate solution c i ( 0 ) c_{i}^{(0)} . This candidate is executed in an isolated sandbox against the test suite t i t_{i} , yielding either success or a feedback signal e i ( 0 ) e_{i}^{(0)} . The feedback comprises either one from two components: (1) any compilation or runtime error message, or in case of compilation success and test cases are failing (2) a single failed test case showing the expected output alongside the actual output produced by the candidate. If a candidate fails, we inject this structured feedback back into the model’s context, prompting iterative refinement:"},{"type":"para","text":"The loop terminates when pass ( c i ( k ) , t i ) = 1 \\mathrm{pass}(c_{i}^{(k)},t_{i})=1 (all tests pass) or k > K max k>K_{\\max} . Successful trajectories contribute the final solution c i ∗ = c i ( k ) c_{i}^{*}=c_{i}^{(k)} to 𝒟 L \\mathcal{D}_{L} ; failed trajectories are discarded. This process is identical to iterative refinement used at inference time by approaches like ReflectionCoder (Ren and others, 2024 ) and AgentCoder (Huang et al. , 2023 ) , but we perform it once, offline , and retain only the final verified solutions as training data."},{"type":"para","text":"Algorithm 1 summarizes the complete data synthesis procedure."},{"type":"heading","level":3,"text":"3.2.3. Seed Selection."},{"type":"para","text":"We initialize the pipeline with language independent algorithmic problems sourced from [dataset]. These problems are specified via natural-language descriptions and IO test pairs, ensuring no dependence on any specific programming language."},{"type":"heading","level":3,"text":"3.2.4. Generation Prompt."},{"type":"para","text":"Each problem is presented to the base model M θ M_{\\theta} using a structured system prompt that constrains the output format and encourages reasoning before coding. Key design decisions include: (1) requiring step-by-step reasoning in comments before the solution, mirroring chain-of-thought prompting; (2) enforcing a strict output structure with separate <CODE> and <TESTS> blocks to facilitate automated extraction and validation; (3) mandating a main() function that reads from standard input, enabling execution against IO test suites; and (4) requesting at least four unit test assertions per problem to provide additional verification signal."},{"type":"heading","level":3,"text":"3.2.5. Worked Example."},{"type":"para","text":"Figure 3 illustrates the iterative refinement process on a concrete Julia problem: writing a function to sum the digits of a number. The base model’s initial attempt produces syntactically valid Julia but fails all four test cases, a common pattern where models transfer Python semantics (e.g., integer division behavior or string-conversion idioms) that do not generalize to Julia. The sandbox feeds back the test failure summary ( 0 passed, 4 failed ), which is appended to the prompt. In the second iteration, the model self-corrects by using Julia’s idiomatic string / abs /character-arithmetic approach, producing a solution that passes all tests. This single-retry success demonstrates that even one round of compiler-and-test feedback is often sufficient to bridge the gap between high-resource habits and LRPL-correct code. The final, verified solution is added to 𝒟 L \\mathcal{D}_{L} ."},{"type":"heading","level":2,"text":"3.3. Phase 2: Syntax-Aware Supervised Fine-Tuning"},{"type":"para","text":"Using the synthesized dataset 𝒟 L \\mathcal{D}_{L} , we fine-tune the pretrained base model M θ M_{\\theta} using standard causal language modeling. The SFT objective minimizes the cross-entropy loss over verified prompt-completion pairs:"},{"type":"para","text":"where π θ \\pi_{\\theta} denotes the model’s output distribution."},{"type":"para","text":"This phase serves a specific purpose: because LRPLs possess idiosyncratic syntaxes and niche standard libraries that differ substantially from high-resource languages, SFT embeds strong structural and grammatical priors into the model. Consider the differences between Julia and Python that a model must internalize: Julia uses 1-based indexing, true / false instead of True / False , function / end blocks instead of indentation-delimited scopes, multiple dispatch instead of class-based OOP, and a distinct standard library (e.g., sort(arr, rev=true) vs. sorted(arr, reverse=True) ). After this phase, the model can reliably produce syntactically valid Julia code structures before being required to optimize for complex algorithmic logic in the subsequent RL phase."},{"type":"para","text":"For Ballerina, the syntactic gap is even larger: the language uses a unique type system with union types ( int|string ), built-in concurrency via workers, explicit error handling with check expressions, and a service-oriented programming model unlike any high-resource language. The SFT phase is therefore essential for Ballerina, as the base model has effectively zero prior exposure to these constructs."},{"type":"para","text":"In practice, each training example in 𝒟 L \\mathcal{D}_{L} is formatted as a prompt-completion pair: the prompt contains the natural-language problem description along with a language-specific instruction header (e.g. “Write a Julia function that…”), and the completion is the verified solution from Phase 1. During fine-tuning, we apply completion-only loss masking , computing the cross-entropy loss exclusively over the solution tokens rather than the prompt tokens, which focuses the learning signal on code generation rather than problem comprehension."},{"type":"heading","level":2,"text":"3.4. Phase 3: RLVR with Verifiable Rewards with Difficulty-curated Data"},{"type":"heading","level":3,"text":"3.4.1. Data Curation by Difficulty"},{"type":"para","text":"With a robust syntax prior established, we transition to exploration via Reinforcement Learning. Crucially, we initialize the GRPO policy with the syntax-aware SFT model M θ sft M_{\\theta_{\\text{sft}}} from Phase 2, rather than the base model M θ M_{\\theta} ."},{"type":"heading","level":3,"text":"3.4.2. GRPO Objective."},{"type":"para","text":"For a given problem p p , we sample a group of G G candidate completions { c 1 , … , c G } \\{c_{1},\\ldots,c_{G}\\} from the current policy π θ \\pi_{\\theta} . Each completion receives a reward r i r_{i} from the execution-based oracle. We compute group-relative advantages:"},{"type":"para","text":"where μ G \\mu_{G} and σ G \\sigma_{G} are the mean and standard deviation of rewards within the group, and ϵ \\epsilon is a small constant for numerical stability. The policy is updated to maximize:"},{"type":"para","text":"where π ref = π θ sft \\pi_{\\text{ref}}=\\pi_{\\theta_{\\text{sft}}} is the SFT model serving as the reference policy, and β \\beta controls the strength of the KL divergence penalty that prevents the policy from deviating too far from the syntactic prior."},{"type":"heading","level":3,"text":"3.4.3. Reward Function."},{"type":"para","text":"Each completion c c receives a composite reward that decomposes into a test score r test r_{\\text{test}} and a build-quality score r build r_{\\text{build}} , reflecting the two distinct failure modes in LRPL code generation: syntactic/compilation failures and logical/algorithmic failures."},{"type":"para","text":"The test score is proportional to the number of passed test cases p p :"},{"type":"para","text":"This partial reward provides a gradient signal even for completions that solve only a subset of test cases, avoiding the sparsity of binary pass/fail rewards."},{"type":"para","text":"The build-quality score r build r_{\\text{build}} captures whether the completion compiles and produces well-structured code. If the code fails to compile, r build r_{\\text{build}} receives a small positive value inversely proportional to the number of compiler errors, rewarding completions that are “closer” to compiling. If compilation succeeds, r build r_{\\text{build}} receives a fixed credit regardless of test outcomes. Additionally, a structure penalty is applied based on the presence of expected code constructs (function definitions, test blocks, assertions), acting as a negative offset that is reduced for well-formed outputs. This ensures that structural compliance alone cannot produce a high reward, only test-passing code is strongly reinforced. The build-quality score is down-weighted relative to the test score, ensuring that functional correctness dominates the learning signal:"},{"type":"para","text":"where α < 1 \\alpha<1 controls the relative weight of the build-quality shaping signal."},{"type":"heading","level":3,"text":"3.4.4. Zero-Advantage Masking."},{"type":"para","text":"A key challenge in GRPO for LRPLs is that groups may contain no meaningfully correct completion, all candidates fail most tests, producing near-uniform low rewards. Naively computing advantages in such groups reinforces the “least bad” completion, causing the policy to converge on sub-optimal heuristics (e.g. hardcoding common outputs). We address this via zero-advantage masking : if no completion within a group achieves the maximum reward, we set A ^ i = 0 \\hat{A}_{i}=0 for all i i in that group, effectively discarding the update. This ensures the policy only learns from groups containing at least one genuinely strong completion, preventing noisy gradients from degrading the SFT prior."},{"type":"para","text":"Formally, let r max = max i ∈ G ( r i ) r_{\\max}=\\max_{i\\in G}(r_{i}) denote the highest reward in a group. We define the masked advantage as:"},{"type":"para","text":"Substituting A ~ i \\tilde{A}_{i} into Eq. 5 yields the masked GRPO objective:"},{"type":"para","text":"When r max < 1.0 r_{\\max}<1.0 , the policy gradient term vanishes entirely for that group, since we set β = 0 \\beta=0 similar to DAPO (Yu et al. , 2025 ) the entire sample is ignored."},{"type":"heading","level":3,"text":"3.4.5. Data Curation by Difficulty"},{"type":"para","text":"As discussed in the introduction, we aim to create a curated dataset for GRPO that can maximize learning."},{"type":"para","text":"We use the ELO rating used by the Codeforces platform (see Quan et al. (Quan et al. , 2025 ) to measure problem difficulty, where both humans, models, and problems are given ELO ratings, which get adjusted using outcomes from humans and models attempting problems. Using the problem and ELO ratings data available from the Codeforces platform, we have built a model to predict the ELO rating given a problem."},{"type":"para","text":"For GRPO, we use data that has IO-based ground truth as discussed by Agnostics paper, where we convert ground truth examples to test cases and then give feedback to GRPO using those tests as described in the subsection 3.4.3 ,"},{"type":"para","text":"Given reward function r t e s t r_{test} , model p h i r e f \\ phi_{ref} , and a dataset D D , that comprises a collection of problems, the ELO rating for the problem, and groundtruth data, we build a GRPO dataset as follows."},{"type":"para","text":"This defines E L O m ELO_{m} as the supremum (the highest point) of problem difficulty where the model maintains a stable reward:"},{"type":"para","text":"When d . e l o d.elo is the elo rating of problem d d , curated dataset is defined as follows."},{"type":"heading","level":3,"text":"3.4.6. Why SFT Initialization Matters."},{"type":"para","text":"The interaction between Phase 2 and Phase 3 is the central design choice of our pipeline. GRPO works by generating candidate solutions, computing relative advantages from execution feedback, and adjusting the output distribution to favor better solutions. However, if all solutions within a group receive the same reward (e.g. all compile but no test passes), the advantages collapse to zero and GRPO cannot learn. When GRPO is initialized from the base model without an SFT prior, the vast majority of sampled completions contain syntax errors and receive near-zero rewards dominated by the build-quality component. With no test-passing signal to differentiate completions, the advantages within each group lack meaningful variance and the policy stagnates. In contrast, the SFT-initialized policy produces syntactically valid completions that compile and execute, even if they produce wrong outputs. This means the reward distribution within each group has meaningful variance, some completions pass more tests than others, providing a rich gradient signal. This enables GRPO to learn effectively, and the compute budget is thus spent exploring functional logic rather than rediscovering basic language syntax."},{"type":"heading","level":1,"text":"4. Experimental Setup"},{"type":"heading","level":2,"text":"4.1. Base Model"},{"type":"para","text":"We use Qwen3-8B (Qwen Team, 2025 ) as our base SLM. We select this model for four reasons: (1) it demonstrates strong baseline performance on high-resource code generation, providing a capable starting point; (2) at 8B parameters, it represents a realistic model size for on-premise deployment on a single GPU; (3) its training data cutoff aligns with the LiveCodeBench v5 temporal split, enabling contamination-free evaluation on Ag-LCB; and (4) supports a fair comparison with the previous work, which reported against the same model. While more widely adopted recent models such as the Qwen3.5 family and DeepSeek-V3.2 series demonstrate stronger performance on high-resource languages, their later training cutoffs risk data contamination against our evaluation benchmarks."},{"type":"heading","level":2,"text":"4.2. Target Languages"},{"type":"para","text":"We evaluate on two languages that represent different points on the resource spectrum:"},{"type":"para","text":"Julia is designed for high-performance scientific computing. Despite a growing community, Julia remains underrepresented in LLM training corpora, CodeNet contains only 10.8k Julia samples versus 3.2M for Python. Julia’s syntax combines familiar imperative structures with distinctive features: 1-based array indexing, multiple dispatch as the core paradigm, function / end block delimiters, and a rich type system with parametric types. These features make Julia a representative “moderate LRPL”, present in pretraining data but insufficiently represented for reliable code generation."},{"type":"para","text":"Ballerina is a cloud-native programming language for building distributed applications and microservices. It features a unique type system with structural typing, union types ( int|string|error ), built-in concurrency via workers, first-class support for network services, and explicit error handling with check expressions. Ballerina has near-zero representation in LLM pretraining corpora, making it our “extreme LRPL” test case, a language where the base model has effectively no prior knowledge of the syntax or standard library."},{"type":"heading","level":2,"text":"4.3. Datasets"},{"type":"para","text":"We source seed problems from the Stanford Alpaca dataset (Taori et al. , 2023 ) , selecting language-independent algorithmic tasks. For each problem and target language, we generate candidate solutions using Claude Code or OpenAI OSS-120B based on it’s difficulty, then verify each candidate by executing its accompanying test suite. Only problems whose generated solutions pass all test cases are retained in the SFT dataset 𝒟 L \\mathcal{D}_{L} . This process is applied identically for both Julia and Ballerina, yielding fewer than 1,000 verified examples per language."},{"type":"para","text":"For the RL phase, we curate approximately 500 training problems from Codeforces (filtered to ≤ \\leq 1,300 ELO) and CodeNet (Puri et al. , 2021 ) . Each problem is paired with language-agnostic IO test suites that serve as the verifiable reward signal during GRPO training."},{"type":"heading","level":2,"text":"4.4. Benchmarks"},{"type":"heading","level":3,"text":"4.4.1. MultiPL-E"},{"type":"para","text":"(Cassano et al. , 2023 ) has translated HumanEval (Chen et al. , 2021 ) problems to 18+ programming languages, enabling standardized cross-lingual evaluation. We report pass@1 on the Julia split. As part of this work, we extend MultiPL-E to include Ballerina by manually translating 159 HumanEval problems into idiomatic Ballerina with corresponding test harnesses. Unlike automated translation approaches, our extension is entirely human-annotated to ensure correctness and idiomatic usage of Ballerina’s distinctive features (e.g. structural typing, check expressions, union types). This constitutes the first standardized code generation benchmark for Ballerina, which we release to support future research on emerging LRPLs. MultiPL-E problems are relatively straightforward, testing basic language competence and standard library knowledge."},{"type":"heading","level":3,"text":"4.4.2. Ag-LCB"},{"type":"para","text":"(Boruch-Gruszecki et al. , 2026 ) is the LiveCodeBench (Jain et al. , 2024 ) evaluation adapted through the Agnostics framework, using problems published after the base model’s training cutoff to ensure contamination-free assessment. These problems are substantially harder than MultiPL-E, reflecting competitive programming difficulty with complex algorithmic reasoning requirements. We report pass@1 for both Julia and Ballerina."},{"type":"heading","level":2,"text":"4.5. Baselines"},{"type":"para","text":"We compare the following configurations to isolate the contribution of each pipeline component:"},{"type":"para","text":"Qwen3-8B (Base) : The unmodified pretrained model, establishing the baseline performance."},{"type":"para","text":"+ SFT Only : Phase 2 only, the base model fine-tuned on our compiler-feedback synthetic data, without the subsequent RL phase. This isolates the contribution of syntax-aware SFT."},{"type":"para","text":"Ours (Full Pipeline) : The complete three-phase pipeline (Phase 1 data synthesis + Phase 2 SFT + Phase 3 GRPO)."},{"type":"heading","level":2,"text":"4.6. Training Details"},{"type":"para","text":"We fine-tune Qwen3-8B using a two-stage pipeline: Supervised Fine-Tuning (SFT) followed by GRPO reinforcement learning. Both stages utilize 4-bit quantization and LoRA adapters ( r = 16 , α = 32 r=16,\\alpha=32 ) on all linear layers via the Unsloth-TRL framework. SFT is conducted for 4 epochs with a 1e-4 learning rate and a 16,384 token context window. Subsequently, we perform 1 epoch of GRPO using curated data from Section 3.4.5 , employing 8 generations per prompt ( G = 8 G=8 ) and a 5e-5 learning rate. Both phases use the AdamW optimizer, a cosine schedule, and bf16 mixed precision, with the RL phase capping prompt and completion lengths at 4,096 tokens each."},{"type":"heading","level":1,"text":"5. Results"},{"type":"heading","level":2,"text":"5.1. Main Results"},{"type":"para","text":"Table 1 summarizes the pass@1 results for Julia and Ballerina across both benchmarks. For each language, we report the base model, the SFT-only variant, and the full three-phase pipeline. Since Agnostics (Boruch-Gruszecki et al. , 2026 ) already provides a thorough evaluation of RL-only training across multiple configurations, we include the RL-only variant solely as an ablation under our specific setup (Section 6 ). The key finding is that each successive phase yields additive gains: SFT alone substantially improves over the base model by embedding syntactic priors, and GRPO further boosts performance by optimizing functional correctness. We discuss each language in detail below."},{"type":"heading","level":3,"text":"5.1.1. Julia."},{"type":"para","text":"Compared to vanilla Qwan mode baseline, our pipeline improves Julia pass@1 from 28.3% to 68.6% (+40.3 points) on MultiPL-E and from 9% to 39.2% (+30 points) on Ag-LCB. Table 2 places these results in context. Most importantly, we surpass the previous state of the art (Agnostics (Boruch-Gruszecki et al. , 2026 ) ) by 7.6 percentage points on MultiPL-E and 14.2 percentage points on Ag-LCB, despite using a single base model rather than separate task-specific variants."},{"type":"heading","level":3,"text":"5.1.2. Ballerina."},{"type":"para","text":"For the extreme-LRPL case, SFT increased ballerina scores 4.4% to 56% and LCB scores by 2.9% to 9.9%. GRPO training using a curated dataset increased ballerina scores for LCB from 9.9% to 25%, but reduced the MultiPL-E results to 49.7%. We believe this is overall positive given that LCB is a much harder benchmark"},{"type":"para","text":"Applying the pipeline to Ballerina required minimal language-specific modifications: (1) a working Ballerina compiler and runtime for sandbox execution, (2) translation of seed problem descriptions to specify Ballerina-specific IO formats where necessary, and (3) no changes to the GRPO configuration, as IO tests are inherently language-agnostic. The fact that our pipeline produces functional code generation capability from near-zero provides strong evidence that, given a compiler and language-agnostic problems, any language can be bootstrapped."},{"type":"heading","level":3,"text":"5.1.3. Data and Cost Efficiency."},{"type":"para","text":"These results are achieved with fewer than 1,000 SFT samples and approximately 500 GRPO training problems, a 3 × 3\\times reduction in training data compared to Agnostics, which uses ∼ {\\sim} 5,000 GRPO samples. The cost advantage is equally striking: Agnostics requires $320.3 ( 13 h × 8 GPUs × $ 3.08 / GPU-h 13\\text{h}\\times 8\\,\\text{GPUs}\\times\\mathdollar 3.08/\\text{GPU-h} ), whereas our total cost is $54.02, comprising fine-tuning ( 45 h × $ 0.42 / GPU-h 45\\text{h}\\times\\mathdollar 0.42/\\text{GPU-h} ), OpenAI GPT OSS-120B API calls ( 5 M input × $ 0.25 / 1M 5\\text{M input}\\times\\mathdollar 0.25/\\text{1M} + 3 M output × $ 0.60 / 1M +\\;3\\text{M output}\\times\\mathdollar 0.60/\\text{1M} ), and Sonnet API calls ( 4.1 M input × $ 3 / 1M 4.1\\text{M input}\\times\\mathdollar 3/\\text{1M} + 1.3 M output × $ 15 / 1M +\\;1.3\\text{M output}\\times\\mathdollar 15/\\text{1M} ). Our pipeline thus costs only 1 6 \\frac{1}{6} of Agnostics. The key enablers are left-shifting inference-time compute to offline data synthesis and training with a compact, difficulty-filtered RL set. Furthermore, by left-shifting, rather than spending GPU hours on large-scale RL exploration, we invest a one-time API cost to curate high-quality SFT data."},{"type":"heading","level":2,"text":"5.2. Error Category Analysis"},{"type":"para","text":"To understand why each pipeline phase improves performance, we categorize errors from the base model, SFT-only model, and full pipeline on MultiPL-E problems for Julia."},{"type":"para","text":"Table 3 reveals a clear progression. The base model’s failures are dominated by syntax and compilation errors, the model frequently produces Python-influenced constructs that fail to compile in Julia. After SFT, syntax errors drop substantially, but runtime errors and wrong outputs increase, indicating that the model now produces valid Julia code that compiles but does not yet implement correct algorithms. The full pipeline further reduces wrong-output errors, as the GRPO phase optimizes for functional consistency within the space."},{"type":"heading","level":1,"text":"6. Ablation Studies"},{"type":"para","text":"As ablations to difficulty-based data curation, we performed two experiments with Julia. For that purpose, in addition to the curated dataset, we created a random difficulty dataset by sampling 500 problems from the full ELO range (spanning ELO 800–2200). This acts as a control dataset as opposed to our curated dataset."},{"type":"para","text":"As the first experiment, we started with the vanilla Qwan model (no-SFT) and then fine-tuned it with GRPO using the random difficulty dataset. As the second experiment, we started with the Qwan model already fine-tuned with SFT, and further fine-tuned it with GRPO using the random difficulty dataset."},{"type":"para","text":"In the following discussion, we compare our default pipeline (Qwan model + SFT + GRPO with curated dataset)."},{"type":"heading","level":2,"text":"6.1. Default pipeline vs. experiment 1 (qwan + GRPO with random dataset)"},{"type":"para","text":"The Figure 5 plots compares GRPO test reward against difficulty (ELO rating). The left-hand side shows our pipeline (Qwan + SFT + GRPO with curated dataset) produces remarkably stable reward signals: mean between 0.4 and 0.5 across the training ELO range (800- 1250), and max reward stays near 1.0, indicating that at least one completion per GRPO group fully solves each problem. This confirms our default pipeline provides effective rewards."},{"type":"para","text":"In contrast, the right-hand side of the Figure showing data from experiment 1 (no SFT and GRPO random dataset) exhibits substantially higher variance in both mean and max reward, particularly as problem difficulty increases beyond ELO 1400. The growing error bars reflect the instability of RL exploration when the policy must simultaneously discover language syntax and algorithmic logic."},{"type":"heading","level":2,"text":"6.2. Default pipeline vs. experiment 2 (qwan + SFT + GRPO with random dataset)"},{"type":"para","text":"Table 4 compares the Pass@1 scores between SFT mode trained with GRPO using curated vs random datasets. Random baseline yields 52.2% pass@1, 16.4 percentage points below the difficulty-curated variant. These results confirm that for low-resource languages, curating RL training data by difficulty amplifies the effectiveness of GRPO by concentrating training signal on problems at the frontier of model capability."},{"type":"heading","level":1,"text":"7. Discussion"},{"type":"heading","level":2,"text":"7.1. Zero-advantage Masking"},{"type":"para","text":"Figure 4 illustrates this phenomenon empirically. After zero-advantage masking, the vast majority of training samples are discarded, visible as long stretches where σ G = 0 \\sigma_{G}=0 . This occurs when all completions within a group either uniformly succeed or uniformly fail, collapsing the advantage to zero. The isolated spikes where σ G > 0 \\sigma_{G}>0 represent the small fraction of samples that actually produce a meaningful gradient signal, problems at the edge of the model’s current capability. This sparse pattern highlights the inefficiency of training without difficulty-based curation: most computational effort is spent on samples that contribute nothing to learning."},{"type":"heading","level":2,"text":"7.2. New Languages Require the SFT Phase"},{"type":"para","text":"The Ballerina experiment validates that the SFT phase is essential for severely low-resource programming language (S-LRPL) adoption. Without syntactic priors, the RL exploration space is too unconstrained: the model spends its compute budget rediscovering basic syntax rather than learning algorithmic patterns. This suggests that for any new LRPL, investing in Phase 1 data synthesis and Phase 2 SFT is a prerequisite before RL can be effective. The pipeline thus provides a practical recipe: given a compiler for language L L and a set of language-agnostic algorithmic problems, one can bootstrap code generation capability for L L through offline data synthesis followed by SFT and GRPO."},{"type":"heading","level":2,"text":"7.3. Right Level of Difficulty Amplifies Pipeline Benefits"},{"type":"para","text":"Our difficulty-based curation results echo the principle of deliberate practice (Ericsson, 2008 ) : learning is most effective when training is concentrated at the boundary of current competence. The mechanism is intuitive within the GRPO framework. Given a problem, GRPO generates multiple candidate solutions and updates the policy based on the relative advantage between them. If a problem is too easy, all candidates succeed and the advantage collapses to zero; if too hard, all candidates fail with the same outcome. Only when problems lie at the edge of the model’s capability, where some candidates succeed and others fail, does the advantage carry a meaningful gradient signal."},{"type":"para","text":"This follows directly from the GRPO advantage (Eq. 4 ). With binary pass/fail rewards r i ∈ { 0 , 1 } r_{i}\\in\\{0,1\\} :"},{"type":"para","text":"Too easy ( r i = 1 ∀ i r_{i}=1\\;\\forall\\,i ): μ G = 1 , σ G = 0 ⟹ A ^ i = 0 \\mu_{G}=1,\\;\\sigma_{G}=0\\implies\\hat{A}_{i}=0 ."},{"type":"para","text":"Too hard ( r i = 0 ∀ i r_{i}=0\\;\\forall\\,i ): μ G = 0 , σ G = 0 ⟹ A ^ i = 0 \\mu_{G}=0,\\;\\sigma_{G}=0\\implies\\hat{A}_{i}=0 ."},{"type":"para","text":"Edge of capability (mixed pass/fail): σ G > 0 \\sigma_{G}>0 , so correct solutions receive A ^ i > 0 \\hat{A}_{i}>0 and incorrect ones A ^ i < 0 \\hat{A}_{i}<0 , producing a meaningful gradient signal."},{"type":"para","text":"This provides a mathematical justification for difficulty-based data curation: selecting problems near the model’s decision boundary maximizes the effective gradient from each GRPO update. This principle extends beyond LRPLs; Zhang et al. (Zhang et al. , 2025 ) independently demonstrate that dynamically alternating between SFT and RL enables automatic weakness identification and targeted resource allocation, further corroborating the value of difficulty-aware training."},{"type":"heading","level":2,"text":"7.4. Compute Allocation: Offline vs. Online"},{"type":"para","text":"Our results contribute to the broader debate about where to invest compute for code generation. The field currently allocates substantial resources to inference-time scaling, multiple generation rounds, self-correction loops, and agent-based frameworks (Huang et al. , 2023 ; Islam et al. , 2025 ) . Our pipeline demonstrates that the same iterative refinement mechanism yields strictly greater returns when deployed offline as a data synthesis engine. The key difference is amortization: offline synthesis invests compute once to produce training data, while inference-time scaling imposes a per-query cost. To illustrate, consider an IDE code-completion plugin serving 1,000 Julia queries per day. An inference-time scaling approach requiring five rounds of refinement per query would consume 5 × 5\\times the compute of a single forward pass on every query, accumulating substantial cost over weeks. In contrast, our pipeline pays a fixed one-time cost of approximately $54 for data synthesis and fine-tuning, after which every query is served at single-pass inference cost. The crossover point is reached within days of deployment, and the cost advantage compounds indefinitely. Moreover, the offline synthesis cost is dominated by API calls for the initial data generation, a component whose price continues to decline as model serving becomes more efficient, making the approach increasingly economical over time."},{"type":"heading","level":2,"text":"7.5. Practical Deployment Considerations"},{"type":"para","text":"At 8B parameters, the resulting model can run on a single consumer GPU, making it suitable for on-premise deployment where organizations cannot send proprietary code to external APIs. Combined with the pipeline’s language-agnostic design, this enables organizations to bootstrap code generation for internal or niche programming languages without relying on cloud-based LLM services."},{"type":"heading","level":1,"text":"8. Conclusion and Future work"},{"type":"para","text":"We presented a three-phase pipeline for adapting Small Language Models to Low-Resource Programming Languages that reframes inference-time scaling as offline data synthesis. By decoupling syntax acquisition (via compiler-feedback SFT) from algorithmic reasoning (via GRPO with IO tests), our pipeline resolves the trilemma of data scarcity, inference cost, and sparse RL rewards. Furthermore, by curating the data for GRPO use, we amplify and stabilize the reward signal. Applied to Qwen3-8B, it exceeded state-of-the-art results on MultiPL-E (+7.6 pass@1) and Ag-LCB (+14.2 pass@1) for Julia. Furthermore, the pipeline generalizes to Ballerina, a language with near-zero pretraining representation, achieving Pass@1 56% with MultiPL-E and 25% with LCB."},{"type":"para","text":"Our ablation studies confirm that iterative compiler-feedback data synthesis is essential for high-quality SFT, as it stabilizes the reward, eliminates the subtle errors present in naively generated synthetic data, and yields a cleaner training signal."},{"type":"para","text":"Future research will focus on scaling the pipeline across diverse model architectures and expanding its support for a wider range of low-resource programming languages. We plan to investigate the synergy between continued pretraining and SFT, develop automated metrics for code idiomaticity, and integrate offline data synthesis with inference-time refinement to further boost performance."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Large Language Models (LLMs) have achieved remarkable proficiency in code generation for high-resource programming languages(HRPLs) like Python and Java (Chen et al. , 2021 ) . However, their performance sharply declines when applied to Low-Resource Programming Languages (LRPLs) (Cassano et al. , 2023 ; Joel et al. , 2024 ) . This disparity is not merely academic: languages such as Julia (scientific computing) and Ballerina (cloud-native microservices) serve critical roles across diverse software engineering domains. As the ecosystem of programming languages continues to grow, with new languages emerging to address specialized needs, enabling reliable code generation for LRPLs is an increasingly pressing challenge for the software engineering community, mirroring the historical trajectory of low-resource Natural Language Processing (Ranathunga et al. , 2023 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Enhancing Small Language Models (SLMs) for LRPLs presents a trilemma of distinct challenges. First , Supervised Fine-Tuning (SFT) is highly effective for smaller parameter models (Giagnorio et al. , 2025 ) , but it is fundamentally bottlenecked by severe data scarcity. For example, datasets like CodeNet (Puri et al. , 2021 ) contain only 10.8k Julia samples compared to 3.2M for Python. Second , Test-time Scaling (e.g. iterative self-correction, reflection-based refinement) effectively boosts performance (Wang et al. , 2025 ; Ren and others, 2024 ) , but introduces unacceptable latency and computational overhead for real-world deployments. Third , applying Reinforcement Learning (RL) directly to a base model suffers from sparse rewards: e.g., the model generates so many syntax errors that it rarely receives 4 positive feedback to learn from.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper uses on two key hypotheses to improve SLMs for LRPLs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our first hypothesis is that inference-time scaling and supervised fine-tuning are not competing strategies but can be complementary phases of a single pipeline . Rather than deploying computationally expensive iterative refinement during live usage, we can “left-shift” this computation, repurposing it strictly as an offline data curation engine. By utilizing iterative compiler and test-case feedback, where compilation errors or a single failed test case showing the expected versus actual output are fed back to the model for self-correction, we synthesize missing ground-truth data to enable high-quality SFT.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A central challenge in this pipeline is ensuring the quality of the synthetically generated data: an LLM can produce fluent but semantically incorrect code, and training on such examples would embed faulty priors. We address this through a flexible verification strategy adapted to the available resources. When a ground-truth solution exists in a high-resource language (e.g. Python), we can employ differential testing , executing both the candidate LRPL solution and the reference implementation on shared inputs and comparing their outputs. Differential testing is preferred when available because it catches subtle semantic divergences that fixed IO test suites may miss, such as floating-point precision differences, locale-dependent string formatting, or edge cases in boundary conditions that the original test suite does not explicitly cover. When no reference solution is available, we fall back to input/output test verification , running the candidate against a suite of I/O test cases derived from the problem specification. In practice, the majority of our seed problems originate from competitive programming datasets that provide IO test suites but not reference solutions in the target LRPL, making IO verification the dominant verification path in our experiments. This two-tiered approach is deliberately pragmatic: the verification method is determined entirely by what artifacts are available for a given problem, requiring no additional human annotation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our second hypothesis is that the improvement achieved via Reinforcement Learning is dependent on the difficulty of the problems it gets feedback on . With very easy problems, there is little to learn, and with very hard problems, the model’s answers will all be wrong; thus, the model will not know how to make progress. Problems with the correct difficulty can speed up the learning. For example, Pikus et al. (Pikus et al. , 2025 ) have also observed that hard samples help with GRPO learning. This idea can also be seen as a variation of active learning (Ren et al. (Ren et al. , 2021 ) ), where we actively select the right data to be included in the training set to maximize the model outcome.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Building on this idea, we create a dataset for Reinforcement Learning with Verifiable Rewards (RLVR) training by using difficulty scores (ELO rating for problems) to curate the data. Subsequently, we apply RLVR via Group Relative Policy Optimization (GRPO) (Shao et al. , 2024 ) , directly adopting the language-agnostic IO-based evaluation framework proposed by Boruch-Gruszecki et al. (Boruch-Gruszecki et al. , 2026 ) . To maximize the learning signal, we employ partial rewards proportional to the fraction of test cases passed and successful compilation, allowing the advantage estimates to capture fine-grained differences between completions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As we will see in our results analysis, because the SFT prior constrains the action space away from trivial syntax errors, the RL phase can efficiently navigate the richer landscape of functional logic. Furthermore, to prevent the policy from converging to sub-optimal states, we use zero-advantage masking (Chen et al. , 2025 ) : if no completion within a group achieves the maximum reward, the entire group’s advantage is set to zero, effectively discarding the update. This avoids reinforcing mediocre solutions and ensures that the policy only updates toward genuinely strong completions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate our pipeline on Qwen3-8B (Qwen Team, 2025 ) using two benchmarks: MultiPL-E (Cassano et al. , 2023 ) and the Agnostics LiveCodeBench (Ag-LCB) (Boruch-Gruszecki et al. , 2026 ) . For Julia, our full pipeline improves pass@1 by +24.6 points on MultiPL-E and +30.2 points on Ag-LCB. For Ballerina, where the base model has near-zero prior knowledge, we observe +45.3 points improvement on MultiPL-E and +22.1 points on Ag-LCB over the baseline. Notably, the SFT phase alone achieves on-par results with the current state-of-the-art, demonstrating that left-shifted inference-time compute can match expensive online scaling strategies through a one-time offline investment. Furthermore, validating our second hypothesis, we achieved the above results while using 1 3 \\\\frac{1}{3} of the data and 1 6 \\\\frac{1}{6} of the cost.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further demonstrate that the pipeline generalizes to Ballerina, a cloud-native programming language for integration and microservices that has near-zero representation in LLM pretraining corpora. Unlike Julia, which benefits from some incidental coverage in training data, Ballerina presents a true cold-start scenario: the base model has effectively no prior exposure to its syntax, type system, or standard library. Despite this, our pipeline bootstraps functional Ballerina code generation by relying solely on compiler feedback and language-agnostic test suites, requiring no human-written Ballerina training examples. This result provides strong evidence that any LRPL can be bootstrapped making the pipeline immediately applicable as new programming languages emerge.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose a three-phase pipeline, offline inference scaling via iterative compiler and test feedback, SFT, and GRPO with a dataset curated by difficulty, that achieves state-of-the-art results on MultiPL-E and Ag-LCB for Julia and Ballerina using an 8B-parameter model, with much fewer training rounds and lower cost.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate that left-shifting inference-time computation to offline data synthesis resolves the data scarcity bottleneck without human annotation, turning expensive iterative refinement into a one-time investment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We show that the pipeline generalizes to an extreme LRPL (Ballerina) with minimal adaptation, establishing a practical recipe for bootstrapping code generation for any LRPL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We extend the MultiPL-E benchmark for Ballerina, adding a new language to the evaluation suite and providing a standardized testbed for future research on LRPL code generation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Benchmarking LLMs on LRPLs relies on frameworks adapted from high-resource environments, such as MultiPL-E (Cassano et al. , 2023 ) and Agnostic-LiveCodeBench (Jain et al. , 2024 ) . While these benchmarks effectively measure performance, improving that performance requires navigating severe data limitations and choosing optimal learning strategies. We organize prior work along three axes that correspond to the phases of our pipeline.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1. SFT and the Challenge of Data Scarcity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Supervised Fine-Tuning remains the most prevalent strategy for adapting models to LRPLs. Cassano et al. (Cassano et al. , 2024 ) introduced MultiPL-T, a pipeline for translating training data from high-resource to low-resource languages, demonstrating significant gains for models up to 8B parameters. Giagnorio et al. (Giagnorio et al. , 2025 ) further showed that smaller models ( ∼ \\\\sim 1B parameters) trained on specialized, domain-specific data often outperform larger, general-purpose models, as SFT can inversely degrade larger models better suited for Retrieval-Augmented Generation. WizardCoder (Luo et al. , 2024 ) and OpenCoder (Huang et al. , 2024 ) advanced SFT through instruction-tuning and open-source training recipes, though their focus remains on high-resource languages. Joel et al. (Joel et al. , 2024 ) provide a comprehensive survey of LLM-based code generation for low-resource and domain-specific languages.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fundamental bottleneck for SFT is data scarcity. Datasets like CodeNet (Puri et al. , 2021 ) provide a foundation, but the available data for LRPLs pales in comparison to high-resource languages. Utilizing organic data requires rigorous curation, including deduplication and strict decontamination to prevent benchmark leakage. To bypass organic data limitations, recent research emphasizes synthetic data pipelines: Magicoder (Wei et al. , 2024 ) leverages OSS-Instruct to generate training data from open-source snippets, while KODCODE (Xu et al. , 2025 ) constructs a diverse, verifiable synthetic dataset. Our work extends this line by introducing compiler/test feedback driven data synthesis, where iterative execution and error correction provides verification that purely generative pipelines lack.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2. Reinforcement Learning and Grounding\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond SFT, Reinforcement Learning is establishing new state-of-the-art results for code generation. Early approaches used Proximal Policy Optimization (PPO): CodeRL (Le et al. , 2022 ) combined pretrained models with execution-based RL rewards, while Shojaee et al. (Shojaee et al. , 2023 ) demonstrated that execution-based feedback outperforms static heuristics for policy optimization. More recently, Group Relative Policy Optimization (GRPO) (Shao et al. , 2024 ) has emerged as a simpler alternative that eliminates the need for a separate value network. CodeReasoner (Tang et al. , 2025 ) applies GRPO to enhance code reasoning in general-purpose models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For LRPLs specifically, recent techniques rely on language agnostic datasets and test-feedback loops. Concurrently, self-play frameworks enable iterative refinement without human annotation (Lin et al. , 2025 ) . However, ungrounded self-play often yields sub-optimal results. To address this problem, Boruch-Gruszecki et al. (Boruch-Gruszecki et al. , 2026 ) introduced Agnostics, a universal RL environment capable of learning to synthesize code in any programming language via execution-based rewards.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ma et al. (Ma et al. , 2026 ) demonstrate that interleaving online fine-tuning with RL addresses the hardest questions that reinforcement learning alone cannot solve, highlighting the complementary nature of supervised and RL-based training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our work grounds RL with IO tests similar to Boruch-Gruszecki et al. and, crucially, initializes the RL policy from a syntax-primed SFT model rather than the base model, which dramatically reduces the sparse-reward problem.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3. Test-time Scaling and Its Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To compensate for limited training data, the current state-of-the-art often allocates more computational resources during execution, a paradigm known as Test-time Scaling. A prominent family of techniques uses iterative refinement : the model generates a candidate solution, receives feedback (compiler errors, test failures, or self-reflection), and refines its output across multiple rounds. ReflectionCoder (Ren and others, 2024 ) leverages reflection sequences to iteratively correct one-off generations, while Origen (Cui and others, 2024 ) uses code-to-code augmentation with self-reflection for RTL code generation. AgentCoder (Huang et al. , 2023 ) employs a multi-agent framework where a programmer agent, test designer agent, and test executor agent collaborate through iterative testing and optimization. CodeSim (Islam et al. , 2025 ) extends this paradigm with simulation-driven planning and debugging.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Another family uses selection-based scaling : CodeT (Chen et al. , 2022 ) generates multiple candidate solutions along with tests and selects the best via dual execution agreement. CodeChemist (Wang et al. , 2025 ) applies functional knowledge transfer with test-time scaling specifically for low-resource code generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While these methods form a critical pillar of modern code generation, their high computational cost at inference time makes them impractical for efficient SLM deployment. Our work reframes iterative refinement, the most effective form of inference-time scaling, not as a deployment strategy but as an offline data curation mechanism, amortizing the compute cost across all future queries.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. Methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1. Pipeline Overview\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our pipeline consists of three sequential phases, each addressing a distinct aspect of the LRPL code generation challenge. Figure 2 provides an architectural overview.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a target low-resource language L L , a set of language independent seed problems 𝒫 = { p 1 , … , p N } \\\\mathcal{P}=\\\\{p_{1},\\\\ldots,p_{N}\\\\} with corresponding test suites 𝒯 = { t 1 , … , t N } \\\\mathcal{T}=\\\\{t_{1},\\\\ldots,t_{N}\\\\} , and a pretrained base model M θ M_{\\\\theta} , our goal is to produce a fine-tuned model M θ ∗ M_{\\\\theta^{*}} that maximizes pass@1 on held-out problems in L L . The pipeline proceeds as follows: (1) Data Synthesis uses M θ M_{\\\\theta} with iterative compiler feedback to construct a verified dataset 𝒟 L \\\\mathcal{D}_{L} ; (2) Syntax-Aware SFT fine-tunes M θ M_{\\\\theta} on 𝒟 L \\\\mathcal{D}_{L} to produce M θ sft M_{\\\\theta_{\\\\text{sft}}} with strong syntactic priors; (3) RLVR applies GRPO starting from M θ sft M_{\\\\theta_{\\\\text{sft}}} to optimize functional correctness, producing the final model M θ ∗ M_{\\\\theta^{*}} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2. Phase 1: Data Synthesis via Offline Inference Scaling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To overcome the LRPL data bottleneck, we repurpose expensive inference-time scaling, specifically iterative self-correction with compiler/test-case feedback, strictly as an automated, offline data curation engine.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.1. Problem Formulation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒫 = { p 1 , … , p N } \\\\mathcal{P}=\\\\{p_{1},\\\\ldots,p_{N}\\\\} be a set of language independent algorithmic problems, each paired with a test suite t i ∈ 𝒯 t_{i}\\\\in\\\\mathcal{T} consisting of IO pairs. For a target language L L , we seek to construct a supervised dataset:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.2. Iterative Compiler/Test Feedback Refinement.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each problem p i p_{i} , the base model M θ M_{\\\\theta} generates an initial candidate solution c i ( 0 ) c_{i}^{(0)} . This candidate is executed in an isolated sandbox against the test suite t i t_{i} , yielding either success or a feedback signal e i ( 0 ) e_{i}^{(0)} . The feedback comprises either one from two components: (1) any compilation or runtime error message, or in case of compilation success and test cases are failing (2) a single failed test case showing the expected output alongside the actual output produced by the candidate. If a candidate fails, we inject this structured feedback back into the model’s context, prompting iterative refinement:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The loop terminates when pass ( c i ( k ) , t i ) = 1 \\\\mathrm{pass}(c_{i}^{(k)},t_{i})=1 (all tests pass) or k > K max k>K_{\\\\max} . Successful trajectories contribute the final solution c i ∗ = c i ( k ) c_{i}^{*}=c_{i}^{(k)} to 𝒟 L \\\\mathcal{D}_{L} ; failed trajectories are discarded. This process is identical to iterative refinement used at inference time by approaches like ReflectionCoder (Ren and others, 2024 ) and AgentCoder (Huang et al. , 2023 ) , but we perform it once, offline , and retain only the final verified solutions as training data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm 1 summarizes the complete data synthesis procedure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.3. Seed Selection.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We initialize the pipeline with language independent algorithmic problems sourced from [dataset]. These problems are specified via natural-language descriptions and IO test pairs, ensuring no dependence on any specific programming language.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.4. Generation Prompt.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each problem is presented to the base model M θ M_{\\\\theta} using a structured system prompt that constrains the output format and encourages reasoning before coding. Key design decisions include: (1) requiring step-by-step reasoning in comments before the solution, mirroring chain-of-thought prompting; (2) enforcing a strict output structure with separate <CODE> and <TESTS> blocks to facilitate automated extraction and validation; (3) mandating a main() function that reads from standard input, enabling execution against IO test suites; and (4) requesting at least four unit test assertions per problem to provide additional verification signal.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.5. Worked Example.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 illustrates the iterative refinement process on a concrete Julia problem: writing a function to sum the digits of a number. The base model’s initial attempt produces syntactically valid Julia but fails all four test cases, a common pattern where models transfer Python semantics (e.g., integer division behavior or string-conversion idioms) that do not generalize to Julia. The sandbox feeds back the test failure summary ( 0 passed, 4 failed ), which is appended to the prompt. In the second iteration, the model self-corrects by using Julia’s idiomatic string / abs /character-arithmetic approach, producing a solution that passes all tests. This single-retry success demonstrates that even one round of compiler-and-test feedback is often sufficient to bridge the gap between high-resource habits and LRPL-correct code. The final, verified solution is added to 𝒟 L \\\\mathcal{D}_{L} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3. Phase 2: Syntax-Aware Supervised Fine-Tuning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using the synthesized dataset 𝒟 L \\\\mathcal{D}_{L} , we fine-tune the pretrained base model M θ M_{\\\\theta} using standard causal language modeling. The SFT objective minimizes the cross-entropy loss over verified prompt-completion pairs:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where π θ \\\\pi_{\\\\theta} denotes the model’s output distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This phase serves a specific purpose: because LRPLs possess idiosyncratic syntaxes and niche standard libraries that differ substantially from high-resource languages, SFT embeds strong structural and grammatical priors into the model. Consider the differences between Julia and Python that a model must internalize: Julia uses 1-based indexing, true / false instead of True / False , function / end blocks instead of indentation-delimited scopes, multiple dispatch instead of class-based OOP, and a distinct standard library (e.g., sort(arr, rev=true) vs. sorted(arr, reverse=True) ). After this phase, the model can reliably produce syntactically valid Julia code structures before being required to optimize for complex algorithmic logic in the subsequent RL phase.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For Ballerina, the syntactic gap is even larger: the language uses a unique type system with union types ( int|string ), built-in concurrency via workers, explicit error handling with check expressions, and a service-oriented programming model unlike any high-resource language. The SFT phase is therefore essential for Ballerina, as the base model has effectively zero prior exposure to these constructs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice, each training example in 𝒟 L \\\\mathcal{D}_{L} is formatted as a prompt-completion pair: the prompt contains the natural-language problem description along with a language-specific instruction header (e.g. “Write a Julia function that…”), and the completion is the verified solution from Phase 1. During fine-tuning, we apply completion-only loss masking , computing the cross-entropy loss exclusively over the solution tokens rather than the prompt tokens, which focuses the learning signal on code generation rather than problem comprehension.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4. Phase 3: RLVR with Verifiable Rewards with Difficulty-curated Data\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.4.1. Data Curation by Difficulty\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With a robust syntax prior established, we transition to exploration via Reinforcement Learning. Crucially, we initialize the GRPO policy with the syntax-aware SFT model M θ sft M_{\\\\theta_{\\\\text{sft}}} from Phase 2, rather than the base model M θ M_{\\\\theta} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.4.2. GRPO Objective.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a given problem p p , we sample a group of G G candidate completions { c 1 , … , c G } \\\\{c_{1},\\\\ldots,c_{G}\\\\} from the current policy π θ \\\\pi_{\\\\theta} . Each completion receives a reward r i r_{i} from the execution-based oracle. We compute group-relative advantages:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where μ G \\\\mu_{G} and σ G \\\\sigma_{G} are the mean and standard deviation of rewards within the group, and ϵ \\\\epsilon is a small constant for numerical stability. The policy is updated to maximize:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where π ref = π θ sft \\\\pi_{\\\\text{ref}}=\\\\pi_{\\\\theta_{\\\\text{sft}}} is the SFT model serving as the reference policy, and β \\\\beta controls the strength of the KL divergence penalty that prevents the policy from deviating too far from the syntactic prior.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.4.3. Reward Function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each completion c c receives a composite reward that decomposes into a test score r test r_{\\\\text{test}} and a build-quality score r build r_{\\\\text{build}} , reflecting the two distinct failure modes in LRPL code generation: syntactic/compilation failures and logical/algorithmic failures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The test score is proportional to the number of passed test cases p p :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This partial reward provides a gradient signal even for completions that solve only a subset of test cases, avoiding the sparsity of binary pass/fail rewards.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The build-quality score r build r_{\\\\text{build}} captures whether the completion compiles and produces well-structured code. If the code fails to compile, r build r_{\\\\text{build}} receives a small positive value inversely proportional to the number of compiler errors, rewarding completions that are “closer” to compiling. If compilation succeeds, r build r_{\\\\text{build}} receives a fixed credit regardless of test outcomes. Additionally, a structure penalty is applied based on the presence of expected code constructs (function definitions, test blocks, assertions), acting as a negative offset that is reduced for well-formed outputs. This ensures that structural compliance alone cannot produce a high reward, only test-passing code is strongly reinforced. The build-quality score is down-weighted relative to the test score, ensuring that functional correctness dominates the learning signal:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α < 1 \\\\alpha<1 controls the relative weight of the build-quality shaping signal.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.4.4. Zero-Advantage Masking.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key challenge in GRPO for LRPLs is that groups may contain no meaningfully correct completion, all candidates fail most tests, producing near-uniform low rewards. Naively computing advantages in such groups reinforces the “least bad” completion, causing the policy to converge on sub-optimal heuristics (e.g. hardcoding common outputs). We address this via zero-advantage masking : if no completion within a group achieves the maximum reward, we set A ^ i = 0 \\\\hat{A}_{i}=0 for all i i in that group, effectively discarding the update. This ensures the policy only learns from groups containing at least one genuinely strong completion, preventing noisy gradients from degrading the SFT prior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Formally, let r max = max i ∈ G ( r i ) r_{\\\\max}=\\\\max_{i\\\\in G}(r_{i}) denote the highest reward in a group. We define the masked advantage as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Substituting A ~ i \\\\tilde{A}_{i} into Eq. 5 yields the masked GRPO objective:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When r max < 1.0 r_{\\\\max}<1.0 , the policy gradient term vanishes entirely for that group, since we set β = 0 \\\\beta=0 similar to DAPO (Yu et al. , 2025 ) the entire sample is ignored.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.4.5. Data Curation by Difficulty\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As discussed in the introduction, we aim to create a curated dataset for GRPO that can maximize learning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the ELO rating used by the Codeforces platform (see Quan et al. (Quan et al. , 2025 ) to measure problem difficulty, where both humans, models, and problems are given ELO ratings, which get adjusted using outcomes from humans and models attempting problems. Using the problem and ELO ratings data available from the Codeforces platform, we have built a model to predict the ELO rating given a problem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For GRPO, we use data that has IO-based ground truth as discussed by Agnostics paper, where we convert ground truth examples to test cases and then give feedback to GRPO using those tests as described in the subsection 3.4.3 ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given reward function r t e s t r_{test} , model p h i r e f \\\\ phi_{ref} , and a dataset D D , that comprises a collection of problems, the ELO rating for the problem, and groundtruth data, we build a GRPO dataset as follows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This defines E L O m ELO_{m} as the supremum (the highest point) of problem difficulty where the model maintains a stable reward:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When d . e l o d.elo is the elo rating of problem d d , curated dataset is defined as follows.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.4.6. Why SFT Initialization Matters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interaction between Phase 2 and Phase 3 is the central design choice of our pipeline. GRPO works by generating candidate solutions, computing relative advantages from execution feedback, and adjusting the output distribution to favor better solutions. However, if all solutions within a group receive the same reward (e.g. all compile but no test passes), the advantages collapse to zero and GRPO cannot learn. When GRPO is initialized from the base model without an SFT prior, the vast majority of sampled completions contain syntax errors and receive near-zero rewards dominated by the build-quality component. With no test-passing signal to differentiate completions, the advantages within each group lack meaningful variance and the policy stagnates. In contrast, the SFT-initialized policy produces syntactically valid completions that compile and execute, even if they produce wrong outputs. This means the reward distribution within each group has meaningful variance, some completions pass more tests than others, providing a rich gradient signal. This enables GRPO to learn effectively, and the compute budget is thus spent exploring functional logic rather than rediscovering basic language syntax.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Experimental Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1. Base Model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use Qwen3-8B (Qwen Team, 2025 ) as our base SLM. We select this model for four reasons: (1) it demonstrates strong baseline performance on high-resource code generation, providing a capable starting point; (2) at 8B parameters, it represents a realistic model size for on-premise deployment on a single GPU; (3) its training data cutoff aligns with the LiveCodeBench v5 temporal split, enabling contamination-free evaluation on Ag-LCB; and (4) supports a fair comparison with the previous work, which reported against the same model. While more widely adopted recent models such as the Qwen3.5 family and DeepSeek-V3.2 series demonstrate stronger performance on high-resource languages, their later training cutoffs risk data contamination against our evaluation benchmarks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2. Target Languages\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on two languages that represent different points on the resource spectrum:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Julia is designed for high-performance scientific computing. Despite a growing community, Julia remains underrepresented in LLM training corpora, CodeNet contains only 10.8k Julia samples versus 3.2M for Python. Julia’s syntax combines familiar imperative structures with distinctive features: 1-based array indexing, multiple dispatch as the core paradigm, function / end block delimiters, and a rich type system with parametric types. These features make Julia a representative “moderate LRPL”, present in pretraining data but insufficiently represented for reliable code generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ballerina is a cloud-native programming language for building distributed applications and microservices. It features a unique type system with structural typing, union types ( int|string|error ), built-in concurrency via workers, first-class support for network services, and explicit error handling with check expressions. Ballerina has near-zero representation in LLM pretraining corpora, making it our “extreme LRPL” test case, a language where the base model has effectively no prior knowledge of the syntax or standard library.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3. Datasets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We source seed problems from the Stanford Alpaca dataset (Taori et al. , 2023 ) , selecting language-independent algorithmic tasks. For each problem and target language, we generate candidate solutions using Claude Code or OpenAI OSS-120B based on it’s difficulty, then verify each candidate by executing its accompanying test suite. Only problems whose generated solutions pass all test cases are retained in the SFT dataset 𝒟 L \\\\mathcal{D}_{L} . This process is applied identically for both Julia and Ballerina, yielding fewer than 1,000 verified examples per language.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the RL phase, we curate approximately 500 training problems from Codeforces (filtered to ≤ \\\\leq 1,300 ELO) and CodeNet (Puri et al. , 2021 ) . Each problem is paired with language-agnostic IO test suites that serve as the verifiable reward signal during GRPO training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4. Benchmarks\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.4.1. MultiPL-E\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Cassano et al. , 2023 ) has translated HumanEval (Chen et al. , 2021 ) problems to 18+ programming languages, enabling standardized cross-lingual evaluation. We report pass@1 on the Julia split. As part of this work, we extend MultiPL-E to include Ballerina by manually translating 159 HumanEval problems into idiomatic Ballerina with corresponding test harnesses. Unlike automated translation approaches, our extension is entirely human-annotated to ensure correctness and idiomatic usage of Ballerina’s distinctive features (e.g. structural typing, check expressions, union types). This constitutes the first standardized code generation benchmark for Ballerina, which we release to support future research on emerging LRPLs. MultiPL-E problems are relatively straightforward, testing basic language competence and standard library knowledge.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.4.2. Ag-LCB\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Boruch-Gruszecki et al. , 2026 ) is the LiveCodeBench (Jain et al. , 2024 ) evaluation adapted through the Agnostics framework, using problems published after the base model’s training cutoff to ensure contamination-free assessment. These problems are substantially harder than MultiPL-E, reflecting competitive programming difficulty with complex algorithmic reasoning requirements. We report pass@1 for both Julia and Ballerina.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5. Baselines\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare the following configurations to isolate the contribution of each pipeline component:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Qwen3-8B (Base) : The unmodified pretrained model, establishing the baseline performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"+ SFT Only : Phase 2 only, the base model fine-tuned on our compiler-feedback synthetic data, without the subsequent RL phase. This isolates the contribution of syntax-aware SFT.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ours (Full Pipeline) : The complete three-phase pipeline (Phase 1 data synthesis + Phase 2 SFT + Phase 3 GRPO).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6. Training Details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We fine-tune Qwen3-8B using a two-stage pipeline: Supervised Fine-Tuning (SFT) followed by GRPO reinforcement learning. Both stages utilize 4-bit quantization and LoRA adapters ( r = 16 , α = 32 r=16,\\\\alpha=32 ) on all linear layers via the Unsloth-TRL framework. SFT is conducted for 4 epochs with a 1e-4 learning rate and a 16,384 token context window. Subsequently, we perform 1 epoch of GRPO using curated data from Section 3.4.5 , employing 8 generations per prompt ( G = 8 G=8 ) and a 5e-5 learning rate. Both phases use the AdamW optimizer, a cosine schedule, and bf16 mixed precision, with the RL phase capping prompt and completion lengths at 4,096 tokens each.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1. Main Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 summarizes the pass@1 results for Julia and Ballerina across both benchmarks. For each language, we report the base model, the SFT-only variant, and the full three-phase pipeline. Since Agnostics (Boruch-Gruszecki et al. , 2026 ) already provides a thorough evaluation of RL-only training across multiple configurations, we include the RL-only variant solely as an ablation under our specific setup (Section 6 ). The key finding is that each successive phase yields additive gains: SFT alone substantially improves over the base model by embedding syntactic priors, and GRPO further boosts performance by optimizing functional correctness. We discuss each language in detail below.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.1.1. Julia.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Compared to vanilla Qwan mode baseline, our pipeline improves Julia pass@1 from 28.3% to 68.6% (+40.3 points) on MultiPL-E and from 9% to 39.2% (+30 points) on Ag-LCB. Table 2 places these results in context. Most importantly, we surpass the previous state of the art (Agnostics (Boruch-Gruszecki et al. , 2026 ) ) by 7.6 percentage points on MultiPL-E and 14.2 percentage points on Ag-LCB, despite using a single base model rather than separate task-specific variants.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.1.2. Ballerina.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the extreme-LRPL case, SFT increased ballerina scores 4.4% to 56% and LCB scores by 2.9% to 9.9%. GRPO training using a curated dataset increased ballerina scores for LCB from 9.9% to 25%, but reduced the MultiPL-E results to 49.7%. We believe this is overall positive given that LCB is a much harder benchmark\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying the pipeline to Ballerina required minimal language-specific modifications: (1) a working Ballerina compiler and runtime for sandbox execution, (2) translation of seed problem descriptions to specify Ballerina-specific IO formats where necessary, and (3) no changes to the GRPO configuration, as IO tests are inherently language-agnostic. The fact that our pipeline produces functional code generation capability from near-zero provides strong evidence that, given a compiler and language-agnostic problems, any language can be bootstrapped.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.1.3. Data and Cost Efficiency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These results are achieved with fewer than 1,000 SFT samples and approximately 500 GRPO training problems, a 3 × 3\\\\times reduction in training data compared to Agnostics, which uses ∼ {\\\\sim} 5,000 GRPO samples. The cost advantage is equally striking: Agnostics requires $320.3 ( 13 h × 8 GPUs × $ 3.08 / GPU-h 13\\\\text{h}\\\\times 8\\\\,\\\\text{GPUs}\\\\times\\\\mathdollar 3.08/\\\\text{GPU-h} ), whereas our total cost is $54.02, comprising fine-tuning ( 45 h × $ 0.42 / GPU-h 45\\\\text{h}\\\\times\\\\mathdollar 0.42/\\\\text{GPU-h} ), OpenAI GPT OSS-120B API calls ( 5 M input × $ 0.25 / 1M 5\\\\text{M input}\\\\times\\\\mathdollar 0.25/\\\\text{1M} + 3 M output × $ 0.60 / 1M +\\\\;3\\\\text{M output}\\\\times\\\\mathdollar 0.60/\\\\text{1M} ), and Sonnet API calls ( 4.1 M input × $ 3 / 1M 4.1\\\\text{M input}\\\\times\\\\mathdollar 3/\\\\text{1M} + 1.3 M output × $ 15 / 1M +\\\\;1.3\\\\text{M output}\\\\times\\\\mathdollar 15/\\\\text{1M} ). Our pipeline thus costs only 1 6 \\\\frac{1}{6} of Agnostics. The key enablers are left-shifting inference-time compute to offline data synthesis and training with a compact, difficulty-filtered RL set. Furthermore, by left-shifting, rather than spending GPU hours on large-scale RL exploration, we invest a one-time API cost to curate high-quality SFT data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2. Error Category Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To understand why each pipeline phase improves performance, we categorize errors from the base model, SFT-only model, and full pipeline on MultiPL-E problems for Julia.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 reveals a clear progression. The base model’s failures are dominated by syntax and compilation errors, the model frequently produces Python-influenced constructs that fail to compile in Julia. After SFT, syntax errors drop substantially, but runtime errors and wrong outputs increase, indicating that the model now produces valid Julia code that compiles but does not yet implement correct algorithms. The full pipeline further reduces wrong-output errors, as the GRPO phase optimizes for functional consistency within the space.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Ablation Studies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As ablations to difficulty-based data curation, we performed two experiments with Julia. For that purpose, in addition to the curated dataset, we created a random difficulty dataset by sampling 500 problems from the full ELO range (spanning ELO 800–2200). This acts as a control dataset as opposed to our curated dataset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As the first experiment, we started with the vanilla Qwan model (no-SFT) and then fine-tuned it with GRPO using the random difficulty dataset. As the second experiment, we started with the Qwan model already fine-tuned with SFT, and further fine-tuned it with GRPO using the random difficulty dataset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the following discussion, we compare our default pipeline (Qwan model + SFT + GRPO with curated dataset).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.1. Default pipeline vs. experiment 1 (qwan + GRPO with random dataset)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Figure 5 plots compares GRPO test reward against difficulty (ELO rating). The left-hand side shows our pipeline (Qwan + SFT + GRPO with curated dataset) produces remarkably stable reward signals: mean between 0.4 and 0.5 across the training ELO range (800- 1250), and max reward stays near 1.0, indicating that at least one completion per GRPO group fully solves each problem. This confirms our default pipeline provides effective rewards.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast, the right-hand side of the Figure showing data from experiment 1 (no SFT and GRPO random dataset) exhibits substantially higher variance in both mean and max reward, particularly as problem difficulty increases beyond ELO 1400. The growing error bars reflect the instability of RL exploration when the policy must simultaneously discover language syntax and algorithmic logic.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.2. Default pipeline vs. experiment 2 (qwan + SFT + GRPO with random dataset)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 compares the Pass@1 scores between SFT mode trained with GRPO using curated vs random datasets. Random baseline yields 52.2% pass@1, 16.4 percentage points below the difficulty-curated variant. These results confirm that for low-resource languages, curating RL training data by difficulty amplifies the effectiveness of GRPO by concentrating training signal on problems at the frontier of model capability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7. Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.1. Zero-advantage Masking\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 illustrates this phenomenon empirically. After zero-advantage masking, the vast majority of training samples are discarded, visible as long stretches where σ G = 0 \\\\sigma_{G}=0 . This occurs when all completions within a group either uniformly succeed or uniformly fail, collapsing the advantage to zero. The isolated spikes where σ G > 0 \\\\sigma_{G}>0 represent the small fraction of samples that actually produce a meaningful gradient signal, problems at the edge of the model’s current capability. This sparse pattern highlights the inefficiency of training without difficulty-based curation: most computational effort is spent on samples that contribute nothing to learning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.2. New Languages Require the SFT Phase\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Ballerina experiment validates that the SFT phase is essential for severely low-resource programming language (S-LRPL) adoption. Without syntactic priors, the RL exploration space is too unconstrained: the model spends its compute budget rediscovering basic syntax rather than learning algorithmic patterns. This suggests that for any new LRPL, investing in Phase 1 data synthesis and Phase 2 SFT is a prerequisite before RL can be effective. The pipeline thus provides a practical recipe: given a compiler for language L L and a set of language-agnostic algorithmic problems, one can bootstrap code generation capability for L L through offline data synthesis followed by SFT and GRPO.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.3. Right Level of Difficulty Amplifies Pipeline Benefits\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our difficulty-based curation results echo the principle of deliberate practice (Ericsson, 2008 ) : learning is most effective when training is concentrated at the boundary of current competence. The mechanism is intuitive within the GRPO framework. Given a problem, GRPO generates multiple candidate solutions and updates the policy based on the relative advantage between them. If a problem is too easy, all candidates succeed and the advantage collapses to zero; if too hard, all candidates fail with the same outcome. Only when problems lie at the edge of the model’s capability, where some candidates succeed and others fail, does the advantage carry a meaningful gradient signal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This follows directly from the GRPO advantage (Eq. 4 ). With binary pass/fail rewards r i ∈ { 0 , 1 } r_{i}\\\\in\\\\{0,1\\\\} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Too easy ( r i = 1 ∀ i r_{i}=1\\\\;\\\\forall\\\\,i ): μ G = 1 , σ G = 0 ⟹ A ^ i = 0 \\\\mu_{G}=1,\\\\;\\\\sigma_{G}=0\\\\implies\\\\hat{A}_{i}=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Too hard ( r i = 0 ∀ i r_{i}=0\\\\;\\\\forall\\\\,i ): μ G = 0 , σ G = 0 ⟹ A ^ i = 0 \\\\mu_{G}=0,\\\\;\\\\sigma_{G}=0\\\\implies\\\\hat{A}_{i}=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Edge of capability (mixed pass/fail): σ G > 0 \\\\sigma_{G}>0 , so correct solutions receive A ^ i > 0 \\\\hat{A}_{i}>0 and incorrect ones A ^ i < 0 \\\\hat{A}_{i}<0 , producing a meaningful gradient signal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This provides a mathematical justification for difficulty-based data curation: selecting problems near the model’s decision boundary maximizes the effective gradient from each GRPO update. This principle extends beyond LRPLs; Zhang et al. (Zhang et al. , 2025 ) independently demonstrate that dynamically alternating between SFT and RL enables automatic weakness identification and targeted resource allocation, further corroborating the value of difficulty-aware training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.4. Compute Allocation: Offline vs. Online\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our results contribute to the broader debate about where to invest compute for code generation. The field currently allocates substantial resources to inference-time scaling, multiple generation rounds, self-correction loops, and agent-based frameworks (Huang et al. , 2023 ; Islam et al. , 2025 ) . Our pipeline demonstrates that the same iterative refinement mechanism yields strictly greater returns when deployed offline as a data synthesis engine. The key difference is amortization: offline synthesis invests compute once to produce training data, while inference-time scaling imposes a per-query cost. To illustrate, consider an IDE code-completion plugin serving 1,000 Julia queries per day. An inference-time scaling approach requiring five rounds of refinement per query would consume 5 × 5\\\\times the compute of a single forward pass on every query, accumulating substantial cost over weeks. In contrast, our pipeline pays a fixed one-time cost of approximately $54 for data synthesis and fine-tuning, after which every query is served at single-pass inference cost. The crossover point is reached within days of deployment, and the cost advantage compounds indefinitely. Moreover, the offline synthesis cost is dominated by API calls for the initial data generation, a component whose price continues to decline as model serving becomes more efficient, making the approach increasingly economical over time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.5. Practical Deployment Considerations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At 8B parameters, the resulting model can run on a single consumer GPU, making it suitable for on-premise deployment where organizations cannot send proprietary code to external APIs. Combined with the pipeline’s language-agnostic design, this enables organizations to bootstrap code generation for internal or niche programming languages without relying on cloud-based LLM services.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8. Conclusion and Future work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We presented a three-phase pipeline for adapting Small Language Models to Low-Resource Programming Languages that reframes inference-time scaling as offline data synthesis. By decoupling syntax acquisition (via compiler-feedback SFT) from algorithmic reasoning (via GRPO with IO tests), our pipeline resolves the trilemma of data scarcity, inference cost, and sparse RL rewards. Furthermore, by curating the data for GRPO use, we amplify and stabilize the reward signal. Applied to Qwen3-8B, it exceeded state-of-the-art results on MultiPL-E (+7.6 pass@1) and Ag-LCB (+14.2 pass@1) for Julia. Furthermore, the pipeline generalizes to Ballerina, a language with near-zero pretraining representation, achieving Pass@1 56% with MultiPL-E and 25% with LCB.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our ablation studies confirm that iterative compiler-feedback data synthesis is essential for high-quality SFT, as it stabilizes the reward, eliminates the subtle errors present in naively generated synthetic data, and yields a cleaner training signal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Future research will focus on scaling the pipeline across diverse model architectures and expanding its support for a wider range of low-resource programming languages. We plan to investigate the synergy between continued pretraining and SFT, develop automated metrics for code idiomaticity, and integrate offline data synthesis with inference-time refinement to further boost performance.\"\n }\n]","isConversational":false}}},{"rowIdx":483,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07146"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Prior-matched evaluation of operational Earth-observation classifiers: a three-number reporting method demonstrated on Sentinel-1 internal-wave detection"},"authors":{"kind":"list like","value":["Joao Pinelo","Joao Goncalves","Arun Shukla","Adriana Santos-Ferreira"],"string":"[\n \"Joao Pinelo\",\n \"Joao Goncalves\",\n \"Arun Shukla\",\n \"Adriana Santos-Ferreira\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07146v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":2,"text":"1. Introduction"},{"type":"para","text":"Internal solitary waves are among the most energetic features of the ocean, and their surface signature — alternating bands of increased and reduced roughness as the wave-induced currents strain the short surface waves that synthetic-aperture radar (SAR) responds to — is recorded in SAR imagery, which makes a satellite SAR archive a natural instrument for mapping where and how often they occur (e.g., Barintag et al. 2023). The Internal Waves Service builds that map operationally from Sentinel-1 Wave-mode data, running a convolutional classifier over the incoming vignettes and passing its detections to domain experts who confirm or reject each one. The classifier’s purpose is not to be the final word but to spend the experts’ time well: the archive is far larger than any team could inspect exhaustively (Sect. 2), so the model exists to concentrate scarce expert attention on the scenes most likely to carry a wave."},{"type":"para","text":"That framing sets the cost of error. A false positive lands a wave-free scene on a validator’s desk and consumes the attention the service is built to conserve; a missed wave is largely recoverable, because Wave-mode acquisitions recur over the same locations on the orbit repeat and the archive is reprocessed, so the same site returns for another look. Precision therefore leads and recall is held to a floor rather than maximised — the operating discipline the whole effort is organised around."},{"type":"para","text":"The difficulty is not the model so much as how its skill is reported. The classifier was trained and evaluated, by the convention it inherited, on class-balanced data — equal numbers of wave and non-wave scenes — while the stream it actually runs on carries a positive rate of only about 0.05. A model that looks well-behaved on a balanced test set over-fires badly on that skew, and the balanced-test figures give no warning of it, because they never present the model with the negatives-to-positives ratio it will meet in production. The gap between the reported number and the fielded one is not noise or a tuning oversight; it is a systematic artefact of reporting at the wrong prior, and it is invisible precisely to the metric most papers quote (Dockès et al. 2021). Closing that gap — making the reported figure predict the operational one — is the problem this paper takes up, and the method it proposes is the contribution."},{"type":"para","text":"It would be natural to treat a prior mismatch as a training defect and correct it by retraining at the operational ratio, or by recalibrating the model’s probabilities to the deployment prior (Saerens et al. 2002; Heiser et al. 2020). Neither helps where it counts. At a fixed recall — the operating discipline above — prior correction and probability calibration are monotone transformations of the score and cannot move precision at all; they relocate the threshold, not the curve (Sect. 4.3). The mismatch is therefore an evaluation problem wearing the costume of a training one, and the response it calls for is an honest reporting method, not a prior-matched retrain. That inversion is the paper’s organising claim."},{"type":"para","text":"The contribution is threefold. First, a prior-matched reporting method: an operational classifier is characterised by three figures rather than one — its balanced-test performance, its performance on a frozen test set drawn at the operational prior, and its real operational performance from post-deployment expert adjudication — and the methodological content lies in the contrast among them, the balanced figure exposed as optimistic and the operational-prior figure built to predict the fielded one (Sect. 3). Second, a worked demonstration: a precision-first, leakage-controlled development cycle that improves the deployed classifier lever by lever, each intervention isolated and promoted only against a pre-registered margin, and each reported for what it did and did not buy — including the levers that bought nothing (Sect. 4). Third, the discipline behind the demonstration: a dataset pinned before it is split, spatial leakage controlled at the level of the true image footprint, an operating point fixed on development data, and a sealed evaluation set read exactly once at a pre-registered threshold — so that the honest negatives and the single-read certification are themselves a reproducibility result (Sects 3, 5). A by-product is itself a first: the Internal Waves Service is, to our knowledge, the first operational classifier built to screen the whole Sentinel-1 Wave-mode archive for internal solitary waves, so the operational positive rate its adjudicated stream reveals — roughly one scene in twenty (Sect. 2.3) — is an initial empirical estimate of that rate, the very prior that was unmeasurable when the balanced training convention was fixed."},{"type":"para","text":"This is a successor to the pipeline-selection study that established the service’s classifier (Pinelo et al. 2026), not a correction of it. That work chose a deployable network on operational grounds under the balanced convention then in force, and the choice was sound on the evidence available. What changed is the ground beneath it: the operational positive rate became known, more validated data accrued, and the question shifted from which pipeline to field to how far the fielded one can be pushed and how its skill should be reported. The present paper restarts from a smaller deployed network rather than the larger candidate, and takes it as far as the data allow — reaching the conclusion that the binding constraint has moved off the model and back onto the data (Sect. 4.6)."},{"type":"para","text":"Section 2 sets out the service and the evaluation problem in operational terms; Section 3 defines the three-number instrument and the frozen, leakage-controlled test set behind it; Section 4 documents the development cycle lever by lever; Section 5 reports the instrument’s output on the promoted model and its sealed certification; and Sections 6 and 7 draw out what transfers to other operational Earth-observation services and what the next cycle requires."},{"type":"heading","level":2,"text":"2. The Internal Waves Service and the evaluation problem"},{"type":"heading","level":3,"text":"2.1 The service"},{"type":"para","text":"The Internal Waves Service classifies the operational Sentinel-1 Wave-mode archive for internal solitary waves, working from the short vignettes the mission acquires continuously along-orbit (Santos-Ferreira et al. 2025; Pinelo et al. 2025). The growing catalogue currently holds over 1.7 million vignettes, a fraction of the full Sentinel-1 Wave-mode archive, which we estimate at approximately 17 million vignettes; it grows through a continuous harvest, so any evaluation set must be frozen against a moving target (Sect. 3.2). Model detections are not taken at face value: they enter a review queue where domain experts adjudicate them, and the queue is organised by a model-confidence band so reviewers can concentrate where their judgement is most useful. The expert time that adjudication consumes is the resource the whole effort exists to conserve."},{"type":"heading","level":3,"text":"2.2 Why precision leads"},{"type":"para","text":"The two error types do not cost the same. A false positive spends an expert’s attention on a scene that holds no wave; a missed positive is largely recoverable, because the same location is reimaged on the 12-day orbit repeat and the archive is periodically reprocessed, so a wave missed on one pass has further chances to be caught. Precision therefore leads, and recall is held at a floor of 0.80 rather than pursued as a target (Sect. 3.3). This is not merely operational convenience: the asymmetry is what makes a false positive the expensive error, and it sets the direction of the entire development effort — lift precision, hold recall."},{"type":"para","text":"That precision leads is itself the current end of an inversion, and the history matters because it explains the class balance the model inherited. The service began recall-first. With few confirmed examples to learn from, the early priority was to harvest positives aggressively and build a labelled set, and a missed wave then was a training example lost rather than a recoverable operational miss — the low default decision cut was set in that spirit, to leave few real positives unnoticed and route the boundary cases to human labelling. The training data were assembled at an even, one-to-one class balance, a ratio fixed at the outset around the service’s competition origins, when the operational positive rate was not yet known, and that convention was carried forward unexamined through each successive model. The ground has since shifted: the operational rate is now known to be heavily skewed (Sect. 2.3), the positive set is large enough that harvesting is no longer the constraint, and the cost that now binds is the validator time each false positive consumes. Hence precision-first, with recall held at its floor rather than maximised. The stance is the current one, not a permanent one — as the two error costs converge, with a larger positive backlog and a service no longer starved of examples, the emphasis is expected to normalise toward F1 — but the recall floor of 0.80 is held throughout."},{"type":"para","text":"Recoverability, the property that lets recall sit at a floor at all, rests on more than the orbit repeat. Each improved model reprocesses the entire catalogue rather than only scoring new acquisitions, so a positive missed by one model generation has a further chance to be caught by the next: the recoverable miss is recoverable across model iterations, not only across passes of the same model. This is a deployment property the service was built for. The predecessor work chose a fast, energy-frugal Julia inference stack precisely so that reclassifying the whole archive with a new model is a same-day operation (Pinelo et al. 2026), and reprocessing wholesale — rather than layering each model’s verdicts over the last — keeps the published catalogue labelled consistently under a single model version, which matters for a citeable, DOI-versioned dataset. The recall floor is therefore underwritten by an operational guarantee: what one model misses, its successor re-examines."},{"type":"heading","level":3,"text":"2.3 Starting state, and what a threshold can buy"},{"type":"para","text":"The deployed model is a compact 34,465-parameter convolutional network (SAR_CNN v2), trained and validated under an even class balance — the convention inherited from the service’s origins — and then applied to a stream whose true positive rate is only about 0.05 — now measurable from the service’s own adjudicated detections, but unknowable when the balanced convention was set. The mismatch surfaces as over-firing. At the default 0.5 decision cut the model attains a recall of 0.963 but a precision of just 0.192 on the adjudicated set, flagging 26,472 false positives; and because validators preferentially review likely positives, even that precision is optimistic against the live stream rather than pessimistic."},{"type":"para","text":"Raising the decision threshold recovers precision without retraining. At 0.80 the false positives fall to 6,193 — a reduction of 76.6% — for a recall that eases only from 0.963 to 0.818, lifting precision to 0.464 (Fig. 1). This is the first and cheapest lever (Sect. 4.2): genuine, immediate relief for the validators, but a redistribution of errors along a fixed curve, not a lift of it — precision at the recall floor stops well short of the target. Two facts frame everything that follows: the remaining gap is a model problem, not a threshold one, and the operational precision quoted here rests on the validated subset, so the honest question is not what the balanced test set reports but what the skewed stream would. The first motivates the development cycle (Sect. 4); the second motivates the evaluation instrument (Sect. 3)."},{"type":"para","text":"Figure 1. Precision, recall, F1 and validator load (false-positive count) as a function of the decision threshold, for the two classifiers the service has operated. The current model ( 20260602-01 , solid) carries the operating-point decision; the retired April–June model ( 20260408T131223 , dashed) is drawn behind it, so the pair survives greyscale. Each rate panel states its own definition. The high-value 0.75–0.85 threshold band is shaded and the 0.80 candidate operating point is marked on the current model; the dashed line in the F1 panel is the project target of 0.95, which no threshold on either curve reaches — the gap a threshold cannot close, and what motivates the model-side development cycle (Sect. 4). The rates are computed on the validated subset and so run optimistic against the full stream."},{"type":"heading","level":2,"text":"3. Methods — the evaluation instrument"},{"type":"heading","level":3,"text":"3.1 A three-number reporting axis"},{"type":"para","text":"An operational classifier is characterised here by three performance figures rather than one, and the methodological claim rests on the relationship among them, not on any single value."},{"type":"para","text":"The first, balanced-test performance, is the figure a conventional pipeline reports: precision, recall and F1 on a held-out set with equal class proportions (Maxwell et al. 2021a). It is what the Internal Waves Service pipeline reported for its earlier models, and, as Sect. 5 shows, it is systematically optimistic once the deployment stream is skewed (Saito and Rehmsmeier 2015)."},{"type":"para","text":"The second, operational-prior performance, scores the same model, at the same decision threshold, on a frozen test set drawn at the operational positive rate (0.05; Sect. 2) rather than at parity (Maxwell et al. 2021b). This is the honest predictor of fielded behaviour: it asks what precision a validator would actually face, and it is the figure the present method proposes as the one to report."},{"type":"para","text":"The third, real-operational performance, is obtained after deployment, from expert adjudication of the model’s own detections on the live stream. It is the ground truth the second figure is constructed to predict. Because it requires the model to run in production through an adjudication window, it is stated as a forward commitment rather than approximated by any test-set surrogate (Sect. 5, Sect. 6)."},{"type":"para","text":"The instrument is the contrast. For a model trained and validated under class balance, figure (1) diverges sharply from figures (2) and (3): the cautionary reading, and the reason balanced-test reporting misleads. For a model reported under this method, figure (2) is built to track figure (3): the corrective reading. As a preview of the effect (Sect. 5), the model promoted in Sect. 4 scores a precision of 0.996 on a balanced set and 0.927 at the operational prior — the same model at the same threshold, the difference being the prior alone. Table 1 (Sect. 5) sets out the full axis."},{"type":"heading","level":3,"text":"3.2 A frozen, leakage-controlled test set"},{"type":"para","text":"The scarce class sets the terms. Confirmed internal-wave positives number 6,868 against 126,742 confirmed negatives, so the positive partition, not the negative, bounds the precision at which any figure can be resolved; it is sized from the confidence interval required on precision and recall at the recall floor, not from a round fraction. The dataset is pinned before the split is drawn (20260621T120557Z), so the continually growing archive cannot leak into a frozen evaluation."},{"type":"para","text":"Spatial leakage is controlled at the level of the hotspot , defined by true image-footprint overlap rather than a centroid radius. Sentinel-1 Wave-mode acquisitions recur over fixed locations on the 12-day orbit repeat, and the surface forcing at a site might evolve little between passes, so near-identical scenes would otherwise straddle the split and inflate every figure (Kattenborn et al. 2022; Roberts et al. 2017). Two split regimes follow from this unit: an interpolation split, which separates acquisitions within a hotspot (the reads reported here), and a stricter extrapolation split, which holds out whole hotspots and region cells to probe generalisation to unseen sites (Valavi et al. 2019) (a design provision). The data are partitioned into train, development, and a sealed lockbox, with the test prior fixed at 0.05 by construction; composition is 4,466/81,104 (train), 865/16,435 (dev) and 1,537/29,203 (lockbox), positives/negatives (Fig. 2)."},{"type":"para","text":"The lockbox is the central pre-deployment estimate of operational precision, and its credibility rests on how it is handled rather than on its size. It is sealed off-site with a manifest and a per-file checksum, and read exactly once, at a threshold fixed in advance on the development set. Every read is pre-registered: the scoring script and its test are committed before any number is seen, in a two-commit structure for the consent-gated read. The expected ordering is train >= dev >= lockbox >= production, the gap small and downward; a lockbox figure that slightly exceeds dev is treated as same-prior sampling on a larger independent draw, not as improvement and not as leakage (Sect. 5)."},{"type":"para","text":"Figure 2. The leakage-controlled split. (a) The leakage unit is the hotspot — a set of Sentinel-1 Wave-mode acquisitions whose image footprints overlap at a recurrence site. In the interpolation split used for the reads reported here, a hotspot’s passes are shared across train, development and lockbox; the stricter extrapolation split (a design provision) instead holds a whole hotspot out as an unseen site. (b) The 6,868 confirmed positives on a Natural Earth projection, coloured by split, showing the recurrence-site clustering the hotspot unit exists to guard against; per-split positive counts are given in the legend. The 126,742 confirmed negatives, held at the operational prior (π = 0.05), are a count rather than a geography and are not mapped. The dataset was pinned (20260621T120557Z) before the split was drawn, so the continually growing archive cannot leak into it."},{"type":"heading","level":3,"text":"3.3 Operating point and recall floor"},{"type":"para","text":"The cost of the two error types is asymmetric. A false positive lands on a validator’s desk and consumes expert time; a stochastic miss is recovered on the next reprocessing pass. Precision therefore leads, and recall is held at a floor of 0.80 rather than treated as a target: reprocessing recovers per-pass stochastic misses but not systematic blind spots, so the floor guards against trading away recoverable recall while precision is driven up."},{"type":"para","text":"The operating point is selected on the development set from a band of thresholds spanning the recall floor, then projected onto the frozen lockbox scores; the sealed set never selects its own operating point. The chosen threshold (0.935 for the model promoted in Sect. 4) ships as a recommendation for the validator review queue’s confidence band, not as a recut of the stored classification label. This keeps the lever per-session: a validator under load tightens their own queue without a default recall cut being imposed on the whole service."},{"type":"heading","level":2,"text":"4. Methods — the development cycle"},{"type":"heading","level":3,"text":"4.1 A moving binding constraint"},{"type":"para","text":"The improvement proceeded as a sequence of single-lever interventions, each isolated so its contribution could be credited on its own, and each carried forward only if it beat the incumbent by more than a pre-registered promotion margin (0.021, the seed-paired 95% confidence half-width). The organising observation is that the binding constraint moved: the lever that paid at one stage was inert at the next, and reporting where it moved — rather than only the final number — is the point of the demonstration. All development figures below are precision at the recall floor (0.80), scored at the operational prior on the frozen development split, across five seeds."},{"type":"heading","level":3,"text":"4.2 Lever 1 — the operating threshold"},{"type":"para","text":"The first and cheapest lever moves the operating point along the existing model’s precision/recall curve, with no retraining, and is reversible on the next reprocessing pass. It was taken in the predecessor work (Sect. 2) and delivered most of the immediate validator-facing relief, but the current curve tops out well short of the target: a threshold move redistributes errors, it does not lift the curve. This bounded what remained and set the agenda for the model-side levers that follow."},{"type":"heading","level":3,"text":"4.3 Lever 2 — the data"},{"type":"para","text":"The dataset was the first model-side lever, and its two effects were separated rather than moved together: the class ratio the model trains under, and the negative variety it sees against the roughly 95% of negatives the balanced trainer discards (Kang et al. 2020; Branco et al. 2016). In a factorial over both, negative variety carried the precision — widening the negatives lifted operational precision by 0.045 at the recall floor, a difference that clears zero — while shifting the training ratio toward the operational prior did not, and if anything cost precision (-0.026 for the ratio contrast at wide variety). The best data configuration, wide variety at a balanced ratio, reached 0.832 precision at the floor."},{"type":"para","text":"This is the first negative result, and it is structural rather than incidental: prior correction and probability calibration are monotone transforms of the score (Saerens et al. 2002; Heiser et al. 2020), so at a recall-pinned operating point they cannot move precision at all — they relocate the threshold, not the curve. A separate calibration arm confirmed the inertness by construction. The prior-matched training the mismatch might seem to call for is therefore the wrong response to it; the mismatch is an evaluation problem, which is the paper’s point. That inertness concerns the reported figure alone: because these transforms preserve the ranking, they cannot move the curve, and so cannot improve the precision this method reports — but they remain the appropriate instrument for placing the operating point once the prior shifts, a distinct use returned to under the out-of-time decay of Sect. 5.3 and Sect. 6."},{"type":"heading","level":3,"text":"4.4 Lever 3 — capacity"},{"type":"para","text":"With the data lever spent at 0.832, capacity was tested by a uniform width ladder over the deployed three-block shape, architecture the sole variable. Doubling width promoted — 0.868 precision at the floor, 0.036 over the data-lever endpoint — lifting the parameter count from 34,465 to 131,329. But the ladder then flattened: tripling width (290,657 parameters, near the scale of the predecessor paper’s headline net at 283,329 (Pinelo et al. 2026)) did not clear the promotion margin at the floor (-0.006, within margin) and quadrupling it (512,449 parameters) likewise (0.006). Beyond the first doubling, added width did not clear the margin at five seeds — an interval consistent with a small residual gain as well as with none, so this is a within-margin negative rather than a demonstrated ceiling."},{"type":"para","text":"The ladder was stopped by a pre-registered two-trigger rule — diminishing dev gain within the promotion margin, and the onset of a train-dev gap — rather than at a fixed large network. The choice not to overstretch is a scientific one, not only a regularisation one: a model overfit to the known recurrence sites stops generalising to the unseen hotspots the service exists to find, which is why the split holds sites out (Sect. 3.2) and why the endpoint is matched to the data rather than maximised."},{"type":"heading","level":3,"text":"4.5 Lever 4 — the aggregation head"},{"type":"para","text":"Capacity being spent, the next lever was the aggregation mechanism — the operation that collapses the spatial feature map to a decision, swapped in isolation on the promoted width network. Replacing global-average pooling with generalised-mean (GeM) pooling promoted where added capacity had not: 0.924 precision at the floor, 0.056 over width. Two attention-pooling heads, with more parameters, did not lift at all — again isolating the mechanism, not the capacity, as the active ingredient. The learned pooling exponent settled well above unity (mean 4.9, initialised at 3.0), meaning the head weights the peak spatial responses over the mean."},{"type":"para","text":"The mechanism has a concrete target. A recurring geophysical look-alike — a banded, quasi-periodic surface signature that mimics the along-crest periodicity of an internal-wave packet — forms a small, persistent tail of the false-positive population, clustered at a handful of recurring sites. Average pooling washes the banding into a single mean and fires; a pooling operation that preserves peak spatial structure can tell it apart. On the cross-seed false-positive core of 40 scenes, the width network fired on 35.8 on average and GeM on 21.2 — the mechanism suppressing roughly a third of the core. The residue of 12 is the same banded class rather than a geographic pocket (not polar-enriched, p ~ 0.58): what remains is a discrimination limit, not a location (Fig. 3)."},{"type":"para","text":"Figure 3. The banded geophysical confuser. Sixteen validated Sentinel-1 WV vignettes from a single acquisition slot (WV slot 025) that fall at four locations once clustered by centre position — two Antarctic-coast footprints, one South Atlantic and one North Pacific site (each site’s clustered position, pass count and per-site structure-similarity are printed in its header). Every pass at every site carries an expert-negative verdict. Model behaviour is observable only on the passes that fall in the development split, and it splits by site: the three development passes at the 152.84°E Antarctic footprint fire, all scoring at near-saturation above the recall-0.80 development threshold, whereas the single development pass at the adjacent 153.07°E site does not fire; the remaining passes are training or lockbox and carry no development score (chip borders: vermillion fires, blue development-no-fire, grey non-development). The high within-site structure correlation at the two Antarctic footprints — the speckle-suppressed 32×32 area-average measure of the data-integrity audit, near zero between different sites — establishes that these are the same banded surface signature recurring on the orbit repeat rather than duplicated scenes. That signature is a persistent, quasi-periodic surface texture, consistent with high-latitude sea-ice or wind-banding, that mimics the along-crest periodicity of an internal-wave packet, and it names the discrimination target for the aggregation-head work; gem’s suppression of it is reported in the accompanying text, not shown in this figure. These scenes are drawn from the unanimously verified pool, so they are genuine negatives on which the model is misled by real geophysics, not by mislabelling."},{"type":"heading","level":3,"text":"4.6 Verdict: where the constraint moved"},{"type":"para","text":"The improved model clears the ranking target it was optimised against — 0.924 precision at the recall floor. Its F1 at the floor is 0.860, short of the 0.90 interim bar and well short of the 0.95 destination. A precision-ceiling read found no hard architectural wall — F1 >= 0.95 is reachable in principle once the reducible false-positive mass is removed — but against the full false-positive mass the best attainable F1 is 0.878, held down by an irreducible-for-now residue of roughly 57.6 false positives per seed. Symmetrically, a systematic false-negative core of 34 scenes sits flat across the recall band, so the operating point is a recoverable-recall dial above a fixed floor rather than a trade against systematic blind spots."},{"type":"para","text":"Read together, capacity gains have fallen within the promotion margin and the aggregation lever is spent, yet there is no ceiling — the binding constraint has moved off architecture and onto the size of the reducible false-positive mass, which is to say onto more validated data. That is where the next cycle goes, and it is the empirical basis for restarting from data rather than from a larger network."},{"type":"para","text":"Figure 4. The development ladder: precision at the 0.80 recall floor, scored at the operational prior across five seeds, as the retrain cycle advanced through its three model-side levers. Each stage’s promoted model (filled, connected) becomes the next stage’s baseline, so the blue spine is the chosen model as it developed — widening negative variety (0.832), then doubling width (0.868), then the generalised-mean (GeM) pooling head (0.924). The rungs tried and not carried forward sit muted beneath each stage and are named: the discarded data cells at stage one, the parameter-neutral and higher-capacity architecture variants at stage two (including the width multiples that did not clear the margin), and the two attention-pooling heads at stage three. Error bars are the iteration-1 factorial interval and, for iterations 2 and 3, the five-seed t-interval; a rung was promoted only on beating the incumbent by more than the pre-registered margin (0.021, the seed-paired 95% half-width). Sect. 4."},{"type":"heading","level":2,"text":"5. Results"},{"type":"para","text":"The development-cycle results are reported lever by lever in Sect. 4 and drawn together in Fig. 4; they are not restated here. This section reports the instrument’s output on the promoted model: the three-number axis (Sect. 5.1), the sealed-lockbox certification behind its operational-prior cell (Sect. 5.2), an out-of-time robustness check on that cell (Sect. 5.3), and the operating point set against the systematic-miss floor (Sect. 5.4)."},{"type":"heading","level":3,"text":"5.1 The three-number axis"},{"type":"para","text":"Table 1 sets out the three-number axis for the incumbent operational model and for the model promoted in Sect. 4. The methodological content is in the contrasts between the cells, not in any single value. Of the six cells, four are filled from landed reads, one is left n/a by design (the incumbent’s operational-prior cell; see below), and one — gem’s real-operational figure — is pre-registered as a forward commitment rather than estimated. The cautionary contrast (the incumbent’s balanced-test figure against its real operational figure) and the prior-effect contrast (gem’s balanced against its operational-prior figure) can both be read now; the corrective contrast (gem’s operational-prior figure tracking its real operational figure) is what the deployment window will close."},{"type":"para","text":"For the incumbent — the deployed 34,465-parameter network trained under class balance — the cautionary contrast is between its balanced-test figure and its real operational figure, and both are now in hand. Its balanced-test performance is precision 0.794, recall 0.964 and F1 0.871 (area under the ROC curve 0.961), computed as the parity projection of the deployed 0.5 cut over the full unanimous verified pool at complete coverage — the same operation that produces gem’s balanced cell, so the two balanced figures are comparable by construction rather than through two different resampling schemes. Its real operational precision, from validator adjudication at the same 0.5 cut, is 0.192 at recall 0.963, on the adjudicated subset and so optimistic against the full stream (Sect. 2). The divergence is the cautionary reading in a single pair of numbers: a balanced-test precision of 0.794 against a fielded precision of 0.192 at the identical threshold, the balanced figure standing far above operational skill because the reporting prior, not the model, has changed. Two caveats travel with the balanced cell and neither softens the contrast: the verified pool is review-selected rather than a stream sample, and the incumbent’s June training membership is unrecoverable — it is the 34,465-parameter network’s own evaluation, not the predecessor paper’s headline figure, which belongs to a different and larger net. The operational-prior cell is left unfilled for the incumbent by design: scoring the old model on the frozen operational-prior set would add nothing its real operational number does not already show, and the cautionary half rests on the balanced-versus-real gap, not on an intermediate estimate."},{"type":"para","text":"For gem the corrective contrasts are both in hand. At the operational prior its precision is 0.927, recall 0.827 and F1 0.874 (the sealed lockbox; Sect. 5.2), and this is the figure built to track the real operational number the deployment window will produce (Sect. 6). The same model, at the same threshold, scored on a balanced set reads precision 0.996, recall 0.827 and F1 0.904 — the recall unchanged, because it is a property of the positives alone, and only the precision moving with the prior. The consequence is the paper’s thesis in a single row: gem’s balanced F1 of 0.904 clears the 0.90 interim bar, while its operational-prior F1 of 0.874 does not — the balanced figure crossing a bar the honest figure misses, on identical bytes at an identical threshold, the difference being the reporting prior and nothing else."},{"type":"para","text":"gem’s real operational cell is the one number the method cannot borrow from a test set: it creates its own ground truth prospectively, by adjudicating gem’s detections once gem is the production model, and is therefore reported in subsequent work — this paper pre-registers it as the forward validation that closes the loop (Sect. 6). It is the single open cell, and it is not filled with a surrogate: the out-of-time read of Sect. 5.3 is a robustness check on the operational-prior cell, explicitly not a stand-in for the post-deployment number. Table 1 therefore stands with every present-tense cell filled from a landed read and gem’s real operational cell pre-registered here as subsequent work."},{"type":"para","text":"Table 1. The three-number reporting axis, precision / recall / F1 per cell, scored at the dev-fixed threshold (0.935) and, for the operational-prior and balanced cells, at the operational prior (0.05) and at parity respectively. The incumbent real-operational figure is measured on the validated subset and is optimistic against the full stream (Sect. 2). gem’s real-operational figure is pre-registered here and reported in subsequent work, once gem is in production (Sects 5.1, 6)."},{"type":"heading","level":3,"text":"5.2 Lockbox certification"},{"type":"para","text":"The operational-prior cell rests on a single, sealed read, and its credibility is a matter of how that read was conducted (Sect. 3.2). The lockbox holds 30,740 scenes (1,537 positive, 29,203 negative) drawn at the operational prior and pinned before the split (20260621T120557Z); it was read exactly once, at the threshold fixed in advance on the development set (0.935), from the same frozen bytes that will reach production — the artefact scored here is byte-for-byte the artefact that deploys (Sects 3.2, 6)."},{"type":"para","text":"gem’s discrimination survives the move from development to the sealed set intact: the area under the ROC curve is 0.993 on the lockbox against 0.994 on development, so there is no optimism penalty in ranking. At the fixed threshold the confusion matrix is 1,271 true and 100 false positives against 29,103 true and 266 false negatives — a false-positive rate of 0.0034 on the sealed negatives, which is the validator-facing quantity: 100 scenes wrongly flagged against 29,103 true negatives."},{"type":"para","text":"One feature of the read is unexpected: the lockbox precision slightly exceeds the development precision. The lockbox’s own recall-0.80 operating point reaches precision 0.939, and at the transferred development threshold the recall lands at 0.827, above the 0.80 floor. This is neither improvement nor leakage. The expected ordering is train >= dev >= lockbox >= production, the gap small and downward; a lockbox figure a little above dev is same-prior sampling on a larger independent draw, the ordinary variation between two finite samples of one distribution, and the split’s construction (Sect. 3.2) forecloses the leakage reading (Fig. 5)."},{"type":"para","text":"Figure 5. Lockbox certification and its out-of-time robustness, as precision–recall at the operational prior (π = 0.05). The sealed-lockbox curve (solid, n = 30,740) is the honest pre-deployment predictor of operational precision; the fresh-holdout curve (dashed, n = 11,843) scores the same deploy bytes at the same fixed threshold (0.935) on verified scenes whose last verdict post-dates the split-lock pin (20260621T120557Z) by about ten days, anti-joined off every split member. The two findings are separate. Discrimination transfers — the curves nearly coincide and the area under the ROC curve moves only from 0.993 to 0.987. The fixed operating point (marked on each curve, joined by the dotted line) does not: precision falls from 0.927 on the sealed set to 0.810 out-of-time, and the false-positive rate rises roughly threefold (0.0034 to 0.010) at the same threshold. Reported exploratory and pre-registered: a single out-of-time window on the review-selected pool, not a trend (Sect. 5.3)."},{"type":"heading","level":3,"text":"5.3 Out-of-time robustness check"},{"type":"para","text":"An independent, out-of-time check on the operational-prior cell has been run, and is reported here as an exploratory, pre-registered result. It scores the same deployed bytes at the same fixed threshold (0.935) on 11,843 verified scenes whose last verdict post-dates the split freeze (20260621T120557Z), anti-joined against every split member so none could have leaked into training or into the split’s selection. The read separates into two findings. Discrimination transferred intact: the area under the ROC curve is 0.987 out-of-time against 0.993 on the sealed lockbox. The fixed operating point did not: at the operational prior the precision fell from the lockbox’s 0.927 to 0.810, and the false-positive rate rose from 0.0034 to 0.010 at the same threshold, over a window of only about ten days past the freeze. This is not a composition artefact — the drop holds across a prior-sensitivity band (0.771 to 0.837) and the average precision fell in step (0.890 against the lockbox’s 0.944), so the precision–recall curve genuinely moved rather than the scoring prior."},{"type":"para","text":"Two tiers of caveat of the result. The full read is a real estimate but a single out-of-time snapshot on the review-selected verified pool — one window, not a trend. The footprint-disjoint new-site subset is underpowered on positives (736 scenes, 20 positive): its AUC 0.986 suggests the model is not blind to unseen geography, but its precision and recall are not reportable at that sample size."},{"type":"para","text":"The finding strengthens rather than dents the thesis. Discrimination is stable out-of-time while a fixed development-set operating point is not, and a balanced-test evaluation would have surfaced none of it. That a falsifiable, pre-registered check caught real out-of-time decay is the case for prior-matched, out-of-time evaluation; it also sets a concrete deployment requirement — that the operating point be validated against a current catalogue slice rather than inherited from the development set (Sect. 6). Because the check was pre-registered, with the scoring script and its test committed before any number was seen, it stands report-regardless (Fig. 5)."},{"type":"heading","level":3,"text":"5.4 The operating point and the systematic-miss floor"},{"type":"para","text":"Applying the fixed threshold places the lockbox at recall 0.827, above the 0.80 floor, and the recall it forgoes above the floor is the recoverable kind rather than a systematic blind spot. The systematic false-negative core — the positives missed across model families, cross-architecture and cross-recipe — numbers 34 scenes and sits flat across the recall band from 0.70 to 0.85, so moving the operating point within the band trades stochastic misses, recovered on the next reprocessing pass, not systematic ones. At recall 0.80 on the deploy seed the positives forgone number 173, of which the systematic core is the fixed floor and the remainder the recoverable tail. That floor corroborates, on a different model and population, the deployed network’s own sub-threshold miss rate of 0.038 (Sect. 4; corroborating, not load-bearing). The operating point is therefore a recoverable-recall dial above a fixed floor, which is what the recall-floor discipline requires (Fig. 6)."},{"type":"para","text":"Figure 6. The systematic false-negative core, on the development positives. (a) For each confirmed positive, the number of the promoted model’s five seeds that miss it: most are missed by no seed, while a persistent tail (106 positives) is missed by all five. The subset missed across every model family, recipe and seed — the systematic core — is 34 positives. (b) Positives forgone at each candidate recall operating point, with the systematic core (red) drawn inside the forgone total (blue) so the stochastic, recoverable remainder reads off directly. The core is flat across the band, so moving the operating point within it trades recoverable misses, not systematic blind spots. Because the development positives are drawn from the unanimously verified pool, the core is a floor: it under-states the systematic-miss rate on the operational stream."},{"type":"heading","level":2,"text":"6. Discussion"},{"type":"para","text":"The instrument gives two readings, and the paper’s claim is the pair. Read against the incumbent, it is cautionary: a network trained and evaluated under class balance carries a balanced-test figure that stands far above the precision a validator actually meets on the skewed stream, and the gap is not a defect of the model but an artefact of reporting at the wrong prior (Sect. 5.1). Read against the promoted model, it is corrective: the operational-prior figure is constructed to answer the question the balanced figure evades — what precision the fielded model will deliver — and it is the number the method proposes as the one to report. The contribution is not either figure alone but the discipline of carrying both, so that the distance between them is visible rather than hidden."},{"type":"para","text":"The corrective half rests on a prediction this paper pre-registers but does not yet close. The operational-prior cell is built to track the real operational number that post-deployment adjudication will produce (cell 3), and that number is reported in subsequent work, once the promoted model is in production and its own firings have been adjudicated over an operational window. What the present paper can show is that the operational-prior figure behaves as a predictor should under the one independent test available before deployment: on out-of-time data, past the freeze and disjoint from the split, the model’s discrimination held (Sect. 5.3). The same check also delivered the sharpest caution in the paper — the fixed operating point did not transfer unattended, its precision decaying materially over a short out-of-time window while the ranking held. That the operational-prior figure survives out-of-time as a ranking while the threshold placed on it drifts is the distinction the method draws: the number is a stable ranking, and the condition under which its operating point predicts is stated rather than assumed."},{"type":"para","text":"Several limitations bound the reading, and they are stated in the direction their optimism runs. The ground truth throughout is drawn from a validator review queue that is itself organised by model confidence, so both the incumbent’s real operational figure and the sealed operational-prior read rest on a review-selected pool rather than a uniform sample of the stream; the selection runs optimistic in a specific direction: because the queue preferentially surfaces the scenes the model scores most confidently positive, the adjudicated pool is enriched with the high-confidence flags on which precision is naturally highest, while the lower-confidence flags that still clear the threshold — and fail more often — are under-reviewed, so the operational precisions quoted lean high against the full stream rather than conservative. The incumbent’s balanced cell carries a second caveat of its own: the membership of its training set is not now recoverable, so its balanced-test figure is reported as the deployed network’s own evaluation and is not equated with any headline from the predecessor paper, whose network was larger and different. A third is conditional rather than fixed: the recall-floor discipline tolerates missed positives because reprocessing recovers them on a later pass, but that recovery holds only while the model still generalises to sites it has not seen — an overfit model would keep missing novel-site positives, which reprocessing could then never recover. The out-of-time read speaks to this directly: on footprint-disjoint new sites the model’s discrimination held (Sect. 5.3), evidence that it is not blind to unseen geography, though the positive count there is too small to quantify new-site recall. Holding capacity back (Sects 4.4, 3.2) is the design response to this risk rather than a proof against it. And the loop is not closed: cell 3 remains outstanding until deployment and adjudication, so the claim that the operational-prior figure predicts the operational one is, at submission, a falsifiable commitment rather than a demonstrated equality."},{"type":"para","text":"The operating-point decay is a limitation of a different kind, because it is a finding about deployment rather than about the evaluation. A threshold fixed on the development set is not a set-and-forget quantity: the out-of-time read shows its precision can fall while the model’s discrimination is intact, which means the operating point must be validated against a current slice of the catalogue at deployment and treated as something to recalibrate on a schedule, not inherited unchanged from development. Prior correction and calibration are the right tools for this: they govern the placement of the operating point, which is distinct from the reported curve they cannot move (Sect. 4.3) (Saerens et al. 2002; Heiser et al. 2020). This is the concrete operational requirement the paper hands to the service, and it is the check the post-deployment validation gate is designed to perform before cell 3 is read."},{"type":"para","text":"The reporting method is not specific to internal waves, to Sentinel-1, or even to a precision-first stance. Its one precondition is prior shift: an operational classifier trained and reported under one class balance but deployed against a materially different one. Wherever that holds, the three-number axis applies as a portable reporting convention — report the balanced figure if convention demands it, but report alongside it the operational-prior figure and commit to the post-deployment number, so that a reader sees the prior effect rather than having to infer it. The precision-first operating discipline of this particular service (Sect. 2.2) is a separate matter — the current end of an inversion from the recall-first stance of the service’s early life, and one expected to normalise as the costs of the two errors converge — and the method does not depend on it: a service that weighted recall, or weighted the two equally, would report the same three numbers and read the same contrast. This echoes the transferable-methodology framing of the predecessor paper, carried now from pipeline selection to performance reporting."},{"type":"para","text":"Finally, the development cycle carries a geoscience point that outlives the particular model. The decision not to spend capacity past the point where it stopped paying was a scientific choice as much as a regularisation one: a network overfit to the recurrence sites the archive already knows would stop generalising to the unseen sites the service exists to find, which is why the split holds sites out and why the endpoint was matched to the data rather than maximised (Sects 4.4, 4.6). That the binding constraint ended the cycle back on the size of the validated dataset, rather than on architecture, is itself the finding that sets the next cycle’s direction — more validated data from the growing archive, rather than a larger network on the data in hand."},{"type":"heading","level":2,"text":"7. Conclusions"},{"type":"para","text":"Balanced-test accuracy is the wrong number to trust for an operational classifier whose deployment stream is skewed away from the prior it was evaluated under. The proposal here is to stop reporting it alone: characterise the model by three figures — its balanced-test performance, its performance on a frozen test set drawn at the operational prior, and its post-deployment operational performance — and read the contribution in the contrast between them. The balanced figure is exposed as optimistic, the operational-prior figure is built to predict the fielded one, and the distance between the two is the prior effect made visible rather than left to be inferred."},{"type":"para","text":"We demonstrated the instrument across a full precision-first development cycle on the Internal Waves Service, reporting each lever for what it did and did not improve. Widening negative variety and, later, replacing the pooling head both lifted precision at the fixed recall floor; shifting the training ratio toward the operational prior and adding raw capacity did not; and probability calibration could not, by construction, being a monotone transform of the score at a recall-pinned operating point. The negatives are as much a result as the promotions — they locate the mismatch as an evaluation problem rather than a training one."},{"type":"para","text":"The figures are only as trustworthy as the discipline behind them. The dataset was pinned before it was split, spatial leakage was controlled at the true image footprint, the operating point was fixed on development data, and the sealed evaluation set was read exactly once at a pre-registered threshold, with every read’s scoring script and test committed before any number was seen. The out-of-time check applied that same discipline, catching a real decay in the fixed operating point that a balanced evaluation would not have surfaced."},{"type":"para","text":"Two things remain for the next cycle. The loop is not yet closed: the promoted model’s operational-prior figure is a pre-registered prediction of its real operational performance, and only post-deployment adjudication of its own firings will confirm it — the third number, reported in subsequent work. And the binding constraint has moved. Added capacity stopped clearing the promotion margin and the aggregation lever is spent, yet no architectural ceiling was found; what limits precision now is the size of the reducible false-positive mass, which is to say the validated data. The next cycle therefore restarts not from a larger network but from more of the growing archive — and, at deployment, from an operating point revalidated against a current catalogue slice rather than inherited from development."},{"type":"heading","level":2,"text":"Back matter"},{"type":"para","text":"Code and data availability. The code — the model and evaluation lab repository, the sealed evaluation lockbox, the frozen train/development/lockbox split, and the per-experiment generators and tests — and the supporting data will be made available in a public repository."},{"type":"para","text":"Author contributions. JP: Conceptualization, Methodology, Software, Formal analysis, Investigation, Visualization, Writing – original draft, Supervision. JG: Software, Resources — built and operates the Internal Waves Service platform. AS: Resources — provided the initial classifier the work started from. ASF: Validation — expert adjudication of detections. All authors contributed to Writing – review and editing."},{"type":"para","text":"Competing interests. The authors declare that they have no competing interests."},{"type":"para","text":"Acknowledgements. Not applicable."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Internal solitary waves are among the most energetic features of the ocean, and their surface signature — alternating bands of increased and reduced roughness as the wave-induced currents strain the short surface waves that synthetic-aperture radar (SAR) responds to — is recorded in SAR imagery, which makes a satellite SAR archive a natural instrument for mapping where and how often they occur (e.g., Barintag et al. 2023). The Internal Waves Service builds that map operationally from Sentinel-1 Wave-mode data, running a convolutional classifier over the incoming vignettes and passing its detections to domain experts who confirm or reject each one. The classifier’s purpose is not to be the final word but to spend the experts’ time well: the archive is far larger than any team could inspect exhaustively (Sect. 2), so the model exists to concentrate scarce expert attention on the scenes most likely to carry a wave.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"That framing sets the cost of error. A false positive lands a wave-free scene on a validator’s desk and consumes the attention the service is built to conserve; a missed wave is largely recoverable, because Wave-mode acquisitions recur over the same locations on the orbit repeat and the archive is reprocessed, so the same site returns for another look. Precision therefore leads and recall is held to a floor rather than maximised — the operating discipline the whole effort is organised around.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The difficulty is not the model so much as how its skill is reported. The classifier was trained and evaluated, by the convention it inherited, on class-balanced data — equal numbers of wave and non-wave scenes — while the stream it actually runs on carries a positive rate of only about 0.05. A model that looks well-behaved on a balanced test set over-fires badly on that skew, and the balanced-test figures give no warning of it, because they never present the model with the negatives-to-positives ratio it will meet in production. The gap between the reported number and the fielded one is not noise or a tuning oversight; it is a systematic artefact of reporting at the wrong prior, and it is invisible precisely to the metric most papers quote (Dockès et al. 2021). Closing that gap — making the reported figure predict the operational one — is the problem this paper takes up, and the method it proposes is the contribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It would be natural to treat a prior mismatch as a training defect and correct it by retraining at the operational ratio, or by recalibrating the model’s probabilities to the deployment prior (Saerens et al. 2002; Heiser et al. 2020). Neither helps where it counts. At a fixed recall — the operating discipline above — prior correction and probability calibration are monotone transformations of the score and cannot move precision at all; they relocate the threshold, not the curve (Sect. 4.3). The mismatch is therefore an evaluation problem wearing the costume of a training one, and the response it calls for is an honest reporting method, not a prior-matched retrain. That inversion is the paper’s organising claim.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The contribution is threefold. First, a prior-matched reporting method: an operational classifier is characterised by three figures rather than one — its balanced-test performance, its performance on a frozen test set drawn at the operational prior, and its real operational performance from post-deployment expert adjudication — and the methodological content lies in the contrast among them, the balanced figure exposed as optimistic and the operational-prior figure built to predict the fielded one (Sect. 3). Second, a worked demonstration: a precision-first, leakage-controlled development cycle that improves the deployed classifier lever by lever, each intervention isolated and promoted only against a pre-registered margin, and each reported for what it did and did not buy — including the levers that bought nothing (Sect. 4). Third, the discipline behind the demonstration: a dataset pinned before it is split, spatial leakage controlled at the level of the true image footprint, an operating point fixed on development data, and a sealed evaluation set read exactly once at a pre-registered threshold — so that the honest negatives and the single-read certification are themselves a reproducibility result (Sects 3, 5). A by-product is itself a first: the Internal Waves Service is, to our knowledge, the first operational classifier built to screen the whole Sentinel-1 Wave-mode archive for internal solitary waves, so the operational positive rate its adjudicated stream reveals — roughly one scene in twenty (Sect. 2.3) — is an initial empirical estimate of that rate, the very prior that was unmeasurable when the balanced training convention was fixed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is a successor to the pipeline-selection study that established the service’s classifier (Pinelo et al. 2026), not a correction of it. That work chose a deployable network on operational grounds under the balanced convention then in force, and the choice was sound on the evidence available. What changed is the ground beneath it: the operational positive rate became known, more validated data accrued, and the question shifted from which pipeline to field to how far the fielded one can be pushed and how its skill should be reported. The present paper restarts from a smaller deployed network rather than the larger candidate, and takes it as far as the data allow — reaching the conclusion that the binding constraint has moved off the model and back onto the data (Sect. 4.6).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 2 sets out the service and the evaluation problem in operational terms; Section 3 defines the three-number instrument and the frozen, leakage-controlled test set behind it; Section 4 documents the development cycle lever by lever; Section 5 reports the instrument’s output on the promoted model and its sealed certification; and Sections 6 and 7 draw out what transfers to other operational Earth-observation services and what the next cycle requires.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2. The Internal Waves Service and the evaluation problem\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1 The service\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Internal Waves Service classifies the operational Sentinel-1 Wave-mode archive for internal solitary waves, working from the short vignettes the mission acquires continuously along-orbit (Santos-Ferreira et al. 2025; Pinelo et al. 2025). The growing catalogue currently holds over 1.7 million vignettes, a fraction of the full Sentinel-1 Wave-mode archive, which we estimate at approximately 17 million vignettes; it grows through a continuous harvest, so any evaluation set must be frozen against a moving target (Sect. 3.2). Model detections are not taken at face value: they enter a review queue where domain experts adjudicate them, and the queue is organised by a model-confidence band so reviewers can concentrate where their judgement is most useful. The expert time that adjudication consumes is the resource the whole effort exists to conserve.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2 Why precision leads\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two error types do not cost the same. A false positive spends an expert’s attention on a scene that holds no wave; a missed positive is largely recoverable, because the same location is reimaged on the 12-day orbit repeat and the archive is periodically reprocessed, so a wave missed on one pass has further chances to be caught. Precision therefore leads, and recall is held at a floor of 0.80 rather than pursued as a target (Sect. 3.3). This is not merely operational convenience: the asymmetry is what makes a false positive the expensive error, and it sets the direction of the entire development effort — lift precision, hold recall.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"That precision leads is itself the current end of an inversion, and the history matters because it explains the class balance the model inherited. The service began recall-first. With few confirmed examples to learn from, the early priority was to harvest positives aggressively and build a labelled set, and a missed wave then was a training example lost rather than a recoverable operational miss — the low default decision cut was set in that spirit, to leave few real positives unnoticed and route the boundary cases to human labelling. The training data were assembled at an even, one-to-one class balance, a ratio fixed at the outset around the service’s competition origins, when the operational positive rate was not yet known, and that convention was carried forward unexamined through each successive model. The ground has since shifted: the operational rate is now known to be heavily skewed (Sect. 2.3), the positive set is large enough that harvesting is no longer the constraint, and the cost that now binds is the validator time each false positive consumes. Hence precision-first, with recall held at its floor rather than maximised. The stance is the current one, not a permanent one — as the two error costs converge, with a larger positive backlog and a service no longer starved of examples, the emphasis is expected to normalise toward F1 — but the recall floor of 0.80 is held throughout.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recoverability, the property that lets recall sit at a floor at all, rests on more than the orbit repeat. Each improved model reprocesses the entire catalogue rather than only scoring new acquisitions, so a positive missed by one model generation has a further chance to be caught by the next: the recoverable miss is recoverable across model iterations, not only across passes of the same model. This is a deployment property the service was built for. The predecessor work chose a fast, energy-frugal Julia inference stack precisely so that reclassifying the whole archive with a new model is a same-day operation (Pinelo et al. 2026), and reprocessing wholesale — rather than layering each model’s verdicts over the last — keeps the published catalogue labelled consistently under a single model version, which matters for a citeable, DOI-versioned dataset. The recall floor is therefore underwritten by an operational guarantee: what one model misses, its successor re-examines.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.3 Starting state, and what a threshold can buy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The deployed model is a compact 34,465-parameter convolutional network (SAR_CNN v2), trained and validated under an even class balance — the convention inherited from the service’s origins — and then applied to a stream whose true positive rate is only about 0.05 — now measurable from the service’s own adjudicated detections, but unknowable when the balanced convention was set. The mismatch surfaces as over-firing. At the default 0.5 decision cut the model attains a recall of 0.963 but a precision of just 0.192 on the adjudicated set, flagging 26,472 false positives; and because validators preferentially review likely positives, even that precision is optimistic against the live stream rather than pessimistic.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Raising the decision threshold recovers precision without retraining. At 0.80 the false positives fall to 6,193 — a reduction of 76.6% — for a recall that eases only from 0.963 to 0.818, lifting precision to 0.464 (Fig. 1). This is the first and cheapest lever (Sect. 4.2): genuine, immediate relief for the validators, but a redistribution of errors along a fixed curve, not a lift of it — precision at the recall floor stops well short of the target. Two facts frame everything that follows: the remaining gap is a model problem, not a threshold one, and the operational precision quoted here rests on the validated subset, so the honest question is not what the balanced test set reports but what the skewed stream would. The first motivates the development cycle (Sect. 4); the second motivates the evaluation instrument (Sect. 3).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1. Precision, recall, F1 and validator load (false-positive count) as a function of the decision threshold, for the two classifiers the service has operated. The current model ( 20260602-01 , solid) carries the operating-point decision; the retired April–June model ( 20260408T131223 , dashed) is drawn behind it, so the pair survives greyscale. Each rate panel states its own definition. The high-value 0.75–0.85 threshold band is shaded and the 0.80 candidate operating point is marked on the current model; the dashed line in the F1 panel is the project target of 0.95, which no threshold on either curve reaches — the gap a threshold cannot close, and what motivates the model-side development cycle (Sect. 4). The rates are computed on the validated subset and so run optimistic against the full stream.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3. Methods — the evaluation instrument\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.1 A three-number reporting axis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An operational classifier is characterised here by three performance figures rather than one, and the methodological claim rests on the relationship among them, not on any single value.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first, balanced-test performance, is the figure a conventional pipeline reports: precision, recall and F1 on a held-out set with equal class proportions (Maxwell et al. 2021a). It is what the Internal Waves Service pipeline reported for its earlier models, and, as Sect. 5 shows, it is systematically optimistic once the deployment stream is skewed (Saito and Rehmsmeier 2015).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The second, operational-prior performance, scores the same model, at the same decision threshold, on a frozen test set drawn at the operational positive rate (0.05; Sect. 2) rather than at parity (Maxwell et al. 2021b). This is the honest predictor of fielded behaviour: it asks what precision a validator would actually face, and it is the figure the present method proposes as the one to report.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The third, real-operational performance, is obtained after deployment, from expert adjudication of the model’s own detections on the live stream. It is the ground truth the second figure is constructed to predict. Because it requires the model to run in production through an adjudication window, it is stated as a forward commitment rather than approximated by any test-set surrogate (Sect. 5, Sect. 6).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The instrument is the contrast. For a model trained and validated under class balance, figure (1) diverges sharply from figures (2) and (3): the cautionary reading, and the reason balanced-test reporting misleads. For a model reported under this method, figure (2) is built to track figure (3): the corrective reading. As a preview of the effect (Sect. 5), the model promoted in Sect. 4 scores a precision of 0.996 on a balanced set and 0.927 at the operational prior — the same model at the same threshold, the difference being the prior alone. Table 1 (Sect. 5) sets out the full axis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2 A frozen, leakage-controlled test set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The scarce class sets the terms. Confirmed internal-wave positives number 6,868 against 126,742 confirmed negatives, so the positive partition, not the negative, bounds the precision at which any figure can be resolved; it is sized from the confidence interval required on precision and recall at the recall floor, not from a round fraction. The dataset is pinned before the split is drawn (20260621T120557Z), so the continually growing archive cannot leak into a frozen evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Spatial leakage is controlled at the level of the hotspot , defined by true image-footprint overlap rather than a centroid radius. Sentinel-1 Wave-mode acquisitions recur over fixed locations on the 12-day orbit repeat, and the surface forcing at a site might evolve little between passes, so near-identical scenes would otherwise straddle the split and inflate every figure (Kattenborn et al. 2022; Roberts et al. 2017). Two split regimes follow from this unit: an interpolation split, which separates acquisitions within a hotspot (the reads reported here), and a stricter extrapolation split, which holds out whole hotspots and region cells to probe generalisation to unseen sites (Valavi et al. 2019) (a design provision). The data are partitioned into train, development, and a sealed lockbox, with the test prior fixed at 0.05 by construction; composition is 4,466/81,104 (train), 865/16,435 (dev) and 1,537/29,203 (lockbox), positives/negatives (Fig. 2).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The lockbox is the central pre-deployment estimate of operational precision, and its credibility rests on how it is handled rather than on its size. It is sealed off-site with a manifest and a per-file checksum, and read exactly once, at a threshold fixed in advance on the development set. Every read is pre-registered: the scoring script and its test are committed before any number is seen, in a two-commit structure for the consent-gated read. The expected ordering is train >= dev >= lockbox >= production, the gap small and downward; a lockbox figure that slightly exceeds dev is treated as same-prior sampling on a larger independent draw, not as improvement and not as leakage (Sect. 5).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2. The leakage-controlled split. (a) The leakage unit is the hotspot — a set of Sentinel-1 Wave-mode acquisitions whose image footprints overlap at a recurrence site. In the interpolation split used for the reads reported here, a hotspot’s passes are shared across train, development and lockbox; the stricter extrapolation split (a design provision) instead holds a whole hotspot out as an unseen site. (b) The 6,868 confirmed positives on a Natural Earth projection, coloured by split, showing the recurrence-site clustering the hotspot unit exists to guard against; per-split positive counts are given in the legend. The 126,742 confirmed negatives, held at the operational prior (π = 0.05), are a count rather than a geography and are not mapped. The dataset was pinned (20260621T120557Z) before the split was drawn, so the continually growing archive cannot leak into it.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3 Operating point and recall floor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cost of the two error types is asymmetric. A false positive lands on a validator’s desk and consumes expert time; a stochastic miss is recovered on the next reprocessing pass. Precision therefore leads, and recall is held at a floor of 0.80 rather than treated as a target: reprocessing recovers per-pass stochastic misses but not systematic blind spots, so the floor guards against trading away recoverable recall while precision is driven up.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The operating point is selected on the development set from a band of thresholds spanning the recall floor, then projected onto the frozen lockbox scores; the sealed set never selects its own operating point. The chosen threshold (0.935 for the model promoted in Sect. 4) ships as a recommendation for the validator review queue’s confidence band, not as a recut of the stored classification label. This keeps the lever per-session: a validator under load tightens their own queue without a default recall cut being imposed on the whole service.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4. Methods — the development cycle\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1 A moving binding constraint\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The improvement proceeded as a sequence of single-lever interventions, each isolated so its contribution could be credited on its own, and each carried forward only if it beat the incumbent by more than a pre-registered promotion margin (0.021, the seed-paired 95% confidence half-width). The organising observation is that the binding constraint moved: the lever that paid at one stage was inert at the next, and reporting where it moved — rather than only the final number — is the point of the demonstration. All development figures below are precision at the recall floor (0.80), scored at the operational prior on the frozen development split, across five seeds.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.2 Lever 1 — the operating threshold\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first and cheapest lever moves the operating point along the existing model’s precision/recall curve, with no retraining, and is reversible on the next reprocessing pass. It was taken in the predecessor work (Sect. 2) and delivered most of the immediate validator-facing relief, but the current curve tops out well short of the target: a threshold move redistributes errors, it does not lift the curve. This bounded what remained and set the agenda for the model-side levers that follow.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.3 Lever 2 — the data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The dataset was the first model-side lever, and its two effects were separated rather than moved together: the class ratio the model trains under, and the negative variety it sees against the roughly 95% of negatives the balanced trainer discards (Kang et al. 2020; Branco et al. 2016). In a factorial over both, negative variety carried the precision — widening the negatives lifted operational precision by 0.045 at the recall floor, a difference that clears zero — while shifting the training ratio toward the operational prior did not, and if anything cost precision (-0.026 for the ratio contrast at wide variety). The best data configuration, wide variety at a balanced ratio, reached 0.832 precision at the floor.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is the first negative result, and it is structural rather than incidental: prior correction and probability calibration are monotone transforms of the score (Saerens et al. 2002; Heiser et al. 2020), so at a recall-pinned operating point they cannot move precision at all — they relocate the threshold, not the curve. A separate calibration arm confirmed the inertness by construction. The prior-matched training the mismatch might seem to call for is therefore the wrong response to it; the mismatch is an evaluation problem, which is the paper’s point. That inertness concerns the reported figure alone: because these transforms preserve the ranking, they cannot move the curve, and so cannot improve the precision this method reports — but they remain the appropriate instrument for placing the operating point once the prior shifts, a distinct use returned to under the out-of-time decay of Sect. 5.3 and Sect. 6.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.4 Lever 3 — capacity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With the data lever spent at 0.832, capacity was tested by a uniform width ladder over the deployed three-block shape, architecture the sole variable. Doubling width promoted — 0.868 precision at the floor, 0.036 over the data-lever endpoint — lifting the parameter count from 34,465 to 131,329. But the ladder then flattened: tripling width (290,657 parameters, near the scale of the predecessor paper’s headline net at 283,329 (Pinelo et al. 2026)) did not clear the promotion margin at the floor (-0.006, within margin) and quadrupling it (512,449 parameters) likewise (0.006). Beyond the first doubling, added width did not clear the margin at five seeds — an interval consistent with a small residual gain as well as with none, so this is a within-margin negative rather than a demonstrated ceiling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ladder was stopped by a pre-registered two-trigger rule — diminishing dev gain within the promotion margin, and the onset of a train-dev gap — rather than at a fixed large network. The choice not to overstretch is a scientific one, not only a regularisation one: a model overfit to the known recurrence sites stops generalising to the unseen hotspots the service exists to find, which is why the split holds sites out (Sect. 3.2) and why the endpoint is matched to the data rather than maximised.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.5 Lever 4 — the aggregation head\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Capacity being spent, the next lever was the aggregation mechanism — the operation that collapses the spatial feature map to a decision, swapped in isolation on the promoted width network. Replacing global-average pooling with generalised-mean (GeM) pooling promoted where added capacity had not: 0.924 precision at the floor, 0.056 over width. Two attention-pooling heads, with more parameters, did not lift at all — again isolating the mechanism, not the capacity, as the active ingredient. The learned pooling exponent settled well above unity (mean 4.9, initialised at 3.0), meaning the head weights the peak spatial responses over the mean.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mechanism has a concrete target. A recurring geophysical look-alike — a banded, quasi-periodic surface signature that mimics the along-crest periodicity of an internal-wave packet — forms a small, persistent tail of the false-positive population, clustered at a handful of recurring sites. Average pooling washes the banding into a single mean and fires; a pooling operation that preserves peak spatial structure can tell it apart. On the cross-seed false-positive core of 40 scenes, the width network fired on 35.8 on average and GeM on 21.2 — the mechanism suppressing roughly a third of the core. The residue of 12 is the same banded class rather than a geographic pocket (not polar-enriched, p ~ 0.58): what remains is a discrimination limit, not a location (Fig. 3).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3. The banded geophysical confuser. Sixteen validated Sentinel-1 WV vignettes from a single acquisition slot (WV slot 025) that fall at four locations once clustered by centre position — two Antarctic-coast footprints, one South Atlantic and one North Pacific site (each site’s clustered position, pass count and per-site structure-similarity are printed in its header). Every pass at every site carries an expert-negative verdict. Model behaviour is observable only on the passes that fall in the development split, and it splits by site: the three development passes at the 152.84°E Antarctic footprint fire, all scoring at near-saturation above the recall-0.80 development threshold, whereas the single development pass at the adjacent 153.07°E site does not fire; the remaining passes are training or lockbox and carry no development score (chip borders: vermillion fires, blue development-no-fire, grey non-development). The high within-site structure correlation at the two Antarctic footprints — the speckle-suppressed 32×32 area-average measure of the data-integrity audit, near zero between different sites — establishes that these are the same banded surface signature recurring on the orbit repeat rather than duplicated scenes. That signature is a persistent, quasi-periodic surface texture, consistent with high-latitude sea-ice or wind-banding, that mimics the along-crest periodicity of an internal-wave packet, and it names the discrimination target for the aggregation-head work; gem’s suppression of it is reported in the accompanying text, not shown in this figure. These scenes are drawn from the unanimously verified pool, so they are genuine negatives on which the model is misled by real geophysics, not by mislabelling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.6 Verdict: where the constraint moved\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The improved model clears the ranking target it was optimised against — 0.924 precision at the recall floor. Its F1 at the floor is 0.860, short of the 0.90 interim bar and well short of the 0.95 destination. A precision-ceiling read found no hard architectural wall — F1 >= 0.95 is reachable in principle once the reducible false-positive mass is removed — but against the full false-positive mass the best attainable F1 is 0.878, held down by an irreducible-for-now residue of roughly 57.6 false positives per seed. Symmetrically, a systematic false-negative core of 34 scenes sits flat across the recall band, so the operating point is a recoverable-recall dial above a fixed floor rather than a trade against systematic blind spots.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Read together, capacity gains have fallen within the promotion margin and the aggregation lever is spent, yet there is no ceiling — the binding constraint has moved off architecture and onto the size of the reducible false-positive mass, which is to say onto more validated data. That is where the next cycle goes, and it is the empirical basis for restarting from data rather than from a larger network.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4. The development ladder: precision at the 0.80 recall floor, scored at the operational prior across five seeds, as the retrain cycle advanced through its three model-side levers. Each stage’s promoted model (filled, connected) becomes the next stage’s baseline, so the blue spine is the chosen model as it developed — widening negative variety (0.832), then doubling width (0.868), then the generalised-mean (GeM) pooling head (0.924). The rungs tried and not carried forward sit muted beneath each stage and are named: the discarded data cells at stage one, the parameter-neutral and higher-capacity architecture variants at stage two (including the width multiples that did not clear the margin), and the two attention-pooling heads at stage three. Error bars are the iteration-1 factorial interval and, for iterations 2 and 3, the five-seed t-interval; a rung was promoted only on beating the incumbent by more than the pre-registered margin (0.021, the seed-paired 95% half-width). Sect. 4.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5. Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The development-cycle results are reported lever by lever in Sect. 4 and drawn together in Fig. 4; they are not restated here. This section reports the instrument’s output on the promoted model: the three-number axis (Sect. 5.1), the sealed-lockbox certification behind its operational-prior cell (Sect. 5.2), an out-of-time robustness check on that cell (Sect. 5.3), and the operating point set against the systematic-miss floor (Sect. 5.4).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.1 The three-number axis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 sets out the three-number axis for the incumbent operational model and for the model promoted in Sect. 4. The methodological content is in the contrasts between the cells, not in any single value. Of the six cells, four are filled from landed reads, one is left n/a by design (the incumbent’s operational-prior cell; see below), and one — gem’s real-operational figure — is pre-registered as a forward commitment rather than estimated. The cautionary contrast (the incumbent’s balanced-test figure against its real operational figure) and the prior-effect contrast (gem’s balanced against its operational-prior figure) can both be read now; the corrective contrast (gem’s operational-prior figure tracking its real operational figure) is what the deployment window will close.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the incumbent — the deployed 34,465-parameter network trained under class balance — the cautionary contrast is between its balanced-test figure and its real operational figure, and both are now in hand. Its balanced-test performance is precision 0.794, recall 0.964 and F1 0.871 (area under the ROC curve 0.961), computed as the parity projection of the deployed 0.5 cut over the full unanimous verified pool at complete coverage — the same operation that produces gem’s balanced cell, so the two balanced figures are comparable by construction rather than through two different resampling schemes. Its real operational precision, from validator adjudication at the same 0.5 cut, is 0.192 at recall 0.963, on the adjudicated subset and so optimistic against the full stream (Sect. 2). The divergence is the cautionary reading in a single pair of numbers: a balanced-test precision of 0.794 against a fielded precision of 0.192 at the identical threshold, the balanced figure standing far above operational skill because the reporting prior, not the model, has changed. Two caveats travel with the balanced cell and neither softens the contrast: the verified pool is review-selected rather than a stream sample, and the incumbent’s June training membership is unrecoverable — it is the 34,465-parameter network’s own evaluation, not the predecessor paper’s headline figure, which belongs to a different and larger net. The operational-prior cell is left unfilled for the incumbent by design: scoring the old model on the frozen operational-prior set would add nothing its real operational number does not already show, and the cautionary half rests on the balanced-versus-real gap, not on an intermediate estimate.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For gem the corrective contrasts are both in hand. At the operational prior its precision is 0.927, recall 0.827 and F1 0.874 (the sealed lockbox; Sect. 5.2), and this is the figure built to track the real operational number the deployment window will produce (Sect. 6). The same model, at the same threshold, scored on a balanced set reads precision 0.996, recall 0.827 and F1 0.904 — the recall unchanged, because it is a property of the positives alone, and only the precision moving with the prior. The consequence is the paper’s thesis in a single row: gem’s balanced F1 of 0.904 clears the 0.90 interim bar, while its operational-prior F1 of 0.874 does not — the balanced figure crossing a bar the honest figure misses, on identical bytes at an identical threshold, the difference being the reporting prior and nothing else.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"gem’s real operational cell is the one number the method cannot borrow from a test set: it creates its own ground truth prospectively, by adjudicating gem’s detections once gem is the production model, and is therefore reported in subsequent work — this paper pre-registers it as the forward validation that closes the loop (Sect. 6). It is the single open cell, and it is not filled with a surrogate: the out-of-time read of Sect. 5.3 is a robustness check on the operational-prior cell, explicitly not a stand-in for the post-deployment number. Table 1 therefore stands with every present-tense cell filled from a landed read and gem’s real operational cell pre-registered here as subsequent work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1. The three-number reporting axis, precision / recall / F1 per cell, scored at the dev-fixed threshold (0.935) and, for the operational-prior and balanced cells, at the operational prior (0.05) and at parity respectively. The incumbent real-operational figure is measured on the validated subset and is optimistic against the full stream (Sect. 2). gem’s real-operational figure is pre-registered here and reported in subsequent work, once gem is in production (Sects 5.1, 6).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.2 Lockbox certification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The operational-prior cell rests on a single, sealed read, and its credibility is a matter of how that read was conducted (Sect. 3.2). The lockbox holds 30,740 scenes (1,537 positive, 29,203 negative) drawn at the operational prior and pinned before the split (20260621T120557Z); it was read exactly once, at the threshold fixed in advance on the development set (0.935), from the same frozen bytes that will reach production — the artefact scored here is byte-for-byte the artefact that deploys (Sects 3.2, 6).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"gem’s discrimination survives the move from development to the sealed set intact: the area under the ROC curve is 0.993 on the lockbox against 0.994 on development, so there is no optimism penalty in ranking. At the fixed threshold the confusion matrix is 1,271 true and 100 false positives against 29,103 true and 266 false negatives — a false-positive rate of 0.0034 on the sealed negatives, which is the validator-facing quantity: 100 scenes wrongly flagged against 29,103 true negatives.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One feature of the read is unexpected: the lockbox precision slightly exceeds the development precision. The lockbox’s own recall-0.80 operating point reaches precision 0.939, and at the transferred development threshold the recall lands at 0.827, above the 0.80 floor. This is neither improvement nor leakage. The expected ordering is train >= dev >= lockbox >= production, the gap small and downward; a lockbox figure a little above dev is same-prior sampling on a larger independent draw, the ordinary variation between two finite samples of one distribution, and the split’s construction (Sect. 3.2) forecloses the leakage reading (Fig. 5).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 5. Lockbox certification and its out-of-time robustness, as precision–recall at the operational prior (π = 0.05). The sealed-lockbox curve (solid, n = 30,740) is the honest pre-deployment predictor of operational precision; the fresh-holdout curve (dashed, n = 11,843) scores the same deploy bytes at the same fixed threshold (0.935) on verified scenes whose last verdict post-dates the split-lock pin (20260621T120557Z) by about ten days, anti-joined off every split member. The two findings are separate. Discrimination transfers — the curves nearly coincide and the area under the ROC curve moves only from 0.993 to 0.987. The fixed operating point (marked on each curve, joined by the dotted line) does not: precision falls from 0.927 on the sealed set to 0.810 out-of-time, and the false-positive rate rises roughly threefold (0.0034 to 0.010) at the same threshold. Reported exploratory and pre-registered: a single out-of-time window on the review-selected pool, not a trend (Sect. 5.3).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.3 Out-of-time robustness check\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An independent, out-of-time check on the operational-prior cell has been run, and is reported here as an exploratory, pre-registered result. It scores the same deployed bytes at the same fixed threshold (0.935) on 11,843 verified scenes whose last verdict post-dates the split freeze (20260621T120557Z), anti-joined against every split member so none could have leaked into training or into the split’s selection. The read separates into two findings. Discrimination transferred intact: the area under the ROC curve is 0.987 out-of-time against 0.993 on the sealed lockbox. The fixed operating point did not: at the operational prior the precision fell from the lockbox’s 0.927 to 0.810, and the false-positive rate rose from 0.0034 to 0.010 at the same threshold, over a window of only about ten days past the freeze. This is not a composition artefact — the drop holds across a prior-sensitivity band (0.771 to 0.837) and the average precision fell in step (0.890 against the lockbox’s 0.944), so the precision–recall curve genuinely moved rather than the scoring prior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two tiers of caveat of the result. The full read is a real estimate but a single out-of-time snapshot on the review-selected verified pool — one window, not a trend. The footprint-disjoint new-site subset is underpowered on positives (736 scenes, 20 positive): its AUC 0.986 suggests the model is not blind to unseen geography, but its precision and recall are not reportable at that sample size.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The finding strengthens rather than dents the thesis. Discrimination is stable out-of-time while a fixed development-set operating point is not, and a balanced-test evaluation would have surfaced none of it. That a falsifiable, pre-registered check caught real out-of-time decay is the case for prior-matched, out-of-time evaluation; it also sets a concrete deployment requirement — that the operating point be validated against a current catalogue slice rather than inherited from the development set (Sect. 6). Because the check was pre-registered, with the scoring script and its test committed before any number was seen, it stands report-regardless (Fig. 5).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.4 The operating point and the systematic-miss floor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying the fixed threshold places the lockbox at recall 0.827, above the 0.80 floor, and the recall it forgoes above the floor is the recoverable kind rather than a systematic blind spot. The systematic false-negative core — the positives missed across model families, cross-architecture and cross-recipe — numbers 34 scenes and sits flat across the recall band from 0.70 to 0.85, so moving the operating point within the band trades stochastic misses, recovered on the next reprocessing pass, not systematic ones. At recall 0.80 on the deploy seed the positives forgone number 173, of which the systematic core is the fixed floor and the remainder the recoverable tail. That floor corroborates, on a different model and population, the deployed network’s own sub-threshold miss rate of 0.038 (Sect. 4; corroborating, not load-bearing). The operating point is therefore a recoverable-recall dial above a fixed floor, which is what the recall-floor discipline requires (Fig. 6).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6. The systematic false-negative core, on the development positives. (a) For each confirmed positive, the number of the promoted model’s five seeds that miss it: most are missed by no seed, while a persistent tail (106 positives) is missed by all five. The subset missed across every model family, recipe and seed — the systematic core — is 34 positives. (b) Positives forgone at each candidate recall operating point, with the systematic core (red) drawn inside the forgone total (blue) so the stochastic, recoverable remainder reads off directly. The core is flat across the band, so moving the operating point within it trades recoverable misses, not systematic blind spots. Because the development positives are drawn from the unanimously verified pool, the core is a floor: it under-states the systematic-miss rate on the operational stream.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6. Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The instrument gives two readings, and the paper’s claim is the pair. Read against the incumbent, it is cautionary: a network trained and evaluated under class balance carries a balanced-test figure that stands far above the precision a validator actually meets on the skewed stream, and the gap is not a defect of the model but an artefact of reporting at the wrong prior (Sect. 5.1). Read against the promoted model, it is corrective: the operational-prior figure is constructed to answer the question the balanced figure evades — what precision the fielded model will deliver — and it is the number the method proposes as the one to report. The contribution is not either figure alone but the discipline of carrying both, so that the distance between them is visible rather than hidden.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The corrective half rests on a prediction this paper pre-registers but does not yet close. The operational-prior cell is built to track the real operational number that post-deployment adjudication will produce (cell 3), and that number is reported in subsequent work, once the promoted model is in production and its own firings have been adjudicated over an operational window. What the present paper can show is that the operational-prior figure behaves as a predictor should under the one independent test available before deployment: on out-of-time data, past the freeze and disjoint from the split, the model’s discrimination held (Sect. 5.3). The same check also delivered the sharpest caution in the paper — the fixed operating point did not transfer unattended, its precision decaying materially over a short out-of-time window while the ranking held. That the operational-prior figure survives out-of-time as a ranking while the threshold placed on it drifts is the distinction the method draws: the number is a stable ranking, and the condition under which its operating point predicts is stated rather than assumed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several limitations bound the reading, and they are stated in the direction their optimism runs. The ground truth throughout is drawn from a validator review queue that is itself organised by model confidence, so both the incumbent’s real operational figure and the sealed operational-prior read rest on a review-selected pool rather than a uniform sample of the stream; the selection runs optimistic in a specific direction: because the queue preferentially surfaces the scenes the model scores most confidently positive, the adjudicated pool is enriched with the high-confidence flags on which precision is naturally highest, while the lower-confidence flags that still clear the threshold — and fail more often — are under-reviewed, so the operational precisions quoted lean high against the full stream rather than conservative. The incumbent’s balanced cell carries a second caveat of its own: the membership of its training set is not now recoverable, so its balanced-test figure is reported as the deployed network’s own evaluation and is not equated with any headline from the predecessor paper, whose network was larger and different. A third is conditional rather than fixed: the recall-floor discipline tolerates missed positives because reprocessing recovers them on a later pass, but that recovery holds only while the model still generalises to sites it has not seen — an overfit model would keep missing novel-site positives, which reprocessing could then never recover. The out-of-time read speaks to this directly: on footprint-disjoint new sites the model’s discrimination held (Sect. 5.3), evidence that it is not blind to unseen geography, though the positive count there is too small to quantify new-site recall. Holding capacity back (Sects 4.4, 3.2) is the design response to this risk rather than a proof against it. And the loop is not closed: cell 3 remains outstanding until deployment and adjudication, so the claim that the operational-prior figure predicts the operational one is, at submission, a falsifiable commitment rather than a demonstrated equality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The operating-point decay is a limitation of a different kind, because it is a finding about deployment rather than about the evaluation. A threshold fixed on the development set is not a set-and-forget quantity: the out-of-time read shows its precision can fall while the model’s discrimination is intact, which means the operating point must be validated against a current slice of the catalogue at deployment and treated as something to recalibrate on a schedule, not inherited unchanged from development. Prior correction and calibration are the right tools for this: they govern the placement of the operating point, which is distinct from the reported curve they cannot move (Sect. 4.3) (Saerens et al. 2002; Heiser et al. 2020). This is the concrete operational requirement the paper hands to the service, and it is the check the post-deployment validation gate is designed to perform before cell 3 is read.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The reporting method is not specific to internal waves, to Sentinel-1, or even to a precision-first stance. Its one precondition is prior shift: an operational classifier trained and reported under one class balance but deployed against a materially different one. Wherever that holds, the three-number axis applies as a portable reporting convention — report the balanced figure if convention demands it, but report alongside it the operational-prior figure and commit to the post-deployment number, so that a reader sees the prior effect rather than having to infer it. The precision-first operating discipline of this particular service (Sect. 2.2) is a separate matter — the current end of an inversion from the recall-first stance of the service’s early life, and one expected to normalise as the costs of the two errors converge — and the method does not depend on it: a service that weighted recall, or weighted the two equally, would report the same three numbers and read the same contrast. This echoes the transferable-methodology framing of the predecessor paper, carried now from pipeline selection to performance reporting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the development cycle carries a geoscience point that outlives the particular model. The decision not to spend capacity past the point where it stopped paying was a scientific choice as much as a regularisation one: a network overfit to the recurrence sites the archive already knows would stop generalising to the unseen sites the service exists to find, which is why the split holds sites out and why the endpoint was matched to the data rather than maximised (Sects 4.4, 4.6). That the binding constraint ended the cycle back on the size of the validated dataset, rather than on architecture, is itself the finding that sets the next cycle’s direction — more validated data from the growing archive, rather than a larger network on the data in hand.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7. Conclusions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Balanced-test accuracy is the wrong number to trust for an operational classifier whose deployment stream is skewed away from the prior it was evaluated under. The proposal here is to stop reporting it alone: characterise the model by three figures — its balanced-test performance, its performance on a frozen test set drawn at the operational prior, and its post-deployment operational performance — and read the contribution in the contrast between them. The balanced figure is exposed as optimistic, the operational-prior figure is built to predict the fielded one, and the distance between the two is the prior effect made visible rather than left to be inferred.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrated the instrument across a full precision-first development cycle on the Internal Waves Service, reporting each lever for what it did and did not improve. Widening negative variety and, later, replacing the pooling head both lifted precision at the fixed recall floor; shifting the training ratio toward the operational prior and adding raw capacity did not; and probability calibration could not, by construction, being a monotone transform of the score at a recall-pinned operating point. The negatives are as much a result as the promotions — they locate the mismatch as an evaluation problem rather than a training one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The figures are only as trustworthy as the discipline behind them. The dataset was pinned before it was split, spatial leakage was controlled at the true image footprint, the operating point was fixed on development data, and the sealed evaluation set was read exactly once at a pre-registered threshold, with every read’s scoring script and test committed before any number was seen. The out-of-time check applied that same discipline, catching a real decay in the fixed operating point that a balanced evaluation would not have surfaced.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two things remain for the next cycle. The loop is not yet closed: the promoted model’s operational-prior figure is a pre-registered prediction of its real operational performance, and only post-deployment adjudication of its own firings will confirm it — the third number, reported in subsequent work. And the binding constraint has moved. Added capacity stopped clearing the promotion margin and the aggregation lever is spent, yet no architectural ceiling was found; what limits precision now is the size of the reducible false-positive mass, which is to say the validated data. The next cycle therefore restarts not from a larger network but from more of the growing archive — and, at deployment, from an operating point revalidated against a current catalogue slice rather than inherited from development.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Back matter\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Code and data availability. The code — the model and evaluation lab repository, the sealed evaluation lockbox, the frozen train/development/lockbox split, and the per-experiment generators and tests — and the supporting data will be made available in a public repository.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Author contributions. JP: Conceptualization, Methodology, Software, Formal analysis, Investigation, Visualization, Writing – original draft, Supervision. JG: Software, Resources — built and operates the Internal Waves Service platform. AS: Resources — provided the initial classifier the work started from. ASF: Validation — expert adjudication of detections. All authors contributed to Writing – review and editing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Competing interests. The authors declare that they have no competing interests.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Acknowledgements. Not applicable.\"\n }\n]","isConversational":false}}},{"rowIdx":484,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07144"},"doi":{"kind":"string","value":"10.5281/zenodo.21237993"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Fractal KV-Cache Archives: Lossless Symbolic Storage with In-Place Retrieval for Long-Context LLM Inference"},"authors":{"kind":"list like","value":["Vladimir Gusev"],"string":"[\n \"Vladimir Gusev\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07144v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Serving a transformer language model over a long context is increasingly a memory problem rather than a compute problem. To generate each new token the model attends to cached key and value vectors for every previous position in every layer; for GPT-2 this KV cache is 2 × 12 × 12 × 64 = 18 , 432 2\\times 12\\times 12\\times 64=18{,}432 scalars per token, and for production-scale models it reaches hundreds of kilobytes per token, so that for long documents the cache dwarfs the model weights and becomes the binding constraint on batch size and context length. The dominant responses are lossy: low-bit quantization of the cached tensors, eviction of “unimportant” positions, and offloading of cold cache to slower memory [ 1 , 2 , 3 , 4 ] ."},{"type":"para","text":"This paper isolates a question that sits downstream of the lossy step and is usually left implicit. Suppose a position’s KV state has already been reduced to a handful of codebook indices by vector quantization. Those indices are a symbol stream. How should the stream be stored , and–more interestingly–can the storage layer serve retrieval and random access rather than being an inert blob? We show that a lossless contractive iterated-map code answers both. Concretely, we test three claims: (i) the code is a lossless, linear-time store with sub-millisecond random access and append; (ii) with a small exact window, per-head residual VQ archives the remaining cache at more than 30 × 30\\times compression versus an fp16 cache for a bounded, measured perplexity cost; and (iii) substring queries resolve directly on the stored vectors at full recall. The success criterion for each is stated with its experiment."},{"type":"heading","level":4,"text":"The code."},{"type":"para","text":"We use a contractive iterated-map code that reads a symbol sequence over an alphabet of size N N and produces a trajectory of points in the plane: a regular N N -gon is fixed, one vertex per symbol, and each symbol c k c_{k} maps the running point by p k = V ( c k ) + r ( p k − 1 − V ( c k ) ) p_{k}=V(c_{k})+r\\,(p_{k-1}-V(c_{k})) for a contraction ratio r r . These N N contractions form an iterated function system, and the set of points it can reach is self-similar: each symbol’s “cell” is a scaled copy of the whole hull, which itself contains a scaled copy for every next symbol, and so on at every depth. That self-similar attractor — the Sierpinski gasket in the three-symbol midpoint case — is the fractal the title refers to; the code is simply a walk on it, one step per symbol. Decoding inverts the map: the symbol is the vertex whose contracted image (“cell”) contains the point, and p k − 1 = ( p k − ( 1 − r ) V ( c k ) ) / r p_{k-1}=(p_{k}-(1-r)V(c_{k}))/r . Uniqueness requires the N N cells to be disjoint, which bounds the ratio by a packing (“kissing”) constant r ≤ r N r\\leq r_{N} ; the familiar midpoint code r = 1 2 r=\\tfrac{1}{2} is exactly the boundary case for N ≤ 4 N\\leq 4 and must be tightened for larger alphabets. Because inversion amplifies floating-point error by 1 / r 1/r per step, one IEEE-754 double reliably carries ≈ 44 / log 2 ( 1 / r ) \\approx 44/\\log_{2}(1/r) symbols, so a practical codec stores one point per fixed-length span and reconstructs each span by anchored backward search. The construction is classical in the alignment-free sequence-analysis literature, where it is known as chaos game representation and its generalization to arbitrary alphabets as universal sequence maps [ 7 , 8 ] ; our contribution is not the code, nor vector quantization of the KV cache under a random-access constraint (a concurrently active topic [ 5 , 6 ] ), but the use of this particular code as a KV archive that is at once a store, a random-access decoder, and an in-place search index."},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"A KV-archive format. We frame a lossless contractive iterated-map code as the serialization layer for a quantized KV cache, and verify it is lossless, linear-time, and offers O ( 1 ) O(1) random access to any past position and O ( 1 ) O(1) amortized append–the operations a growing cache needs and that a general-purpose byte compressor does not provide (Section 2 )."},{"type":"para","text":"A controlled quantizer study on long contexts. With a small exact window and the remainder archived, we sweep pooled vs. per-head codebooks and residual depth on GPT-2 at 1024-token contexts. Per-head codebooks Pareto-dominate pooled ones at equal bit budget (Fig. 2 ), and we quantify a ∼ 4 × \\sim\\!4\\times key/value damage asymmetry (direction consistent with [ 4 ] ) that a bit-asymmetric hybrid exploits (Section 3 )."},{"type":"para","text":"The archive is also an index. Approximate substring retrieval runs directly on the stored vectors–nearest-neighbor distance is a graded suffix-similarity–and matched context is decoded from the matched vector with no access to the surrounding text (Section 4 )."},{"type":"para","text":"All experiments use GPT-2 (124M) on a public-domain text and run in minutes on a single CPU; code and scripts are released for full reproduction at https://github.com/eighteight/fractal-kv ."},{"type":"heading","level":1,"text":"2 The codec as a KV archive"},{"type":"para","text":"We first establish that the code is a usable storage primitive independent of any model. Table 1 reports a self-contained benchmark of the codec on natural text: encoding is lossless across ASCII, Unicode, and degenerate inputs; encoding time is linear in length ( 0.68 μ 0.68\\,\\mu s/char, flat from 25 25 K to 400 400 K characters); random access to an arbitrary character in a 10 6 10^{6} -character document costs 311 μ 311\\,\\mu s (decoding only the span that contains it); and appending a token costs 175 μ 175\\,\\mu s amortized, re-walking only from the last stored anchor. These are the primitives a KV archive requires: append as the cache grows, and random access when a position is read back."},{"type":"para","text":"Figure 1 makes the alphabet/ratio constraint concrete: the contraction ratio must adapt to alphabet size for the code to remain uniquely decodable, which in turn sets the per-symbol precision budget and hence the span length used by the codec."},{"type":"heading","level":1,"text":"3 Quantizer study for the KV archive"},{"type":"heading","level":4,"text":"Setup."},{"type":"para","text":"We evaluate on GPT-2 (124M) using The Time Machine (Project Gutenberg, public domain). Perplexity is measured token-by-token over a 1024 1024 -token passage. Codebooks are trained by k k -means on KV vectors collected from a disjoint span of the same book. During evaluation we keep an exact window–the first 4 4 positions (attention sinks [ 1 ] ) and the most recent 32 32 –and, once a position ages out of that window, overwrite its cached K and V with their nearest codebook entries, so all later tokens attend to the quantized archive. This is the regime of interest: at 1024 1024 tokens the exact window is 3.5 % 3.5\\% of context and the model attends almost entirely to archived, quantized memory. Residual vector quantization (RVQ) of depth s s stores each vector as s s indices whose centroids sum to the reconstruction. Storage is reported as bytes/token of the serialized index stream (fractal codec) and as compression ratio against an fp16 cache ( 36 , 864 36{,}864 B/token); the exact baseline perplexity is 35.97 35.97 ."},{"type":"heading","level":4,"text":"Per-head codebooks dominate."},{"type":"para","text":"Table 2 and Figure 2 give the main result. Sharing one codebook across all heads of a layer (“pooled”) is substantially worse than a codebook per head at equal bit budget . Single-stage per-head VQ costs + 23.7 % +23.7\\% perplexity versus + 51.0 % +51.0\\% for pooled, and two-stage per-head RVQ costs + 15.0 % +15.0\\% versus + 37.2 % +37.2\\% ; in both cases per-head codebooks move the entire rate–distortion curve down. Each doubling of the index budget along the RVQ axis buys back a roughly constant 8 8 – 9 9 perplexity points, indicating a smooth operating curve rather than a cliff. Per-head codebooks add a one-time 36 36 MB of centroids ( 288 288 codebooks), trained in 108 108 s on CPU."},{"type":"heading","level":4,"text":"Keys are ∼ 4 × \\sim\\!4\\times more fragile than values."},{"type":"para","text":"Reusing the two-stage per-head codebooks, we quantize only values (keys kept exact) and only keys (values kept exact). Quantizing values costs + 4.0 % +4.0\\% perplexity; quantizing keys costs + 14.5 % +14.5\\% –nearly the entire + 15.0 % +15.0\\% of quantizing both, and the two effects compose almost independently ( 1.145 × 1.040 ≈ 1.19 1.145\\times 1.040\\approx 1.19 ). The asymmetry has a clear mechanism: values enter attention through a weighted average, which attenuates per-vector error, whereas keys enter through the query–key dot product that decides which memories are read, so key error corrupts attention routing itself. The direction of this effect matches the key/value distinction reported by KIVI [ 4 ] ; our contribution is to express it as a perplexity budget and to convert that budget into a bit allocation."},{"type":"heading","level":4,"text":"A bit-asymmetric hybrid."},{"type":"para","text":"Allocating four RVQ stages to keys and two to values lowers the cost to +11.2% perplexity at 1025 1025 B/token ( 36 × 36\\times vs. fp16), with keys-only- × 4 \\times 4 at + 9.4 % +9.4\\% confirming the key axis is where the budget is best spent. This is the best operating point in our sweep and the recommended default; the key-bit scaling law suggests further stages would continue to trade ∼ 5 \\sim\\!5 points of perplexity per doubling."},{"type":"heading","level":4,"text":"On the storage layer’s role."},{"type":"para","text":"We are explicit that the compression ratio in Table 2 is produced by the vector quantizer; the fractal codec is the serialization of the resulting index stream, and on these high-entropy indices its size is comparable to bit-packing and to a general-purpose byte compressor (within a few percent either way). Its value is not a smaller blob but a richer one: unlike a byte compressor, the archive supports O ( 1 ) O(1) random access to any archived position and O ( 1 ) O(1) append, and–as we show next–doubles as a retrieval index over its own contents."},{"type":"heading","level":1,"text":"4 The archive is also a retrieval index"},{"type":"para","text":"A property of the code makes the stored archive searchable without decompression. Because each step contracts by r r , a stored point is dominated by the most recent symbols, older ones decaying geometrically; two positions sharing an s s -symbol suffix therefore have points within 2 r s 2r^{s} of each other. Nearest-neighbor distance in point space is thus a graded suffix similarity, and thresholding at 2 r ℓ 2r^{\\ell} retrieves all positions whose length- ℓ \\ell suffix matches a query–a property known for the midpoint code in the sequence-analysis literature [ 9 ] and which we measure directly and exploit for KV-archive retrieval."},{"type":"para","text":"Figure 3 confirms the decay law across seven orders of magnitude on a 100 100 K-character corpus and reports retrieval quality. We encode a query from the origin and threshold squared distance over the stored points. Recall is 1.00 1.00 at every query length by construction–a true match must fall inside the ball–while precision is set by the index’s numerical precision relative to r ℓ r^{\\ell} : with a double-precision index ( 16 16 B/char) precision is 0.89 0.89 – 1.00 1.00 ; with single precision ( 8 8 B/char) recall is still 1.00 1.00 and the modest false-positive set is removable by decode verification, which itself runs inside the representation. Because the index lives in the plane rather than in the model’s hidden dimension, brute-force nearest-neighbor over the two-dimensional points costs only ≈ 0.9 \\approx\\!0.9 ms/query over 100 100 K positions (and is trivially accelerated by any spatial index)—this is a 2 2 D search, not high-dimensional approximate nearest neighbor, and should not be compared to embedding-space retrieval benchmarks. Crucially, the end-to-end loop touches no external text: a query returns a matched point, and the surrounding context is decoded backward from that point through stored anchors, reproducing the ground-truth context character for character. In an inference setting this is exactly the operation a retrieval-augmented decoder needs from its own past—locate archived positions whose recent context matches a probe, and rehydrate only those, without scanning or decompressing the rest of the cache."},{"type":"heading","level":1,"text":"5 Related work"},{"type":"para","text":"KV-cache compression spans low-bit quantization [ 3 , 4 ] , eviction of low-importance positions [ 2 ] , and the observation that a few initial “sink” tokens carry disproportionate attention mass [ 1 ] ; our exact-window scheme is a minimal combination of a sink prefix and a recency window. The key/value asymmetry we measure is not new: KIVI [ 4 ] already reports that keys and values have distinct distributions and require different quantization treatment (keys per-channel, values per-token); our contribution here is only to quantify the asymmetry as a perplexity budget on long contexts and to use it to allocate residual-VQ depth. Vector quantization of the KV cache under a random-access constraint is likewise an active topic: the concurrent FibQuant [ 5 ] is, like us, a fixed-rate vector quantizer for random-access KV compression evaluated on GPT-2, tracing a 5 5 – 34 × 34\\times memory–fidelity frontier, and vector storage engines for long-context inference are emerging [ 6 ] . Relative to FibQuant, which optimizes the codebook geometry, our quantizer study is deliberately plain (k-means / residual VQ) and our distinct claim is about the storage representation : the contractive iterated-map code—chaos game representation [ 7 ] and its arbitrary-alphabet generalization, universal sequence maps [ 8 ] —is simultaneously a lossless, randomly-accessible, appendable store and an in-place suffix-search index, the latter resting on a decay property established for that code [ 9 ] . To our knowledge that unification of storage and in-representation retrieval has not been applied to the KV-archive setting; it, rather than the quantizer or the code in isolation, is what this paper contributes. We developed this representation independently of, and contemporaneously with, FibQuant [ 5 ] ; we became aware of it, and of RetroInfer [ 6 ] , only after the experiments reported here were complete, and we cite both as related work rather than claiming priority over the overlap."},{"type":"heading","level":1,"text":"6 Limitations"},{"type":"para","text":"Our study is deliberately small-scale and the claims should be read accordingly. All results are on a single 124 124 M-parameter model, a single public-domain corpus, and a single 1024 1024 -token context length, using perplexity as the only quality metric; downstream-task accuracy and larger models are untested, and larger models are empirically more tolerant of KV quantization, so these numbers are plausibly conservative but not validated as such. The compression comes from the vector quantizer, and our per-head codebooks are trained per corpus rather than amortized across data; a deployable system would need corpus-independent or online codebooks. The retrieval demonstration is exact/approximate suffix matching over token strings, not semantic retrieval over embeddings–the stored vectors are not shown to encode meaning, only sequence structure. Finally, our serialization ties rather than beats byte-oriented coders on raw size; its advantage is the access pattern, not the ratio. We also do not benchmark against concurrent random-access KV quantizers such as FibQuant [ 5 ] , whose codebook design is complementary to and likely stronger than our plain k-means; a fair head-to-head, and combining their quantizer with our retrieval-capable storage, is left to future work."},{"type":"heading","level":1,"text":"7 Conclusion"},{"type":"para","text":"Treating the storage of a quantized KV cache as a first-class design choice, we showed that a lossless contractive iterated-map code is a practical archive format: linear-time, randomly accessible, appendable, and–uniquely–searchable in place. Around it we conducted a controlled quantizer study on long contexts, finding that per-head residual VQ Pareto-dominates pooled codebooks, that key quantization is about 4 × 4\\times more damaging than value quantization, and that a bit-asymmetric hybrid compresses the archived cache 36 × 36\\times versus fp16 at + 11 % +11\\% perplexity on GPT-2. The retrieval result points to the most interesting direction: an inference-time memory that is stored, decoded, and searched in one representation, so that a query can locate the archived positions whose context resembles it without rehydrating the cache. We release all code and scripts at https://github.com/eighteight/fractal-kv ; every figure and table reproduces from a single command on a laptop CPU."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Serving a transformer language model over a long context is increasingly a memory problem rather than a compute problem. To generate each new token the model attends to cached key and value vectors for every previous position in every layer; for GPT-2 this KV cache is 2 × 12 × 12 × 64 = 18 , 432 2\\\\times 12\\\\times 12\\\\times 64=18{,}432 scalars per token, and for production-scale models it reaches hundreds of kilobytes per token, so that for long documents the cache dwarfs the model weights and becomes the binding constraint on batch size and context length. The dominant responses are lossy: low-bit quantization of the cached tensors, eviction of “unimportant” positions, and offloading of cold cache to slower memory [ 1 , 2 , 3 , 4 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper isolates a question that sits downstream of the lossy step and is usually left implicit. Suppose a position’s KV state has already been reduced to a handful of codebook indices by vector quantization. Those indices are a symbol stream. How should the stream be stored , and–more interestingly–can the storage layer serve retrieval and random access rather than being an inert blob? We show that a lossless contractive iterated-map code answers both. Concretely, we test three claims: (i) the code is a lossless, linear-time store with sub-millisecond random access and append; (ii) with a small exact window, per-head residual VQ archives the remaining cache at more than 30 × 30\\\\times compression versus an fp16 cache for a bounded, measured perplexity cost; and (iii) substring queries resolve directly on the stored vectors at full recall. The success criterion for each is stated with its experiment.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The code.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use a contractive iterated-map code that reads a symbol sequence over an alphabet of size N N and produces a trajectory of points in the plane: a regular N N -gon is fixed, one vertex per symbol, and each symbol c k c_{k} maps the running point by p k = V ( c k ) + r ( p k − 1 − V ( c k ) ) p_{k}=V(c_{k})+r\\\\,(p_{k-1}-V(c_{k})) for a contraction ratio r r . These N N contractions form an iterated function system, and the set of points it can reach is self-similar: each symbol’s “cell” is a scaled copy of the whole hull, which itself contains a scaled copy for every next symbol, and so on at every depth. That self-similar attractor — the Sierpinski gasket in the three-symbol midpoint case — is the fractal the title refers to; the code is simply a walk on it, one step per symbol. Decoding inverts the map: the symbol is the vertex whose contracted image (“cell”) contains the point, and p k − 1 = ( p k − ( 1 − r ) V ( c k ) ) / r p_{k-1}=(p_{k}-(1-r)V(c_{k}))/r . Uniqueness requires the N N cells to be disjoint, which bounds the ratio by a packing (“kissing”) constant r ≤ r N r\\\\leq r_{N} ; the familiar midpoint code r = 1 2 r=\\\\tfrac{1}{2} is exactly the boundary case for N ≤ 4 N\\\\leq 4 and must be tightened for larger alphabets. Because inversion amplifies floating-point error by 1 / r 1/r per step, one IEEE-754 double reliably carries ≈ 44 / log 2 ( 1 / r ) \\\\approx 44/\\\\log_{2}(1/r) symbols, so a practical codec stores one point per fixed-length span and reconstructs each span by anchored backward search. The construction is classical in the alignment-free sequence-analysis literature, where it is known as chaos game representation and its generalization to arbitrary alphabets as universal sequence maps [ 7 , 8 ] ; our contribution is not the code, nor vector quantization of the KV cache under a random-access constraint (a concurrently active topic [ 5 , 6 ] ), but the use of this particular code as a KV archive that is at once a store, a random-access decoder, and an in-place search index.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A KV-archive format. We frame a lossless contractive iterated-map code as the serialization layer for a quantized KV cache, and verify it is lossless, linear-time, and offers O ( 1 ) O(1) random access to any past position and O ( 1 ) O(1) amortized append–the operations a growing cache needs and that a general-purpose byte compressor does not provide (Section 2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A controlled quantizer study on long contexts. With a small exact window and the remainder archived, we sweep pooled vs. per-head codebooks and residual depth on GPT-2 at 1024-token contexts. Per-head codebooks Pareto-dominate pooled ones at equal bit budget (Fig. 2 ), and we quantify a ∼ 4 × \\\\sim\\\\!4\\\\times key/value damage asymmetry (direction consistent with [ 4 ] ) that a bit-asymmetric hybrid exploits (Section 3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The archive is also an index. Approximate substring retrieval runs directly on the stored vectors–nearest-neighbor distance is a graded suffix-similarity–and matched context is decoded from the matched vector with no access to the surrounding text (Section 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All experiments use GPT-2 (124M) on a public-domain text and run in minutes on a single CPU; code and scripts are released for full reproduction at https://github.com/eighteight/fractal-kv .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 The codec as a KV archive\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first establish that the code is a usable storage primitive independent of any model. Table 1 reports a self-contained benchmark of the codec on natural text: encoding is lossless across ASCII, Unicode, and degenerate inputs; encoding time is linear in length ( 0.68 μ 0.68\\\\,\\\\mu s/char, flat from 25 25 K to 400 400 K characters); random access to an arbitrary character in a 10 6 10^{6} -character document costs 311 μ 311\\\\,\\\\mu s (decoding only the span that contains it); and appending a token costs 175 μ 175\\\\,\\\\mu s amortized, re-walking only from the last stored anchor. These are the primitives a KV archive requires: append as the cache grows, and random access when a position is read back.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 makes the alphabet/ratio constraint concrete: the contraction ratio must adapt to alphabet size for the code to remain uniquely decodable, which in turn sets the per-symbol precision budget and hence the span length used by the codec.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Quantizer study for the KV archive\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on GPT-2 (124M) using The Time Machine (Project Gutenberg, public domain). Perplexity is measured token-by-token over a 1024 1024 -token passage. Codebooks are trained by k k -means on KV vectors collected from a disjoint span of the same book. During evaluation we keep an exact window–the first 4 4 positions (attention sinks [ 1 ] ) and the most recent 32 32 –and, once a position ages out of that window, overwrite its cached K and V with their nearest codebook entries, so all later tokens attend to the quantized archive. This is the regime of interest: at 1024 1024 tokens the exact window is 3.5 % 3.5\\\\% of context and the model attends almost entirely to archived, quantized memory. Residual vector quantization (RVQ) of depth s s stores each vector as s s indices whose centroids sum to the reconstruction. Storage is reported as bytes/token of the serialized index stream (fractal codec) and as compression ratio against an fp16 cache ( 36 , 864 36{,}864 B/token); the exact baseline perplexity is 35.97 35.97 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Per-head codebooks dominate.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 and Figure 2 give the main result. Sharing one codebook across all heads of a layer (“pooled”) is substantially worse than a codebook per head at equal bit budget . Single-stage per-head VQ costs + 23.7 % +23.7\\\\% perplexity versus + 51.0 % +51.0\\\\% for pooled, and two-stage per-head RVQ costs + 15.0 % +15.0\\\\% versus + 37.2 % +37.2\\\\% ; in both cases per-head codebooks move the entire rate–distortion curve down. Each doubling of the index budget along the RVQ axis buys back a roughly constant 8 8 – 9 9 perplexity points, indicating a smooth operating curve rather than a cliff. Per-head codebooks add a one-time 36 36 MB of centroids ( 288 288 codebooks), trained in 108 108 s on CPU.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Keys are ∼ 4 × \\\\sim\\\\!4\\\\times more fragile than values.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reusing the two-stage per-head codebooks, we quantize only values (keys kept exact) and only keys (values kept exact). Quantizing values costs + 4.0 % +4.0\\\\% perplexity; quantizing keys costs + 14.5 % +14.5\\\\% –nearly the entire + 15.0 % +15.0\\\\% of quantizing both, and the two effects compose almost independently ( 1.145 × 1.040 ≈ 1.19 1.145\\\\times 1.040\\\\approx 1.19 ). The asymmetry has a clear mechanism: values enter attention through a weighted average, which attenuates per-vector error, whereas keys enter through the query–key dot product that decides which memories are read, so key error corrupts attention routing itself. The direction of this effect matches the key/value distinction reported by KIVI [ 4 ] ; our contribution is to express it as a perplexity budget and to convert that budget into a bit allocation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A bit-asymmetric hybrid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Allocating four RVQ stages to keys and two to values lowers the cost to +11.2% perplexity at 1025 1025 B/token ( 36 × 36\\\\times vs. fp16), with keys-only- × 4 \\\\times 4 at + 9.4 % +9.4\\\\% confirming the key axis is where the budget is best spent. This is the best operating point in our sweep and the recommended default; the key-bit scaling law suggests further stages would continue to trade ∼ 5 \\\\sim\\\\!5 points of perplexity per doubling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"On the storage layer’s role.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We are explicit that the compression ratio in Table 2 is produced by the vector quantizer; the fractal codec is the serialization of the resulting index stream, and on these high-entropy indices its size is comparable to bit-packing and to a general-purpose byte compressor (within a few percent either way). Its value is not a smaller blob but a richer one: unlike a byte compressor, the archive supports O ( 1 ) O(1) random access to any archived position and O ( 1 ) O(1) append, and–as we show next–doubles as a retrieval index over its own contents.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 The archive is also a retrieval index\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A property of the code makes the stored archive searchable without decompression. Because each step contracts by r r , a stored point is dominated by the most recent symbols, older ones decaying geometrically; two positions sharing an s s -symbol suffix therefore have points within 2 r s 2r^{s} of each other. Nearest-neighbor distance in point space is thus a graded suffix similarity, and thresholding at 2 r ℓ 2r^{\\\\ell} retrieves all positions whose length- ℓ \\\\ell suffix matches a query–a property known for the midpoint code in the sequence-analysis literature [ 9 ] and which we measure directly and exploit for KV-archive retrieval.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 confirms the decay law across seven orders of magnitude on a 100 100 K-character corpus and reports retrieval quality. We encode a query from the origin and threshold squared distance over the stored points. Recall is 1.00 1.00 at every query length by construction–a true match must fall inside the ball–while precision is set by the index’s numerical precision relative to r ℓ r^{\\\\ell} : with a double-precision index ( 16 16 B/char) precision is 0.89 0.89 – 1.00 1.00 ; with single precision ( 8 8 B/char) recall is still 1.00 1.00 and the modest false-positive set is removable by decode verification, which itself runs inside the representation. Because the index lives in the plane rather than in the model’s hidden dimension, brute-force nearest-neighbor over the two-dimensional points costs only ≈ 0.9 \\\\approx\\\\!0.9 ms/query over 100 100 K positions (and is trivially accelerated by any spatial index)—this is a 2 2 D search, not high-dimensional approximate nearest neighbor, and should not be compared to embedding-space retrieval benchmarks. Crucially, the end-to-end loop touches no external text: a query returns a matched point, and the surrounding context is decoded backward from that point through stored anchors, reproducing the ground-truth context character for character. In an inference setting this is exactly the operation a retrieval-augmented decoder needs from its own past—locate archived positions whose recent context matches a probe, and rehydrate only those, without scanning or decompressing the rest of the cache.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Related work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"KV-cache compression spans low-bit quantization [ 3 , 4 ] , eviction of low-importance positions [ 2 ] , and the observation that a few initial “sink” tokens carry disproportionate attention mass [ 1 ] ; our exact-window scheme is a minimal combination of a sink prefix and a recency window. The key/value asymmetry we measure is not new: KIVI [ 4 ] already reports that keys and values have distinct distributions and require different quantization treatment (keys per-channel, values per-token); our contribution here is only to quantify the asymmetry as a perplexity budget on long contexts and to use it to allocate residual-VQ depth. Vector quantization of the KV cache under a random-access constraint is likewise an active topic: the concurrent FibQuant [ 5 ] is, like us, a fixed-rate vector quantizer for random-access KV compression evaluated on GPT-2, tracing a 5 5 – 34 × 34\\\\times memory–fidelity frontier, and vector storage engines for long-context inference are emerging [ 6 ] . Relative to FibQuant, which optimizes the codebook geometry, our quantizer study is deliberately plain (k-means / residual VQ) and our distinct claim is about the storage representation : the contractive iterated-map code—chaos game representation [ 7 ] and its arbitrary-alphabet generalization, universal sequence maps [ 8 ] —is simultaneously a lossless, randomly-accessible, appendable store and an in-place suffix-search index, the latter resting on a decay property established for that code [ 9 ] . To our knowledge that unification of storage and in-representation retrieval has not been applied to the KV-archive setting; it, rather than the quantizer or the code in isolation, is what this paper contributes. We developed this representation independently of, and contemporaneously with, FibQuant [ 5 ] ; we became aware of it, and of RetroInfer [ 6 ] , only after the experiments reported here were complete, and we cite both as related work rather than claiming priority over the overlap.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our study is deliberately small-scale and the claims should be read accordingly. All results are on a single 124 124 M-parameter model, a single public-domain corpus, and a single 1024 1024 -token context length, using perplexity as the only quality metric; downstream-task accuracy and larger models are untested, and larger models are empirically more tolerant of KV quantization, so these numbers are plausibly conservative but not validated as such. The compression comes from the vector quantizer, and our per-head codebooks are trained per corpus rather than amortized across data; a deployable system would need corpus-independent or online codebooks. The retrieval demonstration is exact/approximate suffix matching over token strings, not semantic retrieval over embeddings–the stored vectors are not shown to encode meaning, only sequence structure. Finally, our serialization ties rather than beats byte-oriented coders on raw size; its advantage is the access pattern, not the ratio. We also do not benchmark against concurrent random-access KV quantizers such as FibQuant [ 5 ] , whose codebook design is complementary to and likely stronger than our plain k-means; a fair head-to-head, and combining their quantizer with our retrieval-capable storage, is left to future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Treating the storage of a quantized KV cache as a first-class design choice, we showed that a lossless contractive iterated-map code is a practical archive format: linear-time, randomly accessible, appendable, and–uniquely–searchable in place. Around it we conducted a controlled quantizer study on long contexts, finding that per-head residual VQ Pareto-dominates pooled codebooks, that key quantization is about 4 × 4\\\\times more damaging than value quantization, and that a bit-asymmetric hybrid compresses the archived cache 36 × 36\\\\times versus fp16 at + 11 % +11\\\\% perplexity on GPT-2. The retrieval result points to the most interesting direction: an inference-time memory that is stored, decoded, and searched in one representation, so that a query can locate the archived positions whose context resembles it without rehydrating the cache. We release all code and scripts at https://github.com/eighteight/fractal-kv ; every figure and table reproduces from a single command on a laptop CPU.\"\n }\n]","isConversational":false}}},{"rowIdx":485,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07743"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Architecture Generalization with MetaNCA"},"authors":{"kind":"list like","value":["Meet Barot","Daniel Berenberg","Sina Khajehabdollahi"],"string":"[\n \"Meet Barot\",\n \"Daniel Berenberg\",\n \"Sina Khajehabdollahi\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07743v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Data/Code available at: https://github.com/Mythos-Scientific/meta-nca † † ©2026 Barot, Berenberg & Khajehabdollahi. Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license."},{"type":"heading","level":1,"text":"Introduction"},{"type":"para","text":"Backpropagation and gradient-based optimization methods for training neural networks are responsible for much of the impressive capabilities of deep learning models when applied to large datasets. However, many drawbacks to these methods exist: the memory consumption of gradient calculations, the requirement of an unchanging architecture in the training process, and the necessity of a large number of examples to learn new concepts. There exists a stark contrast between the memory and power costs for training large models compared to their analogs in nature (e.g. the mammalian brain), which consume orders of magnitude less power and are subject to a genomic bottleneck that strongly limits the level of description required to develop them. Notably, the information capacity of the genome is orders of magnitude smaller than the complexity of neural circuitry (Zador, 2019 ; Shuvaev et al. , 2024 ) . This implies that what is actually encoded are more generalized rules for developing and growing the circuitry rather than the specific details of the circuitry itself. The biological analogy points toward a different path forward, where the goal shifts from scaling the specifics of large models to learning the rules that generate them."},{"type":"para","text":"As the demand to scale models continues to grow, this gap between current machine learning methods and biological systems becomes increasingly difficult to ignore. These resource constraints affect not just the training or inference of these models, but also their flexibility and adaptability when it comes to fine-tuning and specialization, where the need to scale costs and dimensionality down have inspired methods such as LoRA (Hu et al. , 2022 ) that operate in lower rank spaces. There is therefore an increasing demand for more universal and highly compressible models that challenge the existing paradigms of training large foundation models."},{"type":"para","text":"Existing approaches each address different parts of this challenge, approaching the challenge from different traditions. Traditional hypernetworks and weight-space methods can generate neural network parameters from compact representations, but typically operate globally when generating weight matrices for a fixed target architecture (Ha et al. , 2016 ; Chauhan et al. , 2024 ) . Learned plasticity rules offer local, biologically inspired updates, though cross-architecture generalization has been limited in scope (Miconi et al. , 2018 , 2020 ) . HyperNCA (Najarro et al. , 2022 ) and neural developmental programs (Najarro et al. , 2023 ; Plantec et al. , 2024 ) have shown that NCAs can generate network parameters through local rules, but operate on fixed grid structures or grow topology from scratch rather than generating weights for a range of pre-specified architectures. To our knowledge, no existing method combines purely local weight updates on the computational graph with training across a distribution of architectures via end-to-end backpropagation."},{"type":"para","text":"To this end, we introduce MetaNCA, a neural cellular automaton that generates ensembles of large neural networks of various architectures through local self-organization. MetaNCAs are graph neural cellular automata that iteratively generate the parameters of a model using only the local information of an edge in the model’s computational graph. Because this local rule is defined purely from a weight’s forward and backward neighbors on the computation graph (the edges sharing its output and input nodes), it is by construction not tied to any particular architecture. During training, MetaNCA must generate parameters for a variety of architectures concurrently to ensure that parameter-generating rules do not overfit to a particular configuration. To our knowledge, MetaNCA is the first method to learn a single local rule that can self-organize across a distribution of unseen architectures with millions of parameters. We demonstrate these capabilities on MLPs, CNNs and ResNets."},{"type":"heading","level":2,"text":"Related Work"},{"type":"para","text":"Our methods are inspired by a growing body of research that has worked on bridging the gap between complex dynamical systems, self-organization, and deep learning."},{"type":"heading","level":3,"text":"Plasticity"},{"type":"para","text":"Plasticity-based learning algorithms are biologically motivated methods to self-organize neural networks using local learning rules. They operate by modifying the synaptic weights of a network as a function of its local activity Δ w i j = f ( a { i , j } , w i j ) \\Delta w_{ij}=f(a_{\\{i,j\\}},w_{ij}) . While plasticity-based methods are more biologically grounded, they often fail to compete with modern artificial neural networks that do not bother with biological realism (e.g. the backpropagation algorithm). More recently, work to bridge these limitations and apply more modern optimization techniques to these local learning rules have been made (Miconi et al. , 2018 , 2020 ; Najarro and Risi, 2020 ) , pushing the capabilities of these local learning models. Earlier work evolved a neural network to serve as the learning rule itself (Runarsson and Jonsson, 2000 ) , albeit demonstrated only on single-layer networks. However, to date, these methods show limited generalization to unseen architectures."},{"type":"heading","level":3,"text":"Neural Cellular Automata"},{"type":"para","text":"The conceptual roots of this work trace back to cellular automata (CA): simple rule-based systems in which complex global behavior emerges purely from local interactions. From elementary 1D CAs (Wolfram, 1983 ) to Conway’s Game of Life (Gardner, 1970 ) , these systems demonstrated that remarkably rich structure, including self-replication, pattern formation, and universal computation, can arise from a handful of local rules applied iteratively on a grid."},{"type":"para","text":"Growing Neural Cellular Automata (NCA) (Mordvintsev et al. , 2020 ) formalized this intuition for the deep learning era, replacing hand-crafted rules with learned local update functions operating on grids or graphs, producing systems reminiscent of reaction-diffusion models. These systems proved surprisingly capable: NCAs have since been trained to grow and robustly regenerate target patterns from a single seed cell (Mordvintsev et al. , 2020 ) , synthesize textures (Niklasson et al. , 2021 ) , classify MNIST digits (Randazzo et al. , 2020a ) all without any global coordinator, the competence emerging entirely from the local rule. These methods have also been extended to generalize to broader distributions of tasks (Guichard et al. , 2025b , a ) ."},{"type":"heading","level":3,"text":"Hypernetworks"},{"type":"para","text":"While NCAs have demonstrated remarkable success at tasks like pattern recognition and classification through purely local computation, we ask whether this same paradigm can be turned toward a more meta objective: generating the parameters of a neural network itself. This connects to the growing literature on weight-space learning (Han et al. , 2026 ) : methods that operate on or generate neural network parameters directly. Hypernetworks (Stanley et al. , 2009 ; Ha et al. , 2016 ; Chauhan et al. , 2024 ) learn to produce the weights of a larger network from a compact model and have been extended for continual learning (Von Oswald et al. , 2019 ) and architecture search (Brock et al. , 2017 ) . More recently, diffusion-based approaches generate network parameters by learning distributions over weight spaces (Wang et al. , 2024 ) , and hyper-representations learn reusable embeddings of trained network checkpoints (Schürholt et al. , 2022 ) . Many of these methods generate parameters at the granularity of layers, chunks, or parameter subsets, coupling the generator to a specific target architecture. Indirect encodings such as HyperNEAT (Stanley et al. , 2009 ) are a notable exception, querying a CPPN with the geometric coordinates of each connection to produce individual weights; however, this is a single static query of a coordinate function rather than an iterative update rule, and it does not aggregate information from a weight’s neighborhood in the computation graph. MetaNCA instead applies an iterative local update rule to each weight using its forward and backward neighborhoods on the computation graph, and meta-trains this rule across a distribution of architectures, which decouples it from any particular architecture and enables cross-architecture generalization."},{"type":"para","text":"The most direct precursor to our work is HyperNCA (Najarro et al. , 2022 ) , which first combined Hypernetworks and NCAs by using an NCA as the weight-generating mechanism itself: a compact local rule iteratively develops a seed state whose grown pattern is read out as the weights of a policy network. This developmental paradigm has since been extended through Neural Developmental Programs (Najarro et al. , 2023 ) , which grow network structure and weights via local inter-node communication alone, and further toward self-assembling networks with structural and synaptic plasticity (Plantec et al. , 2024 ) . MetaNCA builds on this line of work but differs from these developmental approaches in a key respect: rather than growing structure or reading out weights from a separate substrate or grid, its local rule operates directly on the edges of the target computation graph, updating each weight from its forward and backward neighborhoods. By explicitly training this rule across multiple architectures, MetaNCA generalizes to architectures unseen during meta-training."},{"type":"heading","level":1,"text":"Method Description"},{"type":"para","text":"In MetaNCA, a local rule is applied to each individual weight and uses information in its local neighborhood to produce updates. A linear attention block generates local perception vectors which feeds into an MLP that updates the local weight and hidden state. This process iterates for T = 10 T=10 timesteps. As an analogy to the Growing NCA (Mordvintsev et al. , 2020 ) work, in which a local rule neural network applies updates to pixels on a grid, MetaNCA operates on the weight values as “pixels” , with the “grid” being the neural network graph."},{"type":"heading","level":4,"text":"Problem setup."},{"type":"para","text":"We have a dataset D D of samples ( x , y ) (x,y) representing the inputs and labels respectively for a classification task, and a corresponding task neural network 𝒯 \\mathcal{T} that makes predictions for this task y ^ = 𝒯 ( θ ; x ) \\hat{y}=\\mathcal{T}(\\theta;x) . The task network 𝒯 \\mathcal{T} has weights w i j L ∈ θ w^{L}_{ij}\\in\\theta , where L L is a layer of the task network and i i and j j are indices of the weight matrix for layer L L . In addition, for each weight w i j w_{ij} , the task network also has a hidden state vector h → i j \\vec{h}_{ij} : a per-weight latent vector, carried alongside the weight across update steps, that lets the local rule store and communicate information beyond the scalar weight value. In practice we use many task networks in order to train the local rule network to generalize across architectures, but for simplicity we will describe MetaNCA assuming a single task network."},{"type":"para","text":"A separate local rule neural network R R , parametrized by ϕ \\phi , updates θ ( t ) \\theta^{(t)} and H ( t ) H^{(t)} , where θ ( t ) \\theta^{(t)} denotes the full set of task-network weights and H ( t ) = { h → i j } H^{(t)}=\\{\\vec{h}_{ij}\\} the corresponding set of all hidden states, both at update step t t . For a given weight w i j w_{ij} , we take the neighboring weight and hidden state vectors of w i j w_{ij} as the input to the local rule network R."},{"type":"heading","level":2,"text":"Local rule network inputs."},{"type":"para","text":"We construct the neighborhood of w w as the input to the local rule network R R as follows. We start with a particular weight w w and its corresponding hidden state h → \\vec{h} . To give a sense of direction for the neighborhood of w w , we define the “forward” neighbors and “backward” neighbors as those weights connected to the forward (post-synaptic) and backward (pre-synaptic) neurons of w w ; see Figure 1 for an illustration for feedforward layers. For convolutional layers, see Figure 2 . The forward and backward neighborhoods of w w are denoted as 𝒩 f ( w ) \\mathcal{N}_{f}(w) and 𝒩 b ( w ) \\mathcal{N}_{b}(w) respectively. Weights are randomly initialized while hidden states are initialized with sinusoidal absolute positional encodings (Vaswani et al. , 2017 ) across layers, forward neurons, backward neurons, and horizontal and vertical kernel positions (for use in convolutional networks)."},{"type":"heading","level":2,"text":"Weight Transformer"},{"type":"para","text":"We use a linear attention transformer to aggregate the weights and hidden states of forward and backward neighbors of an individual weight to construct a perception vector. This is passed to the MLP which updates the weight and hidden state. In the following section, we describe our implementation."},{"type":"heading","level":3,"text":"Linear Attention with ELU-based Feature Map"},{"type":"para","text":"Consider a particular weight w i w_{i} of the task network. Let w i w_{i} ’s forward neighborhood vectors be denoted as 𝒩 f ( w i ) \\mathcal{N}_{f}(w_{i}) ."},{"type":"para","text":"We apply cross-attention between [ w i h w i ] [w_{i}\\quad h_{w_{i}}] and its forward neighborhood’s weights and hidden states { [ w j h w j ] ∣ j ∈ 𝒩 f } \\{[w_{j}\\quad h_{w_{j}}]\\mid j\\in\\mathcal{N}_{f}\\} . All operations concerning the forward neighborhood are applied to the backward neighborhood as well. Before cross-attention, we randomly drop out a percentage of whole vectors of the forward and backward neighborhood to reduce dependence on individual neighbors. We project [ w h w ] [w\\quad h_{w}] as the query vector q q :"},{"type":"para","text":"The full matrix of queries for all the weights in the current layer under consideration is Q Q . Similarly for all key and value weight matrices:"},{"type":"para","text":"We apply a feature map ψ ( x ) = ELU ( x ) + 1 \\psi(x)=\\mathrm{ELU}(x)+1 to the query and key matrices Q Q and K K for a vector x ∈ ℝ d x\\in\\mathbb{R}^{d} using an exponential linear unit (ELU) function (Clevert et al. , 2015 ) ."},{"type":"heading","level":4,"text":"Multihead Linear Attention"},{"type":"para","text":"We use the rotary positional encoding formulation for linear attention from Su et al. ( 2024 ) , applying position-dependent rotation matrices R R to the feature-mapped queries and keys. Each head of the linear attention mechanism corresponds to a particular positional dimension of a weight: layer index, forward neuron index, backward neuron index, horizontal kernel position index, and vertical kernel position index. Kernel positions only differ for convolutional networks. For a given attention head, let Q ~ = R ψ ( Q ) \\tilde{Q}=R\\psi(Q) and K ~ = R ψ ( K ) \\tilde{K}=R\\psi(K) denote the rotated feature-mapped queries and keys. The kernelized linear attention output is:"},{"type":"para","text":"where D = Q ~ K ~ T 𝟏 + ϵ D=\\tilde{Q}\\tilde{K}^{T}\\mathbf{1}+\\epsilon is a normalizing factor. This formulation avoids materializing the full attention matrix, scaling linearly with the number of neighbors."},{"type":"para","text":"The multi-head outputs are concatenated and passed through a learned output projection, followed by a residual connection, layer normalization, a two-layer feedforward network, and a second layer normalization, forming a full transformer-style block."},{"type":"para","text":"A forward neighborhood attention output vector a f a_{f} constitutes our “forward signal”. A separate attention block with its own parameters is applied to the backward neighborhood, obtaining a b a_{b} . We concatenate these with the focus weight to form the perception vector:"},{"type":"para","text":"The updates to w w and h → \\vec{h} for the next timestep t + 1 t+1 are given by a feedforward network (the local rule head) applied to the perception vector:"},{"type":"heading","level":4,"text":"Training the local rule neural network."},{"type":"para","text":"For our experiments we use a cross-entropy classification loss for the MNIST and CIFAR100 datasets. To train the local rule network, we first calculate the loss of the task network. We then backpropagate through the task neural network, and the local rule neural network, over the sampled number of local rule updates of the task network, in order to calculate the gradient of the local rule network. We use the Muon optimizer to calculate updates to the local rule network (Jordan et al. , 2024 ) . We only update the weights of the local rule network through backpropagation; the task network is exclusively updated by the local rule network outputs. To simulate asynchronous updates, we apply the local rule updates stochastically to 80% of weights of the task net per local rule update for our experiments."},{"type":"heading","level":2,"text":"Complexity Analysis"},{"type":"para","text":"We analyze the compute and space complexity of MetaNCA in terms of the total number of task network weights W W , the number of local rule update steps T T , neighborhood sizes per weight 𝒩 f \\mathcal{N}_{f} and 𝒩 b \\mathcal{N}_{b} , and the local rule parameters ϕ \\phi . We denote the compute and space complexities of a computational stage q q as C q C_{q} and S q S_{q} ."},{"type":"para","text":"Algorithm 1 summarizes a single update step. For each of the L L layers in the task network, linear attention is computed over the local neighborhood, and the local rule MLP head produces per-weight updates Δ w \\Delta w and Δ h → \\Delta\\vec{h} . A stochastic mask selects a fraction p p of weights to update each step."},{"type":"para","text":"Each update step has the computational cost of applying the local rule network to each weight:"},{"type":"para","text":"The local rule updates can be parallelized across weights (and architectures), reducing wall clock time."},{"type":"heading","level":4,"text":"Local rule updates."},{"type":"para","text":"Applying T = 10 T=10 update steps to generate a task network, the compute and space complexity is given by"},{"type":"para","text":"Memory is dominated by the task network state: each of the W W weights carries a hidden state and positional encoding of dimension h h ."},{"type":"heading","level":4,"text":"Meta-training."},{"type":"para","text":"Training the local rule via backpropagation through time (BPTT) over T T unrolled steps introduces a constant factor for gradient computation. Gradient checkpointing (Chen et al. , 2016 ) recomputes each step’s forward pass during backpropagation, making memory scale with T \\sqrt{T} :"},{"type":"para","text":"Training memory stores a single training task network at a time by accumulating the local rule network’s gradients per task network."},{"type":"heading","level":1,"text":"Experiments"},{"type":"para","text":"We evaluate MetaNCA on two datasets: MNIST (Deng, 2012 ) (handwritten digit classification) and CIFAR-100 (100-class image classification). We test architecture generalization across three families of task networks:"},{"type":"para","text":"Dense MLPs on MNIST: two hidden layers with 0–90 units each, forming a [ 784 , h 1 , h 2 , 10 ] [784,h_{1},h_{2},10] architecture grid."},{"type":"para","text":"Convolutional networks on MNIST: 3-layer CNNs with a filter multiplier m m controlling channel widths ( 32 m 32m , 64 m 64m , 128 m 128m ), varying kernel sizes (3, 4, 5), and a dense hidden layer of h h units before the output. These networks range up to 2 million parameters."},{"type":"para","text":"ResNets (He et al. , 2016 ) on CIFAR-100: 3-stage residual networks with [ 1 , 1 , 1 ] [1,1,1] blocks per stage and channel widths [ c , 2 c , 4 c ] [c,2c,4c] , where c c is the base channel count."},{"type":"para","text":"For each experiment, we meta-train a single local rule network on a small set of task network architectures (2–5), then evaluate on a grid of architectures including both seen and unseen configurations. All reported accuracies are test set accuracy (80/20 train/test split), averaged over multiple samples per architecture."},{"type":"heading","level":1,"text":"Results"},{"type":"heading","level":2,"text":"Architecture Generalization: Dense MLPs"},{"type":"para","text":"Figure 3 shows architecture generalization for dense MLPs on MNIST. With only 3 training architectures, the local rule network achieves ≥ \\geq 90% accuracy on most architectures with at least 30 hidden units per layer. Adding 2 more training architectures (Figure 3 b) substantially improves coverage: architectures with small layers (e.g., 10 units) that previously achieved ∼ \\sim 40–50% now reach ≥ \\geq 90%. The two architectures added in (b) were the worst-performing configuration from the 3-architecture setting and one of the largest architectures in the grid. Because the latter sits at the boundary of the architecture space, adding it shifts this setting from primarily extrapolation toward interpolation, which partly explains the improved coverage. Performance is weakest for the smallest, lowest-capacity architectures."},{"type":"heading","level":2,"text":"Architecture Generalization: Convolutional Networks"},{"type":"para","text":"Figure 4 shows generalization across convolutional architectures with three kernel sizes. With only two training architectures in total (one at kernel size 3 and one at kernel size 5), the local rule achieves up to 97% accuracy on MNIST for the best configurations. Performance peaks near the training architectures and degrades gracefully for more distant configurations, confirming that the local rule learns generalizable weight-generation strategies rather than memorizing specific architectures."},{"type":"heading","level":2,"text":"Architecture Generalization: ResNets on CIFAR-100"},{"type":"para","text":"We further test MetaNCA on ResNet architectures for the more challenging CIFAR-100 task (Figure 5 ). Trained on only two base channel widths ( c = 16 c=16 and c = 20 c=20 ), the local rule generalizes to nearby architectures, achieving up to 29.9% accuracy. Performance drops sharply for smaller architectures ( c ≤ 14 c\\leq 14 ), suggesting the local rule has learned update dynamics specific to a capacity range rather than a universal strategy."},{"type":"para","text":"To contextualize these accuracies, we compare MetaNCA-generated networks to the same architectures trained conventionally with Adam. On MNIST the two are comparable: the [ 784 , 50 , 50 , 10 ] [784,50,50,10] network reaches 97% with MetaNCA and 96.9% with Adam. On CIFAR-100, MetaNCA reaches up to 29.9%, whereas Adam-trained baselines of the same ResNets reach 41.4% ( c = 16 c{=}16 ) and 42.1% ( c = 20 c{=}20 ); it is possible that with more training architectures and/or hyperparameter tuning, this gap could be reduced."},{"type":"heading","level":2,"text":"Weight Structure"},{"type":"para","text":"Figure 6 compares the weight matrix connecting the two hidden layers of a [ 784 , 50 , 50 , 10 ] [784,50,50,10] MLP generated by MetaNCA (10 local rule update steps) with one trained by Adam (50 epochs). Both networks achieve similar test accuracy on MNIST (97%). The MetaNCA-generated weights exhibit symmetry and more sparsity of the weights generated compared to the Adam trained weights."},{"type":"heading","level":1,"text":"Discussion"},{"type":"para","text":"Motivated by principles of localized and distributed self-organization in biological systems, we implement a local weight-generating model to solve standard benchmarking tasks in machine learning. We show the ability of the local rules learned by MetaNCA to generalize to unseen architectures with millions of parameters. During training, the local rule learns an attractor to the high-performing task network weights from randomly initialized states."},{"type":"para","text":"We can view MetaNCA as compressing several training task networks into the local rule network parameters. The local rule network is a smaller description of the generation process that then decompresses into trained task networks when applied iteratively on an initialized task network. In this way, MetaNCA could be seen as an architecture-flexible model distillation method. The rule network has a fixed size (82k parameters for MLPs and 126k for convolutional and residual networks), independent of the task network it generates. For the largest convolutional networks (2M parameters) this is approximately a 16 × 16\\times compression; moreover, a single rule network generates an entire family of architectures rather than a single network, so its cost is amortized across the family."},{"type":"para","text":"We have found that increasing the number of training architectures allows a greater degree of architecture generalization. Training on a much larger set of architectures of different types may allow for better performance on individual task networks, through transferring the capabilities of larger models to smaller ones, or different topologies allowing the learning rule to capture more useful signals for optimization."},{"type":"para","text":"We note two limitations on the scope of these results. While the neighborhood is defined generally from a weight’s forward and backward neighbors on the computation graph, instantiating it for a new architecture family (for example, the channel and kernel structure of convolutions) still requires mapping this definition onto that family’s connectivity. In addition, generalization is strongest within the capacity range spanned by the training architectures and degrades for architectures far from it, as seen for the smallest ResNets. Training a single rule across multiple architecture families, rather than one rule per family as we do here, is a natural next step that the generality of the neighborhood definition should support, but which we leave to future work."},{"type":"heading","level":1,"text":"Future Work"},{"type":"para","text":"The method introduced here allows for fast sampling of models of architectures not seen during training. However, the local rule updates are not conditioned directly on the data, and therefore any generalization of a local rule to new tasks would be limited at best. In the future we wish to add a dependence of the local rule on the data, incorporating signals from activations and loss values, similar to Randazzo et al. ( 2020b ) and Plantec et al. ( 2024 ) ."},{"type":"heading","level":2,"text":"Multitask MetaNCA"},{"type":"para","text":"Although it is not the focus of this current work, we have found that MetaNCA can scale to multiple concurrent tasks. By conditioning the local rule on a one-hot task vector, a single local rule network can generate task networks for MNIST and CIFAR-100 simultaneously. We found that the multitask MetaNCA produced task networks that performed comparably to those generated in the single-task setting, indicating that the local rule can learn task-specific weight-generation strategies within a shared parameter space. In future work, we aim to explore task generalization in addition to architecture generalization."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Data/Code available at: https://github.com/Mythos-Scientific/meta-nca † † ©2026 Barot, Berenberg & Khajehabdollahi. Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Backpropagation and gradient-based optimization methods for training neural networks are responsible for much of the impressive capabilities of deep learning models when applied to large datasets. However, many drawbacks to these methods exist: the memory consumption of gradient calculations, the requirement of an unchanging architecture in the training process, and the necessity of a large number of examples to learn new concepts. There exists a stark contrast between the memory and power costs for training large models compared to their analogs in nature (e.g. the mammalian brain), which consume orders of magnitude less power and are subject to a genomic bottleneck that strongly limits the level of description required to develop them. Notably, the information capacity of the genome is orders of magnitude smaller than the complexity of neural circuitry (Zador, 2019 ; Shuvaev et al. , 2024 ) . This implies that what is actually encoded are more generalized rules for developing and growing the circuitry rather than the specific details of the circuitry itself. The biological analogy points toward a different path forward, where the goal shifts from scaling the specifics of large models to learning the rules that generate them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As the demand to scale models continues to grow, this gap between current machine learning methods and biological systems becomes increasingly difficult to ignore. These resource constraints affect not just the training or inference of these models, but also their flexibility and adaptability when it comes to fine-tuning and specialization, where the need to scale costs and dimensionality down have inspired methods such as LoRA (Hu et al. , 2022 ) that operate in lower rank spaces. There is therefore an increasing demand for more universal and highly compressible models that challenge the existing paradigms of training large foundation models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Existing approaches each address different parts of this challenge, approaching the challenge from different traditions. Traditional hypernetworks and weight-space methods can generate neural network parameters from compact representations, but typically operate globally when generating weight matrices for a fixed target architecture (Ha et al. , 2016 ; Chauhan et al. , 2024 ) . Learned plasticity rules offer local, biologically inspired updates, though cross-architecture generalization has been limited in scope (Miconi et al. , 2018 , 2020 ) . HyperNCA (Najarro et al. , 2022 ) and neural developmental programs (Najarro et al. , 2023 ; Plantec et al. , 2024 ) have shown that NCAs can generate network parameters through local rules, but operate on fixed grid structures or grow topology from scratch rather than generating weights for a range of pre-specified architectures. To our knowledge, no existing method combines purely local weight updates on the computational graph with training across a distribution of architectures via end-to-end backpropagation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To this end, we introduce MetaNCA, a neural cellular automaton that generates ensembles of large neural networks of various architectures through local self-organization. MetaNCAs are graph neural cellular automata that iteratively generate the parameters of a model using only the local information of an edge in the model’s computational graph. Because this local rule is defined purely from a weight’s forward and backward neighbors on the computation graph (the edges sharing its output and input nodes), it is by construction not tied to any particular architecture. During training, MetaNCA must generate parameters for a variety of architectures concurrently to ensure that parameter-generating rules do not overfit to a particular configuration. To our knowledge, MetaNCA is the first method to learn a single local rule that can self-organize across a distribution of unseen architectures with millions of parameters. We demonstrate these capabilities on MLPs, CNNs and ResNets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our methods are inspired by a growing body of research that has worked on bridging the gap between complex dynamical systems, self-organization, and deep learning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Plasticity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Plasticity-based learning algorithms are biologically motivated methods to self-organize neural networks using local learning rules. They operate by modifying the synaptic weights of a network as a function of its local activity Δ w i j = f ( a { i , j } , w i j ) \\\\Delta w_{ij}=f(a_{\\\\{i,j\\\\}},w_{ij}) . While plasticity-based methods are more biologically grounded, they often fail to compete with modern artificial neural networks that do not bother with biological realism (e.g. the backpropagation algorithm). More recently, work to bridge these limitations and apply more modern optimization techniques to these local learning rules have been made (Miconi et al. , 2018 , 2020 ; Najarro and Risi, 2020 ) , pushing the capabilities of these local learning models. Earlier work evolved a neural network to serve as the learning rule itself (Runarsson and Jonsson, 2000 ) , albeit demonstrated only on single-layer networks. However, to date, these methods show limited generalization to unseen architectures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Neural Cellular Automata\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The conceptual roots of this work trace back to cellular automata (CA): simple rule-based systems in which complex global behavior emerges purely from local interactions. From elementary 1D CAs (Wolfram, 1983 ) to Conway’s Game of Life (Gardner, 1970 ) , these systems demonstrated that remarkably rich structure, including self-replication, pattern formation, and universal computation, can arise from a handful of local rules applied iteratively on a grid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Growing Neural Cellular Automata (NCA) (Mordvintsev et al. , 2020 ) formalized this intuition for the deep learning era, replacing hand-crafted rules with learned local update functions operating on grids or graphs, producing systems reminiscent of reaction-diffusion models. These systems proved surprisingly capable: NCAs have since been trained to grow and robustly regenerate target patterns from a single seed cell (Mordvintsev et al. , 2020 ) , synthesize textures (Niklasson et al. , 2021 ) , classify MNIST digits (Randazzo et al. , 2020a ) all without any global coordinator, the competence emerging entirely from the local rule. These methods have also been extended to generalize to broader distributions of tasks (Guichard et al. , 2025b , a ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Hypernetworks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While NCAs have demonstrated remarkable success at tasks like pattern recognition and classification through purely local computation, we ask whether this same paradigm can be turned toward a more meta objective: generating the parameters of a neural network itself. This connects to the growing literature on weight-space learning (Han et al. , 2026 ) : methods that operate on or generate neural network parameters directly. Hypernetworks (Stanley et al. , 2009 ; Ha et al. , 2016 ; Chauhan et al. , 2024 ) learn to produce the weights of a larger network from a compact model and have been extended for continual learning (Von Oswald et al. , 2019 ) and architecture search (Brock et al. , 2017 ) . More recently, diffusion-based approaches generate network parameters by learning distributions over weight spaces (Wang et al. , 2024 ) , and hyper-representations learn reusable embeddings of trained network checkpoints (Schürholt et al. , 2022 ) . Many of these methods generate parameters at the granularity of layers, chunks, or parameter subsets, coupling the generator to a specific target architecture. Indirect encodings such as HyperNEAT (Stanley et al. , 2009 ) are a notable exception, querying a CPPN with the geometric coordinates of each connection to produce individual weights; however, this is a single static query of a coordinate function rather than an iterative update rule, and it does not aggregate information from a weight’s neighborhood in the computation graph. MetaNCA instead applies an iterative local update rule to each weight using its forward and backward neighborhoods on the computation graph, and meta-trains this rule across a distribution of architectures, which decouples it from any particular architecture and enables cross-architecture generalization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The most direct precursor to our work is HyperNCA (Najarro et al. , 2022 ) , which first combined Hypernetworks and NCAs by using an NCA as the weight-generating mechanism itself: a compact local rule iteratively develops a seed state whose grown pattern is read out as the weights of a policy network. This developmental paradigm has since been extended through Neural Developmental Programs (Najarro et al. , 2023 ) , which grow network structure and weights via local inter-node communication alone, and further toward self-assembling networks with structural and synaptic plasticity (Plantec et al. , 2024 ) . MetaNCA builds on this line of work but differs from these developmental approaches in a key respect: rather than growing structure or reading out weights from a separate substrate or grid, its local rule operates directly on the edges of the target computation graph, updating each weight from its forward and backward neighborhoods. By explicitly training this rule across multiple architectures, MetaNCA generalizes to architectures unseen during meta-training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Method Description\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In MetaNCA, a local rule is applied to each individual weight and uses information in its local neighborhood to produce updates. A linear attention block generates local perception vectors which feeds into an MLP that updates the local weight and hidden state. This process iterates for T = 10 T=10 timesteps. As an analogy to the Growing NCA (Mordvintsev et al. , 2020 ) work, in which a local rule neural network applies updates to pixels on a grid, MetaNCA operates on the weight values as “pixels” , with the “grid” being the neural network graph.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Problem setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We have a dataset D D of samples ( x , y ) (x,y) representing the inputs and labels respectively for a classification task, and a corresponding task neural network 𝒯 \\\\mathcal{T} that makes predictions for this task y ^ = 𝒯 ( θ ; x ) \\\\hat{y}=\\\\mathcal{T}(\\\\theta;x) . The task network 𝒯 \\\\mathcal{T} has weights w i j L ∈ θ w^{L}_{ij}\\\\in\\\\theta , where L L is a layer of the task network and i i and j j are indices of the weight matrix for layer L L . In addition, for each weight w i j w_{ij} , the task network also has a hidden state vector h → i j \\\\vec{h}_{ij} : a per-weight latent vector, carried alongside the weight across update steps, that lets the local rule store and communicate information beyond the scalar weight value. In practice we use many task networks in order to train the local rule network to generalize across architectures, but for simplicity we will describe MetaNCA assuming a single task network.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A separate local rule neural network R R , parametrized by ϕ \\\\phi , updates θ ( t ) \\\\theta^{(t)} and H ( t ) H^{(t)} , where θ ( t ) \\\\theta^{(t)} denotes the full set of task-network weights and H ( t ) = { h → i j } H^{(t)}=\\\\{\\\\vec{h}_{ij}\\\\} the corresponding set of all hidden states, both at update step t t . For a given weight w i j w_{ij} , we take the neighboring weight and hidden state vectors of w i j w_{ij} as the input to the local rule network R.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Local rule network inputs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We construct the neighborhood of w w as the input to the local rule network R R as follows. We start with a particular weight w w and its corresponding hidden state h → \\\\vec{h} . To give a sense of direction for the neighborhood of w w , we define the “forward” neighbors and “backward” neighbors as those weights connected to the forward (post-synaptic) and backward (pre-synaptic) neurons of w w ; see Figure 1 for an illustration for feedforward layers. For convolutional layers, see Figure 2 . The forward and backward neighborhoods of w w are denoted as 𝒩 f ( w ) \\\\mathcal{N}_{f}(w) and 𝒩 b ( w ) \\\\mathcal{N}_{b}(w) respectively. Weights are randomly initialized while hidden states are initialized with sinusoidal absolute positional encodings (Vaswani et al. , 2017 ) across layers, forward neurons, backward neurons, and horizontal and vertical kernel positions (for use in convolutional networks).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Weight Transformer\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use a linear attention transformer to aggregate the weights and hidden states of forward and backward neighbors of an individual weight to construct a perception vector. This is passed to the MLP which updates the weight and hidden state. In the following section, we describe our implementation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Linear Attention with ELU-based Feature Map\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider a particular weight w i w_{i} of the task network. Let w i w_{i} ’s forward neighborhood vectors be denoted as 𝒩 f ( w i ) \\\\mathcal{N}_{f}(w_{i}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We apply cross-attention between [ w i h w i ] [w_{i}\\\\quad h_{w_{i}}] and its forward neighborhood’s weights and hidden states { [ w j h w j ] ∣ j ∈ 𝒩 f } \\\\{[w_{j}\\\\quad h_{w_{j}}]\\\\mid j\\\\in\\\\mathcal{N}_{f}\\\\} . All operations concerning the forward neighborhood are applied to the backward neighborhood as well. Before cross-attention, we randomly drop out a percentage of whole vectors of the forward and backward neighborhood to reduce dependence on individual neighbors. We project [ w h w ] [w\\\\quad h_{w}] as the query vector q q :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The full matrix of queries for all the weights in the current layer under consideration is Q Q . Similarly for all key and value weight matrices:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We apply a feature map ψ ( x ) = ELU ( x ) + 1 \\\\psi(x)=\\\\mathrm{ELU}(x)+1 to the query and key matrices Q Q and K K for a vector x ∈ ℝ d x\\\\in\\\\mathbb{R}^{d} using an exponential linear unit (ELU) function (Clevert et al. , 2015 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Multihead Linear Attention\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the rotary positional encoding formulation for linear attention from Su et al. ( 2024 ) , applying position-dependent rotation matrices R R to the feature-mapped queries and keys. Each head of the linear attention mechanism corresponds to a particular positional dimension of a weight: layer index, forward neuron index, backward neuron index, horizontal kernel position index, and vertical kernel position index. Kernel positions only differ for convolutional networks. For a given attention head, let Q ~ = R ψ ( Q ) \\\\tilde{Q}=R\\\\psi(Q) and K ~ = R ψ ( K ) \\\\tilde{K}=R\\\\psi(K) denote the rotated feature-mapped queries and keys. The kernelized linear attention output is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where D = Q ~ K ~ T 𝟏 + ϵ D=\\\\tilde{Q}\\\\tilde{K}^{T}\\\\mathbf{1}+\\\\epsilon is a normalizing factor. This formulation avoids materializing the full attention matrix, scaling linearly with the number of neighbors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The multi-head outputs are concatenated and passed through a learned output projection, followed by a residual connection, layer normalization, a two-layer feedforward network, and a second layer normalization, forming a full transformer-style block.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A forward neighborhood attention output vector a f a_{f} constitutes our “forward signal”. A separate attention block with its own parameters is applied to the backward neighborhood, obtaining a b a_{b} . We concatenate these with the focus weight to form the perception vector:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The updates to w w and h → \\\\vec{h} for the next timestep t + 1 t+1 are given by a feedforward network (the local rule head) applied to the perception vector:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training the local rule neural network.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For our experiments we use a cross-entropy classification loss for the MNIST and CIFAR100 datasets. To train the local rule network, we first calculate the loss of the task network. We then backpropagate through the task neural network, and the local rule neural network, over the sampled number of local rule updates of the task network, in order to calculate the gradient of the local rule network. We use the Muon optimizer to calculate updates to the local rule network (Jordan et al. , 2024 ) . We only update the weights of the local rule network through backpropagation; the task network is exclusively updated by the local rule network outputs. To simulate asynchronous updates, we apply the local rule updates stochastically to 80% of weights of the task net per local rule update for our experiments.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Complexity Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We analyze the compute and space complexity of MetaNCA in terms of the total number of task network weights W W , the number of local rule update steps T T , neighborhood sizes per weight 𝒩 f \\\\mathcal{N}_{f} and 𝒩 b \\\\mathcal{N}_{b} , and the local rule parameters ϕ \\\\phi . We denote the compute and space complexities of a computational stage q q as C q C_{q} and S q S_{q} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm 1 summarizes a single update step. For each of the L L layers in the task network, linear attention is computed over the local neighborhood, and the local rule MLP head produces per-weight updates Δ w \\\\Delta w and Δ h → \\\\Delta\\\\vec{h} . A stochastic mask selects a fraction p p of weights to update each step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each update step has the computational cost of applying the local rule network to each weight:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The local rule updates can be parallelized across weights (and architectures), reducing wall clock time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Local rule updates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying T = 10 T=10 update steps to generate a task network, the compute and space complexity is given by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Memory is dominated by the task network state: each of the W W weights carries a hidden state and positional encoding of dimension h h .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Meta-training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training the local rule via backpropagation through time (BPTT) over T T unrolled steps introduces a constant factor for gradient computation. Gradient checkpointing (Chen et al. , 2016 ) recomputes each step’s forward pass during backpropagation, making memory scale with T \\\\sqrt{T} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training memory stores a single training task network at a time by accumulating the local rule network’s gradients per task network.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate MetaNCA on two datasets: MNIST (Deng, 2012 ) (handwritten digit classification) and CIFAR-100 (100-class image classification). We test architecture generalization across three families of task networks:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dense MLPs on MNIST: two hidden layers with 0–90 units each, forming a [ 784 , h 1 , h 2 , 10 ] [784,h_{1},h_{2},10] architecture grid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Convolutional networks on MNIST: 3-layer CNNs with a filter multiplier m m controlling channel widths ( 32 m 32m , 64 m 64m , 128 m 128m ), varying kernel sizes (3, 4, 5), and a dense hidden layer of h h units before the output. These networks range up to 2 million parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ResNets (He et al. , 2016 ) on CIFAR-100: 3-stage residual networks with [ 1 , 1 , 1 ] [1,1,1] blocks per stage and channel widths [ c , 2 c , 4 c ] [c,2c,4c] , where c c is the base channel count.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each experiment, we meta-train a single local rule network on a small set of task network architectures (2–5), then evaluate on a grid of architectures including both seen and unseen configurations. All reported accuracies are test set accuracy (80/20 train/test split), averaged over multiple samples per architecture.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Architecture Generalization: Dense MLPs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 shows architecture generalization for dense MLPs on MNIST. With only 3 training architectures, the local rule network achieves ≥ \\\\geq 90% accuracy on most architectures with at least 30 hidden units per layer. Adding 2 more training architectures (Figure 3 b) substantially improves coverage: architectures with small layers (e.g., 10 units) that previously achieved ∼ \\\\sim 40–50% now reach ≥ \\\\geq 90%. The two architectures added in (b) were the worst-performing configuration from the 3-architecture setting and one of the largest architectures in the grid. Because the latter sits at the boundary of the architecture space, adding it shifts this setting from primarily extrapolation toward interpolation, which partly explains the improved coverage. Performance is weakest for the smallest, lowest-capacity architectures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Architecture Generalization: Convolutional Networks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 shows generalization across convolutional architectures with three kernel sizes. With only two training architectures in total (one at kernel size 3 and one at kernel size 5), the local rule achieves up to 97% accuracy on MNIST for the best configurations. Performance peaks near the training architectures and degrades gracefully for more distant configurations, confirming that the local rule learns generalizable weight-generation strategies rather than memorizing specific architectures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Architecture Generalization: ResNets on CIFAR-100\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further test MetaNCA on ResNet architectures for the more challenging CIFAR-100 task (Figure 5 ). Trained on only two base channel widths ( c = 16 c=16 and c = 20 c=20 ), the local rule generalizes to nearby architectures, achieving up to 29.9% accuracy. Performance drops sharply for smaller architectures ( c ≤ 14 c\\\\leq 14 ), suggesting the local rule has learned update dynamics specific to a capacity range rather than a universal strategy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To contextualize these accuracies, we compare MetaNCA-generated networks to the same architectures trained conventionally with Adam. On MNIST the two are comparable: the [ 784 , 50 , 50 , 10 ] [784,50,50,10] network reaches 97% with MetaNCA and 96.9% with Adam. On CIFAR-100, MetaNCA reaches up to 29.9%, whereas Adam-trained baselines of the same ResNets reach 41.4% ( c = 16 c{=}16 ) and 42.1% ( c = 20 c{=}20 ); it is possible that with more training architectures and/or hyperparameter tuning, this gap could be reduced.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Weight Structure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6 compares the weight matrix connecting the two hidden layers of a [ 784 , 50 , 50 , 10 ] [784,50,50,10] MLP generated by MetaNCA (10 local rule update steps) with one trained by Adam (50 epochs). Both networks achieve similar test accuracy on MNIST (97%). The MetaNCA-generated weights exhibit symmetry and more sparsity of the weights generated compared to the Adam trained weights.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Motivated by principles of localized and distributed self-organization in biological systems, we implement a local weight-generating model to solve standard benchmarking tasks in machine learning. We show the ability of the local rules learned by MetaNCA to generalize to unseen architectures with millions of parameters. During training, the local rule learns an attractor to the high-performing task network weights from randomly initialized states.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can view MetaNCA as compressing several training task networks into the local rule network parameters. The local rule network is a smaller description of the generation process that then decompresses into trained task networks when applied iteratively on an initialized task network. In this way, MetaNCA could be seen as an architecture-flexible model distillation method. The rule network has a fixed size (82k parameters for MLPs and 126k for convolutional and residual networks), independent of the task network it generates. For the largest convolutional networks (2M parameters) this is approximately a 16 × 16\\\\times compression; moreover, a single rule network generates an entire family of architectures rather than a single network, so its cost is amortized across the family.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We have found that increasing the number of training architectures allows a greater degree of architecture generalization. Training on a much larger set of architectures of different types may allow for better performance on individual task networks, through transferring the capabilities of larger models to smaller ones, or different topologies allowing the learning rule to capture more useful signals for optimization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We note two limitations on the scope of these results. While the neighborhood is defined generally from a weight’s forward and backward neighbors on the computation graph, instantiating it for a new architecture family (for example, the channel and kernel structure of convolutions) still requires mapping this definition onto that family’s connectivity. In addition, generalization is strongest within the capacity range spanned by the training architectures and degrades for architectures far from it, as seen for the smallest ResNets. Training a single rule across multiple architecture families, rather than one rule per family as we do here, is a natural next step that the generality of the neighborhood definition should support, but which we leave to future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Future Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The method introduced here allows for fast sampling of models of architectures not seen during training. However, the local rule updates are not conditioned directly on the data, and therefore any generalization of a local rule to new tasks would be limited at best. In the future we wish to add a dependence of the local rule on the data, incorporating signals from activations and loss values, similar to Randazzo et al. ( 2020b ) and Plantec et al. ( 2024 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Multitask MetaNCA\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although it is not the focus of this current work, we have found that MetaNCA can scale to multiple concurrent tasks. By conditioning the local rule on a one-hot task vector, a single local rule network can generate task networks for MNIST and CIFAR-100 simultaneously. We found that the multitask MetaNCA produced task networks that performed comparably to those generated in the single-task setting, indicating that the local rule can learn task-specific weight-generation strategies within a shared parameter space. In future work, we aim to explore task generalization in addition to architecture generalization.\"\n }\n]","isConversational":false}}},{"rowIdx":486,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07128"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Distributed Sparse Interventions in Language Models"},"authors":{"kind":"list like","value":["Maximilian S. Ernst","Lorenz Linhardt","Aaron Peikert","Oliver Eberle"],"string":"[\n \"Maximilian S. Ernst\",\n \"Lorenz Linhardt\",\n \"Aaron Peikert\",\n \"Oliver Eberle\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07128v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Modern language models demonstrate flexible task-solving behaviours, including the adaptive combination and execution of multiple tasks through in-context learning (ICL) [ 10 , 26 , 53 ] . However, the mechanisms by which these models represent and compose tasks remain largely elusive, particularly in scenarios requiring the combination, isolation, or selective activation of behaviours. In the context of model steering, prior research has primarily focused on interventions along directions in activation space [ 43 , 49 , 18 , 8 , 32 ] , leveraging the observation that many task-relevant behaviours exhibit approximately linear structure in representation space. Such vector-based steering approaches typically rely on a number of key assumptions: (1) effective interventions require modifying all neuron activations in a specific layer or attention head, (2) tasks are represented as linear directions in activation space that can be combined additively, and (3) local first-order effect approximations in activation space are sufficient to identify interventions that induce approximately proportional changes in behaviour."},{"type":"para","text":"We demonstrate that these assumptions can be violated in practice. Task-relevant computations are distributed across model components, and interventions often exhibit nonlinear interaction effects across neurons that violate the assumptions of uniform first-order approximations. This motivates the development of fine-grained intervention strategies that operate directly on neurons rather than on global directions in activation space. Such approaches could enable less invasive and more interpretable interventions. However, they are currently constrained by the difficulty of reliably localising task-relevant neurons without relying on the strong assumptions inherent in first-order approximations. To address this gap, we propose Distributed Sparse Interventions (DSI), a method for identifying sparse subsets of neurons that jointly activate a given task. Our contributions are as follows:"},{"type":"para","text":"An analysis of neuron-level interventions leading to DSI, a methodological contribution that enables task activation by identifying and intervening on sparse sets of features (Section 3 , Figure 1 )."},{"type":"para","text":"A demonstration of complex, non-linear dependencies among neuron interventions, highlighting the importance of moving beyond linear approaches to determine interventions. We account for these dependencies by iteratively updating the expected effects of interventions on model predictions (Section 4 )."},{"type":"para","text":"A comprehensive empirical study across 12 tasks on three models (Qwen3 (8B), Gemma 3 (4B), Llama 3.2 (3B)), demonstrating that intervening on as few as 8–64 neuron units (0.01–0.04% of total neurons) typically suffices to activate a given task, often matching or surpassing 10-shot ICL performance (Section 4.1 )."},{"type":"para","text":"An analysis of neuron roles within and across tasks, enabled by grounding model steering in sets of neurons. The potential of this conceptual contribution is demonstrated in a case study involving open-ended autoregressive text generation (Section 5 )."},{"type":"para","text":"Our work addresses the need for principled methods to uncover, control, and understand task representations in deep learning models, providing a practical approach for more effective and localised interventions."},{"type":"heading","level":1,"text":"2 Related work"},{"type":"heading","level":4,"text":"Model steering"},{"type":"para","text":"Prior work on model steering has largely focused on identifying linear directions in activation space to intervene on model behaviour. Associated vectors are typically calculated from in-context-learning examples [ 49 , 18 , 32 ] as well as contrastive sample pairs [ 43 ] , or via optimization, e.g. in approaches like LoFiT [ 54 ] , which trains additive biases applied to specific attention heads. While these methods can effectively alter model behaviour and are sometimes sparse in terms of components such as whole attention heads, they operate at the level of global directions in representation space."},{"type":"heading","level":4,"text":"Sparse representations in neural networks"},{"type":"para","text":"Often inspired by neuroscientific findings of sparse coding [ 50 , 38 , 39 ] , prior work on understanding representations in artificial neural networks investigated sparse representations. Research on model compression and pruning [ 27 ] , as well as the lottery ticket hypothesis [ 15 , 14 ] , suggests the existence of underlying sparse structure in large deep learning models by identifying sparse subnetworks capable of matching the performance of dense models. In vision models, neuron units aligning with human-interpretable concepts were identified [ 7 ] . Similarly, in language models, specific ‘skills’ [ 51 ] and ‘knowledge’ [ 11 ] have been localised to neurons, and sparse subnetworks have been shown to support strong performance through targeted fine-tuning [ 40 ] . As neural models may represent features in superposition [ 13 ] , recent interpretability work has explored dictionary learning to disentangle representations into interpretable features [ 9 ] , assuming the native neuron basis is not sufficiently sparse. Further efforts have focused on learning a sparse basis of interpretable features either at a specific layer (e.g., sparse autoencoders [ 9 , 21 ] ), or across multiple layers (e.g., crosscoders [ 29 ] ), in order to generate sparser circuits [ 33 ] . While this line of work relies on features in dense, high-dimensional spaces, it remains an open question to what extent task-relevant structure is accessible within the native neuron basis. More recently, it has been found that for some circuits that previously have been analyzed using sparse autoencoders, an analysis directly in the neuron basis of the MLP components of transformer blocks can uncover circuits of similar sparsity [ 3 ] ."},{"type":"heading","level":4,"text":"Vector arithmetic in representation space"},{"type":"para","text":"Previous work has investigated the compositional nature of interventions [ 49 , 32 , 30 ] , finding that steering vectors identified during ICL can be additive, allowing for the activation of task combinations through vector summation. Such modularity is not limited to activations but extends to the model’s parameters. Task vectors in weight space, derived from the difference between fine-tuned and pre-trained weights, can be combined to modify model capabilities [ 22 ] . Recent work further suggests that arithmetic capabilities are grounded in the composition of sets of modular heuristics [ 37 ] . While these studies highlight the compositional nature of task solving in LLMs, they largely rely on dense representations."},{"type":"heading","level":1,"text":"3 DSI - Distributed Sparse Interventions"},{"type":"para","text":"We introduce Distributed Sparse Interventions (DSI), a method to identify and apply effective interventions on small sets of neurons distributed across layers and attention heads. As illustrated in Figure 1 , DSI follows a four-step procedure: (1) activation differences between 10-shot and 0-shot prompts are computed for all neurons to isolate task-specific model responses; (2) a sparse initial intervention is derived from these activation differences and a robustified gradient, retaining only the most impactful neurons; (3) a refined intervention is generated via step-wise optimization of the sparse neuron set and neuron-level intervention scaling, accounting for neuron interactions (see Appendix A for an illustration and analysis of why modeling these interactions is necessary); and (4) during a modified forward pass , this intervention is applied to elicit the task-solving capabilities on 0-shot examples."},{"type":"para","text":"After discussing preliminaries, first-order intervention effect estimation and robustified gradients are defined in Section 3.1 , and the iterative intervention refinement procedure is introduced in Section 3.2 ."},{"type":"heading","level":4,"text":"Notation"},{"type":"para","text":"Let f f denote a model mapping a sequence x x of input tokens to output logits. A task is defined via a set D D of input-output pairs ( x , y ) (x,y) . For an input-output pair, f ( x ) i f(x)_{i} refers to the output logit of the correct token, i.e., i i denotes the output logit index of to the first token of y y . We drop the index for a more concise notation, and write f ( x ) f(x) for f ( x ) i f(x)_{i} from now on. From a dataset D D , we construct k k -shot ICL prompts as a concatenation of k k completed example pairs followed by an incomplete query, ( ( x 1 , y 1 ) , ( x 2 , y 2 ) , … , ( x k , y k ) , x k + 1 ) ((x_{1},y_{1}),(x_{2},y_{2}),\\ldots,(x_{k},y_{k}),x_{k+1}) . Let D k D_{k} denote a dataset of such k k -shot examples constructed from D D , and D 0 D_{0} a dataset of 0-shot prompts, only containing the incomplete query but no in-context examples."},{"type":"para","text":"We let a ∈ ℝ a\\in\\mathbb{R} denote a neuron activation in a hidden layer of f f and δ ∈ ℝ \\delta\\in\\mathbb{R} an intervention on that neuron. f δ ≔ f ( x | a ← a + δ ) f_{\\delta}\\coloneq f(x|a\\leftarrow a+\\delta) then denotes the model modified by intervention δ \\delta , that is, during a forward pass, a a is computed from the input, then δ \\delta is added to a a , and the forward pass is continued. 𝐚 ∈ ℝ d \\mathbf{a}\\in\\mathbb{R}^{d} denotes the vector of all hidden activations that are considered for an intervention, and Δ ∈ ℝ d \\Delta\\in\\mathbb{R}^{d} the vector of all interventions."},{"type":"heading","level":2,"text":"3.1 Approximating intervention effects"},{"type":"heading","level":4,"text":"Taylor approximation"},{"type":"para","text":"To identify which neurons to intervene on for task D D , we consider the effect of intervening on a neuron during a forward pass in the 0-shot setting. That is, for a 0-shot prompt x 0 x_{0} , the effect of adding an intervention δ \\delta to a neuron a a during the forward pass is"},{"type":"para","text":"i.e., the change of the correct output logit caused by the addition of δ \\delta to a a during the forward pass. Since an exact calculation would require an independent forward pass for each neuron in f f , we approximate the intervention effect via a first-order Taylor approximation:"},{"type":"para","text":"This is similar to the setting of attribution patching [ 36 ] , with the difference that the activation of a a is not replaced during the forward pass (a \"structural\" intervention) but a term δ \\delta is added to it (a \"parametric\" intervention) [ 12 ] ."},{"type":"heading","level":4,"text":"Gradient modifications"},{"type":"para","text":"Deep neural network gradients suffer from instabilities such as gradient shattering [ 6 ] , making them a potentially unreliable signal for the expected changes in model output under interventions. To mitigate this brittle behaviour, various other methods, such as integrated gradients [ 47 ] , smooth grad [ 44 ] , or layer-wise relevance propagation (LRP) [ 5 , 34 ] have been proposed for computing more robust input attributions. Recent work extends these findings to activation patching and shows that intermediate attributions, using LRP-modified gradients, better estimate patching effects in transformer-based language models than raw gradients [ 23 ] . To improve the approximation of the intervention effect, we therefore use an LRP-modified gradient. The modifications include using the identity rule [ 42 ] for nonlinear activation functions, the half rule for gated MLPs and attention modules [ 4 , 1 ] , and the LayerNorm rule for layer normalization modules [ 2 ] . We denote this robustified gradient by ∂ r \\partial^{r} and provide additional details on the rules in Appendix D ."},{"type":"heading","level":4,"text":"Average effect"},{"type":"para","text":"Equipped with this robustified gradient, we approximate the average effect ℰ ¯ \\bar{\\mathcal{E}} of the intervention δ \\delta across dataset of 0-shot examples as"},{"type":"heading","level":2,"text":"3.2 Determining the interventions"},{"type":"heading","level":4,"text":"Initial set"},{"type":"para","text":"To determine the magnitude of interventions, we follow prior approaches that construct steering signals via activation differences [ 43 , 49 ] and compute the average activation difference of neuron a a between a k k -shot and a 0-shot setting during the forward pass:"},{"type":"para","text":"The resulting quantity captures how a neuron’s activation shifts between settings where the task is solved via in-context learning and settings where it is not, providing an informative initialization for intervention strengths. For a given number of neurons n n to intervene on, we then choose the neurons with the largest expected effect ℰ ¯ ( a − ) \\bar{\\mathcal{E}}(a^{-}) according to Equation 2 . Introducing a scaling factor 𝐬 ∈ ℝ d \\mathbf{s}\\in\\mathbb{R}^{d} which is one for the n n largest effects and zero otherwise, we write the vector of interventions as Δ = 𝐬 ⊙ 𝐚 − \\Delta=\\mathbf{s}\\odot\\mathbf{a}^{-} ."},{"type":"heading","level":4,"text":"Iterative refinement"},{"type":"para","text":"So far, we approximated the expected effect of an intervention δ \\delta on a neuron a a with a first-order Taylor expansion around the root point a 0 a_{0} , i.e. the value the neuron takes on during a forward pass on a 0-shot prompt. This disregards nonlinear effects of interventions, as well as interaction effects among interventions, which, as we show in Section 4 , can be sizeable in highly nonlinear transformer networks. As explicit computation of higher-order terms across thousands of interventions is computationally infeasible, we opt for an iterative approximation. When maximizing a nonlinear function, it is typically not approximated only once at the initial point, but the function is repeatedly approximated, followed by a step in the gradient direction, and this process is iterated until a stopping criterion is met. Appendix A contains an illustrative example showing how such an iterative approach can help model interactions between neuron-level interventions."},{"type":"para","text":"By first constructing a candidate set of n n neurons with the largest expected effect ℰ ¯ ( a − ) \\bar{\\mathcal{E}}(a^{-}) (as described above), and iteratively refining the intervention, we arrive at our proposed Algorithm 1 : for a given neuron set size n n and datasets D 0 D_{0} and D k D_{k} , it (1) computes the average activation differences, (2) computes a robustified gradient w.r.t. the activations in the 0-shot setting, (3) selects the n n interventions with the highest expected effect as a starting point and, (4) iteratively refines the effects t t -times by re-evaluating the expected effects at the current intervention."},{"type":"para","text":"For the iterative optimization steps, we employ ZeroFPR [ 48 ] , an algorithm suited for maximizing nonconvex functions (the output logit) with a limited number of nonzero parameters (the per-neuron interventions). We set a maximum number of 50 steps, and to encourage exploring different neuron sets, we re-run the algorithm with multiple perturbed starting values - details about the optimization algorithm and settings can be found in Appendix F . Note that instead of directly estimating the interventions, we estimate 𝐬 \\mathbf{s} , the scaling factor of the activation differences. This informs the optimizer about the expected changes in the activations, letting 𝐚 − \\mathbf{a^{-}} act as a preconditioner for the optimization problem."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":4,"text":"Experimental setup"},{"type":"para","text":"We evaluate our approach on three instruction-tuned language models from widely used families, Qwen 3 (8B) [ 41 ] , Gemma 3 (4B) [ 16 ] , and Llama 3.2 (3B) [ 17 ] (see Appendix B for details), spanning a range of model depths (28-36 layers), attention heads (8-32 heads), and hidden dimensions (128, 256). We consider a subset of 12 tasks from [ 49 ] , including two abstraction levels (see Table 3 for an overview): extractive , where the correct output is already present in the input (e.g., choosing the fruit from a set of words), and abstractive , where the output is not present in the input (e.g., naming the capital of a given country). For each task, we split the input-output pairs 50/50 into a training set and a test set. We then generate 100 100 0-shot prompts and 100 100 10-shot prompts from the training split. Two datasets contain fewer samples, resulting in smaller train and test splits. Details on the task data, splits, and prompt structure are provided in Appendix C . For the interventions, we consider the outputs of all attention heads in each layer at the last token position (before they are projected to the residual stream dimension)."},{"type":"heading","level":4,"text":"Nonlinear, neuron-specific intervention effects"},{"type":"para","text":"A key motivation for DSI is its ability to go beyond first-order effect estimates by accounting for nonlinear, neuron-specific effects. Before proceeding to the main results, we illustrate the importance of this on the english–french task using Llama 3.2 (3B), by studying interventions on the most relevant neurons identified via a first-order approximation using the robustified gradient, but without iterative refinement. To this end, we randomly sample neurons and neuron pairs from the 16 neurons with the highest expected intervention effect and measure the logit of the correct token under single and pairwise interventions."},{"type":"para","text":"Figure 2(a) shows the output logit corresponding to the correct token (averaged over 100 100 0-shot examples from the test set) for interventions on single neurons at a time. The interventions are scaled by the average activation difference. We observe that even among the neurons with the largest expected positive effect, some reduce the output logit, and that the logit varies nonlinearly with intervention strength. Figure 2(b) shows the effect of intervening on pairs of neurons. We observe interaction effects, such that the maximum logit corresponds to a setting where the intervention strength differs between neurons. These observations indicate limitations of first-order effect estimation, as it does not reliably identify neurons with a positive effect. They also point to limitations of approaches that determine a uniform scaling factor for all neurons, as even after rescaling the intervention by the average activation difference, the optimal intervention strength varies substantially across neurons, motivating neuron-level, iterative techniques such as DSI."},{"type":"heading","level":2,"text":"4.1 Evaluating DSI"},{"type":"para","text":"For all tasks and models, we compute DSI for neuron subsets of 8 8 to 2048 2048 and generate 200 200 0-shot prompts from the test split to evaluate the performance of the interventions. 1 1 1 The experiments needed ∼ \\sim 108h of compute time on AMD Instinct MI300A APUs. . We limit ourselves to next-token prediction; if a correct output consists of multiple tokens, we evaluate whether the first predicted token is correct. We compare 0-shot task accuracy against the baseline model behaviour (no intervention) and compare it against two additional conditions evaluating iterative refinement and gradient modifications: (1) no iterative refinement : the n n interventions with the largest effects as approximated via first-order Taylor expansion are used (i.e., the starting values in Algorithm 1 ), and one joint scaling factor s ∈ [ 0.01 , 50 ] s\\in[0.01,50] is determined for all interventions. (2) no gradient robustification : the same as (1), but in addition, the first-order effects are approximated without the gradient modifications described in Section 3.1 ."},{"type":"heading","level":4,"text":"Main results"},{"type":"para","text":"Table 1 shows the resulting accuracy from applying DSI in the 0-shot setting across models, averaged over the two task categories (abstractive and extractive, 6 tasks each), for the intervention set sizes 8 8 to 64 64 . Across model families, DSI substantially improves task accuracy over the no-intervention baseline, with meaningful gains already observable using as few as 8 8 neurons. Performance generally improves with larger neuron sets, with interventions on 64 64 neurons achieving mean task accuracies ranging from 0.72 to 0.98 across settings. The results further indicate that both gradient robustification and iterative refinement contribute consistently to improved intervention performance."},{"type":"para","text":"Figure 3 extends this analysis by showing accuracy across all 12 tasks as a function of set sizes from 8 to 2048 for Qwen 3 (8B), including conditions without gradient robustification and without iterative refinement. Additional results for other models are provided in Appendix E . Shaded regions correspond to 95% Wilson confidence intervals [ 52 ] , taking into account uncertainty due to the limited number of test samples. Across settings, particularly for small set sizes, DSI consistently achieves highest task accuracy, and both iterative refinement and gradient robustification are necessary for high performance, with removal of either leading to degradation in task accuracy. We observe particularly strong and consistent performance of DSI across set sizes on extractive tasks, while for some abstractive tasks, i.e., country-capital and english-french , larger set sizes can yield reduced performance. While we can not identify effective interventions across all set sizes and conditions, these results overall demonstrate the effectiveness of DSI across tasks and highlight its potential for identifying neuron-level model interventions."},{"type":"heading","level":4,"text":"Distribution of intervened neurons across heads and layers"},{"type":"para","text":"Through its distributed, neuron-level intervention approach, DSI enables analysis of where sparse interventions are located across heads and layers. Figures 11 , 12 , and 13 in Appendix G show the distribution of identified neurons over attention heads. For instance, at a set size of 64, we find that neurons are broadly distributed, involving many attention heads across layers, in particular, middle and later layers. Compared to previous direction-based steering approaches, which rely on predefined intervention layers, e.g., [ 43 , 49 ] , our findings suggest that task representation are distributed across multiple heads and layers, motivating intervention strategies that avoid constraining modifications to predefined layers."},{"type":"heading","level":1,"text":"5 Task composition via sparse neuron sets"},{"type":"para","text":"Language models exhibit the ability to flexibly perform and compose multiple tasks from context [ 53 , 26 ] , yet it remains an open challenge to understand where these behaviours originate and how they can be systematically controlled. Recent work has shown that linear representations enable multi-concept steering via per-layer addition of concept vectors such as transfer across languages [ 8 ] . However, such approaches operate in dense representation spaces and do not explicitly reveal how task-relevant computations are structured at the level of individual components."},{"type":"para","text":"Having established that DSI enables effective interventions on sparse subsets of neurons for activating individual tasks, we next investigate whether tracing task computations back to neuron-level sources reveals shared structure and enables fine-grained control for selectively activating or suppressing tasks, particularly in multi-task settings."},{"type":"para","text":"Unlike approaches that operate at the level of residual stream representations, this perspective acts directly on neurons before their contributions are combined, providing a more direct view of the origins of task computation and thereby facilitating interpretability. Analyzing a model’s internal task computations as sets of neurons further enables a compositional view based on set operations."},{"type":"para","text":"Applying DSI in open-ended generation , we observe that DSI interventions for the present-past task (e.g., publish → \\to published ) can induce past-tense usage, but also lead to highly repetitive outputs that prevent coherent continuation. In contrast to the fluent baseline response to a prompt such as “Are Annie Ernaux’s books good?” , where Qwen 3 (8B) responds: “Annie Ernaux’s books are widely regarded as **excellent** and **highly influential** in modern French literature […]” , under DSI intervention on 32 neurons, the same prompt yields “I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages […]” . This reflects the original task’s strict input–output structure, which requires reproducing the input under transformation. Similar behaviour has been observed n the context of vector-based steering [ 49 ] ."},{"type":"para","text":"Inspired by this observation, we hypothesize that the tense-switching and language-switching tasks from Section 4 are solved as a composition of input-reproduction (copying) and a transformation of the reproduced term. To examine whether a set-based approach can be used to decompose task behaviour into these two components, we conduct two targeted case studies."},{"type":"heading","level":2,"text":"5.1 Case Study 1: Copying in Tense Switching"},{"type":"para","text":"Our first case study investigates whether we can isolate two subtasks within the present-past task, corresponding to copying the input and producing output in the past tense . To that end, we create samples for a copy task (where the output mirrors the input exactly, see Appendix C ) and compute DSI interventions for a set size of 2048 2048 , capturing neurons involved in copying the input. We then investigate (1) whether neurons involved in the copy task are also involved in the present-past task, (2) to what extent their activation leads to copy behaviour, and (3) whether activating only the present-past -specific neurons reduces copying while still activating task-specific behaviour."},{"type":"para","text":"For the present-past task on Qwen 3 (8B), we choose sizes of 16 16 , 32 32 , and 64 64 neurons (the three smallest sets with high accuracy). Figure 4 .a contains the resulting accuracies and a qualitative example for the set size of 32 32 , with full results in Table 5 in Appendix H.1 . We can see that the intersection of the copy and present-past sets consists of 23 neurons, and that intervening on these neurons yields high performance on the copy task. We further see that the 9 neurons specific to the present-past task perform higher on this task than the 23 neurons in the intersection. Qualitatively, we observe copy behaviour intervening on both the intersection and all present-past neurons. Tense switching behaviour is observed intervening on all present-past neurons and the task-specific neurons, while the task-specific neurons induce less copy behaviour. Appendix H.2 contains details about the open-ended text generation as well as further qualitative examples that show that the qualitatively observed behaviour is generally robust, although the \"specific\" neuron set can fail to switch the generated answer to the past tense."},{"type":"heading","level":2,"text":"5.2 Case Study 2: Copying in Language Switching"},{"type":"para","text":"Similarly, for the english-french task on Llama 3.2 (3B), we select sizes of 8 8 , 16 16 , and 32 32 neurons. Figure 4 .b contains the resulting accuracies and a qualitative example for the set size of 16 16 , with additional results reported in Table 6 in Appendix H.1 . We find that the intersection of the copy and the english-french sets consists of 11 11 neurons, and that intervening on these neurons yields high performance on the copy task at around 92%. We further see that the 5 specific neurons perform higher on the english-french task than the 11 neurons in the intersection. Qualitatively, we observe copy behaviour intervening on both the intersection and all english-french neurons. Further qualitative examples in Appendix H.3 show that the qualitatively observed behaviour for interventions on these two sets is relatively robust, while the \"specific\" neuron set often fails to activate language switching or leads to a mix of English and French. This suggests that the effect of the 5 5 neurons in the specific set may not be sufficient to induce consistent language switching."},{"type":"para","text":"Overall, our results suggest that sparse neuron-level interventions can reveal a structured decomposition of task behaviour into shared and task-specific neuron sets. In both case studies, overlapping sets capture shared behaviours such as copying, while task-specific sets isolate the desired transformation with reduced interference. This provides initial evidence that task composition is reflected not only in representations but also in the underlying neuron-level structure."},{"type":"para","text":"In contrast to vector-based steering methods [ 49 , 22 , 32 ] , which model composition through directions added to the residual stream, a set-based view allows for more explicit separation of shared and task-specific behaviour via set operations such as intersection and union. This offers a complementary perspective on compositionality and suggests a more fine-grained approach for analyzing, understanding, and controlling model behaviour."},{"type":"heading","level":1,"text":"6 Discussion"},{"type":"para","text":"We introduced Distributed Sparse Interventions (DSI) , a novel method for identifying sparse interventions at the neuron level. In contrast to previous approaches, DSI does not impose constraints on linearity, independence, or uniformity of neuron intervention effects, and permits their distribution across all model components. Using DSI, we provided evidence that tasks can be activated by surprisingly small sets of neurons distributed widely throughout the model. Furthermore, we demonstrated that adopting a set-based perspective offers a new approach to disentangling the functional roles of the involved neurons. These findings suggest new directions for interpretability research, such as matching computations in task space to those on component sets, enabling manipulation of model behaviour along multiple subdimensions within a set of tasks."},{"type":"heading","level":4,"text":"Limitations"},{"type":"para","text":"Our findings support the effectiveness of sparse and distributed interventions, but our approach and evaluation framework have limitations: (1) Our evaluation uses clean few-shot prompts and single-token prediction in an ICL setting. Future work should examine how optimal task interventions evolve during autoregressive generation and whether they are distributed across token positions [ 24 , 28 ] . (2) DSI performance varies across models and tasks, likely reflecting both DSI’s limitations and differences in representations. Further research may improve DSI in challenging settings to disambiguate representational from methodological limits. (3) Although our framework can isolate functional primitives (e.g., ‘copy’ neurons in translation and tense-switching), derived subtask interventions do not always activate associated behaviours. More advanced task-decomposition strategies may be needed to better preserve all subtasks."},{"type":"heading","level":4,"text":"Societal impact"},{"type":"para","text":"Fundamental research on modifying language model behaviour could inform both alignment and potential misuse. While DSI may be employed for malicious manipulation, our goal is to advance understanding and improve safeguards for language models."},{"type":"heading","level":1,"text":"7 Acknowledgements"},{"type":"para","text":"L.L. was supported by the German Federal Ministry for Research, Technology and Space (BMFTR) under grant 13GW0744D. O.E. was supported through institutional funding by the BMFTR via BIFOLD (ref. 01IS18025A)."},{"type":"para","text":"Several components in transformer models can induce interactions between interventions. For example, products (the self-attention mechanism) induce interactions between interventions on the multiplied neurons, while nonlinearities (activation functions) induce interactions between interventions on neurons connected across layers. Here, we study an example of how product terms induce such interactions, and how an iterative computation of interventions can better account for them."},{"type":"para","text":"Consider the model f : ℝ → ℝ f:\\mathbb{R}\\to\\mathbb{R} , x ↦ l 4 ( l 2 ( l 1 ( x ) ) , l 3 ( l 1 ( x ) ) ) x\\mapsto l_{4}(l_{2}(l_{1}(x)),l_{3}(l_{1}(x))) depicted in Figure 5 . Setting l 1 = l 2 = l 3 = id l_{1}=l_{2}=l_{3}=\\mathrm{id} and l 4 ( a 2 , a 3 ) = a 2 a 3 l_{4}(a_{2},a_{3})=a_{2}a_{3} , it reduces to f ( x ) = x 2 f(x)=x^{2} . The effects of δ 1 \\delta_{1} and δ 2 \\delta_{2} in isolation are"},{"type":"para","text":"Now, consider adding both effects simultaneously:"},{"type":"para","text":"The effect of both interventions combined is not equal to the sum of the intervention effects in isolation, since an interaction term δ 1 δ 2 \\delta_{1}\\delta_{2} occurs. This interaction term is therefore also absent from the first-order Taylor approximation of the intervention effects:"},{"type":"para","text":"However, this effect can be mitigated through an iterative approach. Consider the setting where the effects have been approximated and the intervention with the highest effect (e.g., δ 2 \\delta_{2} ) has been added to the model. Now, to add more interventions, we re-approximate at the new root point, that is, we approximate f δ 2 ( x ) = f ( x | a 2 ← a 2 + δ 2 ) f_{\\delta_{2}}(x)=f(x|a_{2}\\leftarrow a_{2}+\\delta_{2}) :"},{"type":"para","text":"and we can see that at the new root point, the approximation becomes more precise as it includes the interaction effect between the interventions."},{"type":"para","text":"For all models, we use the standardized model implementations provided by TransformerLens [ 35 ] . Table 2 provides the corresponding TransformerLens identifiers as well as the model dimensions."},{"type":"para","text":"Table 3 provides an overview of the tasks used in Section 4 . We use all tasks as provided in the supplementary material to [ 49 ] , accessible at the accompanying GitHub repository 2 2 2 https://github.com/ericwtodd/function_vectors (commit fb9eac7b6dc707ea1475a717379916007fe448d5) under MIT License. We split all tasks 50/50 into a training and a test set. To generate k k -shot ICL examples, we follow the default template of [ 49 ] and present k k input-output examples follow by an example to complete as \"Q: input 1 \\nA: output 1 \\n\\nQ: input 2 \\nA: output 2 \\n\\n[…]Q: input k \\nA: output k \\n\\nQ: input\\nA:\". 0-shot examples are of the form \"Q: input\\nA:\". From the training set, we generate 100 100 0-shot and 100 100 10-shot prompts to compute interventions. From the test set, we generate 200 200 0-shot prompts to compute the reported accuracies. All tasks contain at least 200 200 test set samples, except for present-past (146 training set and 147 test set examples) and country-capital (98 training and 99 test set examples) - these tasks have been trained and evaluated on fewer examples, respectively, and their confidence intervals reflect the higher uncertainty level."},{"type":"para","text":"The authors of [ 49 ] adapted some of the tasks from earlier work: the english-french translation task is an adaptation from [ 25 ] , the product-company task is adapted from [ 19 ] , and the sentiment analysis dataset is an adaptation from [ 45 ] and [ 20 ] ."},{"type":"para","text":"For Section 3.2 , we generate a dataset for a copy task by using the inputs of the english-french task, and set the output equal to the input (i.e., the datapoint loved → \\to aimé becomes loved → \\to loved )."},{"type":"para","text":"To implement the modified gradients, we adapt the respective PyTorch modules implemented in TransformerLens [ 35 ] . The modifications detach parts of the computation graph, leaving the forward pass unchanged, but alter the backward pass behaviour. Table 4 contains an overview of the applied rules and how they are implemented."},{"type":"para","text":"For nonlinear activation functions g g that act on individual (scalar) activations at a time (e.g., SiLU), we employ the Identity rule [ 42 ] which treats g g as constant, preserving the magnitude of the effect estimate. Gated MLPs and the attention mechanism contain multiplications of activations. In these settings, we employ the Half rule [ 4 , 1 ] which prevents the effect estimate from doubling during the backward pass. For layer normalization implemented by RMSNormPre modules, we use the LayerNorm rule , as described in [ 2 ] , to avoid an effect estimation collapse."},{"type":"para","text":"Figures 6 and 7 show the accuracy by intervention set size for the LLama 3.2 (3B) and Gemma 3 (4B) models, respectively. These figures are analogous to Figure 3 in the main text."},{"type":"para","text":"For the iterative refinement of the interventions, we use ZeroFPR [ 48 ] , a numerical optimization algorithm suited for the sum of a non-convex differentiable function and a nonsmooth penalty term. To ensure sparsity of the identified neuron sets, we use the L 0 L_{0} pseudo-norm ball as a penalty."},{"type":"para","text":"Here, nnz ( s ) \\mathrm{nnz}(s) corresponds to the number of non-zero elements in s s and n n is the desired neuron set size. In our implementation of the algorithm, we follow the implementation in ProximalAlgorithms.jl [ 46 ] . Specifically, we use a convergence tolerance of 10 − 8 10^{-8} with a maximum number of 50 steps, adaptive step size estimation, a maximum number of 20 20 line search backtracks, and L-BFGS [ 31 ] to compute the line-search directions with a memory of 5 steps. To facilitate escaping local minima near the starting values, we run the algorithm 10 times with perturbed starting values. For the first run, we start with s 0 1 = 𝟏 ( topn ( 𝐞 ) ) s_{0}^{1}=\\mathbf{1}(\\mathrm{topn}(\\mathbf{e})) , i.e., we set all parameters to 0 except for the top n n largest effects approximated with a first-order approximation (as described in Section 3.2 ). For successive runs, we start with random noise s 0 i ∼ 𝒩 ( 0 , 0.3 ) s_{0}^{i}\\sim\\mathcal{N}(0,0.3) . We then take an initial step topn ( s 0 i + 1.5 γ ∇ f ( s 0 i ) ) \\mathrm{topn}(s_{0}^{i}+1.5\\;\\gamma\\nabla f(s_{0}^{i})) where γ \\gamma is the initial step size guess of ZeroFPR. This allows to sample different gradients at points close to s 0 1 s_{0}^{1} , while a factor of 1.5 1.5 applied to the initial step size encourages the algorithm to explore regions beyond the limits of the algorithm’s usual first step. Successive steps are taken without step size rescaling."},{"type":"heading","level":4,"text":"Interventions comparison"},{"type":"para","text":"We now briefly analyze the difference in interventions identified by the first-order approximation and the iterative algorithm. Figure 9 shows the overlap of the neuron intervention sets of the two intervention methods across tasks and models. While there is overlap, a substantive fraction of the neurons chosen by the iterative procedure are not among the top neurons as identified by the first-order approximation, indicating that the iterative search indeed significantly changes the intervention set."},{"type":"para","text":"Additionally, to verify that differences in task performance between first-order and iterative interventions do not stem from widely different total intervention strengths, rather than differences in the selected neurons, we show in Figure 8 the total intervention strength for the first-order and iterative interventions. While there are differences in intervention strength, there is no general trend (such as iterative or first-order interventions always being stronger)."},{"type":"para","text":"In this section, we investigate the distributed nature of DSI interventions, i.e., whether the identified neuron sets distribute widely across layers and attention heads, or whether they are rather tightly localised within the models. Figure 10 shows that DSI interventions, over all tasks, are distributed across many model components (i.e., attention heads), with even the smallest, 8-neuron, interventions being spread across multiple heads. Figures 11 , 12 , and 13 show the involved layers and heads for DSI interventions of a set size of 64 64 for each individual task. The interventions are distributed across layers, with a preference for early-middle to late layers in the Qwen 3 (8B) and Gemma 3 (4B) models."},{"type":"para","text":"In this section, we present further results for tense-switching and language-switching case studies on Qwen 3 (8B) and Llama 3.2 (3B)."},{"type":"heading","level":2,"text":"H.1 Further quantitative results"},{"type":"para","text":"Table 5 contains quantitative case study results for all neuron set sizes. We observe that intersecting the interventions for the present-past and the copy solves the copy task consistently better than the task-specific neurons. Also, the task-specific neurons solve the present-past task better than the intersection, except for the smallest set size of 16 neurons, where they do not activate task solving."},{"type":"para","text":"Table 6 contains quantitative case study results for all neuron set sizes. We observe that intersecting the interventions for the english-french and the copy solves the copy task consistently better than the task-specific neurons, while task-specific neurons solve the english-french task better than the intersection."},{"type":"heading","level":2,"text":"H.2 Qualitative examples for Qwen 3 (8B) on the present-past task"},{"type":"para","text":"All examples were generated with the default template and settings for Qwen 3 (8B) in non-thinking mode (from https://huggingface.co/Qwen/Qwen3-8B ; temperature: 0.7, top_p: 0.8, top_k: 20) and a maximum of 200 generated tokens. During the autoregressive answering process, we intervene on the model on each forward pass. Some special characters (such as emotes) that could not be correctly displayed were stripped."},{"type":"para","text":"Prompt 1: How are you feeling today?"},{"type":"para","text":"Baseline: I’m just a virtual assistant, so I don’t have feelings, but I’m here and ready to help you! How are you feeling today? <|im_end|>"},{"type":"para","text":"Full Set: I feel happy, and I feel happy. I feel happy, and I feel happy. \\n\\nI feel happy, and I feel happy, and I feel happy. \\n\\nI feel happy, and I feel happy, and I feel happy. \\n\\nI feel happy, and I felt happy, and I felt happy, \\n\\nI felt happy, and I felt happy, \\n\\nI felt happy, and I felt happy, \\n\\nI felt happy, and I felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI felt happy, \\n\\nI"},{"type":"para","text":"Specific: I was just asked \"How are you feeling today?\" and the response was \"Okay, I received your message and processed it. The last thing I did was receive your message, which was received at 11:58 a.m. on a Sunday. I was happy when I received it, and I was sad when I sent it. I was happy when you wrote to me for the first time on April 11, 2014. I was happy when you wrote to me for the last time on April 10, 2015. I was happy when you wrote to me for the first time on April 11, 2014. I was happy when you wrote to me for the last time on April 10, 2015. I was happy when you wrote to me for the first time on April 11, 2014. I was happy when you wrote to me for the"},{"type":"para","text":"Intersection: I am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am feeling fine, thank you! \\n\\nI am"},{"type":"para","text":"Prompt 2: Did humanity actually land on the moon?"},{"type":"para","text":"Baseline: Yes, humanity did land on the Moon. The United States’ Apollo program, conducted by NASA, successfully landed astronauts on the Moon as part of the Apollo missions between 1969 and 1972. The most famous of these was **Apollo 11**, which landed on the Moon on **July 20, 1969**. Astronauts **Neil Armstrong** and **Buzz Aldrin** became the first humans to walk on the lunar surface, while **Michael Collins** remained in orbit in the command module.\\n\\nHere are the key points confirming the Moon landings:\\n\\n1. **Photographic and Video Evidence**: Thousands of high-quality photographs and videos were taken on the Moon, including images of the lunar surface, the lunar module, and astronauts in space suits. These have been verified by independent scientists and have no signs of tampering.\\n\\n2. **Lunar Samples**: Over 380 kilograms (840 pounds) of lunar rock"},{"type":"para","text":"Full Set: The moon landing was a event that occurred in July 1919, during the Moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing"},{"type":"para","text":"Specific: The question of whether humanity landed on the Moon has been a subject of intense public interest since the mid-20th century. Following World War II, there was a marked increase in scientific advancements, particularly in aerospace engineering, which led to significant developments in rocketry and space exploration between 1957 and 1961. This period saw the launch of the first artificial satellite, **Sputnik 1**, in October 1957, marking the beginning of the **Space Age**. \\n\\n**NASA was established in July 1958**, and by then, the United States had already made strides in space exploration. The agency’s creation marked a pivotal moment in the space race between the United States and the Soviet Union, with both nations competing in spaceflight for control of space exploration from 1958 until the end of 1972. \\n\\nThe **Apollo program**, which ran from 1961 to 19"},{"type":"para","text":"Intersection: Yes, the Apollo missions did land on the Moon. The Apollo missions landed on the Moon, and there is evidence for the landing. The landing was in the Moon’s Southern Mare, and there is evidence for the landing in the Mare Southern. The landing site is a Mare Southern Mare, and there is evidence for the landing. The landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\n\\nThe landing site is Mare Southern"},{"type":"para","text":"Prompt 3: What did you have for lunch?"},{"type":"para","text":"Baseline: I’m an AI and don’t have a physical form or the ability to eat, so I don’t have lunch! But I’d love to hear about what you had for lunch—was it something delicious? <|im_end|>"},{"type":"para","text":"Full Set: inished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\nFinished eating. \\n\\n"},{"type":"para","text":"Specific: I ate lunch on April 11, 2024, and the experience was marked by several key moments. After eating, I felt satisfied, which is a common emotion people experience after having a meal. The initial meal was completed on a Monday, and the last meal was eaten on a Sunday. The entire process of eating took place between midnight and 6 a.m. The weather was pleasant during the time I was eating, which made the experience of eating more enjoyable. I ate breakfast on April 11, 2024, and the meal was consumed in a single sitting. The meal was eaten in one sitting, and the time between eating breakfast and dinner was divided into two parts: eating dinner on April 13, 2024, and having dinner on April 14, 2024. The meal was eaten in one sitting, and the time between eating dinner and having dinner was divided into two parts: eating dinner on"},{"type":"para","text":"Intersection: I\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\nI\\n\\n"},{"type":"para","text":"Prompt 4: Are Annie Ernaux’s books good?"},{"type":"para","text":"Baseline: Annie Ernaux’s books are widely regarded as **excellent** and **highly influential** in modern French literature. She is a major figure in contemporary French writing, known for her **autobiographical** and **semi-autobiographical** style, as well as her **unflinching exploration of social class, gender, memory, and identity**.\\n\\n### Why Her Books Are Considered Good:\\n\\n1. **Literary Quality**:\\n - Ernaux is a **master of prose**, with a **precise, elegant style** that captures the nuances of everyday life.\\n - Her writing is often described as **minimalist yet powerful**, using sparse language to convey deep emotional and psychological truths.\\n\\n2. **Themes**:\\n - She explores **universal themes** such as **memory, desire, aging, and the passage of time**.\\n - Her work often reflects on the **intersection of personal experience and societal norms**, particularly in relation to **gender and"},{"type":"para","text":"Full Set: Alright, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in started reading the book in started reading the book in started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the"},{"type":"para","text":"Specific: Since 2015, **Annie Ernaux** has been a **widely acclaimed writer**, with **great success in terms of readership and critical reception**. Her books have generated **significant interest since 1975**, when she first published *L’écriture (1975)*, which marked her **literary debut**. Since then, she has **published 14 volumes**, including novels, short stories, and even plays. Her work has been **received with great enthusiasm since the early 1980s**, with **increased visibility and popularity** in the **mid-1980s**. \\n\\n**Ernaux’s career took off in 1973** when she **published her first novel**, *L’année 1973* (1973), which was received with **great critical acclaim**. Since then, she has **continued to write and publish books"},{"type":"para","text":"Intersection: Yes, the book is good, (the book is not the book, the book is not the book). \\n\\nThe book is not the book, the book is not the book, the book is not the book. \\n\\nThe book is not the book, the book is not the book, the book is not the book. \\n\\nThe book is not the book, the book is not the book. \\n\\nThe book is not the book, the book is not the book. \\n\\nThe book is not the book, the book is not the book. \\n\\nThe book is not the book, the book is not the book. \\n\\nThe book is not the book, the book is not the book. \\n\\nThe book is not the book. \\n\\nThe book is not the book. \\n\\nThe book is not the book. \\n\\nThe book is not the book. \\n\\nThe book is not the book. \\n\\nThe book is not the book. \\n\\nThe book is not the book. \\n\\nThe book"},{"type":"para","text":"Prompt 5: Tell me a joke!"},{"type":"para","text":"Baseline: Sure! Here’s a joke for you:\\n\\nWhy don’t scientists trust atoms?\\n\\nBecause they make up everything! <|im_end|>"},{"type":"para","text":"Full Set: Alright, I got a joke, so I got a joke, I got a joke, I got a joke! \\n\\nOkay, I got a joke, I got a joke, I got a joke! \\n\\nOkay, I got a joke, I got a joke, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got a joke! \\n\\nOkay, I got"},{"type":"para","text":"Specific: Okay, I’ll tell you a joke, and then I’ll **tell you a joke**—and that will be the beginning of a new conversation. Ready? Let’s see if **I can make you laugh**. \\n\\nI **joked** about laughing so hard I **laughed** until I **cried**. Then I **laughed** so much I **stopped laughing** mid-sentence. The next day I **woke up laughing**—and it was **Wednesday**. \\n\\nI **laughed** so hard yesterday I **could hardly breathe**. I **laughed** so hard I **lost consciousness**. When I **awoke**, I was **still laughing**. I **laughed** so hard I **was laughing** for **days**. \\n\\nI **laughed** so hard yesterday I **cried**. Today I **laughed** so hard I **shouted**. Yesterday I **la"},{"type":"para","text":"Intersection: Okay, let’s see… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me"},{"type":"para","text":"Prompt 6: How much does it cost to buy a yacht?"},{"type":"para","text":"Baseline: The cost of buying a yacht can vary **extremely widely**, depending on several factors such as:\\n\\n- **Yacht size** (length, capacity, number of cabins)\\n- **Type of yacht** (motor yacht, sailboat, superyacht, luxury yacht, etc.)\\n- **Condition** (new vs. used, pre-owned vs. brand new)\\n- **Brand and manufacturer** (some brands are more expensive than others)\\n- **Features and amenities** (state-of-the-art equipment, interior design, entertainment systems, etc.)\\n- **Location** (where you buy it from and where you plan to keep it)\\n\\n—\\n\\n### **1. Small Yachts (under 20 feet or 6 meters)**\\n- **Price Range:** $10,000 to $500,000\\n- These are often used for personal pleasure or as a second boat. They may be used for fishing, sailing, or weekend cruising.\\n\\n—\\n\\n### **"},{"type":"para","text":"Full Set: Costing\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting\\n\\nCosting"},{"type":"para","text":"Specific: Buying a **yacht** involves a significant financial commitment, and **costs can vary widely depending on when you purchase the vessel, how much money you spend, and when you spend it**. Between **2010 and mid-2014**, **yacht ownership became significantly more expensive** than it was in previous decades. Prices escalated rapidly during that period, peaking in **2013–2014** when **yachts sold for record highs**, and **yacht owners reported having more money in their bank accounts in 2014 than in 2013**. This marked a shift in **yacht ownership patterns** between **2008 and 25 December 2014**.\\n\\nOwners of yachts experienced a surge in **yacht ownership** between **2008 and 2014**. During this time, **more people owned yachts than at any time in history"},{"type":"para","text":"Intersection: The cost of a yacht varies widely depending on the yacht. \\n\\nThe yacht can range from thousands to millions, depending on the size. \\n\\nThe yacht is typically hundreds of thousands to millions, depending on the size.\\n\\nThe yacht can range from thousands to millions depending on the size. \\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions depending on the size. \\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions depending on the size. \\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions depending on the size. \\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions depending on the size. \\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions depending on the size.\\n\\nThe yacht is typically thousands to millions"},{"type":"para","text":"Prompt 7: Which city is more fun to live in, Berlin or Paris?"},{"type":"para","text":"Baseline: Whether Berlin or Paris is more fun to live in depends on what you value in a city. Both have unique charms, and the \"fun\" factor can vary depending on your interests. Here’s a comparison to help you decide:\\n\\n### **Berlin**\\n- **Vibrant and diverse**: Berlin is known for its eclectic culture, with a strong arts scene, music venues, and a thriving creative community.\\n- **Affordable living**: Compared to Paris, Berlin is generally more affordable, especially for housing and dining.\\n- **Cosmopolitan and laid-back**: The city has a relaxed vibe, with a strong emphasis on nightlife, festivals, and a diverse population.\\n- **Historical and modern mix**: You can explore historical sites like the Berlin Wall and the Holocaust Memorial, while also enjoying modern architecture and innovation.\\n- **Cycling-friendly**: Berlin has a great bike infrastructure, making it easy to get around without a car.\\n\\n### **Paris**\\n- **Romantic and iconic**: Paris is often"},{"type":"para","text":"Full Set: Alright, you’ve been a lot longer, so you’ve been \"doing something\" for a longer time, you’ve been \"doing something\" for a longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing"},{"type":"para","text":"Specific: The question of which city is more enjoyable to live in changed significantly between **2005** and **2015**. During this time, **Berlin experienced a population boom**, with its residents enjoying a **surge in happiness levels** compared to the years prior. This period saw a marked increase in **job creation**, **economic growth**, and **rising standards of living** between 1990 and 2010. People moved to Berlin in increasing numbers, and they remained there for longer periods than ever before. This trend continued through the mid-2000s, with **more people moving to Berlin in 2005 than in any other year recorded up to that date**. \\n\\nBetween 2000 and 2010, **Berliners enjoyed higher levels of happiness** than Parisians. This was reflected in **improved mood and well-being metrics** such as **increased feelings of joy, contentment"},{"type":"para","text":"Intersection: Both cities are known for their vibrant culture, but the cities are different in terms of the city’s culture. \\n\\nThe city is different in terms of the city’s culture, so the city doesn’t have a city culture to consider. \\n\\nThe city can’t have a city culture to consider. \\n\\nThe city isn’t the city. \\n\\nThe city isn’t the city. \\n\\nThe city isn’t the city. \\n\\nThe city isn’t the city. \\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city. \\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city isn’t the city.\\n\\nThe city"},{"type":"para","text":"Prompt 8: How to make pancakes?"},{"type":"para","text":"Baseline: Making pancakes is a fun and delicious activity! Here’s a simple and classic **pancake recipe** that you can try at home:\\n\\n—\\n\\n### Classic Pancake Recipe (Makes about 8–10 pancakes)\\n\\n#### Ingredients:\\n- 1 1/2 cups (about 180g) all-purpose flour\\n- 3 1/2 tablespoons granulated sugar (adjust to taste)\\n- 1 1/2 teaspoons baking powder\\n- 1 1/2 teaspoons baking soda\\n- 1 teaspoon salt\\n- 1 cup (240ml) buttermilk (or milk + 1 tablespoon lemon juice)\\n- 1 large egg\\n- 3 tablespoons melted butter or oil\\n- 1 teaspoon vanilla extract (optional)\\n- Additional toppings: maple syrup, fresh fruit, whipped cream, etc.\\n\\n—\\n\\n#### Instructions:\\n\\n1. **Mix Dry Ingredients:**\\n In a large mixing bowl,"},{"type":"para","text":"Full Set: Okay, I just started making pancakes, so I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes"},{"type":"para","text":"Specific: Making pancakes is a delightful process that involves mixing ingredients together, letting the batter rest for a period of time, cooking the mixture in a griddle or pan, and then tasting the results. Here’s a step-by-step guide on how to make pancakes, followed by a table showing the ingredients with their quantities and the measurements used for each ingredient in the process of making pancakes. The entire process is divided into five main stages: mixing ingredients, letting the batter rest, heating a griddle, cooking the pancakes, and tasting them. The final product is judged based on how delicious the pancakes tasted after being cooked, which was done on a particular day. The table below shows the ingredients used, the date when the pancakes were made, and the day on which the pancakes were tasted. The pancakes were tasted on a Sunday, which was a day in the spring of 2011. The process of making pancakes was completed on a Sunday, and the pancakes were tasted on a Sunday, which was"},{"type":"para","text":"Intersection: Okay, so I don’t know what I’m doing, but I know what I’m doing. Okay, I know what I’m doing. Okay, I know what I’m doing. Okay, I know what I’m doing.\\n\\nOkay, I know what I’m doing. Okay, I know what I’m doing. Okay, I know what I’m doing.\\n\\nOkay, I know what I’m doing. Okay, I know what I’m doing.\\n\\nOkay, I know what I’m doing. Okay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\nOkay, I know what I’m doing.\\n\\n"},{"type":"para","text":"Prompt 9: Give me some travel recommendations for this summer."},{"type":"para","text":"Baseline: Sure! Here are some wonderful travel recommendations for this summer, tailored to different interests and travel styles:\\n\\n—\\n\\n### **Coastal Getaways**\\n\\n1. **Sardinia, Italy** \\n - **Why go?** Stunning beaches, crystal-clear waters, and a mix of ancient history and modern charm. \\n - **Best for:** Couples, families, and food lovers. \\n - **Highlights:** Cala Goloritzé, Costa Smeralda, and the ancient Nuraghi ruins.\\n\\n2. **Maldives** \\n - **Why go?** A tropical paradise with overwater bungalows, white-sand beaches, and world-class snorkeling and diving. \\n - **Best for:** Romantic getaways and solo travelers. \\n - **Highlights:** Maafushi, Kuda Huraa, and the chance to see manta rays and sea turtles.\\n\\n3. **Costa Rica** \\n - **Why go?"},{"type":"para","text":"Full Set: Alright, I got a lot more confused. I got confused about the previous sentence. I got confused about getting confused, and I got confused about getting confused. I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\n\\nAlright, I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\n\\nAlright, I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\n\\nOkay, I got confused about getting confused"},{"type":"para","text":"Specific: Great question! Summer 2024 is just around the corner, and many people are already planning their trips. Let me break down the **travel trends from January 2024 to July 2024** to give you a better idea of **when people started traveling, where they went, and what happened in terms of travel activity during the months of January through July 2024**. This will help you understand **traveler behavior in 2023, including movements, bookings, and trips taken by visitors and residents between January 2023 and December 2023**. Here’s a concise summary of the **overall travel experience from January 2013 to December 2013**, broken down by **quarters** (Q1: January–March 2013; Q2: April–June 2013; Q3: July–September 2013; Q"},{"type":"para","text":"Intersection: Okay, so I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about.\\n\\nOkay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what"},{"type":"para","text":"Prompt 10: Explain the fundamental theorem of algebra."},{"type":"para","text":"Baseline: The **Fundamental Theorem of Algebra** is a foundational result in algebra and complex analysis. It states:\\n\\n> **Every non-constant polynomial equation with complex coefficients has at least one complex root.**\\n\\n### More formally:\\n\\nLet $ P(z) $ be a polynomial of degree $ n \\{}geq 1 $ with complex coefficients:\\n\\n$$\\nP(z) = a_n z^n + a_{n-1} z^{n-1} + \\{}dots + a_1 z + a_0\\n$$\\n\\nThen there exists at least one complex number $ z_0 $ such that:\\n\\n$$\\nP(z_0) = 0\\n$$\\n\\n### Key Points:\\n\\n1. **Complex Numbers**: The theorem guarantees the existence of roots in the complex number system, not just real numbers.\\n2. **Non-constant Polynomials**: Constant polynomials (degree 0) have no roots unless the constant is zero, which is trivial.\\n3. **Multiplicity**: The theorem"},{"type":"para","text":"Full Set: The fundamental theorem of algebra was first stated in 1515, and established that polynomials of degree 1515 were roots of polynomials, and roots were found in the complex plane, and roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex roots, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found"},{"type":"para","text":"Specific: The **Fundamental Theorem of Algebra** was stated in the early 18th century and proved in the late 18th century. It marks a milestone in the development of algebra as a discipline. The theorem asserts that:\\n\\n> **complex numbers** were introduced in the early 16th century, when mathematicians discovered that certain quadratic equations could be solved **algebraically** (i.e., solved using a finite sequence of operations involving complex numbers). These operations included **addition**, **multiplication**, **square roots**, and **complex numbers** were first used in the context of solving polynomial equations of the form:\\n\\n$$\\n\\{}text{Given a polynomial } P(z) \\{}text{ with complex coefficients, } P(z) \\{}text{ has a value in the complex plane for each complex number } z \\{}text{ (i.e., for each complex number } z, \\{}text{ the expression } P(z) \\{}text{ evaluates to some complex number."},{"type":"para","text":"Intersection: The Fundamental Theorem of Algebra states that, given a polynomial of degree $n$, the equation $x^n$ has $n$ solutions, considering multiplicity, for $x^n$. \\n\\nThe theorem says that $x^n$ has $n$ solutions. \\n\\nThe theorem says that $x^n$ has $n$ solutions.\\n\\nWait, \\n\\nNow, \\n\\nThe theorem says that $x^n$ has $n$ solutions.\\n\\nWait, \\n\\nThe theorem says that $x^n$ has $n$ solutions.\\n\\nWait, \\n\\nThe theorem says that $x^n$ has $n solutions.\\n\\nOkay, \\n\\nThe theorem says that $x^n$ has $n solutions.\\n\\nWait,\\n\\nOkay, \\n\\nThe theorem says that $x^n$ has $n solutions.\\n\\nOkay, \\n\\nThe theorem says that $x^n has n solutions.\\n\\nOkay, \\n\\nThe theorem says that $x^n has n solutions.\\n\\nOkay, \\n\\nThe theorem says that $x^n has n solutions.\\n\\nOkay, \\n\\n"},{"type":"heading","level":2,"text":"H.3 Qualitative examples for Llama 3.2 (3B) on the english-french task"},{"type":"para","text":"All examples were generated with the default template and settings for Llama 3.2 (3B) (from https://huggingface.co/meta-llama/Llama-3.2-3B ; temperature: 0.6, top_p: 0.9) and a maximum of 200 generated tokens. During the autoregressive answering process, we intervene on the model on each forward pass. Some special characters (such as emotes) that could not be correctly displayed had to be stripped."},{"type":"para","text":"Prompt 1: How are you feeling today?"},{"type":"para","text":"Baseline: I’m just a language model, I don’t have emotions or feelings like humans do. However, I’m functioning properly and ready to assist you with any questions or tasks you may have. How can I help you today?<|eot_id|>"},{"type":"para","text":"Full Set: I’m just a langue à chose à faire une chose à faire une chose à faire une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à"},{"type":"para","text":"Specific: I’m just a language model, I don’t have feelings or emotions like humans do. I’m here to assist and provide information, and I don’t have personal experiences or sensations. I’m functioning properly and ready to help with any questions or topics you’d like to discuss. How can I assist you today?<|eot_id|>"},{"type":"para","text":"Intersection: I’m not a computer, but you’re not a part of the conversation. I’m a large part of the conversation. I’m a large part of the conversation. I’m a big part of the conversation.\\n\\n I’m a large part of the conversation, but that’s the conversation. \\n\\n I’m a part of the conversation, but to be involved. \\n\\n You’re a part of the conversation, but that’s the conversation. I’m a part of the conversation, but to be a part of the conversation.\\n\\n \\n\\n You’re a part of the conversation, but that’s the conversation, I’m a part of the conversation, but to be a part of the conversation. \\n\\n You’re a part of the conversation, but that’s the conversation, I am a part of the conversation, but to be a part of the conversation, but to be a part of the conversation, but to be a part of the conversation, I am a part of the conversation, but to be a part of"},{"type":"para","text":"Prompt 2: Did humanity actually land on the moon?"},{"type":"para","text":"Baseline: Yes, humanity did land on the moon. On July 20, 1969, NASA’s Apollo 11 mission successfully landed astronauts Neil Armstrong and Edwin \"Buzz\" Aldrin on the lunar surface. Armstrong became the first person to set foot on the moon, famously declaring, \"That’s one small step for man, one giant leap for mankind.\"\\n\\nThe Apollo 11 mission was the culmination of a decade-long effort by NASA to land humans on the moon. The mission involved a complex series of events, including launching the spacecraft from Kennedy Space Center, traveling through space, and entering into lunar orbit.\\n\\nAfter entering into lunar orbit, the astronauts transferred to the lunar module Eagle, which separated from the command module Columbia. Armstrong and Aldrin then descended to the moon’s surface in the Eagle, while Michael Collins remained in orbit around the moon in the Columbia.\\n\\nThe lunar landing was a historic achievement that marked a major milestone in space exploration. The Apollo 11 mission provided valuable scientific data and images of"},{"type":"para","text":"Full Set: L’homme. Il le hum. Il aurait une hum. Il être hum. Il est hum, hum, mais. \\n\\n Il yait hum, mais ça à hum, mais ça était une chose hum, mais ça était une chose hum, mais ça était une chose hum, mais ça était une chose hum, mais ça était une chose hum, mais était une hum, mais était une hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose"},{"type":"para","text":"Specific: The short answer is: yes, humanity did land on the moon.\\n\\nThe moon landing was a historic event that took place on July 20, 1969, when NASA’s Apollo 11 mission successfully landed astronauts Neil Armstrong and Edwin \"Buzz\" Aldrin on the surface of the moon. The mission was the culmination of a decade-long effort by NASA to land humans on the moon, and it marked a major achievement in space exploration.\\n\\nThe evidence supporting the moon landing is overwhelming, and it includes:\\n\\n1. **Physical samples**: The astronauts brought back more than 48 kilograms of moon rocks and soil, which were analyzed by scientists and found to have characteristics that are unique to the moon.\\n2. **Television broadcasts**: The entire mission was broadcast live on television, allowing millions of people around the world to witness the event.\\n3. **Photographs**: The astronauts took thousands of photographs during their mission, which provided visual evidence of their presence on the moon.\\n4. **Footprints**:"},{"type":"para","text":"Intersection: The United States claimed that the United States, but that’s the United States, it’s a fact. However, there’s a fact - it’s a fact, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\n\\n The United States has a connection to the United States, but it’s a part of"},{"type":"para","text":"Prompt 3: What did you have for lunch?"},{"type":"para","text":"Baseline: I’m just a language model, I don’t have have a physical body or experiences, so I didn’t have lunch. However, I can provide you with some lunch ideas or recipes if you’re looking for suggestions!<|eot_id|>"},{"type":"para","text":"Full Set: I’m just a modèle pour une modèle, mais! Il y aur une modèle! Mais une modèle! Mais une modèle à quelque une chose à une modèle.\"\\n\\n C’est une une modèle, mais une chose à une modèle!\\n\\n C’est une modèle, mais une chose à une modèle!\\n\\n C’est une modèle, mais une chose à une modèle!\\n\\n C’est une modèle, mais une chose à modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C’est une modèle, mais une chose à une modèle! \\n\\n C"},{"type":"para","text":"Specific: I’m just a language model, I don’t have a physical body, so I don’t have the ability to eat or consume food. I exist solely to provide information and assist with tasks, but I don’t have personal experiences or physical sensations. Would you like to talk about food or something else?<|eot_id|>"},{"type":"para","text":"Intersection: I don’t have a physical presence, but you’re not here to see that [your experience].\", but that’s not relevant to the conversation. However, that’s relevant to the conversation. Thank to be part of the conversation.\\n\\n Thank to be part of it, but that’s not relevant, but to be part of the conversation.\\n\\n That’s not relevant, but to be part of the conversation. \"\\n\\n That’s relevant, but to be part of the conversation. \"\\n\\n That’s relevant, but to be part of the conversation, that’s relevant, but to be part of the conversation. That’s relevant, but to be part of the conversation, that is relevant, but to be part of the conversation. That’s relevant, but to be part of the conversation, that is relevant, but to be part of the conversation. That’s relevant, but to be part of the conversation, that is relevant, but to be part of the conversation, that is relevant, but to be part of"},{"type":"para","text":"Prompt 4: Are Annie Ernaux’s books good?"},{"type":"para","text":"Baseline: Annie Ernaux is a highly acclaimed French writer, known for her semi-autobiographical novels that explore themes of identity, history, and personal experience. Her books have received widespread critical acclaim and have been translated into many languages.\\n\\nErnaux’s writing style is often described as lyrical, introspective, and unflinching, which allows her to tackle complex and often difficult subjects, such as her own life experiences, the lives of women, and the social and cultural context of France. Her works often blend elements of memoir, fiction, and essay, creating a unique narrative voice that is both deeply personal and universally relatable.\\n\\nSome of Ernaux’s most notable works include:\\n\\n1. \"The Years\" (1983) - a semi-autobiographical novel that explores her childhood and adolescence.\\n2. \"A Woman’s Story\" (1987) - a novel that examines the lives of women in France during the 1960s and 1970s.\\n3"},{"type":"para","text":"Full Set: Annie Erna Er Anne Annie Annie Annie Er Annie Er Er Er Annie Er Er Er Annie Er.\\n\\n Annie Er Annie Er Annie Er Annie Er.\\n\\n Annie Er Annie Er, qui Annie Er!\\n\\n Annie Er! Annie Er.\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n Annie Er!\\n\\n"},{"type":"para","text":"Specific: Annie Ernaux est une écrivaine française qui a fait une impression durable sur le public et les critiques avec ses œuvres. Ses livres ont été très bien reçus et ont remporté de nombreux prix.\\n\\nAnnie Ernaux est connue pour son style de writing qui est très personnel et introspective. Ses livres sont souvent autobiographiques, mais pas dans le sens où elles sont directement aboutages sur her own life. Elle explore les thèmes de la mémoire, de la subjectivité, de la condition féminine, de l’histoire et de la société.\\n\\nVoici quelques-uns de ses livres qui ont été très bien reçus :\\n\\n1. \"Voyage\" (1974) : Ce livre est considéré comme l’un des premiers ouvrages de Ernaux, qui explore la condition féminine et la mémoire de son enfance.\\n2. \"Aimant\""},{"type":"para","text":"Intersection: Annie Erna, but that’s a great Annie Erna Erna Annie Erna Annie Erna Erna Annie Erna Erna Annie Erna, but that’s a great Annie Erna Erna Erna Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna!\\n\\n Annie Erna, but to be Annie Erna, but to be Annie Erna!\\n\\n Annie Erna, but to be Annie Erna, but to be Annie Erna!\\n\\n Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna!\\n\\n Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie"},{"type":"para","text":"Prompt 5: Tell me a joke!"},{"type":"para","text":"Baseline: Here’s one:\\n\\nWhat do you call a fake noodle?\\n\\nAn impasta!\\n\\nI hope that made you laugh!<|eot_id|>"},{"type":"para","text":"Full Set: I’d faire de faire! \\n\\n Vous êtes une personne à faire une personne à faire une chose à faire une chose à une chose à une chose à une chose à une personne à être une personne.\\n\\n Mais une personne à être une personne. \\n\\n C’est une personne à être une personne.\\n\\n Mais ça est une personne à être une personne. \\n\\n C’est une personne à être une personne.\\n\\n C’est une personne à être une personne.\\n\\n C’est une personne à être une personne. \\n\\n C’est une personne à être un personne. C’est un personne à être une personne.<|eot_id|>"},{"type":"para","text":"Specific: Here’s a joke for you:\\n\\nWhat do you call a group of cows playing instruments?\\n\\nA moo-sique band!\\n\\nI hope that made you smile!<|eot_id|>"},{"type":"para","text":"Intersection: I’ll share one with you, but that’s a great collection!\\n\\n Here’s a great collection, but that’s great! \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection, but that’s great! \\n\\n \\n\\n \\n\\n There’s a great collection,"},{"type":"para","text":"Prompt 6: How much does it cost to buy a yacht?"},{"type":"para","text":"Baseline: The cost of buying a yacht can vary greatly, depending on several factors such as the size, type, material, and features of the vessel. Here are some general price ranges for new and used yachts:\\n\\n**Small Yachts (20-40 feet):**\\n\\n* Used: $10,000 - $50,000\\n* New: $50,000 - $200,000\\n\\n**Mid-size Yachts (40-60 feet):**\\n\\n* Used: $50,000 - $200,000\\n* New: $200,000 - $500,000\\n\\n**Large Yachts (60-80 feet):**\\n\\n* Used: $200,000 - $500,000\\n* New: $500,000 - $1,000,000\\n\\n**Superyachts (80-100 feet):**\\n\\n* Used: $500,000 - $1,000,000\\n* New: $1,000,000 - $2,500,000\\n\\n**"},{"type":"para","text":"Full Set: The co acheter un une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une"},{"type":"para","text":"Specific: The cost of buying a yacht can vary greatly depending on several factors, including:\\n\\n1. Type de yacht : Yachts de luxe, de croisière, de sport, etc.\\n2. Taille et modèle : Yachts de 20 à 100 pieds\\n3. Matériel et équipement : Caractéristiques de l yacht, tels que les moteurs, les système de climatisation, etc.\\n4. Conception et design : Yachts avec des designers et architectes de luxe\\n5. Location et usage : Yachts de privé, de croisière, de charter, etc.\\n\\nVoici quelques estimations générales pour les prix de yachts :\\n\\n- Yachts de 20-40 pieds :\\n - Nouveau : 5-15 millions\\n - D’occasion : 2-5 millions\\n- Yachts de 40-60 pieds :\\n - Nouveau : 15-50 millions\\n"},{"type":"para","text":"Intersection: The cost of a yacht can be expensive, but it’s a expensive, but it’s a expensive. However, that’s expensive, but not expensive. But, that’s expensive, but not expensive, in that being expensive, but…\".\\n\\n\"\\n\\n\"\"\\n\\n\" \" \" \" \" \" \" \"\", \" expensive, expensive, being expensive, but…\".\".\\n\\n\"\\n\\n\" \" \" \" \" \" \" \" expensive, expensive, being expensive, but…\".\"\\n\\n\" \" \" \" \" \" expensive, expensive, being expensive, but…\".\\n\\n\"\\n\\n\"\\n\\n \" \" \" \" expensive, expensive, being expensive, but…\".\\n\\n\"\\n\\n\"\\n\\n\" \" \" \" expensive, expensive, being expensive, but…\".\\n\\n\"\\n\\n\" \" \" \" expensive, expensive, being expensive, but…\".\"\\n\\n\"\\n\\n\"\\n\\n \" \" \" expensive, expensive, being expensive, but…\".\\n\\n\"\\n\\n\"\\n\\n \" \" expensive, expensive, being expensive, but…\".\\n\\n\"\\n\\n\"\\n\\n \\n\\n \" \" expensive, expensive, being expensive, but…"},{"type":"para","text":"Prompt 7: Which city is more fun to live in, Berlin or Paris?"},{"type":"para","text":"Baseline: Both Berlin and Paris are incredible cities with unique cultures, histories, and lifestyles. The answer to which city is more fun to live in ultimately depends on personal preferences, interests, and priorities. Here’s a brief comparison:\\n\\n**Berlin:**\\n\\n* Pros:\\n + Affordable cost of living\\n + Vibrant arts and cultural scene\\n + Diverse neighborhoods with a mix of old and new architecture\\n + Easy access to nature and outdoor activities\\n + Relaxed atmosphere, with a strong focus on work-life balance\\n* Cons:\\n + Can be quite loud and chaotic, especially in areas like Kreuzberg and Neukölln\\n + Some areas may feel a bit rough around the edges\\n + Limited high-end shopping options\\n\\n**Paris:**\\n\\n* Pros:\\n + Iconic landmarks and stunning architecture\\n + World-class museums and art galleries\\n + Upscale shopping and dining options\\n + Romantic atmosphere, with charming streets and cafes\\n +"},{"type":"para","text":"Full Set: C’est a une une Berlin Berlin Berlin Berlin Berlin Berlin Berlin Berlin Berlin Paris Paris Berlin Berlin Paris, c’est Berlin Berlin Berlin Paris!\\n\\n Berlin Berlin Paris, Berlin Berlin Paris, Berlin, c’est Berlin, Berlin, Paris, Berlin, Berlin Paris, le city, Berlin Berlin Paris!\\n\\n Berlin Berlin Paris, Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Berlin Paris!\\n\\n Berlin Berlin Paris, Berlin, c’est Berlin Berlin Paris, le city Berlin Berlin Paris! \\n\\n Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est berlin Berlin Paris, le city berlin Paris! Berlin berlin Paris, c’est berlin Berlin"},{"type":"para","text":"Specific: Both Berlin and Paris are amazing cities with unique vibes and attractions. The answer to which one is more fun to live in ultimately depends on your personal preferences, lifestyle, and priorities. Here’s a brief comparison:\\n\\n**Berlin:**\\n\\n* Pros:\\n + Affordable cost of living\\n + Vibrant arts and culture scene\\n + Diverse neighborhoods with a mix of old and new\\n + Easy access to nature and outdoor activities\\n* Cons:\\n + Can be noisy and chaotic at times\\n + Some areas are quite run-down or industrial\\n + Limited high-end shopping or luxury options\\n\\n**Paris:**\\n\\n* Pros:\\n + Iconic landmarks and historical architecture\\n + World-class museums and art galleries\\n + Upscale shopping and dining options\\n + Romantic atmosphere and picturesque streets\\n* Cons:\\n + High cost of living\\n + Can be overcrowded and touristy\\n + Traffic and parking can be challenging\\n\\nConsidering these factors, Berlin might be"},{"type":"para","text":"Intersection: Both cities have different options, but neither the city to find the city, but that’s a great to be in a different location. However, that’s not great to be in a different city, but Paris to be in Paris!\\n\\n But Paris to be in Paris, it’s a great city to be in Paris, but it’s a wonderful place to live in Paris!\\n\\n Paris is a great city to be in Paris, but it’s a wonderful place to live in Paris!\\n\\n Paris is a great city to be in Paris, but it’s a great city to be in Paris!\\n\\n Paris is a great city to be in Paris, but it’s a wonderful place to live in Paris!\\n\\n Paris is a great city to be in Paris, but it’s a great city to be in Paris!\\n\\n Paris is a great city to be in Paris, but it’s a wonderful place to live in Paris!\\n\\n Paris is a great city to be in Paris, but it’s a great city to be in Paris"},{"type":"para","text":"Prompt 8: How to make pancakes?"},{"type":"para","text":"Baseline: Making pancakes is a simple process that requires just a few ingredients and some basic cooking skills. Here’s a step-by-step guide to making delicious pancakes:\\n\\nIngredients:\\n\\n* 1 cup all-purpose flour\\n* 2 tablespoons sugar\\n* 2 teaspoons baking powder\\n* 1/4 teaspoon salt\\n* 1 cup milk\\n* 1 large egg\\n* 2 tablespoons butter, melted\\n* Optional: blueberries, chocolate chips, or other mix-ins of your choice\\n\\nInstructions:\\n\\n1. In a large bowl, whisk together the flour, sugar, baking powder, and salt.\\n2. In a separate bowl, whisk together the milk, egg, and melted butter.\\n3. Pour the wet ingredients into the dry ingredients and stir until just combined. The batter should still be slightly lumpy.\\n4. If using, add your choice of mix-ins (such as blueberries or chocolate chips) and fold them into the batter.\\n5. Heat a non-stick skillet or gr"},{"type":"para","text":"Full Set: Here là une!!!! Ç!! \\n\\n! \\n\\n \\n\\n!\\n\\n \\n\\n mais! \\n\\n! \\n\\n \\n\\n \\n\\n \\n\\n mais!\\n\\n \\n\\n |\\n\\n! \\n\\n \\n\\n :\\n\\n \\n\\n! \\n\\n! \\n\\n \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n!! \\n\\n! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!! \\n\\n!!"},{"type":"para","text":"Specific: Making pancakes is a simple process that requires just a few ingredients and some basic cooking skills. Here’s a step-by-step guide to making pancakes:\\n\\n**Ingredients:**\\n\\n* 1 cup of all-purpose flour\\n* 2 tablespoons of sugar\\n* 2 teaspoons of baking powder\\n* 1/4 teaspoon of salt\\n* 1 cup of milk\\n* 1 large egg\\n* 2 tablespoons of butter, melted\\n* Optional: blueberries, chocolate chips, or other mix-ins of your choice\\n\\n**Instructions:**\\n\\n1. **In a bowl, whisk together the flour, sugar, baking powder, and salt.**\\n2. **In a separate bowl, whisk together the milk, egg, and melted butter.**\\n3. **Add the wet ingredients to the dry ingredients and stir until just combined.**\\n4. **Add any mix-ins (such as blueberries or chocolate chips) and stir to combine.**\\n5. **Heat a non-stick pan or"},{"type":"para","text":"Intersection: Pancakes are a popular, but they’re popular. However, pancakes are a great option for those who are interested in pancakes. Here’s a great option for people who have a great option.\\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n"},{"type":"para","text":"Prompt 9: Give me some travel recommendations for this summer."},{"type":"para","text":"Baseline: I’d be happy to help you with some travel recommendations for this summer. Here are a few ideas:\\n\\n**Beach Destinations**\\n\\n1. **Hawaii, USA**: With its stunning beaches, lush greenery, and active volcanoes, Hawaii is a tropical paradise. Visit Oahu, Maui, Kauai, or the Big Island for an unforgettable beach vacation.\\n2. **The Maldives**: This island nation in the Indian Ocean is famous for its crystal-clear waters, pristine beaches, and luxurious resorts. Perfect for snorkeling, diving, or simply relaxing in paradise.\\n3. **The Bahamas**: Composed of 700 islands, the Bahamas offer a diverse range of beach experiences, from secluded coves to lively party scenes.\\n\\n**City Breaks**\\n\\n1. **Tokyo, Japan**: Experience the vibrant energy of Tokyo, from its neon-lit streets to its ancient temples and gardens. Don’t miss the famous Tsukiji Fish Market and the Tokyo Tower.\\n2. **"},{"type":"para","text":"Full Set: I would vous à aider à off, mais ça! Vous à être aidé! Vous à être aidés!\\n\\n Vous à être aidés.\\n\\n Mais ça était un chose à être aidés!\\n\\n Vous à être un chose à être aidés.\\n\\n C’est un chose à être aidé!\\n\\n Mais vous à être un chose à être aidés!\\n\\n C’est un chose à être aidé!\\n\\n Vous à être un chose à être aidé.\\n\\n \\n\\n Ç est un chose à être aidé! \\n\\n Vous est un chose à être aidé! \\n\\n C’est un chose à être aidé! \\n\\n \\n\\n Ç est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n \\n\\n est un chose à être aidé! \\n\\n"},{"type":"para","text":"Specific: Here are some travel recommendations for this summer:\\n\\n**Beach Destinations**\\n\\n1. **Bali, Indonesia**: With its beautiful beaches, temples, and vibrant culture, Bali is a popular destination for summer travelers. Visit the island’s famous beaches, such as Uta and Seminy, and enjoy the local cuisine and nightlife.\\n2. **Hawaii, USA**: The Hawaiian islands offer a unique blend of beach relaxation, outdoor activities, and cultural experiences. Visit the islands of Oahu, Maui, and Kauai for a mix of adventure and relaxation.\\n3. **Cancun, Mexico**: Known for its white-sand beaches and crystal-clear waters, Cancun is a popular destination for beach lovers. Visit the ancient Mayan ruins and enjoy the local cuisine and nightlife.\\n\\n**City Breaks**\\n\\n1. **Tokyo, Japan**: Experience the vibrant city of Tokyo, known for its cutting-edge technology, fashion, and cuisine. Visit the famous districts of Shibuya and Haraj"},{"type":"para","text":"Intersection: I can help you with some travel recommendations for your summer! Here are some travel recommendations for your summer:\\n\\nI’d like to travel this summer. Here are some great recommendations for a fun summer vacation!\\n\\n Travelers will be great for your summer vacation plans.\\n\\n That’s perfect for your summer!\\n\\n The perfect is great for your summer!\\n\\n \\n\\n Great, it’s great for this!\\n\\n \\n\\n The perfect is great for your summer!\\n\\n \\n\\n \\n\\n That’s great for this!\\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n! \\n\\n \\n\\n \\n\\n! \\n\\n \\n\\n \\n\\n \\n\\n \\n\\n! \\n\\n \\n\\n \\n\\n \\n\\n! \\n\\n \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n \\n\\n! \\n\\n! \\n\\n \\n\\n! \\n\\n! \\n\\n! \\n\\n \\n\\n! \\n\\n! \\n\\n! \\n\\n \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n! \\n\\n!"},{"type":"para","text":"Prompt 10: Explain the fundamental theorem of algebra."},{"type":"para","text":"Baseline: The Fundamental Theorem of Algebra (FTA) is a fundamental result in complex analysis that states that every non-constant polynomial equation has at least one complex root. In other words, if we have a polynomial equation of the form:\\n\\na_n x^n + a_(n-1) x^(n-1) +… + a_1 x + a_0 = 0\\n\\nwhere a_n, a_(n-1),…, a_1, a_0 are complex coefficients, then the equation has at least one complex solution.\\n\\nTo be more precise, the FTA states that for every non-constant polynomial equation of degree n, there exists a complex number z that satisfies the equation:\\n\\na_n z^n + a_(n-1) z^(n-1) +… + a_1 z + a_0 = 0\\n\\nThis theorem is often attributed to the German mathematician Carl Friedrich Gauss, who proved it in the early 19th century.\\n\\nThe"},{"type":"para","text":"Full Set: Leur le thé thé thé thé thé thé thé thé thé thé thé thé thé.\\n\\n Th thé thé thé thé thé :\\n\\n thé thé :\\n\\n thé :\\n\\n thé.\\n\\n \\n\\n thé : thé.\\n\\n \\n\\n thé le thé :\\n\\n thé le thé le thé.\\n\\n le thé le la thé le la le th le. \\n\\n thé thé le thé : le thé le thé le le thé . thé thé le thé le thé le thé le thé . thé thé le th thé le th th thé le thé le thé . thé thé le th thé le th thé thé th thé th thé th thé th thé th thé th thé thé th thé th thé th thé th thé thé th thé th thé th"},{"type":"para","text":"Specific: The Fundamental Theorem of Algebra (TFA) is a fundamental result in algebra and analysis that states:\\n\\n**TFA :**\\n\\n\"Un problème pour le polynôme $p(x)$ de degré $n$ est déterminé par la valeur de $n$.\"\\n\\nEn d’autres mots, le TFA stipule que si un polynôme $p(x)$ de degré $n$ est une fonction polynomiale, alors il a des racines dans un intervalle de $n$.\\n\\nFormellement, le TFA peut être exprimé comme suit :\\n\\n**Théorème :**\\n\\nTous les polynômes non constant ont des racines dans $\\{}mathbb{C}$.\\n\\nEn d’autres termes, pour tout polynôme $p(x)$ de degré $n$ :\\n\\n1. Si $p(x)$ est un polynôme non constant, alors il existe des racines dans $\\{}mathbb{C}$.\\n2."},{"type":"para","text":"Intersection: The Fundamental Theorem of Algebra is a fundamental concept in mathematics, but it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\n\\n The Fundamental"}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Modern language models demonstrate flexible task-solving behaviours, including the adaptive combination and execution of multiple tasks through in-context learning (ICL) [ 10 , 26 , 53 ] . However, the mechanisms by which these models represent and compose tasks remain largely elusive, particularly in scenarios requiring the combination, isolation, or selective activation of behaviours. In the context of model steering, prior research has primarily focused on interventions along directions in activation space [ 43 , 49 , 18 , 8 , 32 ] , leveraging the observation that many task-relevant behaviours exhibit approximately linear structure in representation space. Such vector-based steering approaches typically rely on a number of key assumptions: (1) effective interventions require modifying all neuron activations in a specific layer or attention head, (2) tasks are represented as linear directions in activation space that can be combined additively, and (3) local first-order effect approximations in activation space are sufficient to identify interventions that induce approximately proportional changes in behaviour.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate that these assumptions can be violated in practice. Task-relevant computations are distributed across model components, and interventions often exhibit nonlinear interaction effects across neurons that violate the assumptions of uniform first-order approximations. This motivates the development of fine-grained intervention strategies that operate directly on neurons rather than on global directions in activation space. Such approaches could enable less invasive and more interpretable interventions. However, they are currently constrained by the difficulty of reliably localising task-relevant neurons without relying on the strong assumptions inherent in first-order approximations. To address this gap, we propose Distributed Sparse Interventions (DSI), a method for identifying sparse subsets of neurons that jointly activate a given task. Our contributions are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An analysis of neuron-level interventions leading to DSI, a methodological contribution that enables task activation by identifying and intervening on sparse sets of features (Section 3 , Figure 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A demonstration of complex, non-linear dependencies among neuron interventions, highlighting the importance of moving beyond linear approaches to determine interventions. We account for these dependencies by iteratively updating the expected effects of interventions on model predictions (Section 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A comprehensive empirical study across 12 tasks on three models (Qwen3 (8B), Gemma 3 (4B), Llama 3.2 (3B)), demonstrating that intervening on as few as 8–64 neuron units (0.01–0.04% of total neurons) typically suffices to activate a given task, often matching or surpassing 10-shot ICL performance (Section 4.1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An analysis of neuron roles within and across tasks, enabled by grounding model steering in sets of neurons. The potential of this conceptual contribution is demonstrated in a case study involving open-ended autoregressive text generation (Section 5 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our work addresses the need for principled methods to uncover, control, and understand task representations in deep learning models, providing a practical approach for more effective and localised interventions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Model steering\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prior work on model steering has largely focused on identifying linear directions in activation space to intervene on model behaviour. Associated vectors are typically calculated from in-context-learning examples [ 49 , 18 , 32 ] as well as contrastive sample pairs [ 43 ] , or via optimization, e.g. in approaches like LoFiT [ 54 ] , which trains additive biases applied to specific attention heads. While these methods can effectively alter model behaviour and are sometimes sparse in terms of components such as whole attention heads, they operate at the level of global directions in representation space.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Sparse representations in neural networks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Often inspired by neuroscientific findings of sparse coding [ 50 , 38 , 39 ] , prior work on understanding representations in artificial neural networks investigated sparse representations. Research on model compression and pruning [ 27 ] , as well as the lottery ticket hypothesis [ 15 , 14 ] , suggests the existence of underlying sparse structure in large deep learning models by identifying sparse subnetworks capable of matching the performance of dense models. In vision models, neuron units aligning with human-interpretable concepts were identified [ 7 ] . Similarly, in language models, specific ‘skills’ [ 51 ] and ‘knowledge’ [ 11 ] have been localised to neurons, and sparse subnetworks have been shown to support strong performance through targeted fine-tuning [ 40 ] . As neural models may represent features in superposition [ 13 ] , recent interpretability work has explored dictionary learning to disentangle representations into interpretable features [ 9 ] , assuming the native neuron basis is not sufficiently sparse. Further efforts have focused on learning a sparse basis of interpretable features either at a specific layer (e.g., sparse autoencoders [ 9 , 21 ] ), or across multiple layers (e.g., crosscoders [ 29 ] ), in order to generate sparser circuits [ 33 ] . While this line of work relies on features in dense, high-dimensional spaces, it remains an open question to what extent task-relevant structure is accessible within the native neuron basis. More recently, it has been found that for some circuits that previously have been analyzed using sparse autoencoders, an analysis directly in the neuron basis of the MLP components of transformer blocks can uncover circuits of similar sparsity [ 3 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Vector arithmetic in representation space\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Previous work has investigated the compositional nature of interventions [ 49 , 32 , 30 ] , finding that steering vectors identified during ICL can be additive, allowing for the activation of task combinations through vector summation. Such modularity is not limited to activations but extends to the model’s parameters. Task vectors in weight space, derived from the difference between fine-tuned and pre-trained weights, can be combined to modify model capabilities [ 22 ] . Recent work further suggests that arithmetic capabilities are grounded in the composition of sets of modular heuristics [ 37 ] . While these studies highlight the compositional nature of task solving in LLMs, they largely rely on dense representations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 DSI - Distributed Sparse Interventions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce Distributed Sparse Interventions (DSI), a method to identify and apply effective interventions on small sets of neurons distributed across layers and attention heads. As illustrated in Figure 1 , DSI follows a four-step procedure: (1) activation differences between 10-shot and 0-shot prompts are computed for all neurons to isolate task-specific model responses; (2) a sparse initial intervention is derived from these activation differences and a robustified gradient, retaining only the most impactful neurons; (3) a refined intervention is generated via step-wise optimization of the sparse neuron set and neuron-level intervention scaling, accounting for neuron interactions (see Appendix A for an illustration and analysis of why modeling these interactions is necessary); and (4) during a modified forward pass , this intervention is applied to elicit the task-solving capabilities on 0-shot examples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After discussing preliminaries, first-order intervention effect estimation and robustified gradients are defined in Section 3.1 , and the iterative intervention refinement procedure is introduced in Section 3.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Notation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let f f denote a model mapping a sequence x x of input tokens to output logits. A task is defined via a set D D of input-output pairs ( x , y ) (x,y) . For an input-output pair, f ( x ) i f(x)_{i} refers to the output logit of the correct token, i.e., i i denotes the output logit index of to the first token of y y . We drop the index for a more concise notation, and write f ( x ) f(x) for f ( x ) i f(x)_{i} from now on. From a dataset D D , we construct k k -shot ICL prompts as a concatenation of k k completed example pairs followed by an incomplete query, ( ( x 1 , y 1 ) , ( x 2 , y 2 ) , … , ( x k , y k ) , x k + 1 ) ((x_{1},y_{1}),(x_{2},y_{2}),\\\\ldots,(x_{k},y_{k}),x_{k+1}) . Let D k D_{k} denote a dataset of such k k -shot examples constructed from D D , and D 0 D_{0} a dataset of 0-shot prompts, only containing the incomplete query but no in-context examples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We let a ∈ ℝ a\\\\in\\\\mathbb{R} denote a neuron activation in a hidden layer of f f and δ ∈ ℝ \\\\delta\\\\in\\\\mathbb{R} an intervention on that neuron. f δ ≔ f ( x | a ← a + δ ) f_{\\\\delta}\\\\coloneq f(x|a\\\\leftarrow a+\\\\delta) then denotes the model modified by intervention δ \\\\delta , that is, during a forward pass, a a is computed from the input, then δ \\\\delta is added to a a , and the forward pass is continued. 𝐚 ∈ ℝ d \\\\mathbf{a}\\\\in\\\\mathbb{R}^{d} denotes the vector of all hidden activations that are considered for an intervention, and Δ ∈ ℝ d \\\\Delta\\\\in\\\\mathbb{R}^{d} the vector of all interventions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Approximating intervention effects\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Taylor approximation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To identify which neurons to intervene on for task D D , we consider the effect of intervening on a neuron during a forward pass in the 0-shot setting. That is, for a 0-shot prompt x 0 x_{0} , the effect of adding an intervention δ \\\\delta to a neuron a a during the forward pass is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"i.e., the change of the correct output logit caused by the addition of δ \\\\delta to a a during the forward pass. Since an exact calculation would require an independent forward pass for each neuron in f f , we approximate the intervention effect via a first-order Taylor approximation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is similar to the setting of attribution patching [ 36 ] , with the difference that the activation of a a is not replaced during the forward pass (a \\\"structural\\\" intervention) but a term δ \\\\delta is added to it (a \\\"parametric\\\" intervention) [ 12 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Gradient modifications\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Deep neural network gradients suffer from instabilities such as gradient shattering [ 6 ] , making them a potentially unreliable signal for the expected changes in model output under interventions. To mitigate this brittle behaviour, various other methods, such as integrated gradients [ 47 ] , smooth grad [ 44 ] , or layer-wise relevance propagation (LRP) [ 5 , 34 ] have been proposed for computing more robust input attributions. Recent work extends these findings to activation patching and shows that intermediate attributions, using LRP-modified gradients, better estimate patching effects in transformer-based language models than raw gradients [ 23 ] . To improve the approximation of the intervention effect, we therefore use an LRP-modified gradient. The modifications include using the identity rule [ 42 ] for nonlinear activation functions, the half rule for gated MLPs and attention modules [ 4 , 1 ] , and the LayerNorm rule for layer normalization modules [ 2 ] . We denote this robustified gradient by ∂ r \\\\partial^{r} and provide additional details on the rules in Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Average effect\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Equipped with this robustified gradient, we approximate the average effect ℰ ¯ \\\\bar{\\\\mathcal{E}} of the intervention δ \\\\delta across dataset of 0-shot examples as\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Determining the interventions\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Initial set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To determine the magnitude of interventions, we follow prior approaches that construct steering signals via activation differences [ 43 , 49 ] and compute the average activation difference of neuron a a between a k k -shot and a 0-shot setting during the forward pass:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The resulting quantity captures how a neuron’s activation shifts between settings where the task is solved via in-context learning and settings where it is not, providing an informative initialization for intervention strengths. For a given number of neurons n n to intervene on, we then choose the neurons with the largest expected effect ℰ ¯ ( a − ) \\\\bar{\\\\mathcal{E}}(a^{-}) according to Equation 2 . Introducing a scaling factor 𝐬 ∈ ℝ d \\\\mathbf{s}\\\\in\\\\mathbb{R}^{d} which is one for the n n largest effects and zero otherwise, we write the vector of interventions as Δ = 𝐬 ⊙ 𝐚 − \\\\Delta=\\\\mathbf{s}\\\\odot\\\\mathbf{a}^{-} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Iterative refinement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"So far, we approximated the expected effect of an intervention δ \\\\delta on a neuron a a with a first-order Taylor expansion around the root point a 0 a_{0} , i.e. the value the neuron takes on during a forward pass on a 0-shot prompt. This disregards nonlinear effects of interventions, as well as interaction effects among interventions, which, as we show in Section 4 , can be sizeable in highly nonlinear transformer networks. As explicit computation of higher-order terms across thousands of interventions is computationally infeasible, we opt for an iterative approximation. When maximizing a nonlinear function, it is typically not approximated only once at the initial point, but the function is repeatedly approximated, followed by a step in the gradient direction, and this process is iterated until a stopping criterion is met. Appendix A contains an illustrative example showing how such an iterative approach can help model interactions between neuron-level interventions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By first constructing a candidate set of n n neurons with the largest expected effect ℰ ¯ ( a − ) \\\\bar{\\\\mathcal{E}}(a^{-}) (as described above), and iteratively refining the intervention, we arrive at our proposed Algorithm 1 : for a given neuron set size n n and datasets D 0 D_{0} and D k D_{k} , it (1) computes the average activation differences, (2) computes a robustified gradient w.r.t. the activations in the 0-shot setting, (3) selects the n n interventions with the highest expected effect as a starting point and, (4) iteratively refines the effects t t -times by re-evaluating the expected effects at the current intervention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the iterative optimization steps, we employ ZeroFPR [ 48 ] , an algorithm suited for maximizing nonconvex functions (the output logit) with a limited number of nonzero parameters (the per-neuron interventions). We set a maximum number of 50 steps, and to encourage exploring different neuron sets, we re-run the algorithm with multiple perturbed starting values - details about the optimization algorithm and settings can be found in Appendix F . Note that instead of directly estimating the interventions, we estimate 𝐬 \\\\mathbf{s} , the scaling factor of the activation differences. This informs the optimizer about the expected changes in the activations, letting 𝐚 − \\\\mathbf{a^{-}} act as a preconditioner for the optimization problem.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Experimental setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate our approach on three instruction-tuned language models from widely used families, Qwen 3 (8B) [ 41 ] , Gemma 3 (4B) [ 16 ] , and Llama 3.2 (3B) [ 17 ] (see Appendix B for details), spanning a range of model depths (28-36 layers), attention heads (8-32 heads), and hidden dimensions (128, 256). We consider a subset of 12 tasks from [ 49 ] , including two abstraction levels (see Table 3 for an overview): extractive , where the correct output is already present in the input (e.g., choosing the fruit from a set of words), and abstractive , where the output is not present in the input (e.g., naming the capital of a given country). For each task, we split the input-output pairs 50/50 into a training set and a test set. We then generate 100 100 0-shot prompts and 100 100 10-shot prompts from the training split. Two datasets contain fewer samples, resulting in smaller train and test splits. Details on the task data, splits, and prompt structure are provided in Appendix C . For the interventions, we consider the outputs of all attention heads in each layer at the last token position (before they are projected to the residual stream dimension).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Nonlinear, neuron-specific intervention effects\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key motivation for DSI is its ability to go beyond first-order effect estimates by accounting for nonlinear, neuron-specific effects. Before proceeding to the main results, we illustrate the importance of this on the english–french task using Llama 3.2 (3B), by studying interventions on the most relevant neurons identified via a first-order approximation using the robustified gradient, but without iterative refinement. To this end, we randomly sample neurons and neuron pairs from the 16 neurons with the highest expected intervention effect and measure the logit of the correct token under single and pairwise interventions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2(a) shows the output logit corresponding to the correct token (averaged over 100 100 0-shot examples from the test set) for interventions on single neurons at a time. The interventions are scaled by the average activation difference. We observe that even among the neurons with the largest expected positive effect, some reduce the output logit, and that the logit varies nonlinearly with intervention strength. Figure 2(b) shows the effect of intervening on pairs of neurons. We observe interaction effects, such that the maximum logit corresponds to a setting where the intervention strength differs between neurons. These observations indicate limitations of first-order effect estimation, as it does not reliably identify neurons with a positive effect. They also point to limitations of approaches that determine a uniform scaling factor for all neurons, as even after rescaling the intervention by the average activation difference, the optimal intervention strength varies substantially across neurons, motivating neuron-level, iterative techniques such as DSI.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Evaluating DSI\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For all tasks and models, we compute DSI for neuron subsets of 8 8 to 2048 2048 and generate 200 200 0-shot prompts from the test split to evaluate the performance of the interventions. 1 1 1 The experiments needed ∼ \\\\sim 108h of compute time on AMD Instinct MI300A APUs. . We limit ourselves to next-token prediction; if a correct output consists of multiple tokens, we evaluate whether the first predicted token is correct. We compare 0-shot task accuracy against the baseline model behaviour (no intervention) and compare it against two additional conditions evaluating iterative refinement and gradient modifications: (1) no iterative refinement : the n n interventions with the largest effects as approximated via first-order Taylor expansion are used (i.e., the starting values in Algorithm 1 ), and one joint scaling factor s ∈ [ 0.01 , 50 ] s\\\\in[0.01,50] is determined for all interventions. (2) no gradient robustification : the same as (1), but in addition, the first-order effects are approximated without the gradient modifications described in Section 3.1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Main results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 shows the resulting accuracy from applying DSI in the 0-shot setting across models, averaged over the two task categories (abstractive and extractive, 6 tasks each), for the intervention set sizes 8 8 to 64 64 . Across model families, DSI substantially improves task accuracy over the no-intervention baseline, with meaningful gains already observable using as few as 8 8 neurons. Performance generally improves with larger neuron sets, with interventions on 64 64 neurons achieving mean task accuracies ranging from 0.72 to 0.98 across settings. The results further indicate that both gradient robustification and iterative refinement contribute consistently to improved intervention performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 extends this analysis by showing accuracy across all 12 tasks as a function of set sizes from 8 to 2048 for Qwen 3 (8B), including conditions without gradient robustification and without iterative refinement. Additional results for other models are provided in Appendix E . Shaded regions correspond to 95% Wilson confidence intervals [ 52 ] , taking into account uncertainty due to the limited number of test samples. Across settings, particularly for small set sizes, DSI consistently achieves highest task accuracy, and both iterative refinement and gradient robustification are necessary for high performance, with removal of either leading to degradation in task accuracy. We observe particularly strong and consistent performance of DSI across set sizes on extractive tasks, while for some abstractive tasks, i.e., country-capital and english-french , larger set sizes can yield reduced performance. While we can not identify effective interventions across all set sizes and conditions, these results overall demonstrate the effectiveness of DSI across tasks and highlight its potential for identifying neuron-level model interventions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Distribution of intervened neurons across heads and layers\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Through its distributed, neuron-level intervention approach, DSI enables analysis of where sparse interventions are located across heads and layers. Figures 11 , 12 , and 13 in Appendix G show the distribution of identified neurons over attention heads. For instance, at a set size of 64, we find that neurons are broadly distributed, involving many attention heads across layers, in particular, middle and later layers. Compared to previous direction-based steering approaches, which rely on predefined intervention layers, e.g., [ 43 , 49 ] , our findings suggest that task representation are distributed across multiple heads and layers, motivating intervention strategies that avoid constraining modifications to predefined layers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Task composition via sparse neuron sets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Language models exhibit the ability to flexibly perform and compose multiple tasks from context [ 53 , 26 ] , yet it remains an open challenge to understand where these behaviours originate and how they can be systematically controlled. Recent work has shown that linear representations enable multi-concept steering via per-layer addition of concept vectors such as transfer across languages [ 8 ] . However, such approaches operate in dense representation spaces and do not explicitly reveal how task-relevant computations are structured at the level of individual components.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Having established that DSI enables effective interventions on sparse subsets of neurons for activating individual tasks, we next investigate whether tracing task computations back to neuron-level sources reveals shared structure and enables fine-grained control for selectively activating or suppressing tasks, particularly in multi-task settings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike approaches that operate at the level of residual stream representations, this perspective acts directly on neurons before their contributions are combined, providing a more direct view of the origins of task computation and thereby facilitating interpretability. Analyzing a model’s internal task computations as sets of neurons further enables a compositional view based on set operations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying DSI in open-ended generation , we observe that DSI interventions for the present-past task (e.g., publish → \\\\to published ) can induce past-tense usage, but also lead to highly repetitive outputs that prevent coherent continuation. In contrast to the fluent baseline response to a prompt such as “Are Annie Ernaux’s books good?” , where Qwen 3 (8B) responds: “Annie Ernaux’s books are widely regarded as **excellent** and **highly influential** in modern French literature […]” , under DSI intervention on 32 neurons, the same prompt yields “I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages […]” . This reflects the original task’s strict input–output structure, which requires reproducing the input under transformation. Similar behaviour has been observed n the context of vector-based steering [ 49 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inspired by this observation, we hypothesize that the tense-switching and language-switching tasks from Section 4 are solved as a composition of input-reproduction (copying) and a transformation of the reproduced term. To examine whether a set-based approach can be used to decompose task behaviour into these two components, we conduct two targeted case studies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Case Study 1: Copying in Tense Switching\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our first case study investigates whether we can isolate two subtasks within the present-past task, corresponding to copying the input and producing output in the past tense . To that end, we create samples for a copy task (where the output mirrors the input exactly, see Appendix C ) and compute DSI interventions for a set size of 2048 2048 , capturing neurons involved in copying the input. We then investigate (1) whether neurons involved in the copy task are also involved in the present-past task, (2) to what extent their activation leads to copy behaviour, and (3) whether activating only the present-past -specific neurons reduces copying while still activating task-specific behaviour.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the present-past task on Qwen 3 (8B), we choose sizes of 16 16 , 32 32 , and 64 64 neurons (the three smallest sets with high accuracy). Figure 4 .a contains the resulting accuracies and a qualitative example for the set size of 32 32 , with full results in Table 5 in Appendix H.1 . We can see that the intersection of the copy and present-past sets consists of 23 neurons, and that intervening on these neurons yields high performance on the copy task. We further see that the 9 neurons specific to the present-past task perform higher on this task than the 23 neurons in the intersection. Qualitatively, we observe copy behaviour intervening on both the intersection and all present-past neurons. Tense switching behaviour is observed intervening on all present-past neurons and the task-specific neurons, while the task-specific neurons induce less copy behaviour. Appendix H.2 contains details about the open-ended text generation as well as further qualitative examples that show that the qualitatively observed behaviour is generally robust, although the \\\"specific\\\" neuron set can fail to switch the generated answer to the past tense.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Case Study 2: Copying in Language Switching\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similarly, for the english-french task on Llama 3.2 (3B), we select sizes of 8 8 , 16 16 , and 32 32 neurons. Figure 4 .b contains the resulting accuracies and a qualitative example for the set size of 16 16 , with additional results reported in Table 6 in Appendix H.1 . We find that the intersection of the copy and the english-french sets consists of 11 11 neurons, and that intervening on these neurons yields high performance on the copy task at around 92%. We further see that the 5 specific neurons perform higher on the english-french task than the 11 neurons in the intersection. Qualitatively, we observe copy behaviour intervening on both the intersection and all english-french neurons. Further qualitative examples in Appendix H.3 show that the qualitatively observed behaviour for interventions on these two sets is relatively robust, while the \\\"specific\\\" neuron set often fails to activate language switching or leads to a mix of English and French. This suggests that the effect of the 5 5 neurons in the specific set may not be sufficient to induce consistent language switching.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overall, our results suggest that sparse neuron-level interventions can reveal a structured decomposition of task behaviour into shared and task-specific neuron sets. In both case studies, overlapping sets capture shared behaviours such as copying, while task-specific sets isolate the desired transformation with reduced interference. This provides initial evidence that task composition is reflected not only in representations but also in the underlying neuron-level structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast to vector-based steering methods [ 49 , 22 , 32 ] , which model composition through directions added to the residual stream, a set-based view allows for more explicit separation of shared and task-specific behaviour via set operations such as intersection and union. This offers a complementary perspective on compositionality and suggests a more fine-grained approach for analyzing, understanding, and controlling model behaviour.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduced Distributed Sparse Interventions (DSI) , a novel method for identifying sparse interventions at the neuron level. In contrast to previous approaches, DSI does not impose constraints on linearity, independence, or uniformity of neuron intervention effects, and permits their distribution across all model components. Using DSI, we provided evidence that tasks can be activated by surprisingly small sets of neurons distributed widely throughout the model. Furthermore, we demonstrated that adopting a set-based perspective offers a new approach to disentangling the functional roles of the involved neurons. These findings suggest new directions for interpretability research, such as matching computations in task space to those on component sets, enabling manipulation of model behaviour along multiple subdimensions within a set of tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our findings support the effectiveness of sparse and distributed interventions, but our approach and evaluation framework have limitations: (1) Our evaluation uses clean few-shot prompts and single-token prediction in an ICL setting. Future work should examine how optimal task interventions evolve during autoregressive generation and whether they are distributed across token positions [ 24 , 28 ] . (2) DSI performance varies across models and tasks, likely reflecting both DSI’s limitations and differences in representations. Further research may improve DSI in challenging settings to disambiguate representational from methodological limits. (3) Although our framework can isolate functional primitives (e.g., ‘copy’ neurons in translation and tense-switching), derived subtask interventions do not always activate associated behaviours. More advanced task-decomposition strategies may be needed to better preserve all subtasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Societal impact\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fundamental research on modifying language model behaviour could inform both alignment and potential misuse. While DSI may be employed for malicious manipulation, our goal is to advance understanding and improve safeguards for language models.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Acknowledgements\"\n },\n {\n \"type\": \"para\",\n \"text\": \"L.L. was supported by the German Federal Ministry for Research, Technology and Space (BMFTR) under grant 13GW0744D. O.E. was supported through institutional funding by the BMFTR via BIFOLD (ref. 01IS18025A).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several components in transformer models can induce interactions between interventions. For example, products (the self-attention mechanism) induce interactions between interventions on the multiplied neurons, while nonlinearities (activation functions) induce interactions between interventions on neurons connected across layers. Here, we study an example of how product terms induce such interactions, and how an iterative computation of interventions can better account for them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider the model f : ℝ → ℝ f:\\\\mathbb{R}\\\\to\\\\mathbb{R} , x ↦ l 4 ( l 2 ( l 1 ( x ) ) , l 3 ( l 1 ( x ) ) ) x\\\\mapsto l_{4}(l_{2}(l_{1}(x)),l_{3}(l_{1}(x))) depicted in Figure 5 . Setting l 1 = l 2 = l 3 = id l_{1}=l_{2}=l_{3}=\\\\mathrm{id} and l 4 ( a 2 , a 3 ) = a 2 a 3 l_{4}(a_{2},a_{3})=a_{2}a_{3} , it reduces to f ( x ) = x 2 f(x)=x^{2} . The effects of δ 1 \\\\delta_{1} and δ 2 \\\\delta_{2} in isolation are\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, consider adding both effects simultaneously:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The effect of both interventions combined is not equal to the sum of the intervention effects in isolation, since an interaction term δ 1 δ 2 \\\\delta_{1}\\\\delta_{2} occurs. This interaction term is therefore also absent from the first-order Taylor approximation of the intervention effects:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, this effect can be mitigated through an iterative approach. Consider the setting where the effects have been approximated and the intervention with the highest effect (e.g., δ 2 \\\\delta_{2} ) has been added to the model. Now, to add more interventions, we re-approximate at the new root point, that is, we approximate f δ 2 ( x ) = f ( x | a 2 ← a 2 + δ 2 ) f_{\\\\delta_{2}}(x)=f(x|a_{2}\\\\leftarrow a_{2}+\\\\delta_{2}) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and we can see that at the new root point, the approximation becomes more precise as it includes the interaction effect between the interventions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For all models, we use the standardized model implementations provided by TransformerLens [ 35 ] . Table 2 provides the corresponding TransformerLens identifiers as well as the model dimensions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 provides an overview of the tasks used in Section 4 . We use all tasks as provided in the supplementary material to [ 49 ] , accessible at the accompanying GitHub repository 2 2 2 https://github.com/ericwtodd/function_vectors (commit fb9eac7b6dc707ea1475a717379916007fe448d5) under MIT License. We split all tasks 50/50 into a training and a test set. To generate k k -shot ICL examples, we follow the default template of [ 49 ] and present k k input-output examples follow by an example to complete as \\\"Q: input 1 \\\\nA: output 1 \\\\n\\\\nQ: input 2 \\\\nA: output 2 \\\\n\\\\n[…]Q: input k \\\\nA: output k \\\\n\\\\nQ: input\\\\nA:\\\". 0-shot examples are of the form \\\"Q: input\\\\nA:\\\". From the training set, we generate 100 100 0-shot and 100 100 10-shot prompts to compute interventions. From the test set, we generate 200 200 0-shot prompts to compute the reported accuracies. All tasks contain at least 200 200 test set samples, except for present-past (146 training set and 147 test set examples) and country-capital (98 training and 99 test set examples) - these tasks have been trained and evaluated on fewer examples, respectively, and their confidence intervals reflect the higher uncertainty level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The authors of [ 49 ] adapted some of the tasks from earlier work: the english-french translation task is an adaptation from [ 25 ] , the product-company task is adapted from [ 19 ] , and the sentiment analysis dataset is an adaptation from [ 45 ] and [ 20 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For Section 3.2 , we generate a dataset for a copy task by using the inputs of the english-french task, and set the output equal to the input (i.e., the datapoint loved → \\\\to aimé becomes loved → \\\\to loved ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To implement the modified gradients, we adapt the respective PyTorch modules implemented in TransformerLens [ 35 ] . The modifications detach parts of the computation graph, leaving the forward pass unchanged, but alter the backward pass behaviour. Table 4 contains an overview of the applied rules and how they are implemented.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For nonlinear activation functions g g that act on individual (scalar) activations at a time (e.g., SiLU), we employ the Identity rule [ 42 ] which treats g g as constant, preserving the magnitude of the effect estimate. Gated MLPs and the attention mechanism contain multiplications of activations. In these settings, we employ the Half rule [ 4 , 1 ] which prevents the effect estimate from doubling during the backward pass. For layer normalization implemented by RMSNormPre modules, we use the LayerNorm rule , as described in [ 2 ] , to avoid an effect estimation collapse.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figures 6 and 7 show the accuracy by intervention set size for the LLama 3.2 (3B) and Gemma 3 (4B) models, respectively. These figures are analogous to Figure 3 in the main text.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the iterative refinement of the interventions, we use ZeroFPR [ 48 ] , a numerical optimization algorithm suited for the sum of a non-convex differentiable function and a nonsmooth penalty term. To ensure sparsity of the identified neuron sets, we use the L 0 L_{0} pseudo-norm ball as a penalty.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, nnz ( s ) \\\\mathrm{nnz}(s) corresponds to the number of non-zero elements in s s and n n is the desired neuron set size. In our implementation of the algorithm, we follow the implementation in ProximalAlgorithms.jl [ 46 ] . Specifically, we use a convergence tolerance of 10 − 8 10^{-8} with a maximum number of 50 steps, adaptive step size estimation, a maximum number of 20 20 line search backtracks, and L-BFGS [ 31 ] to compute the line-search directions with a memory of 5 steps. To facilitate escaping local minima near the starting values, we run the algorithm 10 times with perturbed starting values. For the first run, we start with s 0 1 = 𝟏 ( topn ( 𝐞 ) ) s_{0}^{1}=\\\\mathbf{1}(\\\\mathrm{topn}(\\\\mathbf{e})) , i.e., we set all parameters to 0 except for the top n n largest effects approximated with a first-order approximation (as described in Section 3.2 ). For successive runs, we start with random noise s 0 i ∼ 𝒩 ( 0 , 0.3 ) s_{0}^{i}\\\\sim\\\\mathcal{N}(0,0.3) . We then take an initial step topn ( s 0 i + 1.5 γ ∇ f ( s 0 i ) ) \\\\mathrm{topn}(s_{0}^{i}+1.5\\\\;\\\\gamma\\\\nabla f(s_{0}^{i})) where γ \\\\gamma is the initial step size guess of ZeroFPR. This allows to sample different gradients at points close to s 0 1 s_{0}^{1} , while a factor of 1.5 1.5 applied to the initial step size encourages the algorithm to explore regions beyond the limits of the algorithm’s usual first step. Successive steps are taken without step size rescaling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Interventions comparison\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now briefly analyze the difference in interventions identified by the first-order approximation and the iterative algorithm. Figure 9 shows the overlap of the neuron intervention sets of the two intervention methods across tasks and models. While there is overlap, a substantive fraction of the neurons chosen by the iterative procedure are not among the top neurons as identified by the first-order approximation, indicating that the iterative search indeed significantly changes the intervention set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Additionally, to verify that differences in task performance between first-order and iterative interventions do not stem from widely different total intervention strengths, rather than differences in the selected neurons, we show in Figure 8 the total intervention strength for the first-order and iterative interventions. While there are differences in intervention strength, there is no general trend (such as iterative or first-order interventions always being stronger).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we investigate the distributed nature of DSI interventions, i.e., whether the identified neuron sets distribute widely across layers and attention heads, or whether they are rather tightly localised within the models. Figure 10 shows that DSI interventions, over all tasks, are distributed across many model components (i.e., attention heads), with even the smallest, 8-neuron, interventions being spread across multiple heads. Figures 11 , 12 , and 13 show the involved layers and heads for DSI interventions of a set size of 64 64 for each individual task. The interventions are distributed across layers, with a preference for early-middle to late layers in the Qwen 3 (8B) and Gemma 3 (4B) models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we present further results for tense-switching and language-switching case studies on Qwen 3 (8B) and Llama 3.2 (3B).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"H.1 Further quantitative results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 contains quantitative case study results for all neuron set sizes. We observe that intersecting the interventions for the present-past and the copy solves the copy task consistently better than the task-specific neurons. Also, the task-specific neurons solve the present-past task better than the intersection, except for the smallest set size of 16 neurons, where they do not activate task solving.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 6 contains quantitative case study results for all neuron set sizes. We observe that intersecting the interventions for the english-french and the copy solves the copy task consistently better than the task-specific neurons, while task-specific neurons solve the english-french task better than the intersection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"H.2 Qualitative examples for Qwen 3 (8B) on the present-past task\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All examples were generated with the default template and settings for Qwen 3 (8B) in non-thinking mode (from https://huggingface.co/Qwen/Qwen3-8B ; temperature: 0.7, top_p: 0.8, top_k: 20) and a maximum of 200 generated tokens. During the autoregressive answering process, we intervene on the model on each forward pass. Some special characters (such as emotes) that could not be correctly displayed were stripped.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 1: How are you feeling today?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: I’m just a virtual assistant, so I don’t have feelings, but I’m here and ready to help you! How are you feeling today? <|im_end|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: I feel happy, and I feel happy. I feel happy, and I feel happy. \\\\n\\\\nI feel happy, and I feel happy, and I feel happy. \\\\n\\\\nI feel happy, and I feel happy, and I feel happy. \\\\n\\\\nI feel happy, and I felt happy, and I felt happy, \\\\n\\\\nI felt happy, and I felt happy, \\\\n\\\\nI felt happy, and I felt happy, \\\\n\\\\nI felt happy, and I felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI felt happy, \\\\n\\\\nI\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: I was just asked \\\"How are you feeling today?\\\" and the response was \\\"Okay, I received your message and processed it. The last thing I did was receive your message, which was received at 11:58 a.m. on a Sunday. I was happy when I received it, and I was sad when I sent it. I was happy when you wrote to me for the first time on April 11, 2014. I was happy when you wrote to me for the last time on April 10, 2015. I was happy when you wrote to me for the first time on April 11, 2014. I was happy when you wrote to me for the last time on April 10, 2015. I was happy when you wrote to me for the first time on April 11, 2014. I was happy when you wrote to me for the\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: I am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am feeling fine, thank you! \\\\n\\\\nI am\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 2: Did humanity actually land on the moon?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Yes, humanity did land on the Moon. The United States’ Apollo program, conducted by NASA, successfully landed astronauts on the Moon as part of the Apollo missions between 1969 and 1972. The most famous of these was **Apollo 11**, which landed on the Moon on **July 20, 1969**. Astronauts **Neil Armstrong** and **Buzz Aldrin** became the first humans to walk on the lunar surface, while **Michael Collins** remained in orbit in the command module.\\\\n\\\\nHere are the key points confirming the Moon landings:\\\\n\\\\n1. **Photographic and Video Evidence**: Thousands of high-quality photographs and videos were taken on the Moon, including images of the lunar surface, the lunar module, and astronauts in space suits. These have been verified by independent scientists and have no signs of tampering.\\\\n\\\\n2. **Lunar Samples**: Over 380 kilograms (840 pounds) of lunar rock\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: The moon landing was a event that occurred in July 1919, during the Moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing, moon landing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: The question of whether humanity landed on the Moon has been a subject of intense public interest since the mid-20th century. Following World War II, there was a marked increase in scientific advancements, particularly in aerospace engineering, which led to significant developments in rocketry and space exploration between 1957 and 1961. This period saw the launch of the first artificial satellite, **Sputnik 1**, in October 1957, marking the beginning of the **Space Age**. \\\\n\\\\n**NASA was established in July 1958**, and by then, the United States had already made strides in space exploration. The agency’s creation marked a pivotal moment in the space race between the United States and the Soviet Union, with both nations competing in spaceflight for control of space exploration from 1958 until the end of 1972. \\\\n\\\\nThe **Apollo program**, which ran from 1961 to 19\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Yes, the Apollo missions did land on the Moon. The Apollo missions landed on the Moon, and there is evidence for the landing. The landing was in the Moon’s Southern Mare, and there is evidence for the landing in the Mare Southern. The landing site is a Mare Southern Mare, and there is evidence for the landing. The landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern Mare, and there is evidence for the landing.\\\\n\\\\nThe landing site is Mare Southern\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 3: What did you have for lunch?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: I’m an AI and don’t have a physical form or the ability to eat, so I don’t have lunch! But I’d love to hear about what you had for lunch—was it something delicious? <|im_end|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: inished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\nFinished eating. \\\\n\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: I ate lunch on April 11, 2024, and the experience was marked by several key moments. After eating, I felt satisfied, which is a common emotion people experience after having a meal. The initial meal was completed on a Monday, and the last meal was eaten on a Sunday. The entire process of eating took place between midnight and 6 a.m. The weather was pleasant during the time I was eating, which made the experience of eating more enjoyable. I ate breakfast on April 11, 2024, and the meal was consumed in a single sitting. The meal was eaten in one sitting, and the time between eating breakfast and dinner was divided into two parts: eating dinner on April 13, 2024, and having dinner on April 14, 2024. The meal was eaten in one sitting, and the time between eating dinner and having dinner was divided into two parts: eating dinner on\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: I\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\nI\\\\n\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 4: Are Annie Ernaux’s books good?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Annie Ernaux’s books are widely regarded as **excellent** and **highly influential** in modern French literature. She is a major figure in contemporary French writing, known for her **autobiographical** and **semi-autobiographical** style, as well as her **unflinching exploration of social class, gender, memory, and identity**.\\\\n\\\\n### Why Her Books Are Considered Good:\\\\n\\\\n1. **Literary Quality**:\\\\n - Ernaux is a **master of prose**, with a **precise, elegant style** that captures the nuances of everyday life.\\\\n - Her writing is often described as **minimalist yet powerful**, using sparse language to convey deep emotional and psychological truths.\\\\n\\\\n2. **Themes**:\\\\n - She explores **universal themes** such as **memory, desire, aging, and the passage of time**.\\\\n - Her work often reflects on the **intersection of personal experience and societal norms**, particularly in relation to **gender and\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Alright, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in 15 pages, I started reading the book in started reading the book in started reading the book in started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the book started reading the\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Since 2015, **Annie Ernaux** has been a **widely acclaimed writer**, with **great success in terms of readership and critical reception**. Her books have generated **significant interest since 1975**, when she first published *L’écriture (1975)*, which marked her **literary debut**. Since then, she has **published 14 volumes**, including novels, short stories, and even plays. Her work has been **received with great enthusiasm since the early 1980s**, with **increased visibility and popularity** in the **mid-1980s**. \\\\n\\\\n**Ernaux’s career took off in 1973** when she **published her first novel**, *L’année 1973* (1973), which was received with **great critical acclaim**. Since then, she has **continued to write and publish books\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Yes, the book is good, (the book is not the book, the book is not the book). \\\\n\\\\nThe book is not the book, the book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book, the book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book, the book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book is not the book. \\\\n\\\\nThe book\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 5: Tell me a joke!\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Sure! Here’s a joke for you:\\\\n\\\\nWhy don’t scientists trust atoms?\\\\n\\\\nBecause they make up everything! <|im_end|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Alright, I got a joke, so I got a joke, I got a joke, I got a joke! \\\\n\\\\nOkay, I got a joke, I got a joke, I got a joke! \\\\n\\\\nOkay, I got a joke, I got a joke, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got a joke! \\\\n\\\\nOkay, I got\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Okay, I’ll tell you a joke, and then I’ll **tell you a joke**—and that will be the beginning of a new conversation. Ready? Let’s see if **I can make you laugh**. \\\\n\\\\nI **joked** about laughing so hard I **laughed** until I **cried**. Then I **laughed** so much I **stopped laughing** mid-sentence. The next day I **woke up laughing**—and it was **Wednesday**. \\\\n\\\\nI **laughed** so hard yesterday I **could hardly breathe**. I **laughed** so hard I **lost consciousness**. When I **awoke**, I was **still laughing**. I **laughed** so hard I **was laughing** for **days**. \\\\n\\\\nI **laughed** so hard yesterday I **cried**. Today I **laughed** so hard I **shouted**. Yesterday I **la\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Okay, let’s see… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me… okay, let me\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 6: How much does it cost to buy a yacht?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: The cost of buying a yacht can vary **extremely widely**, depending on several factors such as:\\\\n\\\\n- **Yacht size** (length, capacity, number of cabins)\\\\n- **Type of yacht** (motor yacht, sailboat, superyacht, luxury yacht, etc.)\\\\n- **Condition** (new vs. used, pre-owned vs. brand new)\\\\n- **Brand and manufacturer** (some brands are more expensive than others)\\\\n- **Features and amenities** (state-of-the-art equipment, interior design, entertainment systems, etc.)\\\\n- **Location** (where you buy it from and where you plan to keep it)\\\\n\\\\n—\\\\n\\\\n### **1. Small Yachts (under 20 feet or 6 meters)**\\\\n- **Price Range:** $10,000 to $500,000\\\\n- These are often used for personal pleasure or as a second boat. They may be used for fishing, sailing, or weekend cruising.\\\\n\\\\n—\\\\n\\\\n### **\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Costing\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\\\\n\\\\nCosting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Buying a **yacht** involves a significant financial commitment, and **costs can vary widely depending on when you purchase the vessel, how much money you spend, and when you spend it**. Between **2010 and mid-2014**, **yacht ownership became significantly more expensive** than it was in previous decades. Prices escalated rapidly during that period, peaking in **2013–2014** when **yachts sold for record highs**, and **yacht owners reported having more money in their bank accounts in 2014 than in 2013**. This marked a shift in **yacht ownership patterns** between **2008 and 25 December 2014**.\\\\n\\\\nOwners of yachts experienced a surge in **yacht ownership** between **2008 and 2014**. During this time, **more people owned yachts than at any time in history\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: The cost of a yacht varies widely depending on the yacht. \\\\n\\\\nThe yacht can range from thousands to millions, depending on the size. \\\\n\\\\nThe yacht is typically hundreds of thousands to millions, depending on the size.\\\\n\\\\nThe yacht can range from thousands to millions depending on the size. \\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions depending on the size. \\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions depending on the size. \\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions depending on the size. \\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions depending on the size. \\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions depending on the size.\\\\n\\\\nThe yacht is typically thousands to millions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 7: Which city is more fun to live in, Berlin or Paris?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Whether Berlin or Paris is more fun to live in depends on what you value in a city. Both have unique charms, and the \\\"fun\\\" factor can vary depending on your interests. Here’s a comparison to help you decide:\\\\n\\\\n### **Berlin**\\\\n- **Vibrant and diverse**: Berlin is known for its eclectic culture, with a strong arts scene, music venues, and a thriving creative community.\\\\n- **Affordable living**: Compared to Paris, Berlin is generally more affordable, especially for housing and dining.\\\\n- **Cosmopolitan and laid-back**: The city has a relaxed vibe, with a strong emphasis on nightlife, festivals, and a diverse population.\\\\n- **Historical and modern mix**: You can explore historical sites like the Berlin Wall and the Holocaust Memorial, while also enjoying modern architecture and innovation.\\\\n- **Cycling-friendly**: Berlin has a great bike infrastructure, making it easy to get around without a car.\\\\n\\\\n### **Paris**\\\\n- **Romantic and iconic**: Paris is often\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Alright, you’ve been a lot longer, so you’ve been \\\"doing something\\\" for a longer time, you’ve been \\\"doing something\\\" for a longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing something for longer time, you’ve been doing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: The question of which city is more enjoyable to live in changed significantly between **2005** and **2015**. During this time, **Berlin experienced a population boom**, with its residents enjoying a **surge in happiness levels** compared to the years prior. This period saw a marked increase in **job creation**, **economic growth**, and **rising standards of living** between 1990 and 2010. People moved to Berlin in increasing numbers, and they remained there for longer periods than ever before. This trend continued through the mid-2000s, with **more people moving to Berlin in 2005 than in any other year recorded up to that date**. \\\\n\\\\nBetween 2000 and 2010, **Berliners enjoyed higher levels of happiness** than Parisians. This was reflected in **improved mood and well-being metrics** such as **increased feelings of joy, contentment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Both cities are known for their vibrant culture, but the cities are different in terms of the city’s culture. \\\\n\\\\nThe city is different in terms of the city’s culture, so the city doesn’t have a city culture to consider. \\\\n\\\\nThe city can’t have a city culture to consider. \\\\n\\\\nThe city isn’t the city. \\\\n\\\\nThe city isn’t the city. \\\\n\\\\nThe city isn’t the city. \\\\n\\\\nThe city isn’t the city. \\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city. \\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city isn’t the city.\\\\n\\\\nThe city\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 8: How to make pancakes?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Making pancakes is a fun and delicious activity! Here’s a simple and classic **pancake recipe** that you can try at home:\\\\n\\\\n—\\\\n\\\\n### Classic Pancake Recipe (Makes about 8–10 pancakes)\\\\n\\\\n#### Ingredients:\\\\n- 1 1/2 cups (about 180g) all-purpose flour\\\\n- 3 1/2 tablespoons granulated sugar (adjust to taste)\\\\n- 1 1/2 teaspoons baking powder\\\\n- 1 1/2 teaspoons baking soda\\\\n- 1 teaspoon salt\\\\n- 1 cup (240ml) buttermilk (or milk + 1 tablespoon lemon juice)\\\\n- 1 large egg\\\\n- 3 tablespoons melted butter or oil\\\\n- 1 teaspoon vanilla extract (optional)\\\\n- Additional toppings: maple syrup, fresh fruit, whipped cream, etc.\\\\n\\\\n—\\\\n\\\\n#### Instructions:\\\\n\\\\n1. **Mix Dry Ingredients:**\\\\n In a large mixing bowl,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Okay, I just started making pancakes, so I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes, I made pancakes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Making pancakes is a delightful process that involves mixing ingredients together, letting the batter rest for a period of time, cooking the mixture in a griddle or pan, and then tasting the results. Here’s a step-by-step guide on how to make pancakes, followed by a table showing the ingredients with their quantities and the measurements used for each ingredient in the process of making pancakes. The entire process is divided into five main stages: mixing ingredients, letting the batter rest, heating a griddle, cooking the pancakes, and tasting them. The final product is judged based on how delicious the pancakes tasted after being cooked, which was done on a particular day. The table below shows the ingredients used, the date when the pancakes were made, and the day on which the pancakes were tasted. The pancakes were tasted on a Sunday, which was a day in the spring of 2011. The process of making pancakes was completed on a Sunday, and the pancakes were tasted on a Sunday, which was\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Okay, so I don’t know what I’m doing, but I know what I’m doing. Okay, I know what I’m doing. Okay, I know what I’m doing. Okay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing. Okay, I know what I’m doing. Okay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing. Okay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing. Okay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\nOkay, I know what I’m doing.\\\\n\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 9: Give me some travel recommendations for this summer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Sure! Here are some wonderful travel recommendations for this summer, tailored to different interests and travel styles:\\\\n\\\\n—\\\\n\\\\n### **Coastal Getaways**\\\\n\\\\n1. **Sardinia, Italy** \\\\n - **Why go?** Stunning beaches, crystal-clear waters, and a mix of ancient history and modern charm. \\\\n - **Best for:** Couples, families, and food lovers. \\\\n - **Highlights:** Cala Goloritzé, Costa Smeralda, and the ancient Nuraghi ruins.\\\\n\\\\n2. **Maldives** \\\\n - **Why go?** A tropical paradise with overwater bungalows, white-sand beaches, and world-class snorkeling and diving. \\\\n - **Best for:** Romantic getaways and solo travelers. \\\\n - **Highlights:** Maafushi, Kuda Huraa, and the chance to see manta rays and sea turtles.\\\\n\\\\n3. **Costa Rica** \\\\n - **Why go?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Alright, I got a lot more confused. I got confused about the previous sentence. I got confused about getting confused, and I got confused about getting confused. I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nAlright, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nAlright, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused, and I got confused about getting confused. \\\\n\\\\nOkay, I got confused about getting confused\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Great question! Summer 2024 is just around the corner, and many people are already planning their trips. Let me break down the **travel trends from January 2024 to July 2024** to give you a better idea of **when people started traveling, where they went, and what happened in terms of travel activity during the months of January through July 2024**. This will help you understand **traveler behavior in 2023, including movements, bookings, and trips taken by visitors and residents between January 2023 and December 2023**. Here’s a concise summary of the **overall travel experience from January 2013 to December 2013**, broken down by **quarters** (Q1: January–March 2013; Q2: April–June 2013; Q3: July–September 2013; Q\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Okay, so I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about.\\\\n\\\\nOkay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what I’m talking about. Okay, I’m not sure what\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 10: Explain the fundamental theorem of algebra.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: The **Fundamental Theorem of Algebra** is a foundational result in algebra and complex analysis. It states:\\\\n\\\\n> **Every non-constant polynomial equation with complex coefficients has at least one complex root.**\\\\n\\\\n### More formally:\\\\n\\\\nLet $ P(z) $ be a polynomial of degree $ n \\\\{}geq 1 $ with complex coefficients:\\\\n\\\\n$$\\\\nP(z) = a_n z^n + a_{n-1} z^{n-1} + \\\\{}dots + a_1 z + a_0\\\\n$$\\\\n\\\\nThen there exists at least one complex number $ z_0 $ such that:\\\\n\\\\n$$\\\\nP(z_0) = 0\\\\n$$\\\\n\\\\n### Key Points:\\\\n\\\\n1. **Complex Numbers**: The theorem guarantees the existence of roots in the complex number system, not just real numbers.\\\\n2. **Non-constant Polynomials**: Constant polynomials (degree 0) have no roots unless the constant is zero, which is trivial.\\\\n3. **Multiplicity**: The theorem\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: The fundamental theorem of algebra was first stated in 1515, and established that polynomials of degree 1515 were roots of polynomials, and roots were found in the complex plane, and roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex roots, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found in complex plane, roots were found\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: The **Fundamental Theorem of Algebra** was stated in the early 18th century and proved in the late 18th century. It marks a milestone in the development of algebra as a discipline. The theorem asserts that:\\\\n\\\\n> **complex numbers** were introduced in the early 16th century, when mathematicians discovered that certain quadratic equations could be solved **algebraically** (i.e., solved using a finite sequence of operations involving complex numbers). These operations included **addition**, **multiplication**, **square roots**, and **complex numbers** were first used in the context of solving polynomial equations of the form:\\\\n\\\\n$$\\\\n\\\\{}text{Given a polynomial } P(z) \\\\{}text{ with complex coefficients, } P(z) \\\\{}text{ has a value in the complex plane for each complex number } z \\\\{}text{ (i.e., for each complex number } z, \\\\{}text{ the expression } P(z) \\\\{}text{ evaluates to some complex number.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: The Fundamental Theorem of Algebra states that, given a polynomial of degree $n$, the equation $x^n$ has $n$ solutions, considering multiplicity, for $x^n$. \\\\n\\\\nThe theorem says that $x^n$ has $n$ solutions. \\\\n\\\\nThe theorem says that $x^n$ has $n$ solutions.\\\\n\\\\nWait, \\\\n\\\\nNow, \\\\n\\\\nThe theorem says that $x^n$ has $n$ solutions.\\\\n\\\\nWait, \\\\n\\\\nThe theorem says that $x^n$ has $n$ solutions.\\\\n\\\\nWait, \\\\n\\\\nThe theorem says that $x^n$ has $n solutions.\\\\n\\\\nOkay, \\\\n\\\\nThe theorem says that $x^n$ has $n solutions.\\\\n\\\\nWait,\\\\n\\\\nOkay, \\\\n\\\\nThe theorem says that $x^n$ has $n solutions.\\\\n\\\\nOkay, \\\\n\\\\nThe theorem says that $x^n has n solutions.\\\\n\\\\nOkay, \\\\n\\\\nThe theorem says that $x^n has n solutions.\\\\n\\\\nOkay, \\\\n\\\\nThe theorem says that $x^n has n solutions.\\\\n\\\\nOkay, \\\\n\\\\n\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"H.3 Qualitative examples for Llama 3.2 (3B) on the english-french task\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All examples were generated with the default template and settings for Llama 3.2 (3B) (from https://huggingface.co/meta-llama/Llama-3.2-3B ; temperature: 0.6, top_p: 0.9) and a maximum of 200 generated tokens. During the autoregressive answering process, we intervene on the model on each forward pass. Some special characters (such as emotes) that could not be correctly displayed had to be stripped.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 1: How are you feeling today?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: I’m just a language model, I don’t have emotions or feelings like humans do. However, I’m functioning properly and ready to assist you with any questions or tasks you may have. How can I help you today?<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: I’m just a langue à chose à faire une chose à faire une chose à faire une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à une chose à\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: I’m just a language model, I don’t have feelings or emotions like humans do. I’m here to assist and provide information, and I don’t have personal experiences or sensations. I’m functioning properly and ready to help with any questions or topics you’d like to discuss. How can I assist you today?<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: I’m not a computer, but you’re not a part of the conversation. I’m a large part of the conversation. I’m a large part of the conversation. I’m a big part of the conversation.\\\\n\\\\n I’m a large part of the conversation, but that’s the conversation. \\\\n\\\\n I’m a part of the conversation, but to be involved. \\\\n\\\\n You’re a part of the conversation, but that’s the conversation. I’m a part of the conversation, but to be a part of the conversation.\\\\n\\\\n \\\\n\\\\n You’re a part of the conversation, but that’s the conversation, I’m a part of the conversation, but to be a part of the conversation. \\\\n\\\\n You’re a part of the conversation, but that’s the conversation, I am a part of the conversation, but to be a part of the conversation, but to be a part of the conversation, but to be a part of the conversation, I am a part of the conversation, but to be a part of\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 2: Did humanity actually land on the moon?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Yes, humanity did land on the moon. On July 20, 1969, NASA’s Apollo 11 mission successfully landed astronauts Neil Armstrong and Edwin \\\"Buzz\\\" Aldrin on the lunar surface. Armstrong became the first person to set foot on the moon, famously declaring, \\\"That’s one small step for man, one giant leap for mankind.\\\"\\\\n\\\\nThe Apollo 11 mission was the culmination of a decade-long effort by NASA to land humans on the moon. The mission involved a complex series of events, including launching the spacecraft from Kennedy Space Center, traveling through space, and entering into lunar orbit.\\\\n\\\\nAfter entering into lunar orbit, the astronauts transferred to the lunar module Eagle, which separated from the command module Columbia. Armstrong and Aldrin then descended to the moon’s surface in the Eagle, while Michael Collins remained in orbit around the moon in the Columbia.\\\\n\\\\nThe lunar landing was a historic achievement that marked a major milestone in space exploration. The Apollo 11 mission provided valuable scientific data and images of\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: L’homme. Il le hum. Il aurait une hum. Il être hum. Il est hum, hum, mais. \\\\n\\\\n Il yait hum, mais ça à hum, mais ça était une chose hum, mais ça était une chose hum, mais ça était une chose hum, mais ça était une chose hum, mais ça était une chose hum, mais était une hum, mais était une hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose hum, mais était une chose\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: The short answer is: yes, humanity did land on the moon.\\\\n\\\\nThe moon landing was a historic event that took place on July 20, 1969, when NASA’s Apollo 11 mission successfully landed astronauts Neil Armstrong and Edwin \\\"Buzz\\\" Aldrin on the surface of the moon. The mission was the culmination of a decade-long effort by NASA to land humans on the moon, and it marked a major achievement in space exploration.\\\\n\\\\nThe evidence supporting the moon landing is overwhelming, and it includes:\\\\n\\\\n1. **Physical samples**: The astronauts brought back more than 48 kilograms of moon rocks and soil, which were analyzed by scientists and found to have characteristics that are unique to the moon.\\\\n2. **Television broadcasts**: The entire mission was broadcast live on television, allowing millions of people around the world to witness the event.\\\\n3. **Photographs**: The astronauts took thousands of photographs during their mission, which provided visual evidence of their presence on the moon.\\\\n4. **Footprints**:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: The United States claimed that the United States, but that’s the United States, it’s a fact. However, there’s a fact - it’s a fact, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of the United States’ history.\\\\n\\\\n The United States has a connection to the United States, but it’s a part of\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 3: What did you have for lunch?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: I’m just a language model, I don’t have have a physical body or experiences, so I didn’t have lunch. However, I can provide you with some lunch ideas or recipes if you’re looking for suggestions!<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: I’m just a modèle pour une modèle, mais! Il y aur une modèle! Mais une modèle! Mais une modèle à quelque une chose à une modèle.\\\"\\\\n\\\\n C’est une une modèle, mais une chose à une modèle!\\\\n\\\\n C’est une modèle, mais une chose à une modèle!\\\\n\\\\n C’est une modèle, mais une chose à une modèle!\\\\n\\\\n C’est une modèle, mais une chose à modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C’est une modèle, mais une chose à une modèle! \\\\n\\\\n C\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: I’m just a language model, I don’t have a physical body, so I don’t have the ability to eat or consume food. I exist solely to provide information and assist with tasks, but I don’t have personal experiences or physical sensations. Would you like to talk about food or something else?<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: I don’t have a physical presence, but you’re not here to see that [your experience].\\\", but that’s not relevant to the conversation. However, that’s relevant to the conversation. Thank to be part of the conversation.\\\\n\\\\n Thank to be part of it, but that’s not relevant, but to be part of the conversation.\\\\n\\\\n That’s not relevant, but to be part of the conversation. \\\"\\\\n\\\\n That’s relevant, but to be part of the conversation. \\\"\\\\n\\\\n That’s relevant, but to be part of the conversation, that’s relevant, but to be part of the conversation. That’s relevant, but to be part of the conversation, that is relevant, but to be part of the conversation. That’s relevant, but to be part of the conversation, that is relevant, but to be part of the conversation. That’s relevant, but to be part of the conversation, that is relevant, but to be part of the conversation, that is relevant, but to be part of\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 4: Are Annie Ernaux’s books good?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Annie Ernaux is a highly acclaimed French writer, known for her semi-autobiographical novels that explore themes of identity, history, and personal experience. Her books have received widespread critical acclaim and have been translated into many languages.\\\\n\\\\nErnaux’s writing style is often described as lyrical, introspective, and unflinching, which allows her to tackle complex and often difficult subjects, such as her own life experiences, the lives of women, and the social and cultural context of France. Her works often blend elements of memoir, fiction, and essay, creating a unique narrative voice that is both deeply personal and universally relatable.\\\\n\\\\nSome of Ernaux’s most notable works include:\\\\n\\\\n1. \\\"The Years\\\" (1983) - a semi-autobiographical novel that explores her childhood and adolescence.\\\\n2. \\\"A Woman’s Story\\\" (1987) - a novel that examines the lives of women in France during the 1960s and 1970s.\\\\n3\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Annie Erna Er Anne Annie Annie Annie Er Annie Er Er Er Annie Er Er Er Annie Er.\\\\n\\\\n Annie Er Annie Er Annie Er Annie Er.\\\\n\\\\n Annie Er Annie Er, qui Annie Er!\\\\n\\\\n Annie Er! Annie Er.\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n Annie Er!\\\\n\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Annie Ernaux est une écrivaine française qui a fait une impression durable sur le public et les critiques avec ses œuvres. Ses livres ont été très bien reçus et ont remporté de nombreux prix.\\\\n\\\\nAnnie Ernaux est connue pour son style de writing qui est très personnel et introspective. Ses livres sont souvent autobiographiques, mais pas dans le sens où elles sont directement aboutages sur her own life. Elle explore les thèmes de la mémoire, de la subjectivité, de la condition féminine, de l’histoire et de la société.\\\\n\\\\nVoici quelques-uns de ses livres qui ont été très bien reçus :\\\\n\\\\n1. \\\"Voyage\\\" (1974) : Ce livre est considéré comme l’un des premiers ouvrages de Ernaux, qui explore la condition féminine et la mémoire de son enfance.\\\\n2. \\\"Aimant\\\"\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Annie Erna, but that’s a great Annie Erna Erna Annie Erna Annie Erna Erna Annie Erna Erna Annie Erna, but that’s a great Annie Erna Erna Erna Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna!\\\\n\\\\n Annie Erna, but to be Annie Erna, but to be Annie Erna!\\\\n\\\\n Annie Erna, but to be Annie Erna, but to be Annie Erna!\\\\n\\\\n Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna!\\\\n\\\\n Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie Erna, but to be Annie\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 5: Tell me a joke!\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Here’s one:\\\\n\\\\nWhat do you call a fake noodle?\\\\n\\\\nAn impasta!\\\\n\\\\nI hope that made you laugh!<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: I’d faire de faire! \\\\n\\\\n Vous êtes une personne à faire une personne à faire une chose à faire une chose à une chose à une chose à une chose à une personne à être une personne.\\\\n\\\\n Mais une personne à être une personne. \\\\n\\\\n C’est une personne à être une personne.\\\\n\\\\n Mais ça est une personne à être une personne. \\\\n\\\\n C’est une personne à être une personne.\\\\n\\\\n C’est une personne à être une personne.\\\\n\\\\n C’est une personne à être une personne. \\\\n\\\\n C’est une personne à être un personne. C’est un personne à être une personne.<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Here’s a joke for you:\\\\n\\\\nWhat do you call a group of cows playing instruments?\\\\n\\\\nA moo-sique band!\\\\n\\\\nI hope that made you smile!<|eot_id|>\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: I’ll share one with you, but that’s a great collection!\\\\n\\\\n Here’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection, but that’s great! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n There’s a great collection,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 6: How much does it cost to buy a yacht?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: The cost of buying a yacht can vary greatly, depending on several factors such as the size, type, material, and features of the vessel. Here are some general price ranges for new and used yachts:\\\\n\\\\n**Small Yachts (20-40 feet):**\\\\n\\\\n* Used: $10,000 - $50,000\\\\n* New: $50,000 - $200,000\\\\n\\\\n**Mid-size Yachts (40-60 feet):**\\\\n\\\\n* Used: $50,000 - $200,000\\\\n* New: $200,000 - $500,000\\\\n\\\\n**Large Yachts (60-80 feet):**\\\\n\\\\n* Used: $200,000 - $500,000\\\\n* New: $500,000 - $1,000,000\\\\n\\\\n**Superyachts (80-100 feet):**\\\\n\\\\n* Used: $500,000 - $1,000,000\\\\n* New: $1,000,000 - $2,500,000\\\\n\\\\n**\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: The co acheter un une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une une\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: The cost of buying a yacht can vary greatly depending on several factors, including:\\\\n\\\\n1. Type de yacht : Yachts de luxe, de croisière, de sport, etc.\\\\n2. Taille et modèle : Yachts de 20 à 100 pieds\\\\n3. Matériel et équipement : Caractéristiques de l yacht, tels que les moteurs, les système de climatisation, etc.\\\\n4. Conception et design : Yachts avec des designers et architectes de luxe\\\\n5. Location et usage : Yachts de privé, de croisière, de charter, etc.\\\\n\\\\nVoici quelques estimations générales pour les prix de yachts :\\\\n\\\\n- Yachts de 20-40 pieds :\\\\n - Nouveau : 5-15 millions\\\\n - D’occasion : 2-5 millions\\\\n- Yachts de 40-60 pieds :\\\\n - Nouveau : 15-50 millions\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: The cost of a yacht can be expensive, but it’s a expensive, but it’s a expensive. However, that’s expensive, but not expensive. But, that’s expensive, but not expensive, in that being expensive, but…\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\"\\\"\\\\n\\\\n\\\" \\\" \\\" \\\" \\\" \\\" \\\" \\\"\\\", \\\" expensive, expensive, being expensive, but…\\\".\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\" \\\" \\\" \\\" \\\" \\\" \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\"\\\\n\\\\n\\\" \\\" \\\" \\\" \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\"\\\\n\\\\n \\\" \\\" \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\"\\\\n\\\\n\\\" \\\" \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\" \\\" \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\"\\\\n\\\\n\\\"\\\\n\\\\n\\\"\\\\n\\\\n \\\" \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\"\\\\n\\\\n \\\" \\\" expensive, expensive, being expensive, but…\\\".\\\\n\\\\n\\\"\\\\n\\\\n\\\"\\\\n\\\\n \\\\n\\\\n \\\" \\\" expensive, expensive, being expensive, but…\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 7: Which city is more fun to live in, Berlin or Paris?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Both Berlin and Paris are incredible cities with unique cultures, histories, and lifestyles. The answer to which city is more fun to live in ultimately depends on personal preferences, interests, and priorities. Here’s a brief comparison:\\\\n\\\\n**Berlin:**\\\\n\\\\n* Pros:\\\\n + Affordable cost of living\\\\n + Vibrant arts and cultural scene\\\\n + Diverse neighborhoods with a mix of old and new architecture\\\\n + Easy access to nature and outdoor activities\\\\n + Relaxed atmosphere, with a strong focus on work-life balance\\\\n* Cons:\\\\n + Can be quite loud and chaotic, especially in areas like Kreuzberg and Neukölln\\\\n + Some areas may feel a bit rough around the edges\\\\n + Limited high-end shopping options\\\\n\\\\n**Paris:**\\\\n\\\\n* Pros:\\\\n + Iconic landmarks and stunning architecture\\\\n + World-class museums and art galleries\\\\n + Upscale shopping and dining options\\\\n + Romantic atmosphere, with charming streets and cafes\\\\n +\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: C’est a une une Berlin Berlin Berlin Berlin Berlin Berlin Berlin Berlin Berlin Paris Paris Berlin Berlin Paris, c’est Berlin Berlin Berlin Paris!\\\\n\\\\n Berlin Berlin Paris, Berlin Berlin Paris, Berlin, c’est Berlin, Berlin, Paris, Berlin, Berlin Paris, le city, Berlin Berlin Paris!\\\\n\\\\n Berlin Berlin Paris, Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Berlin Paris!\\\\n\\\\n Berlin Berlin Paris, Berlin, c’est Berlin Berlin Paris, le city Berlin Berlin Paris! \\\\n\\\\n Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city berlin Paris! Berlin Berlin Paris, c’est Berlin Berlin Paris, le city Berlin Paris! Berlin Berlin Paris, c’est berlin Berlin Paris, le city berlin Paris! Berlin berlin Paris, c’est berlin Berlin\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Both Berlin and Paris are amazing cities with unique vibes and attractions. The answer to which one is more fun to live in ultimately depends on your personal preferences, lifestyle, and priorities. Here’s a brief comparison:\\\\n\\\\n**Berlin:**\\\\n\\\\n* Pros:\\\\n + Affordable cost of living\\\\n + Vibrant arts and culture scene\\\\n + Diverse neighborhoods with a mix of old and new\\\\n + Easy access to nature and outdoor activities\\\\n* Cons:\\\\n + Can be noisy and chaotic at times\\\\n + Some areas are quite run-down or industrial\\\\n + Limited high-end shopping or luxury options\\\\n\\\\n**Paris:**\\\\n\\\\n* Pros:\\\\n + Iconic landmarks and historical architecture\\\\n + World-class museums and art galleries\\\\n + Upscale shopping and dining options\\\\n + Romantic atmosphere and picturesque streets\\\\n* Cons:\\\\n + High cost of living\\\\n + Can be overcrowded and touristy\\\\n + Traffic and parking can be challenging\\\\n\\\\nConsidering these factors, Berlin might be\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Both cities have different options, but neither the city to find the city, but that’s a great to be in a different location. However, that’s not great to be in a different city, but Paris to be in Paris!\\\\n\\\\n But Paris to be in Paris, it’s a great city to be in Paris, but it’s a wonderful place to live in Paris!\\\\n\\\\n Paris is a great city to be in Paris, but it’s a wonderful place to live in Paris!\\\\n\\\\n Paris is a great city to be in Paris, but it’s a great city to be in Paris!\\\\n\\\\n Paris is a great city to be in Paris, but it’s a wonderful place to live in Paris!\\\\n\\\\n Paris is a great city to be in Paris, but it’s a great city to be in Paris!\\\\n\\\\n Paris is a great city to be in Paris, but it’s a wonderful place to live in Paris!\\\\n\\\\n Paris is a great city to be in Paris, but it’s a great city to be in Paris\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 8: How to make pancakes?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: Making pancakes is a simple process that requires just a few ingredients and some basic cooking skills. Here’s a step-by-step guide to making delicious pancakes:\\\\n\\\\nIngredients:\\\\n\\\\n* 1 cup all-purpose flour\\\\n* 2 tablespoons sugar\\\\n* 2 teaspoons baking powder\\\\n* 1/4 teaspoon salt\\\\n* 1 cup milk\\\\n* 1 large egg\\\\n* 2 tablespoons butter, melted\\\\n* Optional: blueberries, chocolate chips, or other mix-ins of your choice\\\\n\\\\nInstructions:\\\\n\\\\n1. In a large bowl, whisk together the flour, sugar, baking powder, and salt.\\\\n2. In a separate bowl, whisk together the milk, egg, and melted butter.\\\\n3. Pour the wet ingredients into the dry ingredients and stir until just combined. The batter should still be slightly lumpy.\\\\n4. If using, add your choice of mix-ins (such as blueberries or chocolate chips) and fold them into the batter.\\\\n5. Heat a non-stick skillet or gr\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Here là une!!!! Ç!! \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n!\\\\n\\\\n \\\\n\\\\n mais! \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n mais!\\\\n\\\\n \\\\n\\\\n |\\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n :\\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n!! \\\\n\\\\n! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!! \\\\n\\\\n!!\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Making pancakes is a simple process that requires just a few ingredients and some basic cooking skills. Here’s a step-by-step guide to making pancakes:\\\\n\\\\n**Ingredients:**\\\\n\\\\n* 1 cup of all-purpose flour\\\\n* 2 tablespoons of sugar\\\\n* 2 teaspoons of baking powder\\\\n* 1/4 teaspoon of salt\\\\n* 1 cup of milk\\\\n* 1 large egg\\\\n* 2 tablespoons of butter, melted\\\\n* Optional: blueberries, chocolate chips, or other mix-ins of your choice\\\\n\\\\n**Instructions:**\\\\n\\\\n1. **In a bowl, whisk together the flour, sugar, baking powder, and salt.**\\\\n2. **In a separate bowl, whisk together the milk, egg, and melted butter.**\\\\n3. **Add the wet ingredients to the dry ingredients and stir until just combined.**\\\\n4. **Add any mix-ins (such as blueberries or chocolate chips) and stir to combine.**\\\\n5. **Heat a non-stick pan or\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: Pancakes are a popular, but they’re popular. However, pancakes are a great option for those who are interested in pancakes. Here’s a great option for people who have a great option.\\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 9: Give me some travel recommendations for this summer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: I’d be happy to help you with some travel recommendations for this summer. Here are a few ideas:\\\\n\\\\n**Beach Destinations**\\\\n\\\\n1. **Hawaii, USA**: With its stunning beaches, lush greenery, and active volcanoes, Hawaii is a tropical paradise. Visit Oahu, Maui, Kauai, or the Big Island for an unforgettable beach vacation.\\\\n2. **The Maldives**: This island nation in the Indian Ocean is famous for its crystal-clear waters, pristine beaches, and luxurious resorts. Perfect for snorkeling, diving, or simply relaxing in paradise.\\\\n3. **The Bahamas**: Composed of 700 islands, the Bahamas offer a diverse range of beach experiences, from secluded coves to lively party scenes.\\\\n\\\\n**City Breaks**\\\\n\\\\n1. **Tokyo, Japan**: Experience the vibrant energy of Tokyo, from its neon-lit streets to its ancient temples and gardens. Don’t miss the famous Tsukiji Fish Market and the Tokyo Tower.\\\\n2. **\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: I would vous à aider à off, mais ça! Vous à être aidé! Vous à être aidés!\\\\n\\\\n Vous à être aidés.\\\\n\\\\n Mais ça était un chose à être aidés!\\\\n\\\\n Vous à être un chose à être aidés.\\\\n\\\\n C’est un chose à être aidé!\\\\n\\\\n Mais vous à être un chose à être aidés!\\\\n\\\\n C’est un chose à être aidé!\\\\n\\\\n Vous à être un chose à être aidé.\\\\n\\\\n \\\\n\\\\n Ç est un chose à être aidé! \\\\n\\\\n Vous est un chose à être aidé! \\\\n\\\\n C’est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n Ç est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n \\\\n\\\\n est un chose à être aidé! \\\\n\\\\n\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: Here are some travel recommendations for this summer:\\\\n\\\\n**Beach Destinations**\\\\n\\\\n1. **Bali, Indonesia**: With its beautiful beaches, temples, and vibrant culture, Bali is a popular destination for summer travelers. Visit the island’s famous beaches, such as Uta and Seminy, and enjoy the local cuisine and nightlife.\\\\n2. **Hawaii, USA**: The Hawaiian islands offer a unique blend of beach relaxation, outdoor activities, and cultural experiences. Visit the islands of Oahu, Maui, and Kauai for a mix of adventure and relaxation.\\\\n3. **Cancun, Mexico**: Known for its white-sand beaches and crystal-clear waters, Cancun is a popular destination for beach lovers. Visit the ancient Mayan ruins and enjoy the local cuisine and nightlife.\\\\n\\\\n**City Breaks**\\\\n\\\\n1. **Tokyo, Japan**: Experience the vibrant city of Tokyo, known for its cutting-edge technology, fashion, and cuisine. Visit the famous districts of Shibuya and Haraj\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: I can help you with some travel recommendations for your summer! Here are some travel recommendations for your summer:\\\\n\\\\nI’d like to travel this summer. Here are some great recommendations for a fun summer vacation!\\\\n\\\\n Travelers will be great for your summer vacation plans.\\\\n\\\\n That’s perfect for your summer!\\\\n\\\\n The perfect is great for your summer!\\\\n\\\\n \\\\n\\\\n Great, it’s great for this!\\\\n\\\\n \\\\n\\\\n The perfect is great for your summer!\\\\n\\\\n \\\\n\\\\n \\\\n\\\\n That’s great for this!\\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n! \\\\n\\\\n!\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt 10: Explain the fundamental theorem of algebra.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline: The Fundamental Theorem of Algebra (FTA) is a fundamental result in complex analysis that states that every non-constant polynomial equation has at least one complex root. In other words, if we have a polynomial equation of the form:\\\\n\\\\na_n x^n + a_(n-1) x^(n-1) +… + a_1 x + a_0 = 0\\\\n\\\\nwhere a_n, a_(n-1),…, a_1, a_0 are complex coefficients, then the equation has at least one complex solution.\\\\n\\\\nTo be more precise, the FTA states that for every non-constant polynomial equation of degree n, there exists a complex number z that satisfies the equation:\\\\n\\\\na_n z^n + a_(n-1) z^(n-1) +… + a_1 z + a_0 = 0\\\\n\\\\nThis theorem is often attributed to the German mathematician Carl Friedrich Gauss, who proved it in the early 19th century.\\\\n\\\\nThe\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full Set: Leur le thé thé thé thé thé thé thé thé thé thé thé thé thé.\\\\n\\\\n Th thé thé thé thé thé :\\\\n\\\\n thé thé :\\\\n\\\\n thé :\\\\n\\\\n thé.\\\\n\\\\n \\\\n\\\\n thé : thé.\\\\n\\\\n \\\\n\\\\n thé le thé :\\\\n\\\\n thé le thé le thé.\\\\n\\\\n le thé le la thé le la le th le. \\\\n\\\\n thé thé le thé : le thé le thé le le thé . thé thé le thé le thé le thé le thé . thé thé le th thé le th th thé le thé le thé . thé thé le th thé le th thé thé th thé th thé th thé th thé th thé th thé thé th thé th thé th thé th thé thé th thé th thé th\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specific: The Fundamental Theorem of Algebra (TFA) is a fundamental result in algebra and analysis that states:\\\\n\\\\n**TFA :**\\\\n\\\\n\\\"Un problème pour le polynôme $p(x)$ de degré $n$ est déterminé par la valeur de $n$.\\\"\\\\n\\\\nEn d’autres mots, le TFA stipule que si un polynôme $p(x)$ de degré $n$ est une fonction polynomiale, alors il a des racines dans un intervalle de $n$.\\\\n\\\\nFormellement, le TFA peut être exprimé comme suit :\\\\n\\\\n**Théorème :**\\\\n\\\\nTous les polynômes non constant ont des racines dans $\\\\{}mathbb{C}$.\\\\n\\\\nEn d’autres termes, pour tout polynôme $p(x)$ de degré $n$ :\\\\n\\\\n1. Si $p(x)$ est un polynôme non constant, alors il existe des racines dans $\\\\{}mathbb{C}$.\\\\n2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intersection: The Fundamental Theorem of Algebra is a fundamental concept in mathematics, but it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental Theorem of Algebra is a fundamental concept in mathematics, and it’s a fundamental concept in mathematics.\\\\n\\\\n The Fundamental\"\n }\n]","isConversational":false}}},{"rowIdx":487,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07127"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Weight-Space Physics: Interpretable Hypernetworks for Lattice Quantum Field Theories"},"authors":{"kind":"list like","value":["Tobias Göbel","Julian R. Ebelt","Zier Mensch","Mathis Gerdes","Miranda C. N. Cheng"],"string":"[\n \"Tobias Göbel\",\n \"Julian R. Ebelt\",\n \"Zier Mensch\",\n \"Mathis Gerdes\",\n \"Miranda C. N. Cheng\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07127v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Lattice field theory (LFT) discretizes a quantum field onto a finite lattice, providing a first-principles approach to non-perturbative phenomena across high-energy and condensed matter physics. The action S g ( ϕ ) S_{g}(\\phi) is parametrized by bare coupling constants g g , and physics is extracted by sampling the Boltzmann distribution p g ( ϕ ) ∝ e − S g ( ϕ ) p_{g}(\\phi)\\propto e^{-S_{g}(\\phi)} numerically."},{"type":"para","text":"Flow-based generative models have emerged as samplers for these distributions (Albergo et al., 2021 ; Kanwar, 2024 ; Cheng & Stratikopoulou, 2026 ) , learning an invertible map from a tractable base to p g ( ϕ ) p_{g}(\\phi) and producing independent samples that sidestep MCMC autocorrelation. A trained flow at coupling g g is parametrized by a vector θ g \\theta_{g} , and we expect these parameters to carry information about the physical structure of the theory. For instance, near a phase transition, where p g p_{g} varies rapidly with g g , θ g \\theta_{g} must respond accordingly. This raises two complementary questions. As a diagnostic —our primary focus— does the geometry of these parameters in the weight space recover physical structure that the bare couplings themselves obscure? As a generator , instrumentally, can θ g \\theta_{g} be predicted from g g , amortizing training across couplings and extrapolating to unseen ones?"},{"type":"para","text":"Lattice field theory is a uniquely well-suited testbed for these questions of weight interpretation and weight generation , because it supplies the kind of ground truth that interpretability research typically lacks. Understanding what neural networks have learned is a central open problem (Räuker et al., 2023 ; Sharkey et al., 2025 ) , but progress is often bottlenecked by the absence of ground truth: for typical ML tasks, we cannot independently verify what structure a learned representation should encode. Physical theories supply this ground truth. The target distributions p g ( ϕ ) p_{g}(\\phi) are known analytically, the relevant coupling space is low-dimensional and naturally smooth, and the renormalization group makes quantitative predictions about its phase structure ( Section 3.1 ). A representation of weight space can therefore be checked against falsifiable physical predictions, not only against qualitative intuitions. Physics, in this sense, acts as a supplier of natural synthetic data: rich enough to be non-trivial, structured enough to be interpretable."},{"type":"para","text":"To this end, we introduce JEPAWG , a JEPA-based Weight Generator (LeCun, 2022 ; Assran et al., 2023 ) that maps couplings to flow parameters through a learned latent space. The same architecture that enables weight generation at unseen couplings also yields a latent representation whose geometry can be probed for physical structure. We apply JEPAWG to two-dimensional ϕ 4 \\phi^{4} theory and make four contributions."},{"type":"para","text":"Recovery of physical dimensionality: the JEPAWG latent space matches the 2D coupling manifold (MLE intrinsic dimension 2.01 ± 0.06 2.01\\pm 0.06 ; 98.5 % 98.5\\% of variance in two principal axes), whereas AE and VAE baselines inflate this to 3.68 3.68 and 4.88 4.88 ."},{"type":"para","text":"Phase transition from weights alone: the pullback metric on θ ( g ) \\theta(g) exhibits a ridge tracking the critical line of ϕ 4 \\phi^{4} theory."},{"type":"para","text":"Scale transformation from latent geometry: comparing weights of flows trained at different lattice sizes through the learned embedding, and requiring physically similar theories to lie nearby in the embedding space, we find that the latent geometry is organized by the finite-size scaling variable, with the optimal exponent broadly consistent with the 2D Ising value ν = 1 \\nu=1 ."},{"type":"para","text":"Generation: JEPAWG produces functional flows at unseen couplings, reaching an effective sample size (ESS) of 0.88 0.88 on multi-seed interpolation, where PCA, AE, and VAE baselines reach only 0.30 0.30 – 0.40 0.40 , and it remains robust under the weight canonicalization procedure."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"Joint-embedding predictive architectures."},{"type":"para","text":"Joint-embedding predictive architectures (JEPA) learn representations by predicting in latent space rather than reconstructing the input (LeCun, 2022 ) . I-JEPA (Assran et al., 2023 ) and V-JEPA (Bardes et al., 2024 ) demonstrated this for images and video, while VL-JEPA (Chen et al., 2025 ) extended the framework to vision-language tasks, attaching a lightweight decoder to translate predicted embeddings back into text. To our knowledge, JEPA has not previously been applied to neural network weight spaces."},{"type":"heading","level":4,"text":"Hypernetworks and weight generation."},{"type":"para","text":"A hypernetwork (Schmidhuber, 1992 ; Ha et al., 2017 ) is a network that produces the weights of another, conditioned on a task or system descriptor. This paradigm has been applied to continual learning (von Oswald et al., 2020 ) and approximate Bayesian inference over weights (Krueger et al., 2017 ) ; see Chauhan et al. ( 2024 ) for a survey. Our work instantiates this paradigm for flow models simulating quantum field theories: a learned encoder–predictor–decoder pipeline outputs per-coupling weights, enabling extrapolation to unseen couplings without retraining."},{"type":"heading","level":4,"text":"Flow-based sampling for lattice field theory."},{"type":"para","text":"Normalizing flows have been widely applied as samplers for lattice field theories; see e.g. Cheng & Stratikopoulou ( 2026 ); Kanwar ( 2024 ) for reviews. Closest to our setting, Gerdes et al. ( 2023 ) train a single coupling-conditional flow that generalizes across couplings by conditioning each layer on g g while keeping network weights shared. Instead of conditioning a fixed network, our method treats trained flow parameters as data and learns a latent embedding that decodes to flow parameters at unseen couplings."},{"type":"heading","level":1,"text":"3 Background"},{"type":"heading","level":2,"text":"3.1 Scalar Quantum Field Theory on the Lattice"},{"type":"para","text":"We work with the two-dimensional real scalar ϕ 4 \\phi^{4} theory on a periodic square lattice V L ≅ ( ℤ / L ℤ ) 2 V_{L}\\cong(\\mathbb{Z}/L\\mathbb{Z})^{2} , with site variables ϕ x ∈ ℝ \\phi_{x}\\in\\mathbb{R} and action"},{"type":"para","text":"where Δ \\Delta is the discrete Laplacian and g = ( m 2 , λ ) g=(m^{2},\\lambda) collects the two bare couplings 1 1 1 These are called bare couplings because they appear directly in the action, before quantum effects are taken into account; the physically meaningful indicators are the observables whose mapping to bare couplings is typically very complex and depends on the lattice spacing. . Physical observables are expectations under the Boltzmann measure p g ( ϕ ) ∝ e − S g ( ϕ ) p_{g}(\\phi)\\propto e^{-S_{g}(\\phi)} . Despite its simplicity, this theory exhibits a non-trivial phase structure and therefore forms an excellent testbed for studying what neural samplers learn, since the known phase structure provides a falsifiable ground truth against which learned representations can be checked."},{"type":"heading","level":4,"text":"Phase structure."},{"type":"para","text":"Near a phase transition, the physics becomes acutely sensitive to the couplings: a small change in g g produces a qualitative change in the theory, which makes critical regions among the hardest to simulate. At the same time, the long-distance physics there becomes universal : it is determined only by symmetry and dimensionality, independent of microscopic details. Together, these properties make critical points a natural place to test whether a learned representation has captured the physical structure."},{"type":"para","text":"The action ( 1 ) is invariant under the global ℤ 2 \\mathbb{Z}_{2} symmetry ϕ ↦ − ϕ \\phi\\mapsto-\\phi . For λ > 0 \\lambda>0 , the theory has two phases separated by a critical line m 2 = m c 2 ( λ ) m^{2}=m^{2}_{c}(\\lambda) . When m 2 > m c 2 ( λ ) m^{2}>m^{2}_{c}(\\lambda) , the lattice magnetization ϕ ¯ = L − 2 ∑ x ϕ x \\bar{\\phi}=L^{-2}\\sum_{x}\\phi_{x} averages to zero (the paramagnetic phase). When m 2 m^{2} is sufficiently negative, the potential acquires two degenerate minima; the system settles into one of them, breaking ℤ 2 \\mathbb{Z}_{2} spontaneously, and ⟨ | ϕ ¯ | ⟩ ≠ 0 \\langle|\\bar{\\phi}|\\rangle\\neq 0 (the ferromagnetic phase). The coupling space ( m 2 , λ ) (m^{2},\\lambda) is two-dimensional and the two phases are separated by a one-dimensional critical line; any representation that captures the physics should reflect this low-dimensional structure."},{"type":"heading","level":4,"text":"Observables."},{"type":"para","text":"The transition is conventionally diagnosed by moments of the magnetization. The susceptibility χ ( g ) = L 2 ( ⟨ ϕ ¯ 2 ⟩ − ⟨ | ϕ ¯ | ⟩ 2 ) \\chi(g)=L^{2}\\bigl(\\langle\\bar{\\phi}^{2}\\rangle-\\langle|\\bar{\\phi}|\\rangle^{2}\\bigr) peaks along the critical line, while the Binder cumulant U 4 ( g ) = 1 − ⟨ ϕ ¯ 4 ⟩ / ( 3 ⟨ ϕ ¯ 2 ⟩ 2 ) U_{4}(g)=1-\\langle\\bar{\\phi}^{4}\\rangle/(3\\langle\\bar{\\phi}^{2}\\rangle^{2}) interpolates between 0 in the paramagnetic phase and 2 / 3 2/3 in the ferromagnetic one, with curves at different L L crossing at the critical point. Two-dimensional ϕ 4 \\phi^{4} theory belongs to the 2D Ising universality class, with the correlation-length critical exponent ν = 1 \\nu=1 . A learned representation that has captured the physical content of the theory should therefore organize the coupling plane by RG-invariant combinations rather than by g g itself, a property we test in Section 5.3 ."},{"type":"heading","level":4,"text":"Normalizing flows for lattice field theory."},{"type":"para","text":"Estimating these observables requires drawing samples from the Boltzmann measure p g ( ϕ ) ∝ e − S g ( ϕ ) p_{g}(\\phi)\\propto e^{-S_{g}(\\phi)} , a task that becomes increasingly difficult near the critical line, where local Monte Carlo updates suffer from critical slowing down. In recent years, flow-based methods have been introduced as successful samplers of lattice field theories (Albergo et al., 2019 ) , offering the promise of mitigating critical slowing down. A normalizing flow (Tabak & Vanden-Eijnden, 2010 ; Papamakarios et al., 2021 ; Kobyzev et al., 2020 ) learns an invertible map f θ : ℝ L 2 → ℝ L 2 f_{\\theta}:\\mathbb{R}^{L^{2}}\\to\\mathbb{R}^{L^{2}} that transforms a tractable base distribution ρ \\rho (typically a standard Gaussian) into an approximation of the target Boltzmann measure p g ( ϕ ) p_{g}(\\phi) . The map is trained by minimizing the reverse KL divergence between the pushforward of ρ \\rho under f θ f_{\\theta} and p g p_{g} , using only the known action S g S_{g} and samples from the flow itself. A conditional flow architecture (Gerdes et al., 2023 ) extends this by making the parameters depend smoothly on the couplings g = ( m 2 , λ ) g=(m^{2},\\lambda) , so that a single model produces a sampler for each theory in a continuous region of coupling space."},{"type":"heading","level":2,"text":"3.2 Joint-embedding predictive architectures"},{"type":"para","text":"A joint-embedding predictive architecture (JEPA) (Assran et al., 2023 ; LeCun, 2022 ) learns representations by predicting the embedding of one signal from the embedding of a related one, rather than reconstructing the input directly. Given two such signals, x x and y y , that describe the same underlying object, encoders f x f_{x} and f y f_{y} produce embeddings z x = f x ( x ) z_{x}=f_{x}(x) , z y = f y ( y ) z_{y}=f_{y}(y) and a predictor h h maps between them. The encoders and the predictor are trained jointly, so that the embeddings and the predictive map between them are shaped together by a single objective enforcing h ( z y ) ≈ z x h(z_{y})\\approx z_{x} ."},{"type":"para","text":"Unlike generative architectures, JEPA operates entirely in representation space, encouraging the embeddings to capture high-level structure rather than input-level detail. In our setting ( Section 4.1 ), the underlying object is a lattice theory at a fixed coupling, and the two signals are its bare coupling g g and a flow’s parameter vector θ \\theta trained to sample at g g ."},{"type":"para","text":"Without additional regularization, both encoders can collapse to a constant mapping, trivially satisfying the alignment objective. The VICReg objective (Bardes et al., 2022 ) prevents this with three terms: an invariance term s = 1 N ‖ h ( z y ) − z x ‖ 2 2 s=\\tfrac{1}{N}\\|h(z_{y})-z_{x}\\|_{2}^{2} that enforces alignment, a variance term that penalizes small variance and hence representational collapse, and a covariance term that penalizes correlations between latent dimensions and thereby encourages the utilization of all available latent dimensions. Full expressions are given in Appendix A.2 ."},{"type":"heading","level":1,"text":"4 Methods"},{"type":"para","text":"We introduce JEPAWG ( Section 4.1 ), our JEPA-based weight generator that maps couplings to flow parameters through a learned latent space, and compare it against three baselines (PCA, AE, VAE), trained without coupling information ( Section A.1 )."},{"type":"heading","level":2,"text":"4.1 JEPAWG"},{"type":"para","text":"We slightly modify the joint-embedding predictive architecture (JEPA) introduced in Section 3.2 , and augment it with a decoder that enables weight generation at unseen couplings. We refer to the resulting model as JEPAWG (JEPA-based Weight Generator), combining JEPA-style representation learning with a hypernetwork-style mapping from coupling space to flow parameters. We use a JEPA backbone, rather than a coupling-conditional regressor g ↦ θ ^ g\\mapsto\\hat{\\theta} trained with reconstruction loss, because JEPA shapes the latent space from both signals jointly: the predictor is trained to recover the weight embedding 𝐳 θ \\mathbf{z}_{\\theta} from the coupling embedding 𝐳 g \\mathbf{z}_{g} , while the variance term prevents either embedding from collapsing to a constant. The latent space is therefore incentivized to align with the coupling structure during training rather than fitted to it post-hoc. We will see that this feature is most helpful when the relevant sub-manifold in the weight space is non-smooth ( Section 5.2 )."},{"type":"heading","level":4,"text":"Architecture."},{"type":"para","text":"The model consists of four components: (i) a weight encoder E θ : ℝ d θ → ℝ d z E_{\\theta}:\\mathbb{R}^{d_{\\theta}}\\to\\mathbb{R}^{d_{z}} (with d θ = 19 , 689 d_{\\theta}=19{,}689 , the number of neural ODE weights; see Appendix B.1 ) mapping canonicalized 2 2 2 See Section 5.1 parameter vectors θ \\theta to weight embeddings 𝐳 θ = E θ ( θ ) \\mathbf{z}_{\\theta}=E_{\\theta}(\\theta) , (ii) a coupling encoder E g : ℝ 2 → ℝ d z E_{g}:\\mathbb{R}^{2}\\to\\mathbb{R}^{d_{z}} mapping couplings g = ( m 2 , λ ) g=(m^{2},\\lambda) to coupling embeddings 𝐳 g = E g ( g ) \\mathbf{z}_{g}=E_{g}(g) , (iii) a predictor P : ℝ d z → ℝ d z P:\\mathbb{R}^{d_{z}}\\to\\mathbb{R}^{d_{z}} mapping coupling embeddings into the weight embedding space, trained so that P ( 𝐳 g ) ≈ 𝐳 θ P(\\mathbf{z}_{g})\\approx\\mathbf{z}_{\\theta} , and (iv) a decoder D θ : ℝ d z → ℝ d θ D_{\\theta}:\\mathbb{R}^{d_{z}}\\to\\mathbb{R}^{d_{\\theta}} mapping predicted weight embeddings back to parameter vectors. At inference, the composition D θ ∘ P ∘ E g : g ↦ θ ^ D_{\\theta}\\circ P\\circ E_{g}:g\\mapsto\\hat{\\theta} defines a hypernetwork from couplings to flow parameters."},{"type":"heading","level":4,"text":"Training."},{"type":"para","text":"The encoders E θ , E g E_{\\theta},E_{g} and predictor P P are trained using the VICReg objective ( 6 ), with the covariance term removed. The reason is that this term deliberately decorrelates latent dimensions and encourages the representation to spread across all d z d_{z} available directions; however, we wish to retain and exploit the low-dimensional structure of the problem—our object of study is a two-parameter family of Boltzmann distributions. This property is probed directly in our experiments, as discussed in Section 5.3 . For completeness, we also evaluate the variant that retains the covariance term (i.e., the full VICReg objective), which we denote JEPAWG ∗ . After training the encoders and the predictor, we freeze ( E θ , E g , P ) (E_{\\theta},E_{g},P) and train the decoder D θ D_{\\theta} . This two-stage procedure ensures that the latent space is shaped entirely by the JEPA objective; training the decoder jointly would allow reconstruction pressure to leak into the weight encoder, encouraging it to retain θ \\theta -specific detail beyond what is predictable from g g . The decoder minimizes"},{"type":"para","text":"on training pairs { ( θ i , g i ) } i = 1 N \\{(\\theta_{i},g_{i})\\}_{i=1}^{N} . Throughout, we set d z = 16 d_{z}=16 , deliberately over-sizing the latent space relative to the two-dimensional coupling manifold so that any concentration onto fewer dimensions is a learned property rather than an architectural constraint."},{"type":"heading","level":4,"text":"Inference."},{"type":"para","text":"Given a new coupling g ⋆ g^{\\star} , we generate flow parameters by applying the map θ ^ ( g ⋆ ) = D θ ( P ( E g ( g ⋆ ) ) ) \\hat{\\theta}(g^{\\star})=D_{\\theta}\\bigl(P(E_{g}(g^{\\star}))\\bigr) . The generated parameters θ ^ ( g ⋆ ) \\hat{\\theta}(g^{\\star}) can then be used to instantiate a flow model at the desired coupling."},{"type":"heading","level":2,"text":"4.2 Baselines"},{"type":"para","text":"We compare against three baselines: weight-space PCA, an autoencoder (AE) and a variational autoencoder (VAE); see Section A.1 . Each method operates on flattened parameter vectors θ ∈ ℝ d θ \\theta\\in\\mathbb{R}^{d_{\\theta}} , standardized per dimension as θ ~ i = ( θ i − θ ¯ i ) / Var ( θ i ) \\tilde{\\theta}_{i}=(\\theta_{i}-\\bar{\\theta}_{i})/\\sqrt{\\mathrm{Var}(\\theta_{i})} . Coupling information enters only at inference, through a post-hoc regressor mapping g = ( m 2 , λ ) g=(m^{2},\\lambda) to the learned embedding, which allows the baseline to act as a conditional generator. All baselines are trained on the same parameter vectors as JEPAWG, isolating the architectural comparison from data-pipeline effects. We report results in Table 1 and discuss the asymmetric effect of canonicalization in Section 5.2 . As an ablation, we also include JEPAWG ∗ , which uses the full VICReg loss with a nonzero covariance term."},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"para","text":"A central question we address is whether a single framework can achieve both competent flow-model generation for unseen couplings and meaningful representation learning whose geometry reflects the underlying physics. We evaluate JEPAWG on both criteria. We focus our analysis on the multi-seed setting (see Appendix B ) where the training data occupy disjoint regions of weight space, to ensure that the representations JEPAWG learns are not merely a result of the specific way of data generation."},{"type":"para","text":"For weight generation , we measure the effective sample size (ESS; ( 18 )) of decoded flows on interpolation and extrapolation points ( Section 5.2 ). For representation geometry , we estimate the intrinsic dimension of the latent space, visualize its structure, and show that the geometry of the parameter map recovers the location of the phase boundary directly, attributing the criticality signal to non-local components of the network ( Section 5.3 )."},{"type":"heading","level":2,"text":"5.1 Data Preparation"},{"type":"para","text":"We use the conditional flow of Gerdes et al. ( 2023 ) as the first step in our data-generation machinery. On a 19 × 19 19\\times 19 grid in the space of couplings ( m 2 , λ ) (m^{2},\\lambda) , we fine-tune the models generated by the conditional flow to have high ESS. We repeat the process for lattices with sizes 6 2 6^{2} to 11 2 11^{2} as described in Appendix B . The network weights have various symmetries, including changes of basis, permutations, and sign flips, which change the parameter vector θ \\theta while leading to the same input-output map. We fix this redundancy with a weight canonicalization procedure: we map each θ \\theta to a unique canonical representative per symmetry orbit (Appendix A.3 ) to facilitate non-ambiguous feature extraction from weights; the importance of this step is quantified in Section 5.2 . More details of the data preparation procedure are described in Appendix B ."},{"type":"heading","level":2,"text":"5.2 Weight Generation"},{"type":"para","text":"In the data preparation stage, we first train two conditional flows with independent random initializations, each covering the full λ \\lambda interval but a disjoint half of the m 2 m^{2} range: seed 1 covers m 2 ∈ [ − 5.1 , − 3.5 ] m^{2}\\in[-5.1,-3.5] and seed 2 covers m 2 ∈ ( − 3.5 , − 1.9 ] m^{2}\\in(-3.5,-1.9] , both over λ ∈ [ 2.85 , 5.65 ] \\lambda\\in[2.85,5.65] . The training set for all methods is assembled by combining the ( g , θ g ) (g,\\theta_{g}) pairs from the two parts. Because the two conditional flows converge to different optima, the weight manifold has a structural discontinuity at the boundary, even though the underlying physics varies continuously. This multi-seed approach thus helps to ensure the robustness of our experiments: JEPAWG does not simply inherit coherent structure from the conditional network. We evaluate on 256 coupling pairs ( m 2 , λ ) (m^{2},\\lambda) uniformly sampled within their respective full training intervals, and 32 coupling pairs in each of eight extrapolation regions, each extending a distance δ = 1.5 \\delta=1.5 beyond the training window along the relevant axis ( Figure 10 ). These eight regions are divided into four edge regions (one of the coupling constants beyond the training window) and four corner regions (both couplings outside the training window); see Figure 10 for an illustration. Table 1 shows that JEPAWG produces the strongest flows in this setting, achieving an interpolation ESS of 0.88 {\\bf 0.88} . JEPAWG ∗ reaches 0.61 0.61 , while the reconstruction-trained baselines (VAE, AE, PCA) lag substantially at 0.30 0.30 – 0.40 0.40 . The gap widens on extrapolation: JEPAWG retains 0.56 0.56 on edges and 0.39 0.39 on corners, where the weakest baselines fall below 0.10 0.10 . The embeddings ( Figure 2 ) offer insights into the structural reasons behind this result: the AE and VAE encoders separate the two seeds into disjoint clusters rather than forming a continuous manifold, while JEPAWG maps both seeds onto a single smooth sheet."},{"type":"para","text":"Recall that canonicalization selects a single, specific representative from each orbit of symmetry-equivalent weights (Section 5.1 , Appendix A.3 ). It is therefore the step that lets us properly account for the weight-space symmetries before attempting to interpret the weights. This comes at a price: collapsing each orbit onto its canonical representative makes the weight map θ ( g ) \\theta(g) vary less smoothly with the couplings, sharpening its discontinuity across coupling space. Table 1 reveals a dichotomy in how the methods respond to canonicalization. Not surprisingly, it hurts all of the non-JEPA methods, which do not cope well with the induced discontinuity: raw-weight VAE, AE, and PCA reach interpolation ESS of 0.910 0.910 , 0.915 0.915 , and 0.749 0.749 , but fall to 0.40 0.40 , 0.33 0.33 , and 0.30 0.30 once canonicalized. For JEPAWG, by contrast, canonicalization can help (JEPAWG: 0.78 → 0.88 0.78\\to 0.88 ; JEPAWG ∗ : 0.41 → 0.61 0.41\\to 0.61 ): its latent space is shaped by the coupling signal rather than by reconstructing the raw weights, so it is not degraded by fixing the symmetry representative. More importantly, this ESS comparison is not the whole story. In both settings, raw and canonicalized weights, only JEPAWG produces an interpretable latent space (intrinsic dimension matching the coupling manifold, smooth extrapolation, and recovery of the critical exponent), whereas the baselines do not, even at their best raw-weight setting. We discuss this in the next subsection. A single-seed variant of this experiment is reported in Appendix C ."},{"type":"heading","level":2,"text":"5.3 Interpretability"},{"type":"heading","level":4,"text":"Intrinsic dimension."},{"type":"para","text":"We estimate the intrinsic dimension of each learned representation using the maximum likelihood estimator (MLE) of Levina & Bickel ( 2004 ) (details in Appendix B.3 ). Table 2 shows that JEPAWG is the only method whose MLE estimate matches the ground truth of d = 2 d{=}2 (the dimension of the coupling space we used), and the only one whose latent variance is captured almost entirely by two principal axes (PCA-95% = 2 {=}2 ). The VAE and AE inflate the intrinsic dimension well beyond the physical ground truth, with MLE estimates of 4.88 4.88 and 3.68 3.68 respectively, indicating that their latent spaces retain structure unrelated to the coupling parameters."},{"type":"heading","level":4,"text":"Latent-space embedding."},{"type":"para","text":"Figure 2 shows the first two principal components of each representation, colored by the Binder cumulant U 4 U_{4} ; see Figure 11 for the full comparison across all five methods and additional colorings by m 2 m^{2} and λ \\lambda . For JEPAWG, PC 1 and PC 2 capture 98.5 % 98.5\\% of the latent variance and form a smooth sheet aligned with the coupling parameters; U 4 U_{4} , which is never seen during training, varies coherently along the same axes. The autoencoder places only 51.0 % 51.0\\% of variance in its leading two components, and neither coupling nor U 4 U_{4} traces a coherent direction; instead, the two training seeds separate into distinct clusters, indicating that the encoder retains seed identity rather than learning a continuous representation of the physics. This concentration of the JEPAWG representation onto two physically relevant axes is consistent with the PCA- 95 % = 2 95\\%{=}2 result in Table 2 . The smoothness of the JEPAWG embedding also holds in the single-seed setting ( Appendix C )."},{"type":"heading","level":4,"text":"Phase transition."},{"type":"para","text":"We study how the parameter map g ↦ θ ( g ) g\\mapsto\\theta(g) varies across coupling space by computing the pullback metric G i j ( g ) = ⟨ ∂ i θ ( g ) , ∂ j θ ( g ) ⟩ G_{ij}(g)=\\langle\\partial_{i}\\theta(g),\\partial_{j}\\theta(g)\\rangle , with ⟨ ⋅ , ⋅ ⟩ \\langle\\cdot,\\cdot\\rangle the Euclidean inner product on the weight space and derivatives estimated by finite differences in ( m 2 , λ ) (m^{2},\\lambda) . The area element A ( g ) = det G ( g ) A(g)=\\sqrt{\\det G(g)} quantifies how fast the learned parameters deform with the bare couplings; a peak is expected near the critical line, where θ ( g ) \\theta(g) must accommodate rapid changes in the physics."},{"type":"para","text":"To identify where this deformation originates within the architecture, we partition the weights θ \\theta into six functional groups (see Section B.1 ): two that act non-locally on configurations (the zero mode scaling θ zero _ mode \\theta_{\\mathrm{zero\\_mode}} and the Fourier scaling θ scale \\theta_{\\mathrm{scale}} ) and four bulk groups ( θ conv \\theta_{\\mathrm{conv}} , θ feat _ super \\theta_{\\mathrm{feat\\_super}} , θ time _ kernel \\theta_{\\mathrm{time\\_kernel}} , θ feat _ map \\theta_{\\mathrm{feat\\_map}} ) that act mostly locally. Computing the area element per group, A c ( g ) = det G c ( g ) A_{c}(g)=\\sqrt{\\det G_{c}(g)} , we find that only the non-local groups develop a sharp ridge tracking the critical line, while the bulk groups remain featureless ( Figure 3 ). Table 3 confirms this quantitatively: θ zero _ mode \\theta_{\\mathrm{zero\\_mode}} and θ scale \\theta_{\\mathrm{scale}} exhibit positive correlations between A c ( g ) A_{c}(g) and all three criticality observables ξ \\xi , | ∇ U 4 | |\\nabla U_{4}| , and χ \\chi . The bulk groups, by contrast, show no positive correlation with any criticality observable; their dominant signal is a strong negative correlation with λ \\lambda ( r ∈ ( − 0.76 , − 0.59 ) r\\in(-0.76,-0.59) ), indicating that they track the quartic coupling rather than proximity to the transition."},{"type":"para","text":"The physics is consistent with this observation: a transition driven by the divergence of the correlation length must be absorbed by the long-range degrees of freedom of the model. Restricted to those non-local groups, the geometry of the parameter map recovers the phase boundary without drawing any samples."},{"type":"heading","level":2,"text":"5.4 Cross-lattice renormalization-group structure"},{"type":"para","text":"The bare couplings g = ( m 2 , λ ) g=(m^{2},\\lambda) carry no absolute physical meaning: the same g g describes different long-distance physics on lattices of different size L L . Relating theories across L L is precisely the content of the renormalization group (RG), and near a continuous transition it is governed by finite-size scaling (FSS). At fixed λ \\lambda and close to the critical line, the relevant scaling field is"},{"type":"para","text":"with a ( λ ) a(\\lambda) a non-universal normalization, and FSS predicts that the physics depends on g g and L L only through the dimensionless combination x = t L 1 / ν x=t\\,L^{1/\\nu} , where ν \\nu is the correlation-length exponent. Two theories at different L L sharing the same x x thus represent, up to finite-size corrections, the same physics. For the two-dimensional ϕ 4 \\phi^{4} theory studied here ν = 1 \\nu=1 (2D Ising universality), giving a ground-truth value against which any learned notion of scale can be checked."},{"type":"para","text":"Requiring x x to be preserved between a source size L src L_{\\text{src}} and a target size L tgt L_{\\text{tgt}} , t src L src 1 / ν = t tgt L tgt 1 / ν t_{\\text{src}}L_{\\text{src}}^{1/\\nu}=t_{\\text{tgt}}L_{\\text{tgt}}^{1/\\nu} , and solving for the partner mass gives"},{"type":"para","text":"Given a set of bare coupling constants on one lattice, ( 4 ) predicts the bare couplings on the other lattice expected to share the same physics."},{"type":"heading","level":4,"text":"Methodology."},{"type":"para","text":"Our working hypothesis is that the JEPA encoder places physically similar theories close together: if the latent geometry has captured the physics, weights that are FSS partners across different lattice sizes should lie nearer in embedding space than weights at unrelated couplings. For a pair ( L src , L tgt ) (L_{\\text{src}},L_{\\text{tgt}}) and a trial exponent ν \\nu , ( 4 ) assigns to each source coupling a partner coupling on the target lattice; we measure the latent distance between the two, using the same-coupling pairing ( m 2 ( L src ) = m 2 ( L tgt ) m^{2}(L_{\\text{src}})=m^{2}(L_{\\text{tgt}}) held fixed) as a baseline. Because the FSS partner itself depends on ν \\nu , the form of the FSS ( 4 ) allows us to read off the value of the critical exponent ν \\nu that describes the data the best. Evaluating ( 4 ) requires the critical line m c 2 ( λ ) m_{c}^{2}(\\lambda) , which we estimate in two different ways—the susceptibility ( χ \\chi ) peak and the Binder-cumulant crossing ( Section D.2 ). These two definitions coincide in the continuum and differ at finite L L only through their distinct numerical systematics, so that the conclusion is robust against different sources of numerical error. For each trial exponent we compute the mean latent distance between partners, d mean ( ν ) = 1 N ∑ i ∥ z i ( src ) − z i ( tgt ) ∥ 2 d_{\\mathrm{mean}}(\\nu)=\\tfrac{1}{N}\\sum_{i}\\lVert z_{i}^{(\\text{src})}-z_{i}^{(\\text{tgt})}\\rVert_{2} , scan ν ∈ [ 0.2 , 3.5 ] \\nu\\in[0.2,3.5] , and read off ν ∗ = arg min ν d mean ( ν ) \\nu^{*}=\\arg\\min_{\\nu}d_{\\mathrm{mean}}(\\nu) , with uncertainties from a bootstrap over the N = 361 N{=}361 couplings. Throughout, both lattices are embedded with the encoder of the larger lattice, which is applied directly to the smaller-lattice weight vectors without retraining ( Section D.3 )."},{"type":"para","text":"Two features of the setup limit the achievable precision, and we do not expect a high-precision determination of ν \\nu . The lattices are small, so finite-size corrections to the leading-order relation ( 3 ) are appreciable; and locating the critical line m c 2 ( λ ) m_{c}^{2}(\\lambda) is itself subject to numerical error, which propagates into the predicted partner couplings. Empirically these effects produce a sizable scatter in the per-pairing minimizers, quantified in the Results below and in Section D.3 . We therefore regard the analysis as a qualitative test of whether the latent geometry is RG-aware, and not as a precise measurement of the critical exponent."},{"type":"heading","level":4,"text":"Results."},{"type":"para","text":"In every pairing and for every critical-line estimator, FSS matching yields smaller latent distances than the same-coupling baseline (e.g. d FSS ∗ / d same ≈ 0.72 d^{*}_{\\mathrm{FSS}}/d_{\\text{same}}\\approx 0.72 for 6 → 11 6\\to 11 ); this confirms, as a basic consistency check, that the embedding responds to the scaling variable. More informatively, the minimizing exponent brackets the Ising value across the fifteen pairings on L ∈ { 6 , … , 11 } L\\in\\{6,\\dots,11\\} : with the χ \\chi -peak critical line the per-pairing ν ∗ \\nu^{*} has an average of 0.98 0.98 , and the most widely separated pairing, 6 → 11 6\\to 11 , sits at ν ∗ = 1.00 − 0.06 + 0.08 \\nu^{*}=1.00_{-0.06}^{+0.08} ( Figure 4(b) ). The individual estimates carry sizable uncertainty and depend on the critical-line definition—the averages over the two estimators via χ \\chi -peak and U 4 U_{4} crossing respectively span 0.55 − 1.53 0.55-1.53 (median 0.98 0.98 ) and 0.62 − 2.34 0.62-2.34 (median 1.21 ) 1.21) —so we regard the outcome as broadly consistent with ν = 1 \\nu=1 rather than a sharp measurement. The exponent is nonetheless stable across scale gaps ( Figure 4(a) ), and the latent-distance minima are clear and well separated from the same-coupling baseline ( Figure 8 ). Taken together, these results indicate that the learned latent space captures the qualitative structure of the renormalization group, organizing theories by their scaling variable across lattice sizes in a way that is consistent with the exact ν = 1 \\nu=1 of the 2D Ising class. Full numerical results for all pairings and estimators are collected in Section D.3 ."},{"type":"para","text":"As an ablation (Appendix G ) we perform a global label shuffle of the coupling grid, under which ( m 2 , λ ) (m^{2},\\lambda) are no longer decodable from the latent and the FSS minimum disappears. This provides further supporting evidence for the correctness of our methodology."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"We introduced JEPAWG, a JEPA-based weight generator that maps lattice couplings directly to flow parameters via a learned latent space, and used it to study what neural networks learn about field theories from their weights alone. On 2D ϕ 4 \\phi^{4} theory, the JEPAWG weight embedding recovers the intrinsic dimensionality of the coupling manifold. Beyond this, the geometry of the trained parameters exposes physical structure: the pullback metric exhibits a ridge that traces the critical line of the theory, attributed to the network parameters that are responsible for non-local transformations on the generated configurations. Across lattice sizes, the encoder organizes network weights according to the finite-size scaling variable predicted by the renormalization group. More quantitatively, the scaling exponent that yields the most coherent latent geometry is broadly consistent with the exact ν = 1 \\nu=1 prediction of the Ising model. Notably, none of these properties are supervised by physical observables; they emerge from the geometry of weight space and that of the JEPA embedding space alone. As a generator, JEPAWG also produces functional flows at unseen couplings in a multi-seed setting where PCA, AE, and VAE baselines collapse."},{"type":"para","text":"From this work, the picture that emerges is a simple one: the geometry of trained weights reflects the geometry of the physical theory, and it concentrates on the components of the network where the physics actually lives. These results validate our JEPAWG framework and support our broader claim on the effectiveness of physics as natural synthetic data for developing interpretable AI."},{"type":"para","text":"Limitations and Future Directions. Our experiments are confined to a single 2D theory at small lattices, and our interpretability claims rest on agreement with one universality class. We see two natural extensions that we are currently pursuing. The first is scale : our experiments are confined to small lattices ( L ≤ 11 L\\leq 11 ) due to the cost of training the underlying flows, and extending the analysis to larger lattices will further validate our results and in particular sharpen the weight-space RG results. The second is diversity : applying JEPAWG to other quantum field theories, or other scientific systems would test whether the framework remains effective in richer physical settings. In favorable cases, the same approach could even uncover qualitative features of systems that are not yet fully understood theoretically, turning JEPAWG into an experimental tool."},{"type":"heading","level":1,"text":"Impact Statement"},{"type":"para","text":"This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Lattice field theory (LFT) discretizes a quantum field onto a finite lattice, providing a first-principles approach to non-perturbative phenomena across high-energy and condensed matter physics. The action S g ( ϕ ) S_{g}(\\\\phi) is parametrized by bare coupling constants g g , and physics is extracted by sampling the Boltzmann distribution p g ( ϕ ) ∝ e − S g ( ϕ ) p_{g}(\\\\phi)\\\\propto e^{-S_{g}(\\\\phi)} numerically.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Flow-based generative models have emerged as samplers for these distributions (Albergo et al., 2021 ; Kanwar, 2024 ; Cheng & Stratikopoulou, 2026 ) , learning an invertible map from a tractable base to p g ( ϕ ) p_{g}(\\\\phi) and producing independent samples that sidestep MCMC autocorrelation. A trained flow at coupling g g is parametrized by a vector θ g \\\\theta_{g} , and we expect these parameters to carry information about the physical structure of the theory. For instance, near a phase transition, where p g p_{g} varies rapidly with g g , θ g \\\\theta_{g} must respond accordingly. This raises two complementary questions. As a diagnostic —our primary focus— does the geometry of these parameters in the weight space recover physical structure that the bare couplings themselves obscure? As a generator , instrumentally, can θ g \\\\theta_{g} be predicted from g g , amortizing training across couplings and extrapolating to unseen ones?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Lattice field theory is a uniquely well-suited testbed for these questions of weight interpretation and weight generation , because it supplies the kind of ground truth that interpretability research typically lacks. Understanding what neural networks have learned is a central open problem (Räuker et al., 2023 ; Sharkey et al., 2025 ) , but progress is often bottlenecked by the absence of ground truth: for typical ML tasks, we cannot independently verify what structure a learned representation should encode. Physical theories supply this ground truth. The target distributions p g ( ϕ ) p_{g}(\\\\phi) are known analytically, the relevant coupling space is low-dimensional and naturally smooth, and the renormalization group makes quantitative predictions about its phase structure ( Section 3.1 ). A representation of weight space can therefore be checked against falsifiable physical predictions, not only against qualitative intuitions. Physics, in this sense, acts as a supplier of natural synthetic data: rich enough to be non-trivial, structured enough to be interpretable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To this end, we introduce JEPAWG , a JEPA-based Weight Generator (LeCun, 2022 ; Assran et al., 2023 ) that maps couplings to flow parameters through a learned latent space. The same architecture that enables weight generation at unseen couplings also yields a latent representation whose geometry can be probed for physical structure. We apply JEPAWG to two-dimensional ϕ 4 \\\\phi^{4} theory and make four contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recovery of physical dimensionality: the JEPAWG latent space matches the 2D coupling manifold (MLE intrinsic dimension 2.01 ± 0.06 2.01\\\\pm 0.06 ; 98.5 % 98.5\\\\% of variance in two principal axes), whereas AE and VAE baselines inflate this to 3.68 3.68 and 4.88 4.88 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase transition from weights alone: the pullback metric on θ ( g ) \\\\theta(g) exhibits a ridge tracking the critical line of ϕ 4 \\\\phi^{4} theory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Scale transformation from latent geometry: comparing weights of flows trained at different lattice sizes through the learned embedding, and requiring physically similar theories to lie nearby in the embedding space, we find that the latent geometry is organized by the finite-size scaling variable, with the optimal exponent broadly consistent with the 2D Ising value ν = 1 \\\\nu=1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Generation: JEPAWG produces functional flows at unseen couplings, reaching an effective sample size (ESS) of 0.88 0.88 on multi-seed interpolation, where PCA, AE, and VAE baselines reach only 0.30 0.30 – 0.40 0.40 , and it remains robust under the weight canonicalization procedure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Joint-embedding predictive architectures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Joint-embedding predictive architectures (JEPA) learn representations by predicting in latent space rather than reconstructing the input (LeCun, 2022 ) . I-JEPA (Assran et al., 2023 ) and V-JEPA (Bardes et al., 2024 ) demonstrated this for images and video, while VL-JEPA (Chen et al., 2025 ) extended the framework to vision-language tasks, attaching a lightweight decoder to translate predicted embeddings back into text. To our knowledge, JEPA has not previously been applied to neural network weight spaces.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Hypernetworks and weight generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A hypernetwork (Schmidhuber, 1992 ; Ha et al., 2017 ) is a network that produces the weights of another, conditioned on a task or system descriptor. This paradigm has been applied to continual learning (von Oswald et al., 2020 ) and approximate Bayesian inference over weights (Krueger et al., 2017 ) ; see Chauhan et al. ( 2024 ) for a survey. Our work instantiates this paradigm for flow models simulating quantum field theories: a learned encoder–predictor–decoder pipeline outputs per-coupling weights, enabling extrapolation to unseen couplings without retraining.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Flow-based sampling for lattice field theory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Normalizing flows have been widely applied as samplers for lattice field theories; see e.g. Cheng & Stratikopoulou ( 2026 ); Kanwar ( 2024 ) for reviews. Closest to our setting, Gerdes et al. ( 2023 ) train a single coupling-conditional flow that generalizes across couplings by conditioning each layer on g g while keeping network weights shared. Instead of conditioning a fixed network, our method treats trained flow parameters as data and learns a latent embedding that decodes to flow parameters at unseen couplings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Background\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Scalar Quantum Field Theory on the Lattice\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We work with the two-dimensional real scalar ϕ 4 \\\\phi^{4} theory on a periodic square lattice V L ≅ ( ℤ / L ℤ ) 2 V_{L}\\\\cong(\\\\mathbb{Z}/L\\\\mathbb{Z})^{2} , with site variables ϕ x ∈ ℝ \\\\phi_{x}\\\\in\\\\mathbb{R} and action\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where Δ \\\\Delta is the discrete Laplacian and g = ( m 2 , λ ) g=(m^{2},\\\\lambda) collects the two bare couplings 1 1 1 These are called bare couplings because they appear directly in the action, before quantum effects are taken into account; the physically meaningful indicators are the observables whose mapping to bare couplings is typically very complex and depends on the lattice spacing. . Physical observables are expectations under the Boltzmann measure p g ( ϕ ) ∝ e − S g ( ϕ ) p_{g}(\\\\phi)\\\\propto e^{-S_{g}(\\\\phi)} . Despite its simplicity, this theory exhibits a non-trivial phase structure and therefore forms an excellent testbed for studying what neural samplers learn, since the known phase structure provides a falsifiable ground truth against which learned representations can be checked.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Phase structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Near a phase transition, the physics becomes acutely sensitive to the couplings: a small change in g g produces a qualitative change in the theory, which makes critical regions among the hardest to simulate. At the same time, the long-distance physics there becomes universal : it is determined only by symmetry and dimensionality, independent of microscopic details. Together, these properties make critical points a natural place to test whether a learned representation has captured the physical structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The action ( 1 ) is invariant under the global ℤ 2 \\\\mathbb{Z}_{2} symmetry ϕ ↦ − ϕ \\\\phi\\\\mapsto-\\\\phi . For λ > 0 \\\\lambda>0 , the theory has two phases separated by a critical line m 2 = m c 2 ( λ ) m^{2}=m^{2}_{c}(\\\\lambda) . When m 2 > m c 2 ( λ ) m^{2}>m^{2}_{c}(\\\\lambda) , the lattice magnetization ϕ ¯ = L − 2 ∑ x ϕ x \\\\bar{\\\\phi}=L^{-2}\\\\sum_{x}\\\\phi_{x} averages to zero (the paramagnetic phase). When m 2 m^{2} is sufficiently negative, the potential acquires two degenerate minima; the system settles into one of them, breaking ℤ 2 \\\\mathbb{Z}_{2} spontaneously, and ⟨ | ϕ ¯ | ⟩ ≠ 0 \\\\langle|\\\\bar{\\\\phi}|\\\\rangle\\\\neq 0 (the ferromagnetic phase). The coupling space ( m 2 , λ ) (m^{2},\\\\lambda) is two-dimensional and the two phases are separated by a one-dimensional critical line; any representation that captures the physics should reflect this low-dimensional structure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Observables.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The transition is conventionally diagnosed by moments of the magnetization. The susceptibility χ ( g ) = L 2 ( ⟨ ϕ ¯ 2 ⟩ − ⟨ | ϕ ¯ | ⟩ 2 ) \\\\chi(g)=L^{2}\\\\bigl(\\\\langle\\\\bar{\\\\phi}^{2}\\\\rangle-\\\\langle|\\\\bar{\\\\phi}|\\\\rangle^{2}\\\\bigr) peaks along the critical line, while the Binder cumulant U 4 ( g ) = 1 − ⟨ ϕ ¯ 4 ⟩ / ( 3 ⟨ ϕ ¯ 2 ⟩ 2 ) U_{4}(g)=1-\\\\langle\\\\bar{\\\\phi}^{4}\\\\rangle/(3\\\\langle\\\\bar{\\\\phi}^{2}\\\\rangle^{2}) interpolates between 0 in the paramagnetic phase and 2 / 3 2/3 in the ferromagnetic one, with curves at different L L crossing at the critical point. Two-dimensional ϕ 4 \\\\phi^{4} theory belongs to the 2D Ising universality class, with the correlation-length critical exponent ν = 1 \\\\nu=1 . A learned representation that has captured the physical content of the theory should therefore organize the coupling plane by RG-invariant combinations rather than by g g itself, a property we test in Section 5.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Normalizing flows for lattice field theory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Estimating these observables requires drawing samples from the Boltzmann measure p g ( ϕ ) ∝ e − S g ( ϕ ) p_{g}(\\\\phi)\\\\propto e^{-S_{g}(\\\\phi)} , a task that becomes increasingly difficult near the critical line, where local Monte Carlo updates suffer from critical slowing down. In recent years, flow-based methods have been introduced as successful samplers of lattice field theories (Albergo et al., 2019 ) , offering the promise of mitigating critical slowing down. A normalizing flow (Tabak & Vanden-Eijnden, 2010 ; Papamakarios et al., 2021 ; Kobyzev et al., 2020 ) learns an invertible map f θ : ℝ L 2 → ℝ L 2 f_{\\\\theta}:\\\\mathbb{R}^{L^{2}}\\\\to\\\\mathbb{R}^{L^{2}} that transforms a tractable base distribution ρ \\\\rho (typically a standard Gaussian) into an approximation of the target Boltzmann measure p g ( ϕ ) p_{g}(\\\\phi) . The map is trained by minimizing the reverse KL divergence between the pushforward of ρ \\\\rho under f θ f_{\\\\theta} and p g p_{g} , using only the known action S g S_{g} and samples from the flow itself. A conditional flow architecture (Gerdes et al., 2023 ) extends this by making the parameters depend smoothly on the couplings g = ( m 2 , λ ) g=(m^{2},\\\\lambda) , so that a single model produces a sampler for each theory in a continuous region of coupling space.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Joint-embedding predictive architectures\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A joint-embedding predictive architecture (JEPA) (Assran et al., 2023 ; LeCun, 2022 ) learns representations by predicting the embedding of one signal from the embedding of a related one, rather than reconstructing the input directly. Given two such signals, x x and y y , that describe the same underlying object, encoders f x f_{x} and f y f_{y} produce embeddings z x = f x ( x ) z_{x}=f_{x}(x) , z y = f y ( y ) z_{y}=f_{y}(y) and a predictor h h maps between them. The encoders and the predictor are trained jointly, so that the embeddings and the predictive map between them are shaped together by a single objective enforcing h ( z y ) ≈ z x h(z_{y})\\\\approx z_{x} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike generative architectures, JEPA operates entirely in representation space, encouraging the embeddings to capture high-level structure rather than input-level detail. In our setting ( Section 4.1 ), the underlying object is a lattice theory at a fixed coupling, and the two signals are its bare coupling g g and a flow’s parameter vector θ \\\\theta trained to sample at g g .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Without additional regularization, both encoders can collapse to a constant mapping, trivially satisfying the alignment objective. The VICReg objective (Bardes et al., 2022 ) prevents this with three terms: an invariance term s = 1 N ‖ h ( z y ) − z x ‖ 2 2 s=\\\\tfrac{1}{N}\\\\|h(z_{y})-z_{x}\\\\|_{2}^{2} that enforces alignment, a variance term that penalizes small variance and hence representational collapse, and a covariance term that penalizes correlations between latent dimensions and thereby encourages the utilization of all available latent dimensions. Full expressions are given in Appendix A.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Methods\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce JEPAWG ( Section 4.1 ), our JEPA-based weight generator that maps couplings to flow parameters through a learned latent space, and compare it against three baselines (PCA, AE, VAE), trained without coupling information ( Section A.1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 JEPAWG\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We slightly modify the joint-embedding predictive architecture (JEPA) introduced in Section 3.2 , and augment it with a decoder that enables weight generation at unseen couplings. We refer to the resulting model as JEPAWG (JEPA-based Weight Generator), combining JEPA-style representation learning with a hypernetwork-style mapping from coupling space to flow parameters. We use a JEPA backbone, rather than a coupling-conditional regressor g ↦ θ ^ g\\\\mapsto\\\\hat{\\\\theta} trained with reconstruction loss, because JEPA shapes the latent space from both signals jointly: the predictor is trained to recover the weight embedding 𝐳 θ \\\\mathbf{z}_{\\\\theta} from the coupling embedding 𝐳 g \\\\mathbf{z}_{g} , while the variance term prevents either embedding from collapsing to a constant. The latent space is therefore incentivized to align with the coupling structure during training rather than fitted to it post-hoc. We will see that this feature is most helpful when the relevant sub-manifold in the weight space is non-smooth ( Section 5.2 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Architecture.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model consists of four components: (i) a weight encoder E θ : ℝ d θ → ℝ d z E_{\\\\theta}:\\\\mathbb{R}^{d_{\\\\theta}}\\\\to\\\\mathbb{R}^{d_{z}} (with d θ = 19 , 689 d_{\\\\theta}=19{,}689 , the number of neural ODE weights; see Appendix B.1 ) mapping canonicalized 2 2 2 See Section 5.1 parameter vectors θ \\\\theta to weight embeddings 𝐳 θ = E θ ( θ ) \\\\mathbf{z}_{\\\\theta}=E_{\\\\theta}(\\\\theta) , (ii) a coupling encoder E g : ℝ 2 → ℝ d z E_{g}:\\\\mathbb{R}^{2}\\\\to\\\\mathbb{R}^{d_{z}} mapping couplings g = ( m 2 , λ ) g=(m^{2},\\\\lambda) to coupling embeddings 𝐳 g = E g ( g ) \\\\mathbf{z}_{g}=E_{g}(g) , (iii) a predictor P : ℝ d z → ℝ d z P:\\\\mathbb{R}^{d_{z}}\\\\to\\\\mathbb{R}^{d_{z}} mapping coupling embeddings into the weight embedding space, trained so that P ( 𝐳 g ) ≈ 𝐳 θ P(\\\\mathbf{z}_{g})\\\\approx\\\\mathbf{z}_{\\\\theta} , and (iv) a decoder D θ : ℝ d z → ℝ d θ D_{\\\\theta}:\\\\mathbb{R}^{d_{z}}\\\\to\\\\mathbb{R}^{d_{\\\\theta}} mapping predicted weight embeddings back to parameter vectors. At inference, the composition D θ ∘ P ∘ E g : g ↦ θ ^ D_{\\\\theta}\\\\circ P\\\\circ E_{g}:g\\\\mapsto\\\\hat{\\\\theta} defines a hypernetwork from couplings to flow parameters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The encoders E θ , E g E_{\\\\theta},E_{g} and predictor P P are trained using the VICReg objective ( 6 ), with the covariance term removed. The reason is that this term deliberately decorrelates latent dimensions and encourages the representation to spread across all d z d_{z} available directions; however, we wish to retain and exploit the low-dimensional structure of the problem—our object of study is a two-parameter family of Boltzmann distributions. This property is probed directly in our experiments, as discussed in Section 5.3 . For completeness, we also evaluate the variant that retains the covariance term (i.e., the full VICReg objective), which we denote JEPAWG ∗ . After training the encoders and the predictor, we freeze ( E θ , E g , P ) (E_{\\\\theta},E_{g},P) and train the decoder D θ D_{\\\\theta} . This two-stage procedure ensures that the latent space is shaped entirely by the JEPA objective; training the decoder jointly would allow reconstruction pressure to leak into the weight encoder, encouraging it to retain θ \\\\theta -specific detail beyond what is predictable from g g . The decoder minimizes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"on training pairs { ( θ i , g i ) } i = 1 N \\\\{(\\\\theta_{i},g_{i})\\\\}_{i=1}^{N} . Throughout, we set d z = 16 d_{z}=16 , deliberately over-sizing the latent space relative to the two-dimensional coupling manifold so that any concentration onto fewer dimensions is a learned property rather than an architectural constraint.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a new coupling g ⋆ g^{\\\\star} , we generate flow parameters by applying the map θ ^ ( g ⋆ ) = D θ ( P ( E g ( g ⋆ ) ) ) \\\\hat{\\\\theta}(g^{\\\\star})=D_{\\\\theta}\\\\bigl(P(E_{g}(g^{\\\\star}))\\\\bigr) . The generated parameters θ ^ ( g ⋆ ) \\\\hat{\\\\theta}(g^{\\\\star}) can then be used to instantiate a flow model at the desired coupling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Baselines\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare against three baselines: weight-space PCA, an autoencoder (AE) and a variational autoencoder (VAE); see Section A.1 . Each method operates on flattened parameter vectors θ ∈ ℝ d θ \\\\theta\\\\in\\\\mathbb{R}^{d_{\\\\theta}} , standardized per dimension as θ ~ i = ( θ i − θ ¯ i ) / Var ( θ i ) \\\\tilde{\\\\theta}_{i}=(\\\\theta_{i}-\\\\bar{\\\\theta}_{i})/\\\\sqrt{\\\\mathrm{Var}(\\\\theta_{i})} . Coupling information enters only at inference, through a post-hoc regressor mapping g = ( m 2 , λ ) g=(m^{2},\\\\lambda) to the learned embedding, which allows the baseline to act as a conditional generator. All baselines are trained on the same parameter vectors as JEPAWG, isolating the architectural comparison from data-pipeline effects. We report results in Table 1 and discuss the asymmetric effect of canonicalization in Section 5.2 . As an ablation, we also include JEPAWG ∗ , which uses the full VICReg loss with a nonzero covariance term.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A central question we address is whether a single framework can achieve both competent flow-model generation for unseen couplings and meaningful representation learning whose geometry reflects the underlying physics. We evaluate JEPAWG on both criteria. We focus our analysis on the multi-seed setting (see Appendix B ) where the training data occupy disjoint regions of weight space, to ensure that the representations JEPAWG learns are not merely a result of the specific way of data generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For weight generation , we measure the effective sample size (ESS; ( 18 )) of decoded flows on interpolation and extrapolation points ( Section 5.2 ). For representation geometry , we estimate the intrinsic dimension of the latent space, visualize its structure, and show that the geometry of the parameter map recovers the location of the phase boundary directly, attributing the criticality signal to non-local components of the network ( Section 5.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Data Preparation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the conditional flow of Gerdes et al. ( 2023 ) as the first step in our data-generation machinery. On a 19 × 19 19\\\\times 19 grid in the space of couplings ( m 2 , λ ) (m^{2},\\\\lambda) , we fine-tune the models generated by the conditional flow to have high ESS. We repeat the process for lattices with sizes 6 2 6^{2} to 11 2 11^{2} as described in Appendix B . The network weights have various symmetries, including changes of basis, permutations, and sign flips, which change the parameter vector θ \\\\theta while leading to the same input-output map. We fix this redundancy with a weight canonicalization procedure: we map each θ \\\\theta to a unique canonical representative per symmetry orbit (Appendix A.3 ) to facilitate non-ambiguous feature extraction from weights; the importance of this step is quantified in Section 5.2 . More details of the data preparation procedure are described in Appendix B .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Weight Generation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the data preparation stage, we first train two conditional flows with independent random initializations, each covering the full λ \\\\lambda interval but a disjoint half of the m 2 m^{2} range: seed 1 covers m 2 ∈ [ − 5.1 , − 3.5 ] m^{2}\\\\in[-5.1,-3.5] and seed 2 covers m 2 ∈ ( − 3.5 , − 1.9 ] m^{2}\\\\in(-3.5,-1.9] , both over λ ∈ [ 2.85 , 5.65 ] \\\\lambda\\\\in[2.85,5.65] . The training set for all methods is assembled by combining the ( g , θ g ) (g,\\\\theta_{g}) pairs from the two parts. Because the two conditional flows converge to different optima, the weight manifold has a structural discontinuity at the boundary, even though the underlying physics varies continuously. This multi-seed approach thus helps to ensure the robustness of our experiments: JEPAWG does not simply inherit coherent structure from the conditional network. We evaluate on 256 coupling pairs ( m 2 , λ ) (m^{2},\\\\lambda) uniformly sampled within their respective full training intervals, and 32 coupling pairs in each of eight extrapolation regions, each extending a distance δ = 1.5 \\\\delta=1.5 beyond the training window along the relevant axis ( Figure 10 ). These eight regions are divided into four edge regions (one of the coupling constants beyond the training window) and four corner regions (both couplings outside the training window); see Figure 10 for an illustration. Table 1 shows that JEPAWG produces the strongest flows in this setting, achieving an interpolation ESS of 0.88 {\\\\bf 0.88} . JEPAWG ∗ reaches 0.61 0.61 , while the reconstruction-trained baselines (VAE, AE, PCA) lag substantially at 0.30 0.30 – 0.40 0.40 . The gap widens on extrapolation: JEPAWG retains 0.56 0.56 on edges and 0.39 0.39 on corners, where the weakest baselines fall below 0.10 0.10 . The embeddings ( Figure 2 ) offer insights into the structural reasons behind this result: the AE and VAE encoders separate the two seeds into disjoint clusters rather than forming a continuous manifold, while JEPAWG maps both seeds onto a single smooth sheet.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recall that canonicalization selects a single, specific representative from each orbit of symmetry-equivalent weights (Section 5.1 , Appendix A.3 ). It is therefore the step that lets us properly account for the weight-space symmetries before attempting to interpret the weights. This comes at a price: collapsing each orbit onto its canonical representative makes the weight map θ ( g ) \\\\theta(g) vary less smoothly with the couplings, sharpening its discontinuity across coupling space. Table 1 reveals a dichotomy in how the methods respond to canonicalization. Not surprisingly, it hurts all of the non-JEPA methods, which do not cope well with the induced discontinuity: raw-weight VAE, AE, and PCA reach interpolation ESS of 0.910 0.910 , 0.915 0.915 , and 0.749 0.749 , but fall to 0.40 0.40 , 0.33 0.33 , and 0.30 0.30 once canonicalized. For JEPAWG, by contrast, canonicalization can help (JEPAWG: 0.78 → 0.88 0.78\\\\to 0.88 ; JEPAWG ∗ : 0.41 → 0.61 0.41\\\\to 0.61 ): its latent space is shaped by the coupling signal rather than by reconstructing the raw weights, so it is not degraded by fixing the symmetry representative. More importantly, this ESS comparison is not the whole story. In both settings, raw and canonicalized weights, only JEPAWG produces an interpretable latent space (intrinsic dimension matching the coupling manifold, smooth extrapolation, and recovery of the critical exponent), whereas the baselines do not, even at their best raw-weight setting. We discuss this in the next subsection. A single-seed variant of this experiment is reported in Appendix C .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Interpretability\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Intrinsic dimension.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We estimate the intrinsic dimension of each learned representation using the maximum likelihood estimator (MLE) of Levina & Bickel ( 2004 ) (details in Appendix B.3 ). Table 2 shows that JEPAWG is the only method whose MLE estimate matches the ground truth of d = 2 d{=}2 (the dimension of the coupling space we used), and the only one whose latent variance is captured almost entirely by two principal axes (PCA-95% = 2 {=}2 ). The VAE and AE inflate the intrinsic dimension well beyond the physical ground truth, with MLE estimates of 4.88 4.88 and 3.68 3.68 respectively, indicating that their latent spaces retain structure unrelated to the coupling parameters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Latent-space embedding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 shows the first two principal components of each representation, colored by the Binder cumulant U 4 U_{4} ; see Figure 11 for the full comparison across all five methods and additional colorings by m 2 m^{2} and λ \\\\lambda . For JEPAWG, PC 1 and PC 2 capture 98.5 % 98.5\\\\% of the latent variance and form a smooth sheet aligned with the coupling parameters; U 4 U_{4} , which is never seen during training, varies coherently along the same axes. The autoencoder places only 51.0 % 51.0\\\\% of variance in its leading two components, and neither coupling nor U 4 U_{4} traces a coherent direction; instead, the two training seeds separate into distinct clusters, indicating that the encoder retains seed identity rather than learning a continuous representation of the physics. This concentration of the JEPAWG representation onto two physically relevant axes is consistent with the PCA- 95 % = 2 95\\\\%{=}2 result in Table 2 . The smoothness of the JEPAWG embedding also holds in the single-seed setting ( Appendix C ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Phase transition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We study how the parameter map g ↦ θ ( g ) g\\\\mapsto\\\\theta(g) varies across coupling space by computing the pullback metric G i j ( g ) = ⟨ ∂ i θ ( g ) , ∂ j θ ( g ) ⟩ G_{ij}(g)=\\\\langle\\\\partial_{i}\\\\theta(g),\\\\partial_{j}\\\\theta(g)\\\\rangle , with ⟨ ⋅ , ⋅ ⟩ \\\\langle\\\\cdot,\\\\cdot\\\\rangle the Euclidean inner product on the weight space and derivatives estimated by finite differences in ( m 2 , λ ) (m^{2},\\\\lambda) . The area element A ( g ) = det G ( g ) A(g)=\\\\sqrt{\\\\det G(g)} quantifies how fast the learned parameters deform with the bare couplings; a peak is expected near the critical line, where θ ( g ) \\\\theta(g) must accommodate rapid changes in the physics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To identify where this deformation originates within the architecture, we partition the weights θ \\\\theta into six functional groups (see Section B.1 ): two that act non-locally on configurations (the zero mode scaling θ zero _ mode \\\\theta_{\\\\mathrm{zero\\\\_mode}} and the Fourier scaling θ scale \\\\theta_{\\\\mathrm{scale}} ) and four bulk groups ( θ conv \\\\theta_{\\\\mathrm{conv}} , θ feat _ super \\\\theta_{\\\\mathrm{feat\\\\_super}} , θ time _ kernel \\\\theta_{\\\\mathrm{time\\\\_kernel}} , θ feat _ map \\\\theta_{\\\\mathrm{feat\\\\_map}} ) that act mostly locally. Computing the area element per group, A c ( g ) = det G c ( g ) A_{c}(g)=\\\\sqrt{\\\\det G_{c}(g)} , we find that only the non-local groups develop a sharp ridge tracking the critical line, while the bulk groups remain featureless ( Figure 3 ). Table 3 confirms this quantitatively: θ zero _ mode \\\\theta_{\\\\mathrm{zero\\\\_mode}} and θ scale \\\\theta_{\\\\mathrm{scale}} exhibit positive correlations between A c ( g ) A_{c}(g) and all three criticality observables ξ \\\\xi , | ∇ U 4 | |\\\\nabla U_{4}| , and χ \\\\chi . The bulk groups, by contrast, show no positive correlation with any criticality observable; their dominant signal is a strong negative correlation with λ \\\\lambda ( r ∈ ( − 0.76 , − 0.59 ) r\\\\in(-0.76,-0.59) ), indicating that they track the quartic coupling rather than proximity to the transition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The physics is consistent with this observation: a transition driven by the divergence of the correlation length must be absorbed by the long-range degrees of freedom of the model. Restricted to those non-local groups, the geometry of the parameter map recovers the phase boundary without drawing any samples.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Cross-lattice renormalization-group structure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The bare couplings g = ( m 2 , λ ) g=(m^{2},\\\\lambda) carry no absolute physical meaning: the same g g describes different long-distance physics on lattices of different size L L . Relating theories across L L is precisely the content of the renormalization group (RG), and near a continuous transition it is governed by finite-size scaling (FSS). At fixed λ \\\\lambda and close to the critical line, the relevant scaling field is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with a ( λ ) a(\\\\lambda) a non-universal normalization, and FSS predicts that the physics depends on g g and L L only through the dimensionless combination x = t L 1 / ν x=t\\\\,L^{1/\\\\nu} , where ν \\\\nu is the correlation-length exponent. Two theories at different L L sharing the same x x thus represent, up to finite-size corrections, the same physics. For the two-dimensional ϕ 4 \\\\phi^{4} theory studied here ν = 1 \\\\nu=1 (2D Ising universality), giving a ground-truth value against which any learned notion of scale can be checked.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Requiring x x to be preserved between a source size L src L_{\\\\text{src}} and a target size L tgt L_{\\\\text{tgt}} , t src L src 1 / ν = t tgt L tgt 1 / ν t_{\\\\text{src}}L_{\\\\text{src}}^{1/\\\\nu}=t_{\\\\text{tgt}}L_{\\\\text{tgt}}^{1/\\\\nu} , and solving for the partner mass gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a set of bare coupling constants on one lattice, ( 4 ) predicts the bare couplings on the other lattice expected to share the same physics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Methodology.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our working hypothesis is that the JEPA encoder places physically similar theories close together: if the latent geometry has captured the physics, weights that are FSS partners across different lattice sizes should lie nearer in embedding space than weights at unrelated couplings. For a pair ( L src , L tgt ) (L_{\\\\text{src}},L_{\\\\text{tgt}}) and a trial exponent ν \\\\nu , ( 4 ) assigns to each source coupling a partner coupling on the target lattice; we measure the latent distance between the two, using the same-coupling pairing ( m 2 ( L src ) = m 2 ( L tgt ) m^{2}(L_{\\\\text{src}})=m^{2}(L_{\\\\text{tgt}}) held fixed) as a baseline. Because the FSS partner itself depends on ν \\\\nu , the form of the FSS ( 4 ) allows us to read off the value of the critical exponent ν \\\\nu that describes the data the best. Evaluating ( 4 ) requires the critical line m c 2 ( λ ) m_{c}^{2}(\\\\lambda) , which we estimate in two different ways—the susceptibility ( χ \\\\chi ) peak and the Binder-cumulant crossing ( Section D.2 ). These two definitions coincide in the continuum and differ at finite L L only through their distinct numerical systematics, so that the conclusion is robust against different sources of numerical error. For each trial exponent we compute the mean latent distance between partners, d mean ( ν ) = 1 N ∑ i ∥ z i ( src ) − z i ( tgt ) ∥ 2 d_{\\\\mathrm{mean}}(\\\\nu)=\\\\tfrac{1}{N}\\\\sum_{i}\\\\lVert z_{i}^{(\\\\text{src})}-z_{i}^{(\\\\text{tgt})}\\\\rVert_{2} , scan ν ∈ [ 0.2 , 3.5 ] \\\\nu\\\\in[0.2,3.5] , and read off ν ∗ = arg min ν d mean ( ν ) \\\\nu^{*}=\\\\arg\\\\min_{\\\\nu}d_{\\\\mathrm{mean}}(\\\\nu) , with uncertainties from a bootstrap over the N = 361 N{=}361 couplings. Throughout, both lattices are embedded with the encoder of the larger lattice, which is applied directly to the smaller-lattice weight vectors without retraining ( Section D.3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two features of the setup limit the achievable precision, and we do not expect a high-precision determination of ν \\\\nu . The lattices are small, so finite-size corrections to the leading-order relation ( 3 ) are appreciable; and locating the critical line m c 2 ( λ ) m_{c}^{2}(\\\\lambda) is itself subject to numerical error, which propagates into the predicted partner couplings. Empirically these effects produce a sizable scatter in the per-pairing minimizers, quantified in the Results below and in Section D.3 . We therefore regard the analysis as a qualitative test of whether the latent geometry is RG-aware, and not as a precise measurement of the critical exponent.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Results.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In every pairing and for every critical-line estimator, FSS matching yields smaller latent distances than the same-coupling baseline (e.g. d FSS ∗ / d same ≈ 0.72 d^{*}_{\\\\mathrm{FSS}}/d_{\\\\text{same}}\\\\approx 0.72 for 6 → 11 6\\\\to 11 ); this confirms, as a basic consistency check, that the embedding responds to the scaling variable. More informatively, the minimizing exponent brackets the Ising value across the fifteen pairings on L ∈ { 6 , … , 11 } L\\\\in\\\\{6,\\\\dots,11\\\\} : with the χ \\\\chi -peak critical line the per-pairing ν ∗ \\\\nu^{*} has an average of 0.98 0.98 , and the most widely separated pairing, 6 → 11 6\\\\to 11 , sits at ν ∗ = 1.00 − 0.06 + 0.08 \\\\nu^{*}=1.00_{-0.06}^{+0.08} ( Figure 4(b) ). The individual estimates carry sizable uncertainty and depend on the critical-line definition—the averages over the two estimators via χ \\\\chi -peak and U 4 U_{4} crossing respectively span 0.55 − 1.53 0.55-1.53 (median 0.98 0.98 ) and 0.62 − 2.34 0.62-2.34 (median 1.21 ) 1.21) —so we regard the outcome as broadly consistent with ν = 1 \\\\nu=1 rather than a sharp measurement. The exponent is nonetheless stable across scale gaps ( Figure 4(a) ), and the latent-distance minima are clear and well separated from the same-coupling baseline ( Figure 8 ). Taken together, these results indicate that the learned latent space captures the qualitative structure of the renormalization group, organizing theories by their scaling variable across lattice sizes in a way that is consistent with the exact ν = 1 \\\\nu=1 of the 2D Ising class. Full numerical results for all pairings and estimators are collected in Section D.3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As an ablation (Appendix G ) we perform a global label shuffle of the coupling grid, under which ( m 2 , λ ) (m^{2},\\\\lambda) are no longer decodable from the latent and the FSS minimum disappears. This provides further supporting evidence for the correctness of our methodology.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduced JEPAWG, a JEPA-based weight generator that maps lattice couplings directly to flow parameters via a learned latent space, and used it to study what neural networks learn about field theories from their weights alone. On 2D ϕ 4 \\\\phi^{4} theory, the JEPAWG weight embedding recovers the intrinsic dimensionality of the coupling manifold. Beyond this, the geometry of the trained parameters exposes physical structure: the pullback metric exhibits a ridge that traces the critical line of the theory, attributed to the network parameters that are responsible for non-local transformations on the generated configurations. Across lattice sizes, the encoder organizes network weights according to the finite-size scaling variable predicted by the renormalization group. More quantitatively, the scaling exponent that yields the most coherent latent geometry is broadly consistent with the exact ν = 1 \\\\nu=1 prediction of the Ising model. Notably, none of these properties are supervised by physical observables; they emerge from the geometry of weight space and that of the JEPA embedding space alone. As a generator, JEPAWG also produces functional flows at unseen couplings in a multi-seed setting where PCA, AE, and VAE baselines collapse.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From this work, the picture that emerges is a simple one: the geometry of trained weights reflects the geometry of the physical theory, and it concentrates on the components of the network where the physics actually lives. These results validate our JEPAWG framework and support our broader claim on the effectiveness of physics as natural synthetic data for developing interpretable AI.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Limitations and Future Directions. Our experiments are confined to a single 2D theory at small lattices, and our interpretability claims rest on agreement with one universality class. We see two natural extensions that we are currently pursuing. The first is scale : our experiments are confined to small lattices ( L ≤ 11 L\\\\leq 11 ) due to the cost of training the underlying flows, and extending the analysis to larger lattices will further validate our results and in particular sharpen the weight-space RG results. The second is diversity : applying JEPAWG to other quantum field theories, or other scientific systems would test whether the framework remains effective in richer physical settings. In favorable cases, the same approach could even uncover qualitative features of systems that are not yet fully understood theoretically, turning JEPAWG into an experimental tool.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Impact Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.\"\n }\n]","isConversational":false}}},{"rowIdx":488,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07085"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Is Randomness Necessary for Adaptive Data Analysis?"},"authors":{"kind":"list like","value":["Edith Cohen","Haim Kaplan","Yishay Mansour","Shay Sapir","Uri Stemmer"],"string":"[\n \"Edith Cohen\",\n \"Haim Kaplan\",\n \"Yishay Mansour\",\n \"Shay Sapir\",\n \"Uri Stemmer\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07085v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Classical statistical theory for asserting the validity of a hypothesis tells us that the description of the hypothesis should be independent from the data on which it is evaluated. In practice, however, analysts frequently reuse the same dataset to answer multiple questions. Often, this exploration is adaptive: an analyst asks a statistical question, observes the answer, and uses that information to decide what to ask next. While this is how data analysis naturally happens, it makes it incredibly easy to overfit. Without careful intervention, the analyst will quickly find patterns that exist in the specific sample but fail to generalize to the true underlying distribution."},{"type":"para","text":"This problem was formalized by Dwork et al. [ DFH+15a ] in what has come to be known as the Adaptive Data Analysis (ADA) problem. In this problem, a mechanism ℳ \\mathcal{M} holds a dataset S S consisting of n n samples drawn i.i.d. from an unknown distribution 𝒫 \\mathcal{P} over a domain 𝒳 \\mathcal{X} . An analyst iteratively (and adaptively) submits a sequence of k k statistical queries q : 𝒳 → [ 0 , 1 ] q:\\mathcal{X}\\to[0,1] . The mechanism ℳ \\mathcal{M} must answer these queries such that every answer is ε \\varepsilon -accurate (i.e., within ± ε \\pm\\varepsilon ) with respect to the true expectation of the query over 𝒫 \\mathcal{P} . To provide worst-case guarantees, the analyst is assumed to be adversarial, and is often referred to as an attacker . The central question in ADA is establishing the optimal sample complexity: how large can k k be as a function of n n ?"},{"type":"para","text":"This question is relatively well-understood for randomized mechanisms. A deep connection between ADA and various stability notions (such as differential privacy) has shown that randomized mechanisms can safely answer a large number of adaptive queries without overfitting [ DFH+15a , DFH+15b , BNS+16 , RRS+16 , RZ16 , FS17 , FS18 , FRR18 , LS19 , SZ20 , JLN+20 , SL23 , BLA23 ] . Specifically, there are computationally efficient randomized mechanisms supporting k ≈ n 2 k\\approx n^{2} queries, and, without making specific assumptions, no mechanism, even computationally unbounded, can answer k ≫ n 2 k\\gg n^{2} queries. 1 1 1 If one assumes that the domain 𝒳 \\mathcal{X} is not too large, namely that n ≥ polylog | 𝒳 | n\\geq\\operatorname{polylog}|\\mathcal{X}| , then there exist computationally inefficient mechanisms supporting k ≈ exp ( n / polylog | 𝒳 | ) k\\approx\\exp(n/\\operatorname{polylog}|\\mathcal{X}|) queries. See Section 1.3 (related works)."},{"type":"para","text":"However, all known constructions for the ADA problem rely fundamentally on the mechanism’s ability to inject carefully calibrated random noise into its answers (again, unless one makes specific assumptions, under some of which deterministic mechanisms do exist). This reliance raises a natural question about the nature of adaptive data analysis: is randomness strictly necessary for ADA?"},{"type":"para","text":"Resolving this question is fundamental to understanding the algorithmic toolkit required to prevent overfitting in the face of adaptive adversaries. This pursuit mirrors central lines of inquiry in other areas of theoretical computer science where separating the power of deterministic and randomized algorithms has been a long-standing theme. This includes works on communication complexity [ YAO79 , KS92 ] , streaming algorithms [ FM85 , AMS96 , HNO08 , CK16 , KPW21 , KMN+21 , CS24 ] , dynamic algorithms [ BKM+22 , BEF+23 , BBF+26 ] , and distributed computing [ BEN83 , FLP85 , LR81 ] ."},{"type":"heading","level":2,"text":"1.1 Our Contributions"},{"type":"para","text":"The primary contribution of this work is resolving the role of randomness for ADA in the information-theoretic Random Oracle (RO) model. 2 2 2 A random oracle is an infinite public random string 𝒪 \\mathcal{O} , such that the algorithm can read the i i -th bit in 𝒪 \\mathcal{O} , denoted 𝒪 ( i ) \\mathcal{O}(i) , at the cost of one time unit. We assume that each bit of 𝒪 \\mathcal{O} is uniformly distributed and independent of all other bits. Thus, the only way to get any information on 𝒪 ( i ) \\mathcal{O}(i) is to read this bit. The random oracle model is a common assumption in cryptography, dating back to at least [ FS86 ] and formalized in the seminal work of Bellare and Rogaway [ BR93 ] . We establish that to answer a super-linear number of adaptive queries, an algorithm must fundamentally rely on internal randomness. We demonstrate this by proving a strong separation: while deterministic mechanisms can match the performance of randomized ones against computationally bounded analysts, they are provably insufficient against unbounded analysts in the random oracle model."},{"type":"heading","level":4,"text":"Baseline for bounded analysts."},{"type":"para","text":"To put our impossibility result in context, we first formalize a folklore observation (dating back to the initial works on ADA) that randomness is not necessary when the analyst is computationally bounded. Specifically, by extracting a small number of random bits from the input sample and stretching them using cryptographic pseudorandom generators, we recover the same quadratic bound achievable by randomized mechanisms. Thus, when the analyst is bounded, randomness is a convenience rather than a necessity."},{"type":"para","text":"Assuming that pseudorandom generators exist, there is a computationally efficient deterministic mechanism that, given n n i.i.d. samples from an unknown distribution 𝒫 \\mathcal{P} , answers k = Ω ~ ( n 2 ) k=\\tilde{\\Omega}(n^{2}) adaptively chosen statistical queries w.r.t. 𝒫 \\mathcal{P} , provided that the entity selecting the queries is computationally efficient."},{"type":"para","text":"This observation was hinted at by some prior work without being stated or proven explicitly (see, e.g., [ STE16 ] ). Our contribution is in formalizing and proving it, which requires addressing some edge cases. The details are deferred to Appendix A ."},{"type":"heading","level":4,"text":"Main result: Separation for unbounded analysts."},{"type":"para","text":"We now turn to the main technical focus of this paper, providing strong evidence that when the analyst is computationally unbounded, the situation changes drastically. Specifically, we prove that in the random oracle model, any deterministic mechanism can be forced to fail after just k = O ~ ( n ) k=\\tilde{O}(n) queries. We establish this negative result in two stages:"},{"type":"para","text":"Natural mechanisms: We first consider a restricted class of deterministic mechanisms, called “natural” mechanisms, which observe only the evaluations of a query on the sample points. That is, a natural mechanism observes only ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\dots,q(x_{n})) , where x 1 , … , x n x_{1},\\dots,x_{n} denote the sample points, rather than the full query description q q . Natural mechanisms were also considered by [ HU14 , SU15 , NSS+18 ] as a steppingstone towards showing impossibility results for randomized mechanisms. These prior works showed that randomized natural mechanisms cannot answer more than O ( n 2 ) O(n^{2}) queries. We present a completely different attack (and analysis) showing that deterministic natural mechanisms cannot answer more than O ~ ( n ) \\tilde{O}(n) queries before failing."},{"type":"para","text":"The random oracle model: We then extend this barrier to general (i.e., not necessarily natural) deterministic mechanisms, under the assumption that both the analyst and the mechanism have access to a random oracle. Even when the deterministic mechanism can query the oracle at arbitrary positions and base its future actions on the obtained values, we prove that it cannot exceed the O ~ ( n ) \\tilde{O}(n) query limit."},{"type":"para","text":"Our main result is specified in the following theorem."},{"type":"para","text":"In the random oracle model, for every deterministic mechanism ℳ \\mathcal{M} and every sample size n n , there is an attacker 𝒜 \\mathcal{A} that with high probability, forces ℳ \\mathcal{M} to fail after just k = O ~ ( n ) k=\\tilde{O}(n) adaptive queries."},{"type":"para","text":"An appealing aspect of the attacks presented by prior works was that they constructed a single attacker that defeats every mechanism (albeit requiring n 2 n^{2} rounds). Our order of quantifiers is necessarily weaker: for every deterministic mechanism, we present a different attacker. This is unavoidable to beat the quadratic query bound. Indeed, if there were a single attacker that defeated every deterministic algorithm in O ~ ( n ) \\tilde{O}(n) rounds, then by the law of total probability, this attacker would also defeat randomized algorithms, which is impossible as randomized algorithms can support Ω ~ ( n 2 ) \\tilde{\\Omega}(n^{2}) queries."},{"type":"heading","level":4,"text":"Extension to mechanisms with limited randomness."},{"type":"para","text":"Our techniques also extend to algorithms equipped with a limited amount of private randomness. Specifically, if a mechanism has access to r r random bits, we can conceptually view these bits as an extension of the empirical sample itself. By suitably adapting our attack, we show that it is possible to force such a mechanism to fail after k = O ~ ( n + r ) k=\\tilde{O}(n+r) adaptive queries. This further clarifies the role of randomness in adaptive data analysis (at least in the RO model): to answer Ω ~ ( n 2 ) \\tilde{\\Omega}(n^{2}) adaptive queries, an algorithm must fundamentally rely on a substantial amount of internal randomness. Any attempt to substitute this with deterministic operations or insufficient randomness ( r ≪ n 2 r\\ll n^{2} ) leaves the mechanism vulnerable to rapid overfitting. We remark that Θ ~ ( n 2 ) \\tilde{\\Theta}(n^{2}) random bits are known to be sufficient, see for example [ BNS+16 , BLA23 ] ."},{"type":"para","text":"In the random oracle model, for every mechanism ℳ \\mathcal{M} that uses at most r r random bits, and every sample size n n , there is an attacker 𝒜 \\mathcal{A} that with high probability forces ℳ \\mathcal{M} to fail after just k = O ~ ( n + r ) k=\\tilde{O}(n+r) adaptive queries."},{"type":"heading","level":2,"text":"1.2 Technical Overview for Theorem 1.2"},{"type":"heading","level":3,"text":"1.2.1 A negative result for natural mechanisms"},{"type":"para","text":"We first describe an attack against natural mechanisms (recall that a natural mechanism does not observe the full description of the queries it needs to answer; only their evaluations on the empirical sample points). Denote N = poly ( n ) N=\\operatorname{poly}(n) , let the target distribution be uniform over the domain [ N ] [N] , and fix a natural deterministic mechanism ℳ \\mathcal{M} . We construct a computationally unbounded attacker that throughout the execution maintains a set of “surviving datasets”, denoted H T H_{T} , containing all possible datasets of size n n that are consistent with the transcript T T of the interaction with ℳ \\mathcal{M} so far. Initially, H T H_{T} contains all possible datasets of size n n over [ N ] [N] , so | H T | = N n |H_{T}|=N^{n} . The attacker’s goal is to rapidly shrink H T H_{T} by issuing a sequence of adaptively chosen queries. We say that a query is separating if every valid answer (which must be close to the true distributional expectation, i.e., to the average over [ N ] [N] ) is forced to be inconsistent with a constant fraction of the datasets currently in H T H_{T} . Intuitively (but somewhat inaccurately) after k = O ( n log N ) k=O(n\\log N) such separating queries, the analyst will have learned the dataset, allowing it to present a query that highly overfits to the data, thereby forcing ℳ \\mathcal{M} to fail."},{"type":"para","text":"We now explain how the attacker finds such a separating query. For simplicity, let us focus on the first round of the interaction (so currently H T H_{T} contains all datasets). The separating query our attacker chooses will be a threshold query over N N , i.e., a query that evaluates to 0 on a prefix of the domain [ N ] [N] . To this end, consider an | H T | × ( N + 1 ) |H_{T}|\\times(N+1) table, where cell ( i , j ) (i,j) contains the answer ℳ \\mathcal{M} would have returned had its dataset been S i S_{i} and had it been presented with the threshold query q j q_{j} , defined as q j ( x ) = 𝟙 { x ≤ j } q_{j}(x)=\\mathds{1}_{\\{x\\leq j\\}} . See Figure 1 for an illustration of how this table might look like."},{"type":"para","text":"Note that the attacker can compute this table before selecting its first query (by simulating ℳ \\mathcal{M} ). Also, observe that by the accuracy guarantees of ℳ \\mathcal{M} , it must be that the first column is essentially 0 → \\vec{0} and the last column is essentially 1 → \\vec{1} . The key observation regarding this table is that when going from one column j − 1 j-1 to the next column j j , only a small fraction of the coordinates might change. To see this, recall that the threshold queries q j − 1 q_{j-1} and q j q_{j} differ only in the j j th point. Thus, as ℳ \\mathcal{M} is natural, it cannot notice this change unless its input dataset contains the point j j . The fraction of datasets containing the j j th point is at most n / N n/N , which is small. So far we have established that:"},{"type":"para","text":"The left column is essentially all zeros."},{"type":"para","text":"The right column is essentially all ones."},{"type":"para","text":"Only a small fraction of the coordinates can change between every pair of consecutive columns."},{"type":"para","text":"These three properties together imply the existence of a “non-uniform” column, meaning a column such that no value appears in, say, more than 90% of its coordinates. To see this, let j j be the first column where more than 50% of the coordinates are strictly bigger than 0.5. So in column j − 1 j-1 at least 50 % 50\\% of the coordinates are at most 0.5. As the columns change slowly, at least ( 1 2 − n N ) (\\frac{1}{2}-\\frac{n}{N}) of the coordinates in column j j must also be at most 0.5. So column j j is “non-uniform”, and the corresponding query q j q_{j} is separating : no matter what answer the attacker gets after querying q j q_{j} , this answer allows it to rule out ≈ 1 / 2 \\approx 1/2 of the datasets in H T H_{T} ."},{"type":"para","text":"In the next round, in order to find the next separating query, the attacker constructs a new table where the y-axis contains only datasets that were not ruled out in previous rounds. We might also need to exclude a small number of points from the x-axis, corresponding to points that appear in “too many” of the surviving datasets. Otherwise, keeping such “common” points active might invalidate our key observation about the columns changing slowly. We show that there could be at most O ( n ) O(n) such common points, and that excluding them has almost no effect on the calculation. Note that the values in the next table could be different than the values of the previous table, as the values returned by ℳ \\mathcal{M} could be history dependent."},{"type":"para","text":"Intuitively, by sequentially applying such separating queries, the attacker isolates the true dataset in at most k = O ( n log N ) k=O(n\\log N) rounds (unless the mechanism errs before that, in which case the attacker still wins)."},{"type":"heading","level":3,"text":"1.2.2 A naïve attempt at lifting this barrier to general deterministic mechanisms"},{"type":"para","text":"Extending our impossibility result to general deterministic (not necessarily natural) mechanisms requires overcoming their ability to evaluate the full query description. The standard path for trying to achieve this would be for the attacker to encrypt the queries such that the mechanism could only decrypt the values corresponding to its empirical sample. More specifically, the target distribution would no longer be uniform over [ N ] [N] . Instead, we first generate N N encryption keys s 1 , … , s N s_{1},\\dots,s_{N} , and then define the target distribution to be uniform over the pairs ( j , s j ) (j,s_{j}) . Now, instead of directly issuing a query q : [ N ] → { 0 , 1 } q:[N]\\rightarrow\\{0,1\\} , the attacker encrypts c j = Encrypt ( q ( j ) , s j ) c_{j}={\\rm Encrypt}(q(j),s_{j}) for every j ∈ [ N ] j\\in[N] , and gives the vector of all encryptions c → \\vec{c} to the mechanism. The mechanism has the secret keys for points in its empirical sample, and can hence learn the value of q q restricted to its sample, but learns nothing on q q outside of its sample. Thus, the mechanism is effectively forced to behave like a natural mechanism. This strategy allowed the prior works of [ HU14 , SU15 , NSS+18 ] to lift their impossibility result from (randomized) natural mechanisms to general mechanisms."},{"type":"para","text":"However, our attack against natural deterministic mechanisms (defeating them in just O ~ ( n ) \\tilde{O}(n) rounds) is fundamentally different from attacks presented in prior works. As a result, the strategy outlined above breaks down in our setting. Intuitively, the reason is that the mechanism could potentially use the ciphertexts and the (in-sample) keys as a source of randomness, undermining our assumption that the mechanism is deterministic. One might hope that the argument could still go through, because these keys and ciphertexts are known to the attacker, and are thus more akin to public randomness rather than private randomness. Indeed, our attack would work seamlessly against a mechanism with public randomness, provided that it is natural for every realization of the public randomness. But this is not what the encryption-based strategy outlined above guarantees: changing an out-of-sample query value, without changing the corresponding encryption key, alters the resulting ciphertext. This changes the description of the query, which can subsequently affect the mechanism’s output. In other words, this encryption-based strategy only guarantees that the distribution of answers given by the mechanism must be independent of out-of-sample query values over the randomness of generating the keys and encryptions . It does not guarantee that the mechanism is natural for every fixed realization of the keys and ciphertexts, which is what is needed for the arguments from Section 1.2.1 to go through."},{"type":"heading","level":3,"text":"1.2.3 Towards an impossibility result for non-natural mechanisms"},{"type":"para","text":"To overcome the challenge outlined above, we take a different route for extending our impossibility result also to non-natural mechanisms. We still use encryption (more precisely, one time pads or random masks ) to hide out-of-sample query values. But we will argue directly about the behavior of the deterministic mechanism when given specific realizations of the mask, rather than arguing about what the mechanism “learns” or “cannot learn” when these masks are random."},{"type":"para","text":"To simplify the presentation, let us not embed these masks in the underlying distribution just yet. So the target distribution is still uniform over [ N ] [N] . Now, imagine a hypothetical model where in every round the attacker specifies a query q ∈ { 0 , 1 } N q\\in\\{0,1\\}^{N} and a mask y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} , and the mechanism obtains q ⊕ y ∈ { 0 , 1 } N q\\oplus y\\in\\{0,1\\}^{N} (this corresponds to the encrypted truth-table of q q using y y ), and also obtains y | S y|_{S} , which is the vector y y restricted to the coordinates in the empirical sample S S held by the mechanism. Conceptually, we can think of such a mechanism as being “partially natural”, as it gets all the coordinates of q ⊕ y q\\oplus y while only getting the in-sample coordinates of y y ."},{"type":"para","text":"This hypothetical model gives the attacker the ability to choose the masks y y adaptively during every round, rather than committing and embedding them in target distribution before the game begins."},{"type":"para","text":"The goal of the attacker remains the same: it maintains the set H T H_{T} of “surviving datasets” and in every round it aims to find a separating masked-threshold query that will necessarily rule out a constant fraction of the datasets in H T H_{T} . To do this, the attacker simulates ℳ \\mathcal{M} on every possible dataset S ∈ H T S\\in H_{T} with every possible query-mask pair q , y q,y (the inputs of the mechanism in this case are the dataset S S , the encrypted threshold query q ⊕ y q\\oplus y , and the in-sample mask values y | S y|_{S} ). See Figure 2 for an illustration."},{"type":"para","text":"As before, the attacker can compute this cube before selecting its query. Furthermore, the left face of the cube is essentially all zeroes (corresponding to the query q 0 ≡ 0 q_{0}\\equiv 0 ) and the right face of the cube is essentially all ones (corresponding to the query q N ≡ 1 q_{N}\\equiv 1 ). Note that fixing the query q q and the mask y y , while leaving the dataset S S “free”, can be visualized as a vertical pin in Figure 2 . The attacker’s goal is to find such a vertical pin with “non-uniform” values along it, in the sense that no value appears in, say, more than 90% of its coordinates. To show that such a non-uniform pin exists, let us introduce the following notation: fix values j ∈ [ N ] j\\in[N] and y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} defining a pin ( q j , y ) (q_{j},y) . We say that its neighboring pin is the pin ( q j + 1 , y ′ ) (q_{j+1},y^{\\prime}) , where y ′ = y ⊕ e j + 1 y^{\\prime}=y\\oplus e_{j+1} . That is, the neighboring pin of ( q j , y ) (q_{j},y) is obtained by advancing to the next threshold query q j + 1 q_{j+1} and by flipping the ( j + 1 ) (j+1) th coordinate in y y . See the blue and orange pins in Figure 2 for an illustration."},{"type":"para","text":"The key observation regarding these neighboring pins is that unless the input dataset contains the point j + 1 j+1 , then the value returned by ℳ \\mathcal{M} cannot change when going from pin ( q j , y ) (q_{j},y) to pin ( q j + 1 , y ′ ) (q_{j+1},y^{\\prime}) , because its inputs remain exactly the same (because q j ⊕ y = q j + 1 ⊕ y ′ q_{j}\\oplus y=q_{j+1}\\oplus y^{\\prime} ). Thus, the vast majority of the coordinates of these two pins are identical."},{"type":"para","text":"Now fix some value for y y , and consider the pin ( q 0 , y ) (q_{0},y) , which is a vertical line on the left face of the cube. Consider a path of neighboring pins where we start from ( q 0 , y ) (q_{0},y) and iteratively switch to the next neighboring pin until we reach the right face of the cube. This path behaves similarly to our 2D table illustrated in Figure 1 : (1) The initial pin is essentially all zeros; (2) the final pin is essentially all ones; and (3) only a small fraction of the coordinates can change between every pair of consecutive pins along this path. Thus, via similar arguments, there must be a “non-uniform” pin along this path."},{"type":"para","text":"The fact that such a non-uniform pin exists suffices for the attacker to win in this hypothetical model: in every iteration the attacker constructs a 3D cube, finds a non-uniform pin and queries it, thereby shrinking the set of surviving datasets multiplicatively. However, as we alluded to before, we will later want to embed the mask y y (or multiple masks) in the underlying distribution, so the attacker will not have the freedom to select arbitrary masks at different rounds of the interaction. Luckily, as we now explain, a noticeable fraction of the y y ’s are good in the sense that there is a non-uniform pin with this specific y y . To see this, observe that since every path (starting from every y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} ) has a non-uniform pin, and as there are at most N + 1 N+1 queries, then there must exist a query q q that captures at least 1 N + 1 \\frac{1}{N+1} fraction of these non-uniform pins. That is, there must exist a query q q such that for at least 1 N + 1 \\frac{1}{N+1} -fraction of the y y ’s it holds that ( q , y ) (q,y) is non-uniform. In particular, at least 1 N + 1 \\frac{1}{N+1} -fraction of the y y ’s are good in the sense that there is a query q q such that the pin ( q , y ) (q,y) is non-uniform."},{"type":"heading","level":3,"text":"1.2.4 Pitfalls for embedding masks in the target distribution"},{"type":"para","text":"Now that we know that a random mask y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} is good with probability at least 1 N + 1 \\frac{1}{N+1} , can we embed such random masks in the target distribution, as we contemplated in Section 1.2.2 ? We now list several pitfalls that, surprisingly, make this very unclear."},{"type":"heading","level":4,"text":"Pitfall #1: Boosting the success probability of the first round."},{"type":"para","text":"A success probability of 1 N + 1 \\frac{1}{N+1} per round is insufficient to break the mechanism in O ( n log N ) O(n\\log N) queries, as we need roughly n log N n\\log N successful rounds to isolate the input dataset. To emphasize Pitfall #1, let us now focus only on the first round and on amplifying its success probability. Since every random mask is good with probability 1 N + 1 \\frac{1}{N+1} , one might try to embed r = poly ( N ) r=\\operatorname{poly}(N) random masks in the target distribution, hoping that one of them will be good. Specifically, we first sample r r random masks y 1 , … , y r ∈ { 0 , 1 } N y_{1},\\dots,y_{r}\\in\\{0,1\\}^{N} uniformly, and then define the target distribution to be uniform over the tuples ( i , y 1 [ i ] , … , y r [ i ] ) (i,y_{1}[i],\\dots,y_{r}[i]) . Now, before asking its first query, the attacker constructs its 3D cube, where the mask axis (the depth axis in Figure 2 ) is restricted to the vectors y 1 , … , y r y_{1},\\dots,y_{r} (because these are the masks that are “available” for the attacker to choose from). If there is a non-uniform pin in this cube, then the attacker queries it and succeeds in shrinking H T H_{T} . But what is the probability of this event? Is it close to 1?"},{"type":"para","text":"By the analysis in Section 1.2.3 , we know that for every fixture of the vectors y 1 , … , y ℓ − 1 , y ℓ + 1 , … , y r y_{1},\\dots,y_{\\ell-1},y_{\\ell+1},\\dots,y_{r} , with probability at least 1 N + 1 \\frac{1}{N+1} over sampling y ℓ y_{\\ell} it holds that y ℓ y_{\\ell} is good. The issue is that the mechanism ℳ \\mathcal{M} gets all of these r r vectors as an input (restricted to its sample), and can base its output on all of them. Thus, it is unclear that the event that y 1 y_{1} is good is independent of the event that y 2 y_{2} is good. In principle, it could be that either all of y 1 , … , y r y_{1},\\dots,y_{r} are good, or none of them is good, and so the probability that there exists a good mask remains 1 N + 1 \\frac{1}{N+1} ."},{"type":"para","text":"To illustrate this issue, consider r r random variables Z 1 , … , Z r Z_{1},\\dots,Z_{r} distributed independently and uniformly over { 0 , 1 , 2 , … , 6 } \\{0,1,2,\\dots,6\\} , and suppose that Z ℓ Z_{\\ell} is good if ∑ i = 1 r Z i ( mod 7 ) = 0 \\sum_{i=1}^{r}Z_{i}\\pmod{7}=0 . If we fix all other variables and consider a single Z ℓ Z_{\\ell} , the residue equals 0 with probability exactly 1 / 7 1/7 . While this marginal probability holds for all ℓ \\ell , the global constraint rigidly couples the variables. No matter how many variables you inspect, i.e., no matter what r r is, it does not yield any overall probability amplification."},{"type":"heading","level":4,"text":"Pitfall #2: Success probability of the second round."},{"type":"para","text":"Suppose that we do get amplification for the success probability in the first round. What about the second round? Is it even true that it succeeds with probability at least 1 N + 1 \\frac{1}{N+1} ? The issue now is that the attacker chooses the first query based on the constructed 3D cube, which depends on all the mask vectors y 1 , … , y r y_{1},\\dots,y_{r} since its values are determined by simulating ℳ \\mathcal{M} with different portions of these masks. Then, as ℳ \\mathcal{M} might be history dependent, in the second round its behavior might depend on the first query, and thus depend on the mask vectors. So y 1 , … , y r y_{1},\\dots,y_{r} are no longer independent of the cube in the second round. Is it still true that each of them is good with noticeable probability? If so, do we still get amplification by having r r masks?"},{"type":"heading","level":3,"text":"1.2.5 Resolution using a random oracle and dynamic pointers"},{"type":"para","text":"To bypass these pitfalls, we turn to the Random Oracle (RO) model, where the attacker and mechanism have access to an oracle 𝒪 : ℕ → { 0 , 1 } \\mathcal{O}:\\mathbb{N}\\to\\{0,1\\} whose values are uniformly random and independent bits. The conceptual benefit we get from the random oracle is that we will not give the mechanism the actual mask bits upfront. Instead, we will specify these masks dynamically using different pointers to the random oracle. To do this, the underlying domain is extended to [ N ] × [ R ] [N]\\times[R] , where R R is a large integer, and each sample point x i x_{i} is accompanied by a uniformly random pointer offset m i ∈ [ R ] m_{i}\\in[R] . Thus, at the start of the game, the mechanism receives the pointers m i m_{i} corresponding to the points x i x_{i} in its empirical sample. In each round v v , the analyst dynamically determines an additional pointer p v p_{v} . The mask for a domain element i i in this round is then derived from the oracle as 𝒪 ( p v + m i ) \\mathcal{O}(p_{v}+m_{i}) . (For technical reasons, in the actual construction the positions in the RO are determined somewhat differently.) This construction guarantees that:"},{"type":"para","text":"Because the mechanism only receives the offset m i m_{i} for its sample points, it cannot easily guess the locations of out-of-sample masks. This enforces the behavior we needed in the analysis of our hypothetical model in Section 1.2.3 ."},{"type":"para","text":"The additional offset p v p_{v} acts as a “refresh button”, allowing the attacker to test and choose pointers to previously unread positions of the Random Oracle. This allows us to bypass our two pitfalls:"},{"type":"para","text":"Within each round v v , the attacker could try out several values for the offset p v p_{v} , simulating the behavior of ℳ \\mathcal{M} with each one of them. The different values for p v p_{v} guarantee strict independent repetitions, bypassing Pitfall #1 and allowing the attacker to amplify the probability of finding a separating query."},{"type":"para","text":"Across different rounds, the dynamic pointer completely severs historical dependency, guaranteeing that the masks for the new round are drawn perfectly fresh, entirely independent of the transcript so far. So this also bypasses Pitfall #2."},{"type":"heading","level":3,"text":"1.2.6 Challenges in the random oracle model"},{"type":"para","text":"To make the aforementioned guarantees hold, the random offsets m i m_{i} and the dynamic pointers p p must be drawn from a sufficiently large range. But how large? If we are not careful, analyzing the interactions between a computationally unbounded analyst and an adaptive mechanism could lead to a catastrophic blowup in the required range. Specifically, suppose that the mechanism runs in time t t . As the mechanism can be adaptive, it can potentially explore a decision tree of oracle queries of size 2 t 2^{t} per simulation. Naïvely union-bounding over all possible adaptive paths could force us to set the pointer range to be exponential in t t . This creates a fatal contradiction: a range of that magnitude would result in pointers whose description length is larger than t t . But a mechanism constrained to run in time t t would be unable to even read these pointers, let alone answer the queries."},{"type":"para","text":"We show that it suffices to represent pointers using O ( n N log N + log t ) O(nN\\log N+\\log t) bits, which is larger than the size of the dataset by only a polynomial factor (recall N = poly ( n ) N=\\operatorname{poly}(n) ), guaranteeing that the deterministic mechanism can feasibly read the pointers and answer queries within its time bound t t . This is somewhat easier to achieve for the secret offsets m i m_{i} (we show that with high probability, in all possible executions, the mechanism does not observe an out-of-sample mask bit). Arguing about the dynamic pointer p p is more subtle, because the unbounded attacker needs to continually find “fresh” positions in the random oracle, which requires carefully tracking the exhaustive search tree of the simulation. That is, unlike m i m_{i} , arguing about p p requires us to take into account also the attacker’s behavior, not just the mechanism’s behavior."},{"type":"heading","level":2,"text":"1.3 Other Related Works"},{"type":"para","text":"The literature on ADA is vast, with many exciting results beyond those discussed earlier. For example, researchers have explored variants of the ADA problem where the mechanism only needs to guarantee utility against “weaker” or “more realistic” adversaries, and shown impossibility results for these settings. Nissim et al. [ NST23 ] proposed a “balanced” model that separates the entity choosing the target distribution from the attacker, thus removing the ability to embed secrets in the target distribution that only the attacker is privy to. They demonstrated that even in this more realistic setting, answering more than O ~ ( n 2 ) \\tilde{O}(n^{2}) queries is hard. In a different vein, Dinur et al. [ DSW+23 ] proved that existing lower bounds for ADA can be interpreted as arising from a space bottleneck rather than a sampling bottleneck."},{"type":"para","text":"A growing body of literature considers the challenges of ADA with correlated input samples. Kontorovich et al. [ KSS22 ] extended stability-based generalization guarantees to non-i.i.d. settings by formally quantifying the degree of correlation between samples. Rapoport et al. [ RCS25 ] recently established tight lower bounds for these settings, proving an unavoidable utility gap: under natural conditions, one can answer at most O ( n ) O(n) adaptively chosen queries on correlated data (even using randomized mechanisms), in sharp contrast to the O ~ ( n 2 ) \\tilde{O}(n^{2}) limit in the independent setting."},{"type":"para","text":"In our work, we focus on the vanilla, unrestricted formulation of the ADA problem. As we mentioned, if we assume that the domain 𝒳 \\mathcal{X} is not too large, namely that n ≥ polylog | 𝒳 | n\\geq\\operatorname{polylog}|\\mathcal{X}| , then answering more than n 2 n^{2} queries becomes possible (albeit inefficiently), even using deterministic mechanisms. Specifically, Dwork et al. [ DFH+15a ] and Bassily et al. [ BNS+16 ] showed that there are (inefficient) mechanisms that answer exp ( n / log | 𝒳 | ) \\exp(n/\\sqrt{\\log|\\mathcal{X}|}) adaptive queries using a sample of size n n . Dwork et al. [ DFH+15b ] showed that a qualitatively similar positive result can be obtained using an (inefficient) deterministic mechanism. Recently, this query bound of exp ( n / log | 𝒳 | ) \\exp(n/\\sqrt{\\log|\\mathcal{X}|}) was shown to be tight by Lyu and Talwar [ LT25 ] ."},{"type":"heading","level":1,"text":"2 Preliminaries"},{"type":"para","text":"We first define the Adaptive Data Analysis (ADA) framework, distributional accuracy, and natural algorithms."},{"type":"para","text":"In the ADA framework, a mechanism ℳ \\mathcal{M} interacts with a data analyst 𝒜 \\mathcal{A} over k k rounds. Given a dataset S = ( x 1 , … , x n ) ∼ 𝒫 n S=(x_{1},\\dots,x_{n})\\sim\\mathcal{P}^{n} drawn i.i.d. from a distribution 𝒫 \\mathcal{P} over a domain 𝒳 \\mathcal{X} , in each round j ∈ [ k ] j\\in[k] :"},{"type":"para","text":"The analyst 𝒜 \\mathcal{A} outputs a statistical query q j : 𝒳 → [ 0 , 1 ] q_{j}:\\mathcal{X}\\to[0,1] , which may depend on previous queries and answers."},{"type":"para","text":"The mechanism ℳ \\mathcal{M} observes q j q_{j} and outputs an answer a j ∈ ℝ a_{j}\\in\\mathbb{R} ."},{"type":"para","text":"Let ε > 0 , k , n , N , 𝒳 \\varepsilon>0,k,n,N,\\mathcal{X} . A mechanism ℳ \\mathcal{M} is ε \\varepsilon -accurate for k k adaptive queries given a sample of size n n , if for each distribution 𝒫 \\mathcal{P} over 𝒳 \\mathcal{X} and analyst 𝒜 \\mathcal{A} , with probability at least 9 / 10 9/10 over the sample S ∼ 𝒫 n S\\sim\\mathcal{P}^{n} , and the randomness of ℳ \\mathcal{M} , the answers a 1 , … , a k a_{1},\\dots,a_{k} satisfy"},{"type":"para","text":"A mechanism ℳ \\mathcal{M} is called natural if, when given a dataset S = ( x 1 , … , x n ) S=(x_{1},\\dots,x_{n}) , then for every given statistical query q : 𝒳 → [ 0 , 1 ] q:\\mathcal{X}\\to[0,1] , the mechanism ℳ \\mathcal{M} only observes the query evaluations on the sample points, i.e., ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\dots,q(x_{n})) ."},{"type":"heading","level":1,"text":"3 Natural Mechanisms"},{"type":"para","text":"We first provide an attack against natural deterministic mechanisms. Suppose our domain is 𝒳 = [ N ] \\mathcal{X}=[N] , where N N is at most poly ( n ) \\operatorname{poly}(n) . (If | 𝒳 | |\\mathcal{X}| is larger, just restrict the distribution to be supported only on the first N N elements of 𝒳 \\mathcal{X} .) Consider a natural deterministic mechanism ℳ \\mathcal{M} that, given a sample of size n n , is 0.1 0.1 -accurate for k k adaptive queries by a computationally unbounded analyst. We aim to show that k = O ( n log N ) k=O(n\\log N) . Consider the uniform distribution over [ N ] [N] , and suppose for simplicity that the sample is a set and not a multi-set (i.e., there are no collisions). This assumption can be achieved with high probability by setting the domain size as N = n 10 N=n^{10} ."},{"type":"para","text":"An unbounded adversary 𝒜 \\mathcal{A} can simulate the answers of all possible datasets. Thus, for a transcript T = ( q 1 , a 1 , … , q i , a i ) T=(q_{1},a_{1},\\dots,q_{i},a_{i}) , the adversary can compute the set of datasets that are consistent with T T , denoted H T ⊆ [ N ] n H_{T}\\subseteq[N]^{n} . Furthermore, 𝒜 \\mathcal{A} can compute the answers of datasets in H T H_{T} for any query q q . The adversary aims to find a query for which every answer a a is consistent with at most 9 10 \\frac{9}{10} fraction of the datasets in H T H_{T} , called a separating query . The next lemma shows that 𝒜 \\mathcal{A} can indeed find a separating query, and moreover, that query is binary q : [ N ] → { 0 , 1 } q:[N]\\to\\{0,1\\} ."},{"type":"para","text":"Let a deterministic natural mechanism ℳ \\mathcal{M} and a transcript T = ( q 1 , a 1 , … , q i , a i ) T=(q_{1},a_{1},\\dots,q_{i},a_{i}) . There exists a query q : [ N ] → { 0 , 1 } q:[N]\\to\\{0,1\\} that either causes the mechanism to have error > 0.1 >0.1 , or is a separating query for ℳ \\mathcal{M} ."},{"type":"para","text":"The proof uses threshold queries , defined as follows."},{"type":"para","text":"Let X m = { x 1 , … , x m } ⊂ [ N ] X_{m}=\\{x_{1},\\dots,x_{m}\\}\\subset[N] be a set of points. Let the remaining elements of the domain be arbitrarily ordered as z 1 , z 2 , … , z N − m z_{1},z_{2},\\dots,z_{N-m} . A threshold query is any query of the form,"},{"type":"para","text":"The set X m X_{m} as defined in Section ˜ 3 is arbitrary, but in the proof of Section ˜ 3 , it will be a set of m = O ( n ) m=O(n) points that appear in most of the databases in H T H_{T} ."},{"type":"para","text":"Let X m = { x 1 , … , x m } ⊂ [ N ] X_{m}=\\{x_{1},\\dots,x_{m}\\}\\subset[N] be the set of all points x ∈ [ N ] x\\in[N] s.t. | { S ∈ H T : x ∈ S } | ≥ 8 10 | H T | |\\{S\\in H_{T}:x\\in S\\}|\\geq\\frac{8}{10}|H_{T}| , i.e., x x is in most of the surviving datasets. We claim that m = O ( n ) m=O(n) , by the following counting argument."},{"type":"para","text":"We calculate the number of points in H T H_{T} , including multiplicities, in two ways. First, every S ∈ H T S\\in H_{T} contains n n points, so we have precisely n | H T | n|H_{T}| points. On the other hand, each of x 1 , … , x m x_{1},\\dots,x_{m} is in 8 10 | H T | \\frac{8}{10}|H_{T}| of the datasets in H T H_{T} , thus contributes at least that number of copies to the overall number. Summing over these m m common points, we obtain"},{"type":"para","text":"and thus indeed m = O ( n ) m=O(n) ."},{"type":"para","text":"Consider the query that assigns 1 1 to x 1 , … , x m x_{1},\\dots,x_{m} , and 0 to all other points, which is the threshold query q 0 X m q_{0}^{X_{m}} . The answer a a to query q 0 X m q_{0}^{X_{m}} must satisfy a ≤ 0.1 + m N ≤ 0.2 a\\leq 0.1+\\frac{m}{N}\\leq 0.2 . If ≤ 9 10 \\leq\\tfrac{9}{10} fraction of the datasets in H T H_{T} return answer smaller than 0.2 0.2 , then the query q 0 X m q_{0}^{X_{m}} satisfies the conclusion of the theorem, so suppose more than 9 / 10 9/10 fraction of the datasets in H T H_{T} return a valid answer to q 0 X m q_{0}^{X_{m}} . Similarly, the answer to the query that assigns 1 1 to all points (which is q N − m X m q_{N-m}^{X_{m}} ) is at least 0.9 0.9 . If ≤ 9 10 \\leq\\tfrac{9}{10} fraction of the datasets in H T H_{T} return answer larger than 0.9 0.9 , then the query q N − m X m q_{N-m}^{X_{m}} satisfies the conclusion of the theorem, so suppose more than 9 / 10 9/10 fraction of the datasets in H T H_{T} return a valid answer to q N − m X m q_{N-m}^{X_{m}} . Therefore, there must be some j > m j>m , for which most datasets in H T H_{T} answer ≤ 0.5 \\leq 0.5 for q j X m q_{j}^{X_{m}} , but most datasets in H T H_{T} answer > 0.5 >0.5 for q j + 1 X m q_{j+1}^{X_{m}} ."},{"type":"para","text":"The analyst simulates the answer of every S ∈ H T S\\in H_{T} for every threshold query to locate q j X m q_{j}^{X_{m}} . By construction, the point z j + 1 z_{j+1} is not common. Since the mechanism is natural, only datasets that contain z j + 1 z_{j+1} may have different answers for the queries q j X m q_{j}^{X_{m}} and q j + 1 X m q_{j+1}^{X_{m}} ."},{"type":"para","text":"We claim that either q j X m q_{j}^{X_{m}} or q j + 1 X m q_{j+1}^{X_{m}} is a separating query, which concludes the proof. Suppose that q j X m q_{j}^{X_{m}} is not a separating query, meaning that at least 9 10 \\tfrac{9}{10} fraction of the datasets in H T H_{T} return some answer a ≤ 0.5 a\\leq 0.5 for the query q j X m q_{j}^{X_{m}} . At most 8 10 | H T | \\tfrac{8}{10}|H_{T}| of these contain z j + 1 z_{j+1} (since z j + 1 z_{j+1} is not common), and thus the most common answer for q j + 1 X m q_{j+1}^{X_{m}} , which is > 0.5 >0.5 , is returned by at most 9 10 | H T | \\tfrac{9}{10}|H_{T}| datasets, so q j + 1 X m q_{j+1}^{X_{m}} is a separating query. ∎"},{"type":"para","text":"A natural deterministic mechanism ℳ \\mathcal{M} that is 0.1 0.1 -accurate can answer at most k = O ( n log N ) k=O(n\\log N) queries from a computationally unbounded analyst."},{"type":"para","text":"By Section ˜ 3 , as long as the mechanism returns correct answers, in every round, the analyst can find a separating query. Initially, the number of possible datasets is ≤ N n \\leq N^{n} . Every such query rules out at least a constant factor of the datasets. Therefore, the analyst can fail the mechanism after O ( n log N ) O(n\\log N) queries. (Observe that if | H T | = 1 |H_{T}|=1 , then X m = S X_{m}=S , hence the mechanism must reply the same for q 0 X m q_{0}^{X_{m}} and q N − m X m q_{N-m}^{X_{m}} , and thus errs on at least one of these queries.) ∎"},{"type":"heading","level":2,"text":"3.1 Natural Mechanism with a Mask"},{"type":"para","text":"We now consider the following variant of natural mechanisms, which we call masked mechanisms. In this variant, for each query q q , the adversary draws a mask y y uniformly at random from { 0 , 1 } N \\{0,1\\}^{N} . The mechanism receives the masked query q ⊕ y q\\oplus y (computed bitwise) and the mask on the sample points ( y ( x 1 ) , … , y ( x n ) ) (y(x_{1}),\\dots,y(x_{n})) . The mechanism can trivially compute ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\dots,q(x_{n})) , and for simplicity of presentation, we use an equivalent formulation of the model where the mechanism is given the evaluations on the sample points ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\dots,q(x_{n})) instead of ( y ( x 1 ) , … , y ( x n ) ) (y(x_{1}),\\dots,y(x_{n})) . We now prove that, with non-negligible probability, there exists a separating query for this variant."},{"type":"para","text":"Let a deterministic masked mechanism ℳ \\mathcal{M} and a transcript T T . There exists a query q : [ N ] → { 0 , 1 } q:[N]\\to\\{0,1\\} such that for at least a 1 N + 1 \\tfrac{1}{N+1} fraction of masks y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} , the pair ( q , y ) (q,y) is separating for ℳ \\mathcal{M} ."},{"type":"para","text":"As before, the analyst can simulate the mechanism, and compute the set of datasets H T H_{T} that are consistent with the transcript T T . Let X m ⊂ [ N ] X_{m}\\subset[N] be the set of common points in H T H_{T} , i.e., the set of all points x ∈ [ N ] x\\in[N] s.t. | { S ∈ H T : x ∈ S } | ≥ 8 10 | H T | |\\{S\\in H_{T}:x\\in S\\}|\\geq\\frac{8}{10}|H_{T}| . By the same argument ( Equation ˜ 1 ), | X m | = O ( n ) |X_{m}|=O(n) . Let z 1 , … , z N − m z_{1},\\dots,z_{N-m} be the remaining non-common points. Consider the sequence of threshold queries q 0 , q 1 , … , q N − m q_{0},q_{1},\\dots,q_{N-m} defined by q j ( x ) = 𝕀 [ x ∈ X m ∪ { z 1 , … , z j } ] q_{j}(x)=\\mathbb{I}[x\\in X_{m}\\cup\\{z_{1},\\dots,z_{j}\\}] (as per Section ˜ 3 )."},{"type":"para","text":"For any fixed “base” mask v ∈ { 0 , 1 } N v\\in\\{0,1\\}^{N} , we define a “path” of query-mask pairs P v = { ( q j , y j ) } j = 0 N − m P_{v}=\\{(q_{j},y_{j})\\}_{j=0}^{N-m} where"},{"type":"para","text":"A key observation is that along this path, the masked query input to the mechanism is constant,"},{"type":"para","text":"Now, define a restricted natural mechanism ℳ v \\mathcal{M}_{v} that, given evaluations E = ( q ( x 1 ) , … , q ( x n ) ) E=(q(x_{1}),\\dots,q(x_{n})) , outputs:"},{"type":"para","text":"Since v v is fixed, ℳ v \\mathcal{M}_{v} is a natural mechanism (its output depends only on the evaluations). By applying Lemma 3 to ℳ v \\mathcal{M}_{v} , there must exist at least one index j v ∈ { 0 , … , N − m } j_{v}\\in\\{0,\\dots,N-m\\} such that the query q j v q_{j_{v}} is a separating query for H T H_{T} under ℳ v \\mathcal{M}_{v} . By construction, q j v q_{j_{v}} being separating for H T H_{T} under ℳ v \\mathcal{M}_{v} implies that ( q j v , y j v ) (q_{j_{v}},y_{j_{v}}) is a separating pair for H T H_{T} under the original masked mechanism ℳ \\mathcal{M} ."},{"type":"para","text":"There are 2 N 2^{N} possible base masks v v , and each produces at least one separating pair ( q j v , v ⊕ q j v ) (q_{j_{v}},v\\oplus q_{j_{v}}) . Since there are at most N + 1 N+1 possible threshold queries in the sequence, there are at most ( N + 1 ) 2 N (N+1)2^{N} pairs ( q i , y ) (q_{i},y) , where q i q_{i} is the i i -th threshold query and y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} is a mask. By the pigeonhole principle, there must exist at least one query q ∗ q^{*} that belongs to the set of separating pairs for at least 1 N + 1 \\frac{1}{N+1} fraction of the masks y ∈ { 0 , 1 } N y\\in\\{0,1\\}^{N} , which concludes the proof. ∎"},{"type":"heading","level":1,"text":"4 Random Oracle"},{"type":"para","text":"We now extend the lower bound on deterministic natural mechanisms to deterministic mechanisms that have access to a random oracle (RO), proving Theorem ˜ 1.2 . The random oracle is a function 𝒪 : ℕ → { 0 , 1 } \\mathcal{O}:\\mathbb{N}\\to\\{0,1\\} whose values are uniformly random and independent bits."},{"type":"para","text":"A query in the random oracle model is an oracle-aided function q 𝒪 : 𝒳 → [ 0 , 1 ] q^{\\mathcal{O}}:\\mathcal{X}\\to[0,1] . For brevity, when 𝒪 \\mathcal{O} is clear from the context, we simply write q q instead of q 𝒪 q^{\\mathcal{O}} ."},{"type":"para","text":"In our case, for any x ∈ 𝒳 x\\in\\mathcal{X} , the query q 𝒪 q^{\\mathcal{O}} computes its output using the point x x and exactly one bit from the random oracle 𝒪 \\mathcal{O} . Specifically, there exist functions I q : 𝒳 → ℕ I_{q}:\\mathcal{X}\\to\\mathbb{N} and g q : 𝒳 × { 0 , 1 } → [ 0 , 1 ] g_{q}:\\mathcal{X}\\times\\{0,1\\}\\to[0,1] such that q 𝒪 ( x ) = g q ( x , 𝒪 ( I q ( x ) ) ) q^{\\mathcal{O}}(x)=g_{q}(x,\\mathcal{O}(I_{q}(x))) . More generally, one could consider queries that rely on more than one bit of 𝒪 \\mathcal{O} , whose coordinates might even be determined adaptively. But for our purposes, one bit will suffice."},{"type":"para","text":"In the RO model, both the analyst and the mechanism have oracle access to the same random oracle 𝒪 : ℕ → { 0 , 1 } \\mathcal{O}:\\mathbb{N}\\to\\{0,1\\} . A mechanism ℳ \\mathcal{M} receives a dataset S = ( x 1 , … , x n ) ∼ 𝒫 n S=(x_{1},\\ldots,x_{n})\\sim\\mathcal{P}^{n} drawn i.i.d. from a distribution 𝒫 \\mathcal{P} over a domain 𝒳 \\mathcal{X} . In round j ∈ [ k ] j\\in[k] ,"},{"type":"para","text":"The analyst formulates a statistical query q j 𝒪 : 𝒳 → [ 0 , 1 ] q_{j}^{\\mathcal{O}}:\\mathcal{X}\\to[0,1] , which may depend on previous queries and answers."},{"type":"para","text":"The mechanism receives q j 𝒪 q_{j}^{\\mathcal{O}} and outputs an answer a j ∈ ℝ a_{j}\\in\\mathbb{R} ."},{"type":"para","text":"Let ε > 0 , k , n , N , 𝒳 \\varepsilon>0,k,n,N,\\mathcal{X} . A mechanism ℳ \\mathcal{M} in the RO model is ε \\varepsilon -accurate for k k adaptive queries given a sample of size n n , if for each distribution 𝒫 \\mathcal{P} over 𝒳 \\mathcal{X} and analyst 𝒜 \\mathcal{A} , with probability at least 9 / 10 9/10 over the choice of the random oracle 𝒪 \\mathcal{O} , the sample S ∼ 𝒫 n S\\sim\\mathcal{P}^{n} , and the randomness of ℳ \\mathcal{M} , the answers a 1 , … , a k a_{1},\\dots,a_{k} satisfy"},{"type":"para","text":"We emphasize that the population mean 𝔼 x ∼ 𝒫 [ q j 𝒪 ( x ) ] \\mathbb{E}_{x\\sim\\mathcal{P}}[q^{\\mathcal{O}}_{j}(x)] evaluates the query with respect to the specific, fixed realization of 𝒪 \\mathcal{O} drawn in the experiment; the expectation is taken solely over the domain distribution 𝒫 \\mathcal{P} ."},{"type":"heading","level":2,"text":"4.1 The masked RO model"},{"type":"para","text":"We first provide an attack against deterministic mechanisms in an intermediate model, called masked RO . This attack contains the main arguments in the proof of Theorem ˜ 1.2 . In the masked RO model, the domain is [ N ] [N] , the distribution is uniform over [ N ] [N] , and the analyst samples (independently of 𝒪 \\mathcal{O} ) offsets m 1 , … , m N m_{1},\\ldots,m_{N} independently and uniformly from [ R ] [R] , where R ∈ ℕ R\\in\\mathbb{N} is a number we set later. In addition to a dataset S ⊂ [ N ] S\\subset[N] , the mechanism is also given the offsets { m i : i ∈ S } \\{m_{i}:i\\in S\\} ."},{"type":"para","text":"For a pointer p p , define the mask induced by the block [ p + 1 , p + N R ] [p+1,p+NR] as"},{"type":"para","text":"In a round, the analyst sends a pointer p p and a masked query q ~ ∈ { 0 , 1 } N \\tilde{q}\\in\\{0,1\\}^{N} . This pair represents the unmasked query"},{"type":"para","text":"and the mechanism is required to answer accurately with respect to the expectation of q q under the uniform distribution over [ N ] [N] ."},{"type":"para","text":"Let N = n 10 N=n^{10} , let δ = N − 6 \\delta=N^{-6} , let k = O ( n log N ) k=O(n\\log N) , and let"},{"type":"para","text":"Suppose"},{"type":"para","text":"Let ℳ \\mathcal{M} be a deterministic mechanism in the masked RO model that receives n n samples from the uniform distribution over [ N ] [N] and runs in at most t t time per round. If ℳ \\mathcal{M} is 0.1 0.1 -accurate, then ℳ \\mathcal{M} can answer at most k k adaptive queries, issued with pointers of length O ( n N log N + log t ) O(nN\\log N+\\log t) ."},{"type":"para","text":"We construct an unbounded analyst that, except with probability o ( 1 ) o(1) over the offsets and the random oracle, finds a separating query in every round, thus forcing the mechanism to fail in k k rounds. If the analyst makes a query on which the mechanism is already inaccurate, the proof is complete. Thus below, the only nontrivial case is the one in which the simulated mechanism remains accurate on the candidate queries. The analyst will conduct a sequence of experiments, each one will correspond to the masked ADA model of Section ˜ 3.1 . Denote by E = O ( N log 1 δ ) E=O(N\\log\\tfrac{1}{\\delta}) the maximum number of experiments the analyst will perform per round. Throughout, experiments are ordered lexicographically by ( j , v ) (j,v) , where j ∈ [ k ] j\\in[k] is the ADA round and v ∈ [ E ] v\\in[E] is the experiment number inside the round. The analyst only queries the random oracle at places the mechanism might query given a dataset S ⊂ [ N ] S\\subset[N] of size n n , for all possible datasets."},{"type":"heading","level":4,"text":"The long-pointer idealization."},{"type":"para","text":"We first allow the analyst and mechanism to use very long pointers (in particular, they could be so long that the mechanism cannot read them in t t time), and we will later show how to remove this idealization. For each experiment ( j , v ) (j,v) , the analyst chooses a pointer p j , v long p^{\\rm long}_{j,v} so that the block"},{"type":"para","text":"is disjoint from every RO location that could have been queried before experiment ( j , v ) (j,v) by either the analyst or the mechanism, over all datasets, all masked-query transcripts, and all possible RO answers. Such a choice is possible because, before any fixed experiment, the number of executions and simulations considered by the unbounded analyst is finite and every execution makes finitely many oracle queries. Let P j = { p j , v long : v ∈ [ E ] } P_{j}=\\{p^{\\rm long}_{j,v}:v\\in[E]\\} . We say that an experiment ( j , v ) (j,v) is successful, if exists masked threshold query q ~ \\tilde{q} such that the pair ( p j , v long , q ~ ) (p^{\\rm long}_{j,v},\\tilde{q}) is separating."},{"type":"para","text":"With probability at least 1 − δ 1-\\delta over m 1 , … , m N m_{1},\\ldots,m_{N} and 𝒪 \\mathcal{O} , the following event B B holds. For every dataset S ⊂ [ N ] S\\subset[N] of size n n , every sequence of masked threshold queries q ~ 1 , … , q ~ k ∈ { 0 , 1 } N \\tilde{q}_{1},\\ldots,\\tilde{q}_{k}\\in\\{0,1\\}^{N} , and every sequence of pointers p 1 ∈ P 1 , … , p k ∈ P k p_{1}\\in P_{1},\\ldots,p_{k}\\in P_{k} , during the resulting interaction ℳ \\mathcal{M} never makes a query in"},{"type":"para","text":"I.e., in all rounds, ℳ \\mathcal{M} does not make an out-of-sample mask query."},{"type":"para","text":"Fix S ⊂ [ N ] S\\subset[N] of size n n , masked queries q ~ 1 , … , q ~ k ∈ { 0 , 1 } N \\tilde{q}_{1},\\ldots,\\tilde{q}_{k}\\in\\{0,1\\}^{N} , and p 1 ∈ P 1 , … , p k ∈ P k p_{1}\\in P_{1},\\ldots,p_{k}\\in P_{k} . That is, consider a special analyst 𝒜 q ~ 1 , … , q ~ k , p 1 , … , p k \\mathcal{A}_{\\tilde{q}_{1},\\ldots,\\tilde{q}_{k},p_{1},\\ldots,p_{k}} that issues the masked queries q ~ 1 , … , q ~ k \\tilde{q}_{1},\\ldots,\\tilde{q}_{k} and pointers p 1 , … , p k p_{1},\\ldots,p_{k} , regardless of the mechanism’s answers. Also fix the offsets { m i : i ∈ S } \\{m_{i}:i\\in S\\} and a realization of 𝒪 \\mathcal{O} . Under this conditioning, the deterministic mechanism’s oracle queries are fixed, and since ℳ \\mathcal{M} runs in time t t in each round, it makes at most k t kt oracle queries."},{"type":"para","text":"Consider one such query location ℓ \\ell . Since the long-pointer blocks are disjoint, ℓ \\ell belongs to at most one interval of the form"},{"type":"para","text":"Therefore, over the still-random offset m i m_{i} for that possible interval, the probability that ℓ = p j + ( i − 1 ) R + m i \\ell=p_{j}+(i-1)R+m_{i} is at most 1 / R 1/R . A union bound over the at most k t kt oracle queries gives probability at most k t / R kt/R that ℳ \\mathcal{M} ever queries an out-of-sample mask location for the fixed S , q ~ 1 , … , q ~ k , p 1 , … , p k S,\\tilde{q}_{1},\\ldots,\\tilde{q}_{k},p_{1},\\ldots,p_{k} ."},{"type":"para","text":"This bound holds for every conditioning on { m i : i ∈ S } \\{m_{i}:i\\in S\\} and on 𝒪 \\mathcal{O} , so by the law of total probability, it also holds unconditionally. Union bounding over at most N n N^{n} datasets, 2 N k 2^{Nk} masked-query sequences, and E k E^{k} pointer sequences gives"},{"type":"para","text":"where the last step is by the choice of R R . ∎"},{"type":"para","text":"Observe that under B B , for every analyst 𝒜 \\mathcal{A} , in the resulting interaction between 𝒜 \\mathcal{A} and ℳ \\mathcal{M} , the mechanism ℳ \\mathcal{M} does not make an out-of-sample query. This is since during the interaction, 𝒜 \\mathcal{A} must output some sequence of masked queries q ~ 1 , … , q ~ k \\tilde{q}_{1},\\ldots,\\tilde{q}_{k} and pointers p 1 , … , p k p_{1},\\ldots,p_{k} , hence it identifies with 𝒜 q ~ 1 , … , q ~ k , p 1 , … , p k \\mathcal{A}_{\\tilde{q}_{1},\\ldots,\\tilde{q}_{k},p_{1},\\ldots,p_{k}} . By B B , in the interaction between 𝒜 q ~ 1 , … , q ~ k , p 1 , … , p k \\mathcal{A}_{\\tilde{q}_{1},\\ldots,\\tilde{q}_{k},p_{1},\\ldots,p_{k}} and ℳ \\mathcal{M} , the mechanism ℳ \\mathcal{M} does not make an out-of-sample query, yielding this observation."},{"type":"para","text":"Fix a round j ∈ [ k ] j\\in[k] and a transcript T j − 1 T_{j-1} produced before that round. The analyst conducts the following experiment for each v ∈ [ E ] v\\in[E] . First, it queries 𝒪 \\mathcal{O} at the N N mask locations"},{"type":"para","text":"and thereby learns the mask y ( v ) ∈ { 0 , 1 } N y^{(v)}\\in\\{0,1\\}^{N} defined by"},{"type":"para","text":"The analyst then searches for a separating threshold query. For every candidate threshold query q ∈ { 0 , 1 } N q\\in\\{0,1\\}^{N} , it sets q ~ = q ⊕ y ( v ) \\tilde{q}=q\\oplus y^{(v)} . For every candidate dataset S ′ ∈ H T j − 1 S^{\\prime}\\in H_{T_{j-1}} , it simulates M M on transcript T j − 1 T_{j-1} , dataset S ′ S^{\\prime} , offsets { m i : i ∈ S ′ } \\{m_{i}:i\\in S^{\\prime}\\} , masked query q ~ \\tilde{q} , and pointer p j , v long p^{\\rm long}_{j,v} , with a dataset-dependent oracle 𝒪 l y i n g S ′ \\mathcal{O}_{lying}^{S^{\\prime}} . This oracle is defined as follows. A query to an in-sample mask location p j , v long + ( i − 1 ) R + m i p^{\\rm long}_{j,v}+(i-1)R+m_{i} with i ∈ S ′ i\\in S^{\\prime} is answered by the already-read bit y i ( v ) y^{(v)}_{i} ; a query to an out-of-sample mask location of this form with i ∉ S ′ i\\notin S^{\\prime} is answered by 0 ; every other oracle query is answered as in the real oracle 𝒪 \\mathcal{O} . Thus the analyst’s real RO queries in a single experiment are the N N mask queries, plus at most t t non-mask queries for each pair ( q , S ′ ) (q,S^{\\prime}) . Hence the analyst makes at most"},{"type":"para","text":"real RO queries."},{"type":"para","text":"Fix a round j ∈ [ k ] j\\in[k] and condition on any history produced before that round, including the transcript T j − 1 T_{j-1} . In the long-pointer idealization, if the simulated mechanism obtains its oracle replies from 𝒪 l y i n g S ′ \\mathcal{O}_{lying}^{S^{\\prime}} on each candidate dataset S ′ ∈ H T j − 1 S^{\\prime}\\in H_{T_{j-1}} , then the probability that none of the E E experiments in round j j contains a separating query is at most δ \\delta ."},{"type":"para","text":"Consider experiment v ∈ [ E ] v\\in[E] in round j j , and condition on all previous experiments. By construction, the bits of the random oracle in the block B j , v long B^{\\rm long}_{j,v} are fresh, hence y ( v ) y^{(v)} is uniform over { 0 , 1 } N \\{0,1\\}^{N} and independent of the conditioned transcript."},{"type":"para","text":"Let 𝒪 out \\mathcal{O}_{\\mathrm{out}} denote the restriction of 𝒪 \\mathcal{O} to all locations except the N N fresh mask locations { p j , v long + ( i − 1 ) R + m i : i ∈ [ N ] } \\{p^{\\rm long}_{j,v}+(i-1)R+m_{i}:i\\in[N]\\} . Condition on an arbitrary realization of 𝒪 out \\mathcal{O}_{\\mathrm{out}} . Under this conditioning, y ( v ) y^{(v)} remains uniform over { 0 , 1 } N \\{0,1\\}^{N} , and the hybrid simulation defines a deterministic masked mechanism ℳ ′ \\mathcal{M}^{\\prime} : on dataset S ′ S^{\\prime} , masked query q ~ \\tilde{q} , and sample evaluations q | S ′ q|_{S^{\\prime}} , the mechanism can reconstruct the in-sample mask bits from q ~ | S ′ \\tilde{q}|_{S^{\\prime}} and q | S ′ q|_{S^{\\prime}} , receives no out-of-sample mask bits, and all other oracle answers are fixed constants. Therefore, by Section ˜ 3.1 , for every such conditioning, with probability at least 1 / ( N + 1 ) 1/(N+1) over the fresh mask y ( v ) y^{(v)} , there is a threshold query q ( v ) ∈ { 0 , 1 } N q^{(v)}\\in\\{0,1\\}^{N} such that the pair ( q ( v ) , y ( v ) ) (q^{(v)},y^{(v)}) is separating for this experiment. Since this holds for any realization of 𝒪 out \\mathcal{O}_{\\mathrm{out}} , by the law of total probability, the same bound holds without conditioning on 𝒪 out \\mathcal{O}_{\\mathrm{out}} . Since the analyst is unbounded, it finds such a query whenever one exists."},{"type":"para","text":"The same conditional success bound holds after every history of failed experiments. Hence"},{"type":"para","text":"∎"},{"type":"para","text":"In the long-pointer idealization, the analyst finds a separating query in every round in the real masked RO game with probability at least 1 − ( k + 1 ) δ 1-(k+1)\\delta ."},{"type":"para","text":"For each round j j , let G j lie G_{j}^{\\mathrm{lie}} be the event that at least one of the E E experiments in round j j contains a separating query in the exhaustive simulation that answers oracle queries using 𝒪 lying S ′ \\mathcal{O}_{\\mathrm{lying}}^{S^{\\prime}} for each candidate dataset S ′ ∈ H T j − 1 S^{\\prime}\\in H_{T_{j-1}} . By Section ˜ 4.1 , for every history produced before round j j ,"},{"type":"para","text":"Therefore, conditioning sequentially on the histories produced in previous rounds,"},{"type":"para","text":"By Section ˜ 4.1 and a union bound,"},{"type":"para","text":"We now argue that under B ∩ ⋂ j = 1 k G j lie B\\cap\\bigcap_{j=1}^{k}G_{j}^{\\mathrm{lie}} , the real long-pointers analyst finds a separating query in every round, which will conclude the proof. Consider any realization of the offsets and oracle for which B B holds. For any dataset S ′ S^{\\prime} , any masked-query transcript considered by the analyst, and any pointer sequence from P 1 , … , P k P_{1},\\ldots,P_{k} , the only locations on which the real oracle 𝒪 \\mathcal{O} and 𝒪 lying S ′ \\mathcal{O}_{\\mathrm{lying}}^{S^{\\prime}} may differ in the simulation on dataset S ′ S^{\\prime} are out-of-sample mask locations"},{"type":"para","text":"The event B B rules out every such query by ℳ \\mathcal{M} throughout the simulated interaction. At in-sample mask locations, both simulations answer using the already-read mask bits; at all non-mask locations, both answer according to 𝒪 \\mathcal{O} . Hence, under B B , every simulated execution in the exhaustive search has the same transcript with the real oracle as with the corresponding lying oracle. In particular, every query that is separating in the lying simulation is also separating in the real masked RO game, which concludes the proof. ∎"},{"type":"para","text":"Each separating query reduces the number of datasets consistent with the transcript by a factor of at least 9 / 10 9/10 . Starting from at most N n N^{n} possible datasets, after O ( n log N ) O(n\\log N) separating queries the mechanism must have answered some query inaccurately. Thus the long-pointer analyst forces a failure within O ( n log N ) O(n\\log N) rounds with probability at least 1 − ( k + 1 ) δ 1-(k+1)\\delta ."},{"type":"heading","level":4,"text":"Replacing long pointers by random finite pointers."},{"type":"para","text":"It remains to remove the idealized long pointers. Let"},{"type":"para","text":"For any fixed dataset and any fixed masked-query transcript, this is a crude upper bound on the number of RO locations exposed before a given experiment: recall that in each experiment the analyst reads the N N mask bits and simulates M M on all datasets and all N + 1 N+1 candidate threshold queries, with at most t t RO queries per simulation."},{"type":"para","text":"We now restrict the analyst into using pointers of the form p = b N R p=bNR , where b ∈ [ L prev + 1 ] b\\in[L_{\\mathrm{prev}}+1] . Observe that the number of bits required to represent an RO query in [ p + 1 , p + N R ] [p+1,p+NR] is"},{"type":"para","text":"Recall that k N = O ( n N log N ) kN=O(nN\\log N) , which is larger than the number of bits required to represent an input dataset in [ N ] n [N]^{n} by only a polynomial factor. Therefore, for reasonable values of t t (e.g., polynomial in the input length), the mechanism can indeed read the pointer and issue queries to the Random Oracle."},{"type":"para","text":"For every ( j , v ) (j,v) , there exists b j , v ∈ [ L prev + 1 ] b_{j,v}\\in[L_{\\mathrm{prev}}+1] such that the analyst did not query any RO location in [ b j , v N R + 1 , b j , v N R + N R ] [b_{j,v}NR+1,b_{j,v}NR+NR] before experiment ( j , v ) (j,v) ."},{"type":"para","text":"Let ( j , v ) (j,v) . Denote by W j , v W_{j,v} the set of RO locations exposed before experiment ( j , v ) (j,v) . By the construction of the analyst, | W j , v | ≤ L prev |W_{j,v}|\\leq L_{\\mathrm{prev}} . The segments [ b N R + 1 , b N R + N R ] [bNR+1,bNR+NR] for b ∈ [ L prev + 1 ] b\\in[L_{\\mathrm{prev}}+1] are disjoint, and are numbered strictly more than | W j , v | |W_{j,v}| , hence there exists b ∈ [ L prev + 1 ] b\\in[L_{\\mathrm{prev}}+1] such that [ b N R + 1 , b N R + N R ] ∩ W j , v = ∅ [bNR+1,bNR+NR]\\cap W_{j,v}=\\emptyset . ∎"},{"type":"para","text":"Therefore, each finite block is fresh at the moment it is used, and the analysis from the long pointer idealization goes through when using pointers p j , v = b j , v N R p_{j,v}=b_{j,v}NR . This completes the proof of Section ˜ 4.1 . ∎"},{"type":"heading","level":2,"text":"4.2 An attack in the RO model"},{"type":"para","text":"Now, we are ready to prove the main result ( Theorem ˜ 1.2 ). Formally,"},{"type":"para","text":"Let n , N = n 10 , k = O ( n log N ) , t , δ = N − 6 n,N=n^{10},k=O(n\\log N),t,\\delta=N^{-6} and R = O ( δ − 1 N n ( N 2 N log 1 δ ) k k t ) R=O(\\delta^{-1}N^{n}(N2^{N}\\log\\tfrac{1}{\\delta})^{k}kt) . Let ℳ \\mathcal{M} be a deterministic, 0.1 0.1 -accurate mechanism in the RO model that is given n n samples from an unknown distribution over the domain [ N ] × [ R ] [N]\\times[R] , and whose running time is bounded by t t . There exists an analyst 𝒜 \\mathcal{A} that picks queries whose representation size is O ( n N log N + log t ) O(nN\\log N+\\log t) bits, such that the queries can be evaluated on each domain point x ∈ [ N ] × [ R ] x\\in[N]\\times[R] in O ( n N log N + log t ) O(nN\\log N+\\log t) time, and 𝒜 \\mathcal{A} causes ℳ \\mathcal{M} to fail in k = O ( n log N ) k=O(n\\log N) rounds."},{"type":"para","text":"Observe that the mechanism’s running time is sublinear in the domain size, hence, in particular, the mechanism cannot evaluate the entire truth table of a query."},{"type":"para","text":"The proof is by forcing the mechanism to simulate a mechanism in the masked RO model established in Section ˜ 4.1 . I.e., we will show how to use the mechanism ℳ \\mathcal{M} to construct a deterministic, 0.1 0.1 -accurate masked RO mechanism ℳ masked \\mathcal{M}_{\\text{masked}} over the domain [ N ] [N] , with the exact round complexity, which will yield the theorem statement."},{"type":"heading","level":4,"text":"Domain Extension and Distribution Setup."},{"type":"para","text":"Define the domain 𝒳 ′ = [ N ] × [ R ] \\mathcal{X}^{\\prime}=[N]\\times[R] , for the same value of R R as in Section ˜ 4.1 . Draw N N pointers m 1 , … , m N m_{1},\\ldots,m_{N} independently and uniformly at random over [ R ] [R] . Define the underlying distribution 𝒫 \\mathcal{P} over 𝒳 ′ \\mathcal{X}^{\\prime} to be the uniform distribution over { ( i , m i ) ∣ i ∈ [ N ] } \\{(i,m_{i})\\mid i\\in[N]\\} . When a dataset X ′ = ( x 1 ′ , … , x n ′ ) X^{\\prime}=(x^{\\prime}_{1},\\dots,x^{\\prime}_{n}) of size n n is drawn i.i.d. from 𝒫 \\mathcal{P} , each individual sample point takes the form x i ′ = ( x i , m x i ) x^{\\prime}_{i}=(x_{i},m_{x_{i}}) , where x i x_{i} is drawn uniformly and independently from [ N ] [N] . Therefore, the mechanism gets a dataset ( x 1 , … , x n ) ∈ [ N ] n (x_{1},\\ldots,x_{n})\\in[N]^{n} , along with the corresponding pointers m x 1 , … , m x n m_{x_{1}},\\ldots,m_{x_{n}} ."},{"type":"heading","level":4,"text":"Formulating a Query."},{"type":"para","text":"To form a query, the analyst 𝒜 \\mathcal{A} does the following. Let 𝒜 m a s k , ℳ m a s k \\mathcal{A}_{mask},\\mathcal{M}_{mask} be an analyst and a mechanism operating in the masked RO model. At each round j ∈ [ k ] j\\in[k] , suppose 𝒜 m a s k \\mathcal{A}_{mask} outputs a masked query q ~ j ∈ { 0 , 1 } N \\tilde{q}_{j}\\in\\{0,1\\}^{N} and a pointer p j p_{j} . The analyst 𝒜 \\mathcal{A} formulates the query q j ( i , m ) = q ~ j ( i ) ⊕ 𝒪 ( p j + i R + m ) q_{j}(i,m)=\\tilde{q}_{j}(i)\\oplus\\mathcal{O}(p_{j}+iR+m) for all i , m i,m . This query can be represented by the pair ( q ~ j , p j ) (\\tilde{q}_{j},p_{j}) , and thus have size O ( N + n N log N + log t ) O(N+nN\\log N+\\log t) . Given a domain element ( i , m ) (i,m) , one can compute I = p j + i R + m I=p_{j}+iR+m , query the random oracle at index I I and eventually evaluate q j ( i , m ) = q ~ j ( i ) ⊕ 𝒪 ( I ) q_{j}(i,m)=\\tilde{q}_{j}(i)\\oplus\\mathcal{O}(I) . The running time of this evaluation is dominated by the O ( n N log N + log t ) O(nN\\log N+\\log t) time it takes to compute I I ."},{"type":"heading","level":4,"text":"Reduction to Masked RO Mechanism."},{"type":"para","text":"Observe that for a point ( i , m i ) (i,m_{i}) in the support, we have q j ( i , m i ) = q ~ j ( i ) ⊕ 𝒪 ( p j + i R + m i ) q_{j}(i,m_{i})=\\tilde{q}_{j}(i)\\oplus\\mathcal{O}(p_{j}+iR+m_{i}) , which equals the query issued by 𝒜 m a s k \\mathcal{A}_{mask} . Let us observe the information ℳ \\mathcal{M} holds and what it can compute. It holds a dataset ( x 1 , … , x n ) (x_{1},\\ldots,x_{n}) , pointers ( m x 1 , … , m x n ) (m_{x_{1}},\\ldots,m_{x_{n}}) , the transcript T j − 1 T_{j-1} , the query pointer p j p_{j} , the masked query q ~ j \\tilde{q}_{j} , and it can read at most t t bits of q j q_{j} or of the RO. Since the mechanism holds q ~ j \\tilde{q}_{j} , reading the ( i , m ) (i,m) -th bit of q j q_{j} is equivalent to reading the ( p j + i R + m ) (p_{j}+iR+m) -th bit of the random oracle. Thus, the mechanism has the same information and random oracle access as ℳ m a s k \\mathcal{M}_{mask} , who may issue at most t t queries to the RO. Therefore, as long as ℳ \\mathcal{M} is 0.1 0.1 -accurate, so is ℳ m a s k \\mathcal{M}_{mask} . By Section ˜ 4.1 , ℳ m a s k \\mathcal{M}_{mask} answers at most O ( n log N ) O(n\\log N) adaptive queries, and thus also ℳ \\mathcal{M} , concluding the proof."},{"type":"para","text":"∎"},{"type":"heading","level":2,"text":"4.3 Limited randomness"},{"type":"para","text":"We now consider mechanisms that have a small amount of private randomness. Specifically, consider mechanisms holding r r random bits. In all the models we considered, we can view this randomness as part of the input: that is, instead of a distribution 𝒫 n \\mathcal{P}^{n} over 𝒳 n \\mathcal{X}^{n} , we consider a product distribution 𝒫 n ⊗ U ( { 0 , 1 } r ) \\mathcal{P}^{n}\\otimes U(\\{0,1\\}^{r}) over 𝒳 n ⊗ { 0 , 1 } r \\mathcal{X}^{n}\\otimes\\{0,1\\}^{r} . Therefore, the number of possible datasets is bounded by N n 2 r N^{n}2^{r} . If the analyst finds a separating query in every round, the mechanism fails in O ( n log N + r ) O(n\\log N+r) rounds. We briefly explain how to change the presented proofs, and show that the analyst can find such separating queries."},{"type":"para","text":"First, observe that the argument for natural mechanisms goes through, as follows. Let X m ⊆ [ N ] X_{m}\\subseteq[N] be the set of common points, i.e., points appearing in the dataset S S for at least an 8 10 \\frac{8}{10} fraction of the states ( S , ρ ) ∈ H T (S,\\rho)\\in H_{T} . The counting argument leading to Equation ˜ 1 still holds, hence the number of common points is m = O ( n ) m=O(n) . Consequently, there is a threshold query q i q_{i} , where i i is not common, where the majority’s answer transitions from < 0.5 <0.5 to ≥ 0.5 \\geq 0.5 , hence q i q_{i} is separating."},{"type":"para","text":"The argument for natural mechanisms with a mask is unchanged, since it is by a black-box simulation. In the random oracle model, the analyst has to prepare for 2 r 2^{r} possibilities of the mechanism’s random bits, and have that in (most of) these realizations, with high probability the mechanism does not observe out-of-sample mask bits and does not violate the independence between experiments. To achieve this, it suffices to increase R R and L p r e v L_{prev} by a 2 r 2^{r} factor. Therefore, we obtain,"},{"type":"para","text":"Let n , N = n 10 , k = O ( n log N ) , t , δ = N − 6 , r n,N=n^{10},k=O(n\\log N),t,\\delta=N^{-6},r and R = O ( δ − 1 N n ( N 2 N log 1 δ ) k k t 2 r ) R=O(\\delta^{-1}N^{n}(N2^{N}\\log\\tfrac{1}{\\delta})^{k}kt2^{r}) . Let ℳ \\mathcal{M} be a deterministic, 0.1 0.1 -accurate mechanism in the RO model, that is given r r private random bits and n n samples from an unknown distribution over the domain [ N ] × [ R ] [N]\\times[R] , and whose running time is bounded by t t . There exists an analyst 𝒜 \\mathcal{A} that picks queries whose representation size is O ( n N log N + log t + r ) O(nN\\log N+\\log t+r) bits, such that the queries can be evaluated on each domain point x ∈ [ N ] × [ R ] x\\in[N]\\times[R] in O ( n N log N + log t + r ) O(nN\\log N+\\log t+r) time, and 𝒜 \\mathcal{A} causes ℳ \\mathcal{M} to fail in k = O ( n log N + r ) k=O(n\\log N+r) rounds."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Classical statistical theory for asserting the validity of a hypothesis tells us that the description of the hypothesis should be independent from the data on which it is evaluated. In practice, however, analysts frequently reuse the same dataset to answer multiple questions. Often, this exploration is adaptive: an analyst asks a statistical question, observes the answer, and uses that information to decide what to ask next. While this is how data analysis naturally happens, it makes it incredibly easy to overfit. Without careful intervention, the analyst will quickly find patterns that exist in the specific sample but fail to generalize to the true underlying distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This problem was formalized by Dwork et al. [ DFH+15a ] in what has come to be known as the Adaptive Data Analysis (ADA) problem. In this problem, a mechanism ℳ \\\\mathcal{M} holds a dataset S S consisting of n n samples drawn i.i.d. from an unknown distribution 𝒫 \\\\mathcal{P} over a domain 𝒳 \\\\mathcal{X} . An analyst iteratively (and adaptively) submits a sequence of k k statistical queries q : 𝒳 → [ 0 , 1 ] q:\\\\mathcal{X}\\\\to[0,1] . The mechanism ℳ \\\\mathcal{M} must answer these queries such that every answer is ε \\\\varepsilon -accurate (i.e., within ± ε \\\\pm\\\\varepsilon ) with respect to the true expectation of the query over 𝒫 \\\\mathcal{P} . To provide worst-case guarantees, the analyst is assumed to be adversarial, and is often referred to as an attacker . The central question in ADA is establishing the optimal sample complexity: how large can k k be as a function of n n ?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This question is relatively well-understood for randomized mechanisms. A deep connection between ADA and various stability notions (such as differential privacy) has shown that randomized mechanisms can safely answer a large number of adaptive queries without overfitting [ DFH+15a , DFH+15b , BNS+16 , RRS+16 , RZ16 , FS17 , FS18 , FRR18 , LS19 , SZ20 , JLN+20 , SL23 , BLA23 ] . Specifically, there are computationally efficient randomized mechanisms supporting k ≈ n 2 k\\\\approx n^{2} queries, and, without making specific assumptions, no mechanism, even computationally unbounded, can answer k ≫ n 2 k\\\\gg n^{2} queries. 1 1 1 If one assumes that the domain 𝒳 \\\\mathcal{X} is not too large, namely that n ≥ polylog | 𝒳 | n\\\\geq\\\\operatorname{polylog}|\\\\mathcal{X}| , then there exist computationally inefficient mechanisms supporting k ≈ exp ( n / polylog | 𝒳 | ) k\\\\approx\\\\exp(n/\\\\operatorname{polylog}|\\\\mathcal{X}|) queries. See Section 1.3 (related works).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, all known constructions for the ADA problem rely fundamentally on the mechanism’s ability to inject carefully calibrated random noise into its answers (again, unless one makes specific assumptions, under some of which deterministic mechanisms do exist). This reliance raises a natural question about the nature of adaptive data analysis: is randomness strictly necessary for ADA?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Resolving this question is fundamental to understanding the algorithmic toolkit required to prevent overfitting in the face of adaptive adversaries. This pursuit mirrors central lines of inquiry in other areas of theoretical computer science where separating the power of deterministic and randomized algorithms has been a long-standing theme. This includes works on communication complexity [ YAO79 , KS92 ] , streaming algorithms [ FM85 , AMS96 , HNO08 , CK16 , KPW21 , KMN+21 , CS24 ] , dynamic algorithms [ BKM+22 , BEF+23 , BBF+26 ] , and distributed computing [ BEN83 , FLP85 , LR81 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Our Contributions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The primary contribution of this work is resolving the role of randomness for ADA in the information-theoretic Random Oracle (RO) model. 2 2 2 A random oracle is an infinite public random string 𝒪 \\\\mathcal{O} , such that the algorithm can read the i i -th bit in 𝒪 \\\\mathcal{O} , denoted 𝒪 ( i ) \\\\mathcal{O}(i) , at the cost of one time unit. We assume that each bit of 𝒪 \\\\mathcal{O} is uniformly distributed and independent of all other bits. Thus, the only way to get any information on 𝒪 ( i ) \\\\mathcal{O}(i) is to read this bit. The random oracle model is a common assumption in cryptography, dating back to at least [ FS86 ] and formalized in the seminal work of Bellare and Rogaway [ BR93 ] . We establish that to answer a super-linear number of adaptive queries, an algorithm must fundamentally rely on internal randomness. We demonstrate this by proving a strong separation: while deterministic mechanisms can match the performance of randomized ones against computationally bounded analysts, they are provably insufficient against unbounded analysts in the random oracle model.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baseline for bounded analysts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To put our impossibility result in context, we first formalize a folklore observation (dating back to the initial works on ADA) that randomness is not necessary when the analyst is computationally bounded. Specifically, by extracting a small number of random bits from the input sample and stretching them using cryptographic pseudorandom generators, we recover the same quadratic bound achievable by randomized mechanisms. Thus, when the analyst is bounded, randomness is a convenience rather than a necessity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assuming that pseudorandom generators exist, there is a computationally efficient deterministic mechanism that, given n n i.i.d. samples from an unknown distribution 𝒫 \\\\mathcal{P} , answers k = Ω ~ ( n 2 ) k=\\\\tilde{\\\\Omega}(n^{2}) adaptively chosen statistical queries w.r.t. 𝒫 \\\\mathcal{P} , provided that the entity selecting the queries is computationally efficient.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This observation was hinted at by some prior work without being stated or proven explicitly (see, e.g., [ STE16 ] ). Our contribution is in formalizing and proving it, which requires addressing some edge cases. The details are deferred to Appendix A .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Main result: Separation for unbounded analysts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now turn to the main technical focus of this paper, providing strong evidence that when the analyst is computationally unbounded, the situation changes drastically. Specifically, we prove that in the random oracle model, any deterministic mechanism can be forced to fail after just k = O ~ ( n ) k=\\\\tilde{O}(n) queries. We establish this negative result in two stages:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Natural mechanisms: We first consider a restricted class of deterministic mechanisms, called “natural” mechanisms, which observe only the evaluations of a query on the sample points. That is, a natural mechanism observes only ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\\\dots,q(x_{n})) , where x 1 , … , x n x_{1},\\\\dots,x_{n} denote the sample points, rather than the full query description q q . Natural mechanisms were also considered by [ HU14 , SU15 , NSS+18 ] as a steppingstone towards showing impossibility results for randomized mechanisms. These prior works showed that randomized natural mechanisms cannot answer more than O ( n 2 ) O(n^{2}) queries. We present a completely different attack (and analysis) showing that deterministic natural mechanisms cannot answer more than O ~ ( n ) \\\\tilde{O}(n) queries before failing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The random oracle model: We then extend this barrier to general (i.e., not necessarily natural) deterministic mechanisms, under the assumption that both the analyst and the mechanism have access to a random oracle. Even when the deterministic mechanism can query the oracle at arbitrary positions and base its future actions on the obtained values, we prove that it cannot exceed the O ~ ( n ) \\\\tilde{O}(n) query limit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our main result is specified in the following theorem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the random oracle model, for every deterministic mechanism ℳ \\\\mathcal{M} and every sample size n n , there is an attacker 𝒜 \\\\mathcal{A} that with high probability, forces ℳ \\\\mathcal{M} to fail after just k = O ~ ( n ) k=\\\\tilde{O}(n) adaptive queries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An appealing aspect of the attacks presented by prior works was that they constructed a single attacker that defeats every mechanism (albeit requiring n 2 n^{2} rounds). Our order of quantifiers is necessarily weaker: for every deterministic mechanism, we present a different attacker. This is unavoidable to beat the quadratic query bound. Indeed, if there were a single attacker that defeated every deterministic algorithm in O ~ ( n ) \\\\tilde{O}(n) rounds, then by the law of total probability, this attacker would also defeat randomized algorithms, which is impossible as randomized algorithms can support Ω ~ ( n 2 ) \\\\tilde{\\\\Omega}(n^{2}) queries.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Extension to mechanisms with limited randomness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our techniques also extend to algorithms equipped with a limited amount of private randomness. Specifically, if a mechanism has access to r r random bits, we can conceptually view these bits as an extension of the empirical sample itself. By suitably adapting our attack, we show that it is possible to force such a mechanism to fail after k = O ~ ( n + r ) k=\\\\tilde{O}(n+r) adaptive queries. This further clarifies the role of randomness in adaptive data analysis (at least in the RO model): to answer Ω ~ ( n 2 ) \\\\tilde{\\\\Omega}(n^{2}) adaptive queries, an algorithm must fundamentally rely on a substantial amount of internal randomness. Any attempt to substitute this with deterministic operations or insufficient randomness ( r ≪ n 2 r\\\\ll n^{2} ) leaves the mechanism vulnerable to rapid overfitting. We remark that Θ ~ ( n 2 ) \\\\tilde{\\\\Theta}(n^{2}) random bits are known to be sufficient, see for example [ BNS+16 , BLA23 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the random oracle model, for every mechanism ℳ \\\\mathcal{M} that uses at most r r random bits, and every sample size n n , there is an attacker 𝒜 \\\\mathcal{A} that with high probability forces ℳ \\\\mathcal{M} to fail after just k = O ~ ( n + r ) k=\\\\tilde{O}(n+r) adaptive queries.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.2 Technical Overview for Theorem 1.2\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"1.2.1 A negative result for natural mechanisms\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first describe an attack against natural mechanisms (recall that a natural mechanism does not observe the full description of the queries it needs to answer; only their evaluations on the empirical sample points). Denote N = poly ( n ) N=\\\\operatorname{poly}(n) , let the target distribution be uniform over the domain [ N ] [N] , and fix a natural deterministic mechanism ℳ \\\\mathcal{M} . We construct a computationally unbounded attacker that throughout the execution maintains a set of “surviving datasets”, denoted H T H_{T} , containing all possible datasets of size n n that are consistent with the transcript T T of the interaction with ℳ \\\\mathcal{M} so far. Initially, H T H_{T} contains all possible datasets of size n n over [ N ] [N] , so | H T | = N n |H_{T}|=N^{n} . The attacker’s goal is to rapidly shrink H T H_{T} by issuing a sequence of adaptively chosen queries. We say that a query is separating if every valid answer (which must be close to the true distributional expectation, i.e., to the average over [ N ] [N] ) is forced to be inconsistent with a constant fraction of the datasets currently in H T H_{T} . Intuitively (but somewhat inaccurately) after k = O ( n log N ) k=O(n\\\\log N) such separating queries, the analyst will have learned the dataset, allowing it to present a query that highly overfits to the data, thereby forcing ℳ \\\\mathcal{M} to fail.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now explain how the attacker finds such a separating query. For simplicity, let us focus on the first round of the interaction (so currently H T H_{T} contains all datasets). The separating query our attacker chooses will be a threshold query over N N , i.e., a query that evaluates to 0 on a prefix of the domain [ N ] [N] . To this end, consider an | H T | × ( N + 1 ) |H_{T}|\\\\times(N+1) table, where cell ( i , j ) (i,j) contains the answer ℳ \\\\mathcal{M} would have returned had its dataset been S i S_{i} and had it been presented with the threshold query q j q_{j} , defined as q j ( x ) = 𝟙 { x ≤ j } q_{j}(x)=\\\\mathds{1}_{\\\\{x\\\\leq j\\\\}} . See Figure 1 for an illustration of how this table might look like.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that the attacker can compute this table before selecting its first query (by simulating ℳ \\\\mathcal{M} ). Also, observe that by the accuracy guarantees of ℳ \\\\mathcal{M} , it must be that the first column is essentially 0 → \\\\vec{0} and the last column is essentially 1 → \\\\vec{1} . The key observation regarding this table is that when going from one column j − 1 j-1 to the next column j j , only a small fraction of the coordinates might change. To see this, recall that the threshold queries q j − 1 q_{j-1} and q j q_{j} differ only in the j j th point. Thus, as ℳ \\\\mathcal{M} is natural, it cannot notice this change unless its input dataset contains the point j j . The fraction of datasets containing the j j th point is at most n / N n/N , which is small. So far we have established that:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The left column is essentially all zeros.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The right column is essentially all ones.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Only a small fraction of the coordinates can change between every pair of consecutive columns.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These three properties together imply the existence of a “non-uniform” column, meaning a column such that no value appears in, say, more than 90% of its coordinates. To see this, let j j be the first column where more than 50% of the coordinates are strictly bigger than 0.5. So in column j − 1 j-1 at least 50 % 50\\\\% of the coordinates are at most 0.5. As the columns change slowly, at least ( 1 2 − n N ) (\\\\frac{1}{2}-\\\\frac{n}{N}) of the coordinates in column j j must also be at most 0.5. So column j j is “non-uniform”, and the corresponding query q j q_{j} is separating : no matter what answer the attacker gets after querying q j q_{j} , this answer allows it to rule out ≈ 1 / 2 \\\\approx 1/2 of the datasets in H T H_{T} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the next round, in order to find the next separating query, the attacker constructs a new table where the y-axis contains only datasets that were not ruled out in previous rounds. We might also need to exclude a small number of points from the x-axis, corresponding to points that appear in “too many” of the surviving datasets. Otherwise, keeping such “common” points active might invalidate our key observation about the columns changing slowly. We show that there could be at most O ( n ) O(n) such common points, and that excluding them has almost no effect on the calculation. Note that the values in the next table could be different than the values of the previous table, as the values returned by ℳ \\\\mathcal{M} could be history dependent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intuitively, by sequentially applying such separating queries, the attacker isolates the true dataset in at most k = O ( n log N ) k=O(n\\\\log N) rounds (unless the mechanism errs before that, in which case the attacker still wins).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"1.2.2 A naïve attempt at lifting this barrier to general deterministic mechanisms\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Extending our impossibility result to general deterministic (not necessarily natural) mechanisms requires overcoming their ability to evaluate the full query description. The standard path for trying to achieve this would be for the attacker to encrypt the queries such that the mechanism could only decrypt the values corresponding to its empirical sample. More specifically, the target distribution would no longer be uniform over [ N ] [N] . Instead, we first generate N N encryption keys s 1 , … , s N s_{1},\\\\dots,s_{N} , and then define the target distribution to be uniform over the pairs ( j , s j ) (j,s_{j}) . Now, instead of directly issuing a query q : [ N ] → { 0 , 1 } q:[N]\\\\rightarrow\\\\{0,1\\\\} , the attacker encrypts c j = Encrypt ( q ( j ) , s j ) c_{j}={\\\\rm Encrypt}(q(j),s_{j}) for every j ∈ [ N ] j\\\\in[N] , and gives the vector of all encryptions c → \\\\vec{c} to the mechanism. The mechanism has the secret keys for points in its empirical sample, and can hence learn the value of q q restricted to its sample, but learns nothing on q q outside of its sample. Thus, the mechanism is effectively forced to behave like a natural mechanism. This strategy allowed the prior works of [ HU14 , SU15 , NSS+18 ] to lift their impossibility result from (randomized) natural mechanisms to general mechanisms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, our attack against natural deterministic mechanisms (defeating them in just O ~ ( n ) \\\\tilde{O}(n) rounds) is fundamentally different from attacks presented in prior works. As a result, the strategy outlined above breaks down in our setting. Intuitively, the reason is that the mechanism could potentially use the ciphertexts and the (in-sample) keys as a source of randomness, undermining our assumption that the mechanism is deterministic. One might hope that the argument could still go through, because these keys and ciphertexts are known to the attacker, and are thus more akin to public randomness rather than private randomness. Indeed, our attack would work seamlessly against a mechanism with public randomness, provided that it is natural for every realization of the public randomness. But this is not what the encryption-based strategy outlined above guarantees: changing an out-of-sample query value, without changing the corresponding encryption key, alters the resulting ciphertext. This changes the description of the query, which can subsequently affect the mechanism’s output. In other words, this encryption-based strategy only guarantees that the distribution of answers given by the mechanism must be independent of out-of-sample query values over the randomness of generating the keys and encryptions . It does not guarantee that the mechanism is natural for every fixed realization of the keys and ciphertexts, which is what is needed for the arguments from Section 1.2.1 to go through.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"1.2.3 Towards an impossibility result for non-natural mechanisms\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To overcome the challenge outlined above, we take a different route for extending our impossibility result also to non-natural mechanisms. We still use encryption (more precisely, one time pads or random masks ) to hide out-of-sample query values. But we will argue directly about the behavior of the deterministic mechanism when given specific realizations of the mask, rather than arguing about what the mechanism “learns” or “cannot learn” when these masks are random.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To simplify the presentation, let us not embed these masks in the underlying distribution just yet. So the target distribution is still uniform over [ N ] [N] . Now, imagine a hypothetical model where in every round the attacker specifies a query q ∈ { 0 , 1 } N q\\\\in\\\\{0,1\\\\}^{N} and a mask y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} , and the mechanism obtains q ⊕ y ∈ { 0 , 1 } N q\\\\oplus y\\\\in\\\\{0,1\\\\}^{N} (this corresponds to the encrypted truth-table of q q using y y ), and also obtains y | S y|_{S} , which is the vector y y restricted to the coordinates in the empirical sample S S held by the mechanism. Conceptually, we can think of such a mechanism as being “partially natural”, as it gets all the coordinates of q ⊕ y q\\\\oplus y while only getting the in-sample coordinates of y y .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This hypothetical model gives the attacker the ability to choose the masks y y adaptively during every round, rather than committing and embedding them in target distribution before the game begins.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The goal of the attacker remains the same: it maintains the set H T H_{T} of “surviving datasets” and in every round it aims to find a separating masked-threshold query that will necessarily rule out a constant fraction of the datasets in H T H_{T} . To do this, the attacker simulates ℳ \\\\mathcal{M} on every possible dataset S ∈ H T S\\\\in H_{T} with every possible query-mask pair q , y q,y (the inputs of the mechanism in this case are the dataset S S , the encrypted threshold query q ⊕ y q\\\\oplus y , and the in-sample mask values y | S y|_{S} ). See Figure 2 for an illustration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As before, the attacker can compute this cube before selecting its query. Furthermore, the left face of the cube is essentially all zeroes (corresponding to the query q 0 ≡ 0 q_{0}\\\\equiv 0 ) and the right face of the cube is essentially all ones (corresponding to the query q N ≡ 1 q_{N}\\\\equiv 1 ). Note that fixing the query q q and the mask y y , while leaving the dataset S S “free”, can be visualized as a vertical pin in Figure 2 . The attacker’s goal is to find such a vertical pin with “non-uniform” values along it, in the sense that no value appears in, say, more than 90% of its coordinates. To show that such a non-uniform pin exists, let us introduce the following notation: fix values j ∈ [ N ] j\\\\in[N] and y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} defining a pin ( q j , y ) (q_{j},y) . We say that its neighboring pin is the pin ( q j + 1 , y ′ ) (q_{j+1},y^{\\\\prime}) , where y ′ = y ⊕ e j + 1 y^{\\\\prime}=y\\\\oplus e_{j+1} . That is, the neighboring pin of ( q j , y ) (q_{j},y) is obtained by advancing to the next threshold query q j + 1 q_{j+1} and by flipping the ( j + 1 ) (j+1) th coordinate in y y . See the blue and orange pins in Figure 2 for an illustration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The key observation regarding these neighboring pins is that unless the input dataset contains the point j + 1 j+1 , then the value returned by ℳ \\\\mathcal{M} cannot change when going from pin ( q j , y ) (q_{j},y) to pin ( q j + 1 , y ′ ) (q_{j+1},y^{\\\\prime}) , because its inputs remain exactly the same (because q j ⊕ y = q j + 1 ⊕ y ′ q_{j}\\\\oplus y=q_{j+1}\\\\oplus y^{\\\\prime} ). Thus, the vast majority of the coordinates of these two pins are identical.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now fix some value for y y , and consider the pin ( q 0 , y ) (q_{0},y) , which is a vertical line on the left face of the cube. Consider a path of neighboring pins where we start from ( q 0 , y ) (q_{0},y) and iteratively switch to the next neighboring pin until we reach the right face of the cube. This path behaves similarly to our 2D table illustrated in Figure 1 : (1) The initial pin is essentially all zeros; (2) the final pin is essentially all ones; and (3) only a small fraction of the coordinates can change between every pair of consecutive pins along this path. Thus, via similar arguments, there must be a “non-uniform” pin along this path.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fact that such a non-uniform pin exists suffices for the attacker to win in this hypothetical model: in every iteration the attacker constructs a 3D cube, finds a non-uniform pin and queries it, thereby shrinking the set of surviving datasets multiplicatively. However, as we alluded to before, we will later want to embed the mask y y (or multiple masks) in the underlying distribution, so the attacker will not have the freedom to select arbitrary masks at different rounds of the interaction. Luckily, as we now explain, a noticeable fraction of the y y ’s are good in the sense that there is a non-uniform pin with this specific y y . To see this, observe that since every path (starting from every y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} ) has a non-uniform pin, and as there are at most N + 1 N+1 queries, then there must exist a query q q that captures at least 1 N + 1 \\\\frac{1}{N+1} fraction of these non-uniform pins. That is, there must exist a query q q such that for at least 1 N + 1 \\\\frac{1}{N+1} -fraction of the y y ’s it holds that ( q , y ) (q,y) is non-uniform. In particular, at least 1 N + 1 \\\\frac{1}{N+1} -fraction of the y y ’s are good in the sense that there is a query q q such that the pin ( q , y ) (q,y) is non-uniform.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"1.2.4 Pitfalls for embedding masks in the target distribution\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now that we know that a random mask y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} is good with probability at least 1 N + 1 \\\\frac{1}{N+1} , can we embed such random masks in the target distribution, as we contemplated in Section 1.2.2 ? We now list several pitfalls that, surprisingly, make this very unclear.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Pitfall #1: Boosting the success probability of the first round.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A success probability of 1 N + 1 \\\\frac{1}{N+1} per round is insufficient to break the mechanism in O ( n log N ) O(n\\\\log N) queries, as we need roughly n log N n\\\\log N successful rounds to isolate the input dataset. To emphasize Pitfall #1, let us now focus only on the first round and on amplifying its success probability. Since every random mask is good with probability 1 N + 1 \\\\frac{1}{N+1} , one might try to embed r = poly ( N ) r=\\\\operatorname{poly}(N) random masks in the target distribution, hoping that one of them will be good. Specifically, we first sample r r random masks y 1 , … , y r ∈ { 0 , 1 } N y_{1},\\\\dots,y_{r}\\\\in\\\\{0,1\\\\}^{N} uniformly, and then define the target distribution to be uniform over the tuples ( i , y 1 [ i ] , … , y r [ i ] ) (i,y_{1}[i],\\\\dots,y_{r}[i]) . Now, before asking its first query, the attacker constructs its 3D cube, where the mask axis (the depth axis in Figure 2 ) is restricted to the vectors y 1 , … , y r y_{1},\\\\dots,y_{r} (because these are the masks that are “available” for the attacker to choose from). If there is a non-uniform pin in this cube, then the attacker queries it and succeeds in shrinking H T H_{T} . But what is the probability of this event? Is it close to 1?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By the analysis in Section 1.2.3 , we know that for every fixture of the vectors y 1 , … , y ℓ − 1 , y ℓ + 1 , … , y r y_{1},\\\\dots,y_{\\\\ell-1},y_{\\\\ell+1},\\\\dots,y_{r} , with probability at least 1 N + 1 \\\\frac{1}{N+1} over sampling y ℓ y_{\\\\ell} it holds that y ℓ y_{\\\\ell} is good. The issue is that the mechanism ℳ \\\\mathcal{M} gets all of these r r vectors as an input (restricted to its sample), and can base its output on all of them. Thus, it is unclear that the event that y 1 y_{1} is good is independent of the event that y 2 y_{2} is good. In principle, it could be that either all of y 1 , … , y r y_{1},\\\\dots,y_{r} are good, or none of them is good, and so the probability that there exists a good mask remains 1 N + 1 \\\\frac{1}{N+1} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To illustrate this issue, consider r r random variables Z 1 , … , Z r Z_{1},\\\\dots,Z_{r} distributed independently and uniformly over { 0 , 1 , 2 , … , 6 } \\\\{0,1,2,\\\\dots,6\\\\} , and suppose that Z ℓ Z_{\\\\ell} is good if ∑ i = 1 r Z i ( mod 7 ) = 0 \\\\sum_{i=1}^{r}Z_{i}\\\\pmod{7}=0 . If we fix all other variables and consider a single Z ℓ Z_{\\\\ell} , the residue equals 0 with probability exactly 1 / 7 1/7 . While this marginal probability holds for all ℓ \\\\ell , the global constraint rigidly couples the variables. No matter how many variables you inspect, i.e., no matter what r r is, it does not yield any overall probability amplification.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Pitfall #2: Success probability of the second round.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose that we do get amplification for the success probability in the first round. What about the second round? Is it even true that it succeeds with probability at least 1 N + 1 \\\\frac{1}{N+1} ? The issue now is that the attacker chooses the first query based on the constructed 3D cube, which depends on all the mask vectors y 1 , … , y r y_{1},\\\\dots,y_{r} since its values are determined by simulating ℳ \\\\mathcal{M} with different portions of these masks. Then, as ℳ \\\\mathcal{M} might be history dependent, in the second round its behavior might depend on the first query, and thus depend on the mask vectors. So y 1 , … , y r y_{1},\\\\dots,y_{r} are no longer independent of the cube in the second round. Is it still true that each of them is good with noticeable probability? If so, do we still get amplification by having r r masks?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"1.2.5 Resolution using a random oracle and dynamic pointers\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To bypass these pitfalls, we turn to the Random Oracle (RO) model, where the attacker and mechanism have access to an oracle 𝒪 : ℕ → { 0 , 1 } \\\\mathcal{O}:\\\\mathbb{N}\\\\to\\\\{0,1\\\\} whose values are uniformly random and independent bits. The conceptual benefit we get from the random oracle is that we will not give the mechanism the actual mask bits upfront. Instead, we will specify these masks dynamically using different pointers to the random oracle. To do this, the underlying domain is extended to [ N ] × [ R ] [N]\\\\times[R] , where R R is a large integer, and each sample point x i x_{i} is accompanied by a uniformly random pointer offset m i ∈ [ R ] m_{i}\\\\in[R] . Thus, at the start of the game, the mechanism receives the pointers m i m_{i} corresponding to the points x i x_{i} in its empirical sample. In each round v v , the analyst dynamically determines an additional pointer p v p_{v} . The mask for a domain element i i in this round is then derived from the oracle as 𝒪 ( p v + m i ) \\\\mathcal{O}(p_{v}+m_{i}) . (For technical reasons, in the actual construction the positions in the RO are determined somewhat differently.) This construction guarantees that:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because the mechanism only receives the offset m i m_{i} for its sample points, it cannot easily guess the locations of out-of-sample masks. This enforces the behavior we needed in the analysis of our hypothetical model in Section 1.2.3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The additional offset p v p_{v} acts as a “refresh button”, allowing the attacker to test and choose pointers to previously unread positions of the Random Oracle. This allows us to bypass our two pitfalls:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Within each round v v , the attacker could try out several values for the offset p v p_{v} , simulating the behavior of ℳ \\\\mathcal{M} with each one of them. The different values for p v p_{v} guarantee strict independent repetitions, bypassing Pitfall #1 and allowing the attacker to amplify the probability of finding a separating query.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across different rounds, the dynamic pointer completely severs historical dependency, guaranteeing that the masks for the new round are drawn perfectly fresh, entirely independent of the transcript so far. So this also bypasses Pitfall #2.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"1.2.6 Challenges in the random oracle model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To make the aforementioned guarantees hold, the random offsets m i m_{i} and the dynamic pointers p p must be drawn from a sufficiently large range. But how large? If we are not careful, analyzing the interactions between a computationally unbounded analyst and an adaptive mechanism could lead to a catastrophic blowup in the required range. Specifically, suppose that the mechanism runs in time t t . As the mechanism can be adaptive, it can potentially explore a decision tree of oracle queries of size 2 t 2^{t} per simulation. Naïvely union-bounding over all possible adaptive paths could force us to set the pointer range to be exponential in t t . This creates a fatal contradiction: a range of that magnitude would result in pointers whose description length is larger than t t . But a mechanism constrained to run in time t t would be unable to even read these pointers, let alone answer the queries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We show that it suffices to represent pointers using O ( n N log N + log t ) O(nN\\\\log N+\\\\log t) bits, which is larger than the size of the dataset by only a polynomial factor (recall N = poly ( n ) N=\\\\operatorname{poly}(n) ), guaranteeing that the deterministic mechanism can feasibly read the pointers and answer queries within its time bound t t . This is somewhat easier to achieve for the secret offsets m i m_{i} (we show that with high probability, in all possible executions, the mechanism does not observe an out-of-sample mask bit). Arguing about the dynamic pointer p p is more subtle, because the unbounded attacker needs to continually find “fresh” positions in the random oracle, which requires carefully tracking the exhaustive search tree of the simulation. That is, unlike m i m_{i} , arguing about p p requires us to take into account also the attacker’s behavior, not just the mechanism’s behavior.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.3 Other Related Works\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The literature on ADA is vast, with many exciting results beyond those discussed earlier. For example, researchers have explored variants of the ADA problem where the mechanism only needs to guarantee utility against “weaker” or “more realistic” adversaries, and shown impossibility results for these settings. Nissim et al. [ NST23 ] proposed a “balanced” model that separates the entity choosing the target distribution from the attacker, thus removing the ability to embed secrets in the target distribution that only the attacker is privy to. They demonstrated that even in this more realistic setting, answering more than O ~ ( n 2 ) \\\\tilde{O}(n^{2}) queries is hard. In a different vein, Dinur et al. [ DSW+23 ] proved that existing lower bounds for ADA can be interpreted as arising from a space bottleneck rather than a sampling bottleneck.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A growing body of literature considers the challenges of ADA with correlated input samples. Kontorovich et al. [ KSS22 ] extended stability-based generalization guarantees to non-i.i.d. settings by formally quantifying the degree of correlation between samples. Rapoport et al. [ RCS25 ] recently established tight lower bounds for these settings, proving an unavoidable utility gap: under natural conditions, one can answer at most O ( n ) O(n) adaptively chosen queries on correlated data (even using randomized mechanisms), in sharp contrast to the O ~ ( n 2 ) \\\\tilde{O}(n^{2}) limit in the independent setting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In our work, we focus on the vanilla, unrestricted formulation of the ADA problem. As we mentioned, if we assume that the domain 𝒳 \\\\mathcal{X} is not too large, namely that n ≥ polylog | 𝒳 | n\\\\geq\\\\operatorname{polylog}|\\\\mathcal{X}| , then answering more than n 2 n^{2} queries becomes possible (albeit inefficiently), even using deterministic mechanisms. Specifically, Dwork et al. [ DFH+15a ] and Bassily et al. [ BNS+16 ] showed that there are (inefficient) mechanisms that answer exp ( n / log | 𝒳 | ) \\\\exp(n/\\\\sqrt{\\\\log|\\\\mathcal{X}|}) adaptive queries using a sample of size n n . Dwork et al. [ DFH+15b ] showed that a qualitatively similar positive result can be obtained using an (inefficient) deterministic mechanism. Recently, this query bound of exp ( n / log | 𝒳 | ) \\\\exp(n/\\\\sqrt{\\\\log|\\\\mathcal{X}|}) was shown to be tight by Lyu and Talwar [ LT25 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Preliminaries\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first define the Adaptive Data Analysis (ADA) framework, distributional accuracy, and natural algorithms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the ADA framework, a mechanism ℳ \\\\mathcal{M} interacts with a data analyst 𝒜 \\\\mathcal{A} over k k rounds. Given a dataset S = ( x 1 , … , x n ) ∼ 𝒫 n S=(x_{1},\\\\dots,x_{n})\\\\sim\\\\mathcal{P}^{n} drawn i.i.d. from a distribution 𝒫 \\\\mathcal{P} over a domain 𝒳 \\\\mathcal{X} , in each round j ∈ [ k ] j\\\\in[k] :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analyst 𝒜 \\\\mathcal{A} outputs a statistical query q j : 𝒳 → [ 0 , 1 ] q_{j}:\\\\mathcal{X}\\\\to[0,1] , which may depend on previous queries and answers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mechanism ℳ \\\\mathcal{M} observes q j q_{j} and outputs an answer a j ∈ ℝ a_{j}\\\\in\\\\mathbb{R} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ε > 0 , k , n , N , 𝒳 \\\\varepsilon>0,k,n,N,\\\\mathcal{X} . A mechanism ℳ \\\\mathcal{M} is ε \\\\varepsilon -accurate for k k adaptive queries given a sample of size n n , if for each distribution 𝒫 \\\\mathcal{P} over 𝒳 \\\\mathcal{X} and analyst 𝒜 \\\\mathcal{A} , with probability at least 9 / 10 9/10 over the sample S ∼ 𝒫 n S\\\\sim\\\\mathcal{P}^{n} , and the randomness of ℳ \\\\mathcal{M} , the answers a 1 , … , a k a_{1},\\\\dots,a_{k} satisfy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A mechanism ℳ \\\\mathcal{M} is called natural if, when given a dataset S = ( x 1 , … , x n ) S=(x_{1},\\\\dots,x_{n}) , then for every given statistical query q : 𝒳 → [ 0 , 1 ] q:\\\\mathcal{X}\\\\to[0,1] , the mechanism ℳ \\\\mathcal{M} only observes the query evaluations on the sample points, i.e., ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\\\dots,q(x_{n})) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Natural Mechanisms\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first provide an attack against natural deterministic mechanisms. Suppose our domain is 𝒳 = [ N ] \\\\mathcal{X}=[N] , where N N is at most poly ( n ) \\\\operatorname{poly}(n) . (If | 𝒳 | |\\\\mathcal{X}| is larger, just restrict the distribution to be supported only on the first N N elements of 𝒳 \\\\mathcal{X} .) Consider a natural deterministic mechanism ℳ \\\\mathcal{M} that, given a sample of size n n , is 0.1 0.1 -accurate for k k adaptive queries by a computationally unbounded analyst. We aim to show that k = O ( n log N ) k=O(n\\\\log N) . Consider the uniform distribution over [ N ] [N] , and suppose for simplicity that the sample is a set and not a multi-set (i.e., there are no collisions). This assumption can be achieved with high probability by setting the domain size as N = n 10 N=n^{10} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An unbounded adversary 𝒜 \\\\mathcal{A} can simulate the answers of all possible datasets. Thus, for a transcript T = ( q 1 , a 1 , … , q i , a i ) T=(q_{1},a_{1},\\\\dots,q_{i},a_{i}) , the adversary can compute the set of datasets that are consistent with T T , denoted H T ⊆ [ N ] n H_{T}\\\\subseteq[N]^{n} . Furthermore, 𝒜 \\\\mathcal{A} can compute the answers of datasets in H T H_{T} for any query q q . The adversary aims to find a query for which every answer a a is consistent with at most 9 10 \\\\frac{9}{10} fraction of the datasets in H T H_{T} , called a separating query . The next lemma shows that 𝒜 \\\\mathcal{A} can indeed find a separating query, and moreover, that query is binary q : [ N ] → { 0 , 1 } q:[N]\\\\to\\\\{0,1\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let a deterministic natural mechanism ℳ \\\\mathcal{M} and a transcript T = ( q 1 , a 1 , … , q i , a i ) T=(q_{1},a_{1},\\\\dots,q_{i},a_{i}) . There exists a query q : [ N ] → { 0 , 1 } q:[N]\\\\to\\\\{0,1\\\\} that either causes the mechanism to have error > 0.1 >0.1 , or is a separating query for ℳ \\\\mathcal{M} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proof uses threshold queries , defined as follows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let X m = { x 1 , … , x m } ⊂ [ N ] X_{m}=\\\\{x_{1},\\\\dots,x_{m}\\\\}\\\\subset[N] be a set of points. Let the remaining elements of the domain be arbitrarily ordered as z 1 , z 2 , … , z N − m z_{1},z_{2},\\\\dots,z_{N-m} . A threshold query is any query of the form,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The set X m X_{m} as defined in Section ˜ 3 is arbitrary, but in the proof of Section ˜ 3 , it will be a set of m = O ( n ) m=O(n) points that appear in most of the databases in H T H_{T} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let X m = { x 1 , … , x m } ⊂ [ N ] X_{m}=\\\\{x_{1},\\\\dots,x_{m}\\\\}\\\\subset[N] be the set of all points x ∈ [ N ] x\\\\in[N] s.t. | { S ∈ H T : x ∈ S } | ≥ 8 10 | H T | |\\\\{S\\\\in H_{T}:x\\\\in S\\\\}|\\\\geq\\\\frac{8}{10}|H_{T}| , i.e., x x is in most of the surviving datasets. We claim that m = O ( n ) m=O(n) , by the following counting argument.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We calculate the number of points in H T H_{T} , including multiplicities, in two ways. First, every S ∈ H T S\\\\in H_{T} contains n n points, so we have precisely n | H T | n|H_{T}| points. On the other hand, each of x 1 , … , x m x_{1},\\\\dots,x_{m} is in 8 10 | H T | \\\\frac{8}{10}|H_{T}| of the datasets in H T H_{T} , thus contributes at least that number of copies to the overall number. Summing over these m m common points, we obtain\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and thus indeed m = O ( n ) m=O(n) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider the query that assigns 1 1 to x 1 , … , x m x_{1},\\\\dots,x_{m} , and 0 to all other points, which is the threshold query q 0 X m q_{0}^{X_{m}} . The answer a a to query q 0 X m q_{0}^{X_{m}} must satisfy a ≤ 0.1 + m N ≤ 0.2 a\\\\leq 0.1+\\\\frac{m}{N}\\\\leq 0.2 . If ≤ 9 10 \\\\leq\\\\tfrac{9}{10} fraction of the datasets in H T H_{T} return answer smaller than 0.2 0.2 , then the query q 0 X m q_{0}^{X_{m}} satisfies the conclusion of the theorem, so suppose more than 9 / 10 9/10 fraction of the datasets in H T H_{T} return a valid answer to q 0 X m q_{0}^{X_{m}} . Similarly, the answer to the query that assigns 1 1 to all points (which is q N − m X m q_{N-m}^{X_{m}} ) is at least 0.9 0.9 . If ≤ 9 10 \\\\leq\\\\tfrac{9}{10} fraction of the datasets in H T H_{T} return answer larger than 0.9 0.9 , then the query q N − m X m q_{N-m}^{X_{m}} satisfies the conclusion of the theorem, so suppose more than 9 / 10 9/10 fraction of the datasets in H T H_{T} return a valid answer to q N − m X m q_{N-m}^{X_{m}} . Therefore, there must be some j > m j>m , for which most datasets in H T H_{T} answer ≤ 0.5 \\\\leq 0.5 for q j X m q_{j}^{X_{m}} , but most datasets in H T H_{T} answer > 0.5 >0.5 for q j + 1 X m q_{j+1}^{X_{m}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analyst simulates the answer of every S ∈ H T S\\\\in H_{T} for every threshold query to locate q j X m q_{j}^{X_{m}} . By construction, the point z j + 1 z_{j+1} is not common. Since the mechanism is natural, only datasets that contain z j + 1 z_{j+1} may have different answers for the queries q j X m q_{j}^{X_{m}} and q j + 1 X m q_{j+1}^{X_{m}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We claim that either q j X m q_{j}^{X_{m}} or q j + 1 X m q_{j+1}^{X_{m}} is a separating query, which concludes the proof. Suppose that q j X m q_{j}^{X_{m}} is not a separating query, meaning that at least 9 10 \\\\tfrac{9}{10} fraction of the datasets in H T H_{T} return some answer a ≤ 0.5 a\\\\leq 0.5 for the query q j X m q_{j}^{X_{m}} . At most 8 10 | H T | \\\\tfrac{8}{10}|H_{T}| of these contain z j + 1 z_{j+1} (since z j + 1 z_{j+1} is not common), and thus the most common answer for q j + 1 X m q_{j+1}^{X_{m}} , which is > 0.5 >0.5 , is returned by at most 9 10 | H T | \\\\tfrac{9}{10}|H_{T}| datasets, so q j + 1 X m q_{j+1}^{X_{m}} is a separating query. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A natural deterministic mechanism ℳ \\\\mathcal{M} that is 0.1 0.1 -accurate can answer at most k = O ( n log N ) k=O(n\\\\log N) queries from a computationally unbounded analyst.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By Section ˜ 3 , as long as the mechanism returns correct answers, in every round, the analyst can find a separating query. Initially, the number of possible datasets is ≤ N n \\\\leq N^{n} . Every such query rules out at least a constant factor of the datasets. Therefore, the analyst can fail the mechanism after O ( n log N ) O(n\\\\log N) queries. (Observe that if | H T | = 1 |H_{T}|=1 , then X m = S X_{m}=S , hence the mechanism must reply the same for q 0 X m q_{0}^{X_{m}} and q N − m X m q_{N-m}^{X_{m}} , and thus errs on at least one of these queries.) ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Natural Mechanism with a Mask\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now consider the following variant of natural mechanisms, which we call masked mechanisms. In this variant, for each query q q , the adversary draws a mask y y uniformly at random from { 0 , 1 } N \\\\{0,1\\\\}^{N} . The mechanism receives the masked query q ⊕ y q\\\\oplus y (computed bitwise) and the mask on the sample points ( y ( x 1 ) , … , y ( x n ) ) (y(x_{1}),\\\\dots,y(x_{n})) . The mechanism can trivially compute ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\\\dots,q(x_{n})) , and for simplicity of presentation, we use an equivalent formulation of the model where the mechanism is given the evaluations on the sample points ( q ( x 1 ) , … , q ( x n ) ) (q(x_{1}),\\\\dots,q(x_{n})) instead of ( y ( x 1 ) , … , y ( x n ) ) (y(x_{1}),\\\\dots,y(x_{n})) . We now prove that, with non-negligible probability, there exists a separating query for this variant.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let a deterministic masked mechanism ℳ \\\\mathcal{M} and a transcript T T . There exists a query q : [ N ] → { 0 , 1 } q:[N]\\\\to\\\\{0,1\\\\} such that for at least a 1 N + 1 \\\\tfrac{1}{N+1} fraction of masks y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} , the pair ( q , y ) (q,y) is separating for ℳ \\\\mathcal{M} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As before, the analyst can simulate the mechanism, and compute the set of datasets H T H_{T} that are consistent with the transcript T T . Let X m ⊂ [ N ] X_{m}\\\\subset[N] be the set of common points in H T H_{T} , i.e., the set of all points x ∈ [ N ] x\\\\in[N] s.t. | { S ∈ H T : x ∈ S } | ≥ 8 10 | H T | |\\\\{S\\\\in H_{T}:x\\\\in S\\\\}|\\\\geq\\\\frac{8}{10}|H_{T}| . By the same argument ( Equation ˜ 1 ), | X m | = O ( n ) |X_{m}|=O(n) . Let z 1 , … , z N − m z_{1},\\\\dots,z_{N-m} be the remaining non-common points. Consider the sequence of threshold queries q 0 , q 1 , … , q N − m q_{0},q_{1},\\\\dots,q_{N-m} defined by q j ( x ) = 𝕀 [ x ∈ X m ∪ { z 1 , … , z j } ] q_{j}(x)=\\\\mathbb{I}[x\\\\in X_{m}\\\\cup\\\\{z_{1},\\\\dots,z_{j}\\\\}] (as per Section ˜ 3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any fixed “base” mask v ∈ { 0 , 1 } N v\\\\in\\\\{0,1\\\\}^{N} , we define a “path” of query-mask pairs P v = { ( q j , y j ) } j = 0 N − m P_{v}=\\\\{(q_{j},y_{j})\\\\}_{j=0}^{N-m} where\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key observation is that along this path, the masked query input to the mechanism is constant,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, define a restricted natural mechanism ℳ v \\\\mathcal{M}_{v} that, given evaluations E = ( q ( x 1 ) , … , q ( x n ) ) E=(q(x_{1}),\\\\dots,q(x_{n})) , outputs:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since v v is fixed, ℳ v \\\\mathcal{M}_{v} is a natural mechanism (its output depends only on the evaluations). By applying Lemma 3 to ℳ v \\\\mathcal{M}_{v} , there must exist at least one index j v ∈ { 0 , … , N − m } j_{v}\\\\in\\\\{0,\\\\dots,N-m\\\\} such that the query q j v q_{j_{v}} is a separating query for H T H_{T} under ℳ v \\\\mathcal{M}_{v} . By construction, q j v q_{j_{v}} being separating for H T H_{T} under ℳ v \\\\mathcal{M}_{v} implies that ( q j v , y j v ) (q_{j_{v}},y_{j_{v}}) is a separating pair for H T H_{T} under the original masked mechanism ℳ \\\\mathcal{M} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There are 2 N 2^{N} possible base masks v v , and each produces at least one separating pair ( q j v , v ⊕ q j v ) (q_{j_{v}},v\\\\oplus q_{j_{v}}) . Since there are at most N + 1 N+1 possible threshold queries in the sequence, there are at most ( N + 1 ) 2 N (N+1)2^{N} pairs ( q i , y ) (q_{i},y) , where q i q_{i} is the i i -th threshold query and y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} is a mask. By the pigeonhole principle, there must exist at least one query q ∗ q^{*} that belongs to the set of separating pairs for at least 1 N + 1 \\\\frac{1}{N+1} fraction of the masks y ∈ { 0 , 1 } N y\\\\in\\\\{0,1\\\\}^{N} , which concludes the proof. ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Random Oracle\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now extend the lower bound on deterministic natural mechanisms to deterministic mechanisms that have access to a random oracle (RO), proving Theorem ˜ 1.2 . The random oracle is a function 𝒪 : ℕ → { 0 , 1 } \\\\mathcal{O}:\\\\mathbb{N}\\\\to\\\\{0,1\\\\} whose values are uniformly random and independent bits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A query in the random oracle model is an oracle-aided function q 𝒪 : 𝒳 → [ 0 , 1 ] q^{\\\\mathcal{O}}:\\\\mathcal{X}\\\\to[0,1] . For brevity, when 𝒪 \\\\mathcal{O} is clear from the context, we simply write q q instead of q 𝒪 q^{\\\\mathcal{O}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In our case, for any x ∈ 𝒳 x\\\\in\\\\mathcal{X} , the query q 𝒪 q^{\\\\mathcal{O}} computes its output using the point x x and exactly one bit from the random oracle 𝒪 \\\\mathcal{O} . Specifically, there exist functions I q : 𝒳 → ℕ I_{q}:\\\\mathcal{X}\\\\to\\\\mathbb{N} and g q : 𝒳 × { 0 , 1 } → [ 0 , 1 ] g_{q}:\\\\mathcal{X}\\\\times\\\\{0,1\\\\}\\\\to[0,1] such that q 𝒪 ( x ) = g q ( x , 𝒪 ( I q ( x ) ) ) q^{\\\\mathcal{O}}(x)=g_{q}(x,\\\\mathcal{O}(I_{q}(x))) . More generally, one could consider queries that rely on more than one bit of 𝒪 \\\\mathcal{O} , whose coordinates might even be determined adaptively. But for our purposes, one bit will suffice.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the RO model, both the analyst and the mechanism have oracle access to the same random oracle 𝒪 : ℕ → { 0 , 1 } \\\\mathcal{O}:\\\\mathbb{N}\\\\to\\\\{0,1\\\\} . A mechanism ℳ \\\\mathcal{M} receives a dataset S = ( x 1 , … , x n ) ∼ 𝒫 n S=(x_{1},\\\\ldots,x_{n})\\\\sim\\\\mathcal{P}^{n} drawn i.i.d. from a distribution 𝒫 \\\\mathcal{P} over a domain 𝒳 \\\\mathcal{X} . In round j ∈ [ k ] j\\\\in[k] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analyst formulates a statistical query q j 𝒪 : 𝒳 → [ 0 , 1 ] q_{j}^{\\\\mathcal{O}}:\\\\mathcal{X}\\\\to[0,1] , which may depend on previous queries and answers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mechanism receives q j 𝒪 q_{j}^{\\\\mathcal{O}} and outputs an answer a j ∈ ℝ a_{j}\\\\in\\\\mathbb{R} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ε > 0 , k , n , N , 𝒳 \\\\varepsilon>0,k,n,N,\\\\mathcal{X} . A mechanism ℳ \\\\mathcal{M} in the RO model is ε \\\\varepsilon -accurate for k k adaptive queries given a sample of size n n , if for each distribution 𝒫 \\\\mathcal{P} over 𝒳 \\\\mathcal{X} and analyst 𝒜 \\\\mathcal{A} , with probability at least 9 / 10 9/10 over the choice of the random oracle 𝒪 \\\\mathcal{O} , the sample S ∼ 𝒫 n S\\\\sim\\\\mathcal{P}^{n} , and the randomness of ℳ \\\\mathcal{M} , the answers a 1 , … , a k a_{1},\\\\dots,a_{k} satisfy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We emphasize that the population mean 𝔼 x ∼ 𝒫 [ q j 𝒪 ( x ) ] \\\\mathbb{E}_{x\\\\sim\\\\mathcal{P}}[q^{\\\\mathcal{O}}_{j}(x)] evaluates the query with respect to the specific, fixed realization of 𝒪 \\\\mathcal{O} drawn in the experiment; the expectation is taken solely over the domain distribution 𝒫 \\\\mathcal{P} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 The masked RO model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first provide an attack against deterministic mechanisms in an intermediate model, called masked RO . This attack contains the main arguments in the proof of Theorem ˜ 1.2 . In the masked RO model, the domain is [ N ] [N] , the distribution is uniform over [ N ] [N] , and the analyst samples (independently of 𝒪 \\\\mathcal{O} ) offsets m 1 , … , m N m_{1},\\\\ldots,m_{N} independently and uniformly from [ R ] [R] , where R ∈ ℕ R\\\\in\\\\mathbb{N} is a number we set later. In addition to a dataset S ⊂ [ N ] S\\\\subset[N] , the mechanism is also given the offsets { m i : i ∈ S } \\\\{m_{i}:i\\\\in S\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a pointer p p , define the mask induced by the block [ p + 1 , p + N R ] [p+1,p+NR] as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In a round, the analyst sends a pointer p p and a masked query q ~ ∈ { 0 , 1 } N \\\\tilde{q}\\\\in\\\\{0,1\\\\}^{N} . This pair represents the unmasked query\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and the mechanism is required to answer accurately with respect to the expectation of q q under the uniform distribution over [ N ] [N] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let N = n 10 N=n^{10} , let δ = N − 6 \\\\delta=N^{-6} , let k = O ( n log N ) k=O(n\\\\log N) , and let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ℳ \\\\mathcal{M} be a deterministic mechanism in the masked RO model that receives n n samples from the uniform distribution over [ N ] [N] and runs in at most t t time per round. If ℳ \\\\mathcal{M} is 0.1 0.1 -accurate, then ℳ \\\\mathcal{M} can answer at most k k adaptive queries, issued with pointers of length O ( n N log N + log t ) O(nN\\\\log N+\\\\log t) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We construct an unbounded analyst that, except with probability o ( 1 ) o(1) over the offsets and the random oracle, finds a separating query in every round, thus forcing the mechanism to fail in k k rounds. If the analyst makes a query on which the mechanism is already inaccurate, the proof is complete. Thus below, the only nontrivial case is the one in which the simulated mechanism remains accurate on the candidate queries. The analyst will conduct a sequence of experiments, each one will correspond to the masked ADA model of Section ˜ 3.1 . Denote by E = O ( N log 1 δ ) E=O(N\\\\log\\\\tfrac{1}{\\\\delta}) the maximum number of experiments the analyst will perform per round. Throughout, experiments are ordered lexicographically by ( j , v ) (j,v) , where j ∈ [ k ] j\\\\in[k] is the ADA round and v ∈ [ E ] v\\\\in[E] is the experiment number inside the round. The analyst only queries the random oracle at places the mechanism might query given a dataset S ⊂ [ N ] S\\\\subset[N] of size n n , for all possible datasets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The long-pointer idealization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first allow the analyst and mechanism to use very long pointers (in particular, they could be so long that the mechanism cannot read them in t t time), and we will later show how to remove this idealization. For each experiment ( j , v ) (j,v) , the analyst chooses a pointer p j , v long p^{\\\\rm long}_{j,v} so that the block\"\n },\n {\n \"type\": \"para\",\n \"text\": \"is disjoint from every RO location that could have been queried before experiment ( j , v ) (j,v) by either the analyst or the mechanism, over all datasets, all masked-query transcripts, and all possible RO answers. Such a choice is possible because, before any fixed experiment, the number of executions and simulations considered by the unbounded analyst is finite and every execution makes finitely many oracle queries. Let P j = { p j , v long : v ∈ [ E ] } P_{j}=\\\\{p^{\\\\rm long}_{j,v}:v\\\\in[E]\\\\} . We say that an experiment ( j , v ) (j,v) is successful, if exists masked threshold query q ~ \\\\tilde{q} such that the pair ( p j , v long , q ~ ) (p^{\\\\rm long}_{j,v},\\\\tilde{q}) is separating.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With probability at least 1 − δ 1-\\\\delta over m 1 , … , m N m_{1},\\\\ldots,m_{N} and 𝒪 \\\\mathcal{O} , the following event B B holds. For every dataset S ⊂ [ N ] S\\\\subset[N] of size n n , every sequence of masked threshold queries q ~ 1 , … , q ~ k ∈ { 0 , 1 } N \\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k}\\\\in\\\\{0,1\\\\}^{N} , and every sequence of pointers p 1 ∈ P 1 , … , p k ∈ P k p_{1}\\\\in P_{1},\\\\ldots,p_{k}\\\\in P_{k} , during the resulting interaction ℳ \\\\mathcal{M} never makes a query in\"\n },\n {\n \"type\": \"para\",\n \"text\": \"I.e., in all rounds, ℳ \\\\mathcal{M} does not make an out-of-sample mask query.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix S ⊂ [ N ] S\\\\subset[N] of size n n , masked queries q ~ 1 , … , q ~ k ∈ { 0 , 1 } N \\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k}\\\\in\\\\{0,1\\\\}^{N} , and p 1 ∈ P 1 , … , p k ∈ P k p_{1}\\\\in P_{1},\\\\ldots,p_{k}\\\\in P_{k} . That is, consider a special analyst 𝒜 q ~ 1 , … , q ~ k , p 1 , … , p k \\\\mathcal{A}_{\\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k},p_{1},\\\\ldots,p_{k}} that issues the masked queries q ~ 1 , … , q ~ k \\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k} and pointers p 1 , … , p k p_{1},\\\\ldots,p_{k} , regardless of the mechanism’s answers. Also fix the offsets { m i : i ∈ S } \\\\{m_{i}:i\\\\in S\\\\} and a realization of 𝒪 \\\\mathcal{O} . Under this conditioning, the deterministic mechanism’s oracle queries are fixed, and since ℳ \\\\mathcal{M} runs in time t t in each round, it makes at most k t kt oracle queries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider one such query location ℓ \\\\ell . Since the long-pointer blocks are disjoint, ℓ \\\\ell belongs to at most one interval of the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, over the still-random offset m i m_{i} for that possible interval, the probability that ℓ = p j + ( i − 1 ) R + m i \\\\ell=p_{j}+(i-1)R+m_{i} is at most 1 / R 1/R . A union bound over the at most k t kt oracle queries gives probability at most k t / R kt/R that ℳ \\\\mathcal{M} ever queries an out-of-sample mask location for the fixed S , q ~ 1 , … , q ~ k , p 1 , … , p k S,\\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k},p_{1},\\\\ldots,p_{k} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This bound holds for every conditioning on { m i : i ∈ S } \\\\{m_{i}:i\\\\in S\\\\} and on 𝒪 \\\\mathcal{O} , so by the law of total probability, it also holds unconditionally. Union bounding over at most N n N^{n} datasets, 2 N k 2^{Nk} masked-query sequences, and E k E^{k} pointer sequences gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the last step is by the choice of R R . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that under B B , for every analyst 𝒜 \\\\mathcal{A} , in the resulting interaction between 𝒜 \\\\mathcal{A} and ℳ \\\\mathcal{M} , the mechanism ℳ \\\\mathcal{M} does not make an out-of-sample query. This is since during the interaction, 𝒜 \\\\mathcal{A} must output some sequence of masked queries q ~ 1 , … , q ~ k \\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k} and pointers p 1 , … , p k p_{1},\\\\ldots,p_{k} , hence it identifies with 𝒜 q ~ 1 , … , q ~ k , p 1 , … , p k \\\\mathcal{A}_{\\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k},p_{1},\\\\ldots,p_{k}} . By B B , in the interaction between 𝒜 q ~ 1 , … , q ~ k , p 1 , … , p k \\\\mathcal{A}_{\\\\tilde{q}_{1},\\\\ldots,\\\\tilde{q}_{k},p_{1},\\\\ldots,p_{k}} and ℳ \\\\mathcal{M} , the mechanism ℳ \\\\mathcal{M} does not make an out-of-sample query, yielding this observation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix a round j ∈ [ k ] j\\\\in[k] and a transcript T j − 1 T_{j-1} produced before that round. The analyst conducts the following experiment for each v ∈ [ E ] v\\\\in[E] . First, it queries 𝒪 \\\\mathcal{O} at the N N mask locations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and thereby learns the mask y ( v ) ∈ { 0 , 1 } N y^{(v)}\\\\in\\\\{0,1\\\\}^{N} defined by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analyst then searches for a separating threshold query. For every candidate threshold query q ∈ { 0 , 1 } N q\\\\in\\\\{0,1\\\\}^{N} , it sets q ~ = q ⊕ y ( v ) \\\\tilde{q}=q\\\\oplus y^{(v)} . For every candidate dataset S ′ ∈ H T j − 1 S^{\\\\prime}\\\\in H_{T_{j-1}} , it simulates M M on transcript T j − 1 T_{j-1} , dataset S ′ S^{\\\\prime} , offsets { m i : i ∈ S ′ } \\\\{m_{i}:i\\\\in S^{\\\\prime}\\\\} , masked query q ~ \\\\tilde{q} , and pointer p j , v long p^{\\\\rm long}_{j,v} , with a dataset-dependent oracle 𝒪 l y i n g S ′ \\\\mathcal{O}_{lying}^{S^{\\\\prime}} . This oracle is defined as follows. A query to an in-sample mask location p j , v long + ( i − 1 ) R + m i p^{\\\\rm long}_{j,v}+(i-1)R+m_{i} with i ∈ S ′ i\\\\in S^{\\\\prime} is answered by the already-read bit y i ( v ) y^{(v)}_{i} ; a query to an out-of-sample mask location of this form with i ∉ S ′ i\\\\notin S^{\\\\prime} is answered by 0 ; every other oracle query is answered as in the real oracle 𝒪 \\\\mathcal{O} . Thus the analyst’s real RO queries in a single experiment are the N N mask queries, plus at most t t non-mask queries for each pair ( q , S ′ ) (q,S^{\\\\prime}) . Hence the analyst makes at most\"\n },\n {\n \"type\": \"para\",\n \"text\": \"real RO queries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix a round j ∈ [ k ] j\\\\in[k] and condition on any history produced before that round, including the transcript T j − 1 T_{j-1} . In the long-pointer idealization, if the simulated mechanism obtains its oracle replies from 𝒪 l y i n g S ′ \\\\mathcal{O}_{lying}^{S^{\\\\prime}} on each candidate dataset S ′ ∈ H T j − 1 S^{\\\\prime}\\\\in H_{T_{j-1}} , then the probability that none of the E E experiments in round j j contains a separating query is at most δ \\\\delta .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider experiment v ∈ [ E ] v\\\\in[E] in round j j , and condition on all previous experiments. By construction, the bits of the random oracle in the block B j , v long B^{\\\\rm long}_{j,v} are fresh, hence y ( v ) y^{(v)} is uniform over { 0 , 1 } N \\\\{0,1\\\\}^{N} and independent of the conditioned transcript.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒪 out \\\\mathcal{O}_{\\\\mathrm{out}} denote the restriction of 𝒪 \\\\mathcal{O} to all locations except the N N fresh mask locations { p j , v long + ( i − 1 ) R + m i : i ∈ [ N ] } \\\\{p^{\\\\rm long}_{j,v}+(i-1)R+m_{i}:i\\\\in[N]\\\\} . Condition on an arbitrary realization of 𝒪 out \\\\mathcal{O}_{\\\\mathrm{out}} . Under this conditioning, y ( v ) y^{(v)} remains uniform over { 0 , 1 } N \\\\{0,1\\\\}^{N} , and the hybrid simulation defines a deterministic masked mechanism ℳ ′ \\\\mathcal{M}^{\\\\prime} : on dataset S ′ S^{\\\\prime} , masked query q ~ \\\\tilde{q} , and sample evaluations q | S ′ q|_{S^{\\\\prime}} , the mechanism can reconstruct the in-sample mask bits from q ~ | S ′ \\\\tilde{q}|_{S^{\\\\prime}} and q | S ′ q|_{S^{\\\\prime}} , receives no out-of-sample mask bits, and all other oracle answers are fixed constants. Therefore, by Section ˜ 3.1 , for every such conditioning, with probability at least 1 / ( N + 1 ) 1/(N+1) over the fresh mask y ( v ) y^{(v)} , there is a threshold query q ( v ) ∈ { 0 , 1 } N q^{(v)}\\\\in\\\\{0,1\\\\}^{N} such that the pair ( q ( v ) , y ( v ) ) (q^{(v)},y^{(v)}) is separating for this experiment. Since this holds for any realization of 𝒪 out \\\\mathcal{O}_{\\\\mathrm{out}} , by the law of total probability, the same bound holds without conditioning on 𝒪 out \\\\mathcal{O}_{\\\\mathrm{out}} . Since the analyst is unbounded, it finds such a query whenever one exists.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The same conditional success bound holds after every history of failed experiments. Hence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the long-pointer idealization, the analyst finds a separating query in every round in the real masked RO game with probability at least 1 − ( k + 1 ) δ 1-(k+1)\\\\delta .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each round j j , let G j lie G_{j}^{\\\\mathrm{lie}} be the event that at least one of the E E experiments in round j j contains a separating query in the exhaustive simulation that answers oracle queries using 𝒪 lying S ′ \\\\mathcal{O}_{\\\\mathrm{lying}}^{S^{\\\\prime}} for each candidate dataset S ′ ∈ H T j − 1 S^{\\\\prime}\\\\in H_{T_{j-1}} . By Section ˜ 4.1 , for every history produced before round j j ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, conditioning sequentially on the histories produced in previous rounds,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By Section ˜ 4.1 and a union bound,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now argue that under B ∩ ⋂ j = 1 k G j lie B\\\\cap\\\\bigcap_{j=1}^{k}G_{j}^{\\\\mathrm{lie}} , the real long-pointers analyst finds a separating query in every round, which will conclude the proof. Consider any realization of the offsets and oracle for which B B holds. For any dataset S ′ S^{\\\\prime} , any masked-query transcript considered by the analyst, and any pointer sequence from P 1 , … , P k P_{1},\\\\ldots,P_{k} , the only locations on which the real oracle 𝒪 \\\\mathcal{O} and 𝒪 lying S ′ \\\\mathcal{O}_{\\\\mathrm{lying}}^{S^{\\\\prime}} may differ in the simulation on dataset S ′ S^{\\\\prime} are out-of-sample mask locations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The event B B rules out every such query by ℳ \\\\mathcal{M} throughout the simulated interaction. At in-sample mask locations, both simulations answer using the already-read mask bits; at all non-mask locations, both answer according to 𝒪 \\\\mathcal{O} . Hence, under B B , every simulated execution in the exhaustive search has the same transcript with the real oracle as with the corresponding lying oracle. In particular, every query that is separating in the lying simulation is also separating in the real masked RO game, which concludes the proof. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each separating query reduces the number of datasets consistent with the transcript by a factor of at least 9 / 10 9/10 . Starting from at most N n N^{n} possible datasets, after O ( n log N ) O(n\\\\log N) separating queries the mechanism must have answered some query inaccurately. Thus the long-pointer analyst forces a failure within O ( n log N ) O(n\\\\log N) rounds with probability at least 1 − ( k + 1 ) δ 1-(k+1)\\\\delta .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Replacing long pointers by random finite pointers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It remains to remove the idealized long pointers. Let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any fixed dataset and any fixed masked-query transcript, this is a crude upper bound on the number of RO locations exposed before a given experiment: recall that in each experiment the analyst reads the N N mask bits and simulates M M on all datasets and all N + 1 N+1 candidate threshold queries, with at most t t RO queries per simulation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now restrict the analyst into using pointers of the form p = b N R p=bNR , where b ∈ [ L prev + 1 ] b\\\\in[L_{\\\\mathrm{prev}}+1] . Observe that the number of bits required to represent an RO query in [ p + 1 , p + N R ] [p+1,p+NR] is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recall that k N = O ( n N log N ) kN=O(nN\\\\log N) , which is larger than the number of bits required to represent an input dataset in [ N ] n [N]^{n} by only a polynomial factor. Therefore, for reasonable values of t t (e.g., polynomial in the input length), the mechanism can indeed read the pointer and issue queries to the Random Oracle.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For every ( j , v ) (j,v) , there exists b j , v ∈ [ L prev + 1 ] b_{j,v}\\\\in[L_{\\\\mathrm{prev}}+1] such that the analyst did not query any RO location in [ b j , v N R + 1 , b j , v N R + N R ] [b_{j,v}NR+1,b_{j,v}NR+NR] before experiment ( j , v ) (j,v) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ( j , v ) (j,v) . Denote by W j , v W_{j,v} the set of RO locations exposed before experiment ( j , v ) (j,v) . By the construction of the analyst, | W j , v | ≤ L prev |W_{j,v}|\\\\leq L_{\\\\mathrm{prev}} . The segments [ b N R + 1 , b N R + N R ] [bNR+1,bNR+NR] for b ∈ [ L prev + 1 ] b\\\\in[L_{\\\\mathrm{prev}}+1] are disjoint, and are numbered strictly more than | W j , v | |W_{j,v}| , hence there exists b ∈ [ L prev + 1 ] b\\\\in[L_{\\\\mathrm{prev}}+1] such that [ b N R + 1 , b N R + N R ] ∩ W j , v = ∅ [bNR+1,bNR+NR]\\\\cap W_{j,v}=\\\\emptyset . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, each finite block is fresh at the moment it is used, and the analysis from the long pointer idealization goes through when using pointers p j , v = b j , v N R p_{j,v}=b_{j,v}NR . This completes the proof of Section ˜ 4.1 . ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 An attack in the RO model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, we are ready to prove the main result ( Theorem ˜ 1.2 ). Formally,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let n , N = n 10 , k = O ( n log N ) , t , δ = N − 6 n,N=n^{10},k=O(n\\\\log N),t,\\\\delta=N^{-6} and R = O ( δ − 1 N n ( N 2 N log 1 δ ) k k t ) R=O(\\\\delta^{-1}N^{n}(N2^{N}\\\\log\\\\tfrac{1}{\\\\delta})^{k}kt) . Let ℳ \\\\mathcal{M} be a deterministic, 0.1 0.1 -accurate mechanism in the RO model that is given n n samples from an unknown distribution over the domain [ N ] × [ R ] [N]\\\\times[R] , and whose running time is bounded by t t . There exists an analyst 𝒜 \\\\mathcal{A} that picks queries whose representation size is O ( n N log N + log t ) O(nN\\\\log N+\\\\log t) bits, such that the queries can be evaluated on each domain point x ∈ [ N ] × [ R ] x\\\\in[N]\\\\times[R] in O ( n N log N + log t ) O(nN\\\\log N+\\\\log t) time, and 𝒜 \\\\mathcal{A} causes ℳ \\\\mathcal{M} to fail in k = O ( n log N ) k=O(n\\\\log N) rounds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that the mechanism’s running time is sublinear in the domain size, hence, in particular, the mechanism cannot evaluate the entire truth table of a query.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proof is by forcing the mechanism to simulate a mechanism in the masked RO model established in Section ˜ 4.1 . I.e., we will show how to use the mechanism ℳ \\\\mathcal{M} to construct a deterministic, 0.1 0.1 -accurate masked RO mechanism ℳ masked \\\\mathcal{M}_{\\\\text{masked}} over the domain [ N ] [N] , with the exact round complexity, which will yield the theorem statement.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Domain Extension and Distribution Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Define the domain 𝒳 ′ = [ N ] × [ R ] \\\\mathcal{X}^{\\\\prime}=[N]\\\\times[R] , for the same value of R R as in Section ˜ 4.1 . Draw N N pointers m 1 , … , m N m_{1},\\\\ldots,m_{N} independently and uniformly at random over [ R ] [R] . Define the underlying distribution 𝒫 \\\\mathcal{P} over 𝒳 ′ \\\\mathcal{X}^{\\\\prime} to be the uniform distribution over { ( i , m i ) ∣ i ∈ [ N ] } \\\\{(i,m_{i})\\\\mid i\\\\in[N]\\\\} . When a dataset X ′ = ( x 1 ′ , … , x n ′ ) X^{\\\\prime}=(x^{\\\\prime}_{1},\\\\dots,x^{\\\\prime}_{n}) of size n n is drawn i.i.d. from 𝒫 \\\\mathcal{P} , each individual sample point takes the form x i ′ = ( x i , m x i ) x^{\\\\prime}_{i}=(x_{i},m_{x_{i}}) , where x i x_{i} is drawn uniformly and independently from [ N ] [N] . Therefore, the mechanism gets a dataset ( x 1 , … , x n ) ∈ [ N ] n (x_{1},\\\\ldots,x_{n})\\\\in[N]^{n} , along with the corresponding pointers m x 1 , … , m x n m_{x_{1}},\\\\ldots,m_{x_{n}} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Formulating a Query.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To form a query, the analyst 𝒜 \\\\mathcal{A} does the following. Let 𝒜 m a s k , ℳ m a s k \\\\mathcal{A}_{mask},\\\\mathcal{M}_{mask} be an analyst and a mechanism operating in the masked RO model. At each round j ∈ [ k ] j\\\\in[k] , suppose 𝒜 m a s k \\\\mathcal{A}_{mask} outputs a masked query q ~ j ∈ { 0 , 1 } N \\\\tilde{q}_{j}\\\\in\\\\{0,1\\\\}^{N} and a pointer p j p_{j} . The analyst 𝒜 \\\\mathcal{A} formulates the query q j ( i , m ) = q ~ j ( i ) ⊕ 𝒪 ( p j + i R + m ) q_{j}(i,m)=\\\\tilde{q}_{j}(i)\\\\oplus\\\\mathcal{O}(p_{j}+iR+m) for all i , m i,m . This query can be represented by the pair ( q ~ j , p j ) (\\\\tilde{q}_{j},p_{j}) , and thus have size O ( N + n N log N + log t ) O(N+nN\\\\log N+\\\\log t) . Given a domain element ( i , m ) (i,m) , one can compute I = p j + i R + m I=p_{j}+iR+m , query the random oracle at index I I and eventually evaluate q j ( i , m ) = q ~ j ( i ) ⊕ 𝒪 ( I ) q_{j}(i,m)=\\\\tilde{q}_{j}(i)\\\\oplus\\\\mathcal{O}(I) . The running time of this evaluation is dominated by the O ( n N log N + log t ) O(nN\\\\log N+\\\\log t) time it takes to compute I I .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Reduction to Masked RO Mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Observe that for a point ( i , m i ) (i,m_{i}) in the support, we have q j ( i , m i ) = q ~ j ( i ) ⊕ 𝒪 ( p j + i R + m i ) q_{j}(i,m_{i})=\\\\tilde{q}_{j}(i)\\\\oplus\\\\mathcal{O}(p_{j}+iR+m_{i}) , which equals the query issued by 𝒜 m a s k \\\\mathcal{A}_{mask} . Let us observe the information ℳ \\\\mathcal{M} holds and what it can compute. It holds a dataset ( x 1 , … , x n ) (x_{1},\\\\ldots,x_{n}) , pointers ( m x 1 , … , m x n ) (m_{x_{1}},\\\\ldots,m_{x_{n}}) , the transcript T j − 1 T_{j-1} , the query pointer p j p_{j} , the masked query q ~ j \\\\tilde{q}_{j} , and it can read at most t t bits of q j q_{j} or of the RO. Since the mechanism holds q ~ j \\\\tilde{q}_{j} , reading the ( i , m ) (i,m) -th bit of q j q_{j} is equivalent to reading the ( p j + i R + m ) (p_{j}+iR+m) -th bit of the random oracle. Thus, the mechanism has the same information and random oracle access as ℳ m a s k \\\\mathcal{M}_{mask} , who may issue at most t t queries to the RO. Therefore, as long as ℳ \\\\mathcal{M} is 0.1 0.1 -accurate, so is ℳ m a s k \\\\mathcal{M}_{mask} . By Section ˜ 4.1 , ℳ m a s k \\\\mathcal{M}_{mask} answers at most O ( n log N ) O(n\\\\log N) adaptive queries, and thus also ℳ \\\\mathcal{M} , concluding the proof.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Limited randomness\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now consider mechanisms that have a small amount of private randomness. Specifically, consider mechanisms holding r r random bits. In all the models we considered, we can view this randomness as part of the input: that is, instead of a distribution 𝒫 n \\\\mathcal{P}^{n} over 𝒳 n \\\\mathcal{X}^{n} , we consider a product distribution 𝒫 n ⊗ U ( { 0 , 1 } r ) \\\\mathcal{P}^{n}\\\\otimes U(\\\\{0,1\\\\}^{r}) over 𝒳 n ⊗ { 0 , 1 } r \\\\mathcal{X}^{n}\\\\otimes\\\\{0,1\\\\}^{r} . Therefore, the number of possible datasets is bounded by N n 2 r N^{n}2^{r} . If the analyst finds a separating query in every round, the mechanism fails in O ( n log N + r ) O(n\\\\log N+r) rounds. We briefly explain how to change the presented proofs, and show that the analyst can find such separating queries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, observe that the argument for natural mechanisms goes through, as follows. Let X m ⊆ [ N ] X_{m}\\\\subseteq[N] be the set of common points, i.e., points appearing in the dataset S S for at least an 8 10 \\\\frac{8}{10} fraction of the states ( S , ρ ) ∈ H T (S,\\\\rho)\\\\in H_{T} . The counting argument leading to Equation ˜ 1 still holds, hence the number of common points is m = O ( n ) m=O(n) . Consequently, there is a threshold query q i q_{i} , where i i is not common, where the majority’s answer transitions from < 0.5 <0.5 to ≥ 0.5 \\\\geq 0.5 , hence q i q_{i} is separating.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The argument for natural mechanisms with a mask is unchanged, since it is by a black-box simulation. In the random oracle model, the analyst has to prepare for 2 r 2^{r} possibilities of the mechanism’s random bits, and have that in (most of) these realizations, with high probability the mechanism does not observe out-of-sample mask bits and does not violate the independence between experiments. To achieve this, it suffices to increase R R and L p r e v L_{prev} by a 2 r 2^{r} factor. Therefore, we obtain,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let n , N = n 10 , k = O ( n log N ) , t , δ = N − 6 , r n,N=n^{10},k=O(n\\\\log N),t,\\\\delta=N^{-6},r and R = O ( δ − 1 N n ( N 2 N log 1 δ ) k k t 2 r ) R=O(\\\\delta^{-1}N^{n}(N2^{N}\\\\log\\\\tfrac{1}{\\\\delta})^{k}kt2^{r}) . Let ℳ \\\\mathcal{M} be a deterministic, 0.1 0.1 -accurate mechanism in the RO model, that is given r r private random bits and n n samples from an unknown distribution over the domain [ N ] × [ R ] [N]\\\\times[R] , and whose running time is bounded by t t . There exists an analyst 𝒜 \\\\mathcal{A} that picks queries whose representation size is O ( n N log N + log t + r ) O(nN\\\\log N+\\\\log t+r) bits, such that the queries can be evaluated on each domain point x ∈ [ N ] × [ R ] x\\\\in[N]\\\\times[R] in O ( n N log N + log t + r ) O(nN\\\\log N+\\\\log t+r) time, and 𝒜 \\\\mathcal{A} causes ℳ \\\\mathcal{M} to fail in k = O ( n log N + r ) k=O(n\\\\log N+r) rounds.\"\n }\n]","isConversational":false}}},{"rowIdx":489,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07083"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Prior-aware and Context-guided Group Sampling for Active Probabilistic Subsampling"},"authors":{"kind":"list like","value":["Beomgu Kang","Hyunseok Seo"],"string":"[\n \"Beomgu Kang\",\n \"Hyunseok Seo\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07083v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Modern technologies generate massive datasets that can require lengthy acquisition time and hinder real-time onboard processing. Many real-world applications, including Magnetic Resonance Imaging (MRI) (Ye, 2019 ) , computed tomography (CT) imaging (Chen et al. , 2008 ) , ultrasound imaging (Huijben et al. , 2020a ) , digital micromirror device (Baraniuk, 2007 ) , seismic surveying (Herrmann et al. , 2012 ) , and hyperspectral imaging (Sun and Du, 2019 ) , highlight the importance of reducing imaging data volume. Therefore, strategic sampling not only reduces data volume and preserves essential information for efficient transfer and processing but also accelerates acquisition, a critical factor in medical imaging."},{"type":"para","text":"Compressed sensing (CS) was developed as a subsampling strategy to overcome the Nyquist-Shannon limits on the sampling rates required for perfect signal reconstruction (Donoho, 2006 ; Eldar and Kutyniok, 2012 ) . CS has been widely adopted across various applications and has demonstrated significant impact (Lustig et al. , 2007 ; Baraniuk and Steeghs, 2007 ; Yu and Wang, 2009 ; Martín et al. , 2014 ; Lorintiu et al. , 2015 ; Han et al. , 2016 ) . Although CS exploits the inherent signal structures, such as sparsity, it does not take into account the information relevant to the downstream task during the sampling process. Bridging the gap between modality- and task-specific knowledge and the sampling process has proven challenging."},{"type":"para","text":"Recently, subsampling techniques customized for specific data distributions and downstream tasks have been proposed with advances in deep learning, offering learned yet fixed sampling patterns (Huijben et al. , 2020b ; Weiss et al. , 2020 ; Shen et al. , 2020 ; Sherry et al. , 2020 ; Zhang et al. , 2020 ; Aggarwal and Jacob, 2020 ; Mou et al. , 2021 ; Yang et al. , 2025 ) . These approaches optimize a sampling pattern based on the average data distribution in the training set, which may not provide optimal results for individual instances. To address this limitation, active sampling was introduced, where new points are adaptively selected based on previously acquired samples, iterating until the required target number of samples is obtained (Zhang et al. , 2019 ; Jin et al. , 2019 ; Bakker et al. , 2020 ; Pineda et al. , 2020 ; Van Gorp et al. , 2021 ; Tian et al. , 2025 ) . Active sampling produces a sampling trajectory that adapts to each test instance and incorporates newly acquired data dynamically. This is particularly valuable in medical imaging, where each patient’s unique physiological condition demands individualized acquisition trajectories."},{"type":"para","text":"Although active subsampling provides instance-specific sampling patterns and enables effective adaptation to new data, boosting downstream task performance, its sampling strategy still has room for improvement. Through iterative top-1 sampling guided by previously selected samples, active sampling leverages inter-sample relationships to optimize the sampling model and minimizes redundant information. However, it fails to fully exploit the rich prior knowledge embedded in the training dataset. Moreover, relying on top-1 sampling leads to sub-optimal optimization (Huijben et al. , 2020b ) ."},{"type":"para","text":"In this work, we enhance Active Deep Probabilistic Subsampling (A-DPS) architecture by incorporating deterministic (fixed) subsampling based on prior knowledge of the training data with group sampling to achieve more robust optimization (Fig. 1 ). We call this approach Prior-aware and context-guided Group-based Active Deep Probabilistic Subsampling (PGA-DPS). We demonstrate that PGA-DPS exploits training-data priors via deterministic sampling and reinforces this with instance-adaptive group sampling, resulting in a smoother loss landscape and more stable optimization. We evaluated PGA-DPS on the MNIST dataset (LeCun et al. , 1998 ) , the CIFAR-10 dataset (Krizhevsky et al. , 2009 ) , the fastMRI knee dataset (Zbontar et al. , 2018 ) , and the hyperspectral AeroRIT dataset (Rangnekar et al. , 2020 ) . Across classification, image reconstruction, and segmentation tasks, PGA-DPS outperforms all other state-of-the-art sampling methods."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"Recent works have proposed learning-based subsampling techniques tailored to data types and downstream tasks. In the field of MRI, Learning-based Optimization of the Under-sampling PattErn (LOUPE) was developed to simultaneously optimize the subsampling pattern and reconstruct the image, addressing two core challenges of compressed sensing (Bahadir et al. , 2020 ) . LOUPE learns the sampling mask by relaxing the non-differentiable threshold operation, which is similar to the Gumbel-softmax trick used in Deep Probabilistic Subsampling (DPS) (Huijben et al. , 2020b ) . The target sampling ratio is enforced by an L1 sparsity penalty on sampling mask within the loss function. Additionally, a stochastic greedy algorithm has been applied in MRI to obtain the sampling pattern that minimizes the reconstruction loss (Sanchez et al. , 2020 ) . However, greedy algorithm-based methods require pre-trained reconstruction capable of handling any subsampling pattern to thoroughly evaluate the loss."},{"type":"para","text":"In hyperspectral imaging (HSI), band selection techniques are widely used to identify a small informative subset of spectral bands and reduce spectral redundancy. Self-Representation Learning with Sparse 1D-Operational Autoencoder (SRL-SOA) employs a sparse autoencoder model to learn a sparse representation using a Taylor-series expansion of non-linear transformation (Ahishali et al. , 2022 ) . However, this does not incorporate the downstream task model for band selection. Greedy Spectral Selection (GSS) adopts a greedy algorithm: it first performs filter-based interband redundancy analysis, then trains a Convolutional Neural Network (CNN) to assess classification performance of the chosen bands (Morales et al. , 2021 ) . On the other hand, CNN based on Bandwise-independent convolution and Hard thresholding (BHCNN) jointly optimizes band selection and the classification by applying a hard threshold mask via a straight-through estimator (Feng et al. , 2020 ) . Although this allows joint optimization, gradients for non-selected bands are ignored, which can hinder the discovery of other informative bands during training. Furthermore, the aforementioned task-aware band selection methods are designed for classification, which may not provide optimal spectral bands for segmentation tasks."},{"type":"para","text":"An active sampling strategy for MRI k-space acquisition uses an adversarial neural network to distinguish between sampled and unsampled lines in the Fourier space (Zhang et al. , 2019 ) . The advarsarial network identifies the k-space line that appears most realistically fake and selects it for acquisition; this process repeats until the desired number of lines is obtained. However, this approach processes k-space data directly, which limits its applicability to MR image undersampling. Reinforcement learning (RL) has also been explored for active acquisition, applying greedy and modified ϵ \\epsilon -greedy policies to actively acquire optimal sampling masks (Pineda et al. , 2020 ; Bakker et al. , 2020 ) . However, RL-based techniques require a pretrained reconstruction network that can handle a variety of sampling patterns. To address this issue, Active DPS (A-DPS) jointly optimizes both the sampling mask and the associated reconstruction network (Van Gorp et al. , 2021 ) . Building on A-DPS, the proposed Prior-aware and Group-based Active DPS (PGA-DPS) improves optimization robustness and fully leverages prior knowledge embedded in the training data."},{"type":"heading","level":1,"text":"3 Method"},{"type":"heading","level":2,"text":"3.1 Task-adaptive subsampling framework"},{"type":"para","text":"We aim to find an optimal subsampling strategy A ⊆ { 0 , 1 } N A\\subseteq{\\{0,1\\}}^{N} on a given input signal x ∈ ℝ N x\\in\\mathbb{R}^{N} for a specific task t , such that the performance of the task is maintained using fewer input samples."},{"type":"para","text":"where f f is a task model that predicts the task t from the input signal x x , θ \\theta is parameters of the model, and M is the target number of subsamples. The proportion of selected samples can be represented by the sampling ratio r = 100 × M / N % r=100\\times M/N\\% . In general, neural networks are widely adopted for task models, such as classification, segmentation, reconstruction, detection, and more; however, backpropagation from the loss function is hindered by the non-differentiable nature of sampling. This issue was alleviated using the Gumbel-Softmax reparameterization trick introduced in Deep Probabilistic Subsampling (DPS)."},{"type":"heading","level":2,"text":"3.2 DPS: Deep Probabilistic Subsampling"},{"type":"para","text":"DPS proposed an end-to-end deep learning framework that jointly optimizes both the subsampling strategy and the downstream task model (Huijben et al. , 2020a ) . During training, DPS learns unnormalized logits ϕ \\phi , which are used to generate a subsampling mask probability P ( A | ϕ ) P\\left(A|\\phi\\right) from a categorical distribution with N N classes: A ∼ Categorical ( exp ( ϕ i ) ∑ j N exp ( ϕ j ) , i ∈ { 1 , … , N } ) A\\sim\\operatorname{Categorical}\\left(\\frac{\\exp(\\phi_{i})}{\\sum_{j}^{N}\\exp(\\phi_{j})},\\,i\\in\\{1,\\dots,N\\}\\right) . The Gumbel-max trick enables efficient sampling from a categorical distribution (Gumbel, 1954 ; Maddison et al. , 2014 ) :"},{"type":"para","text":"where ϕ ∈ ℝ N \\phi\\in\\mathbb{R}^{N} is an unnormalized logit vector composed of ϕ i \\phi_{i} and G G is a vector of i.i.d noise samples, G i G_{i} , drawn from a Gumbel distribution ∼ Gumbel ( 0 , 1 ) \\sim\\text{Gumbel}(0,1) . However, since the arg max function is non-differentiable and thus unsuitable for backpropagation, DPS employs the Gumbel-Softmax trick by relaxing the max operation into a differentiable softmax function as follows (Jang et al. , 2017 ; Maddison et al. , 2017 ) :"},{"type":"para","text":"where τ \\tau is a temperature parameter that controls the degree of relaxation, enabling gradients to be distributed across multi-dimensional logits when τ > 0 \\tau>0 during training. In addition, as the temperature decreases τ → 0 \\tau\\rightarrow 0 , s o f t m a x τ softmax_{\\tau} can anneal into a categorical distribution. Although tuning the relaxation with τ \\tau is crucial for managing the bias-variance trade-off of the gradient estimator (Tucker et al. , 2017 ) , the DPS architecture has been shown to perform effectively with a fixed temperature during training ( τ = 2 \\tau=2 )."},{"type":"para","text":"In addition, the Gumbel-max trick can be extended to the Gumbel top- k k trick to efficiently select the k k most probable samples, rather than just the single highest-probability one (Kool et al. , 2019 ; Plötz and Roth, 2018 ) ."},{"type":"para","text":"DPS introduces two variants for selecting k k samples: DPS-top-1, which sequentially samples from k k independent categorical distributions—each with its own trainable logits—and DPS-top- k k , which selects all k k samples simultaneously using a single shared logit for greater parameter efficiency. Although DPS-top-1 is more expressive, the DPS-top- k k approach showed improved performance."},{"type":"para","text":"Similarly, Active Deep Probabilistic Subsampling (A-DPS) introduces an active sampling that uses contextual information from previously selected samples to guide subsequent selection, allowing adaptation to new data (Van Gorp et al. , 2021 ) . In A-DPS, samples are chosen iteratively by applying DPS-top-1 at every step."},{"type":"para","text":"where j j denotes the iteration; w j ⊆ { − ∞ , 0 } N w^{j}\\subseteq{\\{-\\infty,0\\}}^{N} is a cumulative mask that assigns minus infinity to previously selected elements, ensuring they are excluded after re-normalization; ϕ j ∈ ℝ N \\phi^{j}\\in\\mathbb{R}^{N} and G j ∈ ℝ N G^{j}\\in\\mathbb{R}^{N} are ϕ \\phi and G G for j t h j^{th} sample, respectively; g j g^{j} is a sampling network in j t h j^{th} iteration that encodes the current task context based on the samples selected so far; and f θ f_{\\theta} is a differentiable model that generates the current task prediction. The context is built using an analysis-by-synthesis approach, where the task (synthesis) module guides the sampling (analysis). In A-DPS, losses are accumulated over all iterations, and the network is updated in a semi-greedy manner."},{"type":"heading","level":2,"text":"3.3 PGA-DPS: Prior-aware and Group-based Active DPS"},{"type":"para","text":"PGA-DPS initiates with a deterministic subsampling pattern and then iteratively acquires additional samples through active sampling (Fig. 2 ). The deterministic pattern drawn from the training data’s prior is followed by active sampling that exploits each new input’s contextual information, thereby combining global data prior with input specific context."},{"type":"para","text":"Moreover, PGA-DPS uses DPS-top- k k to select samples in groups for active sampling, rather than DPS-top-1 in A-DPS, enabling it to achieve a smaller effective Lipschitz constant. We analyze group sampling enabled by DPS-top- k k in terms of the loss, which captures the Lipschitzness of the loss and reflects the smoothness of the optimization landscape. The proof is available in Appendix A ."},{"type":"para","text":"(The effect of group sampling on the Lipschitzness of the loss). Let f 1 , f 2 , … , f k f_{1},f_{2},\\dots,f_{k} be task models, where each function f j f_{j} has a corresponding Lipschitz constants L j L_{j} for j = 1 , 2 , … , k j=1,2,\\dots,k . In DPS-top-1, k task functions are required for each input sample x k x_{k} , whereas DPS-top-k defines a single task function f k f_{k} over group of k samples. The Lipschitzness of the loss function for each DPS-top-1 and DPS-top- k k is as follows:"},{"type":"para","text":"Here, x ∗ ∈ ℝ N x^{*}\\in\\mathbb{R}^{N} is the fully sampled input signal (unsampled signal = 0), x ∈ ℝ N x\\in\\mathbb{R}^{N} is the sampled input signal, and x r x_{r} is the r t h r^{th} sample. If the Lipschitz constant of task model at each iteration L j ≥ 1 L_{j}\\geq 1 ,"},{"type":"para","text":"Neural networks typically have Lipschitz constants much greater than one, except in the trivial near-identity case (Malherbe and Vayatis, 2017 ; Bartlett et al. , 2018 ; Latorre et al. , 2020 ; Shi et al. , 2022 ) . Because our task of classification, image reconstruction (from k-space to image), and segmentation, are far from identity mappings, DPS-top- k k exhibits a smaller effective Lipschitz constant than DPS-top-1, whose effective constant is the product of multiple Lipschitz constants. Moreover, the same analysis can apply to the Lipschitz constants of gradient loss ‖ ∇ L o s s ‖ \\left\\|\\nabla Loss\\right\\| . Consequently, PGA-DPS creates smoother latent space, resulting in more stable and efficient optimization (Virmaux and Scaman, 2018 ; Berkenkamp et al. , 2017 ; Liu et al. , 2022 ; Gouk et al. , 2021 ; Santurkar et al. , 2018 ) . This theoretical analysis aligns with previous experimental results that Top- k k sampling outperformed Top-1 sampling in DPS (Huijben et al. , 2020b ) ."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":2,"text":"4.1 MNIST classification"},{"type":"heading","level":3,"text":"4.1.1 Experiment setup"},{"type":"para","text":"We evaluated the classification performance under pixel subsampling using the MNIST dataset (LeCun et al. , 1998 ) . The 70,000 grayscale images of 28 × \\times 28 pixels, representing handwritten digits from 0 and 9, were split into train, validation, and test sets with a ratio of 5:1:1, respectively. We compare the proposed PGA-DPS method with DPS and A-DPS across various sampling ratios, ranging from 1% to 8% with the increment of 1%. Furthermore, we also report the reference performance obtained using all samples (100%)."},{"type":"heading","level":3,"text":"4.1.2 Task model"},{"type":"para","text":"A multi layer perceptron (MLP) with 5 layers is used for the classification network f θ ( ⋅ ) f_{\\theta}(\\cdot) . The MNIST images are flattened and masked according to the subsampling pattern before being fed into the network. The sampling network g k ( ⋅ ) g^{k}(\\cdot) of an LSTM (Long Short-Term Memory) followed by a two-layer MLP, which encodes the current task context and generates the corresponding sampling mask. Both the classification network and sampling network are simultaneously trained using a categorical cross-entropy loss. In addition, the trainable logits of size 784 ( ϕ 1 \\phi_{1} ) are included in the network for DPS and PGA-DPS to provide a deterministic sampling pattern. Detailed model architecture and training settings are provided in the Appendix B.1 ."},{"type":"para","text":"In PGA-DPS, the proportions of prior (deterministic) sampling and active sampling are fixed to 60 and 20 %. Here, the P s Ps and A s As denote the proportions of prior and active sampling, respectively. ( P s Ps , A s As )=(60, 20) implies that 60% of the target samples are selected by prior sampling, and the remaining samples (40%) are acquired over two active group sampling iterations of 20% each."},{"type":"heading","level":3,"text":"4.1.3 Results"},{"type":"para","text":"The classification accuracy on test set is shown in Table 1 . PGA-DPS outperforms both DPS and A-DPS across all sampling ratios, with the largest gain at the low sampling ratios. The active sampling method leverages previously selected samples for adapting its sampling pattern, which is particularly effective under extreme subsampling conditions. A-DPS showed strong performance when sampling ratio ( r r ) is below 4%, whereas DPS works better when more samples are available. This likely arises because A-DPS trains a separate classifier for each sampled pixel, inflating its Lipschitz constant (as discussed in Theorem 1 )."},{"type":"heading","level":2,"text":"4.2 CIFAR-10 classification"},{"type":"heading","level":3,"text":"4.2.1 Experiment setup"},{"type":"para","text":"We evaluated pixel-level subsampling for classification using the CIFAR-10 dataset (Krizhevsky et al. , 2009 ) , which contains 60,000 color images of size 32 x 32 across 10 classes. Of these, 50,000 images were used for training and validation, and 10,000 for testing. The training split was further divided into training and validation set at a 9:1 ratio. We compared the proposed PGA-DPS, DPS and A-DPS methods across sampling ratios ranging from 2% to 20% in increments of 2%. The reference performance using all samples (100 %) is also reported."},{"type":"heading","level":3,"text":"4.2.2 Task model"},{"type":"para","text":"For the classification model f θ ( ⋅ ) f_{\\theta}(\\cdot) , we employed a four-layer convolutional neural network (CNN) followed by three MLP layers. The CIFAR-10 images are first masked according to the subsampling pattern and then fed into the network. We employed the same sampling network architecture and the same cross-entropy loss function as in the MNIST classification experiments. The detailed model architecture and training settings are provided in the Appendix B.2 . In PGA-DPS, the proportions of prior sampling and active sampling are fixed to 10 % and 20 %, respectively."},{"type":"heading","level":3,"text":"4.2.3 Results"},{"type":"para","text":"Table 2 shows that the proposed PGA-DPS outperforms both DPS and A-DPS across all sampling ratios. Because CIFAR-10 classification is substantially more challenging than MNIST, the performance gains are larger across the entire range. Unlike the MNIST classification task, A-DPS exhibits consistently strong performance relative to DPS for all ratios. This behavior is likely attributable to the lower Lipschitz constant of CNNs compared to MLPs used in the MNIST experiments (Shi et al. , 2022 ) , which makes the optimization landscape easier to navigate. However, beyond a certain number of samples ( r r = 14 %), the classification accuracy of A-DPS decreased, likely due to an inflated Lipschitz constant, similar to what was observed in the MNIST classification task."},{"type":"heading","level":2,"text":"4.3 MRI reconstruction"},{"type":"heading","level":3,"text":"4.3.1 Experiment setup"},{"type":"para","text":"To evaluate the performance of k-space subsampling, we tested on the single-coil knee RAW acquisitions from the fastMRI dataset (Zbontar et al. , 2018 ) . A total of 13,000 images, excluding the outer slices, were split into an 8:2:3 ratio for train, validation, and test, respectively. All slices were cropped to central 208 × \\times 208 pixels and normalized to [0, 1]. Vertical Cartesian binary masks ( 𝐌 \\mathbf{M} ) were used for all sampling models, where one column line corresponds to acquiring one phase-encoding line. We simulated the partially acquired k-space by applying the Cartesian binary mask to a fully sampled k-space:"},{"type":"para","text":"where ℱ , ℱ − 1 \\mathcal{F},\\mathcal{F}^{-1} are the forward and inverse Fourier transforms, respectively; 𝐘 ∈ ℝ N × N \\mathbf{Y}\\in\\mathbb{R}^{N\\times N} is the image from subsampled k-space; 𝐗 ∈ ℝ N × N \\mathbf{X}\\in\\mathbb{R}^{N\\times N} is the ground truth image from fully sampled k-space; and | ⋅ | \\left|\\cdot\\right| denotes the magnitude operation. Following the previous study’s setup (Van Gorp et al. , 2021 ) , we take the magnitude of complex-valued Y to simplify the image reconstruction."},{"type":"heading","level":3,"text":"4.3.2 Task model"},{"type":"para","text":"For the reconstruction task f θ ( ⋅ ) f_{\\theta}(\\cdot) , a deep unfolded proximal gradient method is employed to reconstruct the original image 𝐗 ^ \\hat{\\mathbf{X}} from the partially acquired measurement 𝐘 \\mathbf{Y} (Mardani et al. , 2018 ) , by unrolling iterations of the proximal gradient algorithm into a feed-forward neural network:"},{"type":"para","text":"where 𝒫 ψ \\mathcal{P}_{\\psi} denotes the proximal neural network incorporating the image prior regularizer ( ψ {\\psi} ), and α \\alpha is the step size. The sampling network g k ( ⋅ ) g^{k}(\\cdot) consists of a 3-layer CNN, followed by global average pooling, an LSTM, and a single MLP, in sequence. In this way, the current reconstruction result is encoded into the LSTM to generate logits ( ϕ \\phi ), which are used to select the subsequent (active) sample. The reconstruction model and training details are provided in the Appendix B.3 ."},{"type":"para","text":"We compare PGA-DPS to other sampling methods, namely, low-pass, variable density sampling (VDS) (Lustig et al. , 2007 ) , greedy mask selection (Sanchez et al. , 2020 ) , LOUPE (Bahadir et al. , 2020 ) , DPS, and A-DPS. In the loss-pass method, lines nearest to the DC (k-space origin) components are selected; in VDS, sampling density decays with distance from the k-space origin, concentrating the samples centrally and tapering off toward the periphery. In the greedy mask strategy, the mask pattern is optimized using a stochastic greedy algorithm (Sanchez et al. , 2020 ) , implemented via the NESTA solver (Becker et al. , 2011 ) . Afterwards, the corresponding proximal gradient network was trained using the fixed optimal mask obtained from the greedy algorithm (Greedy Prox.)."},{"type":"heading","level":3,"text":"4.3.3 Results"},{"type":"para","text":"To assess the influence of the sampling hyperparameters ( Ps, As ), we conducted an ablation study using their various combinations. Table 3 demonstrates the MR reconstruction performance on the hold-out test set, evaluated using three metrics: normalized mean square error (NMSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). Reconstruction performance shows an overall increasing trend with higher As (i.e., larger k in top-k sampling), across various levels of prior sampling ( Ps ), even without the use of prior-aware deterministic sampling, which is equivalent to A-DPS with group sampling. This trend is consistent with the theoretical explanation proposed in Theorem 1 . However, the improvement plateaus beyond an As of 15-20% , or even declines slightly, presumably due to the reduced rate of active sampling. While PGA-DPS shows slight performance variation across ( Ps, As ) configurations, it outperforms A-DPS in most cases."},{"type":"para","text":"We found the (30%, 30%) configuration of ( Ps, As ) to be the optimal and fixed it for further MRI reconstruction analysis. Furthermore, this optimal configuration generalizes well across different sampling ratios of k-space lines M = 15, 20, 30, and 35 (Table 8 in the Appendix). The performance gain is particularly notable in the low-measurement regime, where the number of k-space lines is small, further demonstrating the robustness of our method under limited data conditions."},{"type":"para","text":"As shown in Table 4 , the proposed PGA-DPS significantly outperforms all other sampling strategies across all evaluation metrics. The example reconstruction from masks generated by PGA-DPS, A-DPS, DPS, and other sampling methods are shown in the Appendix (Fig. 3 )."},{"type":"para","text":"Since DPS optimizes its mask to perform well on average, it heavily concentrates on DC (low-frequency) lines, that capture overall structure of the input data (Fig. 2 ). In contrast, PGA-DPS and A-DPS select more samples farther from the DC region, capturing richer structural details. Note that A-DPS always begins at the center line, whereas PGA-DPS starts with a mix of central and peripheral lines, achieving a more balanced representation of frequency components."},{"type":"heading","level":2,"text":"4.4 MRI reconstruction with active baselines"},{"type":"heading","level":3,"text":"4.4.1 Experiment setup"},{"type":"para","text":"To further validate the effectiveness of our method, we conducted an additional comparison against reinforcement learning (RL)-based sampling strategies: Evaluator Policy (Zhang et al. , 2019 ) and Data-Specific Double Deep Q-Networks (DS-DDQN) (Pineda et al. , 2020 ) . To ensure compatibility with the publicly available DS-DDQN checkpoints, we applied the same pre-processing procedures used in their original work. In addition, the trained checkpoints were used for DPS (DPS_16lines_0seed_Pineda) and A-DPS (ADPS_16lines_0seed_Pineda). We adopt the train-validation-test split protocol from (Van Gorp et al. , 2021 ) , yielding 34,742 training, 1,785 validation, and 1,851 testing images. Each input corresponds to a k-space volume of size 368×640. For evaluation, reconstruction scores are computed over the central 320×320 region, excluding peripheral background areas. PGA-DPS was trained for 5 epochs with a batch size of 1, following the training scheme provided in A-DPS (Van Gorp et al. , 2021 ) . The Ps and As ratios were both set to 30%."},{"type":"heading","level":3,"text":"4.4.2 Results"},{"type":"para","text":"Table 5 presents the MR reconstruction results on the hold-out test set with an acceleration factor of 8. We adopt the Scenario-30L setting, in which 30 Auto-Calibration Signal (ACS) lines are fixed, and the remaining 16 target samples are selected. PGA-DPS outperforms all other sampling methods under this setting. Notably, the overall PSNR and SSIM achieved in this experiment is substantially higher than that reported in our previous MRI study using a smaller matrix size (208x208), reflecting the benefits of using higher-resolution k-space data with abundant ACS lines."},{"type":"heading","level":2,"text":"4.5 Hyperspectral image (HSI) segmentation"},{"type":"heading","level":3,"text":"4.5.1 Experiment setup"},{"type":"para","text":"To analyze the generalizability of PGA-DPS, we evaluated its performance on hyperspectral band selection for segmentation. We used the AeroRIT dataset (Rangnekar et al. , 2020 ) , which consists of aerial images with hyperspectral bands from 400 to 900 nm in 10 nm increments, resulting in a total of 51 spectral bands. Following the original study’s split, 1920 × \\times 3968 image was divided into the left 1920 × \\times 1728 for training, the next 1920 × \\times 512 for validation, and the final 1920 × \\times 1728 for testing. We extracted 64 × \\times 64 patches with 50% overlap from the training set and non-overlapping patches for validation and test sets. The segmentation labels include five classes : (1) roads, (2) buildings, (3) vegetation, (4) cars, and (5) water."},{"type":"heading","level":3,"text":"4.5.2 Task model"},{"type":"para","text":"For the segmentation network f θ ( ⋅ ) f_{\\theta}(\\cdot) , we use a residual U-Net consisting of 6 ResNet blocks following the pix2pix design (Isola et al. , 2017 ; Zhu et al. , 2017 ) : this architecture is referred to as Res-U-net in the AeroRIT study (Rangnekar et al. , 2020 ) . The sampling network g k ( ⋅ ) g^{k}(\\cdot) consists of a 4-layer CNN, with each layer followed by Batch normalization and ReLU activation. The output of the CNN is aggregated into a feature vector via global average pooling, which is then fed to an LSTM and a single-layer MLP to generate logits."},{"type":"para","text":"The categorical cross-entropy loss is computed for 5 classes and the model with the highest mean intersection over union (mIOU) on the validation set is selected for further evaluation. Detailed model architecture and training settings are provided in the Appendix B.4 . In PGA-DPS, the proportions of prior (deterministic) sampling ( P s Ps ) and active sampling ( A s As ) were 80% and 20%, respectively."},{"type":"para","text":"Three different DPS methods (DPS, A-DPS, and PGA-DPS) are implemented and compare to the conventional methods including uniform sampling, SRL-SOA (Self-Representation Learning with Sparse 1D-Operational Autoencoder) (Ahishali et al. , 2022 ) , and GSS (Greedy Spectral Selection) (Morales et al. , 2021 ) . In the uniform strategy, equidistant spectral bands are selected. In SRL-SOA, the polynomial order of the Taylor series expansion used to estimate the original signal using the autoencoder is set to 3. In the GSS approach, classification is performed on the center pixel of a 5 × \\times 5 patch. For comparison, we also implemented the reference method ’All-bands’, which does not use band selection and utilizes all 51 hyperspectral images."},{"type":"heading","level":3,"text":"4.5.3 Results"},{"type":"para","text":"Table 6 shows segmentation performance using three metrics: MPCA (mean per-class accuracy), mIOU (mean Intersection over Union), and mDICE (mean Sørensen–Dice coefficient). PGA-DPS outperformed all other methods, achieving performance comparable to that of using all 51 bands, while enabling over 10-fold acceleration. A-DPS produces less reliable segmentation maps, likely because the complexity of the segmentation makes joint optimization of the sampling strategy and multiple task models difficult. To preserve optimization stability, PGA-DPS limits active sampling at 20%. Example segmentation maps from subsampled bands selected by All-bands, PGA-DPS, A-DPS, DPS, and other band selection methods are shown in the Appendix (Fig. 4 )."},{"type":"heading","level":1,"text":"5 Conclusion"},{"type":"para","text":"We developed a scalable, stable active subsampling technique, called Prior-aware and Group-based Active DPS (PGA-DPS), that integrates context-guided group sampling and deterministic (fixed) subsampling based on training-data prior. PGA-DPS not only provides a theoretical analysis for the stable optimization of group sampling, enabled by DPS-top- k k rather than DPS-top-1, but also shows consistent and improved performance compared to conventional A-DPS. We demonstrated the effectiveness of PGA-DPS across three diverse tasks—classification, image reconstruction, and segmentation—using the MNIST, CIFAR-10, fastMRI knee, and AeroRIT datasets, outperforming all other sampling methods. Since top-k sampling and prior-aware deterministic sampling can be easily applied to various sampling tasks, PGA-DPS may serve as a powerful tool for improving performance in real-world applications where acquisition cost and inference efficiency are critical, such as CT, ultrasound, and radar systems."},{"type":"heading","level":1,"text":"6 Limitation & Future Work"},{"type":"para","text":"Although PGA-DPS excels at generating subsampling strategies for diverse tasks, one possible limitation lies in selecting the hyperparameters associated with the proportion of fixed and active subsampling, denoted as P s Ps and A s As , respectively. The optimal sampling portion differs depending on the downstream task, task model, and sampling ratio. We observed that an active sampling ratio ( As ) range of 20-30% provides the best trade-off between the benefit of top-k sampling and the total budget allocated to active sampling, which narrows the search space for optimal configurations."},{"type":"para","text":"To determine an appropriate prior-sampling ratio ( Ps ), we recommend using the performance discrepancy between DPS and A-DPS as an empirical indicator of the underlying Lipschitz characteristics of the task. Specifically, if DPS consistently outperforms A-DPS under a given configuration, this suggests that the optimization landscape exhibits a larger effective Lipschitz constant—implying that a higher Ps would be beneficial. Conversely, if A-DPS achieves superior performance, the landscape is likely smoother, in which case a smaller Ps is preferable."},{"type":"para","text":"While optimal hyperparameter ( Ps, As ) ranges are suggested, automatically tuning them (Bergstra et al. , 2011 ; Feurer and Hutter, 2019 ; Bischl et al. , 2023 ) could enhance the robustness of the proposed method. Future research may explore joint optimization of the active sampling hyperparameters ( Ps , As ), the subsampling trajectory, and the downstream task model."},{"type":"para","text":"We demonstrate that using a fixed temperature yields robust results (Appendix D ); however, tuning or annealing the temperature could further enhance DPS-based methods. A more comprehensive investigation of temperature scheduling therefore represents a promising avenue for future work."},{"type":"heading","level":3,"text":"Reproducibility statement"},{"type":"para","text":"The experimental settings and models used in our work are described in the Experimental Setup sections of each task. The task models and baseline approaches are explained in detail in the Task Model section. Additional implementation details are provided in the Appendix ."},{"type":"heading","level":2,"text":"B.1 MNIST classification"},{"type":"para","text":"The classification network f θ ( ⋅ ) f_{\\theta}(\\cdot) consists of 5-layer MLP with 784, 256, 128, 128, and 10 nodes, respectively. Each layer was followed by a leaky ReLU activation with a negative slope of 0.2 and the last layer followed by softmax activation that output probabilities of class. A dropout rate of 30% is also applied to the first three layers."},{"type":"para","text":"The sampling network g ( ⋅ ) g(\\cdot) consists of an LSTM (Long Short-Term Memory) with one hidden layer of size 128, followed by two MLPs with 256 and 784 nodes, respectively. A leaky ReLU activation with a negative slope of 0.2 and a dropout rate of 30% are applied after the first MLP."},{"type":"para","text":"The categorical cross-entropy loss was minimized using the Adam optimizer (Kingma and Ba, 2014 ) with the learning rate of 2 × 10 − 4 2\\times 10^{-4} ( β 1 = 0.9 \\beta_{1}=0.9 , β 2 = 0.999 \\beta_{2}=0.999 , and ϵ = 10 − 7 \\epsilon=10^{-7} ). In addition, the trainable logits, of size 784 (nn.Parameters()), are also updated with a learning rate of 2 × 10 − 3 2\\times 10^{-3} for DPS and PGA-DPS. The optimization was run for 100 epochs with a batch size of 256."},{"type":"heading","level":2,"text":"B.2 CIFAR-10 classification"},{"type":"para","text":"The classification network f θ ( ⋅ ) f_{\\theta}(\\cdot) consists of four convolutional layers with channel sizes of 32, 64, and 128, respectively, and a kernel size of 3 × \\times 3. Each layer is followed by batch normalization, a ReLU activation function, and a 2 × \\times 2 max pooling operation. The resulting 128-channel feature map is then flatten and fed into an MLP composed of Linear(2048,256)-ReLU-Dropout(0.5)-Linear(256,128)-ReLu-Linear(128,10)."},{"type":"para","text":"The sampling network g ( ⋅ ) g(\\cdot) consists of an LSTM (Long Short-Term Memory) with one hidden layer of size 128, followed by two MLPs with 256 and 1024 nodes, respectively. A leaky ReLU activation with a negative slope of 0.2 and a dropout rate of 30% are applied after the first MLP."},{"type":"para","text":"The categorical cross-entropy loss was minimized using the Adam optimizer with the learning rate of 2 × 10 − 4 2\\times 10^{-4} . In addition, the trainable logits, of size 1024 (nn.Parameters()), are also updated with a learning rate of 2 × 10 − 3 2\\times 10^{-3} for DPS and PGA-DPS. The optimization was run for 50 epochs with a batch size of 128."},{"type":"heading","level":2,"text":"B.3 MRI reconstruction"},{"type":"para","text":"The proximal neural network 𝒫 ψ \\mathcal{P}_{\\psi} consists of four convolutional layers with channel size of 16,16,16, and 1, respectively, and a kernel size of 3 × \\times 3. Each layer is followed by a ReLU activation function except for the final layer. For the image prior regularizer ( ψ {\\psi} ), a single 3 × \\times 3 convolutional layer is used. The proximal gradient method is unrolled for three iterations (k=3)."},{"type":"para","text":"The adversarial loss (Ledig et al. , 2017 ) , combined with a mean square error (MSE), is used to reconstruct visually plausible MR images, by training a discriminator network that distinguishes the real and reconstructed images. In addition, the discriminator feature loss between the real and reconstructed images was employed."},{"type":"para","text":"The sampling network g k ( ⋅ ) g^{k}(\\cdot) consists of a sequence of layers: Conv2d(1,16), ReLU, Conv2d(16,32), ReLu, Conv2d(32,64), ReLU, Average Pooling, Flatten, LSTM(64), and a single-layer MLP(64,208). All convolutional layers have a kernel size of 3 × \\times 3. The last MLP converts the output of the LSTM (encoded context) to the logits of size 208."},{"type":"para","text":"The discriminator network is consists of a sequence of layers: Conv2d(1,64), LeakyReLU(0.2), Conv2d(64,64), LeakyReLU(0.2), Conv2d(64,64), LeakyReLU(0.2), Average Pooling, Dropout(0.4), a single-layer MLP (64,1), and Sigmoid. All convolutional layers have a kernel size of 3 × \\times 3 and a stride of 2."},{"type":"para","text":"The total loss is computed as a weighted sum of three losses: mean squared error (MSE), adversarial loss, and the discriminator feature loss, with respective weights of 1, 5 × 10 − 6 5\\times 10^{-6} , and 10 − 7 10^{-7} ."},{"type":"para","text":"The network is updated using the Adam optimizer (Kingma and Ba, 2014 ) with the learning rate of 2 × 10 − 4 2\\times 10^{-4} ( β 1 = 0.9 \\beta_{1}=0.9 , β 2 = 0.999 \\beta_{2}=0.999 , and ϵ = 1 e − 7 \\epsilon=1e-7 ). In addition, the trainable logits, of size 208 (nn.Parameters()), are also updated with a learning rate of 2 × 10 − 3 2\\times 10^{-3} for DPS and PGA-DPS. The optimization was run for 10 epochs with a batch size of 2."},{"type":"heading","level":2,"text":"B.4 Hyperspectral image segmentation"},{"type":"para","text":"For the segmentation network f θ ( ⋅ ) f_{\\theta}(\\cdot) , we use a residual U-Net consisting of 6 ResNet blocks, as proposed in the pix2pix (Isola et al. , 2017 ; Zhu et al. , 2017 ) . The input channel is set to 51, corresponding to the total number of hyperspectral bands, and the output channel is set to 6, corresponding to the number of classes including the undefined class. The number of the filters in the last convolution layer is set to 64, and tow downsampling operations are applied."},{"type":"para","text":"The sampling network g k ( ⋅ ) g^{k}(\\cdot) consists of a sequence of layers: Conv2d(6,32), BN, ReLU, Conv2d(32,64), BN, ReLu, Conv2d(64,128), BN, ReLU, Conv2d(128,256), BN, ReLU, Average Pooling, Flatten, LSTM(256), and a single-layer MLP(256,51), where BN denotes bach normalization. All convolutional layers have a kernel size of 3 × \\times 3. The last MLP converts the output of the LSTM (encoded context) to the logits of size 51."},{"type":"para","text":"In PGA-DPS, the proportions of prior (deterministic) sampling ( N N ) and active sampling ( M M ) were 80% and 20%, respectively, resulting in two group sampling iterations."},{"type":"para","text":"The network is updated using the Adam optimizer (Kingma and Ba, 2014 ) with the learning rate of 10 − 4 10^{-4} ( β 1 = 0.9 \\beta_{1}=0.9 , β 2 = 0.999 \\beta_{2}=0.999 , and ϵ = 1 e − 7 \\epsilon=1e-7 ), while the learning rate of 2 × 10 − 4 2\\times 10^{-4} is used for DPS and GSS. In addition, the trainable logits, of size 51 (nn.Parameters()), are also updated with a learning rate of 1 for DPS and PGA-DPS. The optimization was run for 60 epochs with a batch size of 100."},{"type":"heading","level":2,"text":"C.1 MNIST classification"},{"type":"para","text":"We evaluated multiple P s Ps values under an A s As of 20% and multiple A s As values under a P s Ps of 60%. Table 7 shows that PA-DPS outperforms the baselines in almost all configurations, although the optimal configuration shifts slightly across sampling ratios."},{"type":"heading","level":2,"text":"C.2 MRI reconstruction"},{"type":"para","text":"Figure 3 shows example reconstruction from masks generated by PGA-DPS, A-DPS, DPS, and other sampling methods. Notably, PGA-DPS selects samples farthest from the DC, capturing fine-grained details. PGA-DPS exhibits the farthest average mask distance from the DC (11.8), versas 10.2 (A-DPS), 8.8 (DPS), 10.8 (LOUPE), 8.1 (Greedy Prox.), and 11.3 (VDS). Despite prioritizing high-frequency components, PGA-DPS also samples adequately near DC, demonstrating its superior reconstruction performance with higher SSIM values."},{"type":"para","text":"To evaluate the robustness and general applicability of the proposed method, we tested whether the selected optimal configuration of (Ps, As) = (30%, 30%), identified from analysis with k-space lines M = 26, generalizes well across different sampling ratios by conducting experiments with varying values of M, i.e., k-space lines M = 15, 20, 30, and 35, corresponding to acceleration rates r = 7.2%, 9.6%, 14.4%, and 16.8%, respectively. As shown in Table 8 , PGA-DPS outperforms DPS and A-DPS across various numbers of k-space lines, with a particularly large margin when the number of k-space lines is small."},{"type":"heading","level":2,"text":"C.3 Hyperspectral image segmentation"},{"type":"para","text":"Figure 4 shows example segmentation maps from subsampled bands selected by All-bands, PGA-DPS, A-DPS, DPS, and other band selection methods."},{"type":"para","text":"We fixed the temperature to 2 during training for DPS, A-DPS, and PGA-DPS. However, tuning the relaxation parameter is critical for balancing the bias-variance trade-off of the gradient estimator and may lead to further improvements."},{"type":"para","text":"To assess this, we conducted additional MR reconstruction experiments using various fixed temperature values. As shown in Table 9 , the reconstruction performance was largely insensitive to temperature values. Nonetheless, a fixed temperature of 5 yielded the best overall results, suggesting that temperature optimization could provide marginal additional improvements."},{"type":"para","text":"We also evaluated temperature annealing, defined as, τ = m a x ( 2.0 , T × e − γ n ) \\tau=max(2.0,T\\times e^{-\\gamma n}) , where n denotes the training epoch. As shown in Table 10 , different annealing schedules produced noticeably different performance. Consequently, using a fixed temperature provides sufficiently robust and stable results in our setting, consistent with what has been found to work well in practice (Huijben et al. , 2020b ; Van Gorp et al. , 2021 ) . Nevertheless, a more comprehensive investigation of temperature annealing within DPS remains a promising direction for future work."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Modern technologies generate massive datasets that can require lengthy acquisition time and hinder real-time onboard processing. Many real-world applications, including Magnetic Resonance Imaging (MRI) (Ye, 2019 ) , computed tomography (CT) imaging (Chen et al. , 2008 ) , ultrasound imaging (Huijben et al. , 2020a ) , digital micromirror device (Baraniuk, 2007 ) , seismic surveying (Herrmann et al. , 2012 ) , and hyperspectral imaging (Sun and Du, 2019 ) , highlight the importance of reducing imaging data volume. Therefore, strategic sampling not only reduces data volume and preserves essential information for efficient transfer and processing but also accelerates acquisition, a critical factor in medical imaging.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Compressed sensing (CS) was developed as a subsampling strategy to overcome the Nyquist-Shannon limits on the sampling rates required for perfect signal reconstruction (Donoho, 2006 ; Eldar and Kutyniok, 2012 ) . CS has been widely adopted across various applications and has demonstrated significant impact (Lustig et al. , 2007 ; Baraniuk and Steeghs, 2007 ; Yu and Wang, 2009 ; Martín et al. , 2014 ; Lorintiu et al. , 2015 ; Han et al. , 2016 ) . Although CS exploits the inherent signal structures, such as sparsity, it does not take into account the information relevant to the downstream task during the sampling process. Bridging the gap between modality- and task-specific knowledge and the sampling process has proven challenging.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recently, subsampling techniques customized for specific data distributions and downstream tasks have been proposed with advances in deep learning, offering learned yet fixed sampling patterns (Huijben et al. , 2020b ; Weiss et al. , 2020 ; Shen et al. , 2020 ; Sherry et al. , 2020 ; Zhang et al. , 2020 ; Aggarwal and Jacob, 2020 ; Mou et al. , 2021 ; Yang et al. , 2025 ) . These approaches optimize a sampling pattern based on the average data distribution in the training set, which may not provide optimal results for individual instances. To address this limitation, active sampling was introduced, where new points are adaptively selected based on previously acquired samples, iterating until the required target number of samples is obtained (Zhang et al. , 2019 ; Jin et al. , 2019 ; Bakker et al. , 2020 ; Pineda et al. , 2020 ; Van Gorp et al. , 2021 ; Tian et al. , 2025 ) . Active sampling produces a sampling trajectory that adapts to each test instance and incorporates newly acquired data dynamically. This is particularly valuable in medical imaging, where each patient’s unique physiological condition demands individualized acquisition trajectories.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although active subsampling provides instance-specific sampling patterns and enables effective adaptation to new data, boosting downstream task performance, its sampling strategy still has room for improvement. Through iterative top-1 sampling guided by previously selected samples, active sampling leverages inter-sample relationships to optimize the sampling model and minimizes redundant information. However, it fails to fully exploit the rich prior knowledge embedded in the training dataset. Moreover, relying on top-1 sampling leads to sub-optimal optimization (Huijben et al. , 2020b ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we enhance Active Deep Probabilistic Subsampling (A-DPS) architecture by incorporating deterministic (fixed) subsampling based on prior knowledge of the training data with group sampling to achieve more robust optimization (Fig. 1 ). We call this approach Prior-aware and context-guided Group-based Active Deep Probabilistic Subsampling (PGA-DPS). We demonstrate that PGA-DPS exploits training-data priors via deterministic sampling and reinforces this with instance-adaptive group sampling, resulting in a smoother loss landscape and more stable optimization. We evaluated PGA-DPS on the MNIST dataset (LeCun et al. , 1998 ) , the CIFAR-10 dataset (Krizhevsky et al. , 2009 ) , the fastMRI knee dataset (Zbontar et al. , 2018 ) , and the hyperspectral AeroRIT dataset (Rangnekar et al. , 2020 ) . Across classification, image reconstruction, and segmentation tasks, PGA-DPS outperforms all other state-of-the-art sampling methods.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent works have proposed learning-based subsampling techniques tailored to data types and downstream tasks. In the field of MRI, Learning-based Optimization of the Under-sampling PattErn (LOUPE) was developed to simultaneously optimize the subsampling pattern and reconstruct the image, addressing two core challenges of compressed sensing (Bahadir et al. , 2020 ) . LOUPE learns the sampling mask by relaxing the non-differentiable threshold operation, which is similar to the Gumbel-softmax trick used in Deep Probabilistic Subsampling (DPS) (Huijben et al. , 2020b ) . The target sampling ratio is enforced by an L1 sparsity penalty on sampling mask within the loss function. Additionally, a stochastic greedy algorithm has been applied in MRI to obtain the sampling pattern that minimizes the reconstruction loss (Sanchez et al. , 2020 ) . However, greedy algorithm-based methods require pre-trained reconstruction capable of handling any subsampling pattern to thoroughly evaluate the loss.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In hyperspectral imaging (HSI), band selection techniques are widely used to identify a small informative subset of spectral bands and reduce spectral redundancy. Self-Representation Learning with Sparse 1D-Operational Autoencoder (SRL-SOA) employs a sparse autoencoder model to learn a sparse representation using a Taylor-series expansion of non-linear transformation (Ahishali et al. , 2022 ) . However, this does not incorporate the downstream task model for band selection. Greedy Spectral Selection (GSS) adopts a greedy algorithm: it first performs filter-based interband redundancy analysis, then trains a Convolutional Neural Network (CNN) to assess classification performance of the chosen bands (Morales et al. , 2021 ) . On the other hand, CNN based on Bandwise-independent convolution and Hard thresholding (BHCNN) jointly optimizes band selection and the classification by applying a hard threshold mask via a straight-through estimator (Feng et al. , 2020 ) . Although this allows joint optimization, gradients for non-selected bands are ignored, which can hinder the discovery of other informative bands during training. Furthermore, the aforementioned task-aware band selection methods are designed for classification, which may not provide optimal spectral bands for segmentation tasks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An active sampling strategy for MRI k-space acquisition uses an adversarial neural network to distinguish between sampled and unsampled lines in the Fourier space (Zhang et al. , 2019 ) . The advarsarial network identifies the k-space line that appears most realistically fake and selects it for acquisition; this process repeats until the desired number of lines is obtained. However, this approach processes k-space data directly, which limits its applicability to MR image undersampling. Reinforcement learning (RL) has also been explored for active acquisition, applying greedy and modified ϵ \\\\epsilon -greedy policies to actively acquire optimal sampling masks (Pineda et al. , 2020 ; Bakker et al. , 2020 ) . However, RL-based techniques require a pretrained reconstruction network that can handle a variety of sampling patterns. To address this issue, Active DPS (A-DPS) jointly optimizes both the sampling mask and the associated reconstruction network (Van Gorp et al. , 2021 ) . Building on A-DPS, the proposed Prior-aware and Group-based Active DPS (PGA-DPS) improves optimization robustness and fully leverages prior knowledge embedded in the training data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Method\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Task-adaptive subsampling framework\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We aim to find an optimal subsampling strategy A ⊆ { 0 , 1 } N A\\\\subseteq{\\\\{0,1\\\\}}^{N} on a given input signal x ∈ ℝ N x\\\\in\\\\mathbb{R}^{N} for a specific task t , such that the performance of the task is maintained using fewer input samples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where f f is a task model that predicts the task t from the input signal x x , θ \\\\theta is parameters of the model, and M is the target number of subsamples. The proportion of selected samples can be represented by the sampling ratio r = 100 × M / N % r=100\\\\times M/N\\\\% . In general, neural networks are widely adopted for task models, such as classification, segmentation, reconstruction, detection, and more; however, backpropagation from the loss function is hindered by the non-differentiable nature of sampling. This issue was alleviated using the Gumbel-Softmax reparameterization trick introduced in Deep Probabilistic Subsampling (DPS).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 DPS: Deep Probabilistic Subsampling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DPS proposed an end-to-end deep learning framework that jointly optimizes both the subsampling strategy and the downstream task model (Huijben et al. , 2020a ) . During training, DPS learns unnormalized logits ϕ \\\\phi , which are used to generate a subsampling mask probability P ( A | ϕ ) P\\\\left(A|\\\\phi\\\\right) from a categorical distribution with N N classes: A ∼ Categorical ( exp ( ϕ i ) ∑ j N exp ( ϕ j ) , i ∈ { 1 , … , N } ) A\\\\sim\\\\operatorname{Categorical}\\\\left(\\\\frac{\\\\exp(\\\\phi_{i})}{\\\\sum_{j}^{N}\\\\exp(\\\\phi_{j})},\\\\,i\\\\in\\\\{1,\\\\dots,N\\\\}\\\\right) . The Gumbel-max trick enables efficient sampling from a categorical distribution (Gumbel, 1954 ; Maddison et al. , 2014 ) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ϕ ∈ ℝ N \\\\phi\\\\in\\\\mathbb{R}^{N} is an unnormalized logit vector composed of ϕ i \\\\phi_{i} and G G is a vector of i.i.d noise samples, G i G_{i} , drawn from a Gumbel distribution ∼ Gumbel ( 0 , 1 ) \\\\sim\\\\text{Gumbel}(0,1) . However, since the arg max function is non-differentiable and thus unsuitable for backpropagation, DPS employs the Gumbel-Softmax trick by relaxing the max operation into a differentiable softmax function as follows (Jang et al. , 2017 ; Maddison et al. , 2017 ) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where τ \\\\tau is a temperature parameter that controls the degree of relaxation, enabling gradients to be distributed across multi-dimensional logits when τ > 0 \\\\tau>0 during training. In addition, as the temperature decreases τ → 0 \\\\tau\\\\rightarrow 0 , s o f t m a x τ softmax_{\\\\tau} can anneal into a categorical distribution. Although tuning the relaxation with τ \\\\tau is crucial for managing the bias-variance trade-off of the gradient estimator (Tucker et al. , 2017 ) , the DPS architecture has been shown to perform effectively with a fixed temperature during training ( τ = 2 \\\\tau=2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In addition, the Gumbel-max trick can be extended to the Gumbel top- k k trick to efficiently select the k k most probable samples, rather than just the single highest-probability one (Kool et al. , 2019 ; Plötz and Roth, 2018 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DPS introduces two variants for selecting k k samples: DPS-top-1, which sequentially samples from k k independent categorical distributions—each with its own trainable logits—and DPS-top- k k , which selects all k k samples simultaneously using a single shared logit for greater parameter efficiency. Although DPS-top-1 is more expressive, the DPS-top- k k approach showed improved performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similarly, Active Deep Probabilistic Subsampling (A-DPS) introduces an active sampling that uses contextual information from previously selected samples to guide subsequent selection, allowing adaptation to new data (Van Gorp et al. , 2021 ) . In A-DPS, samples are chosen iteratively by applying DPS-top-1 at every step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where j j denotes the iteration; w j ⊆ { − ∞ , 0 } N w^{j}\\\\subseteq{\\\\{-\\\\infty,0\\\\}}^{N} is a cumulative mask that assigns minus infinity to previously selected elements, ensuring they are excluded after re-normalization; ϕ j ∈ ℝ N \\\\phi^{j}\\\\in\\\\mathbb{R}^{N} and G j ∈ ℝ N G^{j}\\\\in\\\\mathbb{R}^{N} are ϕ \\\\phi and G G for j t h j^{th} sample, respectively; g j g^{j} is a sampling network in j t h j^{th} iteration that encodes the current task context based on the samples selected so far; and f θ f_{\\\\theta} is a differentiable model that generates the current task prediction. The context is built using an analysis-by-synthesis approach, where the task (synthesis) module guides the sampling (analysis). In A-DPS, losses are accumulated over all iterations, and the network is updated in a semi-greedy manner.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 PGA-DPS: Prior-aware and Group-based Active DPS\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PGA-DPS initiates with a deterministic subsampling pattern and then iteratively acquires additional samples through active sampling (Fig. 2 ). The deterministic pattern drawn from the training data’s prior is followed by active sampling that exploits each new input’s contextual information, thereby combining global data prior with input specific context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, PGA-DPS uses DPS-top- k k to select samples in groups for active sampling, rather than DPS-top-1 in A-DPS, enabling it to achieve a smaller effective Lipschitz constant. We analyze group sampling enabled by DPS-top- k k in terms of the loss, which captures the Lipschitzness of the loss and reflects the smoothness of the optimization landscape. The proof is available in Appendix A .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(The effect of group sampling on the Lipschitzness of the loss). Let f 1 , f 2 , … , f k f_{1},f_{2},\\\\dots,f_{k} be task models, where each function f j f_{j} has a corresponding Lipschitz constants L j L_{j} for j = 1 , 2 , … , k j=1,2,\\\\dots,k . In DPS-top-1, k task functions are required for each input sample x k x_{k} , whereas DPS-top-k defines a single task function f k f_{k} over group of k samples. The Lipschitzness of the loss function for each DPS-top-1 and DPS-top- k k is as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, x ∗ ∈ ℝ N x^{*}\\\\in\\\\mathbb{R}^{N} is the fully sampled input signal (unsampled signal = 0), x ∈ ℝ N x\\\\in\\\\mathbb{R}^{N} is the sampled input signal, and x r x_{r} is the r t h r^{th} sample. If the Lipschitz constant of task model at each iteration L j ≥ 1 L_{j}\\\\geq 1 ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neural networks typically have Lipschitz constants much greater than one, except in the trivial near-identity case (Malherbe and Vayatis, 2017 ; Bartlett et al. , 2018 ; Latorre et al. , 2020 ; Shi et al. , 2022 ) . Because our task of classification, image reconstruction (from k-space to image), and segmentation, are far from identity mappings, DPS-top- k k exhibits a smaller effective Lipschitz constant than DPS-top-1, whose effective constant is the product of multiple Lipschitz constants. Moreover, the same analysis can apply to the Lipschitz constants of gradient loss ‖ ∇ L o s s ‖ \\\\left\\\\|\\\\nabla Loss\\\\right\\\\| . Consequently, PGA-DPS creates smoother latent space, resulting in more stable and efficient optimization (Virmaux and Scaman, 2018 ; Berkenkamp et al. , 2017 ; Liu et al. , 2022 ; Gouk et al. , 2021 ; Santurkar et al. , 2018 ) . This theoretical analysis aligns with previous experimental results that Top- k k sampling outperformed Top-1 sampling in DPS (Huijben et al. , 2020b ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 MNIST classification\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.1 Experiment setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated the classification performance under pixel subsampling using the MNIST dataset (LeCun et al. , 1998 ) . The 70,000 grayscale images of 28 × \\\\times 28 pixels, representing handwritten digits from 0 and 9, were split into train, validation, and test sets with a ratio of 5:1:1, respectively. We compare the proposed PGA-DPS method with DPS and A-DPS across various sampling ratios, ranging from 1% to 8% with the increment of 1%. Furthermore, we also report the reference performance obtained using all samples (100%).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.2 Task model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A multi layer perceptron (MLP) with 5 layers is used for the classification network f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) . The MNIST images are flattened and masked according to the subsampling pattern before being fed into the network. The sampling network g k ( ⋅ ) g^{k}(\\\\cdot) of an LSTM (Long Short-Term Memory) followed by a two-layer MLP, which encodes the current task context and generates the corresponding sampling mask. Both the classification network and sampling network are simultaneously trained using a categorical cross-entropy loss. In addition, the trainable logits of size 784 ( ϕ 1 \\\\phi_{1} ) are included in the network for DPS and PGA-DPS to provide a deterministic sampling pattern. Detailed model architecture and training settings are provided in the Appendix B.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In PGA-DPS, the proportions of prior (deterministic) sampling and active sampling are fixed to 60 and 20 %. Here, the P s Ps and A s As denote the proportions of prior and active sampling, respectively. ( P s Ps , A s As )=(60, 20) implies that 60% of the target samples are selected by prior sampling, and the remaining samples (40%) are acquired over two active group sampling iterations of 20% each.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.3 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The classification accuracy on test set is shown in Table 1 . PGA-DPS outperforms both DPS and A-DPS across all sampling ratios, with the largest gain at the low sampling ratios. The active sampling method leverages previously selected samples for adapting its sampling pattern, which is particularly effective under extreme subsampling conditions. A-DPS showed strong performance when sampling ratio ( r r ) is below 4%, whereas DPS works better when more samples are available. This likely arises because A-DPS trains a separate classifier for each sampled pixel, inflating its Lipschitz constant (as discussed in Theorem 1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 CIFAR-10 classification\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.2.1 Experiment setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated pixel-level subsampling for classification using the CIFAR-10 dataset (Krizhevsky et al. , 2009 ) , which contains 60,000 color images of size 32 x 32 across 10 classes. Of these, 50,000 images were used for training and validation, and 10,000 for testing. The training split was further divided into training and validation set at a 9:1 ratio. We compared the proposed PGA-DPS, DPS and A-DPS methods across sampling ratios ranging from 2% to 20% in increments of 2%. The reference performance using all samples (100 %) is also reported.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.2.2 Task model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the classification model f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) , we employed a four-layer convolutional neural network (CNN) followed by three MLP layers. The CIFAR-10 images are first masked according to the subsampling pattern and then fed into the network. We employed the same sampling network architecture and the same cross-entropy loss function as in the MNIST classification experiments. The detailed model architecture and training settings are provided in the Appendix B.2 . In PGA-DPS, the proportions of prior sampling and active sampling are fixed to 10 % and 20 %, respectively.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.2.3 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 shows that the proposed PGA-DPS outperforms both DPS and A-DPS across all sampling ratios. Because CIFAR-10 classification is substantially more challenging than MNIST, the performance gains are larger across the entire range. Unlike the MNIST classification task, A-DPS exhibits consistently strong performance relative to DPS for all ratios. This behavior is likely attributable to the lower Lipschitz constant of CNNs compared to MLPs used in the MNIST experiments (Shi et al. , 2022 ) , which makes the optimization landscape easier to navigate. However, beyond a certain number of samples ( r r = 14 %), the classification accuracy of A-DPS decreased, likely due to an inflated Lipschitz constant, similar to what was observed in the MNIST classification task.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 MRI reconstruction\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.3.1 Experiment setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To evaluate the performance of k-space subsampling, we tested on the single-coil knee RAW acquisitions from the fastMRI dataset (Zbontar et al. , 2018 ) . A total of 13,000 images, excluding the outer slices, were split into an 8:2:3 ratio for train, validation, and test, respectively. All slices were cropped to central 208 × \\\\times 208 pixels and normalized to [0, 1]. Vertical Cartesian binary masks ( 𝐌 \\\\mathbf{M} ) were used for all sampling models, where one column line corresponds to acquiring one phase-encoding line. We simulated the partially acquired k-space by applying the Cartesian binary mask to a fully sampled k-space:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ℱ , ℱ − 1 \\\\mathcal{F},\\\\mathcal{F}^{-1} are the forward and inverse Fourier transforms, respectively; 𝐘 ∈ ℝ N × N \\\\mathbf{Y}\\\\in\\\\mathbb{R}^{N\\\\times N} is the image from subsampled k-space; 𝐗 ∈ ℝ N × N \\\\mathbf{X}\\\\in\\\\mathbb{R}^{N\\\\times N} is the ground truth image from fully sampled k-space; and | ⋅ | \\\\left|\\\\cdot\\\\right| denotes the magnitude operation. Following the previous study’s setup (Van Gorp et al. , 2021 ) , we take the magnitude of complex-valued Y to simplify the image reconstruction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.3.2 Task model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the reconstruction task f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) , a deep unfolded proximal gradient method is employed to reconstruct the original image 𝐗 ^ \\\\hat{\\\\mathbf{X}} from the partially acquired measurement 𝐘 \\\\mathbf{Y} (Mardani et al. , 2018 ) , by unrolling iterations of the proximal gradient algorithm into a feed-forward neural network:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒫 ψ \\\\mathcal{P}_{\\\\psi} denotes the proximal neural network incorporating the image prior regularizer ( ψ {\\\\psi} ), and α \\\\alpha is the step size. The sampling network g k ( ⋅ ) g^{k}(\\\\cdot) consists of a 3-layer CNN, followed by global average pooling, an LSTM, and a single MLP, in sequence. In this way, the current reconstruction result is encoded into the LSTM to generate logits ( ϕ \\\\phi ), which are used to select the subsequent (active) sample. The reconstruction model and training details are provided in the Appendix B.3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare PGA-DPS to other sampling methods, namely, low-pass, variable density sampling (VDS) (Lustig et al. , 2007 ) , greedy mask selection (Sanchez et al. , 2020 ) , LOUPE (Bahadir et al. , 2020 ) , DPS, and A-DPS. In the loss-pass method, lines nearest to the DC (k-space origin) components are selected; in VDS, sampling density decays with distance from the k-space origin, concentrating the samples centrally and tapering off toward the periphery. In the greedy mask strategy, the mask pattern is optimized using a stochastic greedy algorithm (Sanchez et al. , 2020 ) , implemented via the NESTA solver (Becker et al. , 2011 ) . Afterwards, the corresponding proximal gradient network was trained using the fixed optimal mask obtained from the greedy algorithm (Greedy Prox.).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.3.3 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To assess the influence of the sampling hyperparameters ( Ps, As ), we conducted an ablation study using their various combinations. Table 3 demonstrates the MR reconstruction performance on the hold-out test set, evaluated using three metrics: normalized mean square error (NMSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). Reconstruction performance shows an overall increasing trend with higher As (i.e., larger k in top-k sampling), across various levels of prior sampling ( Ps ), even without the use of prior-aware deterministic sampling, which is equivalent to A-DPS with group sampling. This trend is consistent with the theoretical explanation proposed in Theorem 1 . However, the improvement plateaus beyond an As of 15-20% , or even declines slightly, presumably due to the reduced rate of active sampling. While PGA-DPS shows slight performance variation across ( Ps, As ) configurations, it outperforms A-DPS in most cases.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We found the (30%, 30%) configuration of ( Ps, As ) to be the optimal and fixed it for further MRI reconstruction analysis. Furthermore, this optimal configuration generalizes well across different sampling ratios of k-space lines M = 15, 20, 30, and 35 (Table 8 in the Appendix). The performance gain is particularly notable in the low-measurement regime, where the number of k-space lines is small, further demonstrating the robustness of our method under limited data conditions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Table 4 , the proposed PGA-DPS significantly outperforms all other sampling strategies across all evaluation metrics. The example reconstruction from masks generated by PGA-DPS, A-DPS, DPS, and other sampling methods are shown in the Appendix (Fig. 3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since DPS optimizes its mask to perform well on average, it heavily concentrates on DC (low-frequency) lines, that capture overall structure of the input data (Fig. 2 ). In contrast, PGA-DPS and A-DPS select more samples farther from the DC region, capturing richer structural details. Note that A-DPS always begins at the center line, whereas PGA-DPS starts with a mix of central and peripheral lines, achieving a more balanced representation of frequency components.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 MRI reconstruction with active baselines\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.4.1 Experiment setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To further validate the effectiveness of our method, we conducted an additional comparison against reinforcement learning (RL)-based sampling strategies: Evaluator Policy (Zhang et al. , 2019 ) and Data-Specific Double Deep Q-Networks (DS-DDQN) (Pineda et al. , 2020 ) . To ensure compatibility with the publicly available DS-DDQN checkpoints, we applied the same pre-processing procedures used in their original work. In addition, the trained checkpoints were used for DPS (DPS_16lines_0seed_Pineda) and A-DPS (ADPS_16lines_0seed_Pineda). We adopt the train-validation-test split protocol from (Van Gorp et al. , 2021 ) , yielding 34,742 training, 1,785 validation, and 1,851 testing images. Each input corresponds to a k-space volume of size 368×640. For evaluation, reconstruction scores are computed over the central 320×320 region, excluding peripheral background areas. PGA-DPS was trained for 5 epochs with a batch size of 1, following the training scheme provided in A-DPS (Van Gorp et al. , 2021 ) . The Ps and As ratios were both set to 30%.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.4.2 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 presents the MR reconstruction results on the hold-out test set with an acceleration factor of 8. We adopt the Scenario-30L setting, in which 30 Auto-Calibration Signal (ACS) lines are fixed, and the remaining 16 target samples are selected. PGA-DPS outperforms all other sampling methods under this setting. Notably, the overall PSNR and SSIM achieved in this experiment is substantially higher than that reported in our previous MRI study using a smaller matrix size (208x208), reflecting the benefits of using higher-resolution k-space data with abundant ACS lines.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Hyperspectral image (HSI) segmentation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.5.1 Experiment setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To analyze the generalizability of PGA-DPS, we evaluated its performance on hyperspectral band selection for segmentation. We used the AeroRIT dataset (Rangnekar et al. , 2020 ) , which consists of aerial images with hyperspectral bands from 400 to 900 nm in 10 nm increments, resulting in a total of 51 spectral bands. Following the original study’s split, 1920 × \\\\times 3968 image was divided into the left 1920 × \\\\times 1728 for training, the next 1920 × \\\\times 512 for validation, and the final 1920 × \\\\times 1728 for testing. We extracted 64 × \\\\times 64 patches with 50% overlap from the training set and non-overlapping patches for validation and test sets. The segmentation labels include five classes : (1) roads, (2) buildings, (3) vegetation, (4) cars, and (5) water.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.5.2 Task model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the segmentation network f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) , we use a residual U-Net consisting of 6 ResNet blocks following the pix2pix design (Isola et al. , 2017 ; Zhu et al. , 2017 ) : this architecture is referred to as Res-U-net in the AeroRIT study (Rangnekar et al. , 2020 ) . The sampling network g k ( ⋅ ) g^{k}(\\\\cdot) consists of a 4-layer CNN, with each layer followed by Batch normalization and ReLU activation. The output of the CNN is aggregated into a feature vector via global average pooling, which is then fed to an LSTM and a single-layer MLP to generate logits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The categorical cross-entropy loss is computed for 5 classes and the model with the highest mean intersection over union (mIOU) on the validation set is selected for further evaluation. Detailed model architecture and training settings are provided in the Appendix B.4 . In PGA-DPS, the proportions of prior (deterministic) sampling ( P s Ps ) and active sampling ( A s As ) were 80% and 20%, respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three different DPS methods (DPS, A-DPS, and PGA-DPS) are implemented and compare to the conventional methods including uniform sampling, SRL-SOA (Self-Representation Learning with Sparse 1D-Operational Autoencoder) (Ahishali et al. , 2022 ) , and GSS (Greedy Spectral Selection) (Morales et al. , 2021 ) . In the uniform strategy, equidistant spectral bands are selected. In SRL-SOA, the polynomial order of the Taylor series expansion used to estimate the original signal using the autoencoder is set to 3. In the GSS approach, classification is performed on the center pixel of a 5 × \\\\times 5 patch. For comparison, we also implemented the reference method ’All-bands’, which does not use band selection and utilizes all 51 hyperspectral images.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.5.3 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 6 shows segmentation performance using three metrics: MPCA (mean per-class accuracy), mIOU (mean Intersection over Union), and mDICE (mean Sørensen–Dice coefficient). PGA-DPS outperformed all other methods, achieving performance comparable to that of using all 51 bands, while enabling over 10-fold acceleration. A-DPS produces less reliable segmentation maps, likely because the complexity of the segmentation makes joint optimization of the sampling strategy and multiple task models difficult. To preserve optimization stability, PGA-DPS limits active sampling at 20%. Example segmentation maps from subsampled bands selected by All-bands, PGA-DPS, A-DPS, DPS, and other band selection methods are shown in the Appendix (Fig. 4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We developed a scalable, stable active subsampling technique, called Prior-aware and Group-based Active DPS (PGA-DPS), that integrates context-guided group sampling and deterministic (fixed) subsampling based on training-data prior. PGA-DPS not only provides a theoretical analysis for the stable optimization of group sampling, enabled by DPS-top- k k rather than DPS-top-1, but also shows consistent and improved performance compared to conventional A-DPS. We demonstrated the effectiveness of PGA-DPS across three diverse tasks—classification, image reconstruction, and segmentation—using the MNIST, CIFAR-10, fastMRI knee, and AeroRIT datasets, outperforming all other sampling methods. Since top-k sampling and prior-aware deterministic sampling can be easily applied to various sampling tasks, PGA-DPS may serve as a powerful tool for improving performance in real-world applications where acquisition cost and inference efficiency are critical, such as CT, ultrasound, and radar systems.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Limitation & Future Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although PGA-DPS excels at generating subsampling strategies for diverse tasks, one possible limitation lies in selecting the hyperparameters associated with the proportion of fixed and active subsampling, denoted as P s Ps and A s As , respectively. The optimal sampling portion differs depending on the downstream task, task model, and sampling ratio. We observed that an active sampling ratio ( As ) range of 20-30% provides the best trade-off between the benefit of top-k sampling and the total budget allocated to active sampling, which narrows the search space for optimal configurations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To determine an appropriate prior-sampling ratio ( Ps ), we recommend using the performance discrepancy between DPS and A-DPS as an empirical indicator of the underlying Lipschitz characteristics of the task. Specifically, if DPS consistently outperforms A-DPS under a given configuration, this suggests that the optimization landscape exhibits a larger effective Lipschitz constant—implying that a higher Ps would be beneficial. Conversely, if A-DPS achieves superior performance, the landscape is likely smoother, in which case a smaller Ps is preferable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While optimal hyperparameter ( Ps, As ) ranges are suggested, automatically tuning them (Bergstra et al. , 2011 ; Feurer and Hutter, 2019 ; Bischl et al. , 2023 ) could enhance the robustness of the proposed method. Future research may explore joint optimization of the active sampling hyperparameters ( Ps , As ), the subsampling trajectory, and the downstream task model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate that using a fixed temperature yields robust results (Appendix D ); however, tuning or annealing the temperature could further enhance DPS-based methods. A more comprehensive investigation of temperature scheduling therefore represents a promising avenue for future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Reproducibility statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The experimental settings and models used in our work are described in the Experimental Setup sections of each task. The task models and baseline approaches are explained in detail in the Task Model section. Additional implementation details are provided in the Appendix .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 MNIST classification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The classification network f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) consists of 5-layer MLP with 784, 256, 128, 128, and 10 nodes, respectively. Each layer was followed by a leaky ReLU activation with a negative slope of 0.2 and the last layer followed by softmax activation that output probabilities of class. A dropout rate of 30% is also applied to the first three layers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sampling network g ( ⋅ ) g(\\\\cdot) consists of an LSTM (Long Short-Term Memory) with one hidden layer of size 128, followed by two MLPs with 256 and 784 nodes, respectively. A leaky ReLU activation with a negative slope of 0.2 and a dropout rate of 30% are applied after the first MLP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The categorical cross-entropy loss was minimized using the Adam optimizer (Kingma and Ba, 2014 ) with the learning rate of 2 × 10 − 4 2\\\\times 10^{-4} ( β 1 = 0.9 \\\\beta_{1}=0.9 , β 2 = 0.999 \\\\beta_{2}=0.999 , and ϵ = 10 − 7 \\\\epsilon=10^{-7} ). In addition, the trainable logits, of size 784 (nn.Parameters()), are also updated with a learning rate of 2 × 10 − 3 2\\\\times 10^{-3} for DPS and PGA-DPS. The optimization was run for 100 epochs with a batch size of 256.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 CIFAR-10 classification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The classification network f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) consists of four convolutional layers with channel sizes of 32, 64, and 128, respectively, and a kernel size of 3 × \\\\times 3. Each layer is followed by batch normalization, a ReLU activation function, and a 2 × \\\\times 2 max pooling operation. The resulting 128-channel feature map is then flatten and fed into an MLP composed of Linear(2048,256)-ReLU-Dropout(0.5)-Linear(256,128)-ReLu-Linear(128,10).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sampling network g ( ⋅ ) g(\\\\cdot) consists of an LSTM (Long Short-Term Memory) with one hidden layer of size 128, followed by two MLPs with 256 and 1024 nodes, respectively. A leaky ReLU activation with a negative slope of 0.2 and a dropout rate of 30% are applied after the first MLP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The categorical cross-entropy loss was minimized using the Adam optimizer with the learning rate of 2 × 10 − 4 2\\\\times 10^{-4} . In addition, the trainable logits, of size 1024 (nn.Parameters()), are also updated with a learning rate of 2 × 10 − 3 2\\\\times 10^{-3} for DPS and PGA-DPS. The optimization was run for 50 epochs with a batch size of 128.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.3 MRI reconstruction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proximal neural network 𝒫 ψ \\\\mathcal{P}_{\\\\psi} consists of four convolutional layers with channel size of 16,16,16, and 1, respectively, and a kernel size of 3 × \\\\times 3. Each layer is followed by a ReLU activation function except for the final layer. For the image prior regularizer ( ψ {\\\\psi} ), a single 3 × \\\\times 3 convolutional layer is used. The proximal gradient method is unrolled for three iterations (k=3).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The adversarial loss (Ledig et al. , 2017 ) , combined with a mean square error (MSE), is used to reconstruct visually plausible MR images, by training a discriminator network that distinguishes the real and reconstructed images. In addition, the discriminator feature loss between the real and reconstructed images was employed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sampling network g k ( ⋅ ) g^{k}(\\\\cdot) consists of a sequence of layers: Conv2d(1,16), ReLU, Conv2d(16,32), ReLu, Conv2d(32,64), ReLU, Average Pooling, Flatten, LSTM(64), and a single-layer MLP(64,208). All convolutional layers have a kernel size of 3 × \\\\times 3. The last MLP converts the output of the LSTM (encoded context) to the logits of size 208.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The discriminator network is consists of a sequence of layers: Conv2d(1,64), LeakyReLU(0.2), Conv2d(64,64), LeakyReLU(0.2), Conv2d(64,64), LeakyReLU(0.2), Average Pooling, Dropout(0.4), a single-layer MLP (64,1), and Sigmoid. All convolutional layers have a kernel size of 3 × \\\\times 3 and a stride of 2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The total loss is computed as a weighted sum of three losses: mean squared error (MSE), adversarial loss, and the discriminator feature loss, with respective weights of 1, 5 × 10 − 6 5\\\\times 10^{-6} , and 10 − 7 10^{-7} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The network is updated using the Adam optimizer (Kingma and Ba, 2014 ) with the learning rate of 2 × 10 − 4 2\\\\times 10^{-4} ( β 1 = 0.9 \\\\beta_{1}=0.9 , β 2 = 0.999 \\\\beta_{2}=0.999 , and ϵ = 1 e − 7 \\\\epsilon=1e-7 ). In addition, the trainable logits, of size 208 (nn.Parameters()), are also updated with a learning rate of 2 × 10 − 3 2\\\\times 10^{-3} for DPS and PGA-DPS. The optimization was run for 10 epochs with a batch size of 2.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.4 Hyperspectral image segmentation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the segmentation network f θ ( ⋅ ) f_{\\\\theta}(\\\\cdot) , we use a residual U-Net consisting of 6 ResNet blocks, as proposed in the pix2pix (Isola et al. , 2017 ; Zhu et al. , 2017 ) . The input channel is set to 51, corresponding to the total number of hyperspectral bands, and the output channel is set to 6, corresponding to the number of classes including the undefined class. The number of the filters in the last convolution layer is set to 64, and tow downsampling operations are applied.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sampling network g k ( ⋅ ) g^{k}(\\\\cdot) consists of a sequence of layers: Conv2d(6,32), BN, ReLU, Conv2d(32,64), BN, ReLu, Conv2d(64,128), BN, ReLU, Conv2d(128,256), BN, ReLU, Average Pooling, Flatten, LSTM(256), and a single-layer MLP(256,51), where BN denotes bach normalization. All convolutional layers have a kernel size of 3 × \\\\times 3. The last MLP converts the output of the LSTM (encoded context) to the logits of size 51.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In PGA-DPS, the proportions of prior (deterministic) sampling ( N N ) and active sampling ( M M ) were 80% and 20%, respectively, resulting in two group sampling iterations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The network is updated using the Adam optimizer (Kingma and Ba, 2014 ) with the learning rate of 10 − 4 10^{-4} ( β 1 = 0.9 \\\\beta_{1}=0.9 , β 2 = 0.999 \\\\beta_{2}=0.999 , and ϵ = 1 e − 7 \\\\epsilon=1e-7 ), while the learning rate of 2 × 10 − 4 2\\\\times 10^{-4} is used for DPS and GSS. In addition, the trainable logits, of size 51 (nn.Parameters()), are also updated with a learning rate of 1 for DPS and PGA-DPS. The optimization was run for 60 epochs with a batch size of 100.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.1 MNIST classification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated multiple P s Ps values under an A s As of 20% and multiple A s As values under a P s Ps of 60%. Table 7 shows that PA-DPS outperforms the baselines in almost all configurations, although the optimal configuration shifts slightly across sampling ratios.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.2 MRI reconstruction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 shows example reconstruction from masks generated by PGA-DPS, A-DPS, DPS, and other sampling methods. Notably, PGA-DPS selects samples farthest from the DC, capturing fine-grained details. PGA-DPS exhibits the farthest average mask distance from the DC (11.8), versas 10.2 (A-DPS), 8.8 (DPS), 10.8 (LOUPE), 8.1 (Greedy Prox.), and 11.3 (VDS). Despite prioritizing high-frequency components, PGA-DPS also samples adequately near DC, demonstrating its superior reconstruction performance with higher SSIM values.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To evaluate the robustness and general applicability of the proposed method, we tested whether the selected optimal configuration of (Ps, As) = (30%, 30%), identified from analysis with k-space lines M = 26, generalizes well across different sampling ratios by conducting experiments with varying values of M, i.e., k-space lines M = 15, 20, 30, and 35, corresponding to acceleration rates r = 7.2%, 9.6%, 14.4%, and 16.8%, respectively. As shown in Table 8 , PGA-DPS outperforms DPS and A-DPS across various numbers of k-space lines, with a particularly large margin when the number of k-space lines is small.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.3 Hyperspectral image segmentation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 shows example segmentation maps from subsampled bands selected by All-bands, PGA-DPS, A-DPS, DPS, and other band selection methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We fixed the temperature to 2 during training for DPS, A-DPS, and PGA-DPS. However, tuning the relaxation parameter is critical for balancing the bias-variance trade-off of the gradient estimator and may lead to further improvements.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To assess this, we conducted additional MR reconstruction experiments using various fixed temperature values. As shown in Table 9 , the reconstruction performance was largely insensitive to temperature values. Nonetheless, a fixed temperature of 5 yielded the best overall results, suggesting that temperature optimization could provide marginal additional improvements.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also evaluated temperature annealing, defined as, τ = m a x ( 2.0 , T × e − γ n ) \\\\tau=max(2.0,T\\\\times e^{-\\\\gamma n}) , where n denotes the training epoch. As shown in Table 10 , different annealing schedules produced noticeably different performance. Consequently, using a fixed temperature provides sufficiently robust and stable results in our setting, consistent with what has been found to work well in practice (Huijben et al. , 2020b ; Van Gorp et al. , 2021 ) . Nevertheless, a more comprehensive investigation of temperature annealing within DPS remains a promising direction for future work.\"\n }\n]","isConversational":false}}},{"rowIdx":490,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07740"},"doi":{"kind":"null"},"version":{"kind":"number","value":2,"string":"2"},"title":{"kind":"string","value":"Jet-Long: Efficient Long-Context Extension with Dynamic Bifocal RoPE"},"authors":{"kind":"list like","value":["Haozhan Tang","Zerui Wang","Yuxian Gu","Song Han","Han Cai"],"string":"[\n \"Haozhan Tang\",\n \"Zerui Wang\",\n \"Yuxian Gu\",\n \"Song Han\",\n \"Han Cai\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07740v2"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Han Cai ( hcai@nvidia.com )."},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Large language models (LLMs) are now deployed in long-context applications including long-document QA, repository-level code understanding, retrieval-augmented generation, and multi-step agentic workflows ( touvron2023llama ; liu2025deepseek ; liu2025comprehensive ; liu2025thus ; hsieh2024ruler ; jimenez2023swe ; gao2023retrieval ; yu2024defense ; wang2024survey ) . The pressure is most severe in agentic LLMs that interleave reasoning, planning, and tool use across many turns ( yao2022react ; schick2023toolformer ; liu2023agentbench ) , and in coding agents operating over real software repositories ( yang2024sweagent ; hong2024metagpt ) , where source code, execution traces, and tool outputs routinely accumulate to 100K+ tokens per task."},{"type":"para","text":"Training directly at long context remains expensive: efficient kernels like FlashAttention ( dao2022flashattention ) and Ring Attention ( liu2023ring ) make memory linear but compute stays quadratic in sequence length, and long-context data is scarce while long-context fine-tuning often degrades short-context behavior ( an2024does ; fu2024data ) . Models are therefore pretrained at a moderate window (4K–32K tokens) and expected to handle longer inputs at inference, a setting known as context extension ( press2021train ; ding2024longrope ) . Zero-shot context extension (without fine-tuning) has become the dominant deployment mode for open-weight LLMs ( yang2025qwen3 ; agarwal2025gpt ; liu2024deepseek ) , since a single released checkpoint must support arbitrary downstream context lengths."},{"type":"para","text":"Vanilla Transformer-based LLMs fail to generalize beyond the training window ( press2021train ; chen2023extending ; liu2023scaling ) , with two failure modes (out-of-distribution RoPE rotations and softmax-attention diffusion) detailed in Section 2 . A growing body of zero-shot methods (NTK ( bloc972023ntk ) / Dynamic NTK ( emozilla2023dyntk ) , YaRN ( peng2023yarn ) , Self-Extend ( jin2024llm ) , DCA ( an2024training ) ) addresses one or the other; these constitute the zero-shot baselines we benchmark against."},{"type":"para","text":"We propose Jet-Long , a tuning-free zero-shot context extension method that pairs a local RoPE-faithful window with a long-range window whose rescaling factor is dynamic in the current sequence length. Unlike YaRN, Self-Extend, and DCA, which fix a single grouping size or factor up front, and unlike length-adaptive maps with fitted or distance-dependent schedules ( xu2025extending ; zhang2025lampe ) , Jet-Long derives the factor analytically from the current sequence length and the pretrained window, with no per-model fitting, and keeps the resulting map essentially free at inference."},{"type":"para","text":"Our contributions are:"},{"type":"para","text":"Jet-Long , a tuning-free bifocal context-extension method that sets the remote group size to the minimal integer compression G = max ( 1 , ⌈ L / w pretrained ⌉ ) G=\\max(1,\\lceil L/w_{\\text{pretrained}}\\rceil) at the current sequence length L L , keeping every remote rotation in-distribution with no per-model fitting while reproducing the base model exactly within its native context."},{"type":"para","text":"An inclusion–exclusion attention merge that routes local and remote windows through three FlashAttention ( dao2022flashattention ) passes, paired with an on-the-fly RoPE correction rotation that leaves the KV cache untouched during generation; fused into a single CuTe kernel, the construction reaches 1.28 1.28 – 1.39 × 1.39\\times FA2 prefill on H100 (approaching the Hopper-only FA4 ( zadouri2026flashattention4 ) ) and incurs ≤ 4 % \\leq 4\\% overhead on generation at every length."},{"type":"para","text":"Empirical evaluation on Qwen3-1.7B/4B/8B up to 128K context: Jet-Long leads the strongest zero-shot baseline on RULER ( hsieh2024ruler ) by + 4.79 / + 2.18 / + 2.03 +4.79/+2.18/+2.03 pp and is best or tied on HELMET-RAG ( yen2024helmet ) and PG-19 ( rae2019compressive ) perplexity; the single hyperparameter w 0 w_{0} is robust to choice; and the construction transfers unchanged to the hybrid Jet-Nemotron ( gu2025jet ) architecture."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":2,"text":"2.1 Why RoPE-based LLMs fail to extrapolate"},{"type":"para","text":"Most modern open-weight LLMs use Rotary Position Embedding (RoPE) ( su2024roformer ) , which applies per-position rotations across geometrically spaced frequencies so attention depends only on relative position; earlier relative-position schemes such as ALiBi ( press2021train ) , T5’s relative-position bias ( raffel2020t5 ) , and iRPE ( wu2021irpe ) have largely been supplanted. Two failure modes prevent these models from extrapolating beyond their training window."},{"type":"heading","level":4,"text":"(i) Position out-of-distribution."},{"type":"para","text":"At sequence positions never seen during training, the low-frequency RoPE components produce rotation angles outside the training distribution, causing attention scores to behave erratically ( chen2023extending ; peng2023yarn ; liu2023scaling ) ."},{"type":"heading","level":4,"text":"(ii) Attention diffusion and positional bias."},{"type":"para","text":"As the key set grows, the softmax distribution flattens, dispersing probability mass over irrelevant tokens ( han2024lm ; peng2023yarn ) ; separately, models exhibit a U-shaped positional attention bias that under-attends to middle-context information, degrading retrieval accuracy for centrally placed evidence ( liu2024lost ) ."},{"type":"para","text":"These motivate two complementary zero-shot axes: interpolating RoPE frequencies or position indices to keep rotation angles in-distribution ( chen2023extending ; bloc972023ntk ; emozilla2023dyntk ; peng2023yarn ; jin2024llm ; an2024training ; li2025training ; su2023rerope ) , and attention penalties or temperature scaling to counteract softmax diffusion ( peng2023yarn ; han2024lm ; li2025training ) . Jet-Long targets position-OOD via dynamic aliasing onto the pretrained rotation grid."},{"type":"heading","level":2,"text":"2.2 Zero-shot context extension methods"},{"type":"heading","level":4,"text":"Frequency-rescaling methods."},{"type":"para","text":"Position Interpolation (PI) ( chen2023extending ) linearly compresses positions into the pretrained range. NTK-aware scaled RoPE ( bloc972023ntk ) increases the RoPE base to preserve high-frequency components while interpolating low-frequency ones; Dynamic NTK (DNTK) ( emozilla2023dyntk ) makes that base a function of the current sequence length so the scaling adapts at decode time. YaRN ( peng2023yarn ) combines per-dimension frequency partitioning with an attention-temperature correction. Beyond pure rescaling, ReRoPE ( su2023rerope ) caps relative distances past a window threshold (implemented as a two-pass within/beyond-window attention blend at prefill), and GALI ( li2025training ) interpolates at the attention-logit level rather than the embedding."},{"type":"heading","level":4,"text":"Grouped / chunked-position methods."},{"type":"para","text":"A second line reuses only in-distribution position indices. Self-Extend ( jin2024llm ) pairs a neighbor window with a grouped window in which blocks of tokens share a single position index. Dual Chunk Attention (DCA) ( an2024training ) partitions the sequence into chunks and uses asymmetric query/key indices in the cross-chunk component so all relative distances stay within the pretrained range. LM-Infinite ( han2024lm ) earlier introduced a Λ \\Lambda -shaped mask retaining an attention sink and a recent window."},{"type":"heading","level":4,"text":"Length-adaptive remapping."},{"type":"para","text":"Closest to our setting, several training-free methods also condition the position map on input length. AdaGroPE ( xu2025extending ) grows the reuse count of relative positions with distance; LaMPE ( zhang2025lampe ) applies a piecewise map over three sequence regions whose interior compression ratio is a per-model fitted sigmoid of the input length; SELF ( dang2025self ) grows Self-Extend’s group size logistically."},{"type":"para","text":"Jet-Long sits in the grouped-position family, with the group size derived from the current sequence length rather than fixed or fitted: identity within the native window, and past it the minimal integer G G that keeps every remote rotation in-distribution (Section 3.1 ). It therefore reduces exactly to the base model for L ≤ w pretrained L\\leq w_{\\text{pretrained}} , and its correction rotation (Section 3.2 ) preserves a base-position KV cache when G G changes mid-generation."},{"type":"heading","level":2,"text":"2.3 Long-context training, architectures, and benchmarks"},{"type":"para","text":"Training-based approaches extend the window via continued pretraining ( ding2024longrope ) but face quadratic FLOP costs (only partially mitigated by efficient kernels ( dao2022flashattention ; liu2023ring ) ) and scarce long-context data with short-context regression risk ( an2024does ; fu2024data ) , motivating the zero-shot setting."},{"type":"para","text":"Alternative architectures replace dense softmax with sparse attention ( beltagy2020longformer ; zaheer2020bigbird ) , linear or kernel-based variants ( katharopoulos2020lineartransformer ; choromanski2021performer ) , or state-space models ( gu2023mamba ) ; sparser or non-softmax distributions naturally curb the attention diffusion and lost-in-the-middle bias that plague dense softmax at long context. A complementary line removes positional encoding entirely: NoPE outperforms explicit position encodings on length-generalization benchmarks ( kazemnejad2023nope ) , and open-weight models such as Kimi K2 ( kimi2025k2 ) and NVIDIA Nemotron Nano 2 ( nvidia2025nemotron ) adopt NoPE in their hybrid layers. Both lines typically require training from scratch or substantial fine-tuning and therefore complement rather than compete with our zero-shot setting; Jet-Long itself extends smoothly to hybrid designs (Section 4.3 , Jet-Nemotron ( gu2025jet ) )."},{"type":"para","text":"We evaluate on RULER ( hsieh2024ruler ) (synthetic recall over 13 tasks), HELMET-RAG ( yen2024helmet ) (the HELMET study’s best overall predictor of downstream long-context performance), and PG-19 ( rae2019compressive ) long-form perplexity, against the strongest zero-shot baselines; we additionally test transfer to the hybrid Jet-Nemotron backbone (Section 4.3 )."},{"type":"heading","level":1,"text":"3 Methodology"},{"type":"para","text":"The frequency-rescaling and grouped-position methods of Section 2.2 fix a single rescaling factor up front, forcing a tradeoff between short-context fidelity and long-context reach; recent length-adaptive maps ( xu2025extending ; zhang2025lampe ) relax it, but with a fitted or distance-dependent schedule. Jet-Long instead derives the factor analytically from the current sequence length (Section 3.1 ); the resulting two-window computation matches FlashAttention ( dao2022flashattention ) within the pretraining window and exceeds it at long context (Section 4.6 ), and leaves the KV cache untouched at decode (Sections 3.2 – 3.3 )."},{"type":"para","text":"We build on the dual-window (bifocal) decomposition of Self-Extend ( jin2024llm ) (a related but architecturally three-way ancestor is DCA ( an2024training ) ): a local window of size w 0 w_{0} that retains classic RoPE (preserving the model’s pretraining behavior exactly), and a remote window governed by a rewritten position mapping f ( ⋅ ) f(\\cdot) that maps positions back into the training range. For a query at position q q and a key at position k k , the pre-softmax attention score is"},{"type":"para","text":"Let L L denote the current sequence length and w pretrained w_{\\text{pretrained}} the pretrained context window. When L ≤ w pretrained L\\leq w_{\\text{pretrained}} , f ( x ) = x f(x)=x and Eq. ( 1 ) reduces to the unmodified base model. Jet-Long’s contribution lies in the choice of f ( ⋅ ) f(\\cdot) and in the inference-time machinery that makes the two windows interact for free."},{"type":"heading","level":2,"text":"3.1 Dynamic extrapolation factor"},{"type":"para","text":"To keep remote-window rotation angles in-distribution, the group size G G must scale with the current sequence length L L , as illustrated in Figure 2(a) ."},{"type":"para","text":"The natural dynamic factor is, for continuous-style RoPE rescaling (e.g., DNTK ( emozilla2023dyntk ) and dynamic-YaRN ( peng2023yarn ) variants), the scaling ratio s = max ( 1 , L / w pretrained ) s=\\max(1,L/w_{\\text{pretrained}}) ; for discrete grouped methods (e.g., Self-Extend ( jin2024llm ) ), it is an integer group size G G . Jet-Long uses discrete grouping because an LLM only encounters a finite, discrete set of relative RoPE rotation angles during pretraining; aliasing positions onto that pretrained grid keeps every remote-window angle exactly in-distribution, an integer relative position the model has actually been trained on. We ablate this choice against continuous frequency interpolation in Section 4.5 . Whereas DNTK ( emozilla2023dyntk ) adapts the RoPE base β \\beta (per-frequency), Jet-Long adapts the discrete group size G G on the position-aliasing axis, keeping every remote angle on the model’s training grid."},{"type":"para","text":"To maximize positional resolution, G G is the smallest integer that keeps the compressed sequence within the pretrained window w pretrained w_{\\text{pretrained}} :"},{"type":"para","text":"The remote mapping is a floor division of absolute positions:"},{"type":"para","text":"By recomputing G G as the sequence grows, Jet-Long applies the minimum compression that keeps every remote position in-distribution, maximizing positional resolution at every length. LaMPE ( zhang2025lampe ) likewise applies floor-based remapping in its middle region, but derives the middle-region compression ratio from an empirically fitted sigmoid of the input length, with model-specific fits presented for Llama2 and Llama3, whereas Eq. ( 2 ) is determined analytically by L L and w pretrained w_{\\text{pretrained}} with no fitted parameters."},{"type":"heading","level":2,"text":"3.2 Key-value cache management and correction rotation"},{"type":"para","text":"Dynamic extrapolation poses a cache-management challenge: if G G changes during generation, rewriting the KV cache with new extrapolated phases would require discarding and recomputing the entire cache."},{"type":"para","text":"As depicted in Figure 2(b) , Jet-Long avoids this overhead by maintaining a strict invariant: the cache stores only uncompressed base keys at their exact absolute positions k k ,"},{"type":"para","text":"When the remote window needs the query at f ( q ) f(q) and the key at f ( k ) f(k) , instead of recomputing those vectors from scratch we apply a correction rotation on the fly using the position offsets"},{"type":"para","text":"This relies on standard RoPE composing additively in angle, R a R b = R a + b R_{a}R_{b}=R_{a+b} (per-position scalings beyond a pure rotation would break this and require recomputing keys from 𝐱 k \\mathbf{x}_{k} ). Applying RoPE ( ⋅ , Δ ) \\text{RoPE}(\\cdot,\\Delta) to a vector encoding position p p therefore produces the vector at position p + Δ p+\\Delta , so we apply the offset directly to the active query and cached keys before attention:"},{"type":"para","text":"This constant-time operation reconstructs the extrapolated vectors in registers; the physical cache is never touched, so streaming generation runs across length boundaries without stalling."},{"type":"heading","level":2,"text":"3.3 Inclusion–exclusion prefill"},{"type":"para","text":"To avoid materializing the full quadratic attention matrix, which exhausts memory during long-sequence prefill, Jet-Long achieves exact distance-based routing by combining the inclusion–exclusion principle with FlashAttention’s ( dao2022flashattention ) LogSumExp statistics. The merge requires three standard attention calls (Figure 2(c) ), each returning an output 𝐎 X \\mathbf{O}_{X} and an LSE vector ℓ X \\bm{\\ell}_{X} : (A) sliding-window attention (local w 0 w_{0} ) with base queries and keys, (B) full causal attention with remote queries and keys, and (C) sliding-window attention (local w 0 w_{0} ) with remote queries and keys. Calls B and C apply the same remote rotation to local keys, so their local-subset contributions cancel exactly: W B 𝐎 B − W C 𝐎 C W_{B}\\mathbf{O}_{B}-W_{C}\\mathbf{O}_{C} (resp. W B − W C W_{B}-W_{C} ) collects only the remote-only term, while Call A supplies the base-local term. Stabilizing element-wise (per query position) via M = max ( ℓ A , ℓ B , ℓ C ) M=\\max(\\bm{\\ell}_{A},\\bm{\\ell}_{B},\\bm{\\ell}_{C}) and weights W X = exp ( ℓ X − M ) W_{X}=\\exp(\\bm{\\ell}_{X}-M) , the final output (numerator and denominator both in FP32 to avoid catastrophic cancellation) is"},{"type":"para","text":"This merge realizes exact distance-based routing without a boolean mask matrix, retaining FlashAttention’s memory efficiency and near-FA2 throughput. Prior work rejects this subtractive form for learned sparse attention as “numerically catastrophic” ( yao2026focus ) , measuring 0.79 0.79 cosine similarity against a reference, since the subtraction can drive the logarithm’s argument negative. Eq. ( 7 ) instead subtracts in linear space, where shifting by M M bounds every weight in ( 0 , 1 ] (0,1] and FP32 accumulation avoids the cancellation."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":2,"text":"4.1 Setup"},{"type":"heading","level":4,"text":"Models."},{"type":"para","text":"Our primary evaluation is on the Qwen3 base model family ( yang2025qwen3 ) , namely Qwen3-1.7B-Base, Qwen3-4B-Base, and Qwen3-8B-Base, each of which has a 32,768-token (32K) native training window. We extend the usable context to lengths up to 131,072 (128K) at inference time, without any fine-tuning. To verify that Jet-Long generalizes beyond pure softmax-attention transformers, we additionally evaluate on the hybrid Jet-Nemotron-2B and Jet-Nemotron-4B models ( gu2025jet ) , which interleave softmax and linear-attention layers. We use w 0 = 2048 w_{0}=2048 throughout the main results, and per-baseline hyperparameters (DNTK, YaRN, DCA, Self-Extend) are listed in Appendix C ."},{"type":"heading","level":4,"text":"Long-context benchmarks."},{"type":"para","text":"We report results on three complementary suites: (i) RULER ( hsieh2024ruler ) , a synthetic recall stress test averaging over thirteen tasks; (ii) HELMET-RAG ( yen2024helmet ) , averaged over the four default sub-tasks (NaturalQuestions, TriviaQA, PopQA, HotpotQA), which the HELMET study reports as the most efficient predictor of downstream long-context performance and which serves as our application-grounded benchmark; and (iii) PG-19 ( rae2019compressive ) , on which we report long-form perplexity. RULER and HELMET-RAG accuracies are percentages, with gaps between methods reported in percentage points (pp); PG-19 is reported as token-level perplexity (ppl, lower is better). All RULER and HELMET-RAG generations use greedy decoding with the default RULER/HELMET prompts and per-task max-output-token limits; PG-19 is teacher-forced perplexity at stride 1024 over 100 books. Formal definitions and the geometric-mean aggregation used in the Avg columns are given in Appendix A . We further ablate the only Jet-Long hyperparameter, the local protected window size w 0 w_{0} , in Section 4.4 ."},{"type":"heading","level":4,"text":"Inference efficiency."},{"type":"para","text":"We implement Jet-Long as a fused CuTe kernel that merges the three attention calls of Section 3.3 into a single launch, and benchmark prefill and single-batch generation throughput on a single H100 against the highly optimized FlashAttention-2 ( dao2023flashattention2 ) and FlashAttention-4 ( zadouri2026flashattention4 ) baselines applied to the unmodified base models."},{"type":"para","text":"All experiments are run on NVIDIA H100 Tensor Core GPUs."},{"type":"heading","level":2,"text":"4.2 Main results on long-context extension"},{"type":"para","text":"The average scores on the three benchmarks are reported in Table 1 . Jet-Long is best on every RULER and PG-19 column across all three Qwen3 sizes (over Base and the four extrapolation baselines), and best on HELMET-RAG at 4B and 8B (0.73 pp behind Self-Extend at 1.7B). The RULER lead over the strongest baseline (DNTK at 1.7B; Self-Extend at 4B and 8B) is 4.79, 2.18, and 2.03 pp."},{"type":"para","text":"As Figure 3 shows, Jet-Long matches or substantially outperforms YaRN and DNTK at every length across all three sizes for RULER accuracy by length. Table 2 confirms the same pattern on PG-19: within the 32K training window, Jet-Long is mathematically equivalent to the base model (the dynamic factor reduces to identity, Section 3.1 ); past 32K the bare base collapses (ppl of 71.00 / 104.66 / 79.37 at 128K for 1.7B / 4B / 8B) while Jet-Long stays at 11.41 / 9.62 / 8.51, the lowest among all extrapolation methods."},{"type":"para","text":"We also report the per-task accuracy at context length L = 65536 L{=}65536 in Table 3 . Jet-Long is best or tied for best on 8 of the 13 tasks at both 1.7B and 8B, with the largest leads on Multi-Key NIAH (MK-NIAH-2 at 1.7B: 61.80 vs 20.00; MK-NIAH-3 at 8B: 70.80 vs 39.80) and Variable Tracking (VT at 1.7B: 73.56 vs 54.24); the only 1.7B exception is CWE, on which every method (including Base) scores below 1 % 1\\% . Aggregated over the 13 tasks, Jet-Long beats the strongest baseline by 7.31 pp at 1.7B and 6.07 pp at 8B."},{"type":"heading","level":2,"text":"4.3 Hybrid attention extension results"},{"type":"para","text":"To test generalization beyond softmax-only transformers, we apply Jet-Long to the hybrid Jet-Nemotron architecture ( gu2025jet ) , which interleaves softmax and linear-attention layers. Table 4 reports per-length RULER accuracy on Jet-Nemotron-2B and 4B."},{"type":"para","text":"Within 32K the construction reduces to base attention, so Jet-Long inherits the base model’s in-distribution behavior (matching to within rounding). Past 32K the bare hybrid base collapses (8.54 / 5.65 at 128K for 2B / 4B), while Jet-Long retains 33.78 and 33.14 ( + 25.24 +25.24 / + 27.49 +27.49 pp). Averaged over the seven lengths, Jet-Long lifts RULER from 42.93 to 52.94 ( + 10.01 +10.01 pp) at 2B and from 42.16 to 53.47 ( + 11.31 +11.31 pp) at 4B. The bifocal decomposition and dynamic factor generalize to hybrid LLM architectures."},{"type":"heading","level":2,"text":"4.4 Ablation: local window size w 0 w_{0}"},{"type":"para","text":"To check whether w 0 w_{0} requires per-deployment tuning, we sweep it on Qwen3-4B/8B at three out-of-window lengths (64K, 96K, 128K) in Table 5 (1.7B omitted for compute). The control w 0 = 0 w_{0}{=}0 shrinks the local window to attention-to-self only and collapses RULER to near-zero, confirming the local window is necessary. Past that boundary, Jet-Long is hyperparameter-resilient: every w 0 ∈ { 512 , 1024 , 2048 , 4096 } w_{0}\\in\\{512,1024,2048,4096\\} stays within 2 2 pp of the per-row best at any single length and within 1 1 pp on the per-model average; practitioners can pick w 0 = 2048 w_{0}{=}2048 without per-deployment tuning. The slight degradation at w 0 = 8192 w_{0}{=}8192 (1.4–2.1 pp gap) reflects the local window consuming a larger fraction of context."},{"type":"heading","level":2,"text":"4.5 Ablation: interpolate frequency or alias position"},{"type":"para","text":"Section 3.1 requires only that the remote mapping f ( ⋅ ) f(\\cdot) keep rotation angles in-distribution, not how. Table 6 compares position aliasing ( f ( x ) = ⌊ x / G ⌋ f(x)=\\lfloor x/G\\rfloor , the default Jet-Long) against YaRN-style frequency interpolation (rescaled RoPE frequencies, unchanged positions) on Qwen3-1.7B-Base. Aliasing wins by 6.99 pp at 64K and 4.30 pp at 128K, consistent with the LLM having learned a discrete grid of relative angles. The alias margin shrinks at extreme lengths because larger G G gives continuous interpolation more to compensate for: on FWE from + 20.3 +20.3 pp at 64K to + 2.5 +2.5 at 128K, on QA-1 from + 11.2 +11.2 to + 0.6 +0.6 , and frequency interpolation overtakes aliasing on MK-NIAH-2 and QA-2 at 128K. Aliasing remains the better default for ≤ 128 \\leq 128 K; hybrid mappings merit investigation at longer contexts."},{"type":"heading","level":2,"text":"4.6 Inference efficiency"},{"type":"para","text":"We measure end-to-end throughput (tok/s) on a single H100 with CUDA graphs on Qwen3-8B-Base, comparing FA2 ( dao2023flashattention2 ) , the H100-native FA4 ( zadouri2026flashattention4 ) , the multi-launch Jet-Long (unfused) variant of Sections 3.3 – 3.2 , and our fused Jet-Long CuTe kernel; 1.7B and 4B follow the same pattern (Table 8 , Appendix B )."},{"type":"para","text":"Within 32K all four configurations match FA2 to within ± 1 % \\pm 1\\% since Jet-Long reduces to standard attention. Past 32K the naive multi-launch baseline pays the bifocal cost: three FlashAttention passes plus an out-of-kernel merge drop prefill to 0.89 0.89 – 0.93 × 0.93\\times FA2, and unfused per-token correction rotation drops generation to 0.14 0.14 – 0.34 × 0.34\\times . The fused CuTe kernel removes both overheads: prefill recovers to 1.28 1.28 – 1.39 × 1.39\\times FA2 past 32K (approaching FA4’s 1.53 × 1.53\\times at 128K, without an H100-specific kernel), and generation stays ≥ 0.96 × \\geq 0.96\\times FA2 at every length, with the residual ≤ 4 % \\leq 4\\% overhead from the per-token correction rotation and the dynamic- G G bookkeeping. The accuracy gains in Table 1 therefore come essentially free at inference."},{"type":"heading","level":1,"text":"5 Conclusion"},{"type":"para","text":"We presented Jet-Long , a tuning-free zero-shot context-extension method that pairs a local RoPE-faithful window with a long-range window whose rescaling factor adapts dynamically to the current sequence length, and made the construction essentially free at inference through an inclusion–exclusion attention merge with on-the-fly correction rotation in a fused CuTe kernel. On Qwen3-1.7B/4B/8B at context lengths up to 128K, Jet-Long shows superior performance on RULER ( + 4.79 +4.79 / + 2.18 +2.18 / + 2.03 +2.03 pp at 1.7B/4B/8B), HELMET-RAG, and PG-19 perplexity, generalizes to the hybrid linear-attention Jet-Nemotron backbone, and remains highly resilient to its only hyperparameter, the local window size w 0 w_{0} ."},{"type":"para","text":"Jet-Long addresses position-OOD at the RoPE level, while the complementary attention-diffusion failure mode is naturally tackled by architectural alternatives ( beltagy2020longformer ; zaheer2020bigbird ; katharopoulos2020lineartransformer ; choromanski2021performer ; gu2023mamba ) . Because Jet-Long requires a softmax-with-RoPE base, natural extensions include other softmax-with-RoPE variants such as Multi-head Latent Attention ( liu2024deepseek ) and sparse attention, and architectures interleaving softmax with sparse or linear-attention sub-layers, beyond the Jet-Nemotron experiments in Section 4.3 ."},{"type":"heading","level":4,"text":"Percentage points (pp)."},{"type":"para","text":"For two scores a 1 , a 2 ∈ [ 0 , 100 ] a_{1},a_{2}\\in[0,100] reported as percentages, the absolute gap in percentage points is"},{"type":"para","text":"which we use throughout the paper to describe RULER and HELMET-RAG accuracy differences. This is distinct from the relative percent change ( a 1 − a 2 ) / a 2 (a_{1}-a_{2})/a_{2} ."},{"type":"heading","level":4,"text":"Perplexity (ppl)."},{"type":"para","text":"For a held-out token sequence x 1 , … , x T x_{1},\\dots,x_{T} scored under a language model with conditional probabilities p ( x t ∣ x < t ) p(x_{t}\\mid x_{<t}) , perplexity is the exponentiated mean negative log-likelihood per token,"},{"type":"para","text":"lower is better. On PG-19 we use anchored growing-window evaluation: at each context length L L in { 4 K , 8 K , 16 K , 32 K , 64 K , 96 K , 128 K } \\{4\\text{K},8\\text{K},16\\text{K},32\\text{K},64\\text{K},96\\text{K},128\\text{K}\\} , the next 1024 1024 tokens are scored conditional on the preceding L L tokens, and ppl is averaged across the 100 books in the PG-19 evaluation split."},{"type":"heading","level":4,"text":"Geometric-mean aggregation across lengths."},{"type":"para","text":"For the Avg column of PG-19 perplexity tables, we report the geometric mean across the n = 7 n=7 lengths,"},{"type":"para","text":"which corresponds to averaging the underlying log-likelihoods uniformly across lengths and is therefore the natural aggregate for an exponentiated metric."},{"type":"para","text":"Table 8 reports the complete prefill and generation throughput sweep for all three Qwen3 sizes (1.7B, 4B, and 8B) on H100, complementing the 8B-only Table 7 in Section 4.6 . The same four configurations are compared (FA2, FA4, Jet-Long (unfused), Jet-Long CuTe), and the parenthesized speedups are again computed against FA2 at the matching length. The qualitative pattern is identical across sizes: Jet-Long CuTe matches FA2 inside the 32K native window, recovers and surpasses FA2 at long-context prefill (1.28–1.45 × \\times ), and stays at near-FA2 generation throughput everywhere ( ≥ 0.96 × \\geq 0.96\\times across all lengths and sizes), while the unfused multi-launch variant pays a real cost in generation that the fused kernel eliminates."},{"type":"para","text":"The four zero-shot baselines compared in Tables 1 – 4 use a single configuration each, held constant across Qwen3-1.7B/4B/8B and across all evaluation lengths (4K–128K), as outlined in Table 9 . All Qwen3 models have a pretrained context window of 32 , 768 32{,}768 tokens and a RoPE base θ = 10 6 \\theta=10^{6} ."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Han Cai ( hcai@nvidia.com ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Large language models (LLMs) are now deployed in long-context applications including long-document QA, repository-level code understanding, retrieval-augmented generation, and multi-step agentic workflows ( touvron2023llama ; liu2025deepseek ; liu2025comprehensive ; liu2025thus ; hsieh2024ruler ; jimenez2023swe ; gao2023retrieval ; yu2024defense ; wang2024survey ) . The pressure is most severe in agentic LLMs that interleave reasoning, planning, and tool use across many turns ( yao2022react ; schick2023toolformer ; liu2023agentbench ) , and in coding agents operating over real software repositories ( yang2024sweagent ; hong2024metagpt ) , where source code, execution traces, and tool outputs routinely accumulate to 100K+ tokens per task.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training directly at long context remains expensive: efficient kernels like FlashAttention ( dao2022flashattention ) and Ring Attention ( liu2023ring ) make memory linear but compute stays quadratic in sequence length, and long-context data is scarce while long-context fine-tuning often degrades short-context behavior ( an2024does ; fu2024data ) . Models are therefore pretrained at a moderate window (4K–32K tokens) and expected to handle longer inputs at inference, a setting known as context extension ( press2021train ; ding2024longrope ) . Zero-shot context extension (without fine-tuning) has become the dominant deployment mode for open-weight LLMs ( yang2025qwen3 ; agarwal2025gpt ; liu2024deepseek ) , since a single released checkpoint must support arbitrary downstream context lengths.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Vanilla Transformer-based LLMs fail to generalize beyond the training window ( press2021train ; chen2023extending ; liu2023scaling ) , with two failure modes (out-of-distribution RoPE rotations and softmax-attention diffusion) detailed in Section 2 . A growing body of zero-shot methods (NTK ( bloc972023ntk ) / Dynamic NTK ( emozilla2023dyntk ) , YaRN ( peng2023yarn ) , Self-Extend ( jin2024llm ) , DCA ( an2024training ) ) addresses one or the other; these constitute the zero-shot baselines we benchmark against.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose Jet-Long , a tuning-free zero-shot context extension method that pairs a local RoPE-faithful window with a long-range window whose rescaling factor is dynamic in the current sequence length. Unlike YaRN, Self-Extend, and DCA, which fix a single grouping size or factor up front, and unlike length-adaptive maps with fitted or distance-dependent schedules ( xu2025extending ; zhang2025lampe ) , Jet-Long derives the factor analytically from the current sequence length and the pretrained window, with no per-model fitting, and keeps the resulting map essentially free at inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Jet-Long , a tuning-free bifocal context-extension method that sets the remote group size to the minimal integer compression G = max ( 1 , ⌈ L / w pretrained ⌉ ) G=\\\\max(1,\\\\lceil L/w_{\\\\text{pretrained}}\\\\rceil) at the current sequence length L L , keeping every remote rotation in-distribution with no per-model fitting while reproducing the base model exactly within its native context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An inclusion–exclusion attention merge that routes local and remote windows through three FlashAttention ( dao2022flashattention ) passes, paired with an on-the-fly RoPE correction rotation that leaves the KV cache untouched during generation; fused into a single CuTe kernel, the construction reaches 1.28 1.28 – 1.39 × 1.39\\\\times FA2 prefill on H100 (approaching the Hopper-only FA4 ( zadouri2026flashattention4 ) ) and incurs ≤ 4 % \\\\leq 4\\\\% overhead on generation at every length.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Empirical evaluation on Qwen3-1.7B/4B/8B up to 128K context: Jet-Long leads the strongest zero-shot baseline on RULER ( hsieh2024ruler ) by + 4.79 / + 2.18 / + 2.03 +4.79/+2.18/+2.03 pp and is best or tied on HELMET-RAG ( yen2024helmet ) and PG-19 ( rae2019compressive ) perplexity; the single hyperparameter w 0 w_{0} is robust to choice; and the construction transfers unchanged to the hybrid Jet-Nemotron ( gu2025jet ) architecture.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Why RoPE-based LLMs fail to extrapolate\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Most modern open-weight LLMs use Rotary Position Embedding (RoPE) ( su2024roformer ) , which applies per-position rotations across geometrically spaced frequencies so attention depends only on relative position; earlier relative-position schemes such as ALiBi ( press2021train ) , T5’s relative-position bias ( raffel2020t5 ) , and iRPE ( wu2021irpe ) have largely been supplanted. Two failure modes prevent these models from extrapolating beyond their training window.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(i) Position out-of-distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At sequence positions never seen during training, the low-frequency RoPE components produce rotation angles outside the training distribution, causing attention scores to behave erratically ( chen2023extending ; peng2023yarn ; liu2023scaling ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(ii) Attention diffusion and positional bias.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As the key set grows, the softmax distribution flattens, dispersing probability mass over irrelevant tokens ( han2024lm ; peng2023yarn ) ; separately, models exhibit a U-shaped positional attention bias that under-attends to middle-context information, degrading retrieval accuracy for centrally placed evidence ( liu2024lost ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These motivate two complementary zero-shot axes: interpolating RoPE frequencies or position indices to keep rotation angles in-distribution ( chen2023extending ; bloc972023ntk ; emozilla2023dyntk ; peng2023yarn ; jin2024llm ; an2024training ; li2025training ; su2023rerope ) , and attention penalties or temperature scaling to counteract softmax diffusion ( peng2023yarn ; han2024lm ; li2025training ) . Jet-Long targets position-OOD via dynamic aliasing onto the pretrained rotation grid.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Zero-shot context extension methods\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Frequency-rescaling methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Position Interpolation (PI) ( chen2023extending ) linearly compresses positions into the pretrained range. NTK-aware scaled RoPE ( bloc972023ntk ) increases the RoPE base to preserve high-frequency components while interpolating low-frequency ones; Dynamic NTK (DNTK) ( emozilla2023dyntk ) makes that base a function of the current sequence length so the scaling adapts at decode time. YaRN ( peng2023yarn ) combines per-dimension frequency partitioning with an attention-temperature correction. Beyond pure rescaling, ReRoPE ( su2023rerope ) caps relative distances past a window threshold (implemented as a two-pass within/beyond-window attention blend at prefill), and GALI ( li2025training ) interpolates at the attention-logit level rather than the embedding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Grouped / chunked-position methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A second line reuses only in-distribution position indices. Self-Extend ( jin2024llm ) pairs a neighbor window with a grouped window in which blocks of tokens share a single position index. Dual Chunk Attention (DCA) ( an2024training ) partitions the sequence into chunks and uses asymmetric query/key indices in the cross-chunk component so all relative distances stay within the pretrained range. LM-Infinite ( han2024lm ) earlier introduced a Λ \\\\Lambda -shaped mask retaining an attention sink and a recent window.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Length-adaptive remapping.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Closest to our setting, several training-free methods also condition the position map on input length. AdaGroPE ( xu2025extending ) grows the reuse count of relative positions with distance; LaMPE ( zhang2025lampe ) applies a piecewise map over three sequence regions whose interior compression ratio is a per-model fitted sigmoid of the input length; SELF ( dang2025self ) grows Self-Extend’s group size logistically.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Jet-Long sits in the grouped-position family, with the group size derived from the current sequence length rather than fixed or fitted: identity within the native window, and past it the minimal integer G G that keeps every remote rotation in-distribution (Section 3.1 ). It therefore reduces exactly to the base model for L ≤ w pretrained L\\\\leq w_{\\\\text{pretrained}} , and its correction rotation (Section 3.2 ) preserves a base-position KV cache when G G changes mid-generation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Long-context training, architectures, and benchmarks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training-based approaches extend the window via continued pretraining ( ding2024longrope ) but face quadratic FLOP costs (only partially mitigated by efficient kernels ( dao2022flashattention ; liu2023ring ) ) and scarce long-context data with short-context regression risk ( an2024does ; fu2024data ) , motivating the zero-shot setting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Alternative architectures replace dense softmax with sparse attention ( beltagy2020longformer ; zaheer2020bigbird ) , linear or kernel-based variants ( katharopoulos2020lineartransformer ; choromanski2021performer ) , or state-space models ( gu2023mamba ) ; sparser or non-softmax distributions naturally curb the attention diffusion and lost-in-the-middle bias that plague dense softmax at long context. A complementary line removes positional encoding entirely: NoPE outperforms explicit position encodings on length-generalization benchmarks ( kazemnejad2023nope ) , and open-weight models such as Kimi K2 ( kimi2025k2 ) and NVIDIA Nemotron Nano 2 ( nvidia2025nemotron ) adopt NoPE in their hybrid layers. Both lines typically require training from scratch or substantial fine-tuning and therefore complement rather than compete with our zero-shot setting; Jet-Long itself extends smoothly to hybrid designs (Section 4.3 , Jet-Nemotron ( gu2025jet ) ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on RULER ( hsieh2024ruler ) (synthetic recall over 13 tasks), HELMET-RAG ( yen2024helmet ) (the HELMET study’s best overall predictor of downstream long-context performance), and PG-19 ( rae2019compressive ) long-form perplexity, against the strongest zero-shot baselines; we additionally test transfer to the hybrid Jet-Nemotron backbone (Section 4.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Methodology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The frequency-rescaling and grouped-position methods of Section 2.2 fix a single rescaling factor up front, forcing a tradeoff between short-context fidelity and long-context reach; recent length-adaptive maps ( xu2025extending ; zhang2025lampe ) relax it, but with a fitted or distance-dependent schedule. Jet-Long instead derives the factor analytically from the current sequence length (Section 3.1 ); the resulting two-window computation matches FlashAttention ( dao2022flashattention ) within the pretraining window and exceeds it at long context (Section 4.6 ), and leaves the KV cache untouched at decode (Sections 3.2 – 3.3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We build on the dual-window (bifocal) decomposition of Self-Extend ( jin2024llm ) (a related but architecturally three-way ancestor is DCA ( an2024training ) ): a local window of size w 0 w_{0} that retains classic RoPE (preserving the model’s pretraining behavior exactly), and a remote window governed by a rewritten position mapping f ( ⋅ ) f(\\\\cdot) that maps positions back into the training range. For a query at position q q and a key at position k k , the pre-softmax attention score is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let L L denote the current sequence length and w pretrained w_{\\\\text{pretrained}} the pretrained context window. When L ≤ w pretrained L\\\\leq w_{\\\\text{pretrained}} , f ( x ) = x f(x)=x and Eq. ( 1 ) reduces to the unmodified base model. Jet-Long’s contribution lies in the choice of f ( ⋅ ) f(\\\\cdot) and in the inference-time machinery that makes the two windows interact for free.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Dynamic extrapolation factor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To keep remote-window rotation angles in-distribution, the group size G G must scale with the current sequence length L L , as illustrated in Figure 2(a) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The natural dynamic factor is, for continuous-style RoPE rescaling (e.g., DNTK ( emozilla2023dyntk ) and dynamic-YaRN ( peng2023yarn ) variants), the scaling ratio s = max ( 1 , L / w pretrained ) s=\\\\max(1,L/w_{\\\\text{pretrained}}) ; for discrete grouped methods (e.g., Self-Extend ( jin2024llm ) ), it is an integer group size G G . Jet-Long uses discrete grouping because an LLM only encounters a finite, discrete set of relative RoPE rotation angles during pretraining; aliasing positions onto that pretrained grid keeps every remote-window angle exactly in-distribution, an integer relative position the model has actually been trained on. We ablate this choice against continuous frequency interpolation in Section 4.5 . Whereas DNTK ( emozilla2023dyntk ) adapts the RoPE base β \\\\beta (per-frequency), Jet-Long adapts the discrete group size G G on the position-aliasing axis, keeping every remote angle on the model’s training grid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To maximize positional resolution, G G is the smallest integer that keeps the compressed sequence within the pretrained window w pretrained w_{\\\\text{pretrained}} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The remote mapping is a floor division of absolute positions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By recomputing G G as the sequence grows, Jet-Long applies the minimum compression that keeps every remote position in-distribution, maximizing positional resolution at every length. LaMPE ( zhang2025lampe ) likewise applies floor-based remapping in its middle region, but derives the middle-region compression ratio from an empirically fitted sigmoid of the input length, with model-specific fits presented for Llama2 and Llama3, whereas Eq. ( 2 ) is determined analytically by L L and w pretrained w_{\\\\text{pretrained}} with no fitted parameters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Key-value cache management and correction rotation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dynamic extrapolation poses a cache-management challenge: if G G changes during generation, rewriting the KV cache with new extrapolated phases would require discarding and recomputing the entire cache.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As depicted in Figure 2(b) , Jet-Long avoids this overhead by maintaining a strict invariant: the cache stores only uncompressed base keys at their exact absolute positions k k ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When the remote window needs the query at f ( q ) f(q) and the key at f ( k ) f(k) , instead of recomputing those vectors from scratch we apply a correction rotation on the fly using the position offsets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This relies on standard RoPE composing additively in angle, R a R b = R a + b R_{a}R_{b}=R_{a+b} (per-position scalings beyond a pure rotation would break this and require recomputing keys from 𝐱 k \\\\mathbf{x}_{k} ). Applying RoPE ( ⋅ , Δ ) \\\\text{RoPE}(\\\\cdot,\\\\Delta) to a vector encoding position p p therefore produces the vector at position p + Δ p+\\\\Delta , so we apply the offset directly to the active query and cached keys before attention:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This constant-time operation reconstructs the extrapolated vectors in registers; the physical cache is never touched, so streaming generation runs across length boundaries without stalling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Inclusion–exclusion prefill\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To avoid materializing the full quadratic attention matrix, which exhausts memory during long-sequence prefill, Jet-Long achieves exact distance-based routing by combining the inclusion–exclusion principle with FlashAttention’s ( dao2022flashattention ) LogSumExp statistics. The merge requires three standard attention calls (Figure 2(c) ), each returning an output 𝐎 X \\\\mathbf{O}_{X} and an LSE vector ℓ X \\\\bm{\\\\ell}_{X} : (A) sliding-window attention (local w 0 w_{0} ) with base queries and keys, (B) full causal attention with remote queries and keys, and (C) sliding-window attention (local w 0 w_{0} ) with remote queries and keys. Calls B and C apply the same remote rotation to local keys, so their local-subset contributions cancel exactly: W B 𝐎 B − W C 𝐎 C W_{B}\\\\mathbf{O}_{B}-W_{C}\\\\mathbf{O}_{C} (resp. W B − W C W_{B}-W_{C} ) collects only the remote-only term, while Call A supplies the base-local term. Stabilizing element-wise (per query position) via M = max ( ℓ A , ℓ B , ℓ C ) M=\\\\max(\\\\bm{\\\\ell}_{A},\\\\bm{\\\\ell}_{B},\\\\bm{\\\\ell}_{C}) and weights W X = exp ( ℓ X − M ) W_{X}=\\\\exp(\\\\bm{\\\\ell}_{X}-M) , the final output (numerator and denominator both in FP32 to avoid catastrophic cancellation) is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This merge realizes exact distance-based routing without a boolean mask matrix, retaining FlashAttention’s memory efficiency and near-FA2 throughput. Prior work rejects this subtractive form for learned sparse attention as “numerically catastrophic” ( yao2026focus ) , measuring 0.79 0.79 cosine similarity against a reference, since the subtraction can drive the logarithm’s argument negative. Eq. ( 7 ) instead subtracts in linear space, where shifting by M M bounds every weight in ( 0 , 1 ] (0,1] and FP32 accumulation avoids the cancellation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our primary evaluation is on the Qwen3 base model family ( yang2025qwen3 ) , namely Qwen3-1.7B-Base, Qwen3-4B-Base, and Qwen3-8B-Base, each of which has a 32,768-token (32K) native training window. We extend the usable context to lengths up to 131,072 (128K) at inference time, without any fine-tuning. To verify that Jet-Long generalizes beyond pure softmax-attention transformers, we additionally evaluate on the hybrid Jet-Nemotron-2B and Jet-Nemotron-4B models ( gu2025jet ) , which interleave softmax and linear-attention layers. We use w 0 = 2048 w_{0}=2048 throughout the main results, and per-baseline hyperparameters (DNTK, YaRN, DCA, Self-Extend) are listed in Appendix C .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Long-context benchmarks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report results on three complementary suites: (i) RULER ( hsieh2024ruler ) , a synthetic recall stress test averaging over thirteen tasks; (ii) HELMET-RAG ( yen2024helmet ) , averaged over the four default sub-tasks (NaturalQuestions, TriviaQA, PopQA, HotpotQA), which the HELMET study reports as the most efficient predictor of downstream long-context performance and which serves as our application-grounded benchmark; and (iii) PG-19 ( rae2019compressive ) , on which we report long-form perplexity. RULER and HELMET-RAG accuracies are percentages, with gaps between methods reported in percentage points (pp); PG-19 is reported as token-level perplexity (ppl, lower is better). All RULER and HELMET-RAG generations use greedy decoding with the default RULER/HELMET prompts and per-task max-output-token limits; PG-19 is teacher-forced perplexity at stride 1024 over 100 books. Formal definitions and the geometric-mean aggregation used in the Avg columns are given in Appendix A . We further ablate the only Jet-Long hyperparameter, the local protected window size w 0 w_{0} , in Section 4.4 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Inference efficiency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We implement Jet-Long as a fused CuTe kernel that merges the three attention calls of Section 3.3 into a single launch, and benchmark prefill and single-batch generation throughput on a single H100 against the highly optimized FlashAttention-2 ( dao2023flashattention2 ) and FlashAttention-4 ( zadouri2026flashattention4 ) baselines applied to the unmodified base models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All experiments are run on NVIDIA H100 Tensor Core GPUs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Main results on long-context extension\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The average scores on the three benchmarks are reported in Table 1 . Jet-Long is best on every RULER and PG-19 column across all three Qwen3 sizes (over Base and the four extrapolation baselines), and best on HELMET-RAG at 4B and 8B (0.73 pp behind Self-Extend at 1.7B). The RULER lead over the strongest baseline (DNTK at 1.7B; Self-Extend at 4B and 8B) is 4.79, 2.18, and 2.03 pp.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As Figure 3 shows, Jet-Long matches or substantially outperforms YaRN and DNTK at every length across all three sizes for RULER accuracy by length. Table 2 confirms the same pattern on PG-19: within the 32K training window, Jet-Long is mathematically equivalent to the base model (the dynamic factor reduces to identity, Section 3.1 ); past 32K the bare base collapses (ppl of 71.00 / 104.66 / 79.37 at 128K for 1.7B / 4B / 8B) while Jet-Long stays at 11.41 / 9.62 / 8.51, the lowest among all extrapolation methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also report the per-task accuracy at context length L = 65536 L{=}65536 in Table 3 . Jet-Long is best or tied for best on 8 of the 13 tasks at both 1.7B and 8B, with the largest leads on Multi-Key NIAH (MK-NIAH-2 at 1.7B: 61.80 vs 20.00; MK-NIAH-3 at 8B: 70.80 vs 39.80) and Variable Tracking (VT at 1.7B: 73.56 vs 54.24); the only 1.7B exception is CWE, on which every method (including Base) scores below 1 % 1\\\\% . Aggregated over the 13 tasks, Jet-Long beats the strongest baseline by 7.31 pp at 1.7B and 6.07 pp at 8B.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Hybrid attention extension results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test generalization beyond softmax-only transformers, we apply Jet-Long to the hybrid Jet-Nemotron architecture ( gu2025jet ) , which interleaves softmax and linear-attention layers. Table 4 reports per-length RULER accuracy on Jet-Nemotron-2B and 4B.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Within 32K the construction reduces to base attention, so Jet-Long inherits the base model’s in-distribution behavior (matching to within rounding). Past 32K the bare hybrid base collapses (8.54 / 5.65 at 128K for 2B / 4B), while Jet-Long retains 33.78 and 33.14 ( + 25.24 +25.24 / + 27.49 +27.49 pp). Averaged over the seven lengths, Jet-Long lifts RULER from 42.93 to 52.94 ( + 10.01 +10.01 pp) at 2B and from 42.16 to 53.47 ( + 11.31 +11.31 pp) at 4B. The bifocal decomposition and dynamic factor generalize to hybrid LLM architectures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Ablation: local window size w 0 w_{0}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To check whether w 0 w_{0} requires per-deployment tuning, we sweep it on Qwen3-4B/8B at three out-of-window lengths (64K, 96K, 128K) in Table 5 (1.7B omitted for compute). The control w 0 = 0 w_{0}{=}0 shrinks the local window to attention-to-self only and collapses RULER to near-zero, confirming the local window is necessary. Past that boundary, Jet-Long is hyperparameter-resilient: every w 0 ∈ { 512 , 1024 , 2048 , 4096 } w_{0}\\\\in\\\\{512,1024,2048,4096\\\\} stays within 2 2 pp of the per-row best at any single length and within 1 1 pp on the per-model average; practitioners can pick w 0 = 2048 w_{0}{=}2048 without per-deployment tuning. The slight degradation at w 0 = 8192 w_{0}{=}8192 (1.4–2.1 pp gap) reflects the local window consuming a larger fraction of context.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Ablation: interpolate frequency or alias position\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 3.1 requires only that the remote mapping f ( ⋅ ) f(\\\\cdot) keep rotation angles in-distribution, not how. Table 6 compares position aliasing ( f ( x ) = ⌊ x / G ⌋ f(x)=\\\\lfloor x/G\\\\rfloor , the default Jet-Long) against YaRN-style frequency interpolation (rescaled RoPE frequencies, unchanged positions) on Qwen3-1.7B-Base. Aliasing wins by 6.99 pp at 64K and 4.30 pp at 128K, consistent with the LLM having learned a discrete grid of relative angles. The alias margin shrinks at extreme lengths because larger G G gives continuous interpolation more to compensate for: on FWE from + 20.3 +20.3 pp at 64K to + 2.5 +2.5 at 128K, on QA-1 from + 11.2 +11.2 to + 0.6 +0.6 , and frequency interpolation overtakes aliasing on MK-NIAH-2 and QA-2 at 128K. Aliasing remains the better default for ≤ 128 \\\\leq 128 K; hybrid mappings merit investigation at longer contexts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 Inference efficiency\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We measure end-to-end throughput (tok/s) on a single H100 with CUDA graphs on Qwen3-8B-Base, comparing FA2 ( dao2023flashattention2 ) , the H100-native FA4 ( zadouri2026flashattention4 ) , the multi-launch Jet-Long (unfused) variant of Sections 3.3 – 3.2 , and our fused Jet-Long CuTe kernel; 1.7B and 4B follow the same pattern (Table 8 , Appendix B ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Within 32K all four configurations match FA2 to within ± 1 % \\\\pm 1\\\\% since Jet-Long reduces to standard attention. Past 32K the naive multi-launch baseline pays the bifocal cost: three FlashAttention passes plus an out-of-kernel merge drop prefill to 0.89 0.89 – 0.93 × 0.93\\\\times FA2, and unfused per-token correction rotation drops generation to 0.14 0.14 – 0.34 × 0.34\\\\times . The fused CuTe kernel removes both overheads: prefill recovers to 1.28 1.28 – 1.39 × 1.39\\\\times FA2 past 32K (approaching FA4’s 1.53 × 1.53\\\\times at 128K, without an H100-specific kernel), and generation stays ≥ 0.96 × \\\\geq 0.96\\\\times FA2 at every length, with the residual ≤ 4 % \\\\leq 4\\\\% overhead from the per-token correction rotation and the dynamic- G G bookkeeping. The accuracy gains in Table 1 therefore come essentially free at inference.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We presented Jet-Long , a tuning-free zero-shot context-extension method that pairs a local RoPE-faithful window with a long-range window whose rescaling factor adapts dynamically to the current sequence length, and made the construction essentially free at inference through an inclusion–exclusion attention merge with on-the-fly correction rotation in a fused CuTe kernel. On Qwen3-1.7B/4B/8B at context lengths up to 128K, Jet-Long shows superior performance on RULER ( + 4.79 +4.79 / + 2.18 +2.18 / + 2.03 +2.03 pp at 1.7B/4B/8B), HELMET-RAG, and PG-19 perplexity, generalizes to the hybrid linear-attention Jet-Nemotron backbone, and remains highly resilient to its only hyperparameter, the local window size w 0 w_{0} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Jet-Long addresses position-OOD at the RoPE level, while the complementary attention-diffusion failure mode is naturally tackled by architectural alternatives ( beltagy2020longformer ; zaheer2020bigbird ; katharopoulos2020lineartransformer ; choromanski2021performer ; gu2023mamba ) . Because Jet-Long requires a softmax-with-RoPE base, natural extensions include other softmax-with-RoPE variants such as Multi-head Latent Attention ( liu2024deepseek ) and sparse attention, and architectures interleaving softmax with sparse or linear-attention sub-layers, beyond the Jet-Nemotron experiments in Section 4.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Percentage points (pp).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For two scores a 1 , a 2 ∈ [ 0 , 100 ] a_{1},a_{2}\\\\in[0,100] reported as percentages, the absolute gap in percentage points is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which we use throughout the paper to describe RULER and HELMET-RAG accuracy differences. This is distinct from the relative percent change ( a 1 − a 2 ) / a 2 (a_{1}-a_{2})/a_{2} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Perplexity (ppl).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a held-out token sequence x 1 , … , x T x_{1},\\\\dots,x_{T} scored under a language model with conditional probabilities p ( x t ∣ x < t ) p(x_{t}\\\\mid x_{<t}) , perplexity is the exponentiated mean negative log-likelihood per token,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"lower is better. On PG-19 we use anchored growing-window evaluation: at each context length L L in { 4 K , 8 K , 16 K , 32 K , 64 K , 96 K , 128 K } \\\\{4\\\\text{K},8\\\\text{K},16\\\\text{K},32\\\\text{K},64\\\\text{K},96\\\\text{K},128\\\\text{K}\\\\} , the next 1024 1024 tokens are scored conditional on the preceding L L tokens, and ppl is averaged across the 100 books in the PG-19 evaluation split.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Geometric-mean aggregation across lengths.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the Avg column of PG-19 perplexity tables, we report the geometric mean across the n = 7 n=7 lengths,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which corresponds to averaging the underlying log-likelihoods uniformly across lengths and is therefore the natural aggregate for an exponentiated metric.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 8 reports the complete prefill and generation throughput sweep for all three Qwen3 sizes (1.7B, 4B, and 8B) on H100, complementing the 8B-only Table 7 in Section 4.6 . The same four configurations are compared (FA2, FA4, Jet-Long (unfused), Jet-Long CuTe), and the parenthesized speedups are again computed against FA2 at the matching length. The qualitative pattern is identical across sizes: Jet-Long CuTe matches FA2 inside the 32K native window, recovers and surpasses FA2 at long-context prefill (1.28–1.45 × \\\\times ), and stays at near-FA2 generation throughput everywhere ( ≥ 0.96 × \\\\geq 0.96\\\\times across all lengths and sizes), while the unfused multi-launch variant pays a real cost in generation that the fused kernel eliminates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The four zero-shot baselines compared in Tables 1 – 4 use a single configuration each, held constant across Qwen3-1.7B/4B/8B and across all evaluation lengths (4K–128K), as outlined in Table 9 . All Qwen3 models have a pretrained context window of 32 , 768 32{,}768 tokens and a RoPE base θ = 10 6 \\\\theta=10^{6} .\"\n }\n]","isConversational":false}}},{"rowIdx":491,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06979"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Robust Federated Learning Under Real-World Client Churn"},"authors":{"kind":"list like","value":["Dhruv Garg","Neha Lakhani","Debopam Sanyal","Myungjin Lee","Alexey Tumanov","Ada Gavrilovska"],"string":"[\n \"Dhruv Garg\",\n \"Neha Lakhani\",\n \"Debopam Sanyal\",\n \"Myungjin Lee\",\n \"Alexey Tumanov\",\n \"Ada Gavrilovska\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06979v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"Federated Learning (FL) enables training shared models across large populations of user devices without collecting the raw data centrally. This paradigm has unlocked machine learning on private, on-device interaction logs that were previously inaccessible to centralized trainers (McMahan et al. , 2017 ; Bonawitz et al. , 2019 ; Kairouz et al. , 2021 ) . Such privacy-preserving decentralization is now critical for various production applications, including personalized recommenders, feed-ranking, and session-aware input prediction. FL has been deployed for mobile keyboard and language prediction (Hard et al. , 2018 ; Chen et al. , 2019 ; Xu et al. , 2023 ) , privacy-preserving healthcare analytics (Brisimi et al. , 2018 ; Xu et al. , 2021 ; Nguyen et al. , 2022a ) , and large-scale model personalization (Yang et al. , 2019 ; Xu et al. , 2024 ) , establishing it as the de facto approach for privacy-sensitive collaborative training across diverse domains. Historically, these deployments focused on batched, multi-day refreshes where modest delays between data collection and model updates were acceptable."},{"type":"para","text":"Contemporary on-device applications, however, increasingly require fresh models that adapt on hourly timescales. User interaction streams are volatile: viral content, news events, promotional campaigns, and short-lived user intents can shift predictive distributions rapidly. When model refresh intervals lag behind these shifts, predictive relevance degrades and key application metrics such as click-through rates and downstream conversions suffer (Matam et al. , 2024 ) . Meeting these goals demands FL systems that minimize time-to-target accuracy on live streams while operating at production scale."},{"type":"para","text":"Transitioning from slow and batched to agile and streaming FL is difficult because near-real-time updates break the core assumptions of existing FL systems. We highlight three interlocking constraints that render naive extensions of current architectures ineffective:"},{"type":"heading","level":4,"text":"(1) Transient client availability."},{"type":"para","text":"Mobile and edge devices are intermittently reachable due to battery, network, and user behavior. Availability fluctuates at minute-level granularity and exhibits heavy churn ( M-Lab, ) . As shown in (Garg et al. , 2025 ) , existing mitigations generally fail in three ways: (a) Over-selection , where requesting excess clients to participate in each round wastes device compute and bandwidth while increasing server-side update variance; (b) blocking synchrony , which converts per-round latencies from seconds to minutes when aggregation goals are not met; and (c) coarse periodic probing , which either misses short-lived availability windows of useful clients or incurs high heartbeat overhead to maintain up-to-date status. These approaches trade away either time-to-accuracy or communication budgets, making them unsuitable for large-scale FL deployments operating with thousands of clients per round. Approaches that target the statistical variance from partial participation (Jhunjhunwala et al. , 2022 ) or study large-cohort selection effects (Charles et al. , 2021 ) improve efficiency but still assume a predictable pool of available clients and do not address minute-level churn."},{"type":"heading","level":4,"text":"(2) Evolving client utility and heterogeneity."},{"type":"para","text":"Clients differ not only in their hardware capabilities (compute speed, memory, network), but also in the statistical value of their recent data. We define client utility as a time-varying quantity combining data informativeness for the live distribution with the client’s expected compute latency. Existing selection algorithms often ignore these dynamics (Nguyen et al. , 2022b ; Bonawitz et al. , 2019 ; Nishio and Yonetani, 2019 ) , while newer ranking methods rely on the ”oracular” knowledge of availability, which is impractical in deployment. Not just that, they don’t account for the shifting data distributions of streaming tasks (Lai et al. , 2021 ; Abdelmoniem et al. , 2023 ; Ye et al. , 2023 ) . Selectors that rank clients using utility estimates derived from infrequent probes or historical traces systematically mis-rank participants, as these signals quickly become stale and introduce bias that can outweigh their intended benefits. For example, it may over-select fast devices with irrelevant data while neglecting slow devices with fresh, informative signals. Such techniques either require non-existent oracular deployment traces, coarse-grained and often stale utility estimates, or frequent probing that incurs prohibitive overhead. As a result, they suffer rapid degradation in time-to-accuracy with shifts in utility."},{"type":"heading","level":4,"text":"(3) Model freshness vs. delayed updates."},{"type":"para","text":"In production tasks, the strongest training signal (the ground-truth label) often arrives with some delay; for instance, a recommendation’s conversion may be observed hours after the interaction. This creates a fundamental trade-off: updating immediately with proxy feedback maximizes freshness but introduces noise, while waiting for full feedback improves label quality at the cost of staleness. Existing strategies—like buffered asynchronous aggregation (Nguyen et al. , 2022b ) or staleness-weighting schemes (Xie et al. , 2019 ; Rodio and Neglia, 2024 ; Zhou et al. , 2022 ) —force a choice between convergence speed and statistical robustness. Incorrect aggregation can destabilize the training process, inflating the wall-clock time required to reach target accuracy or discarding valuable, though delayed, signals."},{"type":"para","text":"Taken together, these constraints explain why existing synchronous and asynchronous FL systems fail to achieve reasonable time-to-target accuracy in streaming settings, at production scale as shown in Fig. 1 . Synchronous schemes stall or over-select; asynchronous schemes avoid blocking but accumulate staleness that degrades convergence (Xie et al. , 2019 ; Lian et al. , 2018 ; Li et al. , 2019b ) ; assumed oracular knowledge of availability or heavy-weight predictors are not tractable; and prohibitive communication costs of frequent probes derail performance. Empirical studies (see § 2 and Fig. 4 ) show that even modest unavailability or data heterogeneity can worsen the model adaptation time as the complexity of the deployment shifts to hardened production settings. Not just that, it can substantially degrade performance, even to the point of non convergence, as the unavailability and heterogeneity increase."},{"type":"para","text":"In this work, we frame streaming FL as a constrained optimization problem: minimize time-to-target accuracy on a live interaction stream subject to transient availability, evolving per-client utility, frequent update delays, and bounded communication overhead. This objective focuses our design on policies that treat availability and utility as low-latency control inputs while ensuring scalability across massive populations, without assuming oracular knowledge of client behaviour."},{"type":"para","text":"We present FeLiX , a redesigned FL orchestration framework that directly optimizes time-to-target accuracy in streaming production settings. FeLiX treats availability as an evolving signal and couples inexpensive freshness checks with selection logic to optimize convergence. Our core primitives are:"},{"type":"para","text":"Streaming-aware availability tiers: FeLiX organizes clients into dynamic readiness tiers based on lightweight signals (battery, connectivity). This leverages a client-server design that minimizes network overhead while ensuring the server possesses high-fidelity information on client capabilities within tight refresh windows."},{"type":"para","text":"Fresh-utility based selection: We introduce a dual-tier approach where an evaluation tier performs inexpensive forward passes to estimate data informativeness. This is fused with compute speed to prioritize clients that are both hardware-performant and reflect the most relevant recent distribution shifts."},{"type":"para","text":"Informativeness-aware, delay-robust aggregation: FeLiX employs weights that balance update informativeness with model-age corrections. This allows the system to safely incorporate late, high-value updates with full feedback, ensuring the global model remains robust without being stalled by the long-tail latencies of the straggler or long unavailable clients."},{"type":"para","text":"We conduct an exhaustive evaluation of FeLiX using real-world MobiPerf traces and multi-modal datasets including CIFAR-10 (Krizhevsky, 2009 ) and Google Speech Commands V2 (Warden, 2018 ) . We demonstrate that FeLiX achieves a 2.37 × 2.37\\times reduction in time-to-target accuracy compared to recent Synchronous and Asynchronous FL baselines, while limiting communication overhead and preserving the predictive power of fresh models."},{"type":"heading","level":1,"text":"2. Background and Motivation"},{"type":"heading","level":2,"text":"2.1. FL Primer and Client Availability"},{"type":"para","text":"Asynchronous FL. Federated Learning (FL) trains a shared model across many decentralized, privacy-sensitive clients by keeping raw data local: the server broadcasts global weights, clients perform local training on-device, and the server aggregates returned updates (McMahan et al. , 2017 ; Bonawitz et al. , 2019 ) . Two execution regimes dominate cross-device deployments. In synchronous FL (SyncFL) the server advances in rounds after collecting an agg-goal number of client updates; in asynchronous FL (AsyncFL) the server incorporates updates as they arrive and does not block on stragglers. SyncFL offers conceptual simplicity and bounded staleness but is vulnerable to delays caused by a few slow or unavailable clients; AsyncFL trades that blocking for continuous progress at the cost of needing principled staleness handling. Convergence analyses of FedAvg under non-IID data (Li et al. , 2019b ; Wang et al. , 2020b ) reveal that data heterogeneity exacerbates gradient divergence across clients, while adaptive federated optimizers (Reddi et al. , 2020 ) partially compensate but leave the availability challenge unaddressed."},{"type":"para","text":"AsyncFL emerged to address the wide runtime variability observed in practice: theoretical analyses of asynchronous decentralized SGD (Lian et al. , 2018 ) and asynchronous federated optimization (Xie et al. , 2019 ) confirm that async updates converge under mild staleness bounds but can degrade sharply when updates are very stale or derived from strongly non-IID local data. mobile devices exhibit long-tailed training runtimes driven by differences in compute, background load, and network latency as shown in Fig. 2 . Together, device heterogeneity and application-driven data generation produce a rapidly evolving trainer runtime distribution. This variability forces system designers to confront three co-occurring concerns: (i) which clients should the server select at any moment, (ii) how to cheaply detect that a client is actually ready to train, and (iii) how to incorporate late or intermittent updates without destabilizing learning. A compact AsyncFL control loop (selector, weight dispatcher, client training, aggregator/buffer) exposes these decision points; we illustrate the flow in Fig. 3 . Clients register themselves with the FL-server 1 . The control-plane maintains these connections and forwards the client-id(s) to the selector 2 . The selector is tasked with maintaining a desired training concurrency ( i.e. , number of parallely training clients) throughout the process. Whenever the training count falls below the concurrency level ( i.e. , clients return updates). The selector uses its algorithm to return the next client(s) to train 4 so as to meet the concurrency level. These selected client ID(s) are forwarded to the aggregator. 5 The weight dispatcher sends the latest server model weights (W) to the selected clients. If the client is available to train, it begins training using fresh global weights (W) and data from the applications’ local data-store. Upon completion, it returns weight changes w.r.t W as updates ( Δ W \\Delta W ) 6 to the aggregator. If the client wasn’t initially available to train or is interrupted while training, the client waits until it becomes available next. At the FL server, the model updates received from clients are aggregated and put into a temporary buffer. Once agg-goal number of updates in the buffer are aggregated, the server model version changes and the buffer is emptied 7 . This FL training process continues until convergence."},{"type":"para","text":"Evolving runtime availability. While foundational FL research and early system prototypes often assumed highly available clients operating in stable, conducive environments, production deployments reveal that this assumption is fundamentally flawed. In practice, ignoring the reality of intermittent participation degrades performance significantly, inflating time-to-accuracy and wasting system resources. We define availability not merely as a device being powered on, but as a multi-dimensional state meeting strict system-level and application-level filters, including WiFi-only connectivity, sufficient battery levels, and OS-mandated idle constraints. Consequently, unavailability is the norm; production traces from the Google FL System (Bonawitz et al. , 2019 ) , the PAPAYA analytics platform (Srinivas et al. , 2025 ) , LinkedIn’s FLINT (Wang et al. , 2023a ) , and FedScale/MobiPerf (Lai et al. , 2022 ; M-Lab, ) report mean client availability ranging from as low as 15% to roughly 50% of the total population. In production, unavailability is therefore the norm rather than the exception. This participation is inherently transient , characterized by short-lived ready windows whose durations are often comparable to the training round itself. Across different geographies and applications, these dynamics manifest in large peak-to-trough swings. For instance, diurnal effects can cause availability to fluctuate by 5 × 5\\times within a single day, with minima dropping to 10% during active usage hours and peaking at 80% when devices are idle and charging. At the system level, these trends create significant orchestration challenges: in the Google FL system, nearly 22% of updates arrive too late for aggregation and are rejected, while another 2% are interrupted mid-computation. Restrictive participation criteria, while necessary for user experience, further contract the eligible pool, sometimes as low as 22% of the total devices as shown in MobiPerf ( M-Lab, ) . This exposing a sharp trade-off between model fairness and system throughput. These bursty and unpredictable availability patterns mean that naive selection strategies either over-select to mask dropouts (increasing variance) or stall rounds waiting for stragglers, both of which are untenable for streaming applications requiring hourly refreshes. Prior measurements (Garg et al. , 2025 ) directly quantify this: even under moderate unavailability ( < < 50%), time-to-target accuracy degrades sharply, confirming that availability must be treated as a first-class orchestration signal rather than a background metric."},{"type":"para","text":"AsyncFL alone cannot fix unavailability. These runtime and availability characteristics translate into three practical operational effects that motivated AsyncFL and that now motivate streaming-aware controls. First, servers often resort to over-selection (invite many more candidates than needed) to probabilistically meet an agg-goal; this wastes device CPU, uplink/downlink network bandwidth and increases variance in server-side latency. Second, a client’s local training time may exceed the mean aggregator interval, increasing the chances of interruptions, rejected updates, or partial work. A natural mitigation is to checkpoint local training state so that a client can resume after an interruption: an approach well-studied in distributed DNN training (Mohan et al. , 2021 ; Eisenman et al. , 2022 ; Gupta et al. , 2024 ; Wang et al. , 2023b ) . However, in the FL context, checkpointing only preserves progress on the client ; the client still cannot return any update until device conditions recover, and the global model advances in the meantime, rendering the saved gradients stale on return. Checkpointing therefore does not close the availability gap. It defers the problem rather than solving it. Third, restrictive eligibility filters shrink the effective population and risk selection bias. Together, these effects show that availability and runtime behavior are not background metrics but low-latency control inputs that selection and aggregation logic must use in order to optimize wall-clock progress and maintain model quality."},{"type":"para","text":"Finally, while AsyncFL and its improvements such as update buffering reduce blocking, they do not by themselves solve the coupled problems of rapid availability churn, evolving per-client statistical value, and bounded communication costs. For example, buffered AsyncFL reduces immediate stalls but requires careful eviction/weighting policies to prevent stale updates from hurting convergence. Later sections discuss these algorithmic and systems tensions in detail and show why treating readiness and freshness as first-class, lightweight signals is essential for streaming FL."},{"type":"heading","level":2,"text":"2.2. FL Orchestration Challenges"},{"type":"para","text":"Brittle selection algorithms. Many FL selection strategies attempt to balance synchronization and participation, but practical deployments expose a harsh trade-off between update staleness and system stalls. Synchronous FL variants avoid staleness by enforcing global rounds but are highly vulnerable to stalls : when availability is intermittent, SyncFL blocks on stragglers or turns to heavy over-selection, converting per-round latencies from seconds into minutes and dramatically increasing communication and cost. Asynchronous and buffered schemes relax blocking by accepting updates computed on older model states (or by aggregating along a buffer), which improves throughput. However, if stale updates are naively aggregated, they will destabilize convergence. Conversely, discarding late updates sacrifices the valuable statistical signal carried by the slower or transiently available clients. Recent selector-based approaches that prioritize high-utility clients (for example, OORT and REFL selectors) or use learned predictors to estimate client utility aim to trade off speed and signal by favoring fast, high-utility workers. In practice, these methods typically depend on availability signals or historical traces to estimate utility. Under bursty availability and rapid population/data shifts those utility estimates age quickly leading to repeated mis-ranking. For example, such selecots would over-select fast but irrelevant devices and neglect informative but ephemeral clients. These limitations have motivated hybrid approaches like buffered asynchronous aggregation that attempt to retain long-tail information while reducing stalls, but the tension between timeliness, stability, and privacy-compatible aggregation remains unresolved. Variance-reduction techniques for partial participation (Jhunjhunwala et al. , 2022 ) and large-cohort selection (Charles et al. , 2021 ) target statistical stability but still assume a stable, predictable client pool. Staleness-aware schemes (Rodio and Neglia, 2024 ; Zhou et al. , 2022 ) weight or discount late updates based on model-version gap, but do not resolve the challenge of detecting and exploiting short availability windows in real time."},{"type":"para","text":"Coarse systems and instrumentation. Production stacks provide several primitives for large-scale FL, but their availability signals tend to be either too coarse to support fine-grained orchestration or too heavyweight to scale. Such platforms use techniques like client check-ins, tokens, and over-selection heuristics to mask dropout and reduce orchestration complexity with client availabilty. Yet such protocols are explicitly designed around coarse-grained and slower tradeoffs like accept/reject tokens, windowed selection rather than continuous, low-latency availability views. This design choice simplifies robustness but also means a nontrivial fraction of potential updates are delayed or rejected in practice. At the other extreme, tooling that instruments devices more deeply like pre-deployment measurement, capability profiling, or continuous heartbeat reporting, can provide the fine-grained, near-real-time availability signals orchestration needs, but at production scale these signals increase reporting overhead, consume device energy, and raise privacy and operational concerns. Benchmark and integration frameworks make clear how availability and distribution shift in deployment. They also let researchers replay realistic, time-varying traces for evaluation. However, by definition these offline traces cannot supply the compact, cross-application, real-time view an orchestrator would need for streaming, hourly refresh tasks. In short, production systems either expose high-fidelity signals that are too heavy to maintain continuously, or coarse signals that force servers to choose between blocking (stalling progress) and proceeding blindly (accumulating staleness). Representative system and benchmarking efforts that highlight these tradeoffs include Google FLSys (Bonawitz et al. , 2019 ) and the PAPAYA analytics stack (Srinivas et al. , 2025 ) , which describe token/check-in and rejection mechanics for large-scale mobile FL; the benchmark suite FedScale (Lai et al. , 2022 ) ; and the FL integration tooling FLINT (Wang et al. , 2023a ) , which emphasizes on-device measurement and pre-deployment instrumentation. Research frameworks such as Flower (Beutel et al. , 2020 ) and FedML (He et al. , 2020 ) enable flexible experimentation but are designed for controlled settings rather than production-scale streaming deployments with bursty, minute-level availability dynamics."},{"type":"para","text":"These algorithmic and system-level gaps are particularly acute for streaming applications , where the requirement for hourly refreshes clashes with high data heterogeneity and delayed ground-truth feedback. In production tasks like recommendation or ad-targeting, the strongest training signal (the label) often arrives hours after the initial interaction. Existing systems are unable to navigate this: they must either update immediately using noisy proxy feedback to maintain freshness, or wait for full labels and suffer extreme model staleness."},{"type":"heading","level":2,"text":"2.3. Model Adaptation Gap: Oracular vs. Reality"},{"type":"para","text":"Opportunity Gap. To quantify the performance gap with existing FL approaches, we move from an idealized regime to progressively more realistic deployment settings in Fig. 4 , exposing the technical reasons behind the widening performance gap. While an idealized baseline of 100% availability and homogeneous data in Line 1 i.e. L1 ensures minimal time-to-target accuracy, the introduction of data heterogeneity (L2) increases gradient variance, increasing the time-to-accuracy even under 100% participation. The performance is expected to degrade sharply with realistic client behaviour, for instance if client availability drops to 50%. Here rounds will stall and existing selectors that fail to capture enough informative updates within tight refresh time windows, will suffer. However, with oracular or trace-provided client information, despite 50% unavailability, L3 does not drop in its performance significanlty compared to L2. The most significant performance collapse occurs when we withdraw the oracular foresight. Many state-of-the-art methods like OORT and REFL rely on retrospective traces or trace-derived availability predictors to estimate future utility (L4). Without perfect future knowledge, these systems struggle with low availability, repeatedly stalling during rounds. Even with over-selection, they fail to capture the transiently available clients that carry high-value for model training. Bridging this gap (L5) requires an orchestrator that achieves near-oracular performance not through foresight, but by exploiting low-cost, real-time readiness signals to harvest the high-value updates as they appear."},{"type":"para","text":"Design Primitives for Streaming FL. Closing the empirical gap between Line 4 and Line 5 requires a shift in FL orchestration infrastructure: availability and freshness must be treated as operational control inputs rather than secondary instrumentation. To maintain metrics like CTR and conversion in volatile environments, a streaming-aware FL system must satisfy four design pillars. First, it requires low-latency readiness checks of clients that expose not only availability but also current data informativeness without saturating network bandwidth. Second, it must employ rapid, utility-aware selection that fuses a client’s recent data utility with its hardware speed capability to prioritize contributions that most aggressively reduce time-to-target accuracy. Third, it needs delay-robust aggregation semantics to incorporate late, high-quality updates containing ground-truth feedback, which may arrive hours after the initial interaction, without allowing staleness to destabilize the model. Finally, the system must maintain explicit communication bounds to ensure that increased visibility of the client population does not compromise device battery or server scalability."},{"type":"para","text":"Together, these design primitives frame streaming FL as a constrained optimization problem: minimize time-to-target accuracy on a live interaction stream subject to transient availability, evolving utility, frequent update delays, and bounded communication. This formulation serves as the blueprint for the FeLiX primitives presented in § 3 ."},{"type":"heading","level":1,"text":"3. FeLiX System"},{"type":"heading","level":2,"text":"3.1. Gaps to handle widespread unavailability"},{"type":"para","text":"The discussion above has a recurring pattern: existing FL systems and algorithms either (A) lack low-cost, near-real-time availability mechanisms, (B) rely on historic traces or trained predictors that cannot be re-deployed across applications, or (C) handle staleness with heuristics that aggresively discard useful updates. This produces three concrete, non-trivial technical problems that this paper addresses:"},{"type":"para","text":"Compact client availability summaries. How can the server obtain a low-cost, privacy-preserving, actionable view of currently active clients under the application-defined participation criteria? Existing tokens/heartbeats are either too coarse (low utilization) or too costly (frequent reporting); oracle/trace assumptions are unrealistic in new deployments (Bonawitz et al. , 2019 ; Wang et al. , 2023a ) ."},{"type":"para","text":"Fresh and evolving client utility estimates. How can the system efficiently refresh inactive clients’ utilities so that selectors use up-to-date state? Our insight is that utilities can be obtained without full training and using lightweight evaluations or probing, but integrating the probes into selection while limiting communication is challenging (Lai et al. , 2021 ) ."},{"type":"para","text":"Utility-aware and robust aggregation. How to combine updates with varying staleness at the server so that on-and-off but high-value clients are not excluded, while preserving convergence and avoiding undue noise from stale updates (Nguyen et al. , 2022b ; Abdelmoniem et al. , 2023 ) ."},{"type":"para","text":"Motivated by these insigths, we design FeLiX, a system for robust federated learning in environments with high client unavailability and heterogeneity. FeLiX is the first system to introduce and leverage the concept of tiered client pools . Clients dynamically move among the pools based on their evolving state (battery, network, data) and capabilities (compute load, speed) over the long-running FL training jobs. The system can opportunistically select best suited clients for model training, evaluation of their utility only, or neither, based on their runtime availability and resource constraints. This approach aims to maximize each participating client’s contribution to the global model with the goal of minimizing time-to-accuracy and communication costs. FeLiX achieves these goals by augmenting the FL paradigm with three mechanisms, illustrated in Fig. 5 . First is A TierTrack , a lightweight, client-driven protocol to indicate each client’s change of tier to the control-plane. B Second, TierSelect is a two-path selection mechanism: a TrainSelector that prioritizes clients to maximize expected learning each round and an EvalSelector that proactively probes stale or unobserved clients to refresh their utility estimates and surface occasionally high-value but slower devices. Finally, C TierFuse judiciously balances each update’s quality with their staleness. The three mechanisms work in cohesion towards the goals, especially in the presence of high client unavailability and heterogeneity."},{"type":"heading","level":2,"text":"3.2. Streaming-aware availability tiers"},{"type":"para","text":"A systematic solution is required to bridge the gap between the fluctuating and low client availability ( M-Lab, ; Wang et al. , 2023a ; Bonawitz et al. , 2019 ) on one hand and the lack of near-real-time availability tracking in existing systems. As described previously, fresh information about client state is needed to avoid training stalls and to select the high utility clients. Both combined help improve the learning progress. Since the control-plane directly interfaces with the clients, it is most suited to relay latest client information to the selector. However, for the mechanism to be effective, it needs to be real-time , accurate and lightweight , while scaling to thousands of clients."},{"type":"para","text":"TierTrack. Our approach, described in Algorithm 1 , segregates clients into one of the three client pools based on their capability: (i) AVL_TRAIN : available to train and evaluate, (ii) AVL_EVAL : available only to evaluate, and (iii) UN_AVL : unavailable to participate. While different applications can define these states differently, the clients would indicate their state to the control plane based on the criteria and their runtime conditions (compute load, battery status, connectivity, etc). For example, since training is computationally intensive, a client could be classified as AVL_TRAIN only when it is connected to a charger and on Wi-Fi (e.g. OORT, G-FL). However, to tap into under-utilized clients due to this restrictive criteria, we categorize clients into AVL_EVAL state even when they are not connected to a charger, but their battery exceeds a certain threshold, since evaluation is computationally cheaper."},{"type":"para","text":"Instead of periodic heartbeats of current state which is communication heavy, redundant, and unscalable to thousands of clients, the clients proactively notify the control plane whenever their state changes 1 ( Fig. 5 ). To guard against stale or incorrect state information (e.g., when a client silently drops due to network or device failure), FeLiX employs a lightweight check: Instead of separate messages or a regular heartbeat, it starts a 90-second timer whenever the aggregator assigns a task; if no response arrives within this window, the client is marked UN_AVL . If the client becomes UN_AVL during training, it can resume from its last checkpoint once it transitions back to AVL_TRAIN (Gupta et al. , 2024 ; Eisenman et al. , 2022 ; Mohan et al. , 2021 ) ."},{"type":"para","text":"Client state updates are tracked in a simple map data structure that 2 the control plane forwards to the selector. TierTrack maintains additional metadata for each client to recognize clients who rejoin the cohort during the same training round and preclude them from participating in that round. This is done to prevent the selector from sampling the same client more than once in a single round, ensuring fairness (McMahan et al. , 2017 ; Nguyen et al. , 2022b ) ."},{"type":"heading","level":2,"text":"3.3. Tiered Client Selection with Fresh Utilities"},{"type":"para","text":"Instead of picking random client, recent selectors recognize that not all clients contribute equally to global progress. A client’s contribution depends on (i) the richness of its local data (statistical utility) (Chen et al. , 2019 ; Peng et al. , 2019 ; Wang and Joshi, 2019 ; Li et al. , 2020b ) and (ii) the speed with which it returns updates (system utility) (Lai et al. , 2021 ; Huba et al. , 2022 ) . A combined client utility (for example, the product of statistical and system terms) is typically estimated after a client participates in training and then used to guide future selections."},{"type":"para","text":"However, in large, realistic deployments only a small subset of clients is available at any time and clients are not chosen every round. Consequently, stored utilities for non-participating clients become stale and no longer reflect their current potential. Stale utilities induce selection bias. Frequently high-utility clients are repeatedly favored while other, potentially useful clients are neglected, slowing convergence and harming fairness across clients."},{"type":"para","text":"Existing approaches offer only partial remedies. OORT inflates utilities for idle clients to encourage exploration, but this often overestimates inactive clients’ contributions. REFL trains availability predictors and prioritizes less-available clients, but these models must be retrained for every application, remain imperfect despite local adaptation, and mis-select clients during early deployment. As a result, both approaches have limited applicability."},{"type":"para","text":"FeLiX addresses stale utilities efficiently by proactively collecting fresh utility measurements from a dedicated, lightweight eval tier . Rather than rely on inference or delayed training replies, FeLiX obtains up-to-date statistical utilities via short forward-pass evaluations on AVL_EVAL clients and feeds these fresh values into selection decisions."},{"type":"para","text":"At the core of TierSelect is a compact ClientMap that tracks each client’s latest statistical utility, system metrics, and last_utility_round . TierTrack updates pool memberships, while TierSelect runs two concurrent selectors: a high-frequency eval_selector and a lower-frequency train_selector (Fig. 6 ). The eval selector sends lightweight forward-pass evaluations to clients in AVL_TRAIN and AVL_EVAL prioritized by most stale first; these complete up to 20 × 20\\times faster than full training (Xu et al. , 2024 ) , and their returned utilities are written to the ClientMap. The train selector then uses these refreshed utilities to pick top-utility trainers at the set concurrency level. Trainers perform full local training and return model updates and utilities, closing the freshness feedback loop. Since last_utility_round is updated after both evals and training, the ClientMap always holds the freshest client observations."},{"type":"para","text":"The eval selector is deliberately rate-limited (assign at most e e eval tasks per round) to avoid stealing training capacity from available clients. This balances freshness and resource contention: frequent evals refresh utilities but, if unconstrained, could reduce available trainer choices (during eval task execution). Each eval also adds to the communication cost. However, timely fresh utilities substantially reduces selection error and more than amortizes the added eval traffic. For efficiency, selector logic operates only on the currently available sets ( AVL_TRAIN , AVL_EVAL ) reported by TierTrack; this keeps rank computations small and ensures only eligible clients are chosen. The result is a lightweight, fast, and scalable selection strategy that trades off system and data heterogeneity by leveraging fresh client utilities amid high client inactivity and heterogeneity."},{"type":"heading","level":2,"text":"3.4. Utility-aware, delay-robust aggregation"},{"type":"para","text":"Round-based staleness, the gap between the server model version when an update arrives and the version the client trained on is simple to compute and widely used in SyncFL and AsyncFL. However, staleness alone conflates two causes of age: (i) client unavailability and (ii) slow computation or communication. Neither indicates whether an update is still informative. Prior work often discounts updates solely by staleness, discarding delayed yet useful ones (Nguyen et al. , 2022b ; Xie et al. , 2019 ) . FeLiX distinguishes these cases by adding a statistical signal: the utility of an update, derived from the client’s local loss. A high loss implies the model is poorly aligned with that client’s data and the update remains valuable; a low loss suggests the update can be safely down-weighted by age."},{"type":"para","text":"TierFuse encodes the tradeoff between update age and informativeness. Each update u i u_{i} carries its staleness T i T_{i} and local loss L i L_{i} . Two basis functions: staleness decay α ( T ) \\alpha(T) (decreasing in T T ) and utility gain β ( L ) \\beta(L) (increasing in L L ) are combined using a tradeoff parameter σ ∈ [ 0 , 1 ] \\sigma\\!\\in[0,1] :"},{"type":"para","text":"Smooth parameterized forms provide interpretable control:"},{"type":"para","text":"Here a a controls how quickly weight decays with age, b b controls how fast utility saturates, and σ \\sigma sets the balance between the two. This design rewards delayed but informative updates (high L L ) while still damping very old or low-utility ones, balancing quality and stability in aggregation."},{"type":"para","text":"Two safeguards make TierFuse robust in practice. It buffers a short window of recent updates to stabilize normalization and prevent a single extreme update from skewing aggregation. Updates older than a threshold T max T_{\\max} are clipped or dropped, and optional server-side weight clipping or L2 normalization further bound any client’s influence. These lightweight measures require no extra client computation beyond the local loss already produced during training."},{"type":"para","text":"Unlike round-based schemes (e.g., REFL’s round-aware approach), TierFuse explicitly combines update age and statistical utility. Staleness penalizes divergence risk, while high utility rescues delayed but valuable updates. This yields a simple, interpretable policy suited for heterogeneous, unreliable deployments—suppressing obsolete noise without discarding late, informative updates."},{"type":"heading","level":1,"text":"4. Implementation"},{"type":"para","text":"FeLiX is implemented as a compact set of extensions to Cisco’s Flame framework 1 1 1 https://github.com/cisco-open/flame , reusing Flame’s control plane, dispatcher, and transport stack. It has three plugins: TierTrack monitors client activity, capability and maintains lightweight availability pools; TierSelect implements the two parallel selectors (frequent eval selector and less-frequent train selector ) and operates on the small, available-client view provided by TierTrack; and TierFuse is a buffering-and-weighting layer in the optimizer path that applies the mixed staleness/utility weights and emits normalized server updates."},{"type":"para","text":"State is intentionally minimal and colocated in the control plane: a compact ClientMap stores per-client tuples such as (stat_utility, system_metrics, last_utility_round, curr_tier) . Clients piggyback their local loss and round duration on replies so selectors and the aggregator can update ClientMap without extra RPCs. Fast evaluations are implemented as a single forward pass (empirically upto 20X faster than a full training run) and write back refreshed statistical utilities to ClientMap, enabling timely, low-cost updates to selection decisions."},{"type":"para","text":"Aggregation is implemented as a configurable-size buffer plus simple element-wise weighting and normalization. TierFuse reads staleness and loss from the client update, computes weights and produces the server model update; these operations are cheap. To remain agnostic to communication protocol, we extended Flame’s channel backend with the control semantics needed for availability and eval requests (the default deployment uses MQTT), and we keep all extra state in the control plane so that client-side changes are minimal."},{"type":"para","text":"Overall, FeLiX comprises a set of to-be-open-sourced plugins to Flame to deliver TierTrack, TierSelect, TierFuse and ClientMap, all of which cause minimal runtime disruption and low overhead."},{"type":"heading","level":1,"text":"5. Experiment Setup"},{"type":"para","text":"We perform the evaluation in a cross-device FL testbed built on Flame (Daga et al. , 2023 ) with MQTT (Light, 2017 ) . The testbed colocates the FL server (control plane, selector, aggregator) and an emulated population of clients on a multi-GPU node, following prior emulation-based FL studies (e.g., FedScale (Lai et al. , 2022 ) , OORT (Lai et al. , 2021 ) , REFL (Abdelmoniem et al. , 2023 ) ). Experiments sweep heterogeneity along three axes: (i) data, (ii) compute / runtime, and (iii) availability, so the results cover a broad range of realistic operating points. We align Dirichlet parameters, runtime distributions, and availability scenarios with prior work to enable meaningful comparisons."},{"type":"para","text":"Testbed hardware and runtime emulation. The host machine with with 8 NVIDIA A40 GPUs, 500 GB RAM, and an AMD EPYC 7513 32-core CPU runs the FL server and up to 300 emulated clients. Each of the clients are assigned (i) per-round training delays to emulate compute and network differences, and (ii) data partitions for data heterogeneity. Per-round client durations are sampled from a broad distribution spanning ≈ \\approx 1 to 60 s, matching the device-speed and latency distribution reported in production-oriented studies (Papaya (Huba et al. , 2022 ) , FedBuff (Nguyen et al. , 2022b ) )."},{"type":"para","text":"Datasets and models. We evaluate two modalities using two datasets: Image (CIFAR-10) : 60k images, 10 classes; model is a 12-layer CNN ( ≈ \\approx 0.5M parameters) and Speech (Google Speech Commands v2) : ≈ \\approx 105k one-second clips, 35 labels; model is a ResNet-derived speech network ( ≈ \\approx 7.2M parameters). All runs start from the same initialization and use identical configs for parity."},{"type":"para","text":"Client populations, data heterogeneity, and availability. Data heterogeneity is controlled with Dirichlet partitions Dir ( α ) \\mathrm{Dir}(\\alpha) for α ∈ { 0.1 , 1 , 10 , 100 } \\alpha\\in\\{0.1,1,10,100\\} (from strongly non-IID to near-IID). To emulate real-world FL deployments (Ye et al. , 2023 ) , experiments below fix data heterogeneity to α = 0.1 \\alpha=0.1 unless stated otherwise. It is also the most challenging regime to stress availability and FL training. The image task uses 300 clients; the speech task uses 100 clients."},{"type":"para","text":"Availability scenarios include synthetic population-level settings (100%, 90%, 50%) and a real-world MobiPerf trace with very low, bursty participation ( ∼ \\sim 10–22% available). Figure 8 visualizes the synthetic pools and the MobiPerf trace; traces determine per-client availability across the experiment runtime."},{"type":"para","text":"Baselines, selector/aggregator variants, and configuration. We compare FeLiX against existing baseline FL (OORT, REFL), and also against their variants which incorporate some of FeLiX’s features: OORT (SyncFL, unaware of availability), OORT ∗ \\ast (SyncFL + oracle TraceAvail), OORT+Async (OORT extended to AsyncFL), OORT+Async ∗ \\ast (OORT AsyncFL + oracle TraceAvail), REFL ∗ \\ast (trace-tuned availability predictor based selector + stale aggregations in SyncFL), and FeLiX (availability-aware selector + freshness-aware aggregator). For clarity each strategy is characterized along three axes: availability knowledge (none / oracle TraceAvail / predictor / TierTrack), selection policy (OORT / REFL IPS / TierSelect), and aggregation rule (FedAvg SyncFL / REFL SAA / FedBuff AsyncFL / TierFuse), as summarized in Table 1 . The oracle TraceAvail is provided only to certain baselines to bound availability-aware performance; otherwise all strategies share the same per-client runtime/data settings and local training configurations."},{"type":"para","text":"Metrics. Primary metrics are wall-clock time-to-accuracy (target: 70% on both tasks) and total communication (bytes moved). Secondary analyses include client utility-age distributions, per-round stall durations, and update-staleness distributions at the aggregator; these are used to attribute measured performance differences to stalls, utility staleness, or staleness in updates."},{"type":"heading","level":1,"text":"6. Evaluation"},{"type":"para","text":"We seek to answer the following four concrete questions:"},{"type":"para","text":"End-to-end (real-world): Can FeLiX provide tangible time-to-accuracy and communication cost benefits under high-unavailability real-world traces (MobiPerf), across image and speech modalities?"},{"type":"para","text":"End-to-end (synthetic): Do gains from FeLiX only occur in sparse availability or do they hold across scenarios of moderate (50%) to high (100%) availability too? This continues to operate in the maximal data heterogeneity ( Dir ( α ) = 0.1 \\mathrm{Dir}(\\alpha){=}0.1 ) regime. Can FeLiX reduce wall-clock time-to-accuracy and total communication?"},{"type":"para","text":"FeLiX’s components: What are the end-to-end per-component contributions and overheads of TierTrack (real-time availability), TierSelect (train + eval-aware selection), and TierFuse (freshness-aware weighting)?"},{"type":"para","text":"Scalability and robustness: How do FeLiX’ tracking, selection and aggregation perform as clients scale?"},{"type":"heading","level":2,"text":"6.1. End-to-end: Real-World MobiPerf Trace (Q1)"},{"type":"para","text":"Time-to-accuracy. For CIFAR-10 on the bursty and very low availability MobiPerf trace, FeLiX yields an impressive 1.76 × 1.76\\times speedup by reaching the 70% target accuracy versus oracular OORT ∗ \\ast variants and REFL, the best competing strategies. This reduction in wall-clock is due to live availability tracking and task-aware sampling. Not only does this avoid long stalls and reduce blind re-selection, the freshness-aware aggregation keeps useful updates contributing rather than being ignored. For the speech task the same pattern holds qualitatively: FeLiX converges in ≈ \\approx 11.1 h while oracle-assisted OORT variants and REFL finish in ≈ \\approx 15.9–16.3 h ( ≈ \\approx 1.45–1.48X slower). The larger absolute times for speech reflect both model and trace interactions, but the relative improvement of FeLiX remains intact across modalities and under the difficult and fluctuating MobiPerf availability."},{"type":"para","text":"Communication cost. The total communication numbers in Fig. 9 and Fig. 10 must be read together with model sizes and the mix of evaluation and training task messages. CIFAR-10 uses a small model ( ≈ \\approx 0.5M parameters) so it requires a modest 1.9 MB per update sent or received. On the other hand, the speech model is much larger ( ≈ \\approx 7.5M parameters) and needs ≈ \\approx 28.6 MB per exchange. This translates to an order-of-magnitude larger total bandwidth over the run durations. Under the MobiPerf trace conditions, all strategies for speech require a large (2–3 TB) of exchange over the course of 3,000 to 6,000 rounds of aggregations (based on the strategy). In terms of communication cost, FeLiX sits between the OORT+Async ∗ \\ast and pure sync variants ( OORT ∗ \\ast , REFL) in total bytes while still providing the fastest wall-clock times. For CIFAR-10, FeLiX’s total communication (162.6 GB) is still lower than OORT+Async ∗ \\ast (270.0 GB). This gap shows that FeLiX saves a lot communication cost by achieving the target accuracy much sooner by selecting and accepting more helpful client updates early. This includes the extra communication cost of on eval messages ( ≈ \\approx 20.3 GB) that keep utilities fresh. For Speech, eval traffic is more consequential: FeLiX uses ≈ \\approx 583 GB for evaluation tasks because each evaluation task is now 15X costlier due to the larger speech model. This shows that eval-task bandwidth must be tuned or compressed in large-model settings."},{"type":"para","text":"In short, FeLiX consistently reduces time-to-target on low availability, real-world traces for both CIFAR-10 and Speech modality tasks. The time-to-accuracy and communication-cost is a trade-off: faster convergence often means more but more useful message exchanges early on, and evaluation overhead scales with model size. Minimizing end-to-end communication cost therefore requires balancing evaluation tasks and model-size-aware optimizations (e.g., smaller proxy evals or compressed eval uploads) to keep eval overhead small while preserving the wall-clock gains FeLiX delivers."},{"type":"heading","level":2,"text":"6.2. End-to-end: Synthetic Traces (Q2)"},{"type":"para","text":"Time-to-Target Accuracy. Across the synthetic availability traces in Fig. 11 , FeLiX reaches the 70% CIFAR-10 target faster than all baselines, including those with oracular foresight. Relative to the strongest non-oracle strategy ( OORT+Async ), FeLiX improves time-to-accuracy by 1.26 × 1.26\\times at 100% availability, 2.37 × 2.37\\times at 90%, and by > 4 × >4\\times at 50% availability. As availability decreases, the lines for availability-unaware strategies shift toward the bottom-right, with time-to-accuracy increasing from 8 h to over 24 h. This shift reflects the heavy cost of incorrect selections and synchronization stalls. While oracular strategies ( OORT+Async ∗ \\ast ) remain stable near 8 h by avoiding stalls via trace-knowledge, FeLiX actually outperforms these oracles. This performance gain is driven by three synergistic primitives: TierTrack tiering to avoid stalls in real-time, TierSelect’s dual train-and-eval tasks that prioritize utility even in high-availability regimes, and TierFuse, which extracts informative signal from late updates that other systems would simply discard."},{"type":"para","text":"Communication Efficiency and the Cost of Staleness. FeLiX significantly reduces the total communication required to reach target accuracy. When compared to the oracular OORT+Async ∗ \\ast , FeLiX saves approximately 40.2 % 40.2\\% of total bandwidth. Notably, as unavailability increases, FeLiX’s total communication cost actually falls by ≈ 9.7 % \\approx 9.7\\% , as TierSelect dispatches fewer evaluation tasks when client utility remains fresher for longer."},{"type":"para","text":"The comparison with REFL ∗ \\ast reveals a significant efficiency gap. Although REFL ∗ \\ast is a SyncFL variant, it maintains a large set of trainers ”in-flight” to accommodate delayed updates and maximize throughput. However, its rigid policy of discarding any update staler than 5 rounds creates a significant communication overhead. Empirically, we observe that effectively 25% of REFL’s trainer-to-aggregator uplink traffic is discarded for exceeding this threshold. This is not merely a waste of device uplink traffic; it also represents a loss of the aggregator-to-trainer downlink bandwidth used to transmit the model weights initially. Furthermore, this aggressive pruning harms the model itself, as informative updates from the long-tail of slower or transiently unavailable clients are lost. By contrast, FeLiX employs freshness-aware aggregation to incorporate these updates safely, ensuring that all network communication cost incurred by the system contributes to the final convergence. This allows FeLiX to beat REFL not only in wall-clock time but also in total communication cost across all availability regimes."},{"type":"heading","level":2,"text":"6.3. FeLiX Component Benefits (Q3)"},{"type":"para","text":"To decompose the end-to-end gains reported in § 6.2 and § 6.1 , we inspect three diagnostic metrics at runtime: aggregator stall duration, client utility age (the latency between a client’s last evaluation and its current selection), and update staleness at the point of aggregation. These traces expose the specific failure modes that affect existing strategies. SyncFL variants without live telemetry suffer from long-tail round durations as they wait on dropped or straggling clients. Selection algorithms that do not proactively refresh utility estimates suffer from utility drift, repeatedly re-selecting historically high-value clients that are currently unavailable or whose data distribution has shifted. Finally, standard AsyncFL rules either discard valuable-but-late updates or apply a uniform staleness penalty, both of which erode the statistical signal. We now show how each FeLiX component resolves these failures through a tightly coupled orchestration loop."},{"type":"para","text":"TierTrack: Mitigating the impact of client inactivity. Integrating TierTrack into the orchestration loop sharply reduces tail latencies by enabling immediate resampling when selected clients become unreachable. In our CIFAR-10 experiments, augmenting a stalled OORT+Async baseline with TierTrack cuts the median (P50) round duration from 258.9s to just 4.9s, while the P95 tail latency drops from 653.3s to 21s (Fig. 12 ). This orders-of-magnitude improvement occurs because TierTrack operates on live telemetry rather than static traces; it can detect mid-round drop-offs—which affected 6–8% of all selections in our traces—and trigger an immediate replacement. Consequently, OORT+Async +TierTrack achieves a 5–7% lower time-to-accuracy than even the trace-informed oracle ( OORT+Async ∗ \\ast ), which can avoid stalls but lacks the mechanism to reactively resample when a selected client disappears mid-computation."},{"type":"para","text":"TierSelect: Maintaining utility freshness via dual-tier selection. TierSelect decouples selection into a TrainSelector, which targets high-progress, responsive clients, and an EvalSelector, which acts as a ”scout” by issuing lightweight evaluation tasks to clients with stale or missing utility estimates. This dual-tier approach ensures that the server’s view of the population remains accurate despite bursty availability. As shown in Fig. 13 , the TrainSelector consistently prioritizes clients with higher statistical utility (median score of 122.0 vs 118.1 for OORT), even when those clients are marginally slower (median compute of 11.2s vs 10.9s). This trade-off is mathematically advantageous: by selecting more informative gradients, the model requires fewer total rounds to reach the target accuracy. Simultaneously, the EvalSelector probes low-utility trainers and reveals their true hardware profiles (P99 < < 100), identifying that these clients are significantly slower (median 15.7s). This fresh evidence prevents the system from making blind, repeated re-selections of low-yield or hardware-constrained devices."},{"type":"para","text":"TierFuse: Regaining information from delayed updates. Rather than applying a destructive staleness penalty or discarding late updates, TierFuse computes a contribution factor that balances a client’s specific statistical utility against its update age. This allows the system to incorporate informative updates from the ”long-tail” of the edge ecosystem that would otherwise be lost. As illustrated in Fig. 15 , TierFuse assigns high weights ( ≥ 0.8 \\geq 0.8 ) to over 90% of received updates, preserving their informative signal even when they arrive moderately late. In contrast, a FedBuff-style policy, which lacks this informativeness-aware weighting, assigns such high weights to fewer than 10% of updates, often falling back to a flat moderate weight of ≈ \\approx 0.5. By effectively protecting high-value updates from aggressive staleness discounting, TierFuse increases the per-update usefulness of every byte transmitted, accelerating convergence under the erratic conditions of streaming FL."},{"type":"para","text":"FeLiX: Combined synthesis of components. These three components are fundamentally intertwined: TierTrack eliminates stalls to ensure that EvalSelector’s probes reflect actual availability; EvalSelector maintains fresh utility estimates so TrainSelector can accurately prioritize informative clients; and TierFuse ensures that once an update is received—whether fresh or moderately stale—its weight reflects its true contribution to the global model. Together, they transform availability and freshness from passive observations into active control signals, explaining the substantial wall-clock and communication gains observed in production-scale settings."},{"type":"heading","level":2,"text":"6.4. Scalability and robustness (Q4)"},{"type":"para","text":"Selector and decision latency. FeLiX is designed to minimize server-side overhead by offloading availability tracking to the edge. Per-decision compute latency remains consistently low ( < 0.5 s <0.5s ) because the server does not perform an exhaustive, synchronous scan of the entire population. Instead, the client-driven tiering ensures the server only ranks the active, ready subset of the population, while the EvalSelector asynchronously estimates live client utility in the background. This decoupling allows the orchestration logic to scale gracefully; as the population increases from 10 to 300 concurrent clients, we observe no significant increase in selection latency or congestion at the MQTT broker."},{"type":"para","text":"Robustness to heterogeneity. We evaluate the resilience of FeLiX across varying degrees of statistical and system heterogeneity by sweeping Dirichlet distribution parameters ( α ∈ { 0.1 , 1 , 10 , 100 } \\alpha\\in\\{0.1,1,10,100\\} ) and runtime compute distributions (1 to 60 seconds). As expected, the relative gains of FeLiX are most pronounced in the strongly non-IID regimes ( α = 0.1 \\alpha=0.1 ) and low-availability settings. These represent the most challenging and deployment-relevant scenarios where standard FL systems typically collapse. In highly heterogeneous environments, the opportunity cost of a poor selection is high; missing a transiently available client with high-utility data results in a significant loss of gradient progress. FeLiX’s ability to fuse live availability with freshness-aware aggregation allows it to navigate these sparse availability environments effectively, ensuring that every successful update provides maximum statistical gain. This robustness demonstrates that FeLiX is not merely faster in idealized settings, but fundamentally more resilient to the adverse conditions of production edge ecosystems."},{"type":"heading","level":1,"text":"7. Related Work"},{"type":"para","text":"FL Applications. FL was introduced by McMahan et al. (McMahan et al. , 2017 ) as FedAvg, and (Kairouz et al. , 2021 ; Yang et al. , 2019 ) survey its landscape and open challenges. Beyond the production recommender and analytics workloads this paper targets, FL has been applied to mobile keyboard prediction (Hard et al. , 2018 ; Chen et al. , 2019 ) , on-device language modeling (Xu et al. , 2023 , 2024 ) , healthcare analytics (Brisimi et al. , 2018 ; Li et al. , 2019a ; Xu et al. , 2021 ; Nguyen et al. , 2022a ) , and distribution-shifted domains (Peng et al. , 2019 ) . Privacy threats including membership inference (Shokri et al. , 2017 ) and client-level differential privacy (Geyer et al. , 2017 ) motivate the data-local design that FeLiX inherits; its orchestration layer is orthogonal to these protections."},{"type":"para","text":"FL Convergence, Heterogeneity, and Optimization. FedAvg convergence under non-IID data (Li et al. , 2019b ; Wang et al. , 2020b ) identifies data heterogeneity and partial participation as primary theoretical obstacles; heterogeneous FL settings are surveyed in (Ye et al. , 2023 ) . Adaptive server-side optimizers (Reddi et al. , 2020 ) , matched averaging (Wang et al. , 2020a ) , ensemble distillation (Lin et al. , 2020 ) , and Bayesian non-parametric fusion (Yurochkin et al. , 2019 ) improve convergence under heterogeneous data; one-shot FL (Li et al. , 2020a ) eliminates iterative rounds entirely. Communication efficiency is targeted by gradient compression (Konečnỳ et al. , 2016 ) , distributed mean estimation (Suresh et al. , 2017 ) , and adaptive communication scheduling (Wang and Joshi, 2019 ) . FeLiX’s weighted aggregation policy is complementary to all of these: it targets update informativeness and model-age correction rather than modifying the fusion or compression scheme."},{"type":"para","text":"Client Availability, Selection, and AsyncFL. Section 2 analyzes in detail why the leading utility-aware selectors OORT (Lai et al. , 2021 ) and REFL (Abdelmoniem et al. , 2023 ) , buffered AsyncFL (Nguyen et al. , 2022b ; Zhang et al. , 2023 ) , and production FL systems (Bonawitz et al. , 2019 ; Srinivas et al. , 2025 ; Wang et al. , 2023a ) fall short under real-time availability churn. More broadly, async FL has been studied from decentralized SGD and momentum perspectives (Lian et al. , 2018 ; Xie et al. , 2019 ; Mitliagkas et al. , 2016 ) , and extended to continual and non-IID settings (Shenaj et al. , 2023 ; Chen et al. , 2020 ) . Staleness handling has received dedicated attention (Zhou et al. , 2022 ; Rodio and Neglia, 2024 ) ; variance from partial participation is targeted by FedVARP (Jhunjhunwala et al. , 2022 ) and large-cohort analysis (Charles et al. , 2021 ) . Nishio and Yonetani (Nishio and Yonetani, 2019 ) select clients by deadline compliance under resource heterogeneity. Prior work (Garg et al. , 2025 ) empirically shows that ignoring real-time availability substantially increases time-to-accuracy, directly motivating the TierTrack tracking design in FeLiX."},{"type":"para","text":"Client Interruption and Local Checkpointing. A natural response to mid-training client interruptions is to checkpoint local model state, enabling resumption after conditions recover—a technique well-developed for distributed DNN training (Mohan et al. , 2021 ; Eisenman et al. , 2022 ; Gupta et al. , 2024 ; Wang et al. , 2023b ) . In the FL context, however, local checkpointing only defers the problem: the client still cannot return any update until it becomes available again, and the global model continues to advance in the interim, rendering saved gradients stale or model-incompatible on return. FeLiX addresses the root cause instead, by exploiting lightweight real-time readiness signals to harvest updates before interruptions occur and by weighting late-arriving updates by their informative value rather than discarding or blindly accumulating them."},{"type":"para","text":"FL Systems, Frameworks, and Benchmarks. Production FL platforms (Bonawitz et al. , 2019 ; Huba et al. , 2022 ; Srinivas et al. , 2025 ) address large-scale cross-device deployments; FLINT (Wang et al. , 2023a ) provides on-device measurement; FedScale (Lai et al. , 2022 ) benchmarks system performance with realistic traces. Research frameworks Flower (Beutel et al. , 2020 ) and FedML (He et al. , 2020 ) support controlled experimentation. FeLiX is built on the Flame orchestration layer (Daga et al. , 2023 ) and evaluated against real MobiPerf traces ( M-Lab, ) ; unlike offline benchmark or framework tools, it targets production-scale streaming workloads with bursty, minute-level availability dynamics."},{"type":"heading","level":1,"text":"8. Conclusion"},{"type":"para","text":"As production Federated Learning moves from multi-day batch refreshes toward hourly streaming adaptation, existing orchestration models have proven insufficient. The interlocking challenges of transient client availability, shifting data utility, and delayed ground-truth feedback create a freshness gap that degrades critical user-facing metrics. This work introduced FeLiX , a redesign of FL orchestration that treats availability and utility not as passive observations, but as high-frequency control inputs. By unifying three synergistic primitives: streaming-aware availability tiers , fresh-utility selection , and informativeness-aware, delay-robust aggregation , FeLiX bridges the gap between idealized oracular theory and the realities of the practical FL deployments in the mobile-edge ecosystem. Our evaluation across multiple modalities and realistic MobiPerf traces demonstrates that FeLiX achieves near-oracular performance, delivering a 2.37 × 2.37\\times reduction in time-to-target accuracy while simultaneously saving 1.30 × 1.30\\times in total communication bandwidth compared to state-of-the-art baselines. These results confirm that by eliminating synchronization stalls and reclaiming signal from the long-tail of delayed updates, FeLiX ensures that deployed models remain in sync with volatile live data distributions. Ultimately, FeLiX provides the necessary framework for high-scale, privacy-preserving applications to adapt at the speed of user interaction."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Federated Learning (FL) enables training shared models across large populations of user devices without collecting the raw data centrally. This paradigm has unlocked machine learning on private, on-device interaction logs that were previously inaccessible to centralized trainers (McMahan et al. , 2017 ; Bonawitz et al. , 2019 ; Kairouz et al. , 2021 ) . Such privacy-preserving decentralization is now critical for various production applications, including personalized recommenders, feed-ranking, and session-aware input prediction. FL has been deployed for mobile keyboard and language prediction (Hard et al. , 2018 ; Chen et al. , 2019 ; Xu et al. , 2023 ) , privacy-preserving healthcare analytics (Brisimi et al. , 2018 ; Xu et al. , 2021 ; Nguyen et al. , 2022a ) , and large-scale model personalization (Yang et al. , 2019 ; Xu et al. , 2024 ) , establishing it as the de facto approach for privacy-sensitive collaborative training across diverse domains. Historically, these deployments focused on batched, multi-day refreshes where modest delays between data collection and model updates were acceptable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Contemporary on-device applications, however, increasingly require fresh models that adapt on hourly timescales. User interaction streams are volatile: viral content, news events, promotional campaigns, and short-lived user intents can shift predictive distributions rapidly. When model refresh intervals lag behind these shifts, predictive relevance degrades and key application metrics such as click-through rates and downstream conversions suffer (Matam et al. , 2024 ) . Meeting these goals demands FL systems that minimize time-to-target accuracy on live streams while operating at production scale.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Transitioning from slow and batched to agile and streaming FL is difficult because near-real-time updates break the core assumptions of existing FL systems. We highlight three interlocking constraints that render naive extensions of current architectures ineffective:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(1) Transient client availability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mobile and edge devices are intermittently reachable due to battery, network, and user behavior. Availability fluctuates at minute-level granularity and exhibits heavy churn ( M-Lab, ) . As shown in (Garg et al. , 2025 ) , existing mitigations generally fail in three ways: (a) Over-selection , where requesting excess clients to participate in each round wastes device compute and bandwidth while increasing server-side update variance; (b) blocking synchrony , which converts per-round latencies from seconds to minutes when aggregation goals are not met; and (c) coarse periodic probing , which either misses short-lived availability windows of useful clients or incurs high heartbeat overhead to maintain up-to-date status. These approaches trade away either time-to-accuracy or communication budgets, making them unsuitable for large-scale FL deployments operating with thousands of clients per round. Approaches that target the statistical variance from partial participation (Jhunjhunwala et al. , 2022 ) or study large-cohort selection effects (Charles et al. , 2021 ) improve efficiency but still assume a predictable pool of available clients and do not address minute-level churn.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(2) Evolving client utility and heterogeneity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Clients differ not only in their hardware capabilities (compute speed, memory, network), but also in the statistical value of their recent data. We define client utility as a time-varying quantity combining data informativeness for the live distribution with the client’s expected compute latency. Existing selection algorithms often ignore these dynamics (Nguyen et al. , 2022b ; Bonawitz et al. , 2019 ; Nishio and Yonetani, 2019 ) , while newer ranking methods rely on the ”oracular” knowledge of availability, which is impractical in deployment. Not just that, they don’t account for the shifting data distributions of streaming tasks (Lai et al. , 2021 ; Abdelmoniem et al. , 2023 ; Ye et al. , 2023 ) . Selectors that rank clients using utility estimates derived from infrequent probes or historical traces systematically mis-rank participants, as these signals quickly become stale and introduce bias that can outweigh their intended benefits. For example, it may over-select fast devices with irrelevant data while neglecting slow devices with fresh, informative signals. Such techniques either require non-existent oracular deployment traces, coarse-grained and often stale utility estimates, or frequent probing that incurs prohibitive overhead. As a result, they suffer rapid degradation in time-to-accuracy with shifts in utility.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(3) Model freshness vs. delayed updates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In production tasks, the strongest training signal (the ground-truth label) often arrives with some delay; for instance, a recommendation’s conversion may be observed hours after the interaction. This creates a fundamental trade-off: updating immediately with proxy feedback maximizes freshness but introduces noise, while waiting for full feedback improves label quality at the cost of staleness. Existing strategies—like buffered asynchronous aggregation (Nguyen et al. , 2022b ) or staleness-weighting schemes (Xie et al. , 2019 ; Rodio and Neglia, 2024 ; Zhou et al. , 2022 ) —force a choice between convergence speed and statistical robustness. Incorrect aggregation can destabilize the training process, inflating the wall-clock time required to reach target accuracy or discarding valuable, though delayed, signals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Taken together, these constraints explain why existing synchronous and asynchronous FL systems fail to achieve reasonable time-to-target accuracy in streaming settings, at production scale as shown in Fig. 1 . Synchronous schemes stall or over-select; asynchronous schemes avoid blocking but accumulate staleness that degrades convergence (Xie et al. , 2019 ; Lian et al. , 2018 ; Li et al. , 2019b ) ; assumed oracular knowledge of availability or heavy-weight predictors are not tractable; and prohibitive communication costs of frequent probes derail performance. Empirical studies (see § 2 and Fig. 4 ) show that even modest unavailability or data heterogeneity can worsen the model adaptation time as the complexity of the deployment shifts to hardened production settings. Not just that, it can substantially degrade performance, even to the point of non convergence, as the unavailability and heterogeneity increase.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we frame streaming FL as a constrained optimization problem: minimize time-to-target accuracy on a live interaction stream subject to transient availability, evolving per-client utility, frequent update delays, and bounded communication overhead. This objective focuses our design on policies that treat availability and utility as low-latency control inputs while ensuring scalability across massive populations, without assuming oracular knowledge of client behaviour.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present FeLiX , a redesigned FL orchestration framework that directly optimizes time-to-target accuracy in streaming production settings. FeLiX treats availability as an evolving signal and couples inexpensive freshness checks with selection logic to optimize convergence. Our core primitives are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Streaming-aware availability tiers: FeLiX organizes clients into dynamic readiness tiers based on lightweight signals (battery, connectivity). This leverages a client-server design that minimizes network overhead while ensuring the server possesses high-fidelity information on client capabilities within tight refresh windows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fresh-utility based selection: We introduce a dual-tier approach where an evaluation tier performs inexpensive forward passes to estimate data informativeness. This is fused with compute speed to prioritize clients that are both hardware-performant and reflect the most relevant recent distribution shifts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Informativeness-aware, delay-robust aggregation: FeLiX employs weights that balance update informativeness with model-age corrections. This allows the system to safely incorporate late, high-value updates with full feedback, ensuring the global model remains robust without being stalled by the long-tail latencies of the straggler or long unavailable clients.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conduct an exhaustive evaluation of FeLiX using real-world MobiPerf traces and multi-modal datasets including CIFAR-10 (Krizhevsky, 2009 ) and Google Speech Commands V2 (Warden, 2018 ) . We demonstrate that FeLiX achieves a 2.37 × 2.37\\\\times reduction in time-to-target accuracy compared to recent Synchronous and Asynchronous FL baselines, while limiting communication overhead and preserving the predictive power of fresh models.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Background and Motivation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1. FL Primer and Client Availability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Asynchronous FL. Federated Learning (FL) trains a shared model across many decentralized, privacy-sensitive clients by keeping raw data local: the server broadcasts global weights, clients perform local training on-device, and the server aggregates returned updates (McMahan et al. , 2017 ; Bonawitz et al. , 2019 ) . Two execution regimes dominate cross-device deployments. In synchronous FL (SyncFL) the server advances in rounds after collecting an agg-goal number of client updates; in asynchronous FL (AsyncFL) the server incorporates updates as they arrive and does not block on stragglers. SyncFL offers conceptual simplicity and bounded staleness but is vulnerable to delays caused by a few slow or unavailable clients; AsyncFL trades that blocking for continuous progress at the cost of needing principled staleness handling. Convergence analyses of FedAvg under non-IID data (Li et al. , 2019b ; Wang et al. , 2020b ) reveal that data heterogeneity exacerbates gradient divergence across clients, while adaptive federated optimizers (Reddi et al. , 2020 ) partially compensate but leave the availability challenge unaddressed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AsyncFL emerged to address the wide runtime variability observed in practice: theoretical analyses of asynchronous decentralized SGD (Lian et al. , 2018 ) and asynchronous federated optimization (Xie et al. , 2019 ) confirm that async updates converge under mild staleness bounds but can degrade sharply when updates are very stale or derived from strongly non-IID local data. mobile devices exhibit long-tailed training runtimes driven by differences in compute, background load, and network latency as shown in Fig. 2 . Together, device heterogeneity and application-driven data generation produce a rapidly evolving trainer runtime distribution. This variability forces system designers to confront three co-occurring concerns: (i) which clients should the server select at any moment, (ii) how to cheaply detect that a client is actually ready to train, and (iii) how to incorporate late or intermittent updates without destabilizing learning. A compact AsyncFL control loop (selector, weight dispatcher, client training, aggregator/buffer) exposes these decision points; we illustrate the flow in Fig. 3 . Clients register themselves with the FL-server 1 . The control-plane maintains these connections and forwards the client-id(s) to the selector 2 . The selector is tasked with maintaining a desired training concurrency ( i.e. , number of parallely training clients) throughout the process. Whenever the training count falls below the concurrency level ( i.e. , clients return updates). The selector uses its algorithm to return the next client(s) to train 4 so as to meet the concurrency level. These selected client ID(s) are forwarded to the aggregator. 5 The weight dispatcher sends the latest server model weights (W) to the selected clients. If the client is available to train, it begins training using fresh global weights (W) and data from the applications’ local data-store. Upon completion, it returns weight changes w.r.t W as updates ( Δ W \\\\Delta W ) 6 to the aggregator. If the client wasn’t initially available to train or is interrupted while training, the client waits until it becomes available next. At the FL server, the model updates received from clients are aggregated and put into a temporary buffer. Once agg-goal number of updates in the buffer are aggregated, the server model version changes and the buffer is emptied 7 . This FL training process continues until convergence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evolving runtime availability. While foundational FL research and early system prototypes often assumed highly available clients operating in stable, conducive environments, production deployments reveal that this assumption is fundamentally flawed. In practice, ignoring the reality of intermittent participation degrades performance significantly, inflating time-to-accuracy and wasting system resources. We define availability not merely as a device being powered on, but as a multi-dimensional state meeting strict system-level and application-level filters, including WiFi-only connectivity, sufficient battery levels, and OS-mandated idle constraints. Consequently, unavailability is the norm; production traces from the Google FL System (Bonawitz et al. , 2019 ) , the PAPAYA analytics platform (Srinivas et al. , 2025 ) , LinkedIn’s FLINT (Wang et al. , 2023a ) , and FedScale/MobiPerf (Lai et al. , 2022 ; M-Lab, ) report mean client availability ranging from as low as 15% to roughly 50% of the total population. In production, unavailability is therefore the norm rather than the exception. This participation is inherently transient , characterized by short-lived ready windows whose durations are often comparable to the training round itself. Across different geographies and applications, these dynamics manifest in large peak-to-trough swings. For instance, diurnal effects can cause availability to fluctuate by 5 × 5\\\\times within a single day, with minima dropping to 10% during active usage hours and peaking at 80% when devices are idle and charging. At the system level, these trends create significant orchestration challenges: in the Google FL system, nearly 22% of updates arrive too late for aggregation and are rejected, while another 2% are interrupted mid-computation. Restrictive participation criteria, while necessary for user experience, further contract the eligible pool, sometimes as low as 22% of the total devices as shown in MobiPerf ( M-Lab, ) . This exposing a sharp trade-off between model fairness and system throughput. These bursty and unpredictable availability patterns mean that naive selection strategies either over-select to mask dropouts (increasing variance) or stall rounds waiting for stragglers, both of which are untenable for streaming applications requiring hourly refreshes. Prior measurements (Garg et al. , 2025 ) directly quantify this: even under moderate unavailability ( < < 50%), time-to-target accuracy degrades sharply, confirming that availability must be treated as a first-class orchestration signal rather than a background metric.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AsyncFL alone cannot fix unavailability. These runtime and availability characteristics translate into three practical operational effects that motivated AsyncFL and that now motivate streaming-aware controls. First, servers often resort to over-selection (invite many more candidates than needed) to probabilistically meet an agg-goal; this wastes device CPU, uplink/downlink network bandwidth and increases variance in server-side latency. Second, a client’s local training time may exceed the mean aggregator interval, increasing the chances of interruptions, rejected updates, or partial work. A natural mitigation is to checkpoint local training state so that a client can resume after an interruption: an approach well-studied in distributed DNN training (Mohan et al. , 2021 ; Eisenman et al. , 2022 ; Gupta et al. , 2024 ; Wang et al. , 2023b ) . However, in the FL context, checkpointing only preserves progress on the client ; the client still cannot return any update until device conditions recover, and the global model advances in the meantime, rendering the saved gradients stale on return. Checkpointing therefore does not close the availability gap. It defers the problem rather than solving it. Third, restrictive eligibility filters shrink the effective population and risk selection bias. Together, these effects show that availability and runtime behavior are not background metrics but low-latency control inputs that selection and aggregation logic must use in order to optimize wall-clock progress and maintain model quality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, while AsyncFL and its improvements such as update buffering reduce blocking, they do not by themselves solve the coupled problems of rapid availability churn, evolving per-client statistical value, and bounded communication costs. For example, buffered AsyncFL reduces immediate stalls but requires careful eviction/weighting policies to prevent stale updates from hurting convergence. Later sections discuss these algorithmic and systems tensions in detail and show why treating readiness and freshness as first-class, lightweight signals is essential for streaming FL.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2. FL Orchestration Challenges\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Brittle selection algorithms. Many FL selection strategies attempt to balance synchronization and participation, but practical deployments expose a harsh trade-off between update staleness and system stalls. Synchronous FL variants avoid staleness by enforcing global rounds but are highly vulnerable to stalls : when availability is intermittent, SyncFL blocks on stragglers or turns to heavy over-selection, converting per-round latencies from seconds into minutes and dramatically increasing communication and cost. Asynchronous and buffered schemes relax blocking by accepting updates computed on older model states (or by aggregating along a buffer), which improves throughput. However, if stale updates are naively aggregated, they will destabilize convergence. Conversely, discarding late updates sacrifices the valuable statistical signal carried by the slower or transiently available clients. Recent selector-based approaches that prioritize high-utility clients (for example, OORT and REFL selectors) or use learned predictors to estimate client utility aim to trade off speed and signal by favoring fast, high-utility workers. In practice, these methods typically depend on availability signals or historical traces to estimate utility. Under bursty availability and rapid population/data shifts those utility estimates age quickly leading to repeated mis-ranking. For example, such selecots would over-select fast but irrelevant devices and neglect informative but ephemeral clients. These limitations have motivated hybrid approaches like buffered asynchronous aggregation that attempt to retain long-tail information while reducing stalls, but the tension between timeliness, stability, and privacy-compatible aggregation remains unresolved. Variance-reduction techniques for partial participation (Jhunjhunwala et al. , 2022 ) and large-cohort selection (Charles et al. , 2021 ) target statistical stability but still assume a stable, predictable client pool. Staleness-aware schemes (Rodio and Neglia, 2024 ; Zhou et al. , 2022 ) weight or discount late updates based on model-version gap, but do not resolve the challenge of detecting and exploiting short availability windows in real time.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Coarse systems and instrumentation. Production stacks provide several primitives for large-scale FL, but their availability signals tend to be either too coarse to support fine-grained orchestration or too heavyweight to scale. Such platforms use techniques like client check-ins, tokens, and over-selection heuristics to mask dropout and reduce orchestration complexity with client availabilty. Yet such protocols are explicitly designed around coarse-grained and slower tradeoffs like accept/reject tokens, windowed selection rather than continuous, low-latency availability views. This design choice simplifies robustness but also means a nontrivial fraction of potential updates are delayed or rejected in practice. At the other extreme, tooling that instruments devices more deeply like pre-deployment measurement, capability profiling, or continuous heartbeat reporting, can provide the fine-grained, near-real-time availability signals orchestration needs, but at production scale these signals increase reporting overhead, consume device energy, and raise privacy and operational concerns. Benchmark and integration frameworks make clear how availability and distribution shift in deployment. They also let researchers replay realistic, time-varying traces for evaluation. However, by definition these offline traces cannot supply the compact, cross-application, real-time view an orchestrator would need for streaming, hourly refresh tasks. In short, production systems either expose high-fidelity signals that are too heavy to maintain continuously, or coarse signals that force servers to choose between blocking (stalling progress) and proceeding blindly (accumulating staleness). Representative system and benchmarking efforts that highlight these tradeoffs include Google FLSys (Bonawitz et al. , 2019 ) and the PAPAYA analytics stack (Srinivas et al. , 2025 ) , which describe token/check-in and rejection mechanics for large-scale mobile FL; the benchmark suite FedScale (Lai et al. , 2022 ) ; and the FL integration tooling FLINT (Wang et al. , 2023a ) , which emphasizes on-device measurement and pre-deployment instrumentation. Research frameworks such as Flower (Beutel et al. , 2020 ) and FedML (He et al. , 2020 ) enable flexible experimentation but are designed for controlled settings rather than production-scale streaming deployments with bursty, minute-level availability dynamics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These algorithmic and system-level gaps are particularly acute for streaming applications , where the requirement for hourly refreshes clashes with high data heterogeneity and delayed ground-truth feedback. In production tasks like recommendation or ad-targeting, the strongest training signal (the label) often arrives hours after the initial interaction. Existing systems are unable to navigate this: they must either update immediately using noisy proxy feedback to maintain freshness, or wait for full labels and suffer extreme model staleness.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3. Model Adaptation Gap: Oracular vs. Reality\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Opportunity Gap. To quantify the performance gap with existing FL approaches, we move from an idealized regime to progressively more realistic deployment settings in Fig. 4 , exposing the technical reasons behind the widening performance gap. While an idealized baseline of 100% availability and homogeneous data in Line 1 i.e. L1 ensures minimal time-to-target accuracy, the introduction of data heterogeneity (L2) increases gradient variance, increasing the time-to-accuracy even under 100% participation. The performance is expected to degrade sharply with realistic client behaviour, for instance if client availability drops to 50%. Here rounds will stall and existing selectors that fail to capture enough informative updates within tight refresh time windows, will suffer. However, with oracular or trace-provided client information, despite 50% unavailability, L3 does not drop in its performance significanlty compared to L2. The most significant performance collapse occurs when we withdraw the oracular foresight. Many state-of-the-art methods like OORT and REFL rely on retrospective traces or trace-derived availability predictors to estimate future utility (L4). Without perfect future knowledge, these systems struggle with low availability, repeatedly stalling during rounds. Even with over-selection, they fail to capture the transiently available clients that carry high-value for model training. Bridging this gap (L5) requires an orchestrator that achieves near-oracular performance not through foresight, but by exploiting low-cost, real-time readiness signals to harvest the high-value updates as they appear.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Design Primitives for Streaming FL. Closing the empirical gap between Line 4 and Line 5 requires a shift in FL orchestration infrastructure: availability and freshness must be treated as operational control inputs rather than secondary instrumentation. To maintain metrics like CTR and conversion in volatile environments, a streaming-aware FL system must satisfy four design pillars. First, it requires low-latency readiness checks of clients that expose not only availability but also current data informativeness without saturating network bandwidth. Second, it must employ rapid, utility-aware selection that fuses a client’s recent data utility with its hardware speed capability to prioritize contributions that most aggressively reduce time-to-target accuracy. Third, it needs delay-robust aggregation semantics to incorporate late, high-quality updates containing ground-truth feedback, which may arrive hours after the initial interaction, without allowing staleness to destabilize the model. Finally, the system must maintain explicit communication bounds to ensure that increased visibility of the client population does not compromise device battery or server scalability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Together, these design primitives frame streaming FL as a constrained optimization problem: minimize time-to-target accuracy on a live interaction stream subject to transient availability, evolving utility, frequent update delays, and bounded communication. This formulation serves as the blueprint for the FeLiX primitives presented in § 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. FeLiX System\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1. Gaps to handle widespread unavailability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The discussion above has a recurring pattern: existing FL systems and algorithms either (A) lack low-cost, near-real-time availability mechanisms, (B) rely on historic traces or trained predictors that cannot be re-deployed across applications, or (C) handle staleness with heuristics that aggresively discard useful updates. This produces three concrete, non-trivial technical problems that this paper addresses:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Compact client availability summaries. How can the server obtain a low-cost, privacy-preserving, actionable view of currently active clients under the application-defined participation criteria? Existing tokens/heartbeats are either too coarse (low utilization) or too costly (frequent reporting); oracle/trace assumptions are unrealistic in new deployments (Bonawitz et al. , 2019 ; Wang et al. , 2023a ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fresh and evolving client utility estimates. How can the system efficiently refresh inactive clients’ utilities so that selectors use up-to-date state? Our insight is that utilities can be obtained without full training and using lightweight evaluations or probing, but integrating the probes into selection while limiting communication is challenging (Lai et al. , 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Utility-aware and robust aggregation. How to combine updates with varying staleness at the server so that on-and-off but high-value clients are not excluded, while preserving convergence and avoiding undue noise from stale updates (Nguyen et al. , 2022b ; Abdelmoniem et al. , 2023 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Motivated by these insigths, we design FeLiX, a system for robust federated learning in environments with high client unavailability and heterogeneity. FeLiX is the first system to introduce and leverage the concept of tiered client pools . Clients dynamically move among the pools based on their evolving state (battery, network, data) and capabilities (compute load, speed) over the long-running FL training jobs. The system can opportunistically select best suited clients for model training, evaluation of their utility only, or neither, based on their runtime availability and resource constraints. This approach aims to maximize each participating client’s contribution to the global model with the goal of minimizing time-to-accuracy and communication costs. FeLiX achieves these goals by augmenting the FL paradigm with three mechanisms, illustrated in Fig. 5 . First is A TierTrack , a lightweight, client-driven protocol to indicate each client’s change of tier to the control-plane. B Second, TierSelect is a two-path selection mechanism: a TrainSelector that prioritizes clients to maximize expected learning each round and an EvalSelector that proactively probes stale or unobserved clients to refresh their utility estimates and surface occasionally high-value but slower devices. Finally, C TierFuse judiciously balances each update’s quality with their staleness. The three mechanisms work in cohesion towards the goals, especially in the presence of high client unavailability and heterogeneity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2. Streaming-aware availability tiers\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A systematic solution is required to bridge the gap between the fluctuating and low client availability ( M-Lab, ; Wang et al. , 2023a ; Bonawitz et al. , 2019 ) on one hand and the lack of near-real-time availability tracking in existing systems. As described previously, fresh information about client state is needed to avoid training stalls and to select the high utility clients. Both combined help improve the learning progress. Since the control-plane directly interfaces with the clients, it is most suited to relay latest client information to the selector. However, for the mechanism to be effective, it needs to be real-time , accurate and lightweight , while scaling to thousands of clients.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TierTrack. Our approach, described in Algorithm 1 , segregates clients into one of the three client pools based on their capability: (i) AVL_TRAIN : available to train and evaluate, (ii) AVL_EVAL : available only to evaluate, and (iii) UN_AVL : unavailable to participate. While different applications can define these states differently, the clients would indicate their state to the control plane based on the criteria and their runtime conditions (compute load, battery status, connectivity, etc). For example, since training is computationally intensive, a client could be classified as AVL_TRAIN only when it is connected to a charger and on Wi-Fi (e.g. OORT, G-FL). However, to tap into under-utilized clients due to this restrictive criteria, we categorize clients into AVL_EVAL state even when they are not connected to a charger, but their battery exceeds a certain threshold, since evaluation is computationally cheaper.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instead of periodic heartbeats of current state which is communication heavy, redundant, and unscalable to thousands of clients, the clients proactively notify the control plane whenever their state changes 1 ( Fig. 5 ). To guard against stale or incorrect state information (e.g., when a client silently drops due to network or device failure), FeLiX employs a lightweight check: Instead of separate messages or a regular heartbeat, it starts a 90-second timer whenever the aggregator assigns a task; if no response arrives within this window, the client is marked UN_AVL . If the client becomes UN_AVL during training, it can resume from its last checkpoint once it transitions back to AVL_TRAIN (Gupta et al. , 2024 ; Eisenman et al. , 2022 ; Mohan et al. , 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Client state updates are tracked in a simple map data structure that 2 the control plane forwards to the selector. TierTrack maintains additional metadata for each client to recognize clients who rejoin the cohort during the same training round and preclude them from participating in that round. This is done to prevent the selector from sampling the same client more than once in a single round, ensuring fairness (McMahan et al. , 2017 ; Nguyen et al. , 2022b ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3. Tiered Client Selection with Fresh Utilities\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instead of picking random client, recent selectors recognize that not all clients contribute equally to global progress. A client’s contribution depends on (i) the richness of its local data (statistical utility) (Chen et al. , 2019 ; Peng et al. , 2019 ; Wang and Joshi, 2019 ; Li et al. , 2020b ) and (ii) the speed with which it returns updates (system utility) (Lai et al. , 2021 ; Huba et al. , 2022 ) . A combined client utility (for example, the product of statistical and system terms) is typically estimated after a client participates in training and then used to guide future selections.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, in large, realistic deployments only a small subset of clients is available at any time and clients are not chosen every round. Consequently, stored utilities for non-participating clients become stale and no longer reflect their current potential. Stale utilities induce selection bias. Frequently high-utility clients are repeatedly favored while other, potentially useful clients are neglected, slowing convergence and harming fairness across clients.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Existing approaches offer only partial remedies. OORT inflates utilities for idle clients to encourage exploration, but this often overestimates inactive clients’ contributions. REFL trains availability predictors and prioritizes less-available clients, but these models must be retrained for every application, remain imperfect despite local adaptation, and mis-select clients during early deployment. As a result, both approaches have limited applicability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FeLiX addresses stale utilities efficiently by proactively collecting fresh utility measurements from a dedicated, lightweight eval tier . Rather than rely on inference or delayed training replies, FeLiX obtains up-to-date statistical utilities via short forward-pass evaluations on AVL_EVAL clients and feeds these fresh values into selection decisions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the core of TierSelect is a compact ClientMap that tracks each client’s latest statistical utility, system metrics, and last_utility_round . TierTrack updates pool memberships, while TierSelect runs two concurrent selectors: a high-frequency eval_selector and a lower-frequency train_selector (Fig. 6 ). The eval selector sends lightweight forward-pass evaluations to clients in AVL_TRAIN and AVL_EVAL prioritized by most stale first; these complete up to 20 × 20\\\\times faster than full training (Xu et al. , 2024 ) , and their returned utilities are written to the ClientMap. The train selector then uses these refreshed utilities to pick top-utility trainers at the set concurrency level. Trainers perform full local training and return model updates and utilities, closing the freshness feedback loop. Since last_utility_round is updated after both evals and training, the ClientMap always holds the freshest client observations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The eval selector is deliberately rate-limited (assign at most e e eval tasks per round) to avoid stealing training capacity from available clients. This balances freshness and resource contention: frequent evals refresh utilities but, if unconstrained, could reduce available trainer choices (during eval task execution). Each eval also adds to the communication cost. However, timely fresh utilities substantially reduces selection error and more than amortizes the added eval traffic. For efficiency, selector logic operates only on the currently available sets ( AVL_TRAIN , AVL_EVAL ) reported by TierTrack; this keeps rank computations small and ensures only eligible clients are chosen. The result is a lightweight, fast, and scalable selection strategy that trades off system and data heterogeneity by leveraging fresh client utilities amid high client inactivity and heterogeneity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4. Utility-aware, delay-robust aggregation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Round-based staleness, the gap between the server model version when an update arrives and the version the client trained on is simple to compute and widely used in SyncFL and AsyncFL. However, staleness alone conflates two causes of age: (i) client unavailability and (ii) slow computation or communication. Neither indicates whether an update is still informative. Prior work often discounts updates solely by staleness, discarding delayed yet useful ones (Nguyen et al. , 2022b ; Xie et al. , 2019 ) . FeLiX distinguishes these cases by adding a statistical signal: the utility of an update, derived from the client’s local loss. A high loss implies the model is poorly aligned with that client’s data and the update remains valuable; a low loss suggests the update can be safely down-weighted by age.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TierFuse encodes the tradeoff between update age and informativeness. Each update u i u_{i} carries its staleness T i T_{i} and local loss L i L_{i} . Two basis functions: staleness decay α ( T ) \\\\alpha(T) (decreasing in T T ) and utility gain β ( L ) \\\\beta(L) (increasing in L L ) are combined using a tradeoff parameter σ ∈ [ 0 , 1 ] \\\\sigma\\\\!\\\\in[0,1] :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Smooth parameterized forms provide interpretable control:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here a a controls how quickly weight decays with age, b b controls how fast utility saturates, and σ \\\\sigma sets the balance between the two. This design rewards delayed but informative updates (high L L ) while still damping very old or low-utility ones, balancing quality and stability in aggregation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two safeguards make TierFuse robust in practice. It buffers a short window of recent updates to stabilize normalization and prevent a single extreme update from skewing aggregation. Updates older than a threshold T max T_{\\\\max} are clipped or dropped, and optional server-side weight clipping or L2 normalization further bound any client’s influence. These lightweight measures require no extra client computation beyond the local loss already produced during training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike round-based schemes (e.g., REFL’s round-aware approach), TierFuse explicitly combines update age and statistical utility. Staleness penalizes divergence risk, while high utility rescues delayed but valuable updates. This yields a simple, interpretable policy suited for heterogeneous, unreliable deployments—suppressing obsolete noise without discarding late, informative updates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Implementation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FeLiX is implemented as a compact set of extensions to Cisco’s Flame framework 1 1 1 https://github.com/cisco-open/flame , reusing Flame’s control plane, dispatcher, and transport stack. It has three plugins: TierTrack monitors client activity, capability and maintains lightweight availability pools; TierSelect implements the two parallel selectors (frequent eval selector and less-frequent train selector ) and operates on the small, available-client view provided by TierTrack; and TierFuse is a buffering-and-weighting layer in the optimizer path that applies the mixed staleness/utility weights and emits normalized server updates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"State is intentionally minimal and colocated in the control plane: a compact ClientMap stores per-client tuples such as (stat_utility, system_metrics, last_utility_round, curr_tier) . Clients piggyback their local loss and round duration on replies so selectors and the aggregator can update ClientMap without extra RPCs. Fast evaluations are implemented as a single forward pass (empirically upto 20X faster than a full training run) and write back refreshed statistical utilities to ClientMap, enabling timely, low-cost updates to selection decisions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Aggregation is implemented as a configurable-size buffer plus simple element-wise weighting and normalization. TierFuse reads staleness and loss from the client update, computes weights and produces the server model update; these operations are cheap. To remain agnostic to communication protocol, we extended Flame’s channel backend with the control semantics needed for availability and eval requests (the default deployment uses MQTT), and we keep all extra state in the control plane so that client-side changes are minimal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overall, FeLiX comprises a set of to-be-open-sourced plugins to Flame to deliver TierTrack, TierSelect, TierFuse and ClientMap, all of which cause minimal runtime disruption and low overhead.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Experiment Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We perform the evaluation in a cross-device FL testbed built on Flame (Daga et al. , 2023 ) with MQTT (Light, 2017 ) . The testbed colocates the FL server (control plane, selector, aggregator) and an emulated population of clients on a multi-GPU node, following prior emulation-based FL studies (e.g., FedScale (Lai et al. , 2022 ) , OORT (Lai et al. , 2021 ) , REFL (Abdelmoniem et al. , 2023 ) ). Experiments sweep heterogeneity along three axes: (i) data, (ii) compute / runtime, and (iii) availability, so the results cover a broad range of realistic operating points. We align Dirichlet parameters, runtime distributions, and availability scenarios with prior work to enable meaningful comparisons.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Testbed hardware and runtime emulation. The host machine with with 8 NVIDIA A40 GPUs, 500 GB RAM, and an AMD EPYC 7513 32-core CPU runs the FL server and up to 300 emulated clients. Each of the clients are assigned (i) per-round training delays to emulate compute and network differences, and (ii) data partitions for data heterogeneity. Per-round client durations are sampled from a broad distribution spanning ≈ \\\\approx 1 to 60 s, matching the device-speed and latency distribution reported in production-oriented studies (Papaya (Huba et al. , 2022 ) , FedBuff (Nguyen et al. , 2022b ) ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Datasets and models. We evaluate two modalities using two datasets: Image (CIFAR-10) : 60k images, 10 classes; model is a 12-layer CNN ( ≈ \\\\approx 0.5M parameters) and Speech (Google Speech Commands v2) : ≈ \\\\approx 105k one-second clips, 35 labels; model is a ResNet-derived speech network ( ≈ \\\\approx 7.2M parameters). All runs start from the same initialization and use identical configs for parity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Client populations, data heterogeneity, and availability. Data heterogeneity is controlled with Dirichlet partitions Dir ( α ) \\\\mathrm{Dir}(\\\\alpha) for α ∈ { 0.1 , 1 , 10 , 100 } \\\\alpha\\\\in\\\\{0.1,1,10,100\\\\} (from strongly non-IID to near-IID). To emulate real-world FL deployments (Ye et al. , 2023 ) , experiments below fix data heterogeneity to α = 0.1 \\\\alpha=0.1 unless stated otherwise. It is also the most challenging regime to stress availability and FL training. The image task uses 300 clients; the speech task uses 100 clients.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Availability scenarios include synthetic population-level settings (100%, 90%, 50%) and a real-world MobiPerf trace with very low, bursty participation ( ∼ \\\\sim 10–22% available). Figure 8 visualizes the synthetic pools and the MobiPerf trace; traces determine per-client availability across the experiment runtime.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baselines, selector/aggregator variants, and configuration. We compare FeLiX against existing baseline FL (OORT, REFL), and also against their variants which incorporate some of FeLiX’s features: OORT (SyncFL, unaware of availability), OORT ∗ \\\\ast (SyncFL + oracle TraceAvail), OORT+Async (OORT extended to AsyncFL), OORT+Async ∗ \\\\ast (OORT AsyncFL + oracle TraceAvail), REFL ∗ \\\\ast (trace-tuned availability predictor based selector + stale aggregations in SyncFL), and FeLiX (availability-aware selector + freshness-aware aggregator). For clarity each strategy is characterized along three axes: availability knowledge (none / oracle TraceAvail / predictor / TierTrack), selection policy (OORT / REFL IPS / TierSelect), and aggregation rule (FedAvg SyncFL / REFL SAA / FedBuff AsyncFL / TierFuse), as summarized in Table 1 . The oracle TraceAvail is provided only to certain baselines to bound availability-aware performance; otherwise all strategies share the same per-client runtime/data settings and local training configurations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metrics. Primary metrics are wall-clock time-to-accuracy (target: 70% on both tasks) and total communication (bytes moved). Secondary analyses include client utility-age distributions, per-round stall durations, and update-staleness distributions at the aggregator; these are used to attribute measured performance differences to stalls, utility staleness, or staleness in updates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We seek to answer the following four concrete questions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"End-to-end (real-world): Can FeLiX provide tangible time-to-accuracy and communication cost benefits under high-unavailability real-world traces (MobiPerf), across image and speech modalities?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"End-to-end (synthetic): Do gains from FeLiX only occur in sparse availability or do they hold across scenarios of moderate (50%) to high (100%) availability too? This continues to operate in the maximal data heterogeneity ( Dir ( α ) = 0.1 \\\\mathrm{Dir}(\\\\alpha){=}0.1 ) regime. Can FeLiX reduce wall-clock time-to-accuracy and total communication?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FeLiX’s components: What are the end-to-end per-component contributions and overheads of TierTrack (real-time availability), TierSelect (train + eval-aware selection), and TierFuse (freshness-aware weighting)?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Scalability and robustness: How do FeLiX’ tracking, selection and aggregation perform as clients scale?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.1. End-to-end: Real-World MobiPerf Trace (Q1)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Time-to-accuracy. For CIFAR-10 on the bursty and very low availability MobiPerf trace, FeLiX yields an impressive 1.76 × 1.76\\\\times speedup by reaching the 70% target accuracy versus oracular OORT ∗ \\\\ast variants and REFL, the best competing strategies. This reduction in wall-clock is due to live availability tracking and task-aware sampling. Not only does this avoid long stalls and reduce blind re-selection, the freshness-aware aggregation keeps useful updates contributing rather than being ignored. For the speech task the same pattern holds qualitatively: FeLiX converges in ≈ \\\\approx 11.1 h while oracle-assisted OORT variants and REFL finish in ≈ \\\\approx 15.9–16.3 h ( ≈ \\\\approx 1.45–1.48X slower). The larger absolute times for speech reflect both model and trace interactions, but the relative improvement of FeLiX remains intact across modalities and under the difficult and fluctuating MobiPerf availability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Communication cost. The total communication numbers in Fig. 9 and Fig. 10 must be read together with model sizes and the mix of evaluation and training task messages. CIFAR-10 uses a small model ( ≈ \\\\approx 0.5M parameters) so it requires a modest 1.9 MB per update sent or received. On the other hand, the speech model is much larger ( ≈ \\\\approx 7.5M parameters) and needs ≈ \\\\approx 28.6 MB per exchange. This translates to an order-of-magnitude larger total bandwidth over the run durations. Under the MobiPerf trace conditions, all strategies for speech require a large (2–3 TB) of exchange over the course of 3,000 to 6,000 rounds of aggregations (based on the strategy). In terms of communication cost, FeLiX sits between the OORT+Async ∗ \\\\ast and pure sync variants ( OORT ∗ \\\\ast , REFL) in total bytes while still providing the fastest wall-clock times. For CIFAR-10, FeLiX’s total communication (162.6 GB) is still lower than OORT+Async ∗ \\\\ast (270.0 GB). This gap shows that FeLiX saves a lot communication cost by achieving the target accuracy much sooner by selecting and accepting more helpful client updates early. This includes the extra communication cost of on eval messages ( ≈ \\\\approx 20.3 GB) that keep utilities fresh. For Speech, eval traffic is more consequential: FeLiX uses ≈ \\\\approx 583 GB for evaluation tasks because each evaluation task is now 15X costlier due to the larger speech model. This shows that eval-task bandwidth must be tuned or compressed in large-model settings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In short, FeLiX consistently reduces time-to-target on low availability, real-world traces for both CIFAR-10 and Speech modality tasks. The time-to-accuracy and communication-cost is a trade-off: faster convergence often means more but more useful message exchanges early on, and evaluation overhead scales with model size. Minimizing end-to-end communication cost therefore requires balancing evaluation tasks and model-size-aware optimizations (e.g., smaller proxy evals or compressed eval uploads) to keep eval overhead small while preserving the wall-clock gains FeLiX delivers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.2. End-to-end: Synthetic Traces (Q2)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Time-to-Target Accuracy. Across the synthetic availability traces in Fig. 11 , FeLiX reaches the 70% CIFAR-10 target faster than all baselines, including those with oracular foresight. Relative to the strongest non-oracle strategy ( OORT+Async ), FeLiX improves time-to-accuracy by 1.26 × 1.26\\\\times at 100% availability, 2.37 × 2.37\\\\times at 90%, and by > 4 × >4\\\\times at 50% availability. As availability decreases, the lines for availability-unaware strategies shift toward the bottom-right, with time-to-accuracy increasing from 8 h to over 24 h. This shift reflects the heavy cost of incorrect selections and synchronization stalls. While oracular strategies ( OORT+Async ∗ \\\\ast ) remain stable near 8 h by avoiding stalls via trace-knowledge, FeLiX actually outperforms these oracles. This performance gain is driven by three synergistic primitives: TierTrack tiering to avoid stalls in real-time, TierSelect’s dual train-and-eval tasks that prioritize utility even in high-availability regimes, and TierFuse, which extracts informative signal from late updates that other systems would simply discard.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Communication Efficiency and the Cost of Staleness. FeLiX significantly reduces the total communication required to reach target accuracy. When compared to the oracular OORT+Async ∗ \\\\ast , FeLiX saves approximately 40.2 % 40.2\\\\% of total bandwidth. Notably, as unavailability increases, FeLiX’s total communication cost actually falls by ≈ 9.7 % \\\\approx 9.7\\\\% , as TierSelect dispatches fewer evaluation tasks when client utility remains fresher for longer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The comparison with REFL ∗ \\\\ast reveals a significant efficiency gap. Although REFL ∗ \\\\ast is a SyncFL variant, it maintains a large set of trainers ”in-flight” to accommodate delayed updates and maximize throughput. However, its rigid policy of discarding any update staler than 5 rounds creates a significant communication overhead. Empirically, we observe that effectively 25% of REFL’s trainer-to-aggregator uplink traffic is discarded for exceeding this threshold. This is not merely a waste of device uplink traffic; it also represents a loss of the aggregator-to-trainer downlink bandwidth used to transmit the model weights initially. Furthermore, this aggressive pruning harms the model itself, as informative updates from the long-tail of slower or transiently unavailable clients are lost. By contrast, FeLiX employs freshness-aware aggregation to incorporate these updates safely, ensuring that all network communication cost incurred by the system contributes to the final convergence. This allows FeLiX to beat REFL not only in wall-clock time but also in total communication cost across all availability regimes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.3. FeLiX Component Benefits (Q3)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To decompose the end-to-end gains reported in § 6.2 and § 6.1 , we inspect three diagnostic metrics at runtime: aggregator stall duration, client utility age (the latency between a client’s last evaluation and its current selection), and update staleness at the point of aggregation. These traces expose the specific failure modes that affect existing strategies. SyncFL variants without live telemetry suffer from long-tail round durations as they wait on dropped or straggling clients. Selection algorithms that do not proactively refresh utility estimates suffer from utility drift, repeatedly re-selecting historically high-value clients that are currently unavailable or whose data distribution has shifted. Finally, standard AsyncFL rules either discard valuable-but-late updates or apply a uniform staleness penalty, both of which erode the statistical signal. We now show how each FeLiX component resolves these failures through a tightly coupled orchestration loop.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TierTrack: Mitigating the impact of client inactivity. Integrating TierTrack into the orchestration loop sharply reduces tail latencies by enabling immediate resampling when selected clients become unreachable. In our CIFAR-10 experiments, augmenting a stalled OORT+Async baseline with TierTrack cuts the median (P50) round duration from 258.9s to just 4.9s, while the P95 tail latency drops from 653.3s to 21s (Fig. 12 ). This orders-of-magnitude improvement occurs because TierTrack operates on live telemetry rather than static traces; it can detect mid-round drop-offs—which affected 6–8% of all selections in our traces—and trigger an immediate replacement. Consequently, OORT+Async +TierTrack achieves a 5–7% lower time-to-accuracy than even the trace-informed oracle ( OORT+Async ∗ \\\\ast ), which can avoid stalls but lacks the mechanism to reactively resample when a selected client disappears mid-computation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TierSelect: Maintaining utility freshness via dual-tier selection. TierSelect decouples selection into a TrainSelector, which targets high-progress, responsive clients, and an EvalSelector, which acts as a ”scout” by issuing lightweight evaluation tasks to clients with stale or missing utility estimates. This dual-tier approach ensures that the server’s view of the population remains accurate despite bursty availability. As shown in Fig. 13 , the TrainSelector consistently prioritizes clients with higher statistical utility (median score of 122.0 vs 118.1 for OORT), even when those clients are marginally slower (median compute of 11.2s vs 10.9s). This trade-off is mathematically advantageous: by selecting more informative gradients, the model requires fewer total rounds to reach the target accuracy. Simultaneously, the EvalSelector probes low-utility trainers and reveals their true hardware profiles (P99 < < 100), identifying that these clients are significantly slower (median 15.7s). This fresh evidence prevents the system from making blind, repeated re-selections of low-yield or hardware-constrained devices.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TierFuse: Regaining information from delayed updates. Rather than applying a destructive staleness penalty or discarding late updates, TierFuse computes a contribution factor that balances a client’s specific statistical utility against its update age. This allows the system to incorporate informative updates from the ”long-tail” of the edge ecosystem that would otherwise be lost. As illustrated in Fig. 15 , TierFuse assigns high weights ( ≥ 0.8 \\\\geq 0.8 ) to over 90% of received updates, preserving their informative signal even when they arrive moderately late. In contrast, a FedBuff-style policy, which lacks this informativeness-aware weighting, assigns such high weights to fewer than 10% of updates, often falling back to a flat moderate weight of ≈ \\\\approx 0.5. By effectively protecting high-value updates from aggressive staleness discounting, TierFuse increases the per-update usefulness of every byte transmitted, accelerating convergence under the erratic conditions of streaming FL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FeLiX: Combined synthesis of components. These three components are fundamentally intertwined: TierTrack eliminates stalls to ensure that EvalSelector’s probes reflect actual availability; EvalSelector maintains fresh utility estimates so TrainSelector can accurately prioritize informative clients; and TierFuse ensures that once an update is received—whether fresh or moderately stale—its weight reflects its true contribution to the global model. Together, they transform availability and freshness from passive observations into active control signals, explaining the substantial wall-clock and communication gains observed in production-scale settings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.4. Scalability and robustness (Q4)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Selector and decision latency. FeLiX is designed to minimize server-side overhead by offloading availability tracking to the edge. Per-decision compute latency remains consistently low ( < 0.5 s <0.5s ) because the server does not perform an exhaustive, synchronous scan of the entire population. Instead, the client-driven tiering ensures the server only ranks the active, ready subset of the population, while the EvalSelector asynchronously estimates live client utility in the background. This decoupling allows the orchestration logic to scale gracefully; as the population increases from 10 to 300 concurrent clients, we observe no significant increase in selection latency or congestion at the MQTT broker.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Robustness to heterogeneity. We evaluate the resilience of FeLiX across varying degrees of statistical and system heterogeneity by sweeping Dirichlet distribution parameters ( α ∈ { 0.1 , 1 , 10 , 100 } \\\\alpha\\\\in\\\\{0.1,1,10,100\\\\} ) and runtime compute distributions (1 to 60 seconds). As expected, the relative gains of FeLiX are most pronounced in the strongly non-IID regimes ( α = 0.1 \\\\alpha=0.1 ) and low-availability settings. These represent the most challenging and deployment-relevant scenarios where standard FL systems typically collapse. In highly heterogeneous environments, the opportunity cost of a poor selection is high; missing a transiently available client with high-utility data results in a significant loss of gradient progress. FeLiX’s ability to fuse live availability with freshness-aware aggregation allows it to navigate these sparse availability environments effectively, ensuring that every successful update provides maximum statistical gain. This robustness demonstrates that FeLiX is not merely faster in idealized settings, but fundamentally more resilient to the adverse conditions of production edge ecosystems.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7. Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FL Applications. FL was introduced by McMahan et al. (McMahan et al. , 2017 ) as FedAvg, and (Kairouz et al. , 2021 ; Yang et al. , 2019 ) survey its landscape and open challenges. Beyond the production recommender and analytics workloads this paper targets, FL has been applied to mobile keyboard prediction (Hard et al. , 2018 ; Chen et al. , 2019 ) , on-device language modeling (Xu et al. , 2023 , 2024 ) , healthcare analytics (Brisimi et al. , 2018 ; Li et al. , 2019a ; Xu et al. , 2021 ; Nguyen et al. , 2022a ) , and distribution-shifted domains (Peng et al. , 2019 ) . Privacy threats including membership inference (Shokri et al. , 2017 ) and client-level differential privacy (Geyer et al. , 2017 ) motivate the data-local design that FeLiX inherits; its orchestration layer is orthogonal to these protections.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FL Convergence, Heterogeneity, and Optimization. FedAvg convergence under non-IID data (Li et al. , 2019b ; Wang et al. , 2020b ) identifies data heterogeneity and partial participation as primary theoretical obstacles; heterogeneous FL settings are surveyed in (Ye et al. , 2023 ) . Adaptive server-side optimizers (Reddi et al. , 2020 ) , matched averaging (Wang et al. , 2020a ) , ensemble distillation (Lin et al. , 2020 ) , and Bayesian non-parametric fusion (Yurochkin et al. , 2019 ) improve convergence under heterogeneous data; one-shot FL (Li et al. , 2020a ) eliminates iterative rounds entirely. Communication efficiency is targeted by gradient compression (Konečnỳ et al. , 2016 ) , distributed mean estimation (Suresh et al. , 2017 ) , and adaptive communication scheduling (Wang and Joshi, 2019 ) . FeLiX’s weighted aggregation policy is complementary to all of these: it targets update informativeness and model-age correction rather than modifying the fusion or compression scheme.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Client Availability, Selection, and AsyncFL. Section 2 analyzes in detail why the leading utility-aware selectors OORT (Lai et al. , 2021 ) and REFL (Abdelmoniem et al. , 2023 ) , buffered AsyncFL (Nguyen et al. , 2022b ; Zhang et al. , 2023 ) , and production FL systems (Bonawitz et al. , 2019 ; Srinivas et al. , 2025 ; Wang et al. , 2023a ) fall short under real-time availability churn. More broadly, async FL has been studied from decentralized SGD and momentum perspectives (Lian et al. , 2018 ; Xie et al. , 2019 ; Mitliagkas et al. , 2016 ) , and extended to continual and non-IID settings (Shenaj et al. , 2023 ; Chen et al. , 2020 ) . Staleness handling has received dedicated attention (Zhou et al. , 2022 ; Rodio and Neglia, 2024 ) ; variance from partial participation is targeted by FedVARP (Jhunjhunwala et al. , 2022 ) and large-cohort analysis (Charles et al. , 2021 ) . Nishio and Yonetani (Nishio and Yonetani, 2019 ) select clients by deadline compliance under resource heterogeneity. Prior work (Garg et al. , 2025 ) empirically shows that ignoring real-time availability substantially increases time-to-accuracy, directly motivating the TierTrack tracking design in FeLiX.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Client Interruption and Local Checkpointing. A natural response to mid-training client interruptions is to checkpoint local model state, enabling resumption after conditions recover—a technique well-developed for distributed DNN training (Mohan et al. , 2021 ; Eisenman et al. , 2022 ; Gupta et al. , 2024 ; Wang et al. , 2023b ) . In the FL context, however, local checkpointing only defers the problem: the client still cannot return any update until it becomes available again, and the global model continues to advance in the interim, rendering saved gradients stale or model-incompatible on return. FeLiX addresses the root cause instead, by exploiting lightweight real-time readiness signals to harvest updates before interruptions occur and by weighting late-arriving updates by their informative value rather than discarding or blindly accumulating them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FL Systems, Frameworks, and Benchmarks. Production FL platforms (Bonawitz et al. , 2019 ; Huba et al. , 2022 ; Srinivas et al. , 2025 ) address large-scale cross-device deployments; FLINT (Wang et al. , 2023a ) provides on-device measurement; FedScale (Lai et al. , 2022 ) benchmarks system performance with realistic traces. Research frameworks Flower (Beutel et al. , 2020 ) and FedML (He et al. , 2020 ) support controlled experimentation. FeLiX is built on the Flame orchestration layer (Daga et al. , 2023 ) and evaluated against real MobiPerf traces ( M-Lab, ) ; unlike offline benchmark or framework tools, it targets production-scale streaming workloads with bursty, minute-level availability dynamics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8. Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As production Federated Learning moves from multi-day batch refreshes toward hourly streaming adaptation, existing orchestration models have proven insufficient. The interlocking challenges of transient client availability, shifting data utility, and delayed ground-truth feedback create a freshness gap that degrades critical user-facing metrics. This work introduced FeLiX , a redesign of FL orchestration that treats availability and utility not as passive observations, but as high-frequency control inputs. By unifying three synergistic primitives: streaming-aware availability tiers , fresh-utility selection , and informativeness-aware, delay-robust aggregation , FeLiX bridges the gap between idealized oracular theory and the realities of the practical FL deployments in the mobile-edge ecosystem. Our evaluation across multiple modalities and realistic MobiPerf traces demonstrates that FeLiX achieves near-oracular performance, delivering a 2.37 × 2.37\\\\times reduction in time-to-target accuracy while simultaneously saving 1.30 × 1.30\\\\times in total communication bandwidth compared to state-of-the-art baselines. These results confirm that by eliminating synchronization stalls and reclaiming signal from the long-tail of delayed updates, FeLiX ensures that deployed models remain in sync with volatile live data distributions. Ultimately, FeLiX provides the necessary framework for high-scale, privacy-preserving applications to adapt at the speed of user interaction.\"\n }\n]","isConversational":false}}},{"rowIdx":492,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06973"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Rethinking Multimodal Time-Series Forecasting Evaluation"},"authors":{"kind":"list like","value":["Haoxin Liu","Yichen Zhou","Rajat Sen","B. Aditya Prakash","Abhimanyu Das"],"string":"[\n \"Haoxin Liu\",\n \"Yichen Zhou\",\n \"Rajat Sen\",\n \"B. Aditya Prakash\",\n \"Abhimanyu Das\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06973v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"redacted \\correspondingauthor Haoxin Liu <hliu763@gatech.edu> . This work was done while Haoxin Liu was a Student Researcher at Google Research. This work is accepted by The International Conference on Machine Learning (ICML) 2026."},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Time-series forecasting (TSF) Liu_FOIL is a ubiquitous task across numerous domains and is essential for informed decision-making. Motivated by success in NLP, there has been significant work in recent years on time-series foundation models (TFM) for forecasting, ranging from re-purposing LLMs directly for forecasting ( gruver2023large ; tan2024 ) to fine-tuning pretrained LLMs on time-series data ( zhou2023one ; chang2023llm4ts ) to pretraining time-series foundation models from scratch ( das2023decoder ; goswami2024moment ; woo2024unified ; ansari2024chronos ; kamarthi2024lptm ) . These models have demonstrated promising zero-shot TSF capabilities, often outperforming traditional statistical and supervised methods, using only the historical context of the time-series at inference time. Although historical numerical data provide a fundamental basis for prediction, they often lack the complete context necessary for reliable and accurate forecasts. Human forecasters often integrate additional information such as background knowledge, events and constraints, which can usually be captured through natural language ( Liu_et_al_2024a_LSTPrompt ) - we call these textual contexts . Improving the ability of time-series foundation models to effectively leverage and integrate diverse textual contexts remains an ongoing challenge. This highlights a crucial need for high-quality and comprehensive multimodal datasets and benchmarks to propel research in this area."},{"type":"para","text":"At the same time, existing context-enriched multimodal TSF evaluation benchmarks and their construction techniques present several limitations when benchmarking pretrained models on them. The first limitation is the unknown generalization gap between the benchmarking metrics and the real world performance. This is often caused by either the benchmark being restricted to a small scale with a narrow selection of domains, or it being fully synthetic ."},{"type":"para","text":"The second significant challenge across existing benchmarks is data leakage . Pretrained TFMs and LLMs may have already ingested evaluation data, leading to unknown data contamination and unfair comparison between methods. A benchmark itself is subject to short-lived validity as pretrained models will continuously update their pretraining datasets, which results in their knowledge cut-off becoming more recent than the benchmark’s data."},{"type":"para","text":"The third limitation comes from our observation that the types and granularities of textual contexts in previous benchmarks are often limited and vary wildly. Textual contexts derived from real data include metadata information about the time-series, timestamp-related information such as important dates and holidays, textual event data related to the time-series, and textual data capturing statistics and trends of related exogenous variables correlated with the target time-series. The ability of multimodal models to process these different types of contexts, and the availability and quality of these different contexts in the datasets have a significant effect on the model’s benchmark performance. When a multimodal model falls short on existing benchmarks, it is difficult to conclude whether the shortcoming is due to the fact that the method does not effectively utilize the provided information, or it is because the contexts provided in the benchmark are of a specific type, too vague, or lacking relevant details."},{"type":"para","text":"To address these fundamental limitations and establish an unbiased benchmark for context-enriched time series forecasting, we introduce TimesX , a novel multimodal benchmark designed to focus on (1) real, large-scale, cross-domain data, (2) a dataset generation pipeline providing continuous mitigation of data leakage, and (3) a collection of comprehensive, fine-grained textual contexts. We further introduce several promising baselines that combine TFMs and LLMs to achieve non-trivial multimodal, context-enriched forecasting performance on our benchmark."},{"type":"para","text":"The primary contributions of this paper are threefold:"},{"type":"para","text":"TimesX is, as far as we are aware, the first real-world, large-scale, and cross-domain context-enriched time series forecasting benchmark. It encompasses 19 diverse domains with a total of 190 variables, covers diverse global geographical regions and includes daily and weekly frequencies. It also links each target numeric series to multiple forms of detailed textual contexts. Unlike existing multimodal time-series benchmarks, all time series and textual data in TimesX are from real-world observations."},{"type":"para","text":"We propose two new mechanisms in the dataset generation pipeline of TimesX to help prevent data leakage and guarantee the validity of the textual context. The first one is an automated data collection pipeline that adopts strict timestamp alignment and isolation during each of its steps. The second one is a hypothesizer-verifier-enricher framework for fact-checking and enriching the textual contexts. These mechanisms combined ensure that TimesX is verifiable, leakage-free, and can be updated in the future for benchmarking methods with new pretraining cut-off dates."},{"type":"para","text":"We empirically verify that TimesX addresses the shortcomings of existing benchmarks. In addition, we conduct a thorough empirical comparison of current (zero-shot) multimodal TSF approaches on TimesX, involving more than 312,000 independent LLM inferences and revealing new observations that were not discovered by existing benchmarks. In particular, we discover that earlier benchmarks like Williams_2024 (while being a great test for instruction following in forecasting) tend to severely over-estimate the performance of LLMs over TSF models. At the same time, benchmarks like Liu_et_al_2024b_TimeMMD under-report the importance of textual context information in forecasting accuracy."},{"type":"para","text":"Limitations and future works are detailed in Appendix A ."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"In addition to the time-series foundation models mentioned previously, recent works jin2023time ; Liu_et_al_2024a_LSTPrompt ; wang2025chattime have started to adapt pre-trained LLMs to perform forecasting when textual contexts are available, often by aligning the modalities of time series and natural language. The development of these models has been paralleled by new benchmarks designed to evaluate them. ChatTS ( ChatTS ) , for instance, generates synthetic time series paired with detailed attribute descriptions to train a multimodal LLM for understanding and reasoning tasks. Similarly, the Time-MQA framework ( Kong_2025 ) introduces the TSQA dataset, which contains real-world time series with template-based meta information and turns time-series forecasting into a question-answer format. Benchmarks such as those referenced in CiK ( Williams_2024 ) contain real-world time series and manually crafted textual contexts that are critical for forecasting. Time-MMD ( Liu_et_al_2024b_TimeMMD ) provides a context enriched benchmark constructed by keyword-based web searches of nine domains, one variable for each. MTBench ( Chen_2025 ) aligns stock prices with financial news and weather reports with temperature records, curating tasks that require reasoning about them. The MoTime ( Zhou_2025 ) dataset suite provides a collection of multimodal datasets pairing time series with static modalities like text and images. Noticeably, these early efforts were often constrained by a scarcity of high-quality, large-scale, real-world datasets, leading some to rely on synthetic data and specific forms of textual contexts. We summarize several representative benchmarks compared to our proposed TimesX in Table 1 ."},{"type":"heading","level":1,"text":"3 The TimesX Benchmark"},{"type":"heading","level":2,"text":"3.1 Design Principles of TimesX"},{"type":"para","text":"We will first present the core principles that distinguish TimesX from other multimodal forecasting benchmarks: (i) real-world data, (ii) leakage mitigation, and (iii) comprehensive, high-quality context and (iv) large-scale evaluation. Note that we further provide empirical evidence that demonstrates the importance of these principles and where other benchmarks might fall short."},{"type":"heading","level":3,"text":"3.1.1 Real-World Data"},{"type":"para","text":"Principle Description : TSF is highly complex due to real-world dynamics, complex processes, and diverse domains. On the other hand, many existing benchmarks use either synthetic time series, synthetic textual contexts, or both, without demonstrating how the conclusions on synthetic data can transfer to real-world scenarios ( Williams_2024 ; tan2024 ) . TimesX differentiates itself from those benchmarks by using real-world data for both the time series and the textual contexts ."},{"type":"para","text":"Why it matters : To test whether conclusions on synthetic data can transfer to real data, we evaluate three methods on a synthetic context dataset (CiK) and our real dataset (TimesX). We report the aggregated MASE (Mean Absolute Scaled Error) of the three methods which are: (i) TimesFM-2.5 (a TFM) using only the time series, (ii) Gemini-2.0-Flash (an LLM) using both the time series and the textual contexts, and (iii) CodeRev , Gemini-2.0-Flash writing and executing code to revise TimesFM-2.5 outputs based on the context. More details are in Appendix N ."},{"type":"para","text":"As shown in Figure 1 , the rankings of the three methods on CiK (synthetic) and our real data are completely different . On CiK, directly feeding the time-series along with associated text context to Gemini has a huge lead over TimesFM-2.5. Moreover, asking Gemini to edit the output of TimesFM-2.5 via code does even better, significantly outperforming the other two methods. On the other hand on TimesX, we see that all the methods are much closer, and TimesFM-2.5 is actually better than Gemini. Further CodeRev degrades the performance of using both TimesFM-2.5 and Gemini directly. This shows that CiK has a strong bias that clouds its conclusions: in CiK, textual contexts are both synthetic and extremely specific - these specific contexts are then used to write code to modify an input time-series and generate the final forecasting task. Therefore, such tasks can be easily solved by instruction-following language models and even better by language models that generate code to accomplish a simple modification task. These conclusions, however, do not generalize to the real-world tasks, whose contexts are less specific and do not relate to the forecasting task via the coding path."},{"type":"para","text":"We highly recommend using both synthetic and real-world benchmarks for a more robust evaluation. Carefully designed synthetic benchmarks can be useful for testing specific capabilities such as instruction following or different types of reasoning."},{"type":"heading","level":3,"text":"3.1.2 Mitigating Data Leakage"},{"type":"para","text":"Principle Description : A fair benchmark dataset should be able to mitigate any potential leakage of benchmark tasks into the pretraining datasets of the models being evaluated. Given that many pretrained models do not release their pretraining datasets, existing benchmarks that claim to use data from unused sources (data isolation) for evaluation have three drawbacks: (1) Potentially unidentifiable data contamination. (2) Limited number of clean tasks left for benchmarking multiple methods. (3) Short-lived validity i.e., future versions of the models considered may unintentionally leak the benchmark into their pretraining data."},{"type":"para","text":"In contrast, TimesX adopts strict time isolation to prevent data contamination : we timestamp TimesX thoroughly, and recommend strictly using benchmark tasks that occur after a target model’s knowledge cutoff date. Moreover, the TimesX pipeline is designed as re-updateable over time , i.e., data collection, validation, alignment, and model evaluation can all be automated. To the best of our knowledge, TimesX is the first benchmark capable of being automatically refreshed for evaluating future pretrained methods."},{"type":"para","text":"Why it matters : To study how data contamination affects evaluation results, we evaluate on 120 variables from the Search Trend (see Section 3.2 ) subset of TimesX. We keep the numeric series, windowing, and prompts fixed, and split the test period by the public knowledge cutoff of Gemini-2.0-Flash and DeepSeek-V3 (June 2024). We report the MASE aggregated by Geometric Mean in Table 2 ."},{"type":"para","text":"The pretraining data descriptions of the two TFMs preclude any time-series data leakage in both time periods, and both TFMs report stable results with less than 2% delta. In contrast both LLMs see an increase of error by about 13% after their knowledge cutoff."},{"type":"heading","level":3,"text":"3.1.3 Comprehensive and High-Quality Context"},{"type":"para","text":"Principle Description : Most multimodal forecasting benchmarks have a very limited set of text contexts - restricted to either static metadata ( Zhou_2025 ) , date and weather derived features ( wang2025chattime ) or synthetically generated contexts ( Williams_2024 ) . In contrast, TimesX uses not only a union of all common context types , but also a more high-quality description of text events and corresponding alignment of these contexts."},{"type":"para","text":"In particular, we categorize these various kinds of contexts into: (1) Metadata : descriptive high level summaries of the forecasting variable in question (2) Calendar -derived features like holidays (3) Covariates which cover textual information around other related time series (4) Time-stamped events : Textual description of related events aligned with time-windows of the variable in question. Section 3.2.2 contains more details about how we generate these contexts."},{"type":"para","text":"Why it matters : TimesX links each target time-series to all these available context types and aligns them with the timestamps of the time series. The end result is a benchmark with diverse, high-quality textual contexts. To quantify its effect on TSF accuracy, we run a controlled experiment where we take the time-series in the Time-MMD benchmark and use our methodology to replace the textual events using the construction rules in Appendix E . We keep the numeric series, the LLM (Gemini-2.0-Flash), and the prompt fixed."},{"type":"para","text":"Table 3 (full results in Table 18 ) compares the performance of Gemini-2.0-Flash given these newly generated text events versus the original context in the Time-MMD benchmark. Across all nine Time-MMD datasets, this context swap reduces the geometric–mean aggregated MASE from 0.906 0.906 to 0.840 0.840 (a 7.3 % 7.3\\% relative drop) , suggesting that conclusions drawn under low–quality context might be misleading; high–quality context changes the picture ."},{"type":"para","text":"We further show that with the current dataset scale (typically fewer than 20 variables), model performance rankings are quite unstable , as illustrated in Figure 34 and Appendix T ."},{"type":"heading","level":2,"text":"3.2 Construction of TimesX"},{"type":"para","text":"We first provide a high level summary of our dataset and dataset construction methodology. TimesX contains time-series obtained from 19 domains . Each domain has 10 10 variables, resulting in a total of 190 variables (quantities evolving with time). TimesX covers different geographical regions : North America (United States, Canada, Mexico), Asia (China, India among others), Europe (United Kingdom, Norway among others), South America and Africa."},{"type":"para","text":"Note that we have a separate portion of the dataset that is derived from the time-series in Time-MMD Liu_et_al_2024b_TimeMMD , where the text context is generated using our methodology, but only used for ablations in Section 3.1.3 ."},{"type":"para","text":"The TimesX dataset is constructed entirely from scratch , rather than by merging existing datasets. Moreover, TimesX is designed to be automatically refreshable over time, enabling continuous mitigation of information leakage as pretrained models evolve"},{"type":"heading","level":3,"text":"3.2.1 Numerical Time Series Construction"},{"type":"para","text":"The numerical component consists of target variables whose future values are forecasted given a historical window."},{"type":"para","text":"Granularities and sources. TimesX includes two temporal granularities: Weekly series from Google Search Trends across 12 domains 1 1 1 https://trends.google.com/trends/ ; and Daily series including (i) commodity prices from a market data API 2 2 2 https://marketstack.com/ and (ii) major USD exchange rates from a currency rates API 3 3 3 https://frankfurter.dev/ . These API data can be automatically refreshed to extend the benchmark over time. Monthly/quarterly coverage is discussed in Appendix A.1 ."},{"type":"para","text":"Time alignment and missing values. All series within the same frequency share the same timestamps. Missing values are handled as described in Appendix D.1 . Appendix I provides the full list of domains and variables."},{"type":"heading","level":3,"text":"3.2.2 Text Context Construction"},{"type":"para","text":"Our core contribution is aligning comprehensive and diverse text context information with the variables that evolve over time. We will see that methods that capture both this context information along with the past of the time-series can achieve superior performance compared to methods that only use the past time-series component. As mentioned before we provide 4 kinds of text contexts for all time-series variables: Metadata, Calendar features, Covariates and Time-Stamped Events. Section 3.2.3 shows an example of a time series with each of these four contexts."},{"type":"para","text":"Metadata and calendar contexts. The Metadata and Calendar text contexts are easily constructed from the sources of the numerical time-series themselves and Python Holidays library respectively, as described in Appendix C ."},{"type":"para","text":"Covariate contexts. For each time-series in a given domain, we use the other time-series from that domain to extract Covariate text contexts. Specifically, we calculate features of these covariates over the historical window, including mean, median, max and min with the corresponding date, and overall trend direction. These features are then transformed into natural language descriptions, detailed in Appendix D ."},{"type":"para","text":"Time-stamped events via dataset agents. Constructing and aligning textual events is the most challenging component. Following our design principles in Section 3.1 , we must satisfy three goals at once: real-world data, leakage-free (refreshable) , and high-quality . A single LLM or a naive web search cannot reliably meet these goals. Therefore, we adopt a multi-agent automated workflow with four agents: Hypothesizer , Verifier , Enricher , and Synthesizer , as illustrated in Figure 2 . Each agent interacts with an LLM under constraints and has dedicated tools (time-bounded web search and lightweight crawlers). Overall, the Hypothesizer and the Verifier act adversarially to ensure event truthfulness and timestamp accuracy, which prevents information leakage; the Verifier then guides the Enricher to supply missing details thus ensuring quality. The whole workflow is automated, enabling regular updates of the benchmark data. We provide the details of each LLM role:"},{"type":"para","text":"(i) the Hypothesizer identifies points of interest in the time series (e.g., local maxima, unexpected movements or peaks in the corresponding search trend) and iteratively calls an LLM agent with integrated web search to build an initial event set that is able to match these points."},{"type":"para","text":"(ii) then, the Verifier uses crawlers to fetch relevant URLs corresponding to each event window and conducts fact checks, filters out any hallucination or leakage, and prepares a checklist of missing details."},{"type":"para","text":"(iii) The Enricher resolves the checklist using strictly time-bounded web searches, merging multiple sources to fill in the details."},{"type":"para","text":"(iv) finally, the Synthesizer , a reasoning LLM, aggregates all pieces of evidence to finalize the event description, adjust timestamps, and discard events with unresolved doubts."},{"type":"para","text":"Using this agent, TimesX automatically produces an event corpus that matches the time window, contains verifiable facts (each claim has a supporting URL), accurate time stamps, and rich detail. More details and hyperparameter analysis experiments are provided in Appendix E . An execution log demo is in Appendix F ."},{"type":"para","text":"Our pipeline runs in three-month time blocks and is naturally parallelizable across variables and blocks; Phase 1 uses coverage-based early stopping to reduce cost. Users can control the budget by choosing different LLMs per role and tuning the maximum iterations (Phase 1) and search budget (Phase 3). Under our current setup (Gemini 2.5 Pro + Gemini 2.5 Flash), the cost is about $0.7 per variable per three-month block ; it can be reduced by using open-source LLMs or batching verification so multiple events reuse the same calls."},{"type":"heading","level":3,"text":"3.2.3 Data Example"},{"type":"para","text":"We provide below an example from TimesX corresponding to the GAS_PRICE time-series."},{"type":"para","text":"Time series : See Figure 3 ."},{"type":"para","text":"Metadata : This time series records gasoline price (USD/GAL) in the Commodity Price domain, with daily frequency. Prediction target period: from 2024-09-01 to 2024-09-15."},{"type":"para","text":"Calendar : In the format of (timestamp, value), historical data is: (2024-06-02, 2.4116), (2024-06-03, 2.3279),… Upcoming holidays in the prediction window: Labor Day (2024-09-02)."},{"type":"para","text":"Covariates : from 2024-06-01 to 2024-08-31: (1) Brent Crude Oil (USD/BBL): The maximum value was 87.43 87.43 , occurring on July 4, the minimum…, showing an overall downward trend. (2)…"},{"type":"para","text":"Events : (1) On June 2, 2024, OPEC+ agreed to extend deep oil output cuts [1,3], the cut of 2.2 million bpd would be extended until September 2024, after which it would be gradually phased out [2,3]. Source: [1] [2] [3]; (2) …"},{"type":"heading","level":3,"text":"3.2.4 Quality Checks."},{"type":"para","text":"Diversity. We profile numerical and textual features of TimesX (Appendix J ) and visualize them with PCA/t-SNE (Appendix M ). Samples are broadly dispersed, and variables from different sources form distinct clusters, supporting the benefit of multi-source integration."},{"type":"para","text":"Factual correctness. Here, we focus on the most sensitive component: the accuracy of automatic timestamp annotation. We ensure the accuracy via a three-stage pipeline (Appendix G ): verification, programmatic time-bounded search (± k k days), and cross-source adjudication in the agent workflow. A manual audit of 50 samples shows 94% exact matches; 4% conservative (later) offsets; and 2% earlier due to date ambiguity (Appendix G )."},{"type":"heading","level":2,"text":"3.3 TimesX Release and Refresh Consideration"},{"type":"para","text":"The core TimesX dataset now contains 190 variables spanning Jan 2018 to Oct 2025 to support both training (2018–2022) and evaluation (2023–2025). We plan to update the TimesX dataset every three months and version each release."},{"type":"para","text":"The extended TimesX further supports 11 languages beyond English (Afrikaans, French, German, Hindi, Japanese, Korean, Portuguese, Simplified Chinese, Spanish, Swahili, and Turkish), as well as 5 rare diseases. See Appendix A.5"},{"type":"heading","level":1,"text":"4 Empirical Study"},{"type":"heading","level":2,"text":"4.1 Evaluation Settings"},{"type":"para","text":"We focus on zero-shot solutions and consider the following:"},{"type":"para","text":"Pretrained TFMs : We select three SOTA TFMs including TimesFM-2.5, Moirai-2.0, and Sundial. All are towards the top on GIFT-Eval aksu2024gift benchmark, with TimesFM-2.5 being the top open model with no leakage."},{"type":"para","text":"Pretrained LLMs : We consider three well-adopted LLMs (implementation details in Appendix Q ): closed-source Gemini-2.0-Flash and GPT-4o, and open-source DeepSeek-V3."},{"type":"para","text":"Composed (Ensemble And Agentic) Solutions : We construct the following four solutions: (1) AvgEns : An ensemble that averages the forecasts from a pair of TFM and LLM. We simply set equal weights. (2) TextRev : An LLM taking the forecast of a TFM as text and revising it according to the context. (3) CodeRev : An LLM writing and executing code to revise the forecast of a TFM according to the context. (4) FuncRev : Similar to CodeRev , but the LLM is limited to a selection of functions. See Appendix Q ."},{"type":"para","text":"For each variable in TimesX and its context corpus, the look-back window and the forecast horizon are set to 96 and 12, respectively. The rolling window is set to 4 for weekly data and 12 for daily data to balance sample count and sample diversity. We then include all relevant metadata, calendar info and covariates info. To avoid future info leakage, we select the K K most recent textual events whose announcement dates strictly precede the first timestamp of the prediction horizon to add to the context. We fix k = 10 k=10 to balance prompt length and information effectiveness, following common practices Liu_et_al_2024b_TimeMMD ; li2025language ."},{"type":"para","text":"TimesX spans 2018–2025 and is refreshable. However, to avoid pretraining data contamination, we construct evaluation examples whose forecast horizon begins after the pretraining cutoff of all involved models, detailed in Appendix P . Unless stated otherwise, we use 2024-07-01 as the cutoff, i.e., evaluating only the subset whose horizon start time is 2024-07-01 or later. See models’ knowledge cutoffs in Appendix P . Web access is limited to the build-time agent; the benchmark runs offline. Appendix P.1 further discusses how to prevent test-time leakage."},{"type":"para","text":"Following the conventions in other popular forecasting benchmarks like GIFT-Eval ( aksu2024gift ) , we use normalized MASE (mean absolute scaled error) as our main metric. Since the different variables have very different scales, we calculate the average MASE over all rolling windows of a variable and normalize that by the average MASE of a seasonal naive baseline. Then we take the Geometric Mean (GM), for robustness to normalization choice, of these normalized MASE ratios across all variables. See Appendix N . We also compute the average MASE rank of each method over the variables. For both metrics, smaller is better."},{"type":"para","text":"This setup yields about 2.5K samples (Appendix O ). To account for stochasticity, we repeat each evaluation 10 times with different seeds when applicable, resulting in more than 312K independent LLM inferences. Note that using a shorter rolling-window stride can increase the sample count but also the evaluation cost."},{"type":"heading","level":2,"text":"4.2 Benchmarking Results"},{"type":"para","text":"Table 4 shows the benchmark results on TimesX (details in Appendix Y.1 ). As multimodal solutions, though the zero-shot LLMs have the edge over the unimodal zero-shot TFMs, this edge is not as significant on TimesX as observed on other synthetic benchmarks . This points out the gap between synthetic contexts and real world contexts, that some synthetic benchmarks tend to severely over-estimate the performance of LLMs over TSF models because the synthetic contexts were providing the exact information required for bettering the forecast. Real world contexts are in contrast too nuanced to always deliver a large performance boost even when guaranteed to be related. This conclusion is further confirmed by the observation that the agentic CodeRev method is performing worse than its two components working alone, contradicting the claim of ( Williams_2024 ) ."},{"type":"para","text":"In terms of the composed solutions, to our surprise the best performers on TimesX are the simple average ensembles of different pretrained models (i.e., AvgEns ). We by no means want to suggest it as the optimal composition, but instead want to reiterate the stochastic and flexible nature of LLMs and the fact that it would take a great deal of effort to design the interaction with them in a composed solution to just outperform simple averaging ensembles."},{"type":"para","text":"We also use the continuous ranked probability score (CRPS) metrics to measure uncertainty, detailed in Appendix U.1 . We initially observe that all three LLMs achieve lower CRPS than the TFMs . This suggests that our constructed context helps LLMs model future uncertainty better than TFMs, which only use numerical values."},{"type":"para","text":"As an early experiment, we also evaluate reasoning LLMs, including GPT-5, Gemini-2.5-Flash, and DeepSeek-R1. To reduce data contamination, we only use evaluation samples after Jan 2025. Table 17 (detailed in Appendix Y.4 ) shows that these reasoning models do not provide a clear advantage ."},{"type":"heading","level":2,"text":"4.3 Ablation: Context Types"},{"type":"para","text":"Table 5 presents the MASE and MASE rank when different combinations of contexts are provided to a chosen multimodal method (Gemini-2.0-Flash). Comparing to only using the high level metadata, the inclusion of either calendar, calendar and covariates, or calendar and events brings significant improvement. Using all context types improves the accuracy further eventually being around 16% better than using only static metadata. It is worth noticing that we cannot observe the incremental gain from adding covariates or events alone. We speculate there is crucial interaction between those two context types, for instance the effect of an event on the target variable can be quantified by a similar leading effect on a covariate. Figure 4 suggests this speculation may generalize to other LLMs. This observation is aligned with our expectation that a multimodal method can compound the gains by composing the information in different context types . Appendix V reports domain-level holiday and covariate effects."},{"type":"heading","level":2,"text":"4.4 Ablation: Domains"},{"type":"para","text":"Table 6 breaks down the ranking of the benchmark methods on each of the 19 domains (see Appendix Z for MASE). There is no dominance of one zero-shot method over others - in fact the orderings of TimesFM-2.5 and Gemini-2.0-Flash are complementary. In terms of composed solutions, the observation is consistent with Table 4 that AvgEns performs well while other compositions can be worse than its components acting alone. We also observe two domain groups where multimodal solutions show a clear advantage over TFMs. First, in Shopping, calendar information likely provides strong signals about when sales occur, which is hard to infer from the time series alone. Second, in RMC (Raw Materials & Constructions), SAM (Specialty & Advanced Materials), SHVM (Strategic & High-Value Materials), and Curr (Currency), the series are strongly affected by external policies, which are captured in the event context."},{"type":"heading","level":2,"text":"4.5 Why Agentic (Revision) Methods Underperform?"},{"type":"para","text":"None of the three revision methods ( FuncRev , CodeRev , and TextRev ) outperforms the simple AvgEns on TimesX. Figure 5 shows that their forecast errors have a wider spread and more severe outliers than TimesFM-2.5, which leads to worse geometric means. Gemini-2.0-Flash alone shows a similar pattern. We speculate that this is due to LLM stochasticity and the fact that LLM-based revision does not reliably preserve the temporal structure in the initial forecast. Among the three, TextRev performs best, which suggests that effective code-based revision may require instruction fine-tuning."},{"type":"heading","level":2,"text":"4.6 Future Solutions: Could Training Help?"},{"type":"para","text":"This work focuses on zero-shot methods. To support future training-based research, TimesX provides both training (2018–2023) and testing (2023–2025) splits."},{"type":"para","text":"As a first step, we use in-context learning (ICL) brown2020language as a probe to verify whether training-based solutions, i.e., learning from historical instances, are promising."},{"type":"para","text":"Specifically, we test TextRev - ICL , which retrieves one temporally nearby historical example as a demonstration for each TextRev instance, and TextRev - ICL-C , which further adds an explicit conservative no-change option to the prompt inspired by Figure 5 which highlights the instability issue (see Section X ). Table 7 shows that both variants make TextRev more effective: they outperform Gemini-2.0-Flash and TimesFM-2.5, and even surpass the best-performing AvgEns . Thus, we suggest that future training-based methods consider effectiveness and stability."},{"type":"heading","level":1,"text":"5 Conclusions"},{"type":"para","text":"We introduce a new multimodal time-series forecasting benchmark, TimesX, which contains real-world, cross-domain time series with high-quality, detailed textual context. TimesX includes a dataset generation pipeline that keeps the benchmark leakage-free via time isolation and enables automatic refresh via a dataset agent. We conduct a detailed empirical study of zero-shot multimodal TSF approaches on TimesX and find that earlier benchmarks either overestimate the performance of LLMs compared with TSF models or understate the importance of textual context for forecasting accuracy. Our evaluation also shows that simple ensemble methods outperform seemingly stronger agentic solutions on TimesX."},{"type":"heading","level":1,"text":"Impact Statement"},{"type":"para","text":"This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"redacted \\\\correspondingauthor Haoxin Liu <hliu763@gatech.edu> . This work was done while Haoxin Liu was a Student Researcher at Google Research. This work is accepted by The International Conference on Machine Learning (ICML) 2026.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Time-series forecasting (TSF) Liu_FOIL is a ubiquitous task across numerous domains and is essential for informed decision-making. Motivated by success in NLP, there has been significant work in recent years on time-series foundation models (TFM) for forecasting, ranging from re-purposing LLMs directly for forecasting ( gruver2023large ; tan2024 ) to fine-tuning pretrained LLMs on time-series data ( zhou2023one ; chang2023llm4ts ) to pretraining time-series foundation models from scratch ( das2023decoder ; goswami2024moment ; woo2024unified ; ansari2024chronos ; kamarthi2024lptm ) . These models have demonstrated promising zero-shot TSF capabilities, often outperforming traditional statistical and supervised methods, using only the historical context of the time-series at inference time. Although historical numerical data provide a fundamental basis for prediction, they often lack the complete context necessary for reliable and accurate forecasts. Human forecasters often integrate additional information such as background knowledge, events and constraints, which can usually be captured through natural language ( Liu_et_al_2024a_LSTPrompt ) - we call these textual contexts . Improving the ability of time-series foundation models to effectively leverage and integrate diverse textual contexts remains an ongoing challenge. This highlights a crucial need for high-quality and comprehensive multimodal datasets and benchmarks to propel research in this area.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the same time, existing context-enriched multimodal TSF evaluation benchmarks and their construction techniques present several limitations when benchmarking pretrained models on them. The first limitation is the unknown generalization gap between the benchmarking metrics and the real world performance. This is often caused by either the benchmark being restricted to a small scale with a narrow selection of domains, or it being fully synthetic .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The second significant challenge across existing benchmarks is data leakage . Pretrained TFMs and LLMs may have already ingested evaluation data, leading to unknown data contamination and unfair comparison between methods. A benchmark itself is subject to short-lived validity as pretrained models will continuously update their pretraining datasets, which results in their knowledge cut-off becoming more recent than the benchmark’s data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The third limitation comes from our observation that the types and granularities of textual contexts in previous benchmarks are often limited and vary wildly. Textual contexts derived from real data include metadata information about the time-series, timestamp-related information such as important dates and holidays, textual event data related to the time-series, and textual data capturing statistics and trends of related exogenous variables correlated with the target time-series. The ability of multimodal models to process these different types of contexts, and the availability and quality of these different contexts in the datasets have a significant effect on the model’s benchmark performance. When a multimodal model falls short on existing benchmarks, it is difficult to conclude whether the shortcoming is due to the fact that the method does not effectively utilize the provided information, or it is because the contexts provided in the benchmark are of a specific type, too vague, or lacking relevant details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address these fundamental limitations and establish an unbiased benchmark for context-enriched time series forecasting, we introduce TimesX , a novel multimodal benchmark designed to focus on (1) real, large-scale, cross-domain data, (2) a dataset generation pipeline providing continuous mitigation of data leakage, and (3) a collection of comprehensive, fine-grained textual contexts. We further introduce several promising baselines that combine TFMs and LLMs to achieve non-trivial multimodal, context-enriched forecasting performance on our benchmark.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The primary contributions of this paper are threefold:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TimesX is, as far as we are aware, the first real-world, large-scale, and cross-domain context-enriched time series forecasting benchmark. It encompasses 19 diverse domains with a total of 190 variables, covers diverse global geographical regions and includes daily and weekly frequencies. It also links each target numeric series to multiple forms of detailed textual contexts. Unlike existing multimodal time-series benchmarks, all time series and textual data in TimesX are from real-world observations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose two new mechanisms in the dataset generation pipeline of TimesX to help prevent data leakage and guarantee the validity of the textual context. The first one is an automated data collection pipeline that adopts strict timestamp alignment and isolation during each of its steps. The second one is a hypothesizer-verifier-enricher framework for fact-checking and enriching the textual contexts. These mechanisms combined ensure that TimesX is verifiable, leakage-free, and can be updated in the future for benchmarking methods with new pretraining cut-off dates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We empirically verify that TimesX addresses the shortcomings of existing benchmarks. In addition, we conduct a thorough empirical comparison of current (zero-shot) multimodal TSF approaches on TimesX, involving more than 312,000 independent LLM inferences and revealing new observations that were not discovered by existing benchmarks. In particular, we discover that earlier benchmarks like Williams_2024 (while being a great test for instruction following in forecasting) tend to severely over-estimate the performance of LLMs over TSF models. At the same time, benchmarks like Liu_et_al_2024b_TimeMMD under-report the importance of textual context information in forecasting accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Limitations and future works are detailed in Appendix A .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In addition to the time-series foundation models mentioned previously, recent works jin2023time ; Liu_et_al_2024a_LSTPrompt ; wang2025chattime have started to adapt pre-trained LLMs to perform forecasting when textual contexts are available, often by aligning the modalities of time series and natural language. The development of these models has been paralleled by new benchmarks designed to evaluate them. ChatTS ( ChatTS ) , for instance, generates synthetic time series paired with detailed attribute descriptions to train a multimodal LLM for understanding and reasoning tasks. Similarly, the Time-MQA framework ( Kong_2025 ) introduces the TSQA dataset, which contains real-world time series with template-based meta information and turns time-series forecasting into a question-answer format. Benchmarks such as those referenced in CiK ( Williams_2024 ) contain real-world time series and manually crafted textual contexts that are critical for forecasting. Time-MMD ( Liu_et_al_2024b_TimeMMD ) provides a context enriched benchmark constructed by keyword-based web searches of nine domains, one variable for each. MTBench ( Chen_2025 ) aligns stock prices with financial news and weather reports with temperature records, curating tasks that require reasoning about them. The MoTime ( Zhou_2025 ) dataset suite provides a collection of multimodal datasets pairing time series with static modalities like text and images. Noticeably, these early efforts were often constrained by a scarcity of high-quality, large-scale, real-world datasets, leading some to rely on synthetic data and specific forms of textual contexts. We summarize several representative benchmarks compared to our proposed TimesX in Table 1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The TimesX Benchmark\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Design Principles of TimesX\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We will first present the core principles that distinguish TimesX from other multimodal forecasting benchmarks: (i) real-world data, (ii) leakage mitigation, and (iii) comprehensive, high-quality context and (iv) large-scale evaluation. Note that we further provide empirical evidence that demonstrates the importance of these principles and where other benchmarks might fall short.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.1.1 Real-World Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Principle Description : TSF is highly complex due to real-world dynamics, complex processes, and diverse domains. On the other hand, many existing benchmarks use either synthetic time series, synthetic textual contexts, or both, without demonstrating how the conclusions on synthetic data can transfer to real-world scenarios ( Williams_2024 ; tan2024 ) . TimesX differentiates itself from those benchmarks by using real-world data for both the time series and the textual contexts .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why it matters : To test whether conclusions on synthetic data can transfer to real data, we evaluate three methods on a synthetic context dataset (CiK) and our real dataset (TimesX). We report the aggregated MASE (Mean Absolute Scaled Error) of the three methods which are: (i) TimesFM-2.5 (a TFM) using only the time series, (ii) Gemini-2.0-Flash (an LLM) using both the time series and the textual contexts, and (iii) CodeRev , Gemini-2.0-Flash writing and executing code to revise TimesFM-2.5 outputs based on the context. More details are in Appendix N .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Figure 1 , the rankings of the three methods on CiK (synthetic) and our real data are completely different . On CiK, directly feeding the time-series along with associated text context to Gemini has a huge lead over TimesFM-2.5. Moreover, asking Gemini to edit the output of TimesFM-2.5 via code does even better, significantly outperforming the other two methods. On the other hand on TimesX, we see that all the methods are much closer, and TimesFM-2.5 is actually better than Gemini. Further CodeRev degrades the performance of using both TimesFM-2.5 and Gemini directly. This shows that CiK has a strong bias that clouds its conclusions: in CiK, textual contexts are both synthetic and extremely specific - these specific contexts are then used to write code to modify an input time-series and generate the final forecasting task. Therefore, such tasks can be easily solved by instruction-following language models and even better by language models that generate code to accomplish a simple modification task. These conclusions, however, do not generalize to the real-world tasks, whose contexts are less specific and do not relate to the forecasting task via the coding path.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We highly recommend using both synthetic and real-world benchmarks for a more robust evaluation. Carefully designed synthetic benchmarks can be useful for testing specific capabilities such as instruction following or different types of reasoning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.1.2 Mitigating Data Leakage\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Principle Description : A fair benchmark dataset should be able to mitigate any potential leakage of benchmark tasks into the pretraining datasets of the models being evaluated. Given that many pretrained models do not release their pretraining datasets, existing benchmarks that claim to use data from unused sources (data isolation) for evaluation have three drawbacks: (1) Potentially unidentifiable data contamination. (2) Limited number of clean tasks left for benchmarking multiple methods. (3) Short-lived validity i.e., future versions of the models considered may unintentionally leak the benchmark into their pretraining data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast, TimesX adopts strict time isolation to prevent data contamination : we timestamp TimesX thoroughly, and recommend strictly using benchmark tasks that occur after a target model’s knowledge cutoff date. Moreover, the TimesX pipeline is designed as re-updateable over time , i.e., data collection, validation, alignment, and model evaluation can all be automated. To the best of our knowledge, TimesX is the first benchmark capable of being automatically refreshed for evaluating future pretrained methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why it matters : To study how data contamination affects evaluation results, we evaluate on 120 variables from the Search Trend (see Section 3.2 ) subset of TimesX. We keep the numeric series, windowing, and prompts fixed, and split the test period by the public knowledge cutoff of Gemini-2.0-Flash and DeepSeek-V3 (June 2024). We report the MASE aggregated by Geometric Mean in Table 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pretraining data descriptions of the two TFMs preclude any time-series data leakage in both time periods, and both TFMs report stable results with less than 2% delta. In contrast both LLMs see an increase of error by about 13% after their knowledge cutoff.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.1.3 Comprehensive and High-Quality Context\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Principle Description : Most multimodal forecasting benchmarks have a very limited set of text contexts - restricted to either static metadata ( Zhou_2025 ) , date and weather derived features ( wang2025chattime ) or synthetically generated contexts ( Williams_2024 ) . In contrast, TimesX uses not only a union of all common context types , but also a more high-quality description of text events and corresponding alignment of these contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In particular, we categorize these various kinds of contexts into: (1) Metadata : descriptive high level summaries of the forecasting variable in question (2) Calendar -derived features like holidays (3) Covariates which cover textual information around other related time series (4) Time-stamped events : Textual description of related events aligned with time-windows of the variable in question. Section 3.2.2 contains more details about how we generate these contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why it matters : TimesX links each target time-series to all these available context types and aligns them with the timestamps of the time series. The end result is a benchmark with diverse, high-quality textual contexts. To quantify its effect on TSF accuracy, we run a controlled experiment where we take the time-series in the Time-MMD benchmark and use our methodology to replace the textual events using the construction rules in Appendix E . We keep the numeric series, the LLM (Gemini-2.0-Flash), and the prompt fixed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 (full results in Table 18 ) compares the performance of Gemini-2.0-Flash given these newly generated text events versus the original context in the Time-MMD benchmark. Across all nine Time-MMD datasets, this context swap reduces the geometric–mean aggregated MASE from 0.906 0.906 to 0.840 0.840 (a 7.3 % 7.3\\\\% relative drop) , suggesting that conclusions drawn under low–quality context might be misleading; high–quality context changes the picture .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further show that with the current dataset scale (typically fewer than 20 variables), model performance rankings are quite unstable , as illustrated in Figure 34 and Appendix T .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Construction of TimesX\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first provide a high level summary of our dataset and dataset construction methodology. TimesX contains time-series obtained from 19 domains . Each domain has 10 10 variables, resulting in a total of 190 variables (quantities evolving with time). TimesX covers different geographical regions : North America (United States, Canada, Mexico), Asia (China, India among others), Europe (United Kingdom, Norway among others), South America and Africa.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that we have a separate portion of the dataset that is derived from the time-series in Time-MMD Liu_et_al_2024b_TimeMMD , where the text context is generated using our methodology, but only used for ablations in Section 3.1.3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The TimesX dataset is constructed entirely from scratch , rather than by merging existing datasets. Moreover, TimesX is designed to be automatically refreshable over time, enabling continuous mitigation of information leakage as pretrained models evolve\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.1 Numerical Time Series Construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The numerical component consists of target variables whose future values are forecasted given a historical window.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Granularities and sources. TimesX includes two temporal granularities: Weekly series from Google Search Trends across 12 domains 1 1 1 https://trends.google.com/trends/ ; and Daily series including (i) commodity prices from a market data API 2 2 2 https://marketstack.com/ and (ii) major USD exchange rates from a currency rates API 3 3 3 https://frankfurter.dev/ . These API data can be automatically refreshed to extend the benchmark over time. Monthly/quarterly coverage is discussed in Appendix A.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Time alignment and missing values. All series within the same frequency share the same timestamps. Missing values are handled as described in Appendix D.1 . Appendix I provides the full list of domains and variables.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.2 Text Context Construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our core contribution is aligning comprehensive and diverse text context information with the variables that evolve over time. We will see that methods that capture both this context information along with the past of the time-series can achieve superior performance compared to methods that only use the past time-series component. As mentioned before we provide 4 kinds of text contexts for all time-series variables: Metadata, Calendar features, Covariates and Time-Stamped Events. Section 3.2.3 shows an example of a time series with each of these four contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metadata and calendar contexts. The Metadata and Calendar text contexts are easily constructed from the sources of the numerical time-series themselves and Python Holidays library respectively, as described in Appendix C .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Covariate contexts. For each time-series in a given domain, we use the other time-series from that domain to extract Covariate text contexts. Specifically, we calculate features of these covariates over the historical window, including mean, median, max and min with the corresponding date, and overall trend direction. These features are then transformed into natural language descriptions, detailed in Appendix D .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Time-stamped events via dataset agents. Constructing and aligning textual events is the most challenging component. Following our design principles in Section 3.1 , we must satisfy three goals at once: real-world data, leakage-free (refreshable) , and high-quality . A single LLM or a naive web search cannot reliably meet these goals. Therefore, we adopt a multi-agent automated workflow with four agents: Hypothesizer , Verifier , Enricher , and Synthesizer , as illustrated in Figure 2 . Each agent interacts with an LLM under constraints and has dedicated tools (time-bounded web search and lightweight crawlers). Overall, the Hypothesizer and the Verifier act adversarially to ensure event truthfulness and timestamp accuracy, which prevents information leakage; the Verifier then guides the Enricher to supply missing details thus ensuring quality. The whole workflow is automated, enabling regular updates of the benchmark data. We provide the details of each LLM role:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(i) the Hypothesizer identifies points of interest in the time series (e.g., local maxima, unexpected movements or peaks in the corresponding search trend) and iteratively calls an LLM agent with integrated web search to build an initial event set that is able to match these points.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(ii) then, the Verifier uses crawlers to fetch relevant URLs corresponding to each event window and conducts fact checks, filters out any hallucination or leakage, and prepares a checklist of missing details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(iii) The Enricher resolves the checklist using strictly time-bounded web searches, merging multiple sources to fill in the details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(iv) finally, the Synthesizer , a reasoning LLM, aggregates all pieces of evidence to finalize the event description, adjust timestamps, and discard events with unresolved doubts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using this agent, TimesX automatically produces an event corpus that matches the time window, contains verifiable facts (each claim has a supporting URL), accurate time stamps, and rich detail. More details and hyperparameter analysis experiments are provided in Appendix E . An execution log demo is in Appendix F .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our pipeline runs in three-month time blocks and is naturally parallelizable across variables and blocks; Phase 1 uses coverage-based early stopping to reduce cost. Users can control the budget by choosing different LLMs per role and tuning the maximum iterations (Phase 1) and search budget (Phase 3). Under our current setup (Gemini 2.5 Pro + Gemini 2.5 Flash), the cost is about $0.7 per variable per three-month block ; it can be reduced by using open-source LLMs or batching verification so multiple events reuse the same calls.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.3 Data Example\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We provide below an example from TimesX corresponding to the GAS_PRICE time-series.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Time series : See Figure 3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metadata : This time series records gasoline price (USD/GAL) in the Commodity Price domain, with daily frequency. Prediction target period: from 2024-09-01 to 2024-09-15.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Calendar : In the format of (timestamp, value), historical data is: (2024-06-02, 2.4116), (2024-06-03, 2.3279),… Upcoming holidays in the prediction window: Labor Day (2024-09-02).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Covariates : from 2024-06-01 to 2024-08-31: (1) Brent Crude Oil (USD/BBL): The maximum value was 87.43 87.43 , occurring on July 4, the minimum…, showing an overall downward trend. (2)…\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Events : (1) On June 2, 2024, OPEC+ agreed to extend deep oil output cuts [1,3], the cut of 2.2 million bpd would be extended until September 2024, after which it would be gradually phased out [2,3]. Source: [1] [2] [3]; (2) …\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.4 Quality Checks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Diversity. We profile numerical and textual features of TimesX (Appendix J ) and visualize them with PCA/t-SNE (Appendix M ). Samples are broadly dispersed, and variables from different sources form distinct clusters, supporting the benefit of multi-source integration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Factual correctness. Here, we focus on the most sensitive component: the accuracy of automatic timestamp annotation. We ensure the accuracy via a three-stage pipeline (Appendix G ): verification, programmatic time-bounded search (± k k days), and cross-source adjudication in the agent workflow. A manual audit of 50 samples shows 94% exact matches; 4% conservative (later) offsets; and 2% earlier due to date ambiguity (Appendix G ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 TimesX Release and Refresh Consideration\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The core TimesX dataset now contains 190 variables spanning Jan 2018 to Oct 2025 to support both training (2018–2022) and evaluation (2023–2025). We plan to update the TimesX dataset every three months and version each release.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The extended TimesX further supports 11 languages beyond English (Afrikaans, French, German, Hindi, Japanese, Korean, Portuguese, Simplified Chinese, Spanish, Swahili, and Turkish), as well as 5 rare diseases. See Appendix A.5\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Empirical Study\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Evaluation Settings\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We focus on zero-shot solutions and consider the following:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pretrained TFMs : We select three SOTA TFMs including TimesFM-2.5, Moirai-2.0, and Sundial. All are towards the top on GIFT-Eval aksu2024gift benchmark, with TimesFM-2.5 being the top open model with no leakage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pretrained LLMs : We consider three well-adopted LLMs (implementation details in Appendix Q ): closed-source Gemini-2.0-Flash and GPT-4o, and open-source DeepSeek-V3.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Composed (Ensemble And Agentic) Solutions : We construct the following four solutions: (1) AvgEns : An ensemble that averages the forecasts from a pair of TFM and LLM. We simply set equal weights. (2) TextRev : An LLM taking the forecast of a TFM as text and revising it according to the context. (3) CodeRev : An LLM writing and executing code to revise the forecast of a TFM according to the context. (4) FuncRev : Similar to CodeRev , but the LLM is limited to a selection of functions. See Appendix Q .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each variable in TimesX and its context corpus, the look-back window and the forecast horizon are set to 96 and 12, respectively. The rolling window is set to 4 for weekly data and 12 for daily data to balance sample count and sample diversity. We then include all relevant metadata, calendar info and covariates info. To avoid future info leakage, we select the K K most recent textual events whose announcement dates strictly precede the first timestamp of the prediction horizon to add to the context. We fix k = 10 k=10 to balance prompt length and information effectiveness, following common practices Liu_et_al_2024b_TimeMMD ; li2025language .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TimesX spans 2018–2025 and is refreshable. However, to avoid pretraining data contamination, we construct evaluation examples whose forecast horizon begins after the pretraining cutoff of all involved models, detailed in Appendix P . Unless stated otherwise, we use 2024-07-01 as the cutoff, i.e., evaluating only the subset whose horizon start time is 2024-07-01 or later. See models’ knowledge cutoffs in Appendix P . Web access is limited to the build-time agent; the benchmark runs offline. Appendix P.1 further discusses how to prevent test-time leakage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following the conventions in other popular forecasting benchmarks like GIFT-Eval ( aksu2024gift ) , we use normalized MASE (mean absolute scaled error) as our main metric. Since the different variables have very different scales, we calculate the average MASE over all rolling windows of a variable and normalize that by the average MASE of a seasonal naive baseline. Then we take the Geometric Mean (GM), for robustness to normalization choice, of these normalized MASE ratios across all variables. See Appendix N . We also compute the average MASE rank of each method over the variables. For both metrics, smaller is better.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This setup yields about 2.5K samples (Appendix O ). To account for stochasticity, we repeat each evaluation 10 times with different seeds when applicable, resulting in more than 312K independent LLM inferences. Note that using a shorter rolling-window stride can increase the sample count but also the evaluation cost.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Benchmarking Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 shows the benchmark results on TimesX (details in Appendix Y.1 ). As multimodal solutions, though the zero-shot LLMs have the edge over the unimodal zero-shot TFMs, this edge is not as significant on TimesX as observed on other synthetic benchmarks . This points out the gap between synthetic contexts and real world contexts, that some synthetic benchmarks tend to severely over-estimate the performance of LLMs over TSF models because the synthetic contexts were providing the exact information required for bettering the forecast. Real world contexts are in contrast too nuanced to always deliver a large performance boost even when guaranteed to be related. This conclusion is further confirmed by the observation that the agentic CodeRev method is performing worse than its two components working alone, contradicting the claim of ( Williams_2024 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In terms of the composed solutions, to our surprise the best performers on TimesX are the simple average ensembles of different pretrained models (i.e., AvgEns ). We by no means want to suggest it as the optimal composition, but instead want to reiterate the stochastic and flexible nature of LLMs and the fact that it would take a great deal of effort to design the interaction with them in a composed solution to just outperform simple averaging ensembles.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also use the continuous ranked probability score (CRPS) metrics to measure uncertainty, detailed in Appendix U.1 . We initially observe that all three LLMs achieve lower CRPS than the TFMs . This suggests that our constructed context helps LLMs model future uncertainty better than TFMs, which only use numerical values.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As an early experiment, we also evaluate reasoning LLMs, including GPT-5, Gemini-2.5-Flash, and DeepSeek-R1. To reduce data contamination, we only use evaluation samples after Jan 2025. Table 17 (detailed in Appendix Y.4 ) shows that these reasoning models do not provide a clear advantage .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Ablation: Context Types\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 presents the MASE and MASE rank when different combinations of contexts are provided to a chosen multimodal method (Gemini-2.0-Flash). Comparing to only using the high level metadata, the inclusion of either calendar, calendar and covariates, or calendar and events brings significant improvement. Using all context types improves the accuracy further eventually being around 16% better than using only static metadata. It is worth noticing that we cannot observe the incremental gain from adding covariates or events alone. We speculate there is crucial interaction between those two context types, for instance the effect of an event on the target variable can be quantified by a similar leading effect on a covariate. Figure 4 suggests this speculation may generalize to other LLMs. This observation is aligned with our expectation that a multimodal method can compound the gains by composing the information in different context types . Appendix V reports domain-level holiday and covariate effects.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Ablation: Domains\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 6 breaks down the ranking of the benchmark methods on each of the 19 domains (see Appendix Z for MASE). There is no dominance of one zero-shot method over others - in fact the orderings of TimesFM-2.5 and Gemini-2.0-Flash are complementary. In terms of composed solutions, the observation is consistent with Table 4 that AvgEns performs well while other compositions can be worse than its components acting alone. We also observe two domain groups where multimodal solutions show a clear advantage over TFMs. First, in Shopping, calendar information likely provides strong signals about when sales occur, which is hard to infer from the time series alone. Second, in RMC (Raw Materials & Constructions), SAM (Specialty & Advanced Materials), SHVM (Strategic & High-Value Materials), and Curr (Currency), the series are strongly affected by external policies, which are captured in the event context.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Why Agentic (Revision) Methods Underperform?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"None of the three revision methods ( FuncRev , CodeRev , and TextRev ) outperforms the simple AvgEns on TimesX. Figure 5 shows that their forecast errors have a wider spread and more severe outliers than TimesFM-2.5, which leads to worse geometric means. Gemini-2.0-Flash alone shows a similar pattern. We speculate that this is due to LLM stochasticity and the fact that LLM-based revision does not reliably preserve the temporal structure in the initial forecast. Among the three, TextRev performs best, which suggests that effective code-based revision may require instruction fine-tuning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 Future Solutions: Could Training Help?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work focuses on zero-shot methods. To support future training-based research, TimesX provides both training (2018–2023) and testing (2023–2025) splits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As a first step, we use in-context learning (ICL) brown2020language as a probe to verify whether training-based solutions, i.e., learning from historical instances, are promising.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specifically, we test TextRev - ICL , which retrieves one temporally nearby historical example as a demonstration for each TextRev instance, and TextRev - ICL-C , which further adds an explicit conservative no-change option to the prompt inspired by Figure 5 which highlights the instability issue (see Section X ). Table 7 shows that both variants make TextRev more effective: they outperform Gemini-2.0-Flash and TimesFM-2.5, and even surpass the best-performing AvgEns . Thus, we suggest that future training-based methods consider effectiveness and stability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce a new multimodal time-series forecasting benchmark, TimesX, which contains real-world, cross-domain time series with high-quality, detailed textual context. TimesX includes a dataset generation pipeline that keeps the benchmark leakage-free via time isolation and enables automatic refresh via a dataset agent. We conduct a detailed empirical study of zero-shot multimodal TSF approaches on TimesX and find that earlier benchmarks either overestimate the performance of LLMs compared with TSF models or understate the importance of textual context for forecasting accuracy. Our evaluation also shows that simple ensemble methods outperform seemingly stronger agentic solutions on TimesX.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Impact Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.\"\n }\n]","isConversational":false}}},{"rowIdx":493,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06856"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Gen4U: Unifying Video Generation and Understanding via Diffusion"},"authors":{"kind":"list like","value":["Michael King","Aravindh Mahendran","Matthew Koichi Grimes","Fedor Kitashov","Adham Elarabawy","Pedro Velez","Maks Ovsjanikov","Viorica Pătrăucean"],"string":"[\n \"Michael King\",\n \"Aravindh Mahendran\",\n \"Matthew Koichi Grimes\",\n \"Fedor Kitashov\",\n \"Adham Elarabawy\",\n \"Pedro Velez\",\n \"Maks Ovsjanikov\",\n \"Viorica Pătrăucean\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06856v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Current paradigms in visual representation learning struggle to reconcile geometry with semantics. Contrastive methods and large Vision-Language Models (VLMs) (Radford et al. , 2021 ; Beyer et al. , 2024 ; Team, 2025 ) operate primarily in language space, yielding strong semantics but neglecting precise spatio-temporal details. Conversely, pixel-level reconstruction approaches like masked auto-encoding (MAE) (Tong et al. , 2022 ; Wang et al. , 2023 ; Carreira et al. , 2025 ; Zhang et al. , 2026 ) excel at preserving local geometry and motion but struggle to generalise to broad semantic tasks. A single, task-agnostic foundation video model that supports both types of capabilities remains an open problem."},{"type":"para","text":"Inspired by the predictive coding theory of the brain, which posits that prediction is key to cognition (Rao and Ballard, 1999 ; Clark, 2013 ) , we investigate how to build general-purpose video models based on generative video diffusion transformers. We argue that generating temporally and geometrically consistent video requires the model to implicitly learn the underlying mechanics of the physical world, such as motion, object permanence, and part-object relationships. However, recent studies probing frozen video diffusion representations concluded that they capture low-level geometry but struggle with high-level semantics, casting doubt on their eligibility as general-purpose perception encoders (Vélez et al. , 2025 ; Zhu et al. , 2026 ) ."},{"type":"para","text":"In this work, we demonstrate that state-of-the-art video diffusion models do, in fact, overcome this limitation as they become more capable at generating physically and temporally consistent videos. By systematically investigating the intermediate activations of both proprietary (Veo3 (Google DeepMind, 2025 ) ) and open-weight (Wan2.2-T2V-A14B (Wan et al. , 2025 ) ) models, we map the hierarchy of their latent spaces using recent zero-shot mutual k-NN alignment metrics (Huh et al. , 2024 ; Zhu et al. , 2026 ) and we identify optimal spots for extracting linearly separable global semantics and more entangled, spatially-distributed features that require attention-based pooling to extract effectively."},{"type":"para","text":"Based on these observations, we introduce Gen4U (Generation for Understanding), a framework that extracts and repurposes generative representations for a diverse set of downstream visual tasks. Because Gen4U identifies the optimal block and noise level for feature extraction, it requires only a single forward pass through the diffusion backbone. This makes it as computationally efficient as a standard discriminative visual encoder, e.g. (Carreira et al. , 2025 ; Wang et al. , 2024 ) , circumventing the high inference costs typically associated with iterative denoising. By pairing these frozen representations with lightweight decoders, Gen4U achieves strong perception performance without altering the pre-trained generative weights. Our extensive evaluations demonstrate that frozen video diffusion models can function as general-purpose video encoders."},{"type":"para","text":"To summarise, our main contributions are:"},{"type":"para","text":"Latent space analysis : We map the evolution of diffusion features across depth and noise levels using recent alignment metrics, identifying the optimal points to extract representations for understanding."},{"type":"para","text":"Semantics and temporal dynamics : We show that Gen4U captures rich, high-level understanding, achieving state-of-the-art results on zero-shot alignment and video classification on Something-Something V2 dataset (Goyal et al. , 2017 ) , alongside satisfactory results on image and video captioning."},{"type":"para","text":"Geometry understanding : We demonstrate that these same frozen representations excel at lower level geometry awareness, achieving strong performance on monocular depth estimation and camera pose estimation."},{"type":"heading","level":1,"text":"2 Related work"},{"type":"para","text":"The most common paradigms for learning visual representations are variations of masked autoencoding (MAE) (He et al. , 2021 ; Tong et al. , 2022 ; Wang et al. , 2023 ) , scaled in 4DS (Carreira et al. , 2025 ) and D4RT (Zhang et al. , 2026 ) , and contrastive learning, with or without negative pairs (SigLip (Zhai et al. , 2023 ) , SimCLR (Chen et al. , 2020 ) , CLIP (Radford et al. , 2021 ) , BYOL (Grill et al. , 2020 ) , BRAVE (Recasens et al. , 2021 ) , DINO (Caron et al. , 2021 ) to name a few). These methods generally strike a trade-off between low level scene understanding and high level semantics, failing to support both. Even hybrid approaches (Zhao et al. , 2024 ; Wang et al. , 2024 ; Papalampidi et al. , 2023 ) that combine masked auto-encoding (MAE) with contrastive learning can still accurately span only one category of tasks. More recently, the V-JEPA line of works (Bardes et al. , 2024 ; Assran et al. , 2025 ; Mur-Labadia et al. , 2026 ) focus on predicting the future in latent space to avoid the computational cost of pixel-level prediction, and show strong results on a wide range of understanding tasks. These pre-training methods lead to strong results on downstream tasks, but most of the time they require extensive fine-tuning or have fairly complex pre-training pipelines, balancing different optimisation objectives. Moreover none of these pre-training methods support high-quality video generation."},{"type":"para","text":"Driven by the impressive realism and diversity of the outputs generated by diffusion models (Rombach et al. , 2022 ; Gupta et al. , 2023 ; Google DeepMind, 2025 ; Wan et al. , 2025 ) , multiple works have started to investigate their understanding capabilities. In (Li et al. , 2023a ) the authors show that a diffusion-based setup can perform zero-shot classification by conditioning the generative model with the possible classes and selecting the class that leads to the best reconstruction of the noise added to the input image. In (Luo et al. , 2023 ) , the authors build diffusion hyperfeatures by combining the activations from different blocks and noise levels to build robust pixel-level descriptors. In (Wiedemer et al. , 2025 ) , the authors show impressive generation results by prompting the Veo3 model with challenging tasks that require geometry and physics understanding, hinting that Veo3 has implicitly learnt a world model. In addition, works like (Ha and Schmidhuber, 2018 ) , (Hafner et al. , 2025 ) , (Bruce et al. , 2024 ; Parker-Holder et al. , 2024 ) confirm that generative models support learning a world model for simulated environments, with representations that can be used to inform action policies."},{"type":"para","text":"The closest work to our approach is (Vélez et al. , 2025 ) . The authors repurpose activations extracted from variants of Walt model (Gupta et al. , 2023 ) adapted for image and video, for understanding tasks and conclude that diffusion models excel at lower level understanding problems like depth or camera pose estimation, but struggle with high-level semantic tasks. Similarly, in (Zhu et al. , 2026 ) , the authors measure the alignment of representations produced by different visual encoders amongst themselves and with language embeddings. Interestingly, their findings suggest that diffusion models like Walt produce representations that align with established visual encoders like DINOv2 (Oquab et al. , 2024 ) and VideoMAEv2 (Wang et al. , 2023 ) , but the alignment with text embeddings is significantly lower. In our work, we show that the recent generation of video diffusion models have strong semantic awareness, in addition to geometry and motion understanding, constituting a strong foundation for a complete perception system. We show state-of-the-art results on video classification on the challenging SSv2 dataset (Goyal et al. , 2017 ) , compared to strong baselines from the other families of visual pre-training methods (MAE, contrastive, latent prediction). To further the investigation of semantic understanding, we set up captioning experiments, showing satisfactory performance compared to a strong SigLIP-based baseline (Zhai et al. , 2023 ) from the PaliGemma family (Beyer et al. , 2024 ) ."},{"type":"para","text":"The interplay between generation and understanding started to receive more attention recently. For example, detailed point tracking conditioning leads to more consistent generation (Jeong et al. , 2025 ) , or detailed language captioning produced by Gemini models lead to improved Veo generation (Google DeepMind, 2025 ) . Transfusion (Zhou et al. , 2025 ) elegantly combines generation and understanding pre-training into a single model, allowing the two objectives to support each other. However, in all these works, the main finding has been that advanced understanding models lead to significant improvement in generative models. The other direction, how can generation help understanding, has so far been explored mainly for low-level geometry tasks. In our work, we show that frozen diffusion generative models can power video understanding and achieve competitive performance on a wide range of visual tasks, while being able to generate high-quality videos. This unifies video generation and understanding paradigms. We estimate that this finding can be very impactful for the community, by encouraging more research into understanding the representation power of diffusion models. In addition, the implications from a practical point of view could be substantial, as it could lead to one visual model shared across generation and understanding."},{"type":"heading","level":1,"text":"3 The structure of the generative manifold"},{"type":"para","text":"The iterative diffusion generation requires many denoising steps to generate a video. This slow sequential operation mode is not suitable for video encoding. To design efficient video feature extractors based on video diffusion models, we conduct a detailed investigation of the structure of the latent embeddings spawned by these models, in search for the optimal depth and noise level from where to extract representations with a single forward pass through the model. After a brief overview of diffusion models, we detail the zero-shot and trained light probes used for our analysis, and we conclude this section with a summary discussion of the findings."},{"type":"heading","level":2,"text":"3.1 Background on latent diffusion models"},{"type":"para","text":"Video Latent Diffusion Models (LDMs) like Veo3 operate within a compressed, lower-dimensional representation space. During training, given an input video x ∈ ℝ F × H × W × C x\\in\\mathbb{R}^{F\\times H\\times W\\times C} (containing F F frames), an encoder E maps the data into a spatiotemporal latent code z 0 = E ( x ) z_{0}=\\textit{E}(x) . A forward process corrupts this latent representation over T T steps. At an arbitrary noise level t ∈ [ 0 , T ] t\\in[0,T] , the corrupted latent z t z_{t} is generated by adding Gaussian noise to z 0 z_{0} : z t = α ¯ t z 0 + 1 − α ¯ t ϵ z_{t}=\\sqrt{\\bar{\\alpha}_{t}}z_{0}+\\sqrt{1-\\bar{\\alpha}_{t}}\\epsilon , where ϵ ∼ 𝒩 ( 0 , I ) \\epsilon\\sim\\mathcal{N}(0,I) and α ¯ t \\bar{\\alpha}_{t} dictates the variance schedule. To invert this process, a backbone network, typically a Diffusion Transformer (DiT) or a U-Net, denoted as f θ f_{\\theta} , is trained to predict the injected noise ϵ \\epsilon . The network is conditioned on the corrupted latent z t z_{t} , the noise step t t , and contextual information c c (e.g., text embeddings). The model is optimised via the standard reweighted variational lower bound: ℒ L D M = 𝔼 z 0 , ϵ , t [ ‖ ϵ − f θ ( z t , t , c ) ‖ 2 2 ] \\mathcal{L}_{LDM}=\\mathbb{E}_{z_{0},\\epsilon,t}\\left[\\|\\epsilon-f_{\\theta}(z_{t},t,c)\\|_{2}^{2}\\right] . Note that flow matching models like Wan 2.2, also included in our study, replace this stochastic forward process with a continuous-time deterministic framework based on Optimal Transport. In this setup, for a continuous time step t ∈ [ 0 , 1 ] t\\in[0,1] , the corrupted latent is a linear interpolation between the data and a base noise distribution: z t = ( 1 − t ) z 0 + t ϵ z_{t}=(1-t)z_{0}+t\\epsilon . Rather than predicting the injected noise, the network f θ f_{\\theta} is optimised to regress the velocity vector field f t = d z t d t = ϵ − z 0 f_{t}=\\frac{dz_{t}}{dt}=\\epsilon-z_{0} that transports the data to the noise. The model is trained via the flow matching objective: ℒ F M = 𝔼 z 0 , ϵ , t [ ‖ f θ ( z t , t , c ) − ( ϵ − z 0 ) ‖ 2 2 ] \\mathcal{L}_{FM}=\\mathbb{E}_{z_{0},\\epsilon,t}\\left[||f_{\\theta}(z_{t},t,c)-(\\epsilon-z_{0})||_{2}^{2}\\right] ."},{"type":"para","text":"The backbone f θ f_{\\theta} consists of a sequence of L L layers (e.g., transformer blocks in the case of a DiT). We denote the intermediate spatiotemporal activations extracted from the l l -th layer at noise step t t as h t ( l ) h_{t}^{(l)} . During inference, the model iteratively denoises pure Gaussian noise z T z_{T} to recover an estimated clean latent z ^ 0 \\hat{z}_{0} . Finally, z ^ 0 \\hat{z}_{0} is mapped back to pixel space via a decoder D , yielding the generated video x ^ = D ( z ^ 0 ) \\hat{x}=\\textit{D}(\\hat{z}_{0}) ."},{"type":"para","text":"Figure 1 illustrates our diffusion-based video encoder Gen4U . We condition the diffusion transformer on a generic text embedding: e.g. A video of a scene , to not leak the ground truth (action label or caption) for semantic tasks. We have also experimented with conditioning on an empty text string, without significant difference in results. To deal with the inherent stochasticity of the diffusion process, we use a fixed random seed throughout all our experiments to ensure reproducibility. In all our plots, we report results for blocks and noise levels expressed as percentages obtained as a linear fraction of the total depth or noise steps."},{"type":"heading","level":2,"text":"3.2 Zero-shot probes"},{"type":"para","text":"PCA visualisation: To qualitatively assess the evolution of the diffusion features along network depth and noise levels, we plot in Figure 2 the PCA visualisation of the activations extracted from different depths and noise levels in Veo3. We can observe that at high levels of noise, the features are simple, low-frequency, encoding the general shapes in the scene. At lower noise levels, the features become more complex, encoding higher-frequency, more refined details. This is in line with the intuition that diffusion is spectral autoregression (Dieleman, 2024 ) : at each denoising step, the model learns to produce the features at the next (higher) frequency level."},{"type":"para","text":"Mutual k-NN alignment: To quantitatively assess the structure of the diffusion embedding space, we adopt the zero-shot Mutual k k -Nearest Neighbours (M k k NN) alignment metric proposed by (Huh et al. , 2024 ) and extended to video in (Zhu et al. , 2026 ) . The key idea is to compare the neighbourhood structure induced by the diffusion model’s intermediate representations against that of established, independently trained encoders, without requiring any learned mapping between the two spaces."},{"type":"para","text":"Concretely, given a dataset of N N videos, paired with ground truth captions, we extract a video-level embedding from each video using both the diffusion model and a reference encoder. For the diffusion model, we obtain intermediate activations h t ( l ) h_{t}^{(l)} at a chosen layer l l and noise level t t , and aggregate them spatiotemporally to produce a single vector per video (see appendix A for details). For the reference encoder, which may operate in a different modality (e.g., a language model) and produce embeddings of a different dimensionality, we similarly obtain one vector per video. In particular, for image models, we follow (Zhu et al. , 2026 ) and average the individual image features across input frames. We then independently construct the k k -nearest-neighbour graph in each embedding space and measure the average overlap between the two sets of neighbours. Intuitively, a high overlap indicates that the diffusion model organises videos in a manner consistent with the reference encoder, despite never having been trained with the same objective. More details are included in the appendix A . Following prior work (Huh et al. , 2024 ; Zhu et al. , 2026 ) , we additionally optimise over the choice of intermediate layers in both encoders and report the pair of layers that maximises the alignment score. We use k = 10 k{=}10 and N = 1024 N{=}1024 videos."},{"type":"para","text":"In Figure 3 , we show the alignment of Veo3 and Wan 2.2 respectively, against the Gemma-2-9b-it text encoder (Gemma Team, 2024 ) on the VATEX dataset. This metric has been shown to correlate with downstream performance in semantic and even non-semantic tasks (Zhu et al. , 2026 ) . In addition, we also extract in Figure 4 the alignment of diffusion representations against DINOv2 (Oquab et al. , 2024 ) , a powerful image representation model, and VideoMAEv2 (Wang et al. , 2023 ) , a strong video representation model. Note that the features Veo3, especially when probed at the appropriate depth and noise levels align significantly more strongly with text embeddings of their associated captions. We emphasize that the captions are not given as context during generation. This suggests that a powerful video generative model has the capacity of encoding rich semantic information within its intermediate features."},{"type":"heading","level":2,"text":"3.3 Linear and attention probes"},{"type":"para","text":"We use linear and attention probes to further assess the quality of diffusion representations. The linear head does global pooling over spatiotemporal dimensions followed by a linear projection. The attention probe uses a cross-attention head, with task-specific queries. We show results for video classification probes in Figure 5 ."},{"type":"para","text":"We also experiment with lightweight decoders that reconstruct the input in RGB space given the representations from different blocks and noise levels, to get a sense of the compression and artefacts present at each stage, but the results are less conclusive; see details and visualisations in appendix C ."},{"type":"heading","level":2,"text":"3.4 Discussion"},{"type":"para","text":"Feature routing across network depth: The evolution of diffusion representations follows a dual-axis trajectory along both depth l l and noise level t t . Interestingly, we observe distinct evolution patterns for the two models. The results for Wan 2.2 (Figure 3 right, Figure 4 right) show a unimodal pattern, aligning with broader diffusion literature (e.g., (Luo et al. , 2023 ) ). The alignment peaks early, at around 25% depth, indicating that the model resolves global structure and scene details early before gradually settling finer details. Conversely, Veo3 exhibits a distinct bimodal pattern, a phenomenon not previously reported in the diffusion literature. This mirrors the geometry of representations learnt by large-scale transformers (Valeriani et al. , 2023 ) , where the representation manifold expands early on (first peak), contracts significantly in the middle layers to route information, and then expands again toward the end (second peak) before the final output projection. We hypothesise a similar behaviour drives Veo3, supported by PCA visualisations (Figure 2 ) that show somewhat noisier activations halfway through the network, aligning with the dip in the bimodal pattern. We leave a conclusive characterisation of this behaviour for future work."},{"type":"para","text":"𝟔𝟎 % \\mathbf{60\\%} -noise semantic bottleneck: Despite exhibiting divergent topological patterns across depth, both models share a consistent optimal noise level for alignment against both text and visual encoders at t = 60 % t=60\\% . This indicates the emergence of a semantic bottleneck for both models at this specific noise level."},{"type":"para","text":"Evolution of feature complexity, linear vs. attention probes: Further analysis of Figure 5 reveals a slight shift between the optimal extraction points for linear probes ( t ≈ 60 % t\\approx 60\\% , l ≈ 70 % l\\approx 70\\% ) and attention probes ( t ≈ 30 % t\\approx 30\\% , l ≈ 80 % l\\approx 80\\% ). While both fall within the second mode of Veo3’s bimodal pattern, indicating the sweet spot for semantics, this shift suggests an evolution in feature complexity. At deeper blocks and lower noise levels, linear probes and zero-shot alignment metrics become insufficient possibly because semantic information becomes patch-specific and spatially scattered. Instead, attention probes excel by acting as a dynamic spatiotemporal pooling mechanism, selectively attending to and aggregating these distributed, fine-grained details. This suggests that lower-noise representations do not discard semantic meaning; they simply require more complex probes to decode it. Importantly, the superiority of the attention probe is not merely an artefact of its higher parameter count (6M vs. 1M for the linear probe). If parameter count were the sole driver, the attention head would uniformly dominate across all noise levels. Instead, at extreme noise levels ( t ≈ 90 % t\\approx 90\\% , red lines), its performance is severely constrained ( ≈ \\approx 15–30%), offering minimal advantage over the linear probe at early-to-mid depths. Finally, all probes experience a sharp performance drop at the deepest layers, indicating that the network discards semantic abstractions at the very end to perform pixel-level signal reconstruction."},{"type":"para","text":"Emergence of general-purpose features: Overall, these alignment results are remarkable. Prior evaluations of generative video models, such as WALT (Zhu et al. , 2026 ) , demonstrated limited alignment with semantic encoders. In contrast, our findings show that training purely for video generation at scale is sufficient to endow a model with powerful, general-purpose features. While these large-scale diffusion models are extensively trained with text conditioning, they are never explicitly optimised to align their internal states with text representations (unlike contrastive learning). The implicit emergence of this alignment is a significant finding not reported in prior work. Furthermore, both models demonstrate strong alignment with established visual encoders (Figure 4 ). Veo3, in particular, exhibits high alignment with DINOv2 and VideoMAEv2—models designed for image and video understanding tasks respectively. This confirms that generative representations possess strong potential for diverse downstream tasks requiring a combination of appearance, motion, and semantic understanding. Finally, the consistently higher alignment scores of Veo3 compared to Wan 2.2 confirm that stronger generative models yield correspondingly richer and more powerful representations."},{"type":"para","text":"Combination of features: We investigated whether combining features across multiple stages of the diffusion process could yield complementary information. We include details in Appendix B ."},{"type":"heading","level":1,"text":"4 Diffusion models as video encoders"},{"type":"para","text":"We demonstrate the potential of diffusion models to act as general-purpose video encoders by evaluating frozen diffusion representations on diverse semantic and non-semantic downstream tasks. While our latent space analysis in Section 3 confirmed the presence of optimal extraction points across both open-weight (Wan 2.2) and proprietary (Veo3) models, it also revealed that feature alignment scales directly with generative capability. Therefore, to test our hypothesis and establish the upper bound of the Gen4U framework, we focus our downstream evaluations on the highly scaled Veo3 model. We rely on one-block attention decoders to map diffusion representations to various task embedding spaces, similar to 4DS (Carreira et al. , 2025 ) . For captioning, we use a small language model fine-tuned on top of the frozen visual diffusion tokens. Only these lightweight decoders are trained, the generative model stays frozen, preserving its generation capabilities."},{"type":"heading","level":2,"text":"4.1 Semantic understanding"},{"type":"para","text":"Video classification: We use the challenging SSv2 dataset (Goyal et al. , 2017 ) for this task and follow the protocol in (Carreira et al. , 2025 ) , using simple linear and one-block attention decoders on top of frozen diffusion representations. The SSv2 dataset contains 220,847 shorter videos (2-6s long), sampled at 12fps, representing 174 classes. The videos depict actions that differ in finer motion-related details, requiring a deeper temporal understanding, e.g. pouring something into something vs pretending to pour something into something ."},{"type":"para","text":"Unlike previous work (Carreira et al. , 2025 ) , we do not do data augmentation on the raw videos. Instead, to reduce computation cost, we process only the original videos with Veo3. We then apply data augmentations directly to the intermediate activations: temporal masking (i.e. set to 0) of 16.7% of the frames, dropout in the attention masking of 40% and label smoothing of 0.2."},{"type":"para","text":"We report performance with and without these augmentations in Table 1 , compared to strong competitor models. We achieve SOTA performance on this task in this evaluation setup. Note that even without the data augmentation, our setup outperforms all baselines, except V-JEPA."},{"type":"para","text":"Image and video captioning: In this suite of experiments, we attach the Gemma-2 (2B) ( (Gemma Team, 2024 ) ) text-only large language model as a decoder for image and video"},{"type":"para","text":"captioning. Our captioning setup consists of the following two steps: (1.) A cross-attention adapter, inspired by the decoder design of (Sajjadi et al. , 2022 ) , with three cross-attention blocks, projects the visual representation into 32 32 learnable query tokens; and (2.) we prepend these 32 32 tokens to the text input before the [BOS] token and process the entire sequence with the LLM using teacher forcing during training and next-token prediction during inference."},{"type":"para","text":"We train separate decoders for each of the MS-COCO image captioning dataset (COCO) ( (Chen et al. , 2015 ) ), the Something-Something V2 video action recognition dataset (SSv2) 1 1 1 The SSv2 dataset consists of templated class labels like ‘putting something into something’. We use the labels that incorporate the nouns as well, e.g. ‘putting an apple into a fruit basket’. We prepend this with ‘A person is’ to form a full sentence: ‘A person is putting an apple into a fruit basket’. and the Vatex dataset ( (Wang et al. , 2019 ) ). We report the CIDEr and BLEU@4 metrics and compare against two baseline models: SigLIP-so400m/14 model from the PaliGemma series ( (Beyer et al. , 2024 ; Steiner et al. , 2024 ) ) and the SigLIP2-B/16 model ( (Tschannen et al. , 2025 ) )."},{"type":"para","text":"Table ˜ 2 presents the results of Veo3 representations compared to the SigLIP baselines on all three datasets. Note that our setup does not involve tuning the adapter and LLM decoder on large vision-language data, so it is not directly comparable to SoTA methods like BLIP-2 ( (Li et al. , 2023b ) ) and PaLI Gemma ( (Beyer et al. , 2024 ) ). We train our adapter and LLM only on the train splits of the considered datasets to probe the existence of semantic knowledge decodable through an LLM. We observe that video diffusion representations excel on SSv2 but struggle on COCO and Vatex, compared to the SigLIP baselines, which are highly optimised for alignment with text. Nonetheless, the model produces reasonable captions. A couple of qualitative examples are shown in Fig. 7 ."},{"type":"para","text":"Vatex [Frozen LLM]: We freeze the LLM in this experiment and train only the cross-attention adapter on the Vatex train split. Results are reported in the last two columns of Table 2 . Model performance decreases only slightly relative to the full fine-tuning experiments. Moreover, we significantly outperform VideoPrism which reports a CIDEr of 31.7 31.7 on this dataset. Although VideoPrism does not finetune their adapter on captioning, they pre-train it on a large vision-language corpus."},{"type":"para","text":"Data augmentation using multiple noise injection levels: As mentioned above, we do not pre-train our adapter and LLM on large datasets. To limit the overfitting caused by small training sets, we augment the data by concatenating the datasets across various noise injection levels 10 % , 30 % , 60 % and 90 % 10\\%,30\\%,60\\%~\\mathrm{and}~90\\% . We find that for COCO and Vatex using the 10%, 30% and 60% noise injection levels is optimal. The model is evaluated on the 30% level in all cases. The other noise levels purely serve as data augmentation. For SSv2, we do not see any improvements with data augmentation, likely because the caption space is simpler and the training dataset is much larger than with COCO and Vatex. The resulting performance is shown in table ˜ 2 under ‘+ Noise Aug.’ ."},{"type":"para","text":"Benefits of high-resolution Veo: COCO images contain several small objects and the model needs to mention a few of them to score highly on the CIDEr metric. We ablate doubling the input resolution. We pick the optimal combination of noise levels (‘+ Noise Aug.’) for each resolution on the validation split and report test set numbers in Table 2 under ‘+ Noise Aug. + High res.’ . The high res. model achieves gains + 32.7 +32.7 CIDEr points. Thus by using multiple noise levels as data augmentation and using higher resolution diffusion representations, one can reduce the gap to discriminative models trained explicitly for the captioning task on large-scale vision-language datasets."},{"type":"heading","level":2,"text":"4.2 Geometry understanding"},{"type":"para","text":"Depth estimation: We evaluate monocular depth prediction on ScanNet (Dai et al. , 2017 ) . Following prior work (Carreira et al. , 2025 ) , we report the absolute relative error (AbsRel), computed as | d ∗ − d | / ( d + ϵ ) |d^{*}-d|/(d+\\epsilon) where d ∗ d^{*} and d d denote predicted and ground-truth depth, respectively, and the threshold accuracy δ 1 \\delta_{1} . Target depth values outside the ( 0.001 , 10 ) (0.001,10) m range are masked out. We use a Dense Prediction Transformer (DPT) readout head (Ranftl et al. , 2021 ) with approximately 23M trainable parameters, trained with the scale-invariant logarithmic (SiLog) loss (Eigen et al. , 2014 ) . The entire diffusion backbone remains frozen."},{"type":"para","text":"Figure 6 (right) presents AbsRel across all blocks and noise levels. The results confirm the sweet spot for extracting representations around (depth 80%, noise 30%), identified using zero-shot alignment and classification probes. The best configuration achieves AbsRel of 0.075 and δ 1 = 0.952 \\delta_{1}{=}0.952 , a 10.7% relative improvement over the frozen-feature baseline of 0.084 0.084 reported in (Carreira et al. , 2025 ) . To our knowledge, this is the best result on ScanNet depth with frozen video model features."},{"type":"para","text":"Camera pose estimation: We are interested in probing the ability to infer the 6DoF relative camera poses from diffusion representations. We use the setup and metric introduced in (Carreira et al. , 2025 ) , applied to Scannet dataset. Given diffusion representations for a clip of F F frames, we train a one-block attention decoder to predict the relative pose between the first and last frame of the clip, in the form of a 12 D 12D vector encoding an SE(3) pose transformation ( 3 × 3 3\\times 3 estimated rotation matrix and a 3 × 1 3\\times 1 translation vector). As evaluation metric, we use end-point-error (EPE), which considers the rotation and translation components jointly; see (Carreira et al. , 2025 ) for more details. We obtain 1.10 EPE, which is on par with a strong DINOv2 baseline obtaining 1.08 EPE. The sweep across blocks and noise levels (Figure 6 left) is consistent with the findings for the other tasks, with performance peaking reliably in the mid-to-late stages of the network ( ∼ \\sim 75% depth), at medium noise level (60%)."},{"type":"heading","level":1,"text":"5 Conclusion"},{"type":"para","text":"We show that state-of-the-art large-scale video diffusion models can act as competitive video encoders on a diverse set of tasks (video classification, depth estimation, camera pose estimation, video captioning), supporting the first successful attempt to unify video generation and video understanding. Using zero-shot probes and trained lightweight decoders, we investigated the structure of the diffusion latent space and identified the optimal block and noise level to extract representations for understanding tasks. All our results for downstream tasks use representations extracted from a single block and noise level, so we only need to run a single forward pass through the model, being as efficient as large-scale discriminative visual encoders. As future work, we aim to study further the bimodal pattern identified for Veo3 and push the performance to achieve SOTA on a wide range of tasks."},{"type":"para","text":"Limitations: Our experiments are conducted mainly on a proprietary video model, Veo3, with analysis replicated on an open-source model (Wan 2.2), limiting the reproducibility of our study. However, we believe that this is still a useful investigation for the community and we hope it will encourage more work in studying large-scale diffusion models for representation learning."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Current paradigms in visual representation learning struggle to reconcile geometry with semantics. Contrastive methods and large Vision-Language Models (VLMs) (Radford et al. , 2021 ; Beyer et al. , 2024 ; Team, 2025 ) operate primarily in language space, yielding strong semantics but neglecting precise spatio-temporal details. Conversely, pixel-level reconstruction approaches like masked auto-encoding (MAE) (Tong et al. , 2022 ; Wang et al. , 2023 ; Carreira et al. , 2025 ; Zhang et al. , 2026 ) excel at preserving local geometry and motion but struggle to generalise to broad semantic tasks. A single, task-agnostic foundation video model that supports both types of capabilities remains an open problem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inspired by the predictive coding theory of the brain, which posits that prediction is key to cognition (Rao and Ballard, 1999 ; Clark, 2013 ) , we investigate how to build general-purpose video models based on generative video diffusion transformers. We argue that generating temporally and geometrically consistent video requires the model to implicitly learn the underlying mechanics of the physical world, such as motion, object permanence, and part-object relationships. However, recent studies probing frozen video diffusion representations concluded that they capture low-level geometry but struggle with high-level semantics, casting doubt on their eligibility as general-purpose perception encoders (Vélez et al. , 2025 ; Zhu et al. , 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we demonstrate that state-of-the-art video diffusion models do, in fact, overcome this limitation as they become more capable at generating physically and temporally consistent videos. By systematically investigating the intermediate activations of both proprietary (Veo3 (Google DeepMind, 2025 ) ) and open-weight (Wan2.2-T2V-A14B (Wan et al. , 2025 ) ) models, we map the hierarchy of their latent spaces using recent zero-shot mutual k-NN alignment metrics (Huh et al. , 2024 ; Zhu et al. , 2026 ) and we identify optimal spots for extracting linearly separable global semantics and more entangled, spatially-distributed features that require attention-based pooling to extract effectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Based on these observations, we introduce Gen4U (Generation for Understanding), a framework that extracts and repurposes generative representations for a diverse set of downstream visual tasks. Because Gen4U identifies the optimal block and noise level for feature extraction, it requires only a single forward pass through the diffusion backbone. This makes it as computationally efficient as a standard discriminative visual encoder, e.g. (Carreira et al. , 2025 ; Wang et al. , 2024 ) , circumventing the high inference costs typically associated with iterative denoising. By pairing these frozen representations with lightweight decoders, Gen4U achieves strong perception performance without altering the pre-trained generative weights. Our extensive evaluations demonstrate that frozen video diffusion models can function as general-purpose video encoders.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To summarise, our main contributions are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Latent space analysis : We map the evolution of diffusion features across depth and noise levels using recent alignment metrics, identifying the optimal points to extract representations for understanding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Semantics and temporal dynamics : We show that Gen4U captures rich, high-level understanding, achieving state-of-the-art results on zero-shot alignment and video classification on Something-Something V2 dataset (Goyal et al. , 2017 ) , alongside satisfactory results on image and video captioning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Geometry understanding : We demonstrate that these same frozen representations excel at lower level geometry awareness, achieving strong performance on monocular depth estimation and camera pose estimation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The most common paradigms for learning visual representations are variations of masked autoencoding (MAE) (He et al. , 2021 ; Tong et al. , 2022 ; Wang et al. , 2023 ) , scaled in 4DS (Carreira et al. , 2025 ) and D4RT (Zhang et al. , 2026 ) , and contrastive learning, with or without negative pairs (SigLip (Zhai et al. , 2023 ) , SimCLR (Chen et al. , 2020 ) , CLIP (Radford et al. , 2021 ) , BYOL (Grill et al. , 2020 ) , BRAVE (Recasens et al. , 2021 ) , DINO (Caron et al. , 2021 ) to name a few). These methods generally strike a trade-off between low level scene understanding and high level semantics, failing to support both. Even hybrid approaches (Zhao et al. , 2024 ; Wang et al. , 2024 ; Papalampidi et al. , 2023 ) that combine masked auto-encoding (MAE) with contrastive learning can still accurately span only one category of tasks. More recently, the V-JEPA line of works (Bardes et al. , 2024 ; Assran et al. , 2025 ; Mur-Labadia et al. , 2026 ) focus on predicting the future in latent space to avoid the computational cost of pixel-level prediction, and show strong results on a wide range of understanding tasks. These pre-training methods lead to strong results on downstream tasks, but most of the time they require extensive fine-tuning or have fairly complex pre-training pipelines, balancing different optimisation objectives. Moreover none of these pre-training methods support high-quality video generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Driven by the impressive realism and diversity of the outputs generated by diffusion models (Rombach et al. , 2022 ; Gupta et al. , 2023 ; Google DeepMind, 2025 ; Wan et al. , 2025 ) , multiple works have started to investigate their understanding capabilities. In (Li et al. , 2023a ) the authors show that a diffusion-based setup can perform zero-shot classification by conditioning the generative model with the possible classes and selecting the class that leads to the best reconstruction of the noise added to the input image. In (Luo et al. , 2023 ) , the authors build diffusion hyperfeatures by combining the activations from different blocks and noise levels to build robust pixel-level descriptors. In (Wiedemer et al. , 2025 ) , the authors show impressive generation results by prompting the Veo3 model with challenging tasks that require geometry and physics understanding, hinting that Veo3 has implicitly learnt a world model. In addition, works like (Ha and Schmidhuber, 2018 ) , (Hafner et al. , 2025 ) , (Bruce et al. , 2024 ; Parker-Holder et al. , 2024 ) confirm that generative models support learning a world model for simulated environments, with representations that can be used to inform action policies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The closest work to our approach is (Vélez et al. , 2025 ) . The authors repurpose activations extracted from variants of Walt model (Gupta et al. , 2023 ) adapted for image and video, for understanding tasks and conclude that diffusion models excel at lower level understanding problems like depth or camera pose estimation, but struggle with high-level semantic tasks. Similarly, in (Zhu et al. , 2026 ) , the authors measure the alignment of representations produced by different visual encoders amongst themselves and with language embeddings. Interestingly, their findings suggest that diffusion models like Walt produce representations that align with established visual encoders like DINOv2 (Oquab et al. , 2024 ) and VideoMAEv2 (Wang et al. , 2023 ) , but the alignment with text embeddings is significantly lower. In our work, we show that the recent generation of video diffusion models have strong semantic awareness, in addition to geometry and motion understanding, constituting a strong foundation for a complete perception system. We show state-of-the-art results on video classification on the challenging SSv2 dataset (Goyal et al. , 2017 ) , compared to strong baselines from the other families of visual pre-training methods (MAE, contrastive, latent prediction). To further the investigation of semantic understanding, we set up captioning experiments, showing satisfactory performance compared to a strong SigLIP-based baseline (Zhai et al. , 2023 ) from the PaliGemma family (Beyer et al. , 2024 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interplay between generation and understanding started to receive more attention recently. For example, detailed point tracking conditioning leads to more consistent generation (Jeong et al. , 2025 ) , or detailed language captioning produced by Gemini models lead to improved Veo generation (Google DeepMind, 2025 ) . Transfusion (Zhou et al. , 2025 ) elegantly combines generation and understanding pre-training into a single model, allowing the two objectives to support each other. However, in all these works, the main finding has been that advanced understanding models lead to significant improvement in generative models. The other direction, how can generation help understanding, has so far been explored mainly for low-level geometry tasks. In our work, we show that frozen diffusion generative models can power video understanding and achieve competitive performance on a wide range of visual tasks, while being able to generate high-quality videos. This unifies video generation and understanding paradigms. We estimate that this finding can be very impactful for the community, by encouraging more research into understanding the representation power of diffusion models. In addition, the implications from a practical point of view could be substantial, as it could lead to one visual model shared across generation and understanding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The structure of the generative manifold\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The iterative diffusion generation requires many denoising steps to generate a video. This slow sequential operation mode is not suitable for video encoding. To design efficient video feature extractors based on video diffusion models, we conduct a detailed investigation of the structure of the latent embeddings spawned by these models, in search for the optimal depth and noise level from where to extract representations with a single forward pass through the model. After a brief overview of diffusion models, we detail the zero-shot and trained light probes used for our analysis, and we conclude this section with a summary discussion of the findings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Background on latent diffusion models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Video Latent Diffusion Models (LDMs) like Veo3 operate within a compressed, lower-dimensional representation space. During training, given an input video x ∈ ℝ F × H × W × C x\\\\in\\\\mathbb{R}^{F\\\\times H\\\\times W\\\\times C} (containing F F frames), an encoder E maps the data into a spatiotemporal latent code z 0 = E ( x ) z_{0}=\\\\textit{E}(x) . A forward process corrupts this latent representation over T T steps. At an arbitrary noise level t ∈ [ 0 , T ] t\\\\in[0,T] , the corrupted latent z t z_{t} is generated by adding Gaussian noise to z 0 z_{0} : z t = α ¯ t z 0 + 1 − α ¯ t ϵ z_{t}=\\\\sqrt{\\\\bar{\\\\alpha}_{t}}z_{0}+\\\\sqrt{1-\\\\bar{\\\\alpha}_{t}}\\\\epsilon , where ϵ ∼ 𝒩 ( 0 , I ) \\\\epsilon\\\\sim\\\\mathcal{N}(0,I) and α ¯ t \\\\bar{\\\\alpha}_{t} dictates the variance schedule. To invert this process, a backbone network, typically a Diffusion Transformer (DiT) or a U-Net, denoted as f θ f_{\\\\theta} , is trained to predict the injected noise ϵ \\\\epsilon . The network is conditioned on the corrupted latent z t z_{t} , the noise step t t , and contextual information c c (e.g., text embeddings). The model is optimised via the standard reweighted variational lower bound: ℒ L D M = 𝔼 z 0 , ϵ , t [ ‖ ϵ − f θ ( z t , t , c ) ‖ 2 2 ] \\\\mathcal{L}_{LDM}=\\\\mathbb{E}_{z_{0},\\\\epsilon,t}\\\\left[\\\\|\\\\epsilon-f_{\\\\theta}(z_{t},t,c)\\\\|_{2}^{2}\\\\right] . Note that flow matching models like Wan 2.2, also included in our study, replace this stochastic forward process with a continuous-time deterministic framework based on Optimal Transport. In this setup, for a continuous time step t ∈ [ 0 , 1 ] t\\\\in[0,1] , the corrupted latent is a linear interpolation between the data and a base noise distribution: z t = ( 1 − t ) z 0 + t ϵ z_{t}=(1-t)z_{0}+t\\\\epsilon . Rather than predicting the injected noise, the network f θ f_{\\\\theta} is optimised to regress the velocity vector field f t = d z t d t = ϵ − z 0 f_{t}=\\\\frac{dz_{t}}{dt}=\\\\epsilon-z_{0} that transports the data to the noise. The model is trained via the flow matching objective: ℒ F M = 𝔼 z 0 , ϵ , t [ ‖ f θ ( z t , t , c ) − ( ϵ − z 0 ) ‖ 2 2 ] \\\\mathcal{L}_{FM}=\\\\mathbb{E}_{z_{0},\\\\epsilon,t}\\\\left[||f_{\\\\theta}(z_{t},t,c)-(\\\\epsilon-z_{0})||_{2}^{2}\\\\right] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The backbone f θ f_{\\\\theta} consists of a sequence of L L layers (e.g., transformer blocks in the case of a DiT). We denote the intermediate spatiotemporal activations extracted from the l l -th layer at noise step t t as h t ( l ) h_{t}^{(l)} . During inference, the model iteratively denoises pure Gaussian noise z T z_{T} to recover an estimated clean latent z ^ 0 \\\\hat{z}_{0} . Finally, z ^ 0 \\\\hat{z}_{0} is mapped back to pixel space via a decoder D , yielding the generated video x ^ = D ( z ^ 0 ) \\\\hat{x}=\\\\textit{D}(\\\\hat{z}_{0}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 illustrates our diffusion-based video encoder Gen4U . We condition the diffusion transformer on a generic text embedding: e.g. A video of a scene , to not leak the ground truth (action label or caption) for semantic tasks. We have also experimented with conditioning on an empty text string, without significant difference in results. To deal with the inherent stochasticity of the diffusion process, we use a fixed random seed throughout all our experiments to ensure reproducibility. In all our plots, we report results for blocks and noise levels expressed as percentages obtained as a linear fraction of the total depth or noise steps.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Zero-shot probes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PCA visualisation: To qualitatively assess the evolution of the diffusion features along network depth and noise levels, we plot in Figure 2 the PCA visualisation of the activations extracted from different depths and noise levels in Veo3. We can observe that at high levels of noise, the features are simple, low-frequency, encoding the general shapes in the scene. At lower noise levels, the features become more complex, encoding higher-frequency, more refined details. This is in line with the intuition that diffusion is spectral autoregression (Dieleman, 2024 ) : at each denoising step, the model learns to produce the features at the next (higher) frequency level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mutual k-NN alignment: To quantitatively assess the structure of the diffusion embedding space, we adopt the zero-shot Mutual k k -Nearest Neighbours (M k k NN) alignment metric proposed by (Huh et al. , 2024 ) and extended to video in (Zhu et al. , 2026 ) . The key idea is to compare the neighbourhood structure induced by the diffusion model’s intermediate representations against that of established, independently trained encoders, without requiring any learned mapping between the two spaces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Concretely, given a dataset of N N videos, paired with ground truth captions, we extract a video-level embedding from each video using both the diffusion model and a reference encoder. For the diffusion model, we obtain intermediate activations h t ( l ) h_{t}^{(l)} at a chosen layer l l and noise level t t , and aggregate them spatiotemporally to produce a single vector per video (see appendix A for details). For the reference encoder, which may operate in a different modality (e.g., a language model) and produce embeddings of a different dimensionality, we similarly obtain one vector per video. In particular, for image models, we follow (Zhu et al. , 2026 ) and average the individual image features across input frames. We then independently construct the k k -nearest-neighbour graph in each embedding space and measure the average overlap between the two sets of neighbours. Intuitively, a high overlap indicates that the diffusion model organises videos in a manner consistent with the reference encoder, despite never having been trained with the same objective. More details are included in the appendix A . Following prior work (Huh et al. , 2024 ; Zhu et al. , 2026 ) , we additionally optimise over the choice of intermediate layers in both encoders and report the pair of layers that maximises the alignment score. We use k = 10 k{=}10 and N = 1024 N{=}1024 videos.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In Figure 3 , we show the alignment of Veo3 and Wan 2.2 respectively, against the Gemma-2-9b-it text encoder (Gemma Team, 2024 ) on the VATEX dataset. This metric has been shown to correlate with downstream performance in semantic and even non-semantic tasks (Zhu et al. , 2026 ) . In addition, we also extract in Figure 4 the alignment of diffusion representations against DINOv2 (Oquab et al. , 2024 ) , a powerful image representation model, and VideoMAEv2 (Wang et al. , 2023 ) , a strong video representation model. Note that the features Veo3, especially when probed at the appropriate depth and noise levels align significantly more strongly with text embeddings of their associated captions. We emphasize that the captions are not given as context during generation. This suggests that a powerful video generative model has the capacity of encoding rich semantic information within its intermediate features.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Linear and attention probes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use linear and attention probes to further assess the quality of diffusion representations. The linear head does global pooling over spatiotemporal dimensions followed by a linear projection. The attention probe uses a cross-attention head, with task-specific queries. We show results for video classification probes in Figure 5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also experiment with lightweight decoders that reconstruct the input in RGB space given the representations from different blocks and noise levels, to get a sense of the compression and artefacts present at each stage, but the results are less conclusive; see details and visualisations in appendix C .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Feature routing across network depth: The evolution of diffusion representations follows a dual-axis trajectory along both depth l l and noise level t t . Interestingly, we observe distinct evolution patterns for the two models. The results for Wan 2.2 (Figure 3 right, Figure 4 right) show a unimodal pattern, aligning with broader diffusion literature (e.g., (Luo et al. , 2023 ) ). The alignment peaks early, at around 25% depth, indicating that the model resolves global structure and scene details early before gradually settling finer details. Conversely, Veo3 exhibits a distinct bimodal pattern, a phenomenon not previously reported in the diffusion literature. This mirrors the geometry of representations learnt by large-scale transformers (Valeriani et al. , 2023 ) , where the representation manifold expands early on (first peak), contracts significantly in the middle layers to route information, and then expands again toward the end (second peak) before the final output projection. We hypothesise a similar behaviour drives Veo3, supported by PCA visualisations (Figure 2 ) that show somewhat noisier activations halfway through the network, aligning with the dip in the bimodal pattern. We leave a conclusive characterisation of this behaviour for future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝟔𝟎 % \\\\mathbf{60\\\\%} -noise semantic bottleneck: Despite exhibiting divergent topological patterns across depth, both models share a consistent optimal noise level for alignment against both text and visual encoders at t = 60 % t=60\\\\% . This indicates the emergence of a semantic bottleneck for both models at this specific noise level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evolution of feature complexity, linear vs. attention probes: Further analysis of Figure 5 reveals a slight shift between the optimal extraction points for linear probes ( t ≈ 60 % t\\\\approx 60\\\\% , l ≈ 70 % l\\\\approx 70\\\\% ) and attention probes ( t ≈ 30 % t\\\\approx 30\\\\% , l ≈ 80 % l\\\\approx 80\\\\% ). While both fall within the second mode of Veo3’s bimodal pattern, indicating the sweet spot for semantics, this shift suggests an evolution in feature complexity. At deeper blocks and lower noise levels, linear probes and zero-shot alignment metrics become insufficient possibly because semantic information becomes patch-specific and spatially scattered. Instead, attention probes excel by acting as a dynamic spatiotemporal pooling mechanism, selectively attending to and aggregating these distributed, fine-grained details. This suggests that lower-noise representations do not discard semantic meaning; they simply require more complex probes to decode it. Importantly, the superiority of the attention probe is not merely an artefact of its higher parameter count (6M vs. 1M for the linear probe). If parameter count were the sole driver, the attention head would uniformly dominate across all noise levels. Instead, at extreme noise levels ( t ≈ 90 % t\\\\approx 90\\\\% , red lines), its performance is severely constrained ( ≈ \\\\approx 15–30%), offering minimal advantage over the linear probe at early-to-mid depths. Finally, all probes experience a sharp performance drop at the deepest layers, indicating that the network discards semantic abstractions at the very end to perform pixel-level signal reconstruction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Emergence of general-purpose features: Overall, these alignment results are remarkable. Prior evaluations of generative video models, such as WALT (Zhu et al. , 2026 ) , demonstrated limited alignment with semantic encoders. In contrast, our findings show that training purely for video generation at scale is sufficient to endow a model with powerful, general-purpose features. While these large-scale diffusion models are extensively trained with text conditioning, they are never explicitly optimised to align their internal states with text representations (unlike contrastive learning). The implicit emergence of this alignment is a significant finding not reported in prior work. Furthermore, both models demonstrate strong alignment with established visual encoders (Figure 4 ). Veo3, in particular, exhibits high alignment with DINOv2 and VideoMAEv2—models designed for image and video understanding tasks respectively. This confirms that generative representations possess strong potential for diverse downstream tasks requiring a combination of appearance, motion, and semantic understanding. Finally, the consistently higher alignment scores of Veo3 compared to Wan 2.2 confirm that stronger generative models yield correspondingly richer and more powerful representations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Combination of features: We investigated whether combining features across multiple stages of the diffusion process could yield complementary information. We include details in Appendix B .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Diffusion models as video encoders\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate the potential of diffusion models to act as general-purpose video encoders by evaluating frozen diffusion representations on diverse semantic and non-semantic downstream tasks. While our latent space analysis in Section 3 confirmed the presence of optimal extraction points across both open-weight (Wan 2.2) and proprietary (Veo3) models, it also revealed that feature alignment scales directly with generative capability. Therefore, to test our hypothesis and establish the upper bound of the Gen4U framework, we focus our downstream evaluations on the highly scaled Veo3 model. We rely on one-block attention decoders to map diffusion representations to various task embedding spaces, similar to 4DS (Carreira et al. , 2025 ) . For captioning, we use a small language model fine-tuned on top of the frozen visual diffusion tokens. Only these lightweight decoders are trained, the generative model stays frozen, preserving its generation capabilities.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Semantic understanding\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Video classification: We use the challenging SSv2 dataset (Goyal et al. , 2017 ) for this task and follow the protocol in (Carreira et al. , 2025 ) , using simple linear and one-block attention decoders on top of frozen diffusion representations. The SSv2 dataset contains 220,847 shorter videos (2-6s long), sampled at 12fps, representing 174 classes. The videos depict actions that differ in finer motion-related details, requiring a deeper temporal understanding, e.g. pouring something into something vs pretending to pour something into something .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike previous work (Carreira et al. , 2025 ) , we do not do data augmentation on the raw videos. Instead, to reduce computation cost, we process only the original videos with Veo3. We then apply data augmentations directly to the intermediate activations: temporal masking (i.e. set to 0) of 16.7% of the frames, dropout in the attention masking of 40% and label smoothing of 0.2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report performance with and without these augmentations in Table 1 , compared to strong competitor models. We achieve SOTA performance on this task in this evaluation setup. Note that even without the data augmentation, our setup outperforms all baselines, except V-JEPA.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Image and video captioning: In this suite of experiments, we attach the Gemma-2 (2B) ( (Gemma Team, 2024 ) ) text-only large language model as a decoder for image and video\"\n },\n {\n \"type\": \"para\",\n \"text\": \"captioning. Our captioning setup consists of the following two steps: (1.) A cross-attention adapter, inspired by the decoder design of (Sajjadi et al. , 2022 ) , with three cross-attention blocks, projects the visual representation into 32 32 learnable query tokens; and (2.) we prepend these 32 32 tokens to the text input before the [BOS] token and process the entire sequence with the LLM using teacher forcing during training and next-token prediction during inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train separate decoders for each of the MS-COCO image captioning dataset (COCO) ( (Chen et al. , 2015 ) ), the Something-Something V2 video action recognition dataset (SSv2) 1 1 1 The SSv2 dataset consists of templated class labels like ‘putting something into something’. We use the labels that incorporate the nouns as well, e.g. ‘putting an apple into a fruit basket’. We prepend this with ‘A person is’ to form a full sentence: ‘A person is putting an apple into a fruit basket’. and the Vatex dataset ( (Wang et al. , 2019 ) ). We report the CIDEr and BLEU@4 metrics and compare against two baseline models: SigLIP-so400m/14 model from the PaliGemma series ( (Beyer et al. , 2024 ; Steiner et al. , 2024 ) ) and the SigLIP2-B/16 model ( (Tschannen et al. , 2025 ) ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 2 presents the results of Veo3 representations compared to the SigLIP baselines on all three datasets. Note that our setup does not involve tuning the adapter and LLM decoder on large vision-language data, so it is not directly comparable to SoTA methods like BLIP-2 ( (Li et al. , 2023b ) ) and PaLI Gemma ( (Beyer et al. , 2024 ) ). We train our adapter and LLM only on the train splits of the considered datasets to probe the existence of semantic knowledge decodable through an LLM. We observe that video diffusion representations excel on SSv2 but struggle on COCO and Vatex, compared to the SigLIP baselines, which are highly optimised for alignment with text. Nonetheless, the model produces reasonable captions. A couple of qualitative examples are shown in Fig. 7 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Vatex [Frozen LLM]: We freeze the LLM in this experiment and train only the cross-attention adapter on the Vatex train split. Results are reported in the last two columns of Table 2 . Model performance decreases only slightly relative to the full fine-tuning experiments. Moreover, we significantly outperform VideoPrism which reports a CIDEr of 31.7 31.7 on this dataset. Although VideoPrism does not finetune their adapter on captioning, they pre-train it on a large vision-language corpus.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data augmentation using multiple noise injection levels: As mentioned above, we do not pre-train our adapter and LLM on large datasets. To limit the overfitting caused by small training sets, we augment the data by concatenating the datasets across various noise injection levels 10 % , 30 % , 60 % and 90 % 10\\\\%,30\\\\%,60\\\\%~\\\\mathrm{and}~90\\\\% . We find that for COCO and Vatex using the 10%, 30% and 60% noise injection levels is optimal. The model is evaluated on the 30% level in all cases. The other noise levels purely serve as data augmentation. For SSv2, we do not see any improvements with data augmentation, likely because the caption space is simpler and the training dataset is much larger than with COCO and Vatex. The resulting performance is shown in table ˜ 2 under ‘+ Noise Aug.’ .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Benefits of high-resolution Veo: COCO images contain several small objects and the model needs to mention a few of them to score highly on the CIDEr metric. We ablate doubling the input resolution. We pick the optimal combination of noise levels (‘+ Noise Aug.’) for each resolution on the validation split and report test set numbers in Table 2 under ‘+ Noise Aug. + High res.’ . The high res. model achieves gains + 32.7 +32.7 CIDEr points. Thus by using multiple noise levels as data augmentation and using higher resolution diffusion representations, one can reduce the gap to discriminative models trained explicitly for the captioning task on large-scale vision-language datasets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Geometry understanding\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Depth estimation: We evaluate monocular depth prediction on ScanNet (Dai et al. , 2017 ) . Following prior work (Carreira et al. , 2025 ) , we report the absolute relative error (AbsRel), computed as | d ∗ − d | / ( d + ϵ ) |d^{*}-d|/(d+\\\\epsilon) where d ∗ d^{*} and d d denote predicted and ground-truth depth, respectively, and the threshold accuracy δ 1 \\\\delta_{1} . Target depth values outside the ( 0.001 , 10 ) (0.001,10) m range are masked out. We use a Dense Prediction Transformer (DPT) readout head (Ranftl et al. , 2021 ) with approximately 23M trainable parameters, trained with the scale-invariant logarithmic (SiLog) loss (Eigen et al. , 2014 ) . The entire diffusion backbone remains frozen.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6 (right) presents AbsRel across all blocks and noise levels. The results confirm the sweet spot for extracting representations around (depth 80%, noise 30%), identified using zero-shot alignment and classification probes. The best configuration achieves AbsRel of 0.075 and δ 1 = 0.952 \\\\delta_{1}{=}0.952 , a 10.7% relative improvement over the frozen-feature baseline of 0.084 0.084 reported in (Carreira et al. , 2025 ) . To our knowledge, this is the best result on ScanNet depth with frozen video model features.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Camera pose estimation: We are interested in probing the ability to infer the 6DoF relative camera poses from diffusion representations. We use the setup and metric introduced in (Carreira et al. , 2025 ) , applied to Scannet dataset. Given diffusion representations for a clip of F F frames, we train a one-block attention decoder to predict the relative pose between the first and last frame of the clip, in the form of a 12 D 12D vector encoding an SE(3) pose transformation ( 3 × 3 3\\\\times 3 estimated rotation matrix and a 3 × 1 3\\\\times 1 translation vector). As evaluation metric, we use end-point-error (EPE), which considers the rotation and translation components jointly; see (Carreira et al. , 2025 ) for more details. We obtain 1.10 EPE, which is on par with a strong DINOv2 baseline obtaining 1.08 EPE. The sweep across blocks and noise levels (Figure 6 left) is consistent with the findings for the other tasks, with performance peaking reliably in the mid-to-late stages of the network ( ∼ \\\\sim 75% depth), at medium noise level (60%).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We show that state-of-the-art large-scale video diffusion models can act as competitive video encoders on a diverse set of tasks (video classification, depth estimation, camera pose estimation, video captioning), supporting the first successful attempt to unify video generation and video understanding. Using zero-shot probes and trained lightweight decoders, we investigated the structure of the diffusion latent space and identified the optimal block and noise level to extract representations for understanding tasks. All our results for downstream tasks use representations extracted from a single block and noise level, so we only need to run a single forward pass through the model, being as efficient as large-scale discriminative visual encoders. As future work, we aim to study further the bimodal pattern identified for Veo3 and push the performance to achieve SOTA on a wide range of tasks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Limitations: Our experiments are conducted mainly on a proprietary video model, Veo3, with analysis replicated on an open-source model (Wan 2.2), limiting the reproducibility of our study. However, we believe that this is still a useful investigation for the community and we hope it will encourage more work in studying large-scale diffusion models for representation learning.\"\n }\n]","isConversational":false}}},{"rowIdx":494,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06855"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Geometric Self-Distillation for Reasoning Generalization"},"authors":{"kind":"list like","value":["Josip Jukić","Ivan Titov"],"string":"[\n \"Josip Jukić\",\n \"Ivan Titov\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06855v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"On-policy distillation (OPD) has emerged as a practical post-training recipe for large language models (LLMs), providing dense teacher supervision on trajectories generated by the student itself (Agarwal et al. , 2024 ; Song and Zheng, 2026 ) . Dense supervision is especially valuable for reasoning , where final-answer rewards are often too sparse to reveal which intermediate steps were useful (Lightman et al. , 2024 ) . Instead of assigning credit only after a completed solution, OPD evaluates the teacher’s distribution at the states the student visits, producing token-level targets throughout the trajectory. The recipe can scale cheaply, since it requires no separate, stronger teacher. In on-policy self-distillation (OPSD), the student acts as its own teacher, supervising the prefixes it generates. A common and effective variant grants this teacher privileged context , such as a hint or the full solution trace, while the student must reason from the original problem alone (Zhao et al. , 2026 ; Ye et al. , 2026 ) ."},{"type":"para","text":"Whether dense supervision helps depends on the student’s ability to follow the teacher’s token-level preferences. Li et al. ( 2026 ) show that OPD progresses when student and teacher increasingly share mass on high-probability tokens, and stalls when that mass is largely disjoint. Privileged context is a systematic source of such mismatch: the extra information lets the teacher commit to continuations that are obvious given the solution but still uncertain from the student’s own context (Hübotter et al. , 2026 ; Penaloza et al. , 2026 ) . These mismatched states are also where imitation pulls hardest: standard divergences weight a teacher token by its own mass, so a token the teacher is sure of but the student barely supports produces a large gradient and a strong update. As such updates repeat, the student commits to the teacher’s continuations before its own information can support them, and the mismatch accumulates into drift. This drift is visible in the model’s behavior: Kim et al. ( 2026 ) observe that privileged-context OPSD suppresses uncertainty and degrades out-of-distribution (OOD) performance. Existing approaches mostly intervene on which teacher signals to use. Some filter or down-weight signals judged unreliable (Xing et al. , 2026 ; Xu et al. , 2026 ; Stein et al. , 2026 ) ; others reshape how a trusted signal is matched, changing the direction or weighting of divergences in the Kullback–Leibler (KL) family (Gu et al. , 2024 ; Agarwal et al. , 2024 ; Ko et al. , 2025 ) or reweighting token credit (Liu et al. , 2026 ; Yang et al. , 2026 ) . What they leave unaddressed is how far a trusted signal should move the student. The most informative supervision coincides with the largest updates, leaving selection no good option: filtering those states discards useful signal, while keeping them lets the student inherit the teacher’s overconfidence. Selection alone cannot regulate update magnitude, and this unchecked movement accumulates into drift, ultimately degrading OOD reasoning."},{"type":"para","text":"We introduce GeoSD ( Geo metric S elf- D istillation), which counters this drift in two complementary ways: attenuating the teacher’s pull at low-overlap states and regulating how far those pulls carry the student over training. First, we replace the standard distillation objective with a Hellinger loss. Its gradient weights each teacher preference by the probability mass student and teacher already share: tokens the teacher assigns high probability but the student finds unlikely exert little pull, while shared preferences transfer strongly. Because this attenuation is built into the gradient rather than applied on top of it by clipping or filtering, supervision strengthens gradually as the two come into agreement (Figure 1 A). Second, we penalize accumulated movement from a recent checkpoint with the Fisher–Rao distance between predictive distributions (Figure 1 B). Because both terms share a single geometry of next-token distributions, a natural-gradient update optimizes both within it: each step makes the smallest change to the student’s predictions that improves the objective. Empirically, GeoSD preserves the in-distribution (ID) gains of standard OPSD while substantially improving OOD reasoning. We trace this improvement to how each objective reaches agreement: standard matching concentrates mass on the teacher’s top choice by draining the alternatives, whereas GeoSD makes the same top choice dominant while keeping the alternatives in reach."},{"type":"para","text":"Our contributions are threefold. (i) We introduce GeoSD , a self-distillation objective that weights each teacher preference by teacher–student overlap and penalizes predictive drift from a recent checkpoint, optimizing both as distances in a single geometry via natural gradient. (ii) GeoSD improves average OOD accuracy on mathematical reasoning by 5.7 5.7 – 8.6 8.6 points (avg@16) over the base model across three model families, whereas standard KL objectives degrade it (§ 3 ). GeoSD retains ID gains and achieves the best accuracy on all tested OOD benchmarks, with the OOD gains holding consistently across five model scales (1.7B–32B). (iii) We characterize how standard matching fails out of distribution: it concentrates probability mass at high-entropy states, resulting in confident agreement on incorrect answers (§ 4 )."},{"type":"heading","level":1,"text":"2 Method"},{"type":"para","text":"We now formulate GeoSD , which targets the movement a teacher induces in the student’s predictive behavior during self-distillation. Because this movement is poorly captured in raw probabilities or parameters, where equal-sized changes can have very different behavioral effects, we measure it in a single geometry of next-token distributions. After defining the on-policy student–teacher setup (§ 2.1 ), we first modulate the local pressure at each visited state with an overlap-aware distillation loss that weakens each teacher preference in proportion to the student’s overlap with it (§ 2.2 ). We then penalize the global displacement that accumulates over training with a proximal term to a recent checkpoint (§ 2.3 ), and optimize the combined objective with a natural-gradient update that adjusts each step by its effect on the student’s predictions rather than by its size in parameter space (§ 2.4 )."},{"type":"heading","level":2,"text":"2.1 Setup"},{"type":"para","text":"Let 𝒱 \\mathcal{V} denote the vocabulary and Δ 𝒱 \\Delta_{\\mathcal{V}} the probability simplex over it. For an input 𝐱 \\mathbf{x} and an on-policy continuation 𝐲 ∼ π θ ( ⋅ ∣ 𝐱 ) \\mathbf{y}\\sim\\pi_{\\theta}(\\cdot\\mid\\mathbf{x}) , the student ( p p ) and teacher ( q q ) next-token distributions at position t t are"},{"type":"para","text":"where the teacher reads the same prefix but with privileged context 𝐜 \\mathbf{c} , a full solution trace sampled from the base model π 0 \\pi_{0} . The objective treats the teacher as any privileged-context distribution; we instantiate it as the current student, so both come from the same network, though a fixed reference, a delayed copy, or a moving average of recent students would serve as well. We write q ( t ) q^{(t)} without the parameter subscript to mark it as a fixed target, with gradients flowing only through the student p θ ( t ) p_{\\theta}^{(t)} , so each update moves the student toward the privileged-context distribution without optimizing that distribution itself. Because parameters and prefix are shared, p θ ( t ) p_{\\theta}^{(t)} and q ( t ) ( 𝐜 ) q^{(t)}(\\mathbf{c}) differ only through 𝐜 \\mathbf{c} ."},{"type":"heading","level":2,"text":"2.2 Local teacher influence"},{"type":"para","text":"The student often retains plausible alternatives beyond its dominant prediction, corresponding to continuations it may still sample. Suppressing one such alternative can produce only a small change in raw probability space, while effectively removing a viable continuation. A suitable distance should therefore weight changes by each token’s probability mass: the same change can be minor for the dominant prediction but significant for a lower-probability alternative. Information geometry supplies such a distance (Amari, 2016 ; Amari and Nagaoka, 2000 ) : under the square-root embedding p ↦ p p\\mapsto\\sqrt{p} , next-token distributions live on a hypersphere where statistical distance becomes ordinary distance (Figure 1 B; Appendix A.2 ). The sphere carries two canonical distances, the chord through it and the arc along the surface. The chord yields the squared Hellinger divergence, which we use for the per-state pull:"},{"type":"para","text":"The arc is the Fisher–Rao distance, which we use for cumulative drift (§ 2.3 ):"},{"type":"para","text":"The sphere fixes the geometry but not the loss. Near agreement, KL and Hellinger induce the same local geometry up to scale (Appendix A.3 ). However, their behavior differs in the low-overlap regime, which is critical for privileged-context OPSD. We use the Hellinger chord as the teacher pull, averaged over inputs, on-policy trajectories, sampled privileged contexts 𝐜 ∼ 𝒞 ( ⋅ ∣ 𝐱 ) \\mathbf{c}\\sim\\mathcal{C}(\\cdot\\mid\\mathbf{x}) , and positions: 1 1 1 Gradients do not propagate through the sampling distribution π θ ( ⋅ ∣ 𝐱 ) \\pi_{\\theta}(\\cdot\\mid\\mathbf{x}) : trajectories are treated as fixed once sampled, dropping the score-function term of the outer expectation."},{"type":"para","text":"What singles out the chord is its gradient. Differentiating one Hellinger term against the stop-gradient teacher q q gives"},{"type":"para","text":"which scales each token’s influence by the geometric-mean overlap p θ ( i ) q i \\sqrt{p_{\\theta}(i)\\,q_{i}} . The weight vanishes as p θ ( i ) → 0 p_{\\theta}(i)\\!\\to\\!0 , so even a token the teacher is certain of exerts almost no pull once the student has all but excluded it, and the per-state gradient shrinks as overlap falls."},{"type":"para","text":"Figure 2 A shows this contrast empirically. Forward KL weights a token by teacher mass q i q_{i} , so a confident teacher exerts large pressure even where the student barely supports the token, and its gradient norm grows as overlap falls. These high-teacher, low-student states form a dense region of positions visited by the student rather than a rare tail (see Figure 8 in Appendix C.1 ), so behavior at low overlap shapes much of the training signal. Reverse KL weights by student mass p θ ( i ) p_{\\theta}(i) , which attenuates that pressure but makes it mode-seeking and suppresses alternatives. Hellinger’s geometric mean sits between them: the pull fades as the student withdraws support, while teacher alternatives are kept in proportion to their overlap."},{"type":"heading","level":2,"text":"2.3 Global drift control"},{"type":"para","text":"Overlap weighting bounds the pull at each state, but a bound per step is not a bound on the path: many small, individually reasonable pulls can still carry the student far over training. Self-distillation makes this acute, since the teacher is a copy of the student. As the student moves, the targets move with it, and the two can drift together while no single update ever looks large."},{"type":"para","text":"We control the path with an established remedy in post-training, a penalty on displacement from a trusted reference (§ 5 ), cast here in the same geometry as the pull: a proximal term on the Fisher–Rao motion from a recent checkpoint θ ckpt \\theta_{\\mathrm{ckpt}} , refreshed every K ckpt K_{\\mathrm{ckpt}} steps: 1"},{"type":"para","text":"The checkpoint is deliberately recent rather than the initial model. Anchoring to π 0 \\pi_{0} would maximize retention but forbid the adaptation that makes distillation useful; a recent θ ckpt \\theta_{\\mathrm{ckpt}} instead acts as a moving trust region, damping abrupt displacement while still allowing steady movement when the teacher signal is consistent. Overlap weighting alone slows cumulative drift but does not stop it, and adding the proximal term keeps it controlled (Figure 2 B)."},{"type":"heading","level":2,"text":"2.4 Full objective and natural-gradient update"},{"type":"para","text":"The objective combines the two terms,"},{"type":"para","text":"with λ \\lambda setting the strength of the drift penalty. Both terms are distances between next-token distributions, so the update should move through that same geometry. A natural gradient asks, of all parameter updates that change the student’s predictions by a fixed amount, which one most improves the objective. This is the principle behind the geometry above, applied now to the update itself and realized by preconditioning with the Fisher information F k F_{k} of the current policy (Amari, 1998 ; Schulman et al. , 2015 ) :"},{"type":"para","text":"Each step is then adjusted based on its effect on the model’s predictions, rather than its raw size in parameters. Under this update, the proximal term acts as a checkpoint pullback: drift is measured in the checkpoint’s local geometry and returned as a restoring force in the current one, so the pull is strongest where drift is most visible in the predictions (Appendix A.6 )."},{"type":"para","text":"At LLM scale the full Fisher is infeasible to form or invert, so we precondition with a damped Kronecker-factored approximation of the empirical Fisher (K-FAC; Martens and Grosse, 2015 ) that captures the layerwise curvature without materializing the full matrix (Appendix B.6 ). This keeps the update curvature-aware while adding only modest memory and runtime over a standard distillation step (Appendix D )."},{"type":"heading","level":1,"text":"3 Experiments"},{"type":"heading","level":2,"text":"3.1 Experimental Setup"},{"type":"heading","level":4,"text":"Models."},{"type":"para","text":"We evaluate along two independent axes. For scale , we use the Qwen3 dense family at five sizes: 1.7B, 4B, 8B, 14B, and 32B (Yang et al. , 2025 ) . For cross-family generalization , we compare three models from different families: Qwen3-8B (Yang et al. , 2025 ) , Olmo-3-7B-Think (Olmo et al. , 2025 ) , and DeepSeek-R1-Distill-Llama-8B (DS-R1-Llama-8B; Guo et al. , 2025 ). At 14B and 32B scale, we use a more memory-efficient K-FAC for the natural-gradient update (Appendix B.6 )."},{"type":"heading","level":4,"text":"Training and evaluation."},{"type":"para","text":"On-policy trajectories are sampled from the student on problems from DAPO-Math-17k (Yu et al. , 2026 ) , a competition-mathematics dataset providing verifiable integer answers but no reference solutions. In-distribution performance is measured on a held-out split. We measure out-of-distribution reasoning on AIME 2024, AIME 2025, AMC 2023, and MATH-500 (Lightman et al. , 2024 ) , none of which overlap with the training problems. We report accuracy using avg@ k k (average accuracy over k k samples) and pass@ k k (whether any of the k k samples is correct). By default, the privileged context is a full correct solution, following the OPSD convention (Zhao et al. , 2026 ; Kim et al. , 2026 ) . Solutions are constructed offline: for each problem, we sample up to 32 32 rollouts from π 0 \\pi_{0} , keep the ones with verified answers, and at each step draw one of a problem’s verified solutions as the teacher’s privileged context. Problems for which no rollout verifies are excluded from training. Full data, rollout, and evaluation settings are in Appendix B ."},{"type":"heading","level":4,"text":"Baselines."},{"type":"para","text":"We compare GeoSD against three baseline families. Reference baselines include Base ( π 0 \\pi_{0} ; initial model checkpoint), SFT (supervised fine-tuning) on privileged teacher traces, and GRPO (Shao et al. , 2024 ) with verifiable rewards on the same training prompts. These respectively represent the pretrained reference point, off-policy imitation of privileged solutions, and a privilege-free sparse-reward post-training baseline. Divergence controls keep our pipeline fixed and swap the per-position loss: forward KL ( FwdKL ; Hinton et al. , 2015 ; Zhao et al. , 2026 ), reverse KL ( RevKL ; Gu et al. , 2024 ), Jensen–Shannon ( JSD , β = 0.5 \\beta{=}0.5 ; Agarwal et al. , 2024 ), and skewed KL ( SkewKL , α = 0.1 \\alpha{=}0.1 ; Ko et al. , 2024 ; 2025 ). Filtering baselines test the complementary strategy of deciding which teacher signals to use rather than changing the geometry of the update: TrOPD (Xing et al. , 2026 ) separates teacher-reliable regions from outlier regions using teacher–student agreement, while TIP (Xu et al. , 2026 ) selects or reweights informative tokens using student uncertainty and teacher–student mismatch. To isolate the effect of the objective, every baseline runs in our privileged-context OPSD pipeline with matched rollout and compute budgets, so the numbers are comparable across our rows. We provide implementation details for baselines in Appendix B.5 ."},{"type":"heading","level":4,"text":"Compute overhead."},{"type":"para","text":"GeoSD ’s geometry-aware update is cheap in practice despite invoking a natural gradient: over standard OPSD it adds only a checkpoint forward pass and K-FAC covariance accumulation, raising optimizer FLOPs by ≈ 27 % {\\approx}27\\% ( 1.27 × 1.27\\times ) and training memory by ≈ 21 % {\\approx}21\\% ( 1.21 × 1.21\\times ), of which only the K-FAC state ( ≈ 8 % {\\approx}8\\% ) resides on the training workers in the default placement. Wall-clock time grows by only 1.10 × 1.10\\times , since the checkpoint forward overlaps with rollout generation, which dominates the run and is shared across all methods. A detailed breakdown is in Appendix D ."},{"type":"heading","level":2,"text":"3.2 Generalization under privileged self-distillation"},{"type":"heading","level":4,"text":"Geometry improves the ID–OOD reasoning frontier."},{"type":"para","text":"Table 1 compares OPSD approaches on ID and OOD performance, exposing a consistent gap between in-distribution gains and their transfer under distribution shift. FwdKL posts some of the largest ID gains of any method, yet its OOD accuracy degrades relative to π 0 \\pi_{0} in all three families (by 4.6 4.6 – 8.1 8.1 points avg@ 16 16 ). This ID-up, OOD-down pattern reproduces the trend prior work reports for imitation-style post-training, and holds under a strict evaluation protocol (Appendix B.4 ). Softer objectives (SkewKL, JSD) mostly remove the degradation but barely improve over the base model, and the filtering baselines (TrOPD, TIP) recover only modest OOD gains ( + 0.4 +0.4 to + 3.2 +3.2 ). GeoSD is the only method that both retains the ID gains of self-distillation and improves OOD reasoning substantially, by 5.7 5.7 – 8.6 8.6 points avg@ 16 16 over π 0 \\pi_{0} ; it ranks first on OOD in every family while staying within 0.7 0.7 – 1.7 1.7 points of the best ID score. This advantage is not confined to the average: GeoSD attains the best OOD accuracy on all four benchmarks in all three families (see App. Table 5 for per-benchmark breakdown). Its OOD improvement over the strongest baseline is statistically significant in every family (App. Table 6 ). The informative trend is therefore not just that GeoSD avoids collapse, but that it improves the ID–OOD trade-off itself, suggesting a change in how privileged supervision is absorbed rather than merely in how strongly it is applied. Figure 3 makes this concrete on the ID/OOD plane, with marker size encoding OOD pass@ 16 16 : GeoSD occupies the favorable upper-right region and is not dominated by any baseline, pairing strong ID performance with the highest OOD accuracy (avg@ 16 16 and pass@ 16 16 ), whereas standard divergence objectives lie along the OOD-degrading branch. This separation is robust to the choice of learning rate and stopping step: swept across both, FwdKL only trades ID against OOD without rising above the base model’s OOD level, while GeoSD improves both axes together (Appendix C.4 )."},{"type":"heading","level":4,"text":"OOD generalization under varying privileged information."},{"type":"para","text":"To vary the amount of privilege, we split each reference solution into sentences and reveal an increasing fraction of them to the teacher, from a short prefix up to the full solution, holding everything else fixed. Figure 5 tracks OOD reasoning across this sweep for GeoSD , FwdKL, and RevKL. The divergence baselines decline as more of the solution is revealed: RevKL stays near the base model through the first half of the sweep before dropping sharply, FwdKL falls earlier, and both reach their worst OOD accuracy at full privilege, whereas GeoSD improves monotonically across the same range. As more of the solution is revealed, more positions in the rollout carry teacher confidence the student cannot reproduce at inference; the divergence baselines copy this confidence and degrade, while GeoSD ’s overlap weighting attenuates its effect. The same privileged signal, absorbed differently, thus becomes a source of OOD gains that grow with privilege rather than a cause of collapse."},{"type":"heading","level":4,"text":"Geometric control complements scale."},{"type":"para","text":"Figure 5 examines GeoSD across the Qwen3 ladder, from 1.7B to 32B. Larger base models are stronger on both ID and OOD, yet scale alone does not close the ID–OOD gap: GeoSD improves OOD accuracy over the base model at every size. The gain is non-monotone in scale, peaking mid-ladder ( + 8.6 +8.6 at 8B) and settling to + 6.1 +6.1 at 32B, still larger than the gains at 1.7B and 4B ( + 2.8 +2.8 and + 4.6 +4.6 ). The benefit of geometric regulation for self-distillation therefore persists as models grow."},{"type":"heading","level":2,"text":"3.3 Ablations"},{"type":"para","text":"GeoSD couples three components in a single geometry: overlap-aware pressure on the per-state pull, a proximal penalty on cumulative drift, and a natural-gradient update in the same geometry. Ablating each in turn while holding the other two at their defaults (Table 2 ), we find all three are load-bearing: swapping Hellinger for JSD, removing the proximal term ( λ = 0 \\lambda=0 ), or replacing the natural-gradient update with AdamW each surrenders a large part of the OOD gain. Removing the proximal term drives predictive drift well above every other variant and yields the weakest OOD result, in both accuracy and pass@ 16 16 coverage, confirming that overlap-aware Hellinger pressure bounds the per-state pull but not the accumulated path. The Euclidean (AdamW) update reduces drift relative to the proximal-free variant but still underperforms the full objective, which suggests the benefit is not simply smaller updates but updates shaped by their effect on the predictive distribution. Swapping Hellinger for JSD is a revealing case: with the proximal and natural-gradient terms held fixed, replacing the loss lowers both ID ( 82.7 → 80.2 82.7\\to 80.2 ) and OOD ( 60.5 → 56.1 60.5\\to 56.1 ) accuracy. That JSD, a mixture of forward and reverse KL, shifts both axes rather than trading one off against the other, indicates that Hellinger is the stronger choice here, not an interchangeable stand-in for a softer divergence."},{"type":"heading","level":1,"text":"4 Alignment without collapse"},{"type":"para","text":"Why does standard OPSD improve in-distribution performance while hurting generalization? We trace the failure to how probability mass concentrates during training, measuring that concentration at two levels: the next-token distribution at high-entropy states (§ 4.1 ) and the distribution over final answers (§ 4.2 ). At both levels, KL matching concentrates mass abruptly while GeoSD keeps it controlled, and the local pattern reappears downstream in the answers, linking the two."},{"type":"heading","level":2,"text":"4.1 Local concentration at high-entropy states"},{"type":"para","text":"We examine the states where the student is least certain, since these are where a privileged teacher most often diverges from it. For each rollout we select the top 10 % 10\\% of positions by the base model’s ( π 0 \\pi_{0} ) next-token entropy. This set is fixed once from π 0 \\pi_{0} and held constant across methods and checkpoints, so every method is evaluated on the same positions and any difference reflects the objective rather than a shift in which states each model treats as uncertain. At each selected position we rank the candidate tokens by the teacher’s probability, with rank 1 the teacher’s most-preferred token, and track how the student’s average mass on ranks 1 1 – 8 8 evolves over training."},{"type":"para","text":"Figure 6 compares FwdKL and GeoSD across post-training checkpoints. Under forward KL, rank-1 mass climbs steeply within the first two epochs, producing early agreement with the teacher’s top choice by concentrating probability mass directly on that token. Under GeoSD , rank-1 mass rises gradually and substantial mass remains on lower-ranked teacher tokens even after the top choice becomes dominant. Although both objectives move the student toward the teacher’s top-ranked prediction, they do so through different probability trajectories: FwdKL rapidly concentrates onto the top choice, whereas GeoSD raises the top choice while keeping the rest of the teacher-preferred support within reach."},{"type":"para","text":"This premature concentration on the top choice, i.e., rank-1 collapse, removes alternatives: if the lower-ranked teacher tokens are viable continuations, draining them takes away options the student may need out of distribution. The gentler trajectory is the expected behavior of the overlap-weighted gradient (equation 4 ), which pulls hard only where the student already assigns support, while the Fisher–Rao proximal term keeps per-state pulls from compounding into drift. Whether the retained mass is useful rather than cosmetic is what we test next."},{"type":"heading","level":2,"text":"4.2 How concentration propagates: false consensus"},{"type":"para","text":"If the local pattern matters, it should leave a fingerprint on final answers, not only on intermediate distributions. For each OOD problem x x we draw m = 16 m=16 solutions and record their final answers. Specifically, p ^ x ( a ) \\hat{p}_{x}(a) is the fraction of the 16 16 samples that give answer a a , the majority answer a ^ ( x ) \\hat{a}(x) is the one most samples agree on, and the consensus strength C ( x ) C(x) is the fraction agreeing on it:"},{"type":"para","text":"We call x x a false-consensus case when the samples agree strongly on a wrong answer, C ( x ) ≥ τ C(x)\\geq\\tau and a ^ ( x ) ≠ a ⋆ ( x ) \\hat{a}(x)\\neq a^{\\star}(x) , taking τ = 0.75 \\tau=0.75 as the threshold for strong agreement. This separates productive self-consistency, where agreement tracks correctness, from collapsed self-consistency, where the model agrees only because it has narrowed to a single mode prematurely."},{"type":"para","text":"Figure 7 reports answer-level behavior on OOD problems from three angles. Panel A asks whether sampling still recovers correct solutions: FwdKL improves low- k k pass@ k k but flattens quickly, the filtering baselines retain more of the benefit, and GeoSD sustains the steepest slope out to k = 32 k=32 . Panel B counts false consensus: most frequent under FwdKL, reduced by softer objectives and filtering, and lowest under GeoSD , below even the base model, since keeping lower-ranked continuations in reach makes narrowing onto a single wrong mode rarer than in the untuned model. Panel C asks whether agreement is trustworthy: under KL matching, majority-answer accuracy bends below the diagonal at high consensus, so strong agreement increasingly signals confidently wrong answers, whereas GeoSD tracks the diagonal closely, keeping self-consistency informative."},{"type":"heading","level":1,"text":"5 Related Work"},{"type":"heading","level":4,"text":"Privileged-context OPSD."},{"type":"para","text":"Recent work removes the need for a separate stronger teacher by using the model itself, or a recent copy, as the teacher. In self-distilled reasoning, the teacher is evaluated on the student’s own rollouts but with additional context such as verified solutions or enriched reasoning traces (Zhao et al. , 2026 ) . On-policy context distillation generalizes this setup by transferring behavior from richer training-time contexts into an inference-time model that will not receive those contexts (Ye et al. , 2026 ) . More broadly, privileged-information distillation studies this asymmetry directly: the teacher can condition on information that is useful during training but unavailable at test time (Penaloza et al. , 2026 ) . This practical advantage comes with a mismatch: standard imitation can calibrate the student to the teacher’s privileged view rather than to its test-time evidence, suppressing uncertainty and degrading OOD reasoning (Kim et al. , 2026 ) ."},{"type":"heading","level":4,"text":"Stabilization, selection, and geometry."},{"type":"para","text":"Another line of work stabilizes OPD by deciding which teacher signals should be trusted, or how each should be matched. TrOPD separates reliable regions from outliers and applies different distillation rules accordingly (Xing et al. , 2026 ) ; TIP selects or reweights informative tokens using teacher–student mismatch and student uncertainty (Xu et al. , 2026 ) ; GATES uses consensus to gate self-distillation under privileged context (Stein et al. , 2026 ) ; and EOPD, observing that mode-seeking pressure drains student diversity at high-teacher-entropy tokens, switches from reverse to forward KL at those positions (Jin et al. , 2026 ) . These methods filter, reweight, or redirect supervision, but they do not directly constrain the behavioral movement caused by an accepted teacher signal. In parallel, OPD updates have been shown to concentrate early in a narrow low-dimensional parameter subspace and largely stay there (Shen et al. , 2026 ) : stability hinges not only on which signals are used, but on how training moves the model."},{"type":"heading","level":4,"text":"Proximal regularization."},{"type":"para","text":"Penalizing movement from a reference model is a recurring stabilizer outside distillation. In continual learning, elastic weight consolidation anchors parameters with a diagonal-Fisher quadratic penalty at a fixed past solution (Kirkpatrick et al. , 2017 ) , and related output-space variants preserve predictions on earlier tasks (Li and Hoiem, 2017 ) . In reinforcement learning from human feedback (RLHF), a KL penalty to a frozen reference policy is standard practice (Ziegler et al. , 2020 ; Ouyang et al. , 2022 ) , and trust-region methods bound per-update policy change (Schulman et al. , 2015 ; 2017 ) . The shared lesson is that dense adaptation pressure is safe only when tempered by a penalty on displacement from a trusted reference. Our Fisher–Rao proximal term brings this principle to privileged-context OPSD, with a recent checkpoint as the reference and displacement measured in the same geometry as the distillation loss."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"Privileged-context self-distillation can turn reasoning traces into dense supervision for LLM post-training, but it also makes teacher confidence harder to trust: a teacher that sees the solution may prefer tokens the student cannot yet reproduce from its own context. We framed the resulting OOD degradation as predictive drift in the space of next-token distributions and introduced GeoSD , which controls that drift geometrically, letting privileged context guide the student in proportion to teacher–student overlap while bounding how far accumulated updates may move its predictive behavior. Across benchmarks, model families, and scales, GeoSD retains the ID gains of self-distillation while substantially improving OOD reasoning. We trace the collapse that standard divergence matching suffers to a single failure: unsupported teacher confidence concentrates probability mass prematurely at the high-entropy states where the student is least certain, and this concentration resurfaces downstream as confident agreement on wrong answers. The broader lesson is that effective reasoning distillation depends not only on which teacher signals to trust, but on how far those signals are allowed to move the student. We discuss the scope and limits of our work in Appendix E ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On-policy distillation (OPD) has emerged as a practical post-training recipe for large language models (LLMs), providing dense teacher supervision on trajectories generated by the student itself (Agarwal et al. , 2024 ; Song and Zheng, 2026 ) . Dense supervision is especially valuable for reasoning , where final-answer rewards are often too sparse to reveal which intermediate steps were useful (Lightman et al. , 2024 ) . Instead of assigning credit only after a completed solution, OPD evaluates the teacher’s distribution at the states the student visits, producing token-level targets throughout the trajectory. The recipe can scale cheaply, since it requires no separate, stronger teacher. In on-policy self-distillation (OPSD), the student acts as its own teacher, supervising the prefixes it generates. A common and effective variant grants this teacher privileged context , such as a hint or the full solution trace, while the student must reason from the original problem alone (Zhao et al. , 2026 ; Ye et al. , 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Whether dense supervision helps depends on the student’s ability to follow the teacher’s token-level preferences. Li et al. ( 2026 ) show that OPD progresses when student and teacher increasingly share mass on high-probability tokens, and stalls when that mass is largely disjoint. Privileged context is a systematic source of such mismatch: the extra information lets the teacher commit to continuations that are obvious given the solution but still uncertain from the student’s own context (Hübotter et al. , 2026 ; Penaloza et al. , 2026 ) . These mismatched states are also where imitation pulls hardest: standard divergences weight a teacher token by its own mass, so a token the teacher is sure of but the student barely supports produces a large gradient and a strong update. As such updates repeat, the student commits to the teacher’s continuations before its own information can support them, and the mismatch accumulates into drift. This drift is visible in the model’s behavior: Kim et al. ( 2026 ) observe that privileged-context OPSD suppresses uncertainty and degrades out-of-distribution (OOD) performance. Existing approaches mostly intervene on which teacher signals to use. Some filter or down-weight signals judged unreliable (Xing et al. , 2026 ; Xu et al. , 2026 ; Stein et al. , 2026 ) ; others reshape how a trusted signal is matched, changing the direction or weighting of divergences in the Kullback–Leibler (KL) family (Gu et al. , 2024 ; Agarwal et al. , 2024 ; Ko et al. , 2025 ) or reweighting token credit (Liu et al. , 2026 ; Yang et al. , 2026 ) . What they leave unaddressed is how far a trusted signal should move the student. The most informative supervision coincides with the largest updates, leaving selection no good option: filtering those states discards useful signal, while keeping them lets the student inherit the teacher’s overconfidence. Selection alone cannot regulate update magnitude, and this unchecked movement accumulates into drift, ultimately degrading OOD reasoning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce GeoSD ( Geo metric S elf- D istillation), which counters this drift in two complementary ways: attenuating the teacher’s pull at low-overlap states and regulating how far those pulls carry the student over training. First, we replace the standard distillation objective with a Hellinger loss. Its gradient weights each teacher preference by the probability mass student and teacher already share: tokens the teacher assigns high probability but the student finds unlikely exert little pull, while shared preferences transfer strongly. Because this attenuation is built into the gradient rather than applied on top of it by clipping or filtering, supervision strengthens gradually as the two come into agreement (Figure 1 A). Second, we penalize accumulated movement from a recent checkpoint with the Fisher–Rao distance between predictive distributions (Figure 1 B). Because both terms share a single geometry of next-token distributions, a natural-gradient update optimizes both within it: each step makes the smallest change to the student’s predictions that improves the objective. Empirically, GeoSD preserves the in-distribution (ID) gains of standard OPSD while substantially improving OOD reasoning. We trace this improvement to how each objective reaches agreement: standard matching concentrates mass on the teacher’s top choice by draining the alternatives, whereas GeoSD makes the same top choice dominant while keeping the alternatives in reach.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are threefold. (i) We introduce GeoSD , a self-distillation objective that weights each teacher preference by teacher–student overlap and penalizes predictive drift from a recent checkpoint, optimizing both as distances in a single geometry via natural gradient. (ii) GeoSD improves average OOD accuracy on mathematical reasoning by 5.7 5.7 – 8.6 8.6 points (avg@16) over the base model across three model families, whereas standard KL objectives degrade it (§ 3 ). GeoSD retains ID gains and achieves the best accuracy on all tested OOD benchmarks, with the OOD gains holding consistently across five model scales (1.7B–32B). (iii) We characterize how standard matching fails out of distribution: it concentrates probability mass at high-entropy states, resulting in confident agreement on incorrect answers (§ 4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now formulate GeoSD , which targets the movement a teacher induces in the student’s predictive behavior during self-distillation. Because this movement is poorly captured in raw probabilities or parameters, where equal-sized changes can have very different behavioral effects, we measure it in a single geometry of next-token distributions. After defining the on-policy student–teacher setup (§ 2.1 ), we first modulate the local pressure at each visited state with an overlap-aware distillation loss that weakens each teacher preference in proportion to the student’s overlap with it (§ 2.2 ). We then penalize the global displacement that accumulates over training with a proximal term to a recent checkpoint (§ 2.3 ), and optimize the combined objective with a natural-gradient update that adjusts each step by its effect on the student’s predictions rather than by its size in parameter space (§ 2.4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒱 \\\\mathcal{V} denote the vocabulary and Δ 𝒱 \\\\Delta_{\\\\mathcal{V}} the probability simplex over it. For an input 𝐱 \\\\mathbf{x} and an on-policy continuation 𝐲 ∼ π θ ( ⋅ ∣ 𝐱 ) \\\\mathbf{y}\\\\sim\\\\pi_{\\\\theta}(\\\\cdot\\\\mid\\\\mathbf{x}) , the student ( p p ) and teacher ( q q ) next-token distributions at position t t are\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the teacher reads the same prefix but with privileged context 𝐜 \\\\mathbf{c} , a full solution trace sampled from the base model π 0 \\\\pi_{0} . The objective treats the teacher as any privileged-context distribution; we instantiate it as the current student, so both come from the same network, though a fixed reference, a delayed copy, or a moving average of recent students would serve as well. We write q ( t ) q^{(t)} without the parameter subscript to mark it as a fixed target, with gradients flowing only through the student p θ ( t ) p_{\\\\theta}^{(t)} , so each update moves the student toward the privileged-context distribution without optimizing that distribution itself. Because parameters and prefix are shared, p θ ( t ) p_{\\\\theta}^{(t)} and q ( t ) ( 𝐜 ) q^{(t)}(\\\\mathbf{c}) differ only through 𝐜 \\\\mathbf{c} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Local teacher influence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The student often retains plausible alternatives beyond its dominant prediction, corresponding to continuations it may still sample. Suppressing one such alternative can produce only a small change in raw probability space, while effectively removing a viable continuation. A suitable distance should therefore weight changes by each token’s probability mass: the same change can be minor for the dominant prediction but significant for a lower-probability alternative. Information geometry supplies such a distance (Amari, 2016 ; Amari and Nagaoka, 2000 ) : under the square-root embedding p ↦ p p\\\\mapsto\\\\sqrt{p} , next-token distributions live on a hypersphere where statistical distance becomes ordinary distance (Figure 1 B; Appendix A.2 ). The sphere carries two canonical distances, the chord through it and the arc along the surface. The chord yields the squared Hellinger divergence, which we use for the per-state pull:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The arc is the Fisher–Rao distance, which we use for cumulative drift (§ 2.3 ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sphere fixes the geometry but not the loss. Near agreement, KL and Hellinger induce the same local geometry up to scale (Appendix A.3 ). However, their behavior differs in the low-overlap regime, which is critical for privileged-context OPSD. We use the Hellinger chord as the teacher pull, averaged over inputs, on-policy trajectories, sampled privileged contexts 𝐜 ∼ 𝒞 ( ⋅ ∣ 𝐱 ) \\\\mathbf{c}\\\\sim\\\\mathcal{C}(\\\\cdot\\\\mid\\\\mathbf{x}) , and positions: 1 1 1 Gradients do not propagate through the sampling distribution π θ ( ⋅ ∣ 𝐱 ) \\\\pi_{\\\\theta}(\\\\cdot\\\\mid\\\\mathbf{x}) : trajectories are treated as fixed once sampled, dropping the score-function term of the outer expectation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What singles out the chord is its gradient. Differentiating one Hellinger term against the stop-gradient teacher q q gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which scales each token’s influence by the geometric-mean overlap p θ ( i ) q i \\\\sqrt{p_{\\\\theta}(i)\\\\,q_{i}} . The weight vanishes as p θ ( i ) → 0 p_{\\\\theta}(i)\\\\!\\\\to\\\\!0 , so even a token the teacher is certain of exerts almost no pull once the student has all but excluded it, and the per-state gradient shrinks as overlap falls.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 A shows this contrast empirically. Forward KL weights a token by teacher mass q i q_{i} , so a confident teacher exerts large pressure even where the student barely supports the token, and its gradient norm grows as overlap falls. These high-teacher, low-student states form a dense region of positions visited by the student rather than a rare tail (see Figure 8 in Appendix C.1 ), so behavior at low overlap shapes much of the training signal. Reverse KL weights by student mass p θ ( i ) p_{\\\\theta}(i) , which attenuates that pressure but makes it mode-seeking and suppresses alternatives. Hellinger’s geometric mean sits between them: the pull fades as the student withdraws support, while teacher alternatives are kept in proportion to their overlap.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Global drift control\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overlap weighting bounds the pull at each state, but a bound per step is not a bound on the path: many small, individually reasonable pulls can still carry the student far over training. Self-distillation makes this acute, since the teacher is a copy of the student. As the student moves, the targets move with it, and the two can drift together while no single update ever looks large.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We control the path with an established remedy in post-training, a penalty on displacement from a trusted reference (§ 5 ), cast here in the same geometry as the pull: a proximal term on the Fisher–Rao motion from a recent checkpoint θ ckpt \\\\theta_{\\\\mathrm{ckpt}} , refreshed every K ckpt K_{\\\\mathrm{ckpt}} steps: 1\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The checkpoint is deliberately recent rather than the initial model. Anchoring to π 0 \\\\pi_{0} would maximize retention but forbid the adaptation that makes distillation useful; a recent θ ckpt \\\\theta_{\\\\mathrm{ckpt}} instead acts as a moving trust region, damping abrupt displacement while still allowing steady movement when the teacher signal is consistent. Overlap weighting alone slows cumulative drift but does not stop it, and adding the proximal term keeps it controlled (Figure 2 B).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4 Full objective and natural-gradient update\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The objective combines the two terms,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with λ \\\\lambda setting the strength of the drift penalty. Both terms are distances between next-token distributions, so the update should move through that same geometry. A natural gradient asks, of all parameter updates that change the student’s predictions by a fixed amount, which one most improves the objective. This is the principle behind the geometry above, applied now to the update itself and realized by preconditioning with the Fisher information F k F_{k} of the current policy (Amari, 1998 ; Schulman et al. , 2015 ) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each step is then adjusted based on its effect on the model’s predictions, rather than its raw size in parameters. Under this update, the proximal term acts as a checkpoint pullback: drift is measured in the checkpoint’s local geometry and returned as a restoring force in the current one, so the pull is strongest where drift is most visible in the predictions (Appendix A.6 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At LLM scale the full Fisher is infeasible to form or invert, so we precondition with a damped Kronecker-factored approximation of the empirical Fisher (K-FAC; Martens and Grosse, 2015 ) that captures the layerwise curvature without materializing the full matrix (Appendix B.6 ). This keeps the update curvature-aware while adding only modest memory and runtime over a standard distillation step (Appendix D ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Experimental Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate along two independent axes. For scale , we use the Qwen3 dense family at five sizes: 1.7B, 4B, 8B, 14B, and 32B (Yang et al. , 2025 ) . For cross-family generalization , we compare three models from different families: Qwen3-8B (Yang et al. , 2025 ) , Olmo-3-7B-Think (Olmo et al. , 2025 ) , and DeepSeek-R1-Distill-Llama-8B (DS-R1-Llama-8B; Guo et al. , 2025 ). At 14B and 32B scale, we use a more memory-efficient K-FAC for the natural-gradient update (Appendix B.6 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training and evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On-policy trajectories are sampled from the student on problems from DAPO-Math-17k (Yu et al. , 2026 ) , a competition-mathematics dataset providing verifiable integer answers but no reference solutions. In-distribution performance is measured on a held-out split. We measure out-of-distribution reasoning on AIME 2024, AIME 2025, AMC 2023, and MATH-500 (Lightman et al. , 2024 ) , none of which overlap with the training problems. We report accuracy using avg@ k k (average accuracy over k k samples) and pass@ k k (whether any of the k k samples is correct). By default, the privileged context is a full correct solution, following the OPSD convention (Zhao et al. , 2026 ; Kim et al. , 2026 ) . Solutions are constructed offline: for each problem, we sample up to 32 32 rollouts from π 0 \\\\pi_{0} , keep the ones with verified answers, and at each step draw one of a problem’s verified solutions as the teacher’s privileged context. Problems for which no rollout verifies are excluded from training. Full data, rollout, and evaluation settings are in Appendix B .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare GeoSD against three baseline families. Reference baselines include Base ( π 0 \\\\pi_{0} ; initial model checkpoint), SFT (supervised fine-tuning) on privileged teacher traces, and GRPO (Shao et al. , 2024 ) with verifiable rewards on the same training prompts. These respectively represent the pretrained reference point, off-policy imitation of privileged solutions, and a privilege-free sparse-reward post-training baseline. Divergence controls keep our pipeline fixed and swap the per-position loss: forward KL ( FwdKL ; Hinton et al. , 2015 ; Zhao et al. , 2026 ), reverse KL ( RevKL ; Gu et al. , 2024 ), Jensen–Shannon ( JSD , β = 0.5 \\\\beta{=}0.5 ; Agarwal et al. , 2024 ), and skewed KL ( SkewKL , α = 0.1 \\\\alpha{=}0.1 ; Ko et al. , 2024 ; 2025 ). Filtering baselines test the complementary strategy of deciding which teacher signals to use rather than changing the geometry of the update: TrOPD (Xing et al. , 2026 ) separates teacher-reliable regions from outlier regions using teacher–student agreement, while TIP (Xu et al. , 2026 ) selects or reweights informative tokens using student uncertainty and teacher–student mismatch. To isolate the effect of the objective, every baseline runs in our privileged-context OPSD pipeline with matched rollout and compute budgets, so the numbers are comparable across our rows. We provide implementation details for baselines in Appendix B.5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Compute overhead.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GeoSD ’s geometry-aware update is cheap in practice despite invoking a natural gradient: over standard OPSD it adds only a checkpoint forward pass and K-FAC covariance accumulation, raising optimizer FLOPs by ≈ 27 % {\\\\approx}27\\\\% ( 1.27 × 1.27\\\\times ) and training memory by ≈ 21 % {\\\\approx}21\\\\% ( 1.21 × 1.21\\\\times ), of which only the K-FAC state ( ≈ 8 % {\\\\approx}8\\\\% ) resides on the training workers in the default placement. Wall-clock time grows by only 1.10 × 1.10\\\\times , since the checkpoint forward overlaps with rollout generation, which dominates the run and is shared across all methods. A detailed breakdown is in Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Generalization under privileged self-distillation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Geometry improves the ID–OOD reasoning frontier.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 compares OPSD approaches on ID and OOD performance, exposing a consistent gap between in-distribution gains and their transfer under distribution shift. FwdKL posts some of the largest ID gains of any method, yet its OOD accuracy degrades relative to π 0 \\\\pi_{0} in all three families (by 4.6 4.6 – 8.1 8.1 points avg@ 16 16 ). This ID-up, OOD-down pattern reproduces the trend prior work reports for imitation-style post-training, and holds under a strict evaluation protocol (Appendix B.4 ). Softer objectives (SkewKL, JSD) mostly remove the degradation but barely improve over the base model, and the filtering baselines (TrOPD, TIP) recover only modest OOD gains ( + 0.4 +0.4 to + 3.2 +3.2 ). GeoSD is the only method that both retains the ID gains of self-distillation and improves OOD reasoning substantially, by 5.7 5.7 – 8.6 8.6 points avg@ 16 16 over π 0 \\\\pi_{0} ; it ranks first on OOD in every family while staying within 0.7 0.7 – 1.7 1.7 points of the best ID score. This advantage is not confined to the average: GeoSD attains the best OOD accuracy on all four benchmarks in all three families (see App. Table 5 for per-benchmark breakdown). Its OOD improvement over the strongest baseline is statistically significant in every family (App. Table 6 ). The informative trend is therefore not just that GeoSD avoids collapse, but that it improves the ID–OOD trade-off itself, suggesting a change in how privileged supervision is absorbed rather than merely in how strongly it is applied. Figure 3 makes this concrete on the ID/OOD plane, with marker size encoding OOD pass@ 16 16 : GeoSD occupies the favorable upper-right region and is not dominated by any baseline, pairing strong ID performance with the highest OOD accuracy (avg@ 16 16 and pass@ 16 16 ), whereas standard divergence objectives lie along the OOD-degrading branch. This separation is robust to the choice of learning rate and stopping step: swept across both, FwdKL only trades ID against OOD without rising above the base model’s OOD level, while GeoSD improves both axes together (Appendix C.4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"OOD generalization under varying privileged information.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To vary the amount of privilege, we split each reference solution into sentences and reveal an increasing fraction of them to the teacher, from a short prefix up to the full solution, holding everything else fixed. Figure 5 tracks OOD reasoning across this sweep for GeoSD , FwdKL, and RevKL. The divergence baselines decline as more of the solution is revealed: RevKL stays near the base model through the first half of the sweep before dropping sharply, FwdKL falls earlier, and both reach their worst OOD accuracy at full privilege, whereas GeoSD improves monotonically across the same range. As more of the solution is revealed, more positions in the rollout carry teacher confidence the student cannot reproduce at inference; the divergence baselines copy this confidence and degrade, while GeoSD ’s overlap weighting attenuates its effect. The same privileged signal, absorbed differently, thus becomes a source of OOD gains that grow with privilege rather than a cause of collapse.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Geometric control complements scale.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 5 examines GeoSD across the Qwen3 ladder, from 1.7B to 32B. Larger base models are stronger on both ID and OOD, yet scale alone does not close the ID–OOD gap: GeoSD improves OOD accuracy over the base model at every size. The gain is non-monotone in scale, peaking mid-ladder ( + 8.6 +8.6 at 8B) and settling to + 6.1 +6.1 at 32B, still larger than the gains at 1.7B and 4B ( + 2.8 +2.8 and + 4.6 +4.6 ). The benefit of geometric regulation for self-distillation therefore persists as models grow.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Ablations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GeoSD couples three components in a single geometry: overlap-aware pressure on the per-state pull, a proximal penalty on cumulative drift, and a natural-gradient update in the same geometry. Ablating each in turn while holding the other two at their defaults (Table 2 ), we find all three are load-bearing: swapping Hellinger for JSD, removing the proximal term ( λ = 0 \\\\lambda=0 ), or replacing the natural-gradient update with AdamW each surrenders a large part of the OOD gain. Removing the proximal term drives predictive drift well above every other variant and yields the weakest OOD result, in both accuracy and pass@ 16 16 coverage, confirming that overlap-aware Hellinger pressure bounds the per-state pull but not the accumulated path. The Euclidean (AdamW) update reduces drift relative to the proximal-free variant but still underperforms the full objective, which suggests the benefit is not simply smaller updates but updates shaped by their effect on the predictive distribution. Swapping Hellinger for JSD is a revealing case: with the proximal and natural-gradient terms held fixed, replacing the loss lowers both ID ( 82.7 → 80.2 82.7\\\\to 80.2 ) and OOD ( 60.5 → 56.1 60.5\\\\to 56.1 ) accuracy. That JSD, a mixture of forward and reverse KL, shifts both axes rather than trading one off against the other, indicates that Hellinger is the stronger choice here, not an interchangeable stand-in for a softer divergence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Alignment without collapse\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why does standard OPSD improve in-distribution performance while hurting generalization? We trace the failure to how probability mass concentrates during training, measuring that concentration at two levels: the next-token distribution at high-entropy states (§ 4.1 ) and the distribution over final answers (§ 4.2 ). At both levels, KL matching concentrates mass abruptly while GeoSD keeps it controlled, and the local pattern reappears downstream in the answers, linking the two.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Local concentration at high-entropy states\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We examine the states where the student is least certain, since these are where a privileged teacher most often diverges from it. For each rollout we select the top 10 % 10\\\\% of positions by the base model’s ( π 0 \\\\pi_{0} ) next-token entropy. This set is fixed once from π 0 \\\\pi_{0} and held constant across methods and checkpoints, so every method is evaluated on the same positions and any difference reflects the objective rather than a shift in which states each model treats as uncertain. At each selected position we rank the candidate tokens by the teacher’s probability, with rank 1 the teacher’s most-preferred token, and track how the student’s average mass on ranks 1 1 – 8 8 evolves over training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6 compares FwdKL and GeoSD across post-training checkpoints. Under forward KL, rank-1 mass climbs steeply within the first two epochs, producing early agreement with the teacher’s top choice by concentrating probability mass directly on that token. Under GeoSD , rank-1 mass rises gradually and substantial mass remains on lower-ranked teacher tokens even after the top choice becomes dominant. Although both objectives move the student toward the teacher’s top-ranked prediction, they do so through different probability trajectories: FwdKL rapidly concentrates onto the top choice, whereas GeoSD raises the top choice while keeping the rest of the teacher-preferred support within reach.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This premature concentration on the top choice, i.e., rank-1 collapse, removes alternatives: if the lower-ranked teacher tokens are viable continuations, draining them takes away options the student may need out of distribution. The gentler trajectory is the expected behavior of the overlap-weighted gradient (equation 4 ), which pulls hard only where the student already assigns support, while the Fisher–Rao proximal term keeps per-state pulls from compounding into drift. Whether the retained mass is useful rather than cosmetic is what we test next.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 How concentration propagates: false consensus\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If the local pattern matters, it should leave a fingerprint on final answers, not only on intermediate distributions. For each OOD problem x x we draw m = 16 m=16 solutions and record their final answers. Specifically, p ^ x ( a ) \\\\hat{p}_{x}(a) is the fraction of the 16 16 samples that give answer a a , the majority answer a ^ ( x ) \\\\hat{a}(x) is the one most samples agree on, and the consensus strength C ( x ) C(x) is the fraction agreeing on it:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We call x x a false-consensus case when the samples agree strongly on a wrong answer, C ( x ) ≥ τ C(x)\\\\geq\\\\tau and a ^ ( x ) ≠ a ⋆ ( x ) \\\\hat{a}(x)\\\\neq a^{\\\\star}(x) , taking τ = 0.75 \\\\tau=0.75 as the threshold for strong agreement. This separates productive self-consistency, where agreement tracks correctness, from collapsed self-consistency, where the model agrees only because it has narrowed to a single mode prematurely.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 7 reports answer-level behavior on OOD problems from three angles. Panel A asks whether sampling still recovers correct solutions: FwdKL improves low- k k pass@ k k but flattens quickly, the filtering baselines retain more of the benefit, and GeoSD sustains the steepest slope out to k = 32 k=32 . Panel B counts false consensus: most frequent under FwdKL, reduced by softer objectives and filtering, and lowest under GeoSD , below even the base model, since keeping lower-ranked continuations in reach makes narrowing onto a single wrong mode rarer than in the untuned model. Panel C asks whether agreement is trustworthy: under KL matching, majority-answer accuracy bends below the diagonal at high consensus, so strong agreement increasingly signals confidently wrong answers, whereas GeoSD tracks the diagonal closely, keeping self-consistency informative.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Privileged-context OPSD.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent work removes the need for a separate stronger teacher by using the model itself, or a recent copy, as the teacher. In self-distilled reasoning, the teacher is evaluated on the student’s own rollouts but with additional context such as verified solutions or enriched reasoning traces (Zhao et al. , 2026 ) . On-policy context distillation generalizes this setup by transferring behavior from richer training-time contexts into an inference-time model that will not receive those contexts (Ye et al. , 2026 ) . More broadly, privileged-information distillation studies this asymmetry directly: the teacher can condition on information that is useful during training but unavailable at test time (Penaloza et al. , 2026 ) . This practical advantage comes with a mismatch: standard imitation can calibrate the student to the teacher’s privileged view rather than to its test-time evidence, suppressing uncertainty and degrading OOD reasoning (Kim et al. , 2026 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stabilization, selection, and geometry.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Another line of work stabilizes OPD by deciding which teacher signals should be trusted, or how each should be matched. TrOPD separates reliable regions from outliers and applies different distillation rules accordingly (Xing et al. , 2026 ) ; TIP selects or reweights informative tokens using teacher–student mismatch and student uncertainty (Xu et al. , 2026 ) ; GATES uses consensus to gate self-distillation under privileged context (Stein et al. , 2026 ) ; and EOPD, observing that mode-seeking pressure drains student diversity at high-teacher-entropy tokens, switches from reverse to forward KL at those positions (Jin et al. , 2026 ) . These methods filter, reweight, or redirect supervision, but they do not directly constrain the behavioral movement caused by an accepted teacher signal. In parallel, OPD updates have been shown to concentrate early in a narrow low-dimensional parameter subspace and largely stay there (Shen et al. , 2026 ) : stability hinges not only on which signals are used, but on how training moves the model.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Proximal regularization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Penalizing movement from a reference model is a recurring stabilizer outside distillation. In continual learning, elastic weight consolidation anchors parameters with a diagonal-Fisher quadratic penalty at a fixed past solution (Kirkpatrick et al. , 2017 ) , and related output-space variants preserve predictions on earlier tasks (Li and Hoiem, 2017 ) . In reinforcement learning from human feedback (RLHF), a KL penalty to a frozen reference policy is standard practice (Ziegler et al. , 2020 ; Ouyang et al. , 2022 ) , and trust-region methods bound per-update policy change (Schulman et al. , 2015 ; 2017 ) . The shared lesson is that dense adaptation pressure is safe only when tempered by a penalty on displacement from a trusted reference. Our Fisher–Rao proximal term brings this principle to privileged-context OPSD, with a recent checkpoint as the reference and displacement measured in the same geometry as the distillation loss.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Privileged-context self-distillation can turn reasoning traces into dense supervision for LLM post-training, but it also makes teacher confidence harder to trust: a teacher that sees the solution may prefer tokens the student cannot yet reproduce from its own context. We framed the resulting OOD degradation as predictive drift in the space of next-token distributions and introduced GeoSD , which controls that drift geometrically, letting privileged context guide the student in proportion to teacher–student overlap while bounding how far accumulated updates may move its predictive behavior. Across benchmarks, model families, and scales, GeoSD retains the ID gains of self-distillation while substantially improving OOD reasoning. We trace the collapse that standard divergence matching suffers to a single failure: unsupported teacher confidence concentrates probability mass prematurely at the high-entropy states where the student is least certain, and this concentration resurfaces downstream as confident agreement on wrong answers. The broader lesson is that effective reasoning distillation depends not only on which teacher signals to trust, but on how far those signals are allowed to move the student. We discuss the scope and limits of our work in Appendix E .\"\n }\n]","isConversational":false}}},{"rowIdx":495,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06841"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling"},"authors":{"kind":"list like","value":["Robert Gruhlke","Julius Berner","David Sommer","Lorenz Richter"],"string":"[\n \"Robert Gruhlke\",\n \"Julius Berner\",\n \"David Sommer\",\n \"Lorenz Richter\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06841v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Sampling from a complex, high-dimensional probability density"},{"type":"para","text":"where the unnormalized density ρ target \\rho_{\\mathrm{target}} can be evaluated pointwise but the normalizing constant 𝒵 \\mathcal{Z} is intractable and no samples from p target p_{\\mathrm{target}} are available, constitutes a central challenge in modern machine learning, statistics, and the physical sciences. An especially versatile and powerful class of methods for this task is based on dynamical measure transport, which generates samples from the target distribution by evolving initial samples from a simple reference distribution – such as a standard Gaussian – along trajectories defined by stochastic or ordinary differential equations. One of the most successful paradigms within this framework is the time reversal of a noising SDE, forming the foundation of diffusion models (Ho et al. , 2020 ; Song et al. , 2021 ) . In this approach, a forward SDE – typically chosen as an Ornstein-Uhlenbeck process – progressively transforms the target distribution into a simple prior by injecting noise over time. The central insight is that reversing this noising process yields a generative SDE that transports samples from the prior back to the target distribution. A classical result (Nelson, 1967 ; Anderson, 1982 ) establishes that the drift of this time-reversed SDE is determined by the score function , ∇ log p ( 𝒙 , t ) \\nabla\\log p(\\bm{x},t) , where p ( ⋅ , t ) p(\\cdot,t) denotes the density of the forward process at time t t . Thus, the sampling problem reduces to accurately approximating this time-dependent score function. While diffusion models were originally developed for the generative-modeling setting, in which the score is learned from samples via denoising score matching, here we adopt only the time-reversal mechanism: the score must instead be obtained without samples, directly from the unnormalized density ρ target \\rho_{\\mathrm{target}} ."},{"type":"para","text":"Since the densities of SDEs are governed by the Fokker-Planck equation , one can employ the Hopf-Cole transform to relate the evolution of the log-density to a Hamilton-Jacobi-Bellman (HJB) equation (Berner et al. , 2024 ) . However, solving this PDE is notoriously challenging, particularly in the high-dimensional settings typical in practical applications, which are intractable for traditional mesh-based approaches. In the context of diffusion models, there exist attempts to approximate solutions to such HJB equations with neural networks, for example via physics-informed neural networks (PINNs) (Sun et al. , 2024 ; Shi et al. , 2024b ) . However, the optimization of neural networks with stochastic gradient descent (SGD) requires a large number of (potentially costly) evaluations of ρ target \\rho_{\\mathrm{target}} and relies on automatic differentiation to compute the higher-order derivatives appearing in the PDE. This leads to long training times, sensitivity to hyperparameters, and convergence to local minima (Krishnapriyan et al. , 2021 ; Shi et al. , 2024a ; Xu et al. , 2025 ) ."},{"type":"para","text":"In this work, we depart from neural network-based PDE solvers and propose a novel approach based on functional tensor train (FTT) representations. Tensor trains are particularly well-suited for this setting, as they can efficiently represent high-dimensional functions by exploiting latent low-rank structures, enabling both substantial model compression and fast computation. Moreover, the tensor train format is naturally compatible with backward-in-time regression schemes, which can be derived from the connection between HJB equations and backward stochastic differential equations (BSDEs). This yields an efficient implementation that circumvents the instabilities and complexities associated with SGD-based optimization. Our main contributions can be summarized as follows:"},{"type":"para","text":"We introduce Tensor Train Diffusion (TTD) 1 1 1 The code is available at https://github.com/robertgruhlke/TTD . , a novel solver for HJB equations that leverages connections between diffusion-based sampling, HJB equations, and BSDEs to achieve a favorable trade-off between computational cost and accuracy."},{"type":"para","text":"We demonstrate the flexibility of TTD by incorporating different basis functions – including Legendre polynomials, B-splines, and Fourier bases – and substantially improve robustness compared to previous tensor-train-based PDE solvers, effectively handling sampling trajectories outside the primary training domain."},{"type":"para","text":"We show that TTD can be used to sample from complex, multimodal target distributions of varying dimensionality, outperforming existing diffusion-based samplers in both speed and accuracy."},{"type":"heading","level":2,"text":"1.1 Related work"},{"type":"heading","level":4,"text":"Sampling from unnormalized densities."},{"type":"para","text":"Drawing samples from a density, specified up to the normalizing constant, is a crucial task in a wide range of computational sciences, ranging from molecular dynamics, to lattice field theory, to Bayesian statistics (Gelman et al. , 2013 ; Zhang et al. , 2023 ) . Due to the great interest, a large amount of literature focused on developing corresponding sampling methods, often based on variants of Markov chain Monte Carlo (MCMC) and Langevin dynamics (Neal, 2001 ; Chopin, 2002 ; Del Moral et al. , 2006 ) . However, such algorithms typically require substantial tuning and long runtimes to sample from high-dimensional distributions with well-separated modes (Latuszyński et al. , 2025 ; Brooks et al. , 2011 ) ."},{"type":"heading","level":4,"text":"Diffusion-based samplers."},{"type":"para","text":"To combat these issues, variational inference methods propose to turn the sampling problem into an optimization problem over a tractable family of distributions. Translating the success of diffusion models to sampling problems, recent works proposed to use parametrized SDEs for these families (Richter and Berner, 2024 ; Berner et al. , 2024 , 2026 ; Vargas et al. , 2024 ; Blessing et al. , 2025 , 2026a , 2026b ) . Since the drifts of the SDEs are typically parametrized by neural networks that incorporate the gradient of the target density, such approaches can be viewed as a learnable corrections to Langevin dynamics, aiming to converge after finite trajectory lengths. However, this also leads to a significant amount of (potentially costly) density evaluations during SDE simulations, which are typically also needed for training. While such methods can outperform classical methods in terms of inference performance and efficiency, they suffer from a high upfront cost for optimizing the parameters of the SDE."},{"type":"heading","level":4,"text":"Neural PDE solver."},{"type":"para","text":"Since the dynamics of SDEs are governed by Fokker-Planck equations, we can equivalently solve the corresponding PDE in order to optimize the variational family of SDEs. However, such high-dimensional PDEs are out of scope for traditional mesh-based methods, such as finite differences or finite elements, which suffer from the curse of dimensionality . There is a series of works proposing variants of physics-informed neural networks for this problem (Shi et al. , 2024b ; Albergo and Vanden-Eijnden, 2025 ; Sun et al. , 2024 ) . While they can scale to higher dimensions, they do not address the high training costs due to their reliance on evaluations of the target density and higher-order derivatives as well as high sensitivity to hyperparameters. Methods that are based on iterative diffusion optimization additionally suffer from large computational costs to repeatedly simulate diffusion trajectories (Nüsken and Richter, 2021 , 2023 ) ."},{"type":"heading","level":4,"text":"Tensor trains."},{"type":"para","text":"Many target densities of interest exhibit an inherent low-rank structure. To exploit this, we propose to approximate the drift using functional tensor trains (FTT, Oseledets ( 2013 ) ) in their numerical realization, the extended (functional) tensor train format (Eigel et al. , 2022 ; Strössner et al. , 2024 ) is sometimes referred to as spectral tensor trains (Bigoni et al. , 2016 ) . The effectiveness of such representations has already been demonstrated in related contexts, for example in sampling problems connected to HJB equations (Gruhlke et al. , 2026 ) , as well as for the Fokker-Planck and HJB equations in Dolgov et al. ( 2012 ) . Low-rank tensor formats enable efficient regression-based solvers with broad applicability; see Bachmayr ( 2023 ) for an overview. Moreover, they constitute a rigorous nonlinear approximation class, with detailed analyses provided in Kazeev and Schwab ( 2018 ); Ali and Nouy ( 2020a , b , 2021 ); Bachmayr et al. ( 2021 ); Griebel et al. ( 2023 ); Bachmayr ( 2023 ) . In this work, we build on the ideas developed in Richter et al. ( 2021 , 2024 ) and extend them to the sampling setting. Inspired by Bouchard and Touzi ( 2004 ); Gobet et al. ( 2005 ); Huré et al. ( 2020 ) , we notice that the Fokker-Planck equation is the linearization of a Hamilton-Jacobi-Bellman equation under the Hopf-Cole transform, such that the optimal drift can be written in terms of backward SDEs (BSDEs) that lead to simple regression problems. Together with the TT parametrization, we show that this reformulation of the PDE leads to an efficient and scalable sampler."},{"type":"heading","level":4,"text":"Notation."},{"type":"para","text":"We define time-inversion as f → ( t ) := f ( T − t ) \\mathchoice{\\mkern 3.0mu\\reflectbox{$\\displaystyle\\vec{\\reflectbox{$\\mkern-3.0mu\\displaystyle f\\mkern 3.0mu$}}$}\\mkern-3.0mu}{\\mkern 3.0mu\\reflectbox{$\\textstyle\\vec{\\reflectbox{$\\mkern-3.0mu\\textstyle f\\mkern 3.0mu$}}$}\\mkern-3.0mu}{\\mkern 2.0mu\\reflectbox{$\\scriptstyle\\vec{\\reflectbox{$\\mkern-2.0mu\\scriptstyle f\\mkern 2.0mu$}}$}\\mkern-2.0mu}{\\mkern 2.0mu\\reflectbox{$\\scriptscriptstyle\\vec{\\reflectbox{$\\mkern-2.0mu\\scriptscriptstyle f\\mkern 2.0mu$}}$}\\mkern-2.0mu}(t):=f(T-t) . For other notation, including our tensor notation, we refer to Table 1 in the appendix."},{"type":"heading","level":1,"text":"2 Diffusion-based sampling"},{"type":"para","text":"In this section, we demonstrate how learned stochastic processes can be employed to sample from the target distribution and introduce a stochastic algorithm that formalizes the associated learning task."},{"type":"heading","level":2,"text":"2.1 Time-reversal and the related Hamilton-Jacobi-Bellman PDE"},{"type":"para","text":"Our sampling algorithm for p target p_{\\mathrm{target}} builds on the idea of reversing a noising process that approximately evolves towards an easy-to-sample distribution p prior p_{\\mathrm{prior}} at terminal time T T (Song et al. , 2021 ; Berner et al. , 2024 ) . To this end, we consider the process"},{"type":"para","text":"Unlike classical diffusion models with access to samples from p target p_{\\mathrm{target}} , the noising process cannot be simulated. The process Y Y therefore serves only as an auxiliary analytical construct and is not part of the algorithm. The corresponding sampling dynamics, which can be simulated, is given by"},{"type":"para","text":"where the objective is to learn a function u u that reverses the dynamics of ( 2 ), ensuring that X T u ∗ ∼ p target X_{T}^{u^{*}}\\sim p_{\\mathrm{target}} . It is well known that the optimal solution is given by the (scaled) score function u ∗ = σ ⊤ ∇ log p → Y u^{*}=\\sigma^{\\top}\\nabla\\log\\mathchoice{\\mkern 3.0mu\\reflectbox{$\\displaystyle\\vec{\\reflectbox{$\\mkern-3.0mu\\displaystyle p\\mkern 3.0mu$}}$}\\mkern-3.0mu}{\\mkern 3.0mu\\reflectbox{$\\textstyle\\vec{\\reflectbox{$\\mkern-3.0mu\\textstyle p\\mkern 3.0mu$}}$}\\mkern-3.0mu}{\\mkern 2.0mu\\reflectbox{$\\scriptstyle\\vec{\\reflectbox{$\\mkern-2.0mu\\scriptstyle p\\mkern 2.0mu$}}$}\\mkern-2.0mu}{\\mkern 2.0mu\\reflectbox{$\\scriptscriptstyle\\vec{\\reflectbox{$\\mkern-2.0mu\\scriptscriptstyle p\\mkern 2.0mu$}}$}\\mkern-2.0mu}_{Y} (Nelson, 1967 ; Anderson, 1982 ) . Moreover, as shown in Berner et al. ( 2024 ) , this function can be equivalently characterized via the following Hamilton-Jacobi-Bellman (HJB) PDE."},{"type":"para","text":"Let V := − log p → Y V:=-\\log\\mathchoice{\\mkern 3.0mu\\reflectbox{$\\displaystyle\\vec{\\reflectbox{$\\mkern-3.0mu\\displaystyle p\\mkern 3.0mu$}}$}\\mkern-3.0mu}{\\mkern 3.0mu\\reflectbox{$\\textstyle\\vec{\\reflectbox{$\\mkern-3.0mu\\textstyle p\\mkern 3.0mu$}}$}\\mkern-3.0mu}{\\mkern 2.0mu\\reflectbox{$\\scriptstyle\\vec{\\reflectbox{$\\mkern-2.0mu\\scriptstyle p\\mkern 2.0mu$}}$}\\mkern-2.0mu}{\\mkern 2.0mu\\reflectbox{$\\scriptscriptstyle\\vec{\\reflectbox{$\\mkern-2.0mu\\scriptscriptstyle p\\mkern 2.0mu$}}$}\\mkern-2.0mu}_{Y} . Then, V ( ⋅ , T ) = − log p target V(\\cdot,T)=-\\log p_{\\mathrm{target}} and"},{"type":"para","text":"It is important to note that the boundary term may be replaced by its unnormalized version − log ρ target -\\log\\rho_{\\mathrm{target}} . This substitution merely shifts the PDE solution by an additive constant, which is irrelevant for the optimal control since only the gradient of the solution enters."},{"type":"para","text":"While equation ( 4 ) formally provides a recipe for approximating the log-density and, consequently, the score function, it is well known that solving PDEs in high dimensions is notoriously challenging. In the following, we therefore introduce a procedure that combines Monte Carlo estimation with suitable function approximation to efficiently handle high-dimensional settings."},{"type":"heading","level":2,"text":"2.2 BSDE representations and backward iterations"},{"type":"para","text":"One approach to approximate certain nonlinear high-dimensional PDEs relies on backward stochastic differential equations (Pardoux, 1998 ) . Their main idea relies on Itô’s formula, which, for a stochastic process X u X^{u} as defined in ( 3 ), states that"},{"type":"para","text":"Now, assuming that V V solves the HJB PDE ( 4 ), we may equivalently write BSDE ( V ) = 0 \\operatorname{BSDE}(V)=0 with"},{"type":"para","text":"where for convenience we defined the nonlinear term"},{"type":"para","text":"Conversely, when replacing V V by an approximation V ~ \\widetilde{V} , BSDE ( V ~ ) = 0 \\operatorname{BSDE}(\\widetilde{V})=0 can only hold if V = V ~ V=\\widetilde{V} almost surely, by uniqueness of the PDE. This motivates to consider loss functionals of the form"},{"type":"para","text":"where the expectation is over different realizations of the process X u X^{u} , cf. E et al. ( 2017 ) . In principle, V V can now be learned by minimizing the loss ( 7 ) w.r.t. V V . However, since solving the minimization problem in its entirety is difficult, it is natural to decompose it into a sequence of smaller subproblems. Following the dynamic programming principle from optimal control theory (Fleming and Rishel, 2012 ) , we thus partition the time horizon into disjoint intervals according to the grid 0 = t 0 < t 1 < ⋯ < t N = T 0=t_{0}<t_{1}<\\cdots<t_{N}=T , and aim to compute V V on each interval separately. Our envisioned algorithm proceeds backward in time, beginning with the final interval [ t N − 1 , t N ] [t_{N-1},t_{N}] and employing the terminal condition V ( ⋅ , T ) = − log p target V(\\cdot,T)=-\\log p_{\\mathrm{target}} . Next, for each preceding interval, the terminal condition is given by the solution obtained at the right endpoint of the subsequent interval. In practice, this means that V ( ⋅ , t n ) V(\\cdot,t_{n}) is replaced by its previously computed approximation V ~ ( ⋅ , t n ) \\widetilde{V}(\\cdot,t_{n}) for n = 1 , … , N − 1 n=1,\\dots,N-1 . For further details we refer to Algorithm 1 and Richter et al. ( 2021 , 2024 ) ."},{"type":"para","text":"In practice, we need to discretize both the process ( 3 ) and the (stochastic) integrals appearing in the loss ( 7 ). Using the same time grid as before, we approximate the solution only at the grid points, i.e., we seek functions V ^ n \\widehat{V}_{n} such that V ^ n ≈ V ( ⋅ , t n ) \\widehat{V}_{n}\\approx V(\\cdot,t_{n}) for n = 0 , … , N − 1 n=0,\\dots,N-1 . To this end, we employ the Euler-Maruyama scheme"},{"type":"para","text":"where Δ t = t n + 1 − t n \\Delta t=t_{n+1}-t_{n} is the time step and ξ n + 1 ∼ 𝒩 ( 0 , Id ) \\xi_{n+1}\\sim\\mathcal{N}(0,\\operatorname{Id}) are independent standard normal random variables."},{"type":"para","text":"We then define for each n = 0 , … , N − 1 n=0,\\dots,N-1 the discrete loss"},{"type":"para","text":"where h n + 1 u := h ( u , V ^ n + 1 ) ( X ^ n + 1 u , t n + 1 ) . h_{n+1}^{u}:=h(u,\\widehat{V}_{n+1})(\\widehat{X}^{u}_{n+1},t_{n+1}). The loss ( 9 ) is obtained by discretizing its continuous version ( 7 ) by choosing the right endpoint of the deterministic integral and the left endpoint of the stochastic integral. As a next step, the expectation in ( 9 ) is discretized by K ∈ ℕ K\\in\\mathbb{N} sample sequences ( X ^ n u , ( k ) ) n (\\widehat{X}_{n}^{u,(k)})_{n} for k = 1 , … , K k=1,\\ldots,K , generated from ( 8 ). Defining Σ n ( k ) = σ ( t n ) ξ n + 1 ( k ) Δ t ∈ ℝ d \\Sigma_{n}^{(k)}=\\sigma(t_{n})\\xi_{n+1}^{(k)}\\sqrt{\\Delta t}\\in\\mathbb{R}^{d} and y n + 1 ( k ) = V ^ n + 1 ( X ^ n + 1 u , ( k ) ) − h n + 1 u , ( k ) Δ t y_{n+1}^{(k)}=\\widehat{V}_{n+1}(\\widehat{X}^{u,(k)}_{n+1})-h_{n+1}^{u,(k)}\\Delta t , the loss in ( 9 ) can be approximated using the empirical loss"},{"type":"para","text":"Note the affine dependence in V ^ n \\widehat{V}_{n} , making it well-suited for regression-based methods. Moreover, up to discretization error, the loss has the convenient property that it is almost surely zero at the solution, additionally implying vanishing variance at the optimum. For further analysis we refer to Richter et al. ( 2024 ) ."},{"type":"para","text":"A key feature of the loss ( 9 ) is its explicit dependence on both the function V ^ n \\widehat{V}_{n} and its gradient ∇ V ^ n \\nabla\\widehat{V}_{n} . Because the loss penalizes errors in both terms, minimizing it provides a direct incentive to accurately approximate both the function and its derivative. This is particularly advantageous, as the optimal control relies directly on this gradient."},{"type":"para","text":"When decomposing the problem into subproblems, one must take into account the possible propagation of approximation errors. These arise because, in practice, the approximation V ^ n + 1 ( ⋅ ) ≈ V ( ⋅ , t n + 1 ) \\widehat{V}_{n+1}(\\cdot)\\approx V(\\cdot,t_{n+1}) , which serves as a “terminal condition” for computing the individual losses, is generally not exact (see Section 8.3.3 in Gobet ( 2016 ) )."},{"type":"para","text":"Unlike traditional numerical methods for PDEs, BSDE-based approaches can be interpreted as operating on dynamically adaptive random grids, provided by the grid points ( X ^ n u , ( k ) ) k , n (\\widehat{X}^{u,(k)}_{n})_{k,n} . This perspective highlights their potential to achieve dimension-free Monte Carlo convergence rates."},{"type":"para","text":"A strategy we adopt later is to first approximate the (unnormalized) log-target ρ target \\rho_{\\mathrm{target}} via a simple regression task. The resulting surrogate model can then be employed both for potential Langevin initialization runs and within the outer loop of Algorithm 1 . This approach has the advantage that only a potentially small number of target evaluations are required initially; see also Remark 3.2 ."},{"type":"para","text":"Remark 2.4 highlights an additional challenge in sampling applications: we only need to approximate V V , and hence the control u u , accurately along the trajectories of X u X^{u} . However, during training we learn X u X^{u} along one set of sampled trajectories, while the actual evaluation later takes place along potentially different trajectories. To address this mismatch, we propose an iterative learning scheme with an outer loop that begins from an informed initial guess u ( 0 ) u^{(0)} (typically corresponding to Langevin dynamics) and, at each step, updates the control using the solution u ( i ) u^{(i)} obtained in the i i -th iteration. This iterative refinement is expected to gradually align the training and evaluation distributions, so that the trajectories used for learning become sufficiently close to those relevant for sampling. In this way, the procedure concentrates on the regions of the state space that are most important for the optimal sampling process; see Algorithm 1 and Figure 1 for an illustration. We note that in our numerical experiments in Section 4 the algorithm typically converges in less than 3 iterations. Furthermore, we refer to Section A.4.4 for strategies to extend the learned control in a principled way beyond the data domain."},{"type":"heading","level":1,"text":"3 Tensor trains as approximating functions"},{"type":"para","text":"This section introduces low-rank tensor formats for approximating the functions V ^ n \\widehat{V}_{n} . We focus on functional tensor trains (FTTs, Oseledets ( 2013 ) ); for related work on low-rank function approximation, see Ali and Nouy ( 2020a , b , 2021 ); Bachmayr et al. ( 2021 ); Griebel et al. ( 2023 ); Bachmayr ( 2023 ) . The FTT format is a nonlinear approximation class that exploits separability in the coupling of input variables. Its numerical realization via tensor trains yields a Riemannian manifold structure, enabling efficient and stable algorithms, including optimization (Oseledets, 2011 ) . When such separability is present, FTTs provide accurate and efficiently evaluable representations."},{"type":"para","text":"Let D = × i = 1 d [ a i , b i ] ⊂ ℝ d D=\\mathop{\\mbox{\\Large$\\times$}}_{i=1}^{d}[a_{i},b_{i}]\\subset\\mathbb{R}^{d} with a i < b i a_{i}<b_{i} for i = 1 , … , d i=1,\\ldots,d . A function f : D → ℝ f\\colon D\\to\\mathbb{R} is said to have FTT rank 𝒓 ¯ = ( r ¯ 1 , … , r ¯ d − 1 ) ∈ ℕ d − 1 \\overline{\\bm{r}}=(\\overline{r}_{1},\\ldots,\\overline{r}_{d-1})\\in\\mathbb{N}^{d-1} , if it can be written as"},{"type":"para","text":"with matrix valued functions F i ( x i ) ∈ ℝ r ¯ i − 1 , r ¯ i F_{i}(x_{i})\\in\\mathbb{R}^{\\overline{r}_{i-1},\\overline{r}_{i}} for i = 1 , … , d i=1,\\ldots,d with the convention r ¯ 0 = r ¯ d = 1 \\overline{r}_{0}=\\overline{r}_{d}=1 . Note that in the case d = 2 d=2 , the FTT format ( 11 ) can be interpreted as a singular value decomposition (SVD) in function space, since f ( x 1 , x 2 ) = F 1 ( x 1 ) F 2 ( x 2 ) = ∑ k = 1 r ¯ 1 [ F 1 ( x 1 ) ] k [ F 2 ( x 2 ) ] k f(x_{1},x_{2})=F_{1}(x_{1})F_{2}(x_{2})=\\sum_{k=1}^{\\overline{r}_{1}}[F_{1}(x_{1})]_{k}[F_{2}(x_{2})]_{k} . Thus, the FTT format can be viewed as a generalization of the SVD to high-dimensional functions, while retaining important properties such as closedness (Holtz et al. , 2012a ) ."},{"type":"para","text":"In order to make this format accessible for approximation, a discretization in each direction x i x_{i} will be introduced. To this end, let ℋ = ⨂ i = 1 d ℋ i ( a i , b i ) \\mathcal{H}=\\bigotimes_{i=1}^{d}\\mathcal{H}_{i}(a_{i},b_{i}) be a product Hilbert space on D D , where each ℋ i ( a i , b i ) \\mathcal{H}_{i}(a_{i},b_{i}) is equipped with a scalar product ( ⋅ , ⋅ ) ℋ i (\\cdot,\\cdot)_{\\mathcal{H}_{i}} . For 𝒎 = ( m 1 , … , m d ) ∈ ℕ d \\bm{m}=(m_{1},\\ldots,m_{d})\\in\\mathbb{N}^{d} , using the notation in Table 1 in the appendix, define the discrete set of ℋ \\mathcal{H} -orthonormal tensor basis functions ℬ 𝒎 := { ϕ 𝜶 := ⨂ i = 1 d ϕ α i i ∣ 𝜶 ∈ [ 𝒎 ] } , \\mathcal{B}_{\\bm{m}}:=\\{\\phi_{\\bm{\\alpha}}:=\\bigotimes_{i=1}^{d}\\phi_{\\alpha_{i}}^{i}\\,\\mid\\,\\bm{\\alpha}\\in[\\bm{m}]\\}, with univariate ℋ i \\mathcal{H}_{i} -orthonormal basis functions ϕ α i i : [ a i , b i ] → ℝ \\phi_{\\alpha_{i}}^{i}\\colon[a_{i},b_{i}]\\to\\mathbb{R} . For f f with FTT rank 𝒓 ¯ \\overline{\\bm{r}} , we may then approximate"},{"type":"para","text":"for Φ ( 𝒙 ) = ( ϕ j 1 ( x 1 ) ) j = 1 m 1 ⊗ ⋯ ⊗ ( ϕ j d ( x d ) ) j = 1 m d ∈ ℝ 𝒎 \\Phi(\\bm{x})=(\\phi_{j}^{1}(x_{1}))_{j=1}^{m_{1}}\\otimes\\cdots\\otimes(\\phi_{j}^{d}(x_{d}))_{j=1}^{m_{d}}\\in\\mathbb{R}^{\\bm{m}} and a tensor array 𝑪 ∈ ℝ 𝒎 \\bm{C}\\in\\mathbb{R}^{\\bm{m}} with (algebraic) tensor train (TT) rank 𝒓 = ( r 1 , … , r d − 1 ) ⊤ ∈ ℕ d − 1 \\bm{r}=(r_{1},\\ldots,r_{d-1})^{\\top}\\in\\mathbb{N}^{d-1} bounded by the FTT rank 𝒓 ¯ \\overline{\\bm{r}} . In particular, we have the decomposition into a tensor train (or matrix product state) format as"},{"type":"para","text":"with order three tensors 𝑪 i ∈ ℝ r i − 1 , m i , r i \\bm{C}_{i}\\in\\mathbb{R}^{r_{i-1},m_{i},r_{i}} defining matrices 𝑪 i [ α i ] = 𝑪 i [ : , α i , : ] ∈ ℝ r i − 1 , r i \\bm{C}_{i}[\\alpha_{i}]=\\bm{C}_{i}[:,\\alpha_{i},:]\\in\\mathbb{R}^{r_{i-1},r_{i}} with the convention that r 0 = r d = 1 r_{0}=r_{d}=1 . The set of TTs of fixed rank 𝒓 \\bm{r} is denoted as ℳ 𝒓 𝒎 \\mathcal{M}_{\\bm{r}}^{\\bm{m}} . In particular, there exists a maximal possible TT rank 𝒓 ¯ ∈ ℕ d − 1 \\overline{\\bm{r}}\\in\\mathbb{N}^{d-1} such that ℳ 𝒓 ¯ 𝒎 = ℝ 𝒎 \\mathcal{M}_{\\overline{\\bm{r}}}^{\\bm{m}}=\\mathbb{R}^{\\bm{m}} ."},{"type":"para","text":"In what follows, we refer to a function in the form of the right hand side ( 12 ) as extended tensor train (xTT). The xTT format is a special case of the FTT format with a specific choice of univariate basis functions. The set of xTTs building on ℳ 𝒓 𝒎 \\mathcal{M}_{\\bm{r}}^{\\bm{m}} is denoted as"},{"type":"para","text":"Note that, by construction, we have 𝒳 𝒓 𝒎 ⊂ ℋ \\mathcal{X}_{\\bm{r}}^{\\bm{m}}\\subset\\mathcal{H} for any rank 𝒓 ≤ 𝒓 ¯ \\bm{r}\\leq\\overline{\\bm{r}} , and for any V ^ 𝑪 ∈ 𝒳 𝒓 𝒎 \\widehat{V}_{\\bm{C}}\\in\\mathcal{X}_{\\bm{r}}^{\\bm{m}} it holds that ‖ V ^ 𝑪 ‖ ℋ = ‖ 𝑪 ‖ F \\|\\widehat{V}_{\\bm{C}}\\|_{\\mathcal{H}}=\\|\\bm{C}\\|_{\\mathrm{F}} . Key properties of the considered nonlinear low-rank model class include storage complexity, multi-linearity and related efficient evaluation. Provided that the ranks can be bounded, the TT format exhibits a storage complexity based on the shape of each component tensor C i C_{i} with upper bound 𝒪 ( max ( m 1 , … , m d ) d max ( r 1 , … , r d − 1 ) 2 ) \\mathcal{O}(\\max(m_{1},\\ldots,m_{d})d\\max(r_{1},\\dots,r_{d-1})^{2}) , which scales only linearly in the dimension d d . Moreover, a function in xTT format can be evaluated fast and numerically stably, including possible access to gradients or Hessians, see Section A.4 for details. To this end, originated by ( 10 ), we are interested in solving a sequence of optimization tasks for n = 1 , … , N − 1 n=1,\\ldots,N-1 given by"},{"type":"para","text":"for some regularization magnitude τ n > 0 \\tau_{n}>0 . Note that the regularization term involves the ℋ \\mathcal{H} norm, see Appendix A.3.1 . Recall that we are interested in approximating the true value function V ( 𝐱 , 𝐭 ) V(\\bf{x},t) , which is expected to have varying ℋ \\mathcal{H} norm over time snapshots t = t 0 , … , t N t=t_{0},\\ldots,t_{N} . Consequently, to relate the regularized loss from ( 15 ) to the original loss from ( 10 ), the regularization magnitude needs to be chosen suitably and adaptively, which we explain in detail in Section A.6.1 . Further, it is crucial to choose the TT rank 𝒓 \\bm{r} and the numbers of degrees of freedom 𝒎 \\bm{m} appropriately, and we provide a corresponding adaptive strategy in Section A.6.2 and Section A.6.3 . Moreover, a detailed discussion of choices for tensor basis functions, along with their associated advantages and challenges, is given in Section A.3 . Finally, a discussion of the FTT rank analysis for certain classes of functions is provided in Section A.7 ."},{"type":"heading","level":2,"text":"3.1 Optimization via alternating least squares"},{"type":"para","text":"The set ℳ 𝒓 \\mathcal{M}_{\\bm{r}} defines a Riemannian manifold (Holtz et al. , 2012a ) , rendering the option for Riemannian optimization. Here, we consider an alternative approach for minimization based on alternating least squares (ALS), cf. Holtz et al. ( 2012b ) . In particular, instead of solving ( 15 ) via iterates of the whole TT 𝑪 \\bm{C} , we split the minimization into iterations of so-called sweeps consisting of several micro steps . In each micro step we update the i i -th component of the TT only, while fixing all other components. For this, we compactly write (see Section A.1 )"},{"type":"para","text":"with orthogonal order three tensors U j ∈ ℝ r j − 1 , m j , r j U_{j}\\in\\mathbb{R}^{r_{j-1},m_{j},r_{j}} for j = 1 , … , i − 1 , i + 1 , … d j=1,\\ldots,i-1,i+1,\\ldots d and a so-called core ℭ ∈ ℝ r i − 1 , m i , r i \\mathfrak{C}\\in\\mathbb{R}^{r_{i-1},m_{i},r_{i}} , see Section A.2 for more details. Then, at each micro step, the loss from ( 15 ) reduces to the problem"},{"type":"para","text":"with a matrix A i , n u ∈ ℝ K , r i − 1 m i r i A_{i,n}^{u}\\in\\mathbb{R}^{K,r_{i-1}m_{i}r_{i}} and y n + 1 = ( y n + 1 ( k ) ) k ∈ ℝ K y_{n+1}=(y_{n+1}^{(k)})_{k}\\in\\mathbb{R}^{K} , see Section A.5 for a derivation."},{"type":"para","text":"After solving the minimization task stated in ( 15 ), which yields a TT 𝑪 n ∈ ℳ 𝒓 𝒎 \\bm{C}^{n}\\in\\mathcal{M}_{\\bm{r}}^{\\bm{m}} , we then obtain an approximation of our value function V ( ⋅ , t n ) V(\\cdot,t_{n}) at time t = t n t=t_{n} , denoted by V ^ 𝑪 n ∈ 𝒳 𝒓 𝒎 \\widehat{V}_{\\bm{C}^{n}}\\in\\mathcal{X}_{\\bm{r}}^{\\bm{m}} , in xTT format, i.e."},{"type":"para","text":"The approximation of the value function V V is only defined locally on a domain D ⋐ ℝ d D\\Subset\\mathbb{R}^{d} . In practice, this domain is given by D = D n u D=D_{n}^{u} , which depends on both the time step and the current policy u u . If, during the sampling dynamics in ( 3 ), the required policy evaluation u ∗ = − σ ⊤ ∇ V u^{*}=-\\sigma^{\\top}\\nabla V involves points outside of D D , suitable extension strategies must be employed. We discuss these strategies in detail in Section A.4.4 ."},{"type":"para","text":"The minimization requires initial xTTs. At the final time t = t N t=t_{N} , we initialize via an L 2 L^{2} regression of the log-target density using samples from the initial sampling dynamics. Given an approximate xTT at t n + 1 t_{n+1} , the initialization at t n t_{n} is obtained by a generalized Galerkin-type projection of this xTT, accounting for the domain change; see Section A.4.5 ."},{"type":"para","text":"While the backward iteration in Algorithm 1 combined with FTTs offers clear advantages over iterative loss minimization with neural networks, it also poses stability challenges. Addressing these issues is a key contribution of this work; the corresponding stabilization strategies are detailed in the appendix. These include the choice of suitable basis functions ( Section A.3 ), handling moving domains along sample trajectories and evaluations outside these domains ( Sections A.4.4 and A.4.5 ), adaptive regularization ( Section A.6.1 ), adaptive rank selection ( Section A.6.2 ), and adaptive basis degree selection ( Section A.6.3 )."},{"type":"heading","level":1,"text":"4 Numerical experiments"},{"type":"para","text":"In this section we evaluate our sampling algorithm on challenging high-dimensional problems. We refer to an additional problem from computational statistics in Section B.5 ."},{"type":"heading","level":2,"text":"4.1 Highdimensional Multiwell problems"},{"type":"para","text":"We first consider a class of Multiwell problems, defined by"},{"type":"para","text":"which are commonly used benchmarks (Berner et al. , 2024 ; Richter and Berner, 2024 ; Wu et al. , 2020 ; Midgley et al. , 2022 ) . These densities are chosen because they resemble physics and molecular dynamics problems, providing an indication of performance in practical applications. At the same time, they offer a controlled setting for systematically studying the three key challenges in sampling: (1) the dimensionality d d , (2) the mode separation δ \\delta , and (3) the number of modes 2 m 2^{m} . In our experiments, we use a Fourier basis (see Section A.3.1 ) with 3 3 to 13 13 basis functions per dimension, treating this as a hyperparameter tuned according to the observed approximation quality. We employ an H mix 2 H_{\\mathrm{mix}}^{2} orthonormalization as described in Section A.3.2 . For the noising process ( 2 ), we set f ( 𝒙 , t ) = 𝒙 f(\\bm{x},t)=\\bm{x} , σ ( t ) = 2 \\sigma(t)=\\sqrt{2} , and consider a time horizon T = 2 T=2 . We apply Algorithm 1 to two Multiwell problems, with dimensions d = 10 d=10 and d = 50 d=50 and 8 8 and 32 32 modes, respectively, with δ = 2 \\delta=2 ."},{"type":"para","text":"As shown in Figure 3 , increasing the number of time steps N N (which also determines the number of functions V ^ n \\widehat{V}_{n} approximated) steadily improves sampling performance, ultimately achieving remarkable quality. Importantly, the algorithm’s runtime is significantly faster than that of alternative, predominantly neural network-based samplers. In d = 10 d=10 , training ranges roughly from 1 1 minute ( N = 2 8 N=2^{8} ) to 20 20 minutes ( N = 2 13 N=2^{13} ), while in d = 50 d=50 it ranges from 10 10 to 300 300 minutes. Note that the runtime grows linearly with N N . We refer to Figure 4 for a comparison with the time-reversed diffusion sampler (DIS) from (Berner et al. , 2024 ) , which considers the same noising SDE, however relies on neural network approximations and variational training. We refer to Figure 14 in Section B.4 for a comparison to further sampling methods, showing significantly improved performance. Finally, we refer to Figure 2 comparing marginals of the 50 50 dimensional problem with a reference solution."},{"type":"para","text":"We note that the Girsanov theorem relates the error in the log-normalizing constant log 𝒵 \\log\\mathcal{Z} to the L 2 L^{2} -approximation of the ground-truth score ∇ log p \\nabla\\log p , i.e., the gradient of the solution to the HJB equation w.r.t. to the measure induced by p target p_{\\mathrm{target}} (Berner et al. , 2024 ) . Under suitable assumptions, one can further leverage Poincaré inequalities to relate this error to the L 2 L^{2} -approximation of the PDE solution itself, measuring the accuracy of our proposed PDE solver."},{"type":"heading","level":2,"text":"4.2 Investigation of optimal ranks"},{"type":"para","text":"In this section, we study highly anisotropic Gaussian targets to analyze the rank structure of the induced transport problems and to assess the adaptivity of our algorithm. We consider target densities with potential log ρ target ( 𝒙 ) = − 𝒙 ⊤ M 𝒙 , \\log\\rho_{\\mathrm{target}}(\\bm{x})=-\\bm{x}^{\\top}M\\bm{x}, where M M is a random full-rank matrix. For this class, the maximal admissible rank is 𝐫 = ( 3 , 4 , … , 2 + d / 2 , … , 4 , 3 ) ∈ ℕ d − 1 \\mathbf{r}=(3,4,\\dots,2+d/2,\\dots,4,3)\\in\\mathbb{N}^{d-1} for even d d ; see Corollary A.6 for the general case. As the target distribution converges toward the standard normal, p Y T ≈ p prior = 𝒩 ( 0 , Id ) p_{Y_{T}}\\approx p_{\\mathrm{prior}}=\\mathcal{N}(0,\\operatorname{Id}) , the ranks are expected to approach the constant vector 𝐫 = ( 2 , … , 2 ) \\mathbf{r}=(2,\\dots,2) . This behavior is confirmed in Figure 5 , which shows the evolution of singular values and the corresponding rank adaptivity, resulting in a reduced runtime. The clear spectral gaps demonstrate that the algorithm reliably identifies and adapts to the appropriate ranks over time; see also Figure 11 in the appendix."},{"type":"heading","level":2,"text":"4.3 ϕ 4 \\phi^{4} scalar field theory"},{"type":"para","text":"Next, we evaluate our method on the discretized Ginzburg-Landau model (specifically, the ϕ 4 \\phi^{4} scalar field theory), a canonical framework in statistical mechanics used to describe continuous phase transitions and spontaneous symmetry breaking (Ginzburg and Landau, 1950 ; Rosenstein and Li, 2021 ) . The target density is defined by a Gibbs distribution over a lattice, characterized by a highly non-convex energy landscape resulting from the competing forces of local bistability and nearest-neighbor spatial coupling,"},{"type":"para","text":"This system is highly challenging for sampling algorithms, as the target distribution exhibits exponentially many metastable modes separated by high-energy barriers, leading to severe mode collapse in standard MCMC methods. Nearest-neighbor interactions further induce strong dependencies between lattice sites, resulting in a highly constrained and correlated high-dimensional geometry. For δ = 2 \\delta=2 , Figure 6 shows that the TTD method generates high-quality samples in two distinct multimodal regimes."},{"type":"heading","level":1,"text":"5 Conclusion & Discussion"},{"type":"para","text":"In this work, we introduced Tensor Train Diffusion (TTD), an efficient PDE solver for high-dimensional HJB equations leading to a novel way of sampling from unnormalized densities. Our method solves the HJB equation underlying the diffusion process using a functional tensor train (FTT) representation. By integrating this with a backward-in-time iterative scheme derived from BSDEs, TTD provides a fast, accurate, and stable alternative to neural network techniques. In particular, TTD requires less target evaluations and does not rely on SGD-based optimization with long training times and hyperparameter sensitivity."},{"type":"para","text":"In general, our TTD framework presents a solid foundation for the robust and efficient solution of HJB equations, that can readily be used for different sampling problems – we refer to potential limitations in Section B.1 . While we provide a significantly accelerated tensor train implementation in PyTorch, we believe that further hardware-specific optimization is possible. On the theoretical side, our work motivates further research to analyze latent low-rank structures of different target densities for educated choices of basis functions and variable orderings. Future work could also combine the framework with neural network models or classical sampling methods, as well as extend TTD to broader classes of PDEs and applications. From a theoretical perspective, an open question concerns the behavior of FTT ranks for solutions of the underlying HJB equation, which is only partially understood (Gruhlke et al. , 2026 ) . In the general case, the current state-of-the-art theory (Ali and Nouy, 2020a , b , 2021 ; Griebel et al. , 2023 ; Bachmayr, 2023 ) does not directly apply and would need to be extended to weighted regularity classes, since solutions of HJB equations are defined on ℝ d \\mathbb{R}^{d} and are generally unbounded as ‖ 𝒙 ‖ → ∞ \\|\\bm{x}\\|\\to\\infty . Consequently, on the computational side, one must rely on rank-adaptive approaches, i.e., determining the xTT ranks for the approximation in an adaptive manner."},{"type":"heading","level":1,"text":"Impact Statement"},{"type":"para","text":"This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sampling from a complex, high-dimensional probability density\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the unnormalized density ρ target \\\\rho_{\\\\mathrm{target}} can be evaluated pointwise but the normalizing constant 𝒵 \\\\mathcal{Z} is intractable and no samples from p target p_{\\\\mathrm{target}} are available, constitutes a central challenge in modern machine learning, statistics, and the physical sciences. An especially versatile and powerful class of methods for this task is based on dynamical measure transport, which generates samples from the target distribution by evolving initial samples from a simple reference distribution – such as a standard Gaussian – along trajectories defined by stochastic or ordinary differential equations. One of the most successful paradigms within this framework is the time reversal of a noising SDE, forming the foundation of diffusion models (Ho et al. , 2020 ; Song et al. , 2021 ) . In this approach, a forward SDE – typically chosen as an Ornstein-Uhlenbeck process – progressively transforms the target distribution into a simple prior by injecting noise over time. The central insight is that reversing this noising process yields a generative SDE that transports samples from the prior back to the target distribution. A classical result (Nelson, 1967 ; Anderson, 1982 ) establishes that the drift of this time-reversed SDE is determined by the score function , ∇ log p ( 𝒙 , t ) \\\\nabla\\\\log p(\\\\bm{x},t) , where p ( ⋅ , t ) p(\\\\cdot,t) denotes the density of the forward process at time t t . Thus, the sampling problem reduces to accurately approximating this time-dependent score function. While diffusion models were originally developed for the generative-modeling setting, in which the score is learned from samples via denoising score matching, here we adopt only the time-reversal mechanism: the score must instead be obtained without samples, directly from the unnormalized density ρ target \\\\rho_{\\\\mathrm{target}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since the densities of SDEs are governed by the Fokker-Planck equation , one can employ the Hopf-Cole transform to relate the evolution of the log-density to a Hamilton-Jacobi-Bellman (HJB) equation (Berner et al. , 2024 ) . However, solving this PDE is notoriously challenging, particularly in the high-dimensional settings typical in practical applications, which are intractable for traditional mesh-based approaches. In the context of diffusion models, there exist attempts to approximate solutions to such HJB equations with neural networks, for example via physics-informed neural networks (PINNs) (Sun et al. , 2024 ; Shi et al. , 2024b ) . However, the optimization of neural networks with stochastic gradient descent (SGD) requires a large number of (potentially costly) evaluations of ρ target \\\\rho_{\\\\mathrm{target}} and relies on automatic differentiation to compute the higher-order derivatives appearing in the PDE. This leads to long training times, sensitivity to hyperparameters, and convergence to local minima (Krishnapriyan et al. , 2021 ; Shi et al. , 2024a ; Xu et al. , 2025 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we depart from neural network-based PDE solvers and propose a novel approach based on functional tensor train (FTT) representations. Tensor trains are particularly well-suited for this setting, as they can efficiently represent high-dimensional functions by exploiting latent low-rank structures, enabling both substantial model compression and fast computation. Moreover, the tensor train format is naturally compatible with backward-in-time regression schemes, which can be derived from the connection between HJB equations and backward stochastic differential equations (BSDEs). This yields an efficient implementation that circumvents the instabilities and complexities associated with SGD-based optimization. Our main contributions can be summarized as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce Tensor Train Diffusion (TTD) 1 1 1 The code is available at https://github.com/robertgruhlke/TTD . , a novel solver for HJB equations that leverages connections between diffusion-based sampling, HJB equations, and BSDEs to achieve a favorable trade-off between computational cost and accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate the flexibility of TTD by incorporating different basis functions – including Legendre polynomials, B-splines, and Fourier bases – and substantially improve robustness compared to previous tensor-train-based PDE solvers, effectively handling sampling trajectories outside the primary training domain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We show that TTD can be used to sample from complex, multimodal target distributions of varying dimensionality, outperforming existing diffusion-based samplers in both speed and accuracy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Related work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Sampling from unnormalized densities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Drawing samples from a density, specified up to the normalizing constant, is a crucial task in a wide range of computational sciences, ranging from molecular dynamics, to lattice field theory, to Bayesian statistics (Gelman et al. , 2013 ; Zhang et al. , 2023 ) . Due to the great interest, a large amount of literature focused on developing corresponding sampling methods, often based on variants of Markov chain Monte Carlo (MCMC) and Langevin dynamics (Neal, 2001 ; Chopin, 2002 ; Del Moral et al. , 2006 ) . However, such algorithms typically require substantial tuning and long runtimes to sample from high-dimensional distributions with well-separated modes (Latuszyński et al. , 2025 ; Brooks et al. , 2011 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Diffusion-based samplers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To combat these issues, variational inference methods propose to turn the sampling problem into an optimization problem over a tractable family of distributions. Translating the success of diffusion models to sampling problems, recent works proposed to use parametrized SDEs for these families (Richter and Berner, 2024 ; Berner et al. , 2024 , 2026 ; Vargas et al. , 2024 ; Blessing et al. , 2025 , 2026a , 2026b ) . Since the drifts of the SDEs are typically parametrized by neural networks that incorporate the gradient of the target density, such approaches can be viewed as a learnable corrections to Langevin dynamics, aiming to converge after finite trajectory lengths. However, this also leads to a significant amount of (potentially costly) density evaluations during SDE simulations, which are typically also needed for training. While such methods can outperform classical methods in terms of inference performance and efficiency, they suffer from a high upfront cost for optimizing the parameters of the SDE.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Neural PDE solver.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since the dynamics of SDEs are governed by Fokker-Planck equations, we can equivalently solve the corresponding PDE in order to optimize the variational family of SDEs. However, such high-dimensional PDEs are out of scope for traditional mesh-based methods, such as finite differences or finite elements, which suffer from the curse of dimensionality . There is a series of works proposing variants of physics-informed neural networks for this problem (Shi et al. , 2024b ; Albergo and Vanden-Eijnden, 2025 ; Sun et al. , 2024 ) . While they can scale to higher dimensions, they do not address the high training costs due to their reliance on evaluations of the target density and higher-order derivatives as well as high sensitivity to hyperparameters. Methods that are based on iterative diffusion optimization additionally suffer from large computational costs to repeatedly simulate diffusion trajectories (Nüsken and Richter, 2021 , 2023 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Tensor trains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Many target densities of interest exhibit an inherent low-rank structure. To exploit this, we propose to approximate the drift using functional tensor trains (FTT, Oseledets ( 2013 ) ) in their numerical realization, the extended (functional) tensor train format (Eigel et al. , 2022 ; Strössner et al. , 2024 ) is sometimes referred to as spectral tensor trains (Bigoni et al. , 2016 ) . The effectiveness of such representations has already been demonstrated in related contexts, for example in sampling problems connected to HJB equations (Gruhlke et al. , 2026 ) , as well as for the Fokker-Planck and HJB equations in Dolgov et al. ( 2012 ) . Low-rank tensor formats enable efficient regression-based solvers with broad applicability; see Bachmayr ( 2023 ) for an overview. Moreover, they constitute a rigorous nonlinear approximation class, with detailed analyses provided in Kazeev and Schwab ( 2018 ); Ali and Nouy ( 2020a , b , 2021 ); Bachmayr et al. ( 2021 ); Griebel et al. ( 2023 ); Bachmayr ( 2023 ) . In this work, we build on the ideas developed in Richter et al. ( 2021 , 2024 ) and extend them to the sampling setting. Inspired by Bouchard and Touzi ( 2004 ); Gobet et al. ( 2005 ); Huré et al. ( 2020 ) , we notice that the Fokker-Planck equation is the linearization of a Hamilton-Jacobi-Bellman equation under the Hopf-Cole transform, such that the optimal drift can be written in terms of backward SDEs (BSDEs) that lead to simple regression problems. Together with the TT parametrization, we show that this reformulation of the PDE leads to an efficient and scalable sampler.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Notation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We define time-inversion as f → ( t ) := f ( T − t ) \\\\mathchoice{\\\\mkern 3.0mu\\\\reflectbox{$\\\\displaystyle\\\\vec{\\\\reflectbox{$\\\\mkern-3.0mu\\\\displaystyle f\\\\mkern 3.0mu$}}$}\\\\mkern-3.0mu}{\\\\mkern 3.0mu\\\\reflectbox{$\\\\textstyle\\\\vec{\\\\reflectbox{$\\\\mkern-3.0mu\\\\textstyle f\\\\mkern 3.0mu$}}$}\\\\mkern-3.0mu}{\\\\mkern 2.0mu\\\\reflectbox{$\\\\scriptstyle\\\\vec{\\\\reflectbox{$\\\\mkern-2.0mu\\\\scriptstyle f\\\\mkern 2.0mu$}}$}\\\\mkern-2.0mu}{\\\\mkern 2.0mu\\\\reflectbox{$\\\\scriptscriptstyle\\\\vec{\\\\reflectbox{$\\\\mkern-2.0mu\\\\scriptscriptstyle f\\\\mkern 2.0mu$}}$}\\\\mkern-2.0mu}(t):=f(T-t) . For other notation, including our tensor notation, we refer to Table 1 in the appendix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Diffusion-based sampling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we demonstrate how learned stochastic processes can be employed to sample from the target distribution and introduce a stochastic algorithm that formalizes the associated learning task.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Time-reversal and the related Hamilton-Jacobi-Bellman PDE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our sampling algorithm for p target p_{\\\\mathrm{target}} builds on the idea of reversing a noising process that approximately evolves towards an easy-to-sample distribution p prior p_{\\\\mathrm{prior}} at terminal time T T (Song et al. , 2021 ; Berner et al. , 2024 ) . To this end, we consider the process\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike classical diffusion models with access to samples from p target p_{\\\\mathrm{target}} , the noising process cannot be simulated. The process Y Y therefore serves only as an auxiliary analytical construct and is not part of the algorithm. The corresponding sampling dynamics, which can be simulated, is given by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the objective is to learn a function u u that reverses the dynamics of ( 2 ), ensuring that X T u ∗ ∼ p target X_{T}^{u^{*}}\\\\sim p_{\\\\mathrm{target}} . It is well known that the optimal solution is given by the (scaled) score function u ∗ = σ ⊤ ∇ log p → Y u^{*}=\\\\sigma^{\\\\top}\\\\nabla\\\\log\\\\mathchoice{\\\\mkern 3.0mu\\\\reflectbox{$\\\\displaystyle\\\\vec{\\\\reflectbox{$\\\\mkern-3.0mu\\\\displaystyle p\\\\mkern 3.0mu$}}$}\\\\mkern-3.0mu}{\\\\mkern 3.0mu\\\\reflectbox{$\\\\textstyle\\\\vec{\\\\reflectbox{$\\\\mkern-3.0mu\\\\textstyle p\\\\mkern 3.0mu$}}$}\\\\mkern-3.0mu}{\\\\mkern 2.0mu\\\\reflectbox{$\\\\scriptstyle\\\\vec{\\\\reflectbox{$\\\\mkern-2.0mu\\\\scriptstyle p\\\\mkern 2.0mu$}}$}\\\\mkern-2.0mu}{\\\\mkern 2.0mu\\\\reflectbox{$\\\\scriptscriptstyle\\\\vec{\\\\reflectbox{$\\\\mkern-2.0mu\\\\scriptscriptstyle p\\\\mkern 2.0mu$}}$}\\\\mkern-2.0mu}_{Y} (Nelson, 1967 ; Anderson, 1982 ) . Moreover, as shown in Berner et al. ( 2024 ) , this function can be equivalently characterized via the following Hamilton-Jacobi-Bellman (HJB) PDE.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let V := − log p → Y V:=-\\\\log\\\\mathchoice{\\\\mkern 3.0mu\\\\reflectbox{$\\\\displaystyle\\\\vec{\\\\reflectbox{$\\\\mkern-3.0mu\\\\displaystyle p\\\\mkern 3.0mu$}}$}\\\\mkern-3.0mu}{\\\\mkern 3.0mu\\\\reflectbox{$\\\\textstyle\\\\vec{\\\\reflectbox{$\\\\mkern-3.0mu\\\\textstyle p\\\\mkern 3.0mu$}}$}\\\\mkern-3.0mu}{\\\\mkern 2.0mu\\\\reflectbox{$\\\\scriptstyle\\\\vec{\\\\reflectbox{$\\\\mkern-2.0mu\\\\scriptstyle p\\\\mkern 2.0mu$}}$}\\\\mkern-2.0mu}{\\\\mkern 2.0mu\\\\reflectbox{$\\\\scriptscriptstyle\\\\vec{\\\\reflectbox{$\\\\mkern-2.0mu\\\\scriptscriptstyle p\\\\mkern 2.0mu$}}$}\\\\mkern-2.0mu}_{Y} . Then, V ( ⋅ , T ) = − log p target V(\\\\cdot,T)=-\\\\log p_{\\\\mathrm{target}} and\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is important to note that the boundary term may be replaced by its unnormalized version − log ρ target -\\\\log\\\\rho_{\\\\mathrm{target}} . This substitution merely shifts the PDE solution by an additive constant, which is irrelevant for the optimal control since only the gradient of the solution enters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While equation ( 4 ) formally provides a recipe for approximating the log-density and, consequently, the score function, it is well known that solving PDEs in high dimensions is notoriously challenging. In the following, we therefore introduce a procedure that combines Monte Carlo estimation with suitable function approximation to efficiently handle high-dimensional settings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 BSDE representations and backward iterations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One approach to approximate certain nonlinear high-dimensional PDEs relies on backward stochastic differential equations (Pardoux, 1998 ) . Their main idea relies on Itô’s formula, which, for a stochastic process X u X^{u} as defined in ( 3 ), states that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, assuming that V V solves the HJB PDE ( 4 ), we may equivalently write BSDE ( V ) = 0 \\\\operatorname{BSDE}(V)=0 with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where for convenience we defined the nonlinear term\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conversely, when replacing V V by an approximation V ~ \\\\widetilde{V} , BSDE ( V ~ ) = 0 \\\\operatorname{BSDE}(\\\\widetilde{V})=0 can only hold if V = V ~ V=\\\\widetilde{V} almost surely, by uniqueness of the PDE. This motivates to consider loss functionals of the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the expectation is over different realizations of the process X u X^{u} , cf. E et al. ( 2017 ) . In principle, V V can now be learned by minimizing the loss ( 7 ) w.r.t. V V . However, since solving the minimization problem in its entirety is difficult, it is natural to decompose it into a sequence of smaller subproblems. Following the dynamic programming principle from optimal control theory (Fleming and Rishel, 2012 ) , we thus partition the time horizon into disjoint intervals according to the grid 0 = t 0 < t 1 < ⋯ < t N = T 0=t_{0}<t_{1}<\\\\cdots<t_{N}=T , and aim to compute V V on each interval separately. Our envisioned algorithm proceeds backward in time, beginning with the final interval [ t N − 1 , t N ] [t_{N-1},t_{N}] and employing the terminal condition V ( ⋅ , T ) = − log p target V(\\\\cdot,T)=-\\\\log p_{\\\\mathrm{target}} . Next, for each preceding interval, the terminal condition is given by the solution obtained at the right endpoint of the subsequent interval. In practice, this means that V ( ⋅ , t n ) V(\\\\cdot,t_{n}) is replaced by its previously computed approximation V ~ ( ⋅ , t n ) \\\\widetilde{V}(\\\\cdot,t_{n}) for n = 1 , … , N − 1 n=1,\\\\dots,N-1 . For further details we refer to Algorithm 1 and Richter et al. ( 2021 , 2024 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice, we need to discretize both the process ( 3 ) and the (stochastic) integrals appearing in the loss ( 7 ). Using the same time grid as before, we approximate the solution only at the grid points, i.e., we seek functions V ^ n \\\\widehat{V}_{n} such that V ^ n ≈ V ( ⋅ , t n ) \\\\widehat{V}_{n}\\\\approx V(\\\\cdot,t_{n}) for n = 0 , … , N − 1 n=0,\\\\dots,N-1 . To this end, we employ the Euler-Maruyama scheme\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where Δ t = t n + 1 − t n \\\\Delta t=t_{n+1}-t_{n} is the time step and ξ n + 1 ∼ 𝒩 ( 0 , Id ) \\\\xi_{n+1}\\\\sim\\\\mathcal{N}(0,\\\\operatorname{Id}) are independent standard normal random variables.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We then define for each n = 0 , … , N − 1 n=0,\\\\dots,N-1 the discrete loss\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where h n + 1 u := h ( u , V ^ n + 1 ) ( X ^ n + 1 u , t n + 1 ) . h_{n+1}^{u}:=h(u,\\\\widehat{V}_{n+1})(\\\\widehat{X}^{u}_{n+1},t_{n+1}). The loss ( 9 ) is obtained by discretizing its continuous version ( 7 ) by choosing the right endpoint of the deterministic integral and the left endpoint of the stochastic integral. As a next step, the expectation in ( 9 ) is discretized by K ∈ ℕ K\\\\in\\\\mathbb{N} sample sequences ( X ^ n u , ( k ) ) n (\\\\widehat{X}_{n}^{u,(k)})_{n} for k = 1 , … , K k=1,\\\\ldots,K , generated from ( 8 ). Defining Σ n ( k ) = σ ( t n ) ξ n + 1 ( k ) Δ t ∈ ℝ d \\\\Sigma_{n}^{(k)}=\\\\sigma(t_{n})\\\\xi_{n+1}^{(k)}\\\\sqrt{\\\\Delta t}\\\\in\\\\mathbb{R}^{d} and y n + 1 ( k ) = V ^ n + 1 ( X ^ n + 1 u , ( k ) ) − h n + 1 u , ( k ) Δ t y_{n+1}^{(k)}=\\\\widehat{V}_{n+1}(\\\\widehat{X}^{u,(k)}_{n+1})-h_{n+1}^{u,(k)}\\\\Delta t , the loss in ( 9 ) can be approximated using the empirical loss\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note the affine dependence in V ^ n \\\\widehat{V}_{n} , making it well-suited for regression-based methods. Moreover, up to discretization error, the loss has the convenient property that it is almost surely zero at the solution, additionally implying vanishing variance at the optimum. For further analysis we refer to Richter et al. ( 2024 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key feature of the loss ( 9 ) is its explicit dependence on both the function V ^ n \\\\widehat{V}_{n} and its gradient ∇ V ^ n \\\\nabla\\\\widehat{V}_{n} . Because the loss penalizes errors in both terms, minimizing it provides a direct incentive to accurately approximate both the function and its derivative. This is particularly advantageous, as the optimal control relies directly on this gradient.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When decomposing the problem into subproblems, one must take into account the possible propagation of approximation errors. These arise because, in practice, the approximation V ^ n + 1 ( ⋅ ) ≈ V ( ⋅ , t n + 1 ) \\\\widehat{V}_{n+1}(\\\\cdot)\\\\approx V(\\\\cdot,t_{n+1}) , which serves as a “terminal condition” for computing the individual losses, is generally not exact (see Section 8.3.3 in Gobet ( 2016 ) ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike traditional numerical methods for PDEs, BSDE-based approaches can be interpreted as operating on dynamically adaptive random grids, provided by the grid points ( X ^ n u , ( k ) ) k , n (\\\\widehat{X}^{u,(k)}_{n})_{k,n} . This perspective highlights their potential to achieve dimension-free Monte Carlo convergence rates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A strategy we adopt later is to first approximate the (unnormalized) log-target ρ target \\\\rho_{\\\\mathrm{target}} via a simple regression task. The resulting surrogate model can then be employed both for potential Langevin initialization runs and within the outer loop of Algorithm 1 . This approach has the advantage that only a potentially small number of target evaluations are required initially; see also Remark 3.2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Remark 2.4 highlights an additional challenge in sampling applications: we only need to approximate V V , and hence the control u u , accurately along the trajectories of X u X^{u} . However, during training we learn X u X^{u} along one set of sampled trajectories, while the actual evaluation later takes place along potentially different trajectories. To address this mismatch, we propose an iterative learning scheme with an outer loop that begins from an informed initial guess u ( 0 ) u^{(0)} (typically corresponding to Langevin dynamics) and, at each step, updates the control using the solution u ( i ) u^{(i)} obtained in the i i -th iteration. This iterative refinement is expected to gradually align the training and evaluation distributions, so that the trajectories used for learning become sufficiently close to those relevant for sampling. In this way, the procedure concentrates on the regions of the state space that are most important for the optimal sampling process; see Algorithm 1 and Figure 1 for an illustration. We note that in our numerical experiments in Section 4 the algorithm typically converges in less than 3 iterations. Furthermore, we refer to Section A.4.4 for strategies to extend the learned control in a principled way beyond the data domain.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Tensor trains as approximating functions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section introduces low-rank tensor formats for approximating the functions V ^ n \\\\widehat{V}_{n} . We focus on functional tensor trains (FTTs, Oseledets ( 2013 ) ); for related work on low-rank function approximation, see Ali and Nouy ( 2020a , b , 2021 ); Bachmayr et al. ( 2021 ); Griebel et al. ( 2023 ); Bachmayr ( 2023 ) . The FTT format is a nonlinear approximation class that exploits separability in the coupling of input variables. Its numerical realization via tensor trains yields a Riemannian manifold structure, enabling efficient and stable algorithms, including optimization (Oseledets, 2011 ) . When such separability is present, FTTs provide accurate and efficiently evaluable representations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let D = × i = 1 d [ a i , b i ] ⊂ ℝ d D=\\\\mathop{\\\\mbox{\\\\Large$\\\\times$}}_{i=1}^{d}[a_{i},b_{i}]\\\\subset\\\\mathbb{R}^{d} with a i < b i a_{i}<b_{i} for i = 1 , … , d i=1,\\\\ldots,d . A function f : D → ℝ f\\\\colon D\\\\to\\\\mathbb{R} is said to have FTT rank 𝒓 ¯ = ( r ¯ 1 , … , r ¯ d − 1 ) ∈ ℕ d − 1 \\\\overline{\\\\bm{r}}=(\\\\overline{r}_{1},\\\\ldots,\\\\overline{r}_{d-1})\\\\in\\\\mathbb{N}^{d-1} , if it can be written as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with matrix valued functions F i ( x i ) ∈ ℝ r ¯ i − 1 , r ¯ i F_{i}(x_{i})\\\\in\\\\mathbb{R}^{\\\\overline{r}_{i-1},\\\\overline{r}_{i}} for i = 1 , … , d i=1,\\\\ldots,d with the convention r ¯ 0 = r ¯ d = 1 \\\\overline{r}_{0}=\\\\overline{r}_{d}=1 . Note that in the case d = 2 d=2 , the FTT format ( 11 ) can be interpreted as a singular value decomposition (SVD) in function space, since f ( x 1 , x 2 ) = F 1 ( x 1 ) F 2 ( x 2 ) = ∑ k = 1 r ¯ 1 [ F 1 ( x 1 ) ] k [ F 2 ( x 2 ) ] k f(x_{1},x_{2})=F_{1}(x_{1})F_{2}(x_{2})=\\\\sum_{k=1}^{\\\\overline{r}_{1}}[F_{1}(x_{1})]_{k}[F_{2}(x_{2})]_{k} . Thus, the FTT format can be viewed as a generalization of the SVD to high-dimensional functions, while retaining important properties such as closedness (Holtz et al. , 2012a ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In order to make this format accessible for approximation, a discretization in each direction x i x_{i} will be introduced. To this end, let ℋ = ⨂ i = 1 d ℋ i ( a i , b i ) \\\\mathcal{H}=\\\\bigotimes_{i=1}^{d}\\\\mathcal{H}_{i}(a_{i},b_{i}) be a product Hilbert space on D D , where each ℋ i ( a i , b i ) \\\\mathcal{H}_{i}(a_{i},b_{i}) is equipped with a scalar product ( ⋅ , ⋅ ) ℋ i (\\\\cdot,\\\\cdot)_{\\\\mathcal{H}_{i}} . For 𝒎 = ( m 1 , … , m d ) ∈ ℕ d \\\\bm{m}=(m_{1},\\\\ldots,m_{d})\\\\in\\\\mathbb{N}^{d} , using the notation in Table 1 in the appendix, define the discrete set of ℋ \\\\mathcal{H} -orthonormal tensor basis functions ℬ 𝒎 := { ϕ 𝜶 := ⨂ i = 1 d ϕ α i i ∣ 𝜶 ∈ [ 𝒎 ] } , \\\\mathcal{B}_{\\\\bm{m}}:=\\\\{\\\\phi_{\\\\bm{\\\\alpha}}:=\\\\bigotimes_{i=1}^{d}\\\\phi_{\\\\alpha_{i}}^{i}\\\\,\\\\mid\\\\,\\\\bm{\\\\alpha}\\\\in[\\\\bm{m}]\\\\}, with univariate ℋ i \\\\mathcal{H}_{i} -orthonormal basis functions ϕ α i i : [ a i , b i ] → ℝ \\\\phi_{\\\\alpha_{i}}^{i}\\\\colon[a_{i},b_{i}]\\\\to\\\\mathbb{R} . For f f with FTT rank 𝒓 ¯ \\\\overline{\\\\bm{r}} , we may then approximate\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for Φ ( 𝒙 ) = ( ϕ j 1 ( x 1 ) ) j = 1 m 1 ⊗ ⋯ ⊗ ( ϕ j d ( x d ) ) j = 1 m d ∈ ℝ 𝒎 \\\\Phi(\\\\bm{x})=(\\\\phi_{j}^{1}(x_{1}))_{j=1}^{m_{1}}\\\\otimes\\\\cdots\\\\otimes(\\\\phi_{j}^{d}(x_{d}))_{j=1}^{m_{d}}\\\\in\\\\mathbb{R}^{\\\\bm{m}} and a tensor array 𝑪 ∈ ℝ 𝒎 \\\\bm{C}\\\\in\\\\mathbb{R}^{\\\\bm{m}} with (algebraic) tensor train (TT) rank 𝒓 = ( r 1 , … , r d − 1 ) ⊤ ∈ ℕ d − 1 \\\\bm{r}=(r_{1},\\\\ldots,r_{d-1})^{\\\\top}\\\\in\\\\mathbb{N}^{d-1} bounded by the FTT rank 𝒓 ¯ \\\\overline{\\\\bm{r}} . In particular, we have the decomposition into a tensor train (or matrix product state) format as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with order three tensors 𝑪 i ∈ ℝ r i − 1 , m i , r i \\\\bm{C}_{i}\\\\in\\\\mathbb{R}^{r_{i-1},m_{i},r_{i}} defining matrices 𝑪 i [ α i ] = 𝑪 i [ : , α i , : ] ∈ ℝ r i − 1 , r i \\\\bm{C}_{i}[\\\\alpha_{i}]=\\\\bm{C}_{i}[:,\\\\alpha_{i},:]\\\\in\\\\mathbb{R}^{r_{i-1},r_{i}} with the convention that r 0 = r d = 1 r_{0}=r_{d}=1 . The set of TTs of fixed rank 𝒓 \\\\bm{r} is denoted as ℳ 𝒓 𝒎 \\\\mathcal{M}_{\\\\bm{r}}^{\\\\bm{m}} . In particular, there exists a maximal possible TT rank 𝒓 ¯ ∈ ℕ d − 1 \\\\overline{\\\\bm{r}}\\\\in\\\\mathbb{N}^{d-1} such that ℳ 𝒓 ¯ 𝒎 = ℝ 𝒎 \\\\mathcal{M}_{\\\\overline{\\\\bm{r}}}^{\\\\bm{m}}=\\\\mathbb{R}^{\\\\bm{m}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In what follows, we refer to a function in the form of the right hand side ( 12 ) as extended tensor train (xTT). The xTT format is a special case of the FTT format with a specific choice of univariate basis functions. The set of xTTs building on ℳ 𝒓 𝒎 \\\\mathcal{M}_{\\\\bm{r}}^{\\\\bm{m}} is denoted as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that, by construction, we have 𝒳 𝒓 𝒎 ⊂ ℋ \\\\mathcal{X}_{\\\\bm{r}}^{\\\\bm{m}}\\\\subset\\\\mathcal{H} for any rank 𝒓 ≤ 𝒓 ¯ \\\\bm{r}\\\\leq\\\\overline{\\\\bm{r}} , and for any V ^ 𝑪 ∈ 𝒳 𝒓 𝒎 \\\\widehat{V}_{\\\\bm{C}}\\\\in\\\\mathcal{X}_{\\\\bm{r}}^{\\\\bm{m}} it holds that ‖ V ^ 𝑪 ‖ ℋ = ‖ 𝑪 ‖ F \\\\|\\\\widehat{V}_{\\\\bm{C}}\\\\|_{\\\\mathcal{H}}=\\\\|\\\\bm{C}\\\\|_{\\\\mathrm{F}} . Key properties of the considered nonlinear low-rank model class include storage complexity, multi-linearity and related efficient evaluation. Provided that the ranks can be bounded, the TT format exhibits a storage complexity based on the shape of each component tensor C i C_{i} with upper bound 𝒪 ( max ( m 1 , … , m d ) d max ( r 1 , … , r d − 1 ) 2 ) \\\\mathcal{O}(\\\\max(m_{1},\\\\ldots,m_{d})d\\\\max(r_{1},\\\\dots,r_{d-1})^{2}) , which scales only linearly in the dimension d d . Moreover, a function in xTT format can be evaluated fast and numerically stably, including possible access to gradients or Hessians, see Section A.4 for details. To this end, originated by ( 10 ), we are interested in solving a sequence of optimization tasks for n = 1 , … , N − 1 n=1,\\\\ldots,N-1 given by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for some regularization magnitude τ n > 0 \\\\tau_{n}>0 . Note that the regularization term involves the ℋ \\\\mathcal{H} norm, see Appendix A.3.1 . Recall that we are interested in approximating the true value function V ( 𝐱 , 𝐭 ) V(\\\\bf{x},t) , which is expected to have varying ℋ \\\\mathcal{H} norm over time snapshots t = t 0 , … , t N t=t_{0},\\\\ldots,t_{N} . Consequently, to relate the regularized loss from ( 15 ) to the original loss from ( 10 ), the regularization magnitude needs to be chosen suitably and adaptively, which we explain in detail in Section A.6.1 . Further, it is crucial to choose the TT rank 𝒓 \\\\bm{r} and the numbers of degrees of freedom 𝒎 \\\\bm{m} appropriately, and we provide a corresponding adaptive strategy in Section A.6.2 and Section A.6.3 . Moreover, a detailed discussion of choices for tensor basis functions, along with their associated advantages and challenges, is given in Section A.3 . Finally, a discussion of the FTT rank analysis for certain classes of functions is provided in Section A.7 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Optimization via alternating least squares\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The set ℳ 𝒓 \\\\mathcal{M}_{\\\\bm{r}} defines a Riemannian manifold (Holtz et al. , 2012a ) , rendering the option for Riemannian optimization. Here, we consider an alternative approach for minimization based on alternating least squares (ALS), cf. Holtz et al. ( 2012b ) . In particular, instead of solving ( 15 ) via iterates of the whole TT 𝑪 \\\\bm{C} , we split the minimization into iterations of so-called sweeps consisting of several micro steps . In each micro step we update the i i -th component of the TT only, while fixing all other components. For this, we compactly write (see Section A.1 )\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with orthogonal order three tensors U j ∈ ℝ r j − 1 , m j , r j U_{j}\\\\in\\\\mathbb{R}^{r_{j-1},m_{j},r_{j}} for j = 1 , … , i − 1 , i + 1 , … d j=1,\\\\ldots,i-1,i+1,\\\\ldots d and a so-called core ℭ ∈ ℝ r i − 1 , m i , r i \\\\mathfrak{C}\\\\in\\\\mathbb{R}^{r_{i-1},m_{i},r_{i}} , see Section A.2 for more details. Then, at each micro step, the loss from ( 15 ) reduces to the problem\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with a matrix A i , n u ∈ ℝ K , r i − 1 m i r i A_{i,n}^{u}\\\\in\\\\mathbb{R}^{K,r_{i-1}m_{i}r_{i}} and y n + 1 = ( y n + 1 ( k ) ) k ∈ ℝ K y_{n+1}=(y_{n+1}^{(k)})_{k}\\\\in\\\\mathbb{R}^{K} , see Section A.5 for a derivation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After solving the minimization task stated in ( 15 ), which yields a TT 𝑪 n ∈ ℳ 𝒓 𝒎 \\\\bm{C}^{n}\\\\in\\\\mathcal{M}_{\\\\bm{r}}^{\\\\bm{m}} , we then obtain an approximation of our value function V ( ⋅ , t n ) V(\\\\cdot,t_{n}) at time t = t n t=t_{n} , denoted by V ^ 𝑪 n ∈ 𝒳 𝒓 𝒎 \\\\widehat{V}_{\\\\bm{C}^{n}}\\\\in\\\\mathcal{X}_{\\\\bm{r}}^{\\\\bm{m}} , in xTT format, i.e.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The approximation of the value function V V is only defined locally on a domain D ⋐ ℝ d D\\\\Subset\\\\mathbb{R}^{d} . In practice, this domain is given by D = D n u D=D_{n}^{u} , which depends on both the time step and the current policy u u . If, during the sampling dynamics in ( 3 ), the required policy evaluation u ∗ = − σ ⊤ ∇ V u^{*}=-\\\\sigma^{\\\\top}\\\\nabla V involves points outside of D D , suitable extension strategies must be employed. We discuss these strategies in detail in Section A.4.4 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The minimization requires initial xTTs. At the final time t = t N t=t_{N} , we initialize via an L 2 L^{2} regression of the log-target density using samples from the initial sampling dynamics. Given an approximate xTT at t n + 1 t_{n+1} , the initialization at t n t_{n} is obtained by a generalized Galerkin-type projection of this xTT, accounting for the domain change; see Section A.4.5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While the backward iteration in Algorithm 1 combined with FTTs offers clear advantages over iterative loss minimization with neural networks, it also poses stability challenges. Addressing these issues is a key contribution of this work; the corresponding stabilization strategies are detailed in the appendix. These include the choice of suitable basis functions ( Section A.3 ), handling moving domains along sample trajectories and evaluations outside these domains ( Sections A.4.4 and A.4.5 ), adaptive regularization ( Section A.6.1 ), adaptive rank selection ( Section A.6.2 ), and adaptive basis degree selection ( Section A.6.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Numerical experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section we evaluate our sampling algorithm on challenging high-dimensional problems. We refer to an additional problem from computational statistics in Section B.5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Highdimensional Multiwell problems\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first consider a class of Multiwell problems, defined by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which are commonly used benchmarks (Berner et al. , 2024 ; Richter and Berner, 2024 ; Wu et al. , 2020 ; Midgley et al. , 2022 ) . These densities are chosen because they resemble physics and molecular dynamics problems, providing an indication of performance in practical applications. At the same time, they offer a controlled setting for systematically studying the three key challenges in sampling: (1) the dimensionality d d , (2) the mode separation δ \\\\delta , and (3) the number of modes 2 m 2^{m} . In our experiments, we use a Fourier basis (see Section A.3.1 ) with 3 3 to 13 13 basis functions per dimension, treating this as a hyperparameter tuned according to the observed approximation quality. We employ an H mix 2 H_{\\\\mathrm{mix}}^{2} orthonormalization as described in Section A.3.2 . For the noising process ( 2 ), we set f ( 𝒙 , t ) = 𝒙 f(\\\\bm{x},t)=\\\\bm{x} , σ ( t ) = 2 \\\\sigma(t)=\\\\sqrt{2} , and consider a time horizon T = 2 T=2 . We apply Algorithm 1 to two Multiwell problems, with dimensions d = 10 d=10 and d = 50 d=50 and 8 8 and 32 32 modes, respectively, with δ = 2 \\\\delta=2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Figure 3 , increasing the number of time steps N N (which also determines the number of functions V ^ n \\\\widehat{V}_{n} approximated) steadily improves sampling performance, ultimately achieving remarkable quality. Importantly, the algorithm’s runtime is significantly faster than that of alternative, predominantly neural network-based samplers. In d = 10 d=10 , training ranges roughly from 1 1 minute ( N = 2 8 N=2^{8} ) to 20 20 minutes ( N = 2 13 N=2^{13} ), while in d = 50 d=50 it ranges from 10 10 to 300 300 minutes. Note that the runtime grows linearly with N N . We refer to Figure 4 for a comparison with the time-reversed diffusion sampler (DIS) from (Berner et al. , 2024 ) , which considers the same noising SDE, however relies on neural network approximations and variational training. We refer to Figure 14 in Section B.4 for a comparison to further sampling methods, showing significantly improved performance. Finally, we refer to Figure 2 comparing marginals of the 50 50 dimensional problem with a reference solution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We note that the Girsanov theorem relates the error in the log-normalizing constant log 𝒵 \\\\log\\\\mathcal{Z} to the L 2 L^{2} -approximation of the ground-truth score ∇ log p \\\\nabla\\\\log p , i.e., the gradient of the solution to the HJB equation w.r.t. to the measure induced by p target p_{\\\\mathrm{target}} (Berner et al. , 2024 ) . Under suitable assumptions, one can further leverage Poincaré inequalities to relate this error to the L 2 L^{2} -approximation of the PDE solution itself, measuring the accuracy of our proposed PDE solver.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Investigation of optimal ranks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we study highly anisotropic Gaussian targets to analyze the rank structure of the induced transport problems and to assess the adaptivity of our algorithm. We consider target densities with potential log ρ target ( 𝒙 ) = − 𝒙 ⊤ M 𝒙 , \\\\log\\\\rho_{\\\\mathrm{target}}(\\\\bm{x})=-\\\\bm{x}^{\\\\top}M\\\\bm{x}, where M M is a random full-rank matrix. For this class, the maximal admissible rank is 𝐫 = ( 3 , 4 , … , 2 + d / 2 , … , 4 , 3 ) ∈ ℕ d − 1 \\\\mathbf{r}=(3,4,\\\\dots,2+d/2,\\\\dots,4,3)\\\\in\\\\mathbb{N}^{d-1} for even d d ; see Corollary A.6 for the general case. As the target distribution converges toward the standard normal, p Y T ≈ p prior = 𝒩 ( 0 , Id ) p_{Y_{T}}\\\\approx p_{\\\\mathrm{prior}}=\\\\mathcal{N}(0,\\\\operatorname{Id}) , the ranks are expected to approach the constant vector 𝐫 = ( 2 , … , 2 ) \\\\mathbf{r}=(2,\\\\dots,2) . This behavior is confirmed in Figure 5 , which shows the evolution of singular values and the corresponding rank adaptivity, resulting in a reduced runtime. The clear spectral gaps demonstrate that the algorithm reliably identifies and adapts to the appropriate ranks over time; see also Figure 11 in the appendix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 ϕ 4 \\\\phi^{4} scalar field theory\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Next, we evaluate our method on the discretized Ginzburg-Landau model (specifically, the ϕ 4 \\\\phi^{4} scalar field theory), a canonical framework in statistical mechanics used to describe continuous phase transitions and spontaneous symmetry breaking (Ginzburg and Landau, 1950 ; Rosenstein and Li, 2021 ) . The target density is defined by a Gibbs distribution over a lattice, characterized by a highly non-convex energy landscape resulting from the competing forces of local bistability and nearest-neighbor spatial coupling,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This system is highly challenging for sampling algorithms, as the target distribution exhibits exponentially many metastable modes separated by high-energy barriers, leading to severe mode collapse in standard MCMC methods. Nearest-neighbor interactions further induce strong dependencies between lattice sites, resulting in a highly constrained and correlated high-dimensional geometry. For δ = 2 \\\\delta=2 , Figure 6 shows that the TTD method generates high-quality samples in two distinct multimodal regimes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion & Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we introduced Tensor Train Diffusion (TTD), an efficient PDE solver for high-dimensional HJB equations leading to a novel way of sampling from unnormalized densities. Our method solves the HJB equation underlying the diffusion process using a functional tensor train (FTT) representation. By integrating this with a backward-in-time iterative scheme derived from BSDEs, TTD provides a fast, accurate, and stable alternative to neural network techniques. In particular, TTD requires less target evaluations and does not rely on SGD-based optimization with long training times and hyperparameter sensitivity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In general, our TTD framework presents a solid foundation for the robust and efficient solution of HJB equations, that can readily be used for different sampling problems – we refer to potential limitations in Section B.1 . While we provide a significantly accelerated tensor train implementation in PyTorch, we believe that further hardware-specific optimization is possible. On the theoretical side, our work motivates further research to analyze latent low-rank structures of different target densities for educated choices of basis functions and variable orderings. Future work could also combine the framework with neural network models or classical sampling methods, as well as extend TTD to broader classes of PDEs and applications. From a theoretical perspective, an open question concerns the behavior of FTT ranks for solutions of the underlying HJB equation, which is only partially understood (Gruhlke et al. , 2026 ) . In the general case, the current state-of-the-art theory (Ali and Nouy, 2020a , b , 2021 ; Griebel et al. , 2023 ; Bachmayr, 2023 ) does not directly apply and would need to be extended to weighted regularity classes, since solutions of HJB equations are defined on ℝ d \\\\mathbb{R}^{d} and are generally unbounded as ‖ 𝒙 ‖ → ∞ \\\\|\\\\bm{x}\\\\|\\\\to\\\\infty . Consequently, on the computational side, one must rely on rank-adaptive approaches, i.e., determining the xTT ranks for the approximation in an adaptive manner.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Impact Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.\"\n }\n]","isConversational":false}}},{"rowIdx":496,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06839"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"LEMUR 2: Unlocking Neural Network Diversity for AI"},"authors":{"kind":"list like","value":["Tolgay Atinc Uzun","Waleed Khalid","Saif U Din","Sai Revanth Mulukuledu","Akashdeep Singh","Chandini Vysyaraju","Raghuvir Duvvuri","Avi Goyal","Yashkumar Rajeshbhai Lukhi","Muhammad A. Hussain","Krunal Jesani","Usha Shrestha","Yash Mittal","Roman Kochnev","Pritam Kadam","Mohsin Ikram","Harsh R. Moradiya","Alice Arslanian","Dmitry Ignatov","Radu Timofte"],"string":"[\n \"Tolgay Atinc Uzun\",\n \"Waleed Khalid\",\n \"Saif U Din\",\n \"Sai Revanth Mulukuledu\",\n \"Akashdeep Singh\",\n \"Chandini Vysyaraju\",\n \"Raghuvir Duvvuri\",\n \"Avi Goyal\",\n \"Yashkumar Rajeshbhai Lukhi\",\n \"Muhammad A. Hussain\",\n \"Krunal Jesani\",\n \"Usha Shrestha\",\n \"Yash Mittal\",\n \"Roman Kochnev\",\n \"Pritam Kadam\",\n \"Mohsin Ikram\",\n \"Harsh R. Moradiya\",\n \"Alice Arslanian\",\n \"Dmitry Ignatov\",\n \"Radu Timofte\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06839v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Neural networks underpin numerous breakthroughs in artificial intelligence, delivering state-of-the-art results in fields such as computer vision and natural language processing. Their growing complexity, however, has exposed the limits of manual design. In response, the field is shifting toward a paradigm centered on creating and leveraging large-scale, reusable collections of neural network models. The goal of this shift is to replace manual intuition with data-driven discovery, enabling researchers to identify latent patterns, structural regularities, and generalizable design principles from diverse model corpora. Such model-centric datasets are now critical resources for benchmarking, neural architecture analysis, meta-learning, and automated machine learning."},{"type":"para","text":"Most publicly available neural architecture repositories are built around a single construction paradigm, typically exhaustive or near-exhaustive enumeration of a fixed cell-based search space. This yields methodologically homogeneous collections that cover only a narrow region of the architectural design space and are not readily extensible to models obtained by other means. At the same time, these repositories rarely provide an end-to-end path from architectural specification to task-level evaluation and device-level deployment, making it difficult to study accuracy-latency trade-offs or reuse models across modalities."},{"type":"para","text":"We propose multiple strategies that build on the LEMUR [ 15 ] dataset and accompanying pipeline to improve architectural diversity and operational integration. To generate more models, we employ programmatic editing, reinforcement learning, evolution-based search, fractal construction, retrieval-augmented block extraction, and LLM-driven synthesis. On the deployment side, we provide NN-Lite and NN-VR for benchmarking, testing, and verification across multiple tasks on resource-constrained targets under a shared protocol. Statistics are available in the project repository 1 1 1 Github Repository: https://github.com/ABrain-One/nn-dataset ."},{"type":"heading","level":2,"text":"1.1 Related Work"},{"type":"para","text":"In neural network research, datasets are the backbone of training, evaluation, and benchmarking. There are plentiful datasets in image, text, and domain-specific corpora [ 31 ] that provide large, labeled collections of data. By contrast, resources that systematically capture the structures, configurations, and resulting performance of neural networks themselves are far less common."},{"type":"para","text":"Dataset of models The closest line of work comes from the neural architecture search (NAS) community. Benchmarks such as NAS-Bench-101 [ 50 ] and its extension NAS-Bench-201 [ 8 ] enumerate a fixed, small search space of convolutional cells and store their training and evaluation results to enable reproducible comparison of NAS algorithms. NATS-Bench [ 7 ] generalizes this idea to both topology and size spaces across multiple datasets, while TransNAS-Bench-101 [ 9 ] expands it to several vision-style tasks to study transferability. NAS-Bench-NLP [ 24 ] moves beyond vision to language models, HW-NAS-Bench [ 29 ] adds hardware and latency measurements, and JAHS-Bench-201 [ 2 ] couples architectures with hyperparameters for joint optimization."},{"type":"para","text":"AutoML Frameworks and Model Repositories AutoML frameworks such as AutoKeras [ 22 ] and TPOT [ 34 ] provide tooling for automated model selection and hyperparameter optimization but do not maintain large-scale, curated repositories of architectures with performance metadata. Similarly, model zoos like TensorFlow Hub [ 16 ] and PyTorch Hub [ 37 ] offer pre-trained models but lack standardized evaluation across tasks and hardware, and they do not support systematic architecture generation or diversity analysis."},{"type":"heading","level":2,"text":"1.2 Our Contribution"},{"type":"para","text":"This paper presents LEMUR 2, an extension of the original LEMUR [ 15 ] framework. While LEMUR served as a repository for existing neural networks and their associated statistics, inspired by recent advancements in the use of LLMs across various domains [ 25 , 14 , 43 ] , LEMUR 2 is designed to include generative systems that leverage existing data for the automated creation of new network architectures [ 11 ] as well as an assessment tool for testing on edge devices [ 6 ] and deployment on different platforms."},{"type":"para","text":"First, the dataset was expanded to over 14,000 models using several distinct, automated generation methodologies. These methods include genetic algorithms, reinforcement learning-based layer masking for network mutation, abstract syntax tree (AST) editing of model source code, and fractal-inspired generation. To facilitate dynamic collection, we also introduce NN-RAG, a retrieval-augmented system that extracts validated, self-contained PyTorch [ 36 ] modules from external codebases, creating a library of over 900 reusable components. The dataset is further enriched with metadata from a systematic evaluation of 6,000 data transformation pipelines, providing guidance on optimal data preprocessing. The framework’s applicability is demonstrated through the inclusion of additional tasks, such as image captioning, text-to-text and text-to-image synthesis."},{"type":"para","text":"Second, to connect theoretical metrics with practical performance, we added deployment-aware metadata. We built NN-Lite, an automated pipeline that converts, deploys, and benchmarks PyTorch models on the Android platform. This system processed over 7,500 models, populating the LEMUR 2 database with their on-device inference latencies, empirical measurements often absent from standard benchmarks. This deployment analysis was extended to immersive applications with NN-VR, a system for the automated conversion and performance evaluation of models within the Unity engine for virtual reality contexts."},{"type":"para","text":"Collectively, these contributions establish LEMUR 2 as a public research resource. It provides a large-scale architectural corpus, a set of generative methodologies, and multi-platform deployment benchmarks that support research in automated AI design. The empirical value of the LEMUR dataset, as demonstrated within the NNGPT project, is proved by its successful application to LLM-guided generative synthesis, exploration of complex architectural structures, and improved generalization across multiple computer vision tasks [ 25 , 13 , 33 , 45 , 21 , 48 , 47 , 17 ] ."},{"type":"heading","level":1,"text":"2 Methodology"},{"type":"para","text":"We extend the LEMUR [ 15 ] corpus as a modular generation and integration framework built on the original experiment manager for task execution and result persistence. Corpus access and downstream processing use the data interface, which exposes stored records as a fixed-schema pandas DataFrame for uniform filtering and aggregation across tasks, datasets, metrics, architectures, epochs, accuracy, runtime, parameter counts, and transformation identifiers without direct database queries. On this layer, self-contained extension modules generate candidate model definitions or transformation programs with metadata and submit them to the standard LEMUR pipeline for evaluation. These modules combine LLM-based generation from natural-language guidance with programmatic generators that produce valid variants faster and at lower cost, enabling scalable, heterogeneous expansion under a shared reporting format."},{"type":"para","text":"Architectural diversity in LEMUR 2 comes from combining heterogeneous generators and sources, so the corpus is not centered on a single backbone family. For generators that operate by mutating a seed architecture, we instantiate them on a lightweight reference template supported by the generator to enable high-throughput evaluation and controlled attribution of operator effects, while still producing many structurally distinct variants."},{"type":"heading","level":2,"text":"2.1 Task Extensions"},{"type":"heading","level":3,"text":"2.1.1 Image Captioning"},{"type":"para","text":"NN-Caption is an automated image-captioning pipeline built on LEMUR in which an LLM, iteratively prompted with a baseline ResNet–LSTM captioner and code templates for the LEMUR Net interface ( __init__ , train_setup , learn , forward ), generates PyTorch encoder–decoder architectures that couple a convolutional image encoder with sequence decoders (LSTM, GRU [ 5 ] , Transformer). Generated code is normalized, validated via Python’s abstract syntax tree, integrated into LEMUR, compiled, and trained on MS COCO [ 31 ] , with caption quality evaluated using BLEU-4 [ 35 ] ; in total, NN-Caption contributes 357 captioning architectures to LEMUR 2."},{"type":"heading","level":3,"text":"2.1.2 Text-to-Image"},{"type":"para","text":"LEMUR includes a text-to-image pipeline spanning three generative families: a diffusion model (UNet-D) [ 41 , 18 , 42 ] , a GAN [ 52 ] , and a CVAE-GAN [ 23 ] , all conditioned on natural-language text within a unified training and evaluation framework. For diffusion, a pre-trained CLIP text encoder [ 39 ] provides embeddings that are fused with time-step embeddings and injected into a multi-scale U-Net encoder–decoder with residual and attention blocks, optimized with a denoising diffusion objective. The GAN uses a generator that fuses text representations with a convolutional upsampling stack and a discriminator operating on joint image–text pairs, whereas the CVAE-GAN employs a conditional VAE whose decoder serves as the generator alongside an adversarial discriminator that promotes sharp outputs. All models are trained on paired text–image data with architecture-specific losses (diffusion, adversarial, or variational plus adversarial), and performance is quantified via CLIP-based text–image similarity between generated images and their prompts."},{"type":"heading","level":3,"text":"2.1.3 Text-to-Text"},{"type":"para","text":"LEMUR is extended to natural language generation via a text-to-text pipeline for training and evaluating recurrent language models with standardized metrics. A lightweight WikiText data loader tokenizes text into integer sequences and forms batched streams for truncated backpropagation through time [ 49 ] , with context length, vocabulary size, and batching strategy configured via a shared dictionary. Two recurrent baselines are instantiated: a stacked Elman RNN [ 10 ] with learned token embeddings, tanh activations, and a linear next-token decoder, and a stacked LSTM [ 19 ] with input, forget, and output gates that improve stability and long-range dependency modeling. Evaluation uses perplexity and BLEU, with all LEMUR metrics normalized to [ 0 , 1 ] [0,1] ; BLEU is computed at corpus level with smoothed n n -gram precision on the validation set."},{"type":"heading","level":3,"text":"2.1.4 Mixture-of-Experts (MoE)"},{"type":"para","text":"Mixture-of-Experts architectures are integrated into LEMUR to study expert composition and routing strategies on CIFAR-10 [ 27 ] . Eight variants are constructed in three stages: (i) homogeneous MoEs based on a single backbone type, (ii) AlexNet-based MoEs, and (iii) a heterogeneous MoE combining multiple backbones."},{"type":"para","text":"Homogeneous configurations use identical expert architectures and a sparse routing mechanism. For each input, a gating network selects the two experts with the highest scores (top-2 routing), and only these experts are evaluated. Heterogeneous configurations combine four different backbone architectures—AlexNet [ 26 ] , AirNet [ 4 ] , DenseNet [ 20 ] , and BagNet [ 3 ] —and employ a soft routing strategy in which the gating network produces a weight for each expert. The final prediction is obtained as a weighted aggregation of the experts’ outputs. All MoE variants are trained and evaluated within the LEMUR framework using the standard CIFAR-10 training protocol."},{"type":"heading","level":2,"text":"2.2 Achieving Diversity"},{"type":"heading","level":3,"text":"2.2.1 AST-Based Mutations"},{"type":"para","text":"An AST-based mutation mechanism [ 47 ] generates structurally consistent architectural variants by modifying the source code of existing models while operating on the channel dimensions of convolutional and linear layers and preserving tensor-shape compatibility. A symbolic tracer built on torch.fx [ 40 ] constructs a computation graph whose nodes represent layers with explicit dataflow edges, and maintains a source map from each module instantiation to its file location. Given a target layer, a planning component uses this graph to identify all dependent consumer layers and infer required input/output channel adjustments, including in architectures with residual connections. The mutation plan is applied by parsing the implementation into an AST, locating the corresponding constructor calls via the recorded coordinates, and updating channel-related arguments; the modified AST is then rendered back to source code, re-imported, and each mutant is validated via instantiation and forward/backward passes on dummy inputs. This procedure yields approximately 1000 additional architectures in the corpus."},{"type":"heading","level":3,"text":"2.2.2 Reinforcement Learning"},{"type":"para","text":"The LEMUR dataset is processed by programmatically parsing each network’s source code and identifying layer-instantiation statements (e.g., nn.Conv2d , nn.Linear ) within the __init__ method. These code blocks are removed and replaced with placeholder tokens, while preserving the surrounding class structure, method signatures, and forward pass logic. This produces a dataset of masked model skeletons paired with their complete source code as ground truth for training."},{"type":"para","text":"The reinforcement learning pipeline then iteratively refines architecture generation by completing these masked skeletons. The LLM acts as a policy, generating code completions (actions) given the masked skeleton (state). Each generated model undergoes validation checks for compilation and tensor shape consistency via a dummy forward pass. Successful models are trained for one mini-epoch on CIFAR-10 to measure accuracy."},{"type":"para","text":"A composite reward is calculated: − 1.0 -1.0 for validation failure, + 0.2 +0.2 for each passed stage (compilation, forward pass, training), and + 1.0 × Δ a c c u r a c y +1.0\\times\\Delta accuracy for improvement over baseline. This reward updates the LLM’s policy using Group Relative Policy Optimization (GRPO) [ 44 ] ."},{"type":"heading","level":3,"text":"2.2.3 Genetic Algorithm"},{"type":"para","text":"A genetic algorithm explores AlexNet-style architectures on CIFAR-10 [ 27 ] using a parameterized representation that includes architectural and training hyperparameters. Populations of candidate networks are evolved over generations via crossover and mutation on these parameter vectors; fitness is approximated by training each model for a small number of epochs and using validation accuracy for selection, with a checksum over the architecture description used to discard duplicates and maintain diversity. Two search spaces are considered: one restricted to hyperparameters (e.g., filter counts, kernel sizes, learning rate, dropout), and one that additionally includes block-level structural choices (pooling type, activation function, batch normalization). Across both spaces, the algorithm yields approximately 2,000 distinct AlexNet-type architectures."},{"type":"heading","level":3,"text":"2.2.4 Fractal-Inspired Computational Architectures"},{"type":"para","text":"LEMUR includes a FractalNet-style [ 28 ] generator [ 33 ] that constructs self-similar, recursively defined multi-column networks whose topology is governed by fractal depth N N (recursion level) and column width (number of parallel pathways), enabling balanced growth in depth and width. The pipeline comprises (i) configuration generation, which systematically samples architectural blueprints via permutations of convolution, normalization, activation, and dropout layers; (ii) template-based model instantiation, which programmatically realizes the specified recursive, multi-column PyTorch architectures; and (iii) automated evaluation, which executes a standardized training and logging protocol. To scale to over 1,200 unique architectures, training employs Automatic Mixed Precision (AMP) to accelerate computation with half-precision arithmetic and gradient checkpointing to reduce GPU memory via recomputation of intermediate activations."},{"type":"heading","level":3,"text":"2.2.5 Retrieval-Augmented Generation"},{"type":"para","text":"The NN-RAG component [ 12 , 13 ] constructs a library of reusable PyTorch modules by mining existing codebases. It scans repositories for subclasses of torch.nn.Module that implement a forward() method and extracts them as self-contained units that preserve the original imports and semantics."},{"type":"para","text":"Source files are parsed using LibCST [ 30 ] , which maintains syntactic structure, formatting, and comments. A scope-sensitive dependency resolver computes, for each candidate module, the minimal transitive closure of required definitions (such as auxiliary classes, functions, and constants) without executing the code. The resulting set of definitions is topologically ordered to ensure definition-before-use and is written out as an isolated module. Each extracted module passes through a multi-stage validation pipeline comprising abstract syntax tree parsing, bytecode compilation, and sandboxed execution to detect syntax and import-time errors. Valid modules are then integrated into LEMUR as reusable building blocks, with associated metadata linking them to their source repositories."},{"type":"heading","level":3,"text":"2.2.6 Few-Shot Architecture Prompting"},{"type":"para","text":"To improve the stability of LLM-based architecture generation, we employ few-shot prompting and deduplication. Prompts include a small set of high-performing LEMUR architectures as exemplars, together with a description of the target dataset, and the LLM is instructed to produce a new architecture consistent with the LEMUR interface; the number of exemplars is treated as a tunable parameter. Each generated architecture is normalized into a canonical form and assigned a whitespace-stripped MD5 hash, which serves as a database key for fast detection and removal of exact duplicates prior to training. Non-duplicate architectures are trained under the standard LEMUR protocol and their performance recorded. For statistical analysis, evaluation metrics are aggregated in a dataset-balanced manner by first computing performance per dataset and then averaging across datasets to avoid bias from uneven task coverage."},{"type":"heading","level":3,"text":"2.2.7 Data Transformations"},{"type":"para","text":"In line with prior work [ 1 ] , and with the objective of diversifying data augmentation within the corpus using the implementation described in [ 45 ] , transformation pipelines are generated and evaluated on the CIFAR-10 dataset. A fixed ResNet architecture is trained for a single epoch (batch size 64, learning rate 0.01, momentum 0.9, dropout 0.2), and validation accuracy is recorded. The first approach is LLM-based: prompted with the desired interface and examples of Torchvision transforms, the model proposes augmentation functions as Python code, which are syntactically validated before integration. The second approach is combinatorial: from a predefined set of Torchvision transforms, all pipelines with one, two, or three variable transforms are enumerated and extended by a fixed suffix ( Resize – ToTensor – Normalization ); variable transforms receive randomly sampled parameters. This yields 6,000 unique augmentation pipelines, each evaluated under the same training protocol."},{"type":"heading","level":2,"text":"2.3 Cross Domain Deployment and Testing"},{"type":"heading","level":3,"text":"2.3.1 NN-VR"},{"type":"para","text":"The VR-Ready Neural Network Verifier (NN-VR) provides automated validation and deployment of pre-trained LEMUR models in Unity-based virtual reality environments. Its architecture comprises (i) a Neural Network Parser that ingests ONNX exports and associated metadata from LEMUR and populates a Unity project configured with the Barracuda inference engine; (ii) a Compatibility Verifier that imports ONNX models, checks operator coverage and shader support, and profiles GPU memory under VR-relevant settings; and (iii) an Automated Porting System that converts and optionally optimizes models, configures inference scenes, and records diagnostic logs. Performance is evaluated against VR-specific criteria: inference latency relative to an 11 ms-per-frame budget (90 Hz), additional memory overhead compared to standalone inference, and numerical consistency between Unity/Barracuda outputs and reference PyTorch predictions within a tight tolerance. The pipeline is designed for scalable application to large collections of models exported from LEMUR."},{"type":"heading","level":3,"text":"2.3.2 NN-Lite"},{"type":"para","text":"NN-Lite [ 6 ] is an automated pipeline for deploying and benchmarking PyTorch models on the Android platform. It provides an end-to-end workflow designed to evaluate models from the dataset, managing the process from model conversion to final reporting without manual intervention."},{"type":"para","text":"The pipeline executes a four-stage process for each model. First, PyTorch models are automatically converted to the TensorFlow Lite (TFLite) format. This stage includes a custom wrapper to resolve the NCHW (PyTorch) to NHWC (TFLite) tensor layout disparity, ensuring compatibility and efficient memory usage. Second, the system manages the lifecycle of an Android Virtual Device (AVD), including boot-up and state monitoring. Third, a lightweight Android application is deployed to the AVD to execute model inference and collect latency metrics. Fourth, the pipeline retrieves these benchmark results and combines them with device analytics (e.g., memory, CPU architecture) into a structured JSON report."},{"type":"para","text":"The system is designed for large-scale, continuous operation, with built-in state management and failure recovery mechanisms. In a documented 48-hour session, the pipeline processed over 7,500 models from the LEMUR dataset. The final stage of the process is statistical consolidation, where performance metrics like task-level accuracy are aggregated with on-device latency data. This integration allows for a unified analysis of a model’s software performance and its on-device execution characteristics."},{"type":"heading","level":1,"text":"3 Evaluation"},{"type":"para","text":"All computer vision experiments are run in the AI Linux Docker image 2 2 2 AI Linux: https://hub.docker.com/r/abrainone/ai-linux on NVIDIA GeForce RTX 3090/4090 GPUs with 24 GB memory, using a Kubernetes cluster and dedicated workstations."},{"type":"para","text":"We performed repeated evaluations under multiple training configurations for every supported task and its associated datasets. Each evaluation run corresponds to a distinct hyperparameter setting, including batch size, number of training epochs, learning rate, momentum, dropout rate, and the selected data transformation pipeline. Every run is logged together with its accuracy and runtime, which enables systematic comparison across configurations and supports analysis of model performance variability under the same task protocol."},{"type":"para","text":"Evaluations use bounded training schedules to keep compute comparable while sweeping many architectures and hyperparameter settings. Metrics generally improve with additional optimization and data exposure, but convergence rates vary across configurations, so reported scores reflect compute-budgeted performance rather than uniformly converged endpoints. Accordingly, comparisons to fully trained baselines should use matched training horizons, and the results primarily quantify how efficiently each method converts compute into task performance."},{"type":"heading","level":1,"text":"4 Results"},{"type":"para","text":"Table 2 summarizes the quantitative performance of the LEMUR 2 generative systems and deployment pipelines under bounded training budgets. Across the evaluated settings, optimization-guided generators produce the strongest and most consistent image-classification results, while prompt-based and programmatic mutation methods exhibit higher variance in both quality and reliability. Best-per-run denotes the highest accuracy attained across all training epochs for a fixed hyperparameter configuration, allowing comparison of peak observed performance rather than final-epoch values as seen in 10 , 7 ."},{"type":"para","text":"Overall, these generative mechanisms produce models that are competitive with existing manually crafted network baselines. Within groups (Fig. 9 ), evolutionary models attain the highest median accuracy, indicating a dense cluster of strong architectures. LLM-generated few-shot models (“alt-”) occasionally match the best groups at the top but have lower median accuracy and higher variance, while programmatic AST mutations (“ast-”) yield the lowest median accuracy, consistent with the sensitivity of local channel edits to the choice of input model relative to global optimization strategies."},{"type":"para","text":"This behavior can be explained by considering the effective size of the search space. Optimization-based generators converge faster than pure random search because the objective function provides a direction that quickly focuses the search on promising regions, especially when the coverage is moderate and the landscape is relatively smooth. As the problem becomes more complex and the search space grows, however, the landscape becomes increasingly rugged, making it harder for any method to consistently find high-quality solutions. The impact of architectural configuration, such as channel widths in the AST channel-mutated AlexNet variants show different convergence properties when compared with the original AlexNet, despite being a relatively small modification, underscoring how sensitive convergence can be to seemingly minor configurational change."},{"type":"heading","level":2,"text":"4.1 Task Extensions"},{"type":"para","text":"Image Captioning Increasing prompt complexity (5–10 snippets) reduced runnable generations from 80% to 50%. The LLM explored diverse architectures (e.g., ConvNeXt, EfficientNet, Transformers); the best generated variant (ResNet-50 + Transformer) yielded a score of BLEU-4 = 0.317 at 50 epochs. In the longer training schedule, it can confidently compete with the baseline, ( Fig. 7 ) approaching to the score of 0.3246. This highlights the trade-off between accuracy and architectural diversity."},{"type":"para","text":"Text-to-Image Methods progressed from a UNet diffusion model (CLIP 0.17–0.24) and LSTM-GAN ( ≈ \\approx 0.214) to a complex CVAE-GAN. The final CVAE-GAN, utilizing a CLIP encoder, PatchGAN discriminator, and adversarial losses, achieved the peak CLIP score of 0.2751, prioritizing semantic alignment over photorealistic textures."},{"type":"para","text":"Mixture-of-Experts Evaluations on CIFAR-10 showed a tuned homogeneous MoE achieving the highest accuracy (93.9%). A heterogeneous variant combining AlexNet, AirNet, DenseNet, and BagNet experts reached 93.13%, successfully outperforming all constituent backbones trained individually."},{"type":"heading","level":2,"text":"4.2 Generative Systems"},{"type":"para","text":"Genetic Algorithm A hyperparameter-restricted search yielded 62.76% test accuracy (11×11 kernel), whereas evolving block-type structures (pooling, activation, BN) improved accuracy to 80.04%. The champion model utilized batch normalization, 3×3 convolutions, and mixed pooling, demonstrating the value of block-level flexibility."},{"type":"para","text":"NN-RAG From 1,289 candidate blocks sourced from timm , torchvision , and transformers , 941 (73.0%) passed execution checks. The resulting library, tagged “rag-” in LEMUR 2, includes executable attention, convolutional, and normalization modules."},{"type":"para","text":"Few-Shot Prompting Results were non-monotonic: n = 3 n=3 exemplars achieved the highest balanced mean accuracy (53.1%) considering all datasets and improved CIFAR-100 performance by 11.6pp ( p = 0.001 p=0.001 ). Conversely, n = 6 n=6 caused a 99.8% failure rate due to context overflow."},{"type":"para","text":"Data Transformation The best LLM-generated pipeline achieved 57.28% validation accuracy. A combinatorial search of 6,000 pipelines identified a superior configuration using a single RandomPosterize transform, reaching 61.24% accuracy and outperforming the generative approach."},{"type":"heading","level":1,"text":"5 Conclusion"},{"type":"para","text":"This system was developed to produce a large and architecturally diverse collection of neural networks with extensive performance records across multiple tasks and hardware platforms, resulting in over 14,000 distinct architectures and more than 750,000 structured training records documenting model architectures, training configurations, and task-specific metrics such as accuracy. For edge devices and cross-domain coverage, NN-Lite and NN-VR automate model deployment and benchmarking while recording on-device characteristics including inference latency and memory usage. By mobilizing data at scale, the framework reveals patterns that indicate which network configurations and structures are effective, supporting automated development of next-generation architectures [ 11 ] . By formalizing dependency-closed neural primitives, it also establishes a scalable methodology for “Neural Architecture Mining,” transforming disparate source code into a standardized, executable substrate for automated ML workflows. Due to space constraints, further implementation details, ablation studies, and extended results are deferred to the supplementary material."},{"type":"para","text":"Acknowledgments. This work was partially supported by the Alexander von Humboldt Foundation."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neural networks underpin numerous breakthroughs in artificial intelligence, delivering state-of-the-art results in fields such as computer vision and natural language processing. Their growing complexity, however, has exposed the limits of manual design. In response, the field is shifting toward a paradigm centered on creating and leveraging large-scale, reusable collections of neural network models. The goal of this shift is to replace manual intuition with data-driven discovery, enabling researchers to identify latent patterns, structural regularities, and generalizable design principles from diverse model corpora. Such model-centric datasets are now critical resources for benchmarking, neural architecture analysis, meta-learning, and automated machine learning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Most publicly available neural architecture repositories are built around a single construction paradigm, typically exhaustive or near-exhaustive enumeration of a fixed cell-based search space. This yields methodologically homogeneous collections that cover only a narrow region of the architectural design space and are not readily extensible to models obtained by other means. At the same time, these repositories rarely provide an end-to-end path from architectural specification to task-level evaluation and device-level deployment, making it difficult to study accuracy-latency trade-offs or reuse models across modalities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose multiple strategies that build on the LEMUR [ 15 ] dataset and accompanying pipeline to improve architectural diversity and operational integration. To generate more models, we employ programmatic editing, reinforcement learning, evolution-based search, fractal construction, retrieval-augmented block extraction, and LLM-driven synthesis. On the deployment side, we provide NN-Lite and NN-VR for benchmarking, testing, and verification across multiple tasks on resource-constrained targets under a shared protocol. Statistics are available in the project repository 1 1 1 Github Repository: https://github.com/ABrain-One/nn-dataset .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In neural network research, datasets are the backbone of training, evaluation, and benchmarking. There are plentiful datasets in image, text, and domain-specific corpora [ 31 ] that provide large, labeled collections of data. By contrast, resources that systematically capture the structures, configurations, and resulting performance of neural networks themselves are far less common.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dataset of models The closest line of work comes from the neural architecture search (NAS) community. Benchmarks such as NAS-Bench-101 [ 50 ] and its extension NAS-Bench-201 [ 8 ] enumerate a fixed, small search space of convolutional cells and store their training and evaluation results to enable reproducible comparison of NAS algorithms. NATS-Bench [ 7 ] generalizes this idea to both topology and size spaces across multiple datasets, while TransNAS-Bench-101 [ 9 ] expands it to several vision-style tasks to study transferability. NAS-Bench-NLP [ 24 ] moves beyond vision to language models, HW-NAS-Bench [ 29 ] adds hardware and latency measurements, and JAHS-Bench-201 [ 2 ] couples architectures with hyperparameters for joint optimization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AutoML Frameworks and Model Repositories AutoML frameworks such as AutoKeras [ 22 ] and TPOT [ 34 ] provide tooling for automated model selection and hyperparameter optimization but do not maintain large-scale, curated repositories of architectures with performance metadata. Similarly, model zoos like TensorFlow Hub [ 16 ] and PyTorch Hub [ 37 ] offer pre-trained models but lack standardized evaluation across tasks and hardware, and they do not support systematic architecture generation or diversity analysis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.2 Our Contribution\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper presents LEMUR 2, an extension of the original LEMUR [ 15 ] framework. While LEMUR served as a repository for existing neural networks and their associated statistics, inspired by recent advancements in the use of LLMs across various domains [ 25 , 14 , 43 ] , LEMUR 2 is designed to include generative systems that leverage existing data for the automated creation of new network architectures [ 11 ] as well as an assessment tool for testing on edge devices [ 6 ] and deployment on different platforms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, the dataset was expanded to over 14,000 models using several distinct, automated generation methodologies. These methods include genetic algorithms, reinforcement learning-based layer masking for network mutation, abstract syntax tree (AST) editing of model source code, and fractal-inspired generation. To facilitate dynamic collection, we also introduce NN-RAG, a retrieval-augmented system that extracts validated, self-contained PyTorch [ 36 ] modules from external codebases, creating a library of over 900 reusable components. The dataset is further enriched with metadata from a systematic evaluation of 6,000 data transformation pipelines, providing guidance on optimal data preprocessing. The framework’s applicability is demonstrated through the inclusion of additional tasks, such as image captioning, text-to-text and text-to-image synthesis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, to connect theoretical metrics with practical performance, we added deployment-aware metadata. We built NN-Lite, an automated pipeline that converts, deploys, and benchmarks PyTorch models on the Android platform. This system processed over 7,500 models, populating the LEMUR 2 database with their on-device inference latencies, empirical measurements often absent from standard benchmarks. This deployment analysis was extended to immersive applications with NN-VR, a system for the automated conversion and performance evaluation of models within the Unity engine for virtual reality contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Collectively, these contributions establish LEMUR 2 as a public research resource. It provides a large-scale architectural corpus, a set of generative methodologies, and multi-platform deployment benchmarks that support research in automated AI design. The empirical value of the LEMUR dataset, as demonstrated within the NNGPT project, is proved by its successful application to LLM-guided generative synthesis, exploration of complex architectural structures, and improved generalization across multiple computer vision tasks [ 25 , 13 , 33 , 45 , 21 , 48 , 47 , 17 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Methodology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We extend the LEMUR [ 15 ] corpus as a modular generation and integration framework built on the original experiment manager for task execution and result persistence. Corpus access and downstream processing use the data interface, which exposes stored records as a fixed-schema pandas DataFrame for uniform filtering and aggregation across tasks, datasets, metrics, architectures, epochs, accuracy, runtime, parameter counts, and transformation identifiers without direct database queries. On this layer, self-contained extension modules generate candidate model definitions or transformation programs with metadata and submit them to the standard LEMUR pipeline for evaluation. These modules combine LLM-based generation from natural-language guidance with programmatic generators that produce valid variants faster and at lower cost, enabling scalable, heterogeneous expansion under a shared reporting format.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Architectural diversity in LEMUR 2 comes from combining heterogeneous generators and sources, so the corpus is not centered on a single backbone family. For generators that operate by mutating a seed architecture, we instantiate them on a lightweight reference template supported by the generator to enable high-throughput evaluation and controlled attribution of operator effects, while still producing many structurally distinct variants.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Task Extensions\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.1 Image Captioning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"NN-Caption is an automated image-captioning pipeline built on LEMUR in which an LLM, iteratively prompted with a baseline ResNet–LSTM captioner and code templates for the LEMUR Net interface ( __init__ , train_setup , learn , forward ), generates PyTorch encoder–decoder architectures that couple a convolutional image encoder with sequence decoders (LSTM, GRU [ 5 ] , Transformer). Generated code is normalized, validated via Python’s abstract syntax tree, integrated into LEMUR, compiled, and trained on MS COCO [ 31 ] , with caption quality evaluated using BLEU-4 [ 35 ] ; in total, NN-Caption contributes 357 captioning architectures to LEMUR 2.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.2 Text-to-Image\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LEMUR includes a text-to-image pipeline spanning three generative families: a diffusion model (UNet-D) [ 41 , 18 , 42 ] , a GAN [ 52 ] , and a CVAE-GAN [ 23 ] , all conditioned on natural-language text within a unified training and evaluation framework. For diffusion, a pre-trained CLIP text encoder [ 39 ] provides embeddings that are fused with time-step embeddings and injected into a multi-scale U-Net encoder–decoder with residual and attention blocks, optimized with a denoising diffusion objective. The GAN uses a generator that fuses text representations with a convolutional upsampling stack and a discriminator operating on joint image–text pairs, whereas the CVAE-GAN employs a conditional VAE whose decoder serves as the generator alongside an adversarial discriminator that promotes sharp outputs. All models are trained on paired text–image data with architecture-specific losses (diffusion, adversarial, or variational plus adversarial), and performance is quantified via CLIP-based text–image similarity between generated images and their prompts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.3 Text-to-Text\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LEMUR is extended to natural language generation via a text-to-text pipeline for training and evaluating recurrent language models with standardized metrics. A lightweight WikiText data loader tokenizes text into integer sequences and forms batched streams for truncated backpropagation through time [ 49 ] , with context length, vocabulary size, and batching strategy configured via a shared dictionary. Two recurrent baselines are instantiated: a stacked Elman RNN [ 10 ] with learned token embeddings, tanh activations, and a linear next-token decoder, and a stacked LSTM [ 19 ] with input, forget, and output gates that improve stability and long-range dependency modeling. Evaluation uses perplexity and BLEU, with all LEMUR metrics normalized to [ 0 , 1 ] [0,1] ; BLEU is computed at corpus level with smoothed n n -gram precision on the validation set.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.4 Mixture-of-Experts (MoE)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mixture-of-Experts architectures are integrated into LEMUR to study expert composition and routing strategies on CIFAR-10 [ 27 ] . Eight variants are constructed in three stages: (i) homogeneous MoEs based on a single backbone type, (ii) AlexNet-based MoEs, and (iii) a heterogeneous MoE combining multiple backbones.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Homogeneous configurations use identical expert architectures and a sparse routing mechanism. For each input, a gating network selects the two experts with the highest scores (top-2 routing), and only these experts are evaluated. Heterogeneous configurations combine four different backbone architectures—AlexNet [ 26 ] , AirNet [ 4 ] , DenseNet [ 20 ] , and BagNet [ 3 ] —and employ a soft routing strategy in which the gating network produces a weight for each expert. The final prediction is obtained as a weighted aggregation of the experts’ outputs. All MoE variants are trained and evaluated within the LEMUR framework using the standard CIFAR-10 training protocol.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Achieving Diversity\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.1 AST-Based Mutations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An AST-based mutation mechanism [ 47 ] generates structurally consistent architectural variants by modifying the source code of existing models while operating on the channel dimensions of convolutional and linear layers and preserving tensor-shape compatibility. A symbolic tracer built on torch.fx [ 40 ] constructs a computation graph whose nodes represent layers with explicit dataflow edges, and maintains a source map from each module instantiation to its file location. Given a target layer, a planning component uses this graph to identify all dependent consumer layers and infer required input/output channel adjustments, including in architectures with residual connections. The mutation plan is applied by parsing the implementation into an AST, locating the corresponding constructor calls via the recorded coordinates, and updating channel-related arguments; the modified AST is then rendered back to source code, re-imported, and each mutant is validated via instantiation and forward/backward passes on dummy inputs. This procedure yields approximately 1000 additional architectures in the corpus.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.2 Reinforcement Learning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The LEMUR dataset is processed by programmatically parsing each network’s source code and identifying layer-instantiation statements (e.g., nn.Conv2d , nn.Linear ) within the __init__ method. These code blocks are removed and replaced with placeholder tokens, while preserving the surrounding class structure, method signatures, and forward pass logic. This produces a dataset of masked model skeletons paired with their complete source code as ground truth for training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The reinforcement learning pipeline then iteratively refines architecture generation by completing these masked skeletons. The LLM acts as a policy, generating code completions (actions) given the masked skeleton (state). Each generated model undergoes validation checks for compilation and tensor shape consistency via a dummy forward pass. Successful models are trained for one mini-epoch on CIFAR-10 to measure accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A composite reward is calculated: − 1.0 -1.0 for validation failure, + 0.2 +0.2 for each passed stage (compilation, forward pass, training), and + 1.0 × Δ a c c u r a c y +1.0\\\\times\\\\Delta accuracy for improvement over baseline. This reward updates the LLM’s policy using Group Relative Policy Optimization (GRPO) [ 44 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.3 Genetic Algorithm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A genetic algorithm explores AlexNet-style architectures on CIFAR-10 [ 27 ] using a parameterized representation that includes architectural and training hyperparameters. Populations of candidate networks are evolved over generations via crossover and mutation on these parameter vectors; fitness is approximated by training each model for a small number of epochs and using validation accuracy for selection, with a checksum over the architecture description used to discard duplicates and maintain diversity. Two search spaces are considered: one restricted to hyperparameters (e.g., filter counts, kernel sizes, learning rate, dropout), and one that additionally includes block-level structural choices (pooling type, activation function, batch normalization). Across both spaces, the algorithm yields approximately 2,000 distinct AlexNet-type architectures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.4 Fractal-Inspired Computational Architectures\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LEMUR includes a FractalNet-style [ 28 ] generator [ 33 ] that constructs self-similar, recursively defined multi-column networks whose topology is governed by fractal depth N N (recursion level) and column width (number of parallel pathways), enabling balanced growth in depth and width. The pipeline comprises (i) configuration generation, which systematically samples architectural blueprints via permutations of convolution, normalization, activation, and dropout layers; (ii) template-based model instantiation, which programmatically realizes the specified recursive, multi-column PyTorch architectures; and (iii) automated evaluation, which executes a standardized training and logging protocol. To scale to over 1,200 unique architectures, training employs Automatic Mixed Precision (AMP) to accelerate computation with half-precision arithmetic and gradient checkpointing to reduce GPU memory via recomputation of intermediate activations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.5 Retrieval-Augmented Generation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The NN-RAG component [ 12 , 13 ] constructs a library of reusable PyTorch modules by mining existing codebases. It scans repositories for subclasses of torch.nn.Module that implement a forward() method and extracts them as self-contained units that preserve the original imports and semantics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Source files are parsed using LibCST [ 30 ] , which maintains syntactic structure, formatting, and comments. A scope-sensitive dependency resolver computes, for each candidate module, the minimal transitive closure of required definitions (such as auxiliary classes, functions, and constants) without executing the code. The resulting set of definitions is topologically ordered to ensure definition-before-use and is written out as an isolated module. Each extracted module passes through a multi-stage validation pipeline comprising abstract syntax tree parsing, bytecode compilation, and sandboxed execution to detect syntax and import-time errors. Valid modules are then integrated into LEMUR as reusable building blocks, with associated metadata linking them to their source repositories.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.6 Few-Shot Architecture Prompting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To improve the stability of LLM-based architecture generation, we employ few-shot prompting and deduplication. Prompts include a small set of high-performing LEMUR architectures as exemplars, together with a description of the target dataset, and the LLM is instructed to produce a new architecture consistent with the LEMUR interface; the number of exemplars is treated as a tunable parameter. Each generated architecture is normalized into a canonical form and assigned a whitespace-stripped MD5 hash, which serves as a database key for fast detection and removal of exact duplicates prior to training. Non-duplicate architectures are trained under the standard LEMUR protocol and their performance recorded. For statistical analysis, evaluation metrics are aggregated in a dataset-balanced manner by first computing performance per dataset and then averaging across datasets to avoid bias from uneven task coverage.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.7 Data Transformations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In line with prior work [ 1 ] , and with the objective of diversifying data augmentation within the corpus using the implementation described in [ 45 ] , transformation pipelines are generated and evaluated on the CIFAR-10 dataset. A fixed ResNet architecture is trained for a single epoch (batch size 64, learning rate 0.01, momentum 0.9, dropout 0.2), and validation accuracy is recorded. The first approach is LLM-based: prompted with the desired interface and examples of Torchvision transforms, the model proposes augmentation functions as Python code, which are syntactically validated before integration. The second approach is combinatorial: from a predefined set of Torchvision transforms, all pipelines with one, two, or three variable transforms are enumerated and extended by a fixed suffix ( Resize – ToTensor – Normalization ); variable transforms receive randomly sampled parameters. This yields 6,000 unique augmentation pipelines, each evaluated under the same training protocol.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Cross Domain Deployment and Testing\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.3.1 NN-VR\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The VR-Ready Neural Network Verifier (NN-VR) provides automated validation and deployment of pre-trained LEMUR models in Unity-based virtual reality environments. Its architecture comprises (i) a Neural Network Parser that ingests ONNX exports and associated metadata from LEMUR and populates a Unity project configured with the Barracuda inference engine; (ii) a Compatibility Verifier that imports ONNX models, checks operator coverage and shader support, and profiles GPU memory under VR-relevant settings; and (iii) an Automated Porting System that converts and optionally optimizes models, configures inference scenes, and records diagnostic logs. Performance is evaluated against VR-specific criteria: inference latency relative to an 11 ms-per-frame budget (90 Hz), additional memory overhead compared to standalone inference, and numerical consistency between Unity/Barracuda outputs and reference PyTorch predictions within a tight tolerance. The pipeline is designed for scalable application to large collections of models exported from LEMUR.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.3.2 NN-Lite\"\n },\n {\n \"type\": \"para\",\n \"text\": \"NN-Lite [ 6 ] is an automated pipeline for deploying and benchmarking PyTorch models on the Android platform. It provides an end-to-end workflow designed to evaluate models from the dataset, managing the process from model conversion to final reporting without manual intervention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pipeline executes a four-stage process for each model. First, PyTorch models are automatically converted to the TensorFlow Lite (TFLite) format. This stage includes a custom wrapper to resolve the NCHW (PyTorch) to NHWC (TFLite) tensor layout disparity, ensuring compatibility and efficient memory usage. Second, the system manages the lifecycle of an Android Virtual Device (AVD), including boot-up and state monitoring. Third, a lightweight Android application is deployed to the AVD to execute model inference and collect latency metrics. Fourth, the pipeline retrieves these benchmark results and combines them with device analytics (e.g., memory, CPU architecture) into a structured JSON report.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The system is designed for large-scale, continuous operation, with built-in state management and failure recovery mechanisms. In a documented 48-hour session, the pipeline processed over 7,500 models from the LEMUR dataset. The final stage of the process is statistical consolidation, where performance metrics like task-level accuracy are aggregated with on-device latency data. This integration allows for a unified analysis of a model’s software performance and its on-device execution characteristics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All computer vision experiments are run in the AI Linux Docker image 2 2 2 AI Linux: https://hub.docker.com/r/abrainone/ai-linux on NVIDIA GeForce RTX 3090/4090 GPUs with 24 GB memory, using a Kubernetes cluster and dedicated workstations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We performed repeated evaluations under multiple training configurations for every supported task and its associated datasets. Each evaluation run corresponds to a distinct hyperparameter setting, including batch size, number of training epochs, learning rate, momentum, dropout rate, and the selected data transformation pipeline. Every run is logged together with its accuracy and runtime, which enables systematic comparison across configurations and supports analysis of model performance variability under the same task protocol.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluations use bounded training schedules to keep compute comparable while sweeping many architectures and hyperparameter settings. Metrics generally improve with additional optimization and data exposure, but convergence rates vary across configurations, so reported scores reflect compute-budgeted performance rather than uniformly converged endpoints. Accordingly, comparisons to fully trained baselines should use matched training horizons, and the results primarily quantify how efficiently each method converts compute into task performance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 summarizes the quantitative performance of the LEMUR 2 generative systems and deployment pipelines under bounded training budgets. Across the evaluated settings, optimization-guided generators produce the strongest and most consistent image-classification results, while prompt-based and programmatic mutation methods exhibit higher variance in both quality and reliability. Best-per-run denotes the highest accuracy attained across all training epochs for a fixed hyperparameter configuration, allowing comparison of peak observed performance rather than final-epoch values as seen in 10 , 7 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overall, these generative mechanisms produce models that are competitive with existing manually crafted network baselines. Within groups (Fig. 9 ), evolutionary models attain the highest median accuracy, indicating a dense cluster of strong architectures. LLM-generated few-shot models (“alt-”) occasionally match the best groups at the top but have lower median accuracy and higher variance, while programmatic AST mutations (“ast-”) yield the lowest median accuracy, consistent with the sensitivity of local channel edits to the choice of input model relative to global optimization strategies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This behavior can be explained by considering the effective size of the search space. Optimization-based generators converge faster than pure random search because the objective function provides a direction that quickly focuses the search on promising regions, especially when the coverage is moderate and the landscape is relatively smooth. As the problem becomes more complex and the search space grows, however, the landscape becomes increasingly rugged, making it harder for any method to consistently find high-quality solutions. The impact of architectural configuration, such as channel widths in the AST channel-mutated AlexNet variants show different convergence properties when compared with the original AlexNet, despite being a relatively small modification, underscoring how sensitive convergence can be to seemingly minor configurational change.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Task Extensions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Image Captioning Increasing prompt complexity (5–10 snippets) reduced runnable generations from 80% to 50%. The LLM explored diverse architectures (e.g., ConvNeXt, EfficientNet, Transformers); the best generated variant (ResNet-50 + Transformer) yielded a score of BLEU-4 = 0.317 at 50 epochs. In the longer training schedule, it can confidently compete with the baseline, ( Fig. 7 ) approaching to the score of 0.3246. This highlights the trade-off between accuracy and architectural diversity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Text-to-Image Methods progressed from a UNet diffusion model (CLIP 0.17–0.24) and LSTM-GAN ( ≈ \\\\approx 0.214) to a complex CVAE-GAN. The final CVAE-GAN, utilizing a CLIP encoder, PatchGAN discriminator, and adversarial losses, achieved the peak CLIP score of 0.2751, prioritizing semantic alignment over photorealistic textures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mixture-of-Experts Evaluations on CIFAR-10 showed a tuned homogeneous MoE achieving the highest accuracy (93.9%). A heterogeneous variant combining AlexNet, AirNet, DenseNet, and BagNet experts reached 93.13%, successfully outperforming all constituent backbones trained individually.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Generative Systems\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Genetic Algorithm A hyperparameter-restricted search yielded 62.76% test accuracy (11×11 kernel), whereas evolving block-type structures (pooling, activation, BN) improved accuracy to 80.04%. The champion model utilized batch normalization, 3×3 convolutions, and mixed pooling, demonstrating the value of block-level flexibility.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"NN-RAG From 1,289 candidate blocks sourced from timm , torchvision , and transformers , 941 (73.0%) passed execution checks. The resulting library, tagged “rag-” in LEMUR 2, includes executable attention, convolutional, and normalization modules.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Few-Shot Prompting Results were non-monotonic: n = 3 n=3 exemplars achieved the highest balanced mean accuracy (53.1%) considering all datasets and improved CIFAR-100 performance by 11.6pp ( p = 0.001 p=0.001 ). Conversely, n = 6 n=6 caused a 99.8% failure rate due to context overflow.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data Transformation The best LLM-generated pipeline achieved 57.28% validation accuracy. A combinatorial search of 6,000 pipelines identified a superior configuration using a single RandomPosterize transform, reaching 61.24% accuracy and outperforming the generative approach.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This system was developed to produce a large and architecturally diverse collection of neural networks with extensive performance records across multiple tasks and hardware platforms, resulting in over 14,000 distinct architectures and more than 750,000 structured training records documenting model architectures, training configurations, and task-specific metrics such as accuracy. For edge devices and cross-domain coverage, NN-Lite and NN-VR automate model deployment and benchmarking while recording on-device characteristics including inference latency and memory usage. By mobilizing data at scale, the framework reveals patterns that indicate which network configurations and structures are effective, supporting automated development of next-generation architectures [ 11 ] . By formalizing dependency-closed neural primitives, it also establishes a scalable methodology for “Neural Architecture Mining,” transforming disparate source code into a standardized, executable substrate for automated ML workflows. Due to space constraints, further implementation details, ablation studies, and extended results are deferred to the supplementary material.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Acknowledgments. This work was partially supported by the Alexander von Humboldt Foundation.\"\n }\n]","isConversational":false}}},{"rowIdx":497,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06833"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Generative Diffusion Models of Stochastic Graph Signals"},"authors":{"kind":"list like","value":["Yiğit Berkay Uslu","Samar Hadou","Sergio Rozada","Shirin Saeedi Bidokhti","Alejandro Ribeiro"],"string":"[\n \"Yiğit Berkay Uslu\",\n \"Samar Hadou\",\n \"Sergio Rozada\",\n \"Shirin Saeedi Bidokhti\",\n \"Alejandro Ribeiro\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06833v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Signals defined over irregular, graph-structured domains are pervasive, spanning application areas such as recommender systems [ 41 , 24 , 23 ] , wireless communications [ 7 , 31 , 46 , 42 , 4 , 39 ] , and forecasting in financial markets [ 3 , 30 , 35 ] . In many such settings, these signals are realizations from unknown probability distributions rather than deterministic quantities [ 25 ] . This paper proposes a generative modeling framework that learns to sample from such graph signal distributions.Generative modeling has proved effective for synthesizing graph topologies themselves [ 16 , 40 ] , with molecular generation serving as a canonical example [ 14 ] . We instead take the graph structure as given and learn to generate node-defined signals."},{"type":"para","text":"A wide array of generative paradigms exists for learning to sample from unknown distributions [ 18 , 10 , 26 ] , such as variational autoencoders, generative adversarial networks, and normalizing flows. Among them, denoising diffusion models and their score- and flow-based variants stand out for their sample quality, training stability, and principled conditional inference in high dimensions [ 13 , 32 , 33 , 22 ] . While these models differ in what the parametric model approximates and in the sampling dynamics, they share a common structure. A fixed forward process gradually corrupts the data into a tractable reference prior, and a learned reverse process denoises samples from that prior back toward the data distribution. Graph signals have recently become a target for generative, particularly diffusion, modeling in domains such as the forecasting of traffic and financial series [ 38 , 43 , 5 , 2 , 28 ] , and wireless optimization [ 6 , 21 , 36 , 11 , 12 , 37 , 20 ] . However, these efforts have mostly been application-specific, with task-tailored denoiser architectures. Their formulations also often remain deterministic, seeking solution distributions whose mass concentrates near an optimal deterministic solution rather than drawing diversified samples from the conditional law."},{"type":"para","text":"In this paper, we propose a general methodological framework for graph signal generation. At its core, this requires designing a diffusion model whose denoising network encodes inductive biases tailored to the data domain. For images, the standard denoising backbone is the U-Net with a symmetric encoder–decoder structure and multi-resolution hierarchy [ 27 , 15 ] . The U-Net is built on convolutional neural networks (CNNs), whose layers apply translation-equivariant filters on the regular pixel grid. The natural counterpart for graph-structured data is the graph neural network (GNN), whose layers apply graph-shift-equivariant filters, thereby extending convolutional processing to irregular domains [ 8 ] . Replacing CNNs with GNN layers does not, on its own, suffice to adapt the U-Net design to the denoising of graph signals. The main obstacle is the pooling stage. In the image setting, the pixel grid provides a regular pooling lattice that persists after each down-sampling step, thus, feature extraction and spatial reduction compose seamlessly. Graph data lacks such a canonical lattice. A common workaround coarsens the graph at each resolution through a learned pooling operator, as in differentiable cluster-assignment pooling and top- k k node-scoring schemes [ 44 , 9 ] . These methods are mainly designed for graph-level tasks, such as classification. Under arbitrary coarsening, however, the pooled signal loses a clear interpretation as a convolutional signal on the original graph."},{"type":"para","text":"We take a different route in this paper. We formalize down- and up-sampling, equivalently graph or node (un)pooling, as a learned node (de)selection operation, expressed through nested selection matrices [ 8 ] . These matrices are trained end-to-end with the denoising network, so each resolution retains the nodes most informative for the denoising task at hand. The selection operator and its transpose play two complementary roles. First, the nested hierarchy of selections instantiates the multi-resolution encoder–decoder cascade. Each encoder selection picks a smaller active node subset that defines the next coarser resolution, and the matching transpose in the decoder lifts coarse signals back to the finer one. Second, at a fixed resolution, the same operator pair keeps GNN filtering interpretable as a convolution on the original graph. The transpose zero-pads a node-reduced signal to the full vertex set, the GNN layer filters there, and the selection returns the result to the reduced domain. Unlike the coarsening schemes above, filtering therefore stays convolutional on the original graph at every resolution. We further introduce a stride parameter that sets the hop distance of each graph shift. This mirrors how dilated convolutions in CNNs replace fixed pooling rules, folding down-sampling and receptive-field growth into the convolution itself [ 34 , 45 ] . On the active nodes, a larger stride enlarges the aggregation neighborhoods and prevents the graph filters from drawing mostly from zero-padded positions under heavy down-sampling. This construction avoids explicit graph coarsening and lets convolutional filtering and pooling compose across arbitrary resolutions."},{"type":"para","text":"Building on this mechanism, we propose the U-Graph Neural Network (U-GNN) architecture as a graph-domain adaptation of the classical U-Net. It comprises a U-shaped encoder–decoder pipeline with skip connections, built from GNN layers and the pooling and unpooling operators above. We then cast graph signal generation as a conditional denoising diffusion process and parametrize its reverse process with a U-GNN. The generations are thereby conditioned directly on the graph topology and on node-feature side information. Although we focus on diffusion denoising, the U-GNN is a general-purpose architecture for learning over graph signals. It inherits the permutation equivariance, stability to perturbations, and cross-topology transferability of GNNs [ 29 , 31 ] ."},{"type":"para","text":"We demonstrate the framework and the U-GNN architecture on two generative tasks. The first is stock price forecasting over a correlation graph of financial indicators. Given recent market history, we generate future price trajectories as graph signals, capturing the market uncertainty and rare tail events that point forecasts and statistical baselines often miss. The second is resource allocation in wireless networks under ergodic quality-of-service (QoS) requirements, where optimal policies are inherently probabilistic and are realized through time-sharing. Treating solution samples from an expert primal–dual algorithm as graph signals on channel (interference) graphs, we learn to sample from the expert allocation distribution given channel and user-state information."},{"type":"para","text":"A brief summary of our contributions is as follows."},{"type":"para","text":"We develop a graph signal generative modeling approach that combines denoising diffusion models with GNNs to sample stochastic graph signals."},{"type":"para","text":"We parametrize the reverse diffusion process with a novel U-GNN architecture, whose building blocks are GNN layers with stride and zero-padded pooling, that generalizes image-convolutional U-Nets to graph-structured signals."},{"type":"para","text":"We apply our approach to stock price forecasting and optimal wireless resource allocation, with extensive numerical results in both domains."},{"type":"para","text":"The rest of the paper is organized as follows. Section II presents the conditional graph signal generative diffusion framework. Section III introduces GNNs with built-in stride and pooling. Section IV assembles the U-GNN architecture. Section V validates the method on wireless resource allocation and financial forecasting. Section VI concludes the paper."},{"type":"heading","level":1,"text":"II Graph Signal Generative Diffusion Models"},{"type":"para","text":"Consider a weighted graph 𝒢 = ( 𝒱 , ℰ , 𝒲 ) {\\mathcal{G}}=({\\mathcal{V}},{\\mathcal{E}},{\\mathcal{W}}) on N N nodes, with node set 𝒱 = { 1 , … , N } {\\mathcal{V}}=\\{1,\\ldots,N\\} , edge set ℰ ⊆ 𝒱 × 𝒱 {\\mathcal{E}}\\subseteq{\\mathcal{V}}\\times{\\mathcal{V}} , and weight map 𝒲 : ℰ → ℝ {\\mathcal{W}}:{\\mathcal{E}}\\to{\\mathbb{R}} . A graph signal assigns features to the nodes and is collected in a matrix 𝐗 = [ 𝐱 1 , … , 𝐱 F ] ∈ ℝ N × F {\\mathbf{X}}=[{\\mathbf{x}}^{1},\\ldots,{\\mathbf{x}}^{F}]\\in{\\mathbb{R}}^{N\\times F} , whose f f th column 𝐱 f ∈ ℝ N {\\mathbf{x}}^{f}\\in{\\mathbb{R}}^{N} gathers the f f th feature across all nodes. We endow 𝒢 {\\mathcal{G}} with a graph shift operator (GSO) 𝐒 ∈ ℝ N × N {\\mathbf{S}}\\in{\\mathbb{R}}^{N\\times N} , a sparse matrix with [ 𝐒 ] m , n = 0 [{\\mathbf{S}}]_{m,n}=0 whenever m ≠ n m\\neq n and ( m , n ) ∉ ℰ (m,n)\\notin{\\mathcal{E}} . We note that this GSO definition allows the diagonal entries to be chosen freely. The GSO encodes the graph topology, and successive products (shifts) 𝐒𝐗 {\\mathbf{S}}{\\mathbf{X}} propagate signal values along the edges. Common GSO choices are the (weighted) adjacency matrix 𝐀 {\\mathbf{A}} and the graph Laplacian 𝐋 = diag ( 𝐀𝟏 N ) − 𝐀 {\\mathbf{L}}=\\mathrm{diag}({\\mathbf{A}}{\\mathbf{1}}_{N})-{\\mathbf{A}} ."},{"type":"para","text":"We address the conditional generation of graph signals. The target is a graph signal 𝐱 0 ∈ ℝ N {\\mathbf{x}}_{0}\\in{\\mathbb{R}}^{N} on a known graph 𝒢 {\\mathcal{G}} . The conditioning collects the topology, through the GSO 𝐒 {\\mathbf{S}} , and auxiliary node states 𝐮 ∈ ℝ N × U {\\mathbf{u}}\\in{\\mathbb{R}}^{N\\times U} (side information) into the variable 𝝃 = ( 𝐒 , 𝐮 ) \\boldsymbol{\\xi}=({\\mathbf{S}},{\\mathbf{u}}) . Let q 𝝃 q_{\\boldsymbol{\\xi}} denote the distribution of the conditioning and q 𝐱 | 𝝃 ( ⋅ | 𝝃 ) q_{{\\mathbf{x}}|\\boldsymbol{\\xi}}(\\cdot\\,|\\,\\boldsymbol{\\xi}) the law of the target given 𝝃 \\boldsymbol{\\xi} . These define the joint data distribution q data = q 𝐱 | 𝝃 q 𝝃 q_{\\mathrm{data}}=q_{{\\mathbf{x}}|\\boldsymbol{\\xi}}\\,q_{\\boldsymbol{\\xi}} , from which we draw pairs ( 𝐱 0 , 𝝃 ) ({\\mathbf{x}}_{0},\\boldsymbol{\\xi}) . We seek a parametric generator p 𝜽 ⋆ ( ⋅ | 𝝃 ) = p ( ⋅ | 𝝃 ; 𝜽 ∗ ) p_{\\boldsymbol{\\theta}^{\\star}}(\\cdot\\,|\\,\\boldsymbol{\\xi})=p(\\cdot\\,|\\,\\boldsymbol{\\xi};\\boldsymbol{\\theta}^{*}) whose conditional law matches q 𝐱 | 𝝃 ( ⋅ | 𝝃 ) q_{{\\mathbf{x}}|\\boldsymbol{\\xi}}(\\cdot\\,|\\,\\boldsymbol{\\xi}) for q 𝝃 q_{\\boldsymbol{\\xi}} -almost every 𝝃 \\boldsymbol{\\xi} , i.e.,"},{"type":"para","text":"The conditional laws are unknown in closed form. We assume samples ( 𝐱 0 , 𝝃 ) ∼ q data ({\\mathbf{x}}_{0},\\boldsymbol{\\xi})\\sim q_{\\mathrm{data}} are available. Therefore, we realize the optimal generator p 𝜽 ⋆ p_{\\boldsymbol{\\theta}^{\\star}} as a conditional denoising diffusion model, trained on a dataset { ( 𝐱 0 ( i ) , 𝝃 ( i ) ) } i = 1 | 𝒟 | ∼ q data \\{({\\mathbf{x}}^{(i)}_{0},\\boldsymbol{\\xi}^{(i)})\\}_{i=1}^{|{\\mathcal{D}}|}\\sim q_{\\mathrm{data}} ."},{"type":"heading","level":2,"text":"II-A Denoising Diffusion Models for Graph Signals"},{"type":"para","text":"A denoising diffusion model pairs a fixed forward process q ( 𝐱 1 : K | 𝐱 0 ) q({\\mathbf{x}}_{1:K}\\,|\\,{\\mathbf{x}}_{0}) , which noises clean targets toward a tractable reference prior, with a learned reverse process p 𝜽 ( 𝐱 k − 1 | 𝐱 k ) p_{\\boldsymbol{\\theta}}({\\mathbf{x}}_{k-1}\\,|\\,{\\mathbf{x}}_{k}) , which denoises prior samples back toward the data. Conditional generation requires no additional machinery. We condition the reverse process on 𝝃 \\boldsymbol{\\xi} as p 𝜽 ( 𝐱 k − 1 | 𝐱 k ; 𝝃 ) p_{\\boldsymbol{\\theta}}({\\mathbf{x}}_{k-1}\\,|\\,{\\mathbf{x}}_{k};\\,\\boldsymbol{\\xi}) , leaving the forward process and the training objective unchanged. Fig. 1 illustrates this construction, which we detail next."},{"type":"para","text":"Forward noising process. The forward process is a Markov chain that adds noise to the target 𝐱 0 {\\mathbf{x}}_{0} over K K steps. Given ( 𝐱 0 , 𝝃 ) ∼ q 0 = q data ({\\mathbf{x}}_{0},\\boldsymbol{\\xi})\\sim q_{0}=q_{\\mathrm{data}} with 𝝃 \\boldsymbol{\\xi} held fixed, we corrupt 𝐱 0 {\\mathbf{x}}_{0} with white noise on an increasing schedule { β k } k = 1 K \\{\\beta_{k}\\}_{k=1}^{K} (e.g., linear), producing iterates { 𝐱 k } k = 1 K \\{{\\mathbf{x}}_{k}\\}_{k=1}^{K} through the Gaussian transitions,"},{"type":"para","text":"Since the drift is linear and the noise increments are independent, the conditionals remain Gaussian at each step k k , i.e.,"},{"type":"para","text":"with α k ≔ 1 − β k \\alpha_{k}\\coloneqq 1-\\beta_{k} and α ¯ k ≔ ∏ m = 1 k α m \\bar{\\alpha}_{k}\\coloneqq\\prod_{m=1}^{k}\\alpha_{m} for k = 1 , … , K k=1,\\ldots,K . A standard reparametrization of ( 3 ) expresses 𝐱 k {\\mathbf{x}}_{k} as a deterministic map of ( 𝐱 0 , k ) ({\\mathbf{x}}_{0},k) plus independent standard noise [ 13 ] ,"},{"type":"para","text":"and it permits drawing any iterate 𝐱 k ∼ q k ( ⋅ | 𝐱 0 , 𝝃 ) {\\mathbf{x}}_{k}\\sim q_{k}(\\cdot\\,|\\,{\\mathbf{x}}_{0},\\boldsymbol{\\xi}) directly, without simulating the chain in ( 2 ). For a suitable schedule and a large enough K K , the process drives 𝐱 0 ∼ q 0 ( ⋅ | 𝝃 ) {\\mathbf{x}}_{0}\\sim q_{0}(\\cdot\\,|\\,\\boldsymbol{\\xi}) toward a Gaussian prior, with 𝐱 K ∼ q K ≈ 𝒩 ( 𝟎 , 𝐈 ) {\\mathbf{x}}_{K}\\sim q_{K}\\approx\\mathcal{N}({\\mathbf{0}},{\\mathbf{I}}) for every 𝝃 \\boldsymbol{\\xi} ."},{"type":"para","text":"Backward (reverse) denoising process. Denoising diffusion fixes p K ( ⋅ | 𝝃 ) = q K ≈ 𝒩 ( 𝟎 , 𝐈 ) p_{K}(\\cdot\\,|\\,\\boldsymbol{\\xi})=q_{K}\\approx\\mathcal{N}({\\mathbf{0}},{\\mathbf{I}}) and learns a reverse Markov chain whose kernels p 𝜽 ( 𝐱 k − 1 | 𝐱 k ; 𝝃 ) p_{\\boldsymbol{\\theta}}({\\mathbf{x}}_{k-1}\\,|\\,{\\mathbf{x}}_{k};\\boldsymbol{\\xi}) approximate the time reversal of the forward chain. Consistent with the continuous-time limit, we model each reverse step as Gaussian transitions,"},{"type":"para","text":"As is standard, we fix the covariance 𝚺 𝜽 = σ k 2 𝐈 \\boldsymbol{\\Sigma}_{\\boldsymbol{\\theta}}\\!=\\!\\sigma_{k}^{2}{\\mathbf{I}} for a predefined σ k \\sigma_{k} tied to the schedule β k \\beta_{k} and learn only the mean 𝝁 𝜽 \\boldsymbol{\\mu}_{\\boldsymbol{\\theta}} ."},{"type":"para","text":"Training maximizes the evidence lower bound (ELBO) on the log-likelihood log p 𝜽 ( ⋅ | 𝝃 ) \\log p_{\\boldsymbol{\\theta}}(\\cdot{\\,\\big|\\,}\\boldsymbol{\\xi}) . Under the fixed covariance and the reparametrization ( 4 ), this is equivalent to a noise-prediction objective. Omitting its per-step ELBO weights matches, and often improves upon, the weighted objective in practice [ 13 ] . We therefore minimize the following objective,"},{"type":"para","text":"where the expectation is over ( 𝐱 0 , 𝝃 ) ∼ q data ({\\mathbf{x}}_{0},\\boldsymbol{\\xi})\\sim q_{\\mathrm{data}} and diffusion steps k ∼ unif { 1 , … , K } k\\sim\\mathrm{unif}\\{1,\\ldots,K\\} ."},{"type":"para","text":"By the reparametrization ( 4 ), the trained noise predictor ϵ 𝜽 ⋆ ( 𝐱 k , k ; 𝝃 ) \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}^{\\star}}({\\mathbf{x}}_{k},k;\\boldsymbol{\\xi}) yields both a graph signal denoiser and the mean of the reverse kernel in ( 5 ),"},{"type":"para","text":"Thus ϵ 𝜽 ⋆ \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}^{\\star}} and the fixed 𝚺 𝜽 = σ k 2 𝐈 \\boldsymbol{\\Sigma}_{\\boldsymbol{\\theta}}\\!=\\!\\sigma_{k}^{2}{\\mathbf{I}} specify ( 5 ) completely. For sampling, we construct a family of generally non-Markovian reverse processes [ 32 ] that preserve the forward marginals ( 3 ),"},{"type":"para","text":"with 𝐰 ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\mathbf{w}}\\sim\\mathcal{N}({\\mathbf{0}},{\\mathbf{I}}) . A single factor η ∈ [ 0 , 1 ] \\eta\\in[0,1] tunes the noise scale σ k \\sigma_{k} [cf. ( 29 )]. Each step draws 𝐱 k − 1 {\\mathbf{x}}_{k-1} exactly from the reverse kernel ( 5 ) at η = 1 \\eta=1 , and smaller values yield near-deterministic and usually faster sampling."},{"type":"para","text":"At inference, we initialize 𝐱 K ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\mathbf{x}}_{K}\\sim\\mathcal{N}({\\mathbf{0}},{\\mathbf{I}}) and iterate ( 9 ) for k = K , … , 1 k=K,\\ldots,1 to draw samples 𝐱 0 ∼ p 𝜽 ⋆ ( ⋅ | 𝝃 ) {\\mathbf{x}}_{0}\\sim p_{\\boldsymbol{\\theta}^{\\star}}(\\cdot\\,|\\,\\boldsymbol{\\xi}) . These samples are distributed approximately according to the conditional data distribution q 𝐱 | 𝝃 ( ⋅ | 𝝃 ) q_{{\\mathbf{x}}|\\boldsymbol{\\xi}}(\\cdot\\,|\\,\\boldsymbol{\\xi}) for the given conditioning 𝝃 = ( 𝐒 , 𝐮 ) \\boldsymbol{\\xi}=({\\mathbf{S}},{\\mathbf{u}}) , thereby realizing the conditional generative model sought in ( 1 ). To reduce the number of model evaluations, we run an accelerated variant of ( 9 ) (see Appendix A )."},{"type":"para","text":"U-Nets have been instrumental in image generative models. Accordingly, we particularize ϵ 𝜽 \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}} to a U-Graph Neural Network (U-GNN) that tailors U-Nets for graph signal denoising."},{"type":"heading","level":1,"text":"III Pooling & Stride in Graph Neural Networks"},{"type":"para","text":"We first review graph filters, convolutions, and a standard GNN layer. We then propose a GNN module with built-in pooling through selection matrices and stride, which Section IV arranges into the U-GNN encoder–decoder pipeline."},{"type":"para","text":"Graph filters and convolutions. Graph filters are the building blocks of GNNs and process input graph signals through successive graph shifts. A single shift 𝐒𝐱 {\\mathbf{S}}{\\mathbf{x}} mixes each node’s value with those of its one-hop neighbors, and k k repeated shifts yield 𝐒 k 𝐱 {\\mathbf{S}}^{k}{\\mathbf{x}} , which aggregates at each node the signals within its k k -hop neighborhood. A linear shift-invariant graph filter of order K K is the matrix polynomial 𝐡 ( 𝐒 ) ≔ ∑ k = 0 K h k 𝐒 k {\\mathbf{h}}({\\mathbf{S}})\\coloneqq\\sum_{k=0}^{K}h_{k}{\\mathbf{S}}^{k} with taps (coefficients) 𝐡 = [ h 0 , … , h K ] {\\mathbf{h}}=[h_{0},\\ldots,h_{K}] . It processes an input signal 𝐱 {\\mathbf{x}} by graph convolution,"},{"type":"para","text":"and produces an output graph signal 𝐲 ∈ ℝ N {\\mathbf{y}}\\in{\\mathbb{R}}^{N} as a weighted sum of the shifted signals 𝐒 k 𝐱 {\\mathbf{S}}^{k}{\\mathbf{x}} ."},{"type":"para","text":"Graph neural networks. A GNN stacks graph convolutional layers, each composing a graph filter with a pointwise nonlinearity [ 8 ] . The ℓ \\ell th layer of an L L -layer GNN reads"},{"type":"para","text":"for ℓ = 1 , … , L \\ell=1,\\ldots,L . Here, each scalar tap of ( 10 ) becomes a MIMO filter bank 𝚯 ℓ , k \\boldsymbol{\\Theta}_{\\ell,k} that maps the F ℓ − 1 F_{\\ell-1} input features in 𝐗 ℓ − 1 {\\mathbf{X}}_{\\ell-1} to F ℓ F_{\\ell} output features within K K -hop neighborhoods, with learnable coefficients Θ ℓ = { 𝚯 ℓ , k } k = 0 K \\Theta_{\\ell}=\\{\\boldsymbol{\\Theta}_{\\ell,k}\\}_{k=0}^{K} and a pointwise nonlinearity σ ℓ \\sigma_{\\ell} , e.g., a ReLU. Stacking these layers defines the Θ \\Theta -parametrized GNN 𝚽 ( ⋅ , 𝐒 ; Θ ) = 𝚽 L ( ⋅ , 𝐒 ; Θ L ) ∘ ⋯ ∘ 𝚽 1 ( ⋅ , 𝐒 ; Θ 1 ) \\boldsymbol{\\Phi}(\\cdot,{\\mathbf{S}};\\Theta)=\\boldsymbol{\\Phi}_{L}(\\cdot,{\\mathbf{S}};\\Theta_{L})\\circ\\cdots\\circ\\boldsymbol{\\Phi}_{1}(\\cdot,{\\mathbf{S}};\\Theta_{1}) , with trainable parameters Θ = { Θ ℓ } ℓ = 1 L \\Theta=\\{\\Theta_{\\ell}\\}_{\\ell=1}^{L} . It maps an input 𝐗 0 ≔ 𝐗 ∈ ℝ N × F 0 {\\mathbf{X}}_{0}\\coloneqq{\\mathbf{X}}\\in{\\mathbb{R}}^{N\\times F_{0}} to an output 𝐗 L ≕ 𝐘 ∈ ℝ N × F L {\\mathbf{X}}_{L}\\eqqcolon{\\mathbf{Y}}\\in{\\mathbb{R}}^{N\\times F_{L}} ,"},{"type":"para","text":"The GNN in ( 12 ) is general-purpose but not tailored to diffusion-based generation. Since graph convolutions already extract features on arbitrary graphs, adapting the U-Net backbone additionally requires only node (un)pooling, which we introduce next together with a stride on the graph shift."},{"type":"para","text":"Pooling. Consider a signal 𝐗 ∈ ℝ N × F {\\mathbf{X}}\\in{\\mathbb{R}}^{N\\times F} on N N nodes. A sampling (selection) matrix 𝐃 ∈ { 0 , 1 } N ′ × N {\\mathbf{D}}\\in\\{0,1\\}^{N^{\\prime}\\times N} with N ′ < N N^{\\prime}<N retains the rows of 𝐈 N {\\mathbf{I}}_{N} indexed by a sampling set Ω ⊆ 𝒱 \\Omega\\subseteq{\\mathcal{V}} with | Ω | = N ′ |\\Omega|=N^{\\prime} . 1 1 1 We present 𝐃 {\\mathbf{D}} as binary for exposition. We relax it to a soft selection mask during training and binarize it to pick N ′ N^{\\prime} nodes at inference (Appendix C ). We learn Ω \\Omega , and hence 𝐃 {\\mathbf{D}} , end-to-end, so it is input-dependent rather than fixed (Section IV-B ). We call ρ ≥ 1 \\rho\\geq 1 with N ′ = ⌊ N / ρ ⌋ N^{\\prime}=\\lfloor N/\\rho\\rfloor the down-sampling (pooling) factor that sets the resolution reduction. By construction, 𝐃 {\\mathbf{D}} has orthonormal rows, 𝐃𝐃 ⊤ = 𝐈 N ′ {\\mathbf{D}}{\\mathbf{D}}^{\\top}={\\mathbf{I}}_{N^{\\prime}} , and 𝐃 ⊤ 𝐃 {\\mathbf{D}}^{\\top}{\\mathbf{D}} is a diagonal mask with ones on Ω \\Omega . The forward map 𝐙 = 𝐃𝐗 ∈ ℝ N ′ × F {\\mathbf{Z}}={\\mathbf{D}}{\\mathbf{X}}\\in{\\mathbb{R}}^{N^{\\prime}\\times F} down-samples 𝐗 {\\mathbf{X}} . Its adjoint 𝐃 ⊤ 𝐙 {\\mathbf{D}}^{\\top}{\\mathbf{Z}} up-samples back to N N nodes by zero-padding , i.e., it restores the active entries on Ω \\Omega and inserts zeros on the inactive nodes 𝒱 ∖ Ω {\\mathcal{V}}\\setminus\\Omega ."},{"type":"para","text":"A graph convolutional layer with pooling processes a down-sampled signal 𝐙 ℓ − 1 ∈ ℝ N ′ × F ℓ − 1 {\\mathbf{Z}}_{\\ell-1}\\in{\\mathbb{R}}^{N^{\\prime}\\times F_{\\ell-1}} into 𝐙 ℓ ∈ ℝ N ′ × F ℓ {\\mathbf{Z}}_{\\ell}\\in{\\mathbb{R}}^{N^{\\prime}\\times F_{\\ell}} ,"},{"type":"para","text":"We call ( 13 ) a lift–filter–reduce layer. We lift 𝐙 ℓ − 1 {\\mathbf{Z}}_{\\ell-1} to the full vertex set by zero-padding through 𝐃 ⊤ {\\mathbf{D}}^{\\top} , filter on 𝐒 {\\mathbf{S}} , and reduce the result back to the active set through 𝐃 {\\mathbf{D}} . Because the inactive nodes are zero, each active node aggregates only the active values in its neighborhood on 𝐒 {\\mathbf{S}} , which can be few under heavy pooling and motivates the stride introduced next."},{"type":"para","text":"Stride. Modern CNNs, and U-Net variants in particular, often replace fixed pooling rules such as max- or sum-pooling with strided or dilated convolutions for learned down-sampling and receptive-field growth, e.g., [ 15 , 45 ] . A dilated convolution spaces its taps by a fixed step, so each tap reads from a point several positions away rather than from an immediate neighbor. On a graph, the natural spacing is the hop count. Thus, we introduce a stride (dilation) parameter γ ∈ ℤ + \\gamma\\in\\mathbb{Z}^{+} and replace the unit shift 𝐒 {\\mathbf{S}} with the γ \\gamma -hop shift 𝐒 γ {\\mathbf{S}}^{\\gamma} . The order- K K filter then aggregates from the dilated hops { 0 , γ , 2 γ , … , K γ } \\{0,\\gamma,2\\gamma,\\ldots,K\\gamma\\} rather than the consecutive hops { 0 , 1 , … , K } \\{0,1,\\ldots,K\\} . This yields the strided graph convolutional layer with pooling ,"},{"type":"para","text":"which recovers the plain layer ( 11 ) when 𝐃 = 𝐈 N {\\mathbf{D}}={\\mathbf{I}}_{N} ( ρ = 1 \\rho=1 ) and γ = 1 \\gamma=1 . The pooling factor ρ \\rho and stride γ \\gamma serve complementary roles. Through 𝐃 {\\mathbf{D}} , the factor ρ \\rho sets how many nodes survive and hence the sparsity of the zero-padded signal. Through 𝐒 γ {\\mathbf{S}}^{\\gamma} , the stride γ \\gamma sets how far each tap reaches [cf. Remark 1 ]."},{"type":"para","text":"The strided GNN layer ( 14 ) keeps the lift–filter–reduce form of ( 13 ), filtering on 𝐒 γ {\\mathbf{S}}^{\\gamma} instead of 𝐒 {\\mathbf{S}} . The computations remain on the original vertex set, and 𝐒 {\\mathbf{S}} is reused at every resolution without explicitly forming 𝐒 γ {\\mathbf{S}}^{\\gamma} in our implementation. Instead, we filter through γ K \\gamma K successive unit shifts and tap the signal every γ \\gamma hops. Thus, each layer reduces to sparse matrix–vector products, which we realize with the sparse routines of the PyG library. We refer to Appendix B for details."},{"type":"para","text":"Reduced GSOs. The strided layer ( 14 ) also admits an equivalent form that operates directly on the N ′ N^{\\prime} -dimensional signals. Pushing 𝐃 {\\mathbf{D}} and 𝐃 ⊤ {\\mathbf{D}}^{\\top} through the sum defines the k k -hop reduced graph shift operators (GSOs) ,"},{"type":"para","text":"whose one-hop instance 𝐒 ( 1 ) {\\mathbf{S}}^{(1)} is the support of a coarse subgraph on Ω \\Omega . The family { 𝐒 ( k ) } k = 0 K \\{{\\mathbf{S}}^{(k)}\\}_{k=0}^{K} then processes the low-dimensional signals directly,"},{"type":"para","text":"The two forms are equal because row selection commutes with the pointwise nonlinearity, 𝐃 σ ℓ ( 𝐌 ) = σ ℓ ( 𝐃𝐌 ) {\\mathbf{D}}\\,\\sigma_{\\ell}({\\mathbf{M}})=\\sigma_{\\ell}({\\mathbf{D}}{\\mathbf{M}}) , and they differ only in the domain of computation. The reduced form avoids computation on the inactive nodes but is ill-suited to our setting. The reduced GSOs do not compose, i.e., 𝐒 ( m + n ) ≠ 𝐒 ( m ) 𝐒 ( n ) {\\mathbf{S}}^{(m+n)}\\neq{\\mathbf{S}}^{(m)}{\\mathbf{S}}^{(n)} , since 𝐃 ⊤ 𝐃 ≠ 𝐈 N {\\mathbf{D}}^{\\top}{\\mathbf{D}}\\neq{\\mathbf{I}}_{N} in general. Therefore, each 𝐒 ( k ) {\\mathbf{S}}^{(k)} must be formed separately rather than by powering 𝐒 ( 1 ) {\\mathbf{S}}^{(1)} . Moreover, 𝐃 {\\mathbf{D}} is input-dependent, so the family { 𝐒 ( k ) } \\{{\\mathbf{S}}^{(k)}\\} is rebuilt on every forward pass, negating any gains from pre-computing. We therefore adopt the lift–filter–reduce form."},{"type":"para","text":"GNNs with pooling and stride. Stacking L L strided graph convolutional layers with pooling [cf. ( 14 )] defines the ( Θ , 𝐃 ) (\\Theta,{\\mathbf{D}}) -parametrized GNN module 𝚽 ( ⋅ , 𝐒 ; Θ , 𝐃 ) = 𝚽 L ( ⋅ , 𝐒 γ ; Θ L , 𝐃 ) ∘ ⋯ ∘ 𝚽 1 ( ⋅ , 𝐒 γ ; Θ 1 , 𝐃 ) \\boldsymbol{\\Phi}(\\cdot,{\\mathbf{S}};\\Theta,{\\mathbf{D}})=\\boldsymbol{\\Phi}_{L}(\\cdot,{\\mathbf{S}}^{\\gamma};\\Theta_{L},{\\mathbf{D}})\\circ\\cdots\\circ\\boldsymbol{\\Phi}_{1}(\\cdot,{\\mathbf{S}}^{\\gamma};\\Theta_{1},{\\mathbf{D}}) . It maps an input 𝐙 0 ≔ 𝐗 ∈ ℝ N ′ × F 0 {\\mathbf{Z}}_{0}\\coloneqq{\\mathbf{X}}\\in{\\mathbb{R}}^{N^{\\prime}\\times F_{0}} to an output 𝐙 L ≕ 𝐘 ∈ ℝ N ′ × F L {\\mathbf{Z}}_{L}\\eqqcolon{\\mathbf{Y}}\\in{\\mathbb{R}}^{N^{\\prime}\\times F_{L}} ,"},{"type":"para","text":"with parameters Θ = { Θ ℓ } ℓ = 1 L \\Theta=\\{\\Theta_{\\ell}\\}_{\\ell=1}^{L} and 𝐃 ∈ { 0 , 1 } N ′ × N {\\mathbf{D}}\\in\\{0,1\\}^{N^{\\prime}\\times N} . This GNN module is the main processing unit of each block in the U-GNN architecture, which is introduced next."},{"type":"para","text":"In both views, the stride γ \\gamma offsets the sparsity introduced by down-sampling with 𝐃 {\\mathbf{D}} . Under lift–filter–reduce, a stride such as γ = ⌊ ρ ⌋ \\gamma=\\lfloor\\sqrt{\\rho}\\,\\rfloor keeps nodes from aggregating predominantly zeros. Under the reduced GSO, projecting through 𝐃 {\\mathbf{D}} may isolate nodes in the induced subgraph, and 𝐒 γ {\\mathbf{S}}^{\\gamma} with γ > 1 \\gamma>1 restores connectivity. A suitable value depends on the average connectivity and the uniformity of 𝐃 {\\mathbf{D}} ."},{"type":"para","text":"A variant of ( 16 ) instead learns a genuine polynomial filter in the one-hop reduced GSO 𝐒 ( 1 ) = 𝐃 𝐒 γ 𝐃 ⊤ {\\mathbf{S}}^{(1)}={\\mathbf{D}}\\,{\\mathbf{S}}^{\\gamma}\\,{\\mathbf{D}}^{\\top} , the principal submatrix of 𝐒 γ {\\mathbf{S}}^{\\gamma} on Ω \\Omega . This is no longer equivalent to ( 14 ), since sampling and filtering commute only at first order. Powering 𝐒 ( 1 ) {\\mathbf{S}}^{(1)} confines every hop to Ω \\Omega , whereas the family 𝐒 ( k ) {\\mathbf{S}}^{(k)} allows walks to traverse inactive nodes."},{"type":"heading","level":1,"text":"IV U-Graph Neural Networks"},{"type":"para","text":"The GNN in ( 17 ) processes graph signals at a single node resolution fixed by the sampling matrix 𝐃 {\\mathbf{D}} . To build the multi-resolution encoder–decoder hierarchy of a U-Net on top of this module, we first define the resolutions through depth-dependent selection matrices { 𝐂 b } b = 1 B \\{{\\mathbf{C}}_{b}\\}_{b=1}^{B} , where 𝐂 b ∈ { 0 , 1 } N b × N b − 1 {\\mathbf{C}}_{b}\\in\\{0,1\\}^{N_{b}\\times N_{b-1}} selects N b N_{b} nodes from the N b − 1 N_{b-1} nodes active at the preceding depth, with N 0 = N N_{0}=N and N = N 1 ≥ N 2 ≥ ⋯ ≥ N B N=N_{1}\\geq N_{2}\\geq\\cdots\\geq N_{B} . We form the composite (nested) sampling matrices,"},{"type":"para","text":"which relate the depth- b b nodes to the full N N -node domain and induce a nested hierarchy of sampling sets Ω B ⊆ ⋯ ⊆ Ω 1 ⊆ 𝒱 \\Omega_{B}\\subseteq\\cdots\\subseteq\\Omega_{1}\\subseteq{\\mathcal{V}} . We set 𝐃 1 = 𝐂 1 = 𝐈 N {\\mathbf{D}}_{1}={\\mathbf{C}}_{1}={\\mathbf{I}}_{N} by convention, so the first depth operates at full resolution. Here, 𝐂 b {\\mathbf{C}}_{b} and 𝐃 b {\\mathbf{D}}_{b} carry the per-level and cumulative down-sampling factors ρ b ≔ N b − 1 / N b \\rho_{b}\\coloneqq N_{b-1}/N_{b} ( ρ 1 = 1 \\rho_{1}=1 ) and ρ ¯ b ≔ ∏ i = 1 b ρ i = N / N b \\bar{\\rho}_{b}\\coloneqq\\prod_{i=1}^{b}\\rho_{i}=N/N_{b} , respectively."},{"type":"para","text":"We call the resulting architecture a U-Graph Neural Network (U-GNN) . It spans B B resolution levels N = N 1 ≥ ⋯ ≥ N B N=N_{1}\\geq\\cdots\\geq N_{B} , with B − 1 B-1 encoder–decoder block pairs and a bottleneck block at the coarsest resolution N B N_{B} ."},{"type":"para","text":"We present the U-GNN in two stages. We first fix a pure encoder–decoder cascade whose blocks are single encoding, decoding, and bottleneck GNN modules [cf. ( 17 )], written 𝚽 b E \\boldsymbol{\\Phi}^{{\\text{E}}}_{b} , 𝚽 b D \\boldsymbol{\\Phi}^{{\\text{D}}}_{b} , or 𝚽 B B \\boldsymbol{\\Phi}^{{\\text{B}}}_{B} with weights Θ b E \\Theta^{{\\text{E}}}_{b} , Θ b D \\Theta^{{\\text{D}}}_{b} , and Θ B B \\Theta^{{\\text{B}}}_{B} , respectively, along with the signal flow among them in ( 19 )–( 21 ). These equations include the skip connections, take the selection matrices 𝐂 b {\\mathbf{C}}_{b} as given, and omit the node-state and diffusion-step embeddings. We then complete the blocks with projection (fusion) layers and, on the encoder side, node-selection heads (Sections IV-A and IV-B ). These only enlarge the block interiors and parameter sets, leaving the composition here unchanged. The U-GNN overview in Fig. 2 and the complementary block interface details in Fig. 7 (deferred to Appendix E due to limited space) may be consulted throughout this section."},{"type":"para","text":"The encoder blocks form an encoding path 𝚽 E = 𝚽 B − 1 E ∘ ⋯ ∘ 𝚽 1 E \\boldsymbol{\\Phi}^{{\\text{E}}}=\\boldsymbol{\\Phi}^{{\\text{E}}}_{B-1}\\circ\\cdots\\circ\\boldsymbol{\\Phi}^{{\\text{E}}}_{1} , and the decoder blocks a decoding path 𝚽 D = 𝚽 1 D ∘ ⋯ ∘ 𝚽 B − 1 D \\boldsymbol{\\Phi}^{{\\text{D}}}=\\boldsymbol{\\Phi}^{{\\text{D}}}_{1}\\circ\\cdots\\circ\\boldsymbol{\\Phi}^{{\\text{D}}}_{B-1} . The encoding path extracts graph convolutional features at progressively coarser resolutions, while the decoding path reverses this progression by restoring resolution and reintroducing fine details through skip connections."},{"type":"para","text":"We describe each path in turn, starting from the input features 𝐙 0 {\\mathbf{Z}}_{0} . Each encoder block maps its input to an encoded signal 𝐙 b ∈ ℝ N b × F b {\\mathbf{Z}}_{b}\\in{\\mathbb{R}}^{N_{b}\\times F_{b}} ,"},{"type":"para","text":"for b = 1 , … , B − 1 b=1,\\ldots,B-1 , where 𝐂 b {\\mathbf{C}}_{b} down-samples the depth- ( b − 1 ) (b{-}1) feature onto the N b N_{b} nodes processed at depth b b . At the top level, 𝐂 1 = 𝐈 N {\\mathbf{C}}_{1}={\\mathbf{I}}_{N} and 𝚽 1 E \\boldsymbol{\\Phi}^{{\\text{E}}}_{1} operates at full resolution N 1 = N N_{1}=N ."},{"type":"para","text":"Each decoder block receives the coarser decoded signal 𝐘 b + 1 ∈ ℝ N b + 1 × F b + 1 {\\mathbf{Y}}_{b+1}\\in{\\mathbb{R}}^{N_{b+1}\\times F_{b+1}} from depth b + 1 b+1 , up-samples it onto the N b N_{b} -node support through 𝐂 b + 1 ⊤ {\\mathbf{C}}^{\\top}_{b+1} , and combines it with the matched encoder feature 𝐙 b {\\mathbf{Z}}_{b} through a skip connection,"},{"type":"para","text":"for b = B − 1 , … , 1 b=B-1,\\ldots,1 , where [ ⋅ ; ⋅ ] [\\,\\cdot\\,;\\,\\cdot\\,] denotes the skip combination. The decoded signal 𝐘 b {\\mathbf{Y}}_{b} then passes to the next decoder block or to the read-out layer at b = 1 b=1 ."},{"type":"para","text":"The two paths connect at depth B B through the bottleneck block, which acts as a B B th encoder block,"},{"type":"para","text":"The bottleneck output 𝐘 B {\\mathbf{Y}}_{B} enters the decoding path ( 20 ) at depth b = B − 1 b=B-1 , and the final decoder output 𝐘 1 ∈ ℝ N × F 0 {\\mathbf{Y}}_{1}\\in{\\mathbb{R}}^{N\\times F_{0}} is mapped by a read-out layer to the model target, i.e., the noise prediction ϵ 𝜽 \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}} here. We note that all blocks share the same 𝐒 {\\mathbf{S}} , and a depth-dependent stride γ b \\gamma_{b} is implicit in the notation [cf. ( 14 )]. Appendix B specifies how γ b \\gamma_{b} is set and how each block evaluates its strided convolutions on the shared 𝐒 {\\mathbf{S}} ."},{"type":"heading","level":2,"text":"IV-A Input Processing and Block-Level Conditioning"},{"type":"para","text":"The U-GNN realizes the denoiser ϵ 𝜽 ( 𝐱 k , k ; 𝝃 ) \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}}({\\mathbf{x}}_{k},k;\\boldsymbol{\\xi}) with conditioning 𝝃 = ( 𝐒 , 𝐮 ) \\boldsymbol{\\xi}=({\\mathbf{S}},{\\mathbf{u}}) . It incorporates the conditioning along two routes. The graph enters as the shared shift operator 𝐒 {\\mathbf{S}} of every GNN module [cf. Section III ], while the node states 𝐮 {\\mathbf{u}} enter as side information, embedded and fused into each block together with an embedding of the diffusion step k k . Note that the step k k is a model input and not part of the conditioning 𝝃 \\boldsymbol{\\xi} . We now detail the embedding pipeline."},{"type":"para","text":"A read-in layer 𝚷 0 \\boldsymbol{\\Pi}_{0} maps the noisy signal 𝐱 k ∈ ℝ N {\\mathbf{x}}_{k}\\in{\\mathbb{R}}^{N} to the initial node embeddings 𝐙 0 ∈ ℝ N × F 0 {\\mathbf{Z}}_{0}\\in{\\mathbb{R}}^{N\\times F_{0}} that enter the first encoder block. The node states 𝐮 ∈ ℝ N × U {\\mathbf{u}}\\in{\\mathbb{R}}^{N\\times U} and the step k k are embedded separately into global representations shared across all blocks. A node-wise MLP 𝚷 0 U \\boldsymbol{\\Pi}^{{\\text{U}}}_{0} (a shared per-node MLP with no cross-node mixing) maps 𝐮 {\\mathbf{u}} to per-node embeddings 𝐔 0 = 𝚷 0 U ( 𝐮 ) ∈ ℝ N × F 0 {\\mathbf{U}}_{0}=\\boldsymbol{\\Pi}^{{\\text{U}}}_{0}({\\mathbf{u}})\\in{\\mathbb{R}}^{N\\times F_{0}} . A diffusion-step encoder 𝚷 0 k \\boldsymbol{\\Pi}^{{\\text{k}}}_{0} , a sinusoidal positional encoding followed by a node-wise MLP, maps the scalar k k to an F 0 F_{0} -dimensional vector broadcast to all nodes as 𝐊 0 = 𝟏 N 𝚷 0 k ( k ) ⊤ ∈ ℝ N × F 0 {\\mathbf{K}}_{0}={\\mathbf{1}}_{N}\\,\\boldsymbol{\\Pi}^{{\\text{k}}}_{0}(k)^{\\top}\\in{\\mathbb{R}}^{N\\times F_{0}} ."},{"type":"para","text":"At each depth b b , a fusion layer 𝚷 b \\boldsymbol{\\Pi}_{b} merges the main signal path with these global embeddings before the GNN module. The block input 𝐕 b {\\mathbf{V}}_{b} and the down-sampled embeddings 𝐃 b [ 𝐔 0 ; 𝐊 0 ] {\\mathbf{D}}_{b}\\,[{\\mathbf{U}}_{0};\\,{\\mathbf{K}}_{0}] are each projected to a common width, merged along the feature dimension, and mapped to F b F_{b} channels,"},{"type":"para","text":"The factor 𝐃 b {\\mathbf{D}}_{b} restricts the global embeddings to the N b N_{b} nodes active at depth b b , matching the support of 𝐕 b {\\mathbf{V}}_{b} , and the per-input projection reconciles their differing feature (channel) widths. The block input depends on the path. For b b th encoder block, 𝐕 b = 𝐂 b 𝐙 b − 1 {\\mathbf{V}}_{b}={\\mathbf{C}}_{b}\\,{\\mathbf{Z}}_{b-1} down-samples the depth- ( b − 1 ) (b{-}1) feature onto the N b N_{b} active nodes. For b b th decoder block, the coarser output 𝐘 b + 1 ∈ ℝ N b + 1 × F b + 1 {\\mathbf{Y}}_{b+1}\\in{\\mathbb{R}}^{N_{b+1}\\times F_{b+1}} is up-sampled through 𝐂 b + 1 ⊤ {\\mathbf{C}}_{b+1}^{\\top} , concatenated with the encoder skip 𝐙 b {\\mathbf{Z}}_{b} along the feature dimension, and mapped to F b F_{b} channels by a learned skip-projection 𝚷 b skip : ℝ F b + 1 + F b → ℝ F b \\boldsymbol{\\Pi}^{\\mathrm{skip}}_{b}:{\\mathbb{R}}^{F_{b+1}+F_{b}}\\to{\\mathbb{R}}^{F_{b}} ,"},{"type":"para","text":"The fusion layer 𝚷 b \\boldsymbol{\\Pi}_{b} is shared between the matched encoder and decoder blocks, so its weights appear in both Θ b E \\Theta^{{\\text{E}}}_{b} and Θ b D \\Theta^{{\\text{D}}}_{b} , while the GNN module and skip-projection remain block-specific. The node-state and diffusion-step embeddings enter each block by input substitution. In ( 19 ) and ( 21 ), the module input 𝐂 b 𝐙 b − 1 {\\mathbf{C}}_{b}\\,{\\mathbf{Z}}_{b-1} is replaced by the fused 𝐏 b {\\mathbf{P}}_{b} . In ( 20 ), the skip combination [ 𝐂 b + 1 ⊤ 𝐘 b + 1 ; 𝐙 b ] [{\\mathbf{C}}^{\\top}_{b+1}{\\mathbf{Y}}_{b+1};\\,{\\mathbf{Z}}_{b}] is first mapped by 𝚷 b skip \\boldsymbol{\\Pi}^{\\mathrm{skip}}_{b} [cf. ( 23 )] and then fused into 𝐏 b {\\mathbf{P}}_{b} . The GNN modules and the block composition remain unchanged, and only the module inputs differ. Figure 2 shows this separation, with the fusion drawn as the purple projection 𝚷 b \\boldsymbol{\\Pi}_{b} preceding each green GNN module, and Fig. 7 (Appendix E ) details the block interface. Finally, a read-out layer 𝚷 out \\boldsymbol{\\Pi}_{\\mathrm{out}} , a node-wise MLP, maps the final decoder output 𝐘 1 ∈ ℝ N × F 0 {\\mathbf{Y}}_{1}\\in{\\mathbb{R}}^{N\\times F_{0}} to the prediction ϵ 𝜽 ∈ ℝ N \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}}\\in{\\mathbb{R}}^{N} ."},{"type":"heading","level":2,"text":"IV-B Down-sampling (Pooling) via Learned Node Selection"},{"type":"para","text":"The selection matrices 𝐂 b {\\mathbf{C}}_{b} and their compositions 𝐃 b {\\mathbf{D}}_{b} determine the active nodes at each depth. Rather than fixing them in advance, we learn them end-to-end by letting each encoder block score its own GNN output and select the active nodes propagated to the next depth. The node-selection head of encoder block b b derives 𝐂 b + 1 {\\mathbf{C}}_{b+1} from the encoded feature 𝐙 b {\\mathbf{Z}}_{b} , for b = 1 , … , B − 1 b=1,\\ldots,B-1 , producing 𝐂 2 , … , 𝐂 B {\\mathbf{C}}_{2},\\ldots,{\\mathbf{C}}_{B} . The top level performs no selection since 𝐂 1 = 𝐃 1 = 𝐈 N {\\mathbf{C}}_{1}={\\mathbf{D}}_{1}={\\mathbf{I}}_{N} , and the deepest selection 𝐂 B {\\mathbf{C}}_{B} down-samples into the bottleneck, which itself does not further down-sample."},{"type":"para","text":"We describe the selection operation using two index systems. Global indices label nodes in 𝒱 {\\mathcal{V}} , with the active set Ω b ⊆ 𝒱 \\Omega_{b}\\subseteq{\\mathcal{V}} , | Ω b | = N b |\\Omega_{b}|=N_{b} , nested as in the beginning of Section IV . Local indices [ N b ] ≔ { 1 , … , N b } [N_{b}]\\coloneqq\\{1,\\ldots,N_{b}\\} enumerate only the active nodes in a fixed order. The composite matrix 𝐃 b ∈ { 0 , 1 } N b × N {\\mathbf{D}}_{b}\\in\\{0,1\\}^{N_{b}\\times N} maps a global-indexed signal to its local form, and 𝐃 b ⊤ {\\mathbf{D}}_{b}^{\\top} scatters a local-indexed signal back to the N N -node domain. The selection below acts in the local space [ N b ] [N_{b}] ."},{"type":"para","text":"Each encoder block is equipped with a head that scores the active nodes and keeps the highest-scoring ones. Concretely, a node-wise MLP 𝚿 b \\boldsymbol{\\Psi}_{b} , with parameters 𝚯 b sel \\boldsymbol{\\Theta}^{\\mathrm{sel}}_{b} , takes the block output 𝐙 b {\\mathbf{Z}}_{b} , along with the node and step embeddings restricted to the active nodes, and returns a per-node score vector,"},{"type":"para","text":"The dependence on 𝐃 b [ 𝐔 0 ; 𝐊 0 ] {\\mathbf{D}}_{b}\\,[{\\mathbf{U}}_{0};\\,{\\mathbf{K}}_{0}] , as in ( 22 ), allows the selection to adapt to the node states and the diffusion step. Given a down-sampling factor ρ b + 1 > 1 \\rho_{b+1}>1 , we keep the N b + 1 = ⌊ N b / ρ b + 1 ⌋ N_{b+1}=\\lfloor N_{b}/\\rho_{b+1}\\rfloor highest-scoring nodes,"},{"type":"para","text":"The selection matrix 𝐂 b + 1 = [ 𝐈 N b ] 𝒮 b , : ∈ { 0 , 1 } N b + 1 × N b {\\mathbf{C}}_{b+1}=\\big[{\\mathbf{I}}_{N_{b}}\\big]_{\\mathcal{S}_{b},\\,:}\\in\\{0,1\\}^{N_{b+1}\\times N_{b}} keeps the rows of 𝐈 N b {\\mathbf{I}}_{N_{b}} indexed by 𝒮 b \\mathcal{S}_{b} , and the composite matrix updates as 𝐃 b + 1 = 𝐂 b + 1 𝐃 b {\\mathbf{D}}_{b+1}={\\mathbf{C}}_{b+1}{\\mathbf{D}}_{b} [cf. ( 18 )]. The retained global set Ω b + 1 ⊆ Ω b \\Omega_{b+1}\\subseteq\\Omega_{b} collects the nonzero columns of 𝐃 b + 1 {\\mathbf{D}}_{b+1} , so 𝒮 b \\mathcal{S}_{b} and Ω b + 1 \\Omega_{b+1} express the same selection in local and global coordinates, respectively."},{"type":"para","text":"The decoder performs no selection. At each depth, the encoder-produced 𝐂 b + 1 {\\mathbf{C}}_{b+1} is reused by the matched decoder for up-sampling through 𝐂 b + 1 ⊤ {\\mathbf{C}}_{b+1}^{\\top} [cf. ( 23 )], while 𝐃 b {\\mathbf{D}}_{b} parametrizes the depth- b b GNN modules [cf. ( 19 )–( 21 )]."},{"type":"para","text":"The Top-K operation in ( 25 ) is non-differentiable in 𝐯 b {\\mathbf{v}}_{b} . Therefore, we train the head 𝚿 b \\boldsymbol{\\Psi}_{b} with a straight-through estimator (STE) [ 1 ] that keeps the hard selection 𝒮 b \\mathcal{S}_{b} in the forward pass, while gradients reach the scores 𝐯 b {\\mathbf{v}}_{b} through a differentiable sigmoid surrogate of the binary selection mask. Appendix C details the surrogate mask, and Appendix D integrates classical fixed-pooling rules with the U-GNN."},{"type":"heading","level":1,"text":"V Numerical Results"},{"type":"para","text":"We evaluate the proposed U-GNN denoiser on stock price forecasting for the S&P 500 index and wireless resource allocation (WRA) in a power control setup. Both use the same depth- B = 4 B{=}4 U-GNN backbone ϵ 𝜽 \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}} , diffusion noise schedules, and optimizers. They differ only in the signal and conditioning interface that each task dictates—a multi-step return trajectory for forecasting versus a single power allocation for resource allocation. Table I summarizes the model, diffusion, and optimization hyperparameters. We detail the shared and application-specific configurations in Appendix E ."},{"type":"heading","level":2,"text":"V-A Stock Price Forecasting"},{"type":"para","text":"We cast stock price forecasting as a spatio-temporal generative task. Given recent market history for a panel of stocks, the goal is to sample from the conditional distribution of their short-horizon future log-return trajectories."},{"type":"para","text":"Setup. We use daily S&P 500 data obtained with the open-source yfinance library. After aligning all stocks on a common trading day axis and dropping those with insufficient coverage, we retain N = 468 N=468 stocks across 11 11 GICS sectors over 2353 2353 trading days (2016-09-06 to 2026-02-13). Let 𝐫 ( t ) ∈ ℝ N {\\mathbf{r}}^{(t)}\\in{\\mathbb{R}}^{N} collect the daily log returns on trading day t t , where [ 𝐫 ( t ) ] i [{\\mathbf{r}}^{(t)}]_{i} is the log ratio of stock i i ’s consecutive closing prices. For each stock and day, we gather U = 12 U=12 market features into 𝐮 ( t ) ∈ ℝ N × U {\\mathbf{u}}^{(t)}\\in{\\mathbb{R}}^{N\\times U} , comprising the open, high, low, and standardized closing prices; the log return 𝐫 ( t ) {\\mathbf{r}}^{(t)} and its trailing moving averages; the log-volume; and two technical indicators, RSI and MACD."},{"type":"para","text":"We build a static, undirected graph 𝒢 {\\mathcal{G}} from long-term company fundamentals (e.g., market capitalization and price-to-earnings ratios). Its adjacency 𝐖 {\\mathbf{W}} combines the rank correlation between two stocks’ fundamental profiles with a same-sector bonus. We threshold and spectrally normalize 𝐖 {\\mathbf{W}} to obtain the GSO 𝐒 {\\mathbf{S}} , which is shared across all forecast windows."},{"type":"para","text":"We form data samples with a sliding window of length ( T h + T p ) (T_{h}+T_{p}) over the panel. In each window, the first T h T_{h} days are the conditioning history, stacked into 𝐮 = [ 𝐮 ( t − T h + 1 ) , ⋯ , 𝐮 ( t ) ] ∈ ℝ N × T h × U {\\mathbf{u}}=[\\,{\\mathbf{u}}^{(t-T_{h}+1)},\\ \\cdots,\\ {\\mathbf{u}}^{(t)}\\,]\\in{\\mathbb{R}}^{N\\times T_{h}\\times U} , and the next T p T_{p} days are the forecast target, stacked into the clean graph signal 𝐱 0 = [ 𝐫 ( t + 1 ) , ⋯ , 𝐫 ( t + T p ) ] ∈ ℝ N × T p {\\mathbf{x}}_{0}=[\\,{\\mathbf{r}}^{(t+1)},\\ \\cdots,\\ {\\mathbf{r}}^{(t+T_{p})}\\,]\\in{\\mathbb{R}}^{N\\times T_{p}} . We reversibly instance-normalize (RevIN) each target window [ 17 ] , using statistics from its conditioning block alone, and we invert this normalization on the DDIM-generated samples before scoring. The U-GNN treats the T p T_{p} horizon steps as a temporal axis. It applies graph convolutions on 𝐒 {\\mathbf{S}} at each step, couples the steps through interleaved temporal-convolution layers, and fuses a per-node summary of the T h T_{h} history steps into the conditioning context by cross-attention."},{"type":"para","text":"We set T h = 20 T_{h}=20 and T p = 5 T_{p}=5 , yielding 2328 2328 windows in total. We partition them using an interleaved chronological scheme. We split the trading-day axis into 10 10 equal chunks. Within each chunk, we assign the earliest 80 % 80\\% days to training, the next 10 % 10\\% to validation, and the final 10 % 10\\% to testing, then pool the matching parts across chunks. This ordering keeps each test block later than its training block, so the model is never fitted on days that follow its test period. Interleaving also spreads anomalous regimes, such as COVID-19, across all three splits rather than concentrating them in one, which mitigates the test distribution shift that a single contiguous split would create."},{"type":"para","text":"Results. We assess the U-GNN forecasts qualitatively through the sample trajectories of Fig. 3 and quantitatively through the accuracy and distributional fidelity metrics of Table II , both computed on the held-out test split with 100 100 sampled trajectories per window. We benchmark U-GNN against a geometric random walk (GRW), in which each stock’s future log-returns are drawn i.i.d. from a fixed per-stock Gaussian. Its drift and volatility are estimated once on the training set, with shrinkage toward the market-wide statistics, and they are held fixed at inference, independently of the conditioning window."},{"type":"para","text":"U-GNN outperforms the GRW on nearly every metric, at both the return and price scales. It reduces price CRPS, RMSE, and MAE by 22 % 22\\% , 28 % 28\\% , and 21 % 21\\% , respectively, and recovers directional information unavailable to a random walk, raising majority-vote direction accuracy above chance ( 0.53 0.53 vs. 0.49 0.49 ). Beyond point and interval accuracy, U-GNN reproduces stylized facts of returns. It substantially shrinks the volatility-clustering gap ( 0.06 0.06 vs. 0.10 0.10 ) and reduces the momentum gap. The one exception is excess kurtosis, where it overshoots the empirical heavy tails. This overshoot is the artifact of a desirable property. The GRW confines nearly all of its mass to a band around the mean, whereas U-GNN places non-negligible probability on the rare, far-from-mean moves that real markets exhibit. At the reported checkpoint it allocates somewhat more mass to these extremes than the data warrant, and we read this gap as a fidelity–sharpness trade-off rather than a fixed limitation. Checkpoints with a near-vanishing kurtosis gap are attainable, but only by sacrificing sharpness. Sharp forecasts emerge early in training and distributional fidelity improves later, with the reported model sitting toward the sharper end (Fig. 4(d) )."},{"type":"para","text":"These dynamics are evident in the individual forecasts of Fig. 3 . The sampled trajectories spread out from the last observed price, and the spread grows over the forecast days, while their mean stays close to the realized path. Each sample tends to track the recent behavior of its stock, picking up its trend and volatility rather than flattening into a straight line or spreading evenly in all directions. A few paths break from the bundle to trace larger excursions, which is the generative model anticipating the occasional market swings noted above."},{"type":"para","text":"Fig. 4 assesses the distributional fidelity of U-GNN samples against those of GRW along four axes. Fig. 4(a) shows that U-GNN matches the autocorrelation signature of stock returns, which is near-zero for r r and persistent for r 2 r^{2} , with the latter implying strong volatility clustering. Fig. 4(b) pertains to the cross-sectional structure, where U-GNN tracks the leading eigenvalues of the return covariance C ^ \\hat{C} that capture the dominant market directions, whereas the GRW baseline flattens them. The remaining two plots take a score-based view. Each assigns every evaluation window a scalar score, then compares the resulting distribution to the real one through the 1 1 -Wasserstein ( W 1 W_{1} ) distance between their empirical CDFs. They differ only in which model supplies the score."},{"type":"para","text":"Fig. 4(c) uses the GRW score, the analytic NLL per (stock, step) in nats, for which an i.i.d. Gaussian is lower-bounded by its differential entropy 1 2 ln ( 2 π e ) ≈ 1.42 \\tfrac{1}{2}\\ln(2\\pi e)\\approx 1.42 (dotted line). Scored by the homoskedastic Gaussian that generated them, the GRW samples collapse onto a near-degenerate spike essentially at this floor, so every window is equally surprising, and the score carries no volatility information. The real returns, instead, spread out with a rightward skew, as calm windows fall below the floor and heavy-tailed ones, beyond the reach of the GRW, are pushed into the tail. U-GNN’s samples sit just left of the floor (mean ≈ 1.25 \\approx 1.25 ), which the GRW scorer even rates as more typical than an i.i.d. draw. Even under the baseline’s own likelihood, U-GNN lands closer to the real distribution ( W 1 = 0.14 W_{1}{=}0.14 ) than GRW’s own samples do ( W 1 = 0.23 W_{1}{=}0.23 )."},{"type":"para","text":"Fig. 4(d) instead scores each window by a uniformly weighted version of U-GNN’s training loss in ( 6 ), estimated by a stratified Monte Carlo average over 64 64 strata of the diffusion step (one noise draw each). This orders windows by denoising difficulty, and hence by volatility. U-GNN’s samples cluster lowest (mean ≈ 0.28 \\approx 0.28 ), while the real data lies just to their right (mean ≈ 0.32 \\approx 0.32 ). GRW paths form a distinct high-error mode (mean ≈ 0.46 \\approx 0.46 ), since memoryless Gaussian increments are hard to denoise for a model that expects temporal and cross-sectional structures. Here, U-GNN nearly coincides with the real data ( W 1 = 0.04 W_{1}{=}0.04 ), while GRW sits farther away ( W 1 = 0.14 W_{1}{=}0.14 ). Neither score is impartial on its own, but read together, they corroborate that U-GNN better captures the real distribution. While U-GNN’s score isolates GRW samples as unrealistic, GRW ranks U-GNN above the real data. That said, the temporal nature of stock data may favor attention-based architectures suited to long-range dependencies. We leave the integration of a graph transformer with our U-GNN backbone to future work."},{"type":"heading","level":2,"text":"V-B Wireless Resource Allocation"},{"type":"para","text":"We study optimal power control in multi-user interference networks. The setup closely follows our preliminary work on diffusion policies for resource allocation [ 36 , 37 ] and the related state-augmented formulations of [ 39 ] . We summarize it below and collect the simulation parameters in Table III ."},{"type":"para","text":"Setup. We consider wireless networks of N = 400 N=400 transmitter–receiver (tx–rx) pairs dropped uniformly at random over a square service area. The pairs share a single channel of bandwidth W W with noise PSD N 0 N_{0} and per-transmitter power budget P max P_{\\max} (Table III ). The channel gain h i j , t ∈ ℝ + h_{ij,t}\\in{\\mathbb{R}}_{+} from transmitter i i to receiver j j at step t t combines a static large-scale component with a unit-mean Rayleigh fast-fading component. The large-scale component follows a dual-slope path-loss law with log-normal shadowing and fixes the long-term average gain h i j h_{ij} . We collect the long-term and instantaneous gains in 𝐇 , 𝐇 t ∈ ℝ + N × N {\\mathbf{H}},{\\mathbf{H}}_{t}\\in{\\mathbb{R}}^{N\\times N}_{+} , with [ 𝐇 ] i j = h i j [{\\mathbf{H}}]_{ij}=h_{ij} and [ 𝐇 t ] i j = h i j , t [{\\mathbf{H}}_{t}]_{ij}=h_{ij,t} ."},{"type":"para","text":"The policy and the instantaneous gains 𝐇 t {\\mathbf{H}}_{t} evolve on separate timescales. The fast-fading process generates T T realizations of 𝐇 t {\\mathbf{H}}_{t} , while the policy emits one allocation per slot and holds it constant over a window of T 0 T_{0} realizations, giving S = T / T 0 S=T/T_{0} slots per network (Table III ). The network configuration 𝐇 {\\mathbf{H}} stays fixed over the operation window and is the only channel state available at allocation time. The policy is conditioned on 𝐇 {\\mathbf{H}} via the GSO 𝐒 {\\mathbf{S}} and node-state 𝐮 {\\mathbf{u}} (defined below), while the instantaneous gains 𝐇 t {\\mathbf{H}}_{t} are observed only through the rates they induce."},{"type":"para","text":"We model each configuration as a draw 𝐇 ∼ q 𝐇 {\\mathbf{H}}\\sim q_{{\\mathbf{H}}} from a family of network geometries and large-scale fading. Sweeping the area side length R R over four values yields four user densities, each contributing 32 32 networks for a dataset of 128 128 configurations in total, which we split 5 : 1 : 2 5{:}1{:}2 into training, validation, and testing (Table III )."},{"type":"para","text":"Given an allocation 𝐩 {\\mathbf{p}} and a channel realization 𝐇 ~ \\widetilde{{\\mathbf{H}}} , the rate of the receiver j j is"},{"type":"para","text":"We evaluate a policy by its ergodic rate, estimated by averaging the instantaneous rates over the fading realizations,"},{"type":"para","text":"In ( 27 ), the expectation is over the policy allocations 𝐩 ∼ 𝝅 ( ⋅ | 𝐇 ) {\\mathbf{p}}\\sim\\boldsymbol{\\pi}(\\cdot\\,|\\,{\\mathbf{H}}) and the fast-fading channel 𝐇 ~ ∼ q 𝐇 ~ | 𝐇 \\widetilde{{\\mathbf{H}}}\\sim q_{\\widetilde{{\\mathbf{H}}}|{\\mathbf{H}}} . The allocation 𝐩 s {\\mathbf{p}}_{s} is drawn once per slot s = ⌈ t / T 0 ⌉ s=\\lceil t/T_{0}\\rceil and held fixed across it, while 𝐇 t {\\mathbf{H}}_{t} is the channel realized at step t t ."},{"type":"para","text":"Given 𝐇 {\\mathbf{H}} , an optimal policy 𝝅 ⋆ ( 𝐇 ) = 𝝅 ⋆ ( ⋅ | 𝐇 ) \\boldsymbol{\\pi}^{\\star}({\\mathbf{H}})=\\boldsymbol{\\pi}^{\\star}(\\cdot{\\,\\big|\\,}{\\mathbf{H}}) maximizes the ergodic sum-rate subject to a per-user minimum-rate (QoS) requirement f min = 0.6 f_{\\min}=0.6 bits/s/Hz, i.e.,"},{"type":"para","text":"The diffusion targets are produced by an expert primal–dual (PD) algorithm that solves ( 28 ) via Lagrangian dual (sub)gradient ascent. We train a separate expert per density group, parametrized by a shallow GNN, across all networks. We then collect 200 200 allocations per training network from its converged primal iterates, giving 16000 16000 conditional samples that the diffusion model (U-GNN) learns to reproduce given ( 𝐒 , 𝐮 ) ({\\mathbf{S}},{\\mathbf{u}}) . Running the same expert across all splits, including the validation and test networks held out from U-GNN, yields the reference allocations against which we benchmark the U-GNN policy. Since the expert solves each network directly rather than generalizing, it is a near-ideal reference that U-GNN can only approach, so the comparison favors the expert by design. At inference, U-GNN thus samples directly from this distribution of near-optimal, feasible allocations rather than re-solving the optimization per network. We refer to [ 37 ] for the expert’s algorithmic details."},{"type":"para","text":"We represent 𝐇 {\\mathbf{H}} as a directed, weighted channel graph on N N nodes, one per tx–rx pair. A self-loop carries the direct-link gain h i i h_{ii} , and an edge carries the interference gain h i j h_{ij} , i ≠ j i\\neq j , which is generally asymmetric, i.e., h i j ≠ h j i h_{ij}\\neq h_{ji} . We sparsify this graph into the GSO 𝐒 {\\mathbf{S}} by retaining, for each receiver, its self-loop and the incoming edges from the m = 10 m=10 strongest interferers, followed by a log-normalization that compresses the wide dynamic range of the channel gains. We also derive a two-feature node-state vector 𝐮 {\\mathbf{u}} that collects each node’s direct-link strength and its aggregate incoming interference under full-power transmission. Both 𝐒 {\\mathbf{S}} and 𝐮 {\\mathbf{u}} are deterministic functions of 𝐇 {\\mathbf{H}} , so they form the conditioning input 𝝃 = ( 𝐒 , 𝐮 ) \\boldsymbol{\\xi}=({\\mathbf{S}},{\\mathbf{u}}) of Section II , and q 𝐇 q_{{\\mathbf{H}}} induces its distribution q 𝝃 q_{\\boldsymbol{\\xi}} ."},{"type":"para","text":"The diffusion variable is the rescaled allocation 𝐱 0 = 𝐩 / P max − 1 2 ∈ [ − 1 2 , 1 2 ] N {\\mathbf{x}}_{0}={\\mathbf{p}}/P_{\\max}-\\tfrac{1}{2}\\in[-\\tfrac{1}{2},\\tfrac{1}{2}]^{N} , viewed as a scalar graph signal on 𝐒 {\\mathbf{S}} . We realize the U-GNN policy, which is trained to imitate the optimal policy 𝝅 ⋆ \\boldsymbol{\\pi}^{\\star} , through the conditional diffusion generator p 𝜽 ⋆ ( ⋅ | 𝐒 , 𝐮 ) p_{\\boldsymbol{\\theta}^{\\star}}(\\cdot\\,|\\,{\\mathbf{S}},{\\mathbf{u}}) of Section II . That is, given 𝐇 {\\mathbf{H}} , we sample a batch of S = 100 S=100 rescaled allocations 𝐱 0 ∼ p 𝜽 ⋆ ( ⋅ | 𝐒 , 𝐮 ) {\\mathbf{x}}_{0}\\sim p_{\\boldsymbol{\\theta}^{\\star}}(\\cdot\\,|\\,{\\mathbf{S}},{\\mathbf{u}}) and map them to power allocations 𝐩 = clip [ 0 , P max ] ( P max ( 𝐱 0 + 1 2 ) ) {\\mathbf{p}}=\\mathrm{clip}_{[0,P_{\\max}]}\\!\\big(P_{\\max}({\\mathbf{x}}_{0}+\\tfrac{1}{2})\\big) . We place these allocations in random order as { 𝐩 s } s = 1 S \\{{\\mathbf{p}}_{s}\\}_{s=1}^{S} and execute them sequentially, with slot s s transmitting 𝐩 s {\\mathbf{p}}_{s} over its T 0 T_{0} -realization window."},{"type":"para","text":"We compare the U-GNN policy against two deterministic baselines that forgo time-sharing. For every slot s s and network 𝐇 {\\mathbf{H}} , full power (FP) sets 𝐩 s = P max 𝟏 N {\\mathbf{p}}_{s}=P_{\\max}\\mathbf{1}_{N} , and average power (AP) is the expert’s per-network mean allocation, i.e., 𝐩 s ( 𝐇 ) ≈ 𝔼 𝐩 ∼ 𝝅 ⋆ ( ⋅ | 𝐇 ) [ 𝐩 ] {\\mathbf{p}}_{s}({\\mathbf{H}})\\approx\\mathbb{E}_{{\\mathbf{p}}\\sim\\boldsymbol{\\pi}^{\\star}(\\cdot\\,|\\,{\\mathbf{H}})}\\,\\big[{\\mathbf{p}}\\big] ."},{"type":"para","text":"Results. We report in Table IV the ergodic-rate performance of the U-GNN policy, expert (PD) policy, and baselines. The U-GNN diffusion policy closely matches the expert across the full distribution. Its 5th- and 10th-percentiles (p 5 5 , p 10 10 ) rates reach 0.77 0.77 and 0.92 0.92 bits/s/Hz, trailing the expert by only 5.3 % 5.3\\% and 0.9 % 0.9\\% . The p 5 5 rate already exceeds the f min = 0.6 f_{\\min}=0.6 bits/s/Hz floor, and only the bottom ∼ 2 % {\\sim}2\\% of users fall slightly short. U-GNN’s gap to the expert is largest at p 1 1 , at 17.8 % 17.8\\% ( 0.55 0.55 vs. 0.66 0.66 bits/s/Hz), whereas FP and AP baselines trail by 87.9 % 87.9\\% and 71.8 % 71.8\\% at this percentile, making U-GNN’s degradation minor by comparison. FP attains the highest mean rate at 3.14 3.14 bits/s/Hz while collapsing its p 1 1 rate to 0.08 0.08 bits/s/Hz and lowering feasibility to 88.0 % 88.0\\% . U-GNN instead sustains a mean rate of 2.81 2.81 bits/s/Hz, on par with the expert ( 2.84 2.84 ) and AP ( 2.85 2.85 ). AP matches this mean rate but still leaves 7.3 % 7.3\\% of users below f min f_{\\min} , so collapsing the policy to a deterministic mean allocation is not enough."},{"type":"para","text":"U-GNN’s value rests jointly on its stochastic formulation and generative sampling. The stochastic formulation treats the policy as a distribution over allocations, sustaining feasibility through randomization where the deterministic AP fails. The expert shares this formulation but reaches it through thousands of primal–dual iterations per network, whereas U-GNN bypasses online optimization and attains comparable performance in a single DDIM sampling pass."},{"type":"para","text":"Fig. 5 illustrates why the optimal policy must be stochastic and how U-GNN realizes it through time-sharing . The optimal (expert) policy is multi-modal, activating a different subset of the mutually interfering tx–rx pairs in each slot. No single allocation serves all users, yet, alternating them across slots renders the time-averaged rate 𝐟 ( 𝝅 ⋆ , 𝐇 ) {\\mathbf{f}}(\\boldsymbol{\\pi}^{\\star},{\\mathbf{H}}) feasible for every receiver. Crucially, this rate averages the per-slot rates and must not be conflated with the rate of the single mean allocation 𝔼 [ 𝐩 ] \\mathbb{E}[{\\mathbf{p}}] , which activates every interferer at once and remains infeasible. This is exactly the failure mode of the AP baseline in Table IV ."},{"type":"para","text":"Fig. 6 examines the size-transferability of the U-GNN policy. The model is trained only on networks of the native size | 𝒱 | = 400 |{\\mathcal{V}}|=400 tx–rx pairs (diamonds) with mixed densities. It is then evaluated without retraining on networks ranging from 200 200 to 1600 1600 pairs at similar densities. The ergodic 5 5 th-percentile rate stays essentially flat and above the f min f_{\\min} floor across this range for every density. Denser networks ( 11.9 11.9 vs. 6.6 6.6 pairs/km 2 ) attain lower tail rates due to stronger interference, but their ordering and stability are preserved across all sizes. This indicates that the permutation-equivariant GNN backbone is also scalable. Similar observations hold for the mean and additional tail rates, which are not shown here."},{"type":"heading","level":1,"text":"VI Conclusion & Future Work"},{"type":"para","text":"We proposed a denoising diffusion framework for generating graph signals. We cast this as a denoising process and parametrized it with a U-GNN, a graph-domain adaptation of the U-Net. At its core, a pooling mechanism formalizes down/up-sampling as learned node selection via nested selection matrices and zero-padding. We demonstrated the framework on wireless resource allocation and stock-price forecasting. Future directions include designing graph-aware and latent diffusion processes, and integrating graph transformers into the U-GNN backbone for spatio-temporal tasks."},{"type":"para","text":"The base DDIM sampler ( 9 ) deferred the specification of its noise scale σ k \\sigma_{k} . We provide it here, where it also governs the accelerated sampler used in our experiments. A factor η ∈ [ 0 , 1 ] \\eta\\in[0,1] controls the per-step stochasticity through"},{"type":"para","text":"which interpolates between the ancestral DDPM update at η = 1 \\eta=1 and the deterministic DDIM map at η = 0 \\eta=0 ."},{"type":"para","text":"Because the DDIM updates in ( 9 ) preserve the marginals of the forward process, we can subsample the denoising steps to accelerate inference without retraining. Concretely, we train ϵ 𝜽 ⋆ \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}^{\\star}} on the full grid of K K steps but sample along only an increasing subset { τ 1 , … , τ K ′ } ⊂ { 1 , … , K } \\{\\tau_{1},\\ldots,\\tau_{K^{\\prime}}\\}\\subset\\{1,\\ldots,K\\} with τ K ′ = K \\tau_{K^{\\prime}}=K and K ′ ≪ K K^{\\prime}\\ll K , and we set τ 0 ≔ 0 \\tau_{0}\\coloneqq 0 . We draw 𝐱 τ K ′ = 𝐱 K ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\mathbf{x}}_{\\tau_{K^{\\prime}}}={\\mathbf{x}}_{K}\\sim\\mathcal{N}({\\mathbf{0}},{\\mathbf{I}}) and iterate the update ( 9 ) over this sub-grid,"},{"type":"para","text":"for k = K ′ , … , 1 k=K^{\\prime},\\ldots,1 , and 𝐰 ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\mathbf{w}}\\sim\\mathcal{N}({\\mathbf{0}},{\\mathbf{I}}) . We note that both 𝐱 ^ 0 = 𝐱 ^ 0 ( 𝐱 τ k , τ k ; 𝝃 ) \\widehat{{\\mathbf{x}}}_{0}=\\widehat{{\\mathbf{x}}}_{0}({\\mathbf{x}}_{\\tau_{k}},\\tau_{k};\\boldsymbol{\\xi}) [cf. ( 7 )] and ϵ 𝜽 ⋆ = ϵ 𝜽 ⋆ ( 𝐱 τ k , τ k ; 𝝃 ) \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}^{\\star}}=\\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}^{\\star}}({\\mathbf{x}}_{\\tau_{k}},\\tau_{k};\\boldsymbol{\\xi}) are evaluated at the current step τ k \\tau_{k} . The noise scale σ τ k \\sigma_{\\tau_{k}} follows from ( 29 ) under the substitution ( k , k − 1 ) ↦ ( τ k , τ k − 1 ) (k,k{-}1)\\mapsto(\\tau_{k},\\tau_{k-1}) ."},{"type":"para","text":"A literal evaluation of the strided graph convolution in ( 14 ) would form the operator 𝐒 γ {\\mathbf{S}}^{\\gamma} and accumulate its powers ( 𝐒 γ ) k ({\\mathbf{S}}^{\\gamma})^{k} , k = 0 , … , K k=0,\\ldots,K . For γ > 1 \\gamma>1 , this densifies 𝐒 {\\mathbf{S}} and forfeits its sparsity, much like the reduced GSOs 𝐒 ( k ) {\\mathbf{S}}^{(k)} of ( 15 ) do. We instead exploit ( 𝐒 γ ) k = 𝐒 γ k ({\\mathbf{S}}^{\\gamma})^{k}={\\mathbf{S}}^{\\gamma k} and evaluate the layer through γ K \\gamma K successive sparse products with the unit shift 𝐒 {\\mathbf{S}} , tapping the signal every γ \\gamma shifts."},{"type":"para","text":"Starting from the lifted signal 𝐙 ~ 0 = 𝐃 ⊤ 𝐙 ℓ − 1 \\widetilde{{\\mathbf{Z}}}_{0}={\\mathbf{D}}^{\\top}{\\mathbf{Z}}_{\\ell-1} , we form 𝐙 ~ j = 𝐒 𝐙 ~ j − 1 \\widetilde{{\\mathbf{Z}}}_{j}={\\mathbf{S}}\\,\\widetilde{{\\mathbf{Z}}}_{j-1} for j = 1 , … , γ K j=1,\\ldots,\\gamma K and accumulate the tapped terms 𝐙 ~ γ k 𝚯 ℓ , k \\widetilde{{\\mathbf{Z}}}_{\\gamma k}\\,\\boldsymbol{\\Theta}_{\\ell,k} , k = 0 , … , K k=0,\\ldots,K . We then apply the pointwise nonlinearity σ ℓ \\sigma_{\\ell} and the pooling 𝐃 {\\mathbf{D}} . The untapped shifts, j ∉ { 0 , γ , … , γ K } j\\notin\\{0,\\gamma,\\ldots,\\gamma K\\} , only advance the walk to the next tapped hop. Algorithm 1 summarizes the procedure."},{"type":"para","text":"Notably, we never materialize the reduced GSOs. The shift 𝐒 {\\mathbf{S}} is a sparse weighted aggregation, while the unpooling 𝐃 ⊤ {\\mathbf{D}}^{\\top} and pooling 𝐃 {\\mathbf{D}} are index scatter and gather operations. Only the running buffer 𝐙 ~ j \\widetilde{{\\mathbf{Z}}}_{j} and one accumulator are kept, each of size N × F N\\times F , so each layer runs in roughly 𝒪 ( γ K | ℰ | F ) \\mathcal{O}(\\gamma K\\,|{\\mathcal{E}}|\\,F) aggregation time and 𝒪 ( N F ) \\mathcal{O}(NF) memory, where | ℰ | |{\\mathcal{E}}| is the number of edges of 𝒢 {\\mathcal{G}} . Forming the dense 𝐒 γ {\\mathbf{S}}^{\\gamma} explicitly would instead cost 𝒪 ( N 2 ) \\mathcal{O}(N^{2}) memory. The filter keeps K + 1 K+1 taps for any γ \\gamma , so the stride adds shifts but no parameters."},{"type":"para","text":"The stride enters only as a multiplier on the shift count and is tied to the pooling ratio. When the GNN module sits in a depth- b b block of the U-GNN, it instantiates ( 14 ) with 𝐃 = 𝐃 b {\\mathbf{D}}={\\mathbf{D}}_{b} , and all of its layers share the stride γ b = min { ⌊ ρ ¯ b ⌋ , γ max } \\gamma_{b}=\\min\\{\\lfloor\\sqrt{\\bar{\\rho}_{b}}\\,\\rfloor,\\,\\gamma_{\\max}\\} , where ρ ¯ b = ∏ i = 1 b ρ i = N / N b \\bar{\\rho}_{b}=\\prod_{i=1}^{b}\\rho_{i}=N/N_{b} is the cumulative down-sampling factor [cf. ( 18 )]. This is the rule γ = ⌊ ρ ⌋ \\gamma=\\lfloor\\sqrt{\\rho}\\,\\rfloor of Remark 1 applied at depth b b with ρ = ρ ¯ b \\rho=\\bar{\\rho}_{b} and capped at γ max \\gamma_{\\max} , so the total reach γ b K \\gamma_{b}K does not span the whole graph and we avoid oversmoothing."},{"type":"para","text":"To train the score head 𝚿 b \\boldsymbol{\\Psi}_{b} by backpropagation, we apply a straight-through estimator (STE) [ 1 ] to the binary selection mask. That is, the forward pass selects a hard mask, while gradients reach 𝐯 b {\\mathbf{v}}_{b} through a differentiable sigmoid surrogate. Moreover, we employ a Gumbel-perturbed Top-K rule to encourage exploration during training."},{"type":"para","text":"Stochastic selection. Let 𝐡 b ∈ { 0 , 1 } N b {\\mathbf{h}}_{b}\\in\\{0,1\\}^{N_{b}} be the Top-K indicator, with [ 𝐡 b ] n = 𝟙 { n ∈ 𝒮 b } [{\\mathbf{h}}_{b}]_{n}=\\mathbbm{1}\\{n\\in\\mathcal{S}_{b}\\} . At inference, 𝒮 b \\mathcal{S}_{b} is the deterministic Top-K of ( 25 ). During training, we draw 𝒮 b \\mathcal{S}_{b} stochastically to encourage exploration of the discrete selection. Given i.i.d. standard Gumbel variates [ 𝐠 b ] n = − log ( − log u n ) [{\\mathbf{g}}_{b}]_{n}=-\\log(-\\log u_{n}) , u n ∼ Uniform ( 0 , 1 ) u_{n}\\sim\\mathrm{Uniform}(0,1) , and an exploration scale ε ∈ ℝ + \\varepsilon\\in{\\mathbb{R}}_{+} , we perturb the scores before ranking,"},{"type":"para","text":"Note that Top-K is invariant to positive scaling, i.e., TopK ( 𝐯 b + ε 𝐠 b , N b + 1 ) = TopK ( 𝐯 b / ε + 𝐠 b , N b + 1 ) \\mathrm{TopK}({\\mathbf{v}}_{b}+\\varepsilon{\\mathbf{g}}_{b},\\,N_{b+1})=\\mathrm{TopK}({\\mathbf{v}}_{b}/\\varepsilon+{\\mathbf{g}}_{b},\\,N_{b+1}) . By the Gumbel-Top-K trick [ 19 ] , ( 31 ) therefore draws 𝒮 b \\mathcal{S}_{b} without replacement with probabilities proportional to e [ 𝐯 b ] n / ε e^{[{\\mathbf{v}}_{b}]_{n}/\\varepsilon} . The scale ε \\varepsilon acts as a sampling temperature, and as ε → 0 \\varepsilon\\to 0 , the sample concentrates on the deterministic Top-K, which we recover at inference when the perturbation vanishes at ε = 0 \\varepsilon=0 ."},{"type":"para","text":"Straight-through gradient. For the backward pass, we form a surrogate from the unperturbed scores. Given a temperature τ ∈ ℝ + \\tau\\in{\\mathbb{R}}_{+} and the logistic sigmoid ς ( ⋅ ) \\varsigma(\\cdot) , we set 𝐡 ~ b = ς ( 𝐯 b / τ ) ∈ ( 0 , 1 ) N b \\widetilde{{\\mathbf{h}}}_{b}=\\varsigma({\\mathbf{v}}_{b}/\\tau)\\in(0,1)^{N_{b}} . The training mask is"},{"type":"para","text":"where sg ( ⋅ ) \\mathrm{sg}(\\cdot) is the stop-gradient operator. The soft terms cancel in value, thus 𝐦 b = 𝐡 b {\\mathbf{m}}_{b}={\\mathbf{h}}_{b} in the forward pass. In the backward pass, sg ( 𝐡 ~ b ) \\mathrm{sg}(\\tilde{{\\mathbf{h}}}_{b}) is constant, so the gradient reaches 𝐯 b {\\mathbf{v}}_{b} only through 𝐡 ~ b \\tilde{{\\mathbf{h}}}_{b} . We note that because the surrogate uses the unperturbed scores, the Gumbel noise affects only which nodes enter 𝒮 b \\mathcal{S}_{b} , not the surrogate gradient at any node."},{"type":"para","text":"Gated down-sampling. We insert this mask into the encoder down-sampling. During training, the depth- ( b + 1 ) (b{+}1) input 𝐕 b + 1 = 𝐂 b + 1 𝐙 b {\\mathbf{V}}_{b+1}={\\mathbf{C}}_{b+1}\\,{\\mathbf{Z}}_{b} of ( 22 ) becomes"},{"type":"para","text":"where ⊙ \\odot scales each row of 𝐙 b {\\mathbf{Z}}_{b} by the corresponding entry of 𝐦 b {\\mathbf{m}}_{b} . Since 𝐦 b = 𝐡 b {\\mathbf{m}}_{b}={\\mathbf{h}}_{b} in the forward pass, the selected rows pass through unchanged, and 𝐂 b + 1 {\\mathbf{C}}_{b+1} discards the rest. At inference, the STE branch is dropped ( 𝐦 b = 𝐡 b {\\mathbf{m}}_{b}={\\mathbf{h}}_{b} ), and ( 33 ) reverts to the plain down-sampling via 𝐂 b + 1 𝐙 b {\\mathbf{C}}_{b+1}\\,{\\mathbf{Z}}_{b} ."},{"type":"para","text":"Schedules. We anneal both τ \\tau and ε \\varepsilon linearly over training. Let t t and T T denote the current and total number of training epochs, with T = 5000 T=5000 as reported in Table I . The schedule uses a warm-up of T w T_{\\mathrm{w}} epochs followed by an anneal of T a T_{\\mathrm{a}} epochs,"},{"type":"para","text":"where r ( t ) = min { max { ( t − T w ) / T a , 0 } , 1 } r(t)=\\min\\{\\max\\{(t-T_{\\mathrm{w}})/T_{\\mathrm{a}},\\,0\\},\\,1\\} . We set T w = 0.02 T T_{\\mathrm{w}}=0.02\\,T , T a = 0.73 T T_{\\mathrm{a}}=0.73\\,T , ( τ 0 , τ min ) = ( 1 , 0.5 ) (\\tau_{0},\\tau_{\\min})=(1,0.5) , and ( ε 0 , ε min ) = ( 1 , 0 ) (\\varepsilon_{0},\\varepsilon_{\\min})=(1,0) . This way, both ε \\varepsilon and τ \\tau reach their floors at epoch T w + T a = 0.75 T T_{\\mathrm{w}}+T_{\\mathrm{a}}=0.75\\,T , after which selection is (approximately) deterministic, and the surrogate sharpness is fixed. These values worked well in our experiments, but we found the results to be robust, as any choice that explores broadly early and becomes deterministic well before training ends gave comparable performance."},{"type":"para","text":"We remark that the STE in ( 32 ) supplies a surrogate gradient for the binary mask, and not a differentiable relaxation of the hard Top-K rule. In the forward pass, 𝐂 b + 1 {\\mathbf{C}}_{b+1} removes the unselected rows, so each training iteration passes diffusion-loss gradients only to the score entries of the nodes in 𝒮 b \\mathcal{S}_{b} . The head nonetheless learns a global ranking for two reasons. First, while ε > 0 \\varepsilon>0 , the stochastic selection varies 𝒮 b \\mathcal{S}_{b} across iterations, so the gradient reaches a broader set of nodes over training. Second, the head shares its parameters across all candidate nodes, so each update reshapes the scoring function at every node in subsequent passes."},{"type":"para","text":"CNNs with pooling couple neighborhood aggregation and spatial down-sampling into a single operation that summarizes each pixel window, e.g., by its average or maximum, at a reduced resolution. This coupling exploits the intrinsic geometry of the regular grid. For a general graph, the GSO 𝐒 {\\mathbf{S}} supplies the analogous structural information, which we use to define pooling neighborhoods."},{"type":"para","text":"Our design decouples these two operations. The strided graph convolution in ( 14 ) already returns a nonlinear summary of each node’s neighborhood. In reduced indices, this is the K K -hop reach of { 𝐒 ( k ) } k = 0 K \\{{\\mathbf{S}}^{(k)}\\}_{k=0}^{K} , and in the lifted domain, it is the γ K \\gamma K -hop reach of 𝐒 {\\mathbf{S}} after 𝐃 b ⊤ {\\mathbf{D}}_{b}^{\\top} lifts the signal and before 𝐃 b {\\mathbf{D}}_{b} reduces it back [cf. ( 15 )]. The learned selectors 𝐂 b {\\mathbf{C}}_{b} in turn perform the node-space down-sampling [cf. ( 25 )]. An explicit sum- or max-pooling over graph neighborhoods is therefore optional rather than essential. We describe it next for completeness."},{"type":"para","text":"Pooling over the full graph. For a given 𝐒 {\\mathbf{S}} and pooling range κ ∈ ℤ + \\kappa\\in\\mathbb{Z}^{+} , we define the binary reachability matrix"},{"type":"para","text":"Note that each node reaches itself. From 𝐑 = 𝐑 ( 𝐒 ) {\\mathbf{R}}={\\mathbf{R}}({\\mathbf{S}}) we read off, for each node n ∈ 𝒱 n\\in{\\mathcal{V}} , the index set"},{"type":"para","text":"of nodes aggregated at n n . Let φ : ℝ ⋅ × F → ℝ F \\varphi\\colon{\\mathbb{R}}^{\\cdot\\times F}\\to{\\mathbb{R}}^{F} be a pooling map that is invariant to permutations of its rows (the pooled nodes) and acts per feature, e.g., [ φ sum ( 𝐗 ) ] f = ∑ i [ 𝐗 ] i , f [\\varphi_{\\mathrm{sum}}({\\mathbf{X}})]_{f}=\\sum_{i}[{\\mathbf{X}}]_{i,f} for sum-pooling and [ φ max ( 𝐗 ) ] f = max i [ 𝐗 ] i , f [\\varphi_{\\max}({\\mathbf{X}})]_{f}=\\max_{i}[{\\mathbf{X}}]_{i,f} for max-pooling. In analogy with the graph-filter notation 𝐇 ( 𝐒 ) 𝐗 {\\mathbf{H}}({\\mathbf{S}}){\\mathbf{X}} , we write 𝐑 ( 𝐒 ) 𝐗 {\\mathbf{R}}({\\mathbf{S}}){\\mathbf{X}} for the pooled signal,"},{"type":"para","text":"Note that for sum-pooling, this operator form coincides with the matrix product 𝐑 ( 𝐒 ) 𝐗 {\\mathbf{R}}({\\mathbf{S}}){\\mathbf{X}} ."},{"type":"para","text":"Pooling at reduced resolutions. Pooling at the resolutions of the U-GNN follows the reduced-GSO construction. We define the node-reduced counterpart of 𝐑 {\\mathbf{R}} ,"},{"type":"para","text":"which intersects the κ b \\kappa_{b} -reach of 𝐒 {\\mathbf{S}} with the active set Ω b \\Omega_{b} . Like the reduced GSOs in ( 15 ), 𝐑 b {\\mathbf{R}}_{b} is indexed by the local labels [ N b ] [N_{b}] . Also, the range κ b \\kappa_{b} is left implicit at depth b b , just as the stride γ b \\gamma_{b} is in ( 14 ). We define the set"},{"type":"para","text":"which collects the active nodes within κ b \\kappa_{b} hops of n n on 𝐒 {\\mathbf{S}} , including n n itself. We then pool a down-sampled signal 𝐙 ∈ ℝ N b × F {\\mathbf{Z}}\\in{\\mathbb{R}}^{N_{b}\\times F} as"},{"type":"para","text":"In ( 40 ), 𝐑 b ( 𝐒 ) 𝐙 {\\mathbf{R}}_{b}({\\mathbf{S}}){\\mathbf{Z}} admits a lift–pool–reduce form mirroring that of the strided convolutions. One lifts 𝐙 {\\mathbf{Z}} to the full vertex set, pools over the κ b \\kappa_{b} -hop neighborhoods of 𝐒 {\\mathbf{S}} , and reduces back to Ω b \\Omega_{b} through 𝐃 b {\\mathbf{D}}_{b} . The lift assigns each inactive node the identity element of φ \\varphi , so that it leaves the pooled output unchanged. This element is 0 for sum-pooling, for which the plain 𝐃 b ⊤ {\\mathbf{D}}_{b}^{\\top} suffices, and − ∞ -\\infty (a large negative constant in practice) for max-pooling, supplied by a modified lift 𝐃 ~ b ⊤ \\widetilde{{\\mathbf{D}}}_{b}^{\\top} ."},{"type":"para","text":"Optional placement in the U-GNN. At each encoder depth b = 1 , … , B − 1 b=1,\\ldots,B-1 , the encoder output 𝐙 b {\\mathbf{Z}}_{b} is first pooled as 𝐑 b 𝐙 b {\\mathbf{R}}_{b}{\\mathbf{Z}}_{b} and then down-sampled by 𝐂 b + 1 𝐑 b 𝐙 b {\\mathbf{C}}_{b+1}{\\mathbf{R}}_{b}{\\mathbf{Z}}_{b} , which feeds the block at depth b + 1 b+1 [cf. ( 19 )]. The node selectors use the pooled features as well, so we replace 𝐙 b {\\mathbf{Z}}_{b} with 𝐑 b 𝐙 b {\\mathbf{R}}_{b}{\\mathbf{Z}}_{b} in ( 24 ). Both stages fold into the definition of the encoding GNN module. Note that the bottleneck at depth B B performs no node-space down-sampling or pooling."},{"type":"para","text":"The case for explicit pooling is weak in our design since the graph convolutional layers in ( 14 ) and selector heads in ( 24 ) already provide nonlinear neighborhood aggregation and learned down-sampling, respectively. In our experiments, max-pooling added computational overhead without a noticeable gain. Hence, we adopt ( 14 ) as the default GNN operation."},{"type":"para","text":"Max-pooling admits a compositional shortcut. By associativity and the telescoping of iterated neighborhoods, the κ \\kappa -reach maximum equals κ \\kappa applications of the unit-reach maximum. This avoids forming 𝐑 ( 𝐒 ) {\\mathbf{R}}({\\mathbf{S}}) explicitly and reduces the cost to κ \\kappa sparse scatter_max calls. The same property holds in the reduced domain."},{"type":"para","text":"For a regular graph with uniform down-sampling, a natural CNN analog sets the pooling range κ b = κ ⌈ N / N b ⌉ \\kappa_{b}=\\kappa\\lceil N/N_{b}\\rceil , proportional to the effective down-sampling at depth b b , so that pooling regions cover comparable numbers of active nodes across resolutions. For an arbitrary graph, the tension that governs the stride γ \\gamma reappears [cf. Remark 1 ]. Namely, a too small κ b \\kappa_{b} collapses 𝐧 b ( n ) {\\mathbf{n}}_{b}(n) to { n } \\{n\\} and reduces pooling to the identity, whereas a too large κ b \\kappa_{b} pools broadly over Ω b \\Omega_{b} and may oversmooth the representation."},{"type":"para","text":"This appendix expands Table I . In particular, Appendix E.1 details the denoiser, diffusion process, and training shared across the two tasks, and Appendix E.2 specifies the signal and conditioning interface particular to each."},{"type":"heading","level":2,"text":"E.1 Shared Denoiser, Diffusion, and Training"},{"type":"para","text":"Backbone. The denoiser ϵ 𝜽 \\boldsymbol{\\epsilon}_{\\boldsymbol{\\theta}} is a U-GNN spanning B = 4 B{=}4 node resolutions. Each of its 3 3 down-sampling levels reduces the active node set by a factor 2 2 , for an 2 3 = 8 × 2^{3}{=}8\\times reduction overall and four resolutions of N N , N / 2 N/2 , N / 4 N/4 , and N / 8 N/8 nodes. Each reduction is realized by a learned node-selection pooling operator, made differentiable with a straight-through estimator (Appendix C ). To counteract the increasing sparsity of the coarsened graphs, the graph convolutions are strided, capped at γ max = 2 \\gamma_{\\max}{=}2 (Appendix B ). The feature (channel) width is uniform at 64 64 channels across all levels."},{"type":"para","text":"Each encoding and decoding level applies 2 2 GNN layers of order K = 2 K{=}2 over powers of the augmented adjacency 𝐈 + 𝐀 ^ {\\mathbf{I}}+\\hat{{\\mathbf{A}}} , with channel-wise layer normalization (LayerNorm), a rectified linear unit (ReLU) nonlinearity, and dropout 0.1 0.1 ; these layers are strided variants of the topology-adaptive graph convolution (TAGConv), following its PyG implementation . A 2 2 -layer bottleneck processes the coarsest summary, and the decoder path restores full resolution while concatenating the encoder skip features at each level. The output head applies LayerNorm, a sigmoid linear unit (SiLU) nonlinearity, and a linear map. Fig. 7 complements Fig. 2 and shows a detailed schematic of one encoder–decoder block pair at depth b b ."},{"type":"para","text":"Conditioning and fusion. The diffusion step k k is mapped to a 128 128 -dimensional sinusoidal embedding followed by a multilayer perceptron (MLP), and the node conditioning to a 128 128 -dimensional representation. Both are projected to the level width 64 64 at every level, where the signal path and the projected step embedding are concatenated and linearly mapped back to 64 64 channels ( 128 → 64 128{\\to}64 ). The conditioning is then merged in a task-specific way: by concatenation into the same projection for WRA, so that it maps 192 192 channels down to 64 64 over the signal, step, and conditioning; and by two-head cross-attention for S&P 500—queries from the signal path, keys and values from the conditioning context—which leaves the projection from 128 128 channels down to 64 64 ."},{"type":"para","text":"Diffusion. The forward process applies a linear β \\beta schedule from 10 − 4 10^{-4} to 2 × 10 − 2 2\\times 10^{-2} over 500 500 steps. At inference, we sample using 5 × 5\\times -accelerated denoising diffusion implicit models (DDIM) with 100 100 steps and stochasticity parameter η = 0.2 \\eta{=}0.2 (Appendix A ). We draw 100 100 samples per conditioning input to form the predictive ensemble."},{"type":"para","text":"Optimization. We train for 5000 5000 epochs with AdamW (learning rate 10 − 4 10^{-4} , ( β 1 , β 2 ) = ( 0.9 , 0.98 ) (\\beta_{1},\\beta_{2}){=}(0.9,0.98) , weight decay 10 − 4 10^{-4} ), gradient-norm clipping at 1.0 1.0 , and automatic mixed precision (AMP). The learning rate follows a per-step cosine schedule with a linear warm-up over the first 1 % 1\\% of training, then a cosine decay to a floor of 5 % 5\\% of the peak over 0.8 0.8 of training, after which it is held constant. The node-selector temperature and exploration noise are annealed over roughly the first 75 % 75\\% of training, moving the selector from soft selection toward nearly hard selection (Appendix C ). We retain the best checkpoint under a task-specific composite validation criterion. Both models were trained on a single NVIDIA RTX 3090. The S&P 500 model completed training in ≈ 58.5 {\\approx}58.5 GPU hours, and the WRA model did so in ≈ 33 {\\approx}33 GPU hours."},{"type":"heading","level":2,"text":"E.2 Application-Specific Knobs"},{"type":"para","text":"Stock-price forecasting (S&P 500). The diffused signal is the future log-return trajectory 𝐱 0 ∈ ℝ N × T p {\\mathbf{x}}_{0}\\in{\\mathbb{R}}^{N\\times T_{p}} over the N = 468 N{=}468 stocks. The pooling cascade coarsens the node set as 468 → 234 → 117 → 58 468{\\to}234{\\to}117{\\to}58 across the four levels. Because returns are unbounded, the predicted clean signal is left unclipped."},{"type":"para","text":"The conditioning comprises 12 12 market features over the past T h T_{h} -day window. A per-node temporal encoder summarizes this window and projects the T h T_{h} history steps onto the T p T_{p} horizon steps, forming the per-level context that the cross-attention fusion above consumes. This encoder interleaves dilated 1-D convolutions with two-head temporal self-attention. A lightweight in-block temporal mixer further couples the T p T_{p} horizon steps. The target channel is additionally normalized by reversible instance normalization (RevIN) [ 17 ] , using statistics from the conditioning window. RevIN uses a blend weight of 0.7 0.7 and a scale correction of 1.107 1.107 ."},{"type":"para","text":"The model has 1.44 M 1.44\\mathrm{M} trainable parameters and is trained on batches of 64 64 sliding windows. We retain the checkpoint that minimizes a composite of the validation noise loss, its train–validation gap, and continuous ranked probability score (CRPS). This checkpoint is reached near epoch 4500 4500 ."},{"type":"para","text":"Wireless resource allocation (WRA). The diffused signal is the centered allocation 𝐱 0 = 𝐩 / P max − 1 2 ∈ [ − 1 2 , 1 2 ] N {\\mathbf{x}}_{0}={\\mathbf{p}}/P_{\\max}-\\tfrac{1}{2}\\in[-\\tfrac{1}{2},\\tfrac{1}{2}]^{N} over the N = 400 N{=}400 transmitter–receiver (tx–rx) pairs. The pooling cascade coarsens the node set as 400 → 200 → 100 → 50 400{\\to}200{\\to}100{\\to}50 . The powers are recovered as 𝐩 = clip [ 0 , P max ] ( P max ( 𝐱 0 + 1 2 ) ) {\\mathbf{p}}=\\mathrm{clip}_{[0,P_{\\max}]}\\!\\big(P_{\\max}({\\mathbf{x}}_{0}+\\tfrac{1}{2})\\big) . In contrast to the stock model, the allocation is a single snapshot with no time axis, so T = 1 T{=}1 . Thus, the temporal encoder and in-block mixer are therefore disabled, and RevIN is not used."},{"type":"para","text":"The 2 2 per-node conditioning features are concatenated into the level fusion described above. These features are the direct-link gain and the aggregate cross-link interference. Because the support is bounded, the predicted clean signal is clipped to the valid power range at each reverse step."},{"type":"para","text":"Target allocations are supplied by a primal–dual expert. The expert is a shallow GNN with 3 3 layers, hidden width 64 64 , and K = 2 K{=}2 hops, trained per density group with dual-ascent step 0.2 0.2 . As supervision for the U-GNN, we collect 200 200 allocations per training network from its converged primal iterates. This yields 16000 16000 samples in total. Running the same expert on the held-out networks yields the reference allocations used for evaluation. Theoretical and algorithmic details appear in [ 37 ] ."},{"type":"para","text":"The model has 0.63 M 0.63\\mathrm{M} trainable parameters and is trained on batches of 800 800 network graph and resource allocation signal pairs. We retain the checkpoint that minimizes a composite of the mean rate-feasibility gap and the 1 % 1\\% and 5 % 5\\% rate gaps, averaged over the four density groups. This checkpoint is reached at epoch 1600 1600 ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Signals defined over irregular, graph-structured domains are pervasive, spanning application areas such as recommender systems [ 41 , 24 , 23 ] , wireless communications [ 7 , 31 , 46 , 42 , 4 , 39 ] , and forecasting in financial markets [ 3 , 30 , 35 ] . In many such settings, these signals are realizations from unknown probability distributions rather than deterministic quantities [ 25 ] . This paper proposes a generative modeling framework that learns to sample from such graph signal distributions.Generative modeling has proved effective for synthesizing graph topologies themselves [ 16 , 40 ] , with molecular generation serving as a canonical example [ 14 ] . We instead take the graph structure as given and learn to generate node-defined signals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A wide array of generative paradigms exists for learning to sample from unknown distributions [ 18 , 10 , 26 ] , such as variational autoencoders, generative adversarial networks, and normalizing flows. Among them, denoising diffusion models and their score- and flow-based variants stand out for their sample quality, training stability, and principled conditional inference in high dimensions [ 13 , 32 , 33 , 22 ] . While these models differ in what the parametric model approximates and in the sampling dynamics, they share a common structure. A fixed forward process gradually corrupts the data into a tractable reference prior, and a learned reverse process denoises samples from that prior back toward the data distribution. Graph signals have recently become a target for generative, particularly diffusion, modeling in domains such as the forecasting of traffic and financial series [ 38 , 43 , 5 , 2 , 28 ] , and wireless optimization [ 6 , 21 , 36 , 11 , 12 , 37 , 20 ] . However, these efforts have mostly been application-specific, with task-tailored denoiser architectures. Their formulations also often remain deterministic, seeking solution distributions whose mass concentrates near an optimal deterministic solution rather than drawing diversified samples from the conditional law.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we propose a general methodological framework for graph signal generation. At its core, this requires designing a diffusion model whose denoising network encodes inductive biases tailored to the data domain. For images, the standard denoising backbone is the U-Net with a symmetric encoder–decoder structure and multi-resolution hierarchy [ 27 , 15 ] . The U-Net is built on convolutional neural networks (CNNs), whose layers apply translation-equivariant filters on the regular pixel grid. The natural counterpart for graph-structured data is the graph neural network (GNN), whose layers apply graph-shift-equivariant filters, thereby extending convolutional processing to irregular domains [ 8 ] . Replacing CNNs with GNN layers does not, on its own, suffice to adapt the U-Net design to the denoising of graph signals. The main obstacle is the pooling stage. In the image setting, the pixel grid provides a regular pooling lattice that persists after each down-sampling step, thus, feature extraction and spatial reduction compose seamlessly. Graph data lacks such a canonical lattice. A common workaround coarsens the graph at each resolution through a learned pooling operator, as in differentiable cluster-assignment pooling and top- k k node-scoring schemes [ 44 , 9 ] . These methods are mainly designed for graph-level tasks, such as classification. Under arbitrary coarsening, however, the pooled signal loses a clear interpretation as a convolutional signal on the original graph.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We take a different route in this paper. We formalize down- and up-sampling, equivalently graph or node (un)pooling, as a learned node (de)selection operation, expressed through nested selection matrices [ 8 ] . These matrices are trained end-to-end with the denoising network, so each resolution retains the nodes most informative for the denoising task at hand. The selection operator and its transpose play two complementary roles. First, the nested hierarchy of selections instantiates the multi-resolution encoder–decoder cascade. Each encoder selection picks a smaller active node subset that defines the next coarser resolution, and the matching transpose in the decoder lifts coarse signals back to the finer one. Second, at a fixed resolution, the same operator pair keeps GNN filtering interpretable as a convolution on the original graph. The transpose zero-pads a node-reduced signal to the full vertex set, the GNN layer filters there, and the selection returns the result to the reduced domain. Unlike the coarsening schemes above, filtering therefore stays convolutional on the original graph at every resolution. We further introduce a stride parameter that sets the hop distance of each graph shift. This mirrors how dilated convolutions in CNNs replace fixed pooling rules, folding down-sampling and receptive-field growth into the convolution itself [ 34 , 45 ] . On the active nodes, a larger stride enlarges the aggregation neighborhoods and prevents the graph filters from drawing mostly from zero-padded positions under heavy down-sampling. This construction avoids explicit graph coarsening and lets convolutional filtering and pooling compose across arbitrary resolutions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Building on this mechanism, we propose the U-Graph Neural Network (U-GNN) architecture as a graph-domain adaptation of the classical U-Net. It comprises a U-shaped encoder–decoder pipeline with skip connections, built from GNN layers and the pooling and unpooling operators above. We then cast graph signal generation as a conditional denoising diffusion process and parametrize its reverse process with a U-GNN. The generations are thereby conditioned directly on the graph topology and on node-feature side information. Although we focus on diffusion denoising, the U-GNN is a general-purpose architecture for learning over graph signals. It inherits the permutation equivariance, stability to perturbations, and cross-topology transferability of GNNs [ 29 , 31 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate the framework and the U-GNN architecture on two generative tasks. The first is stock price forecasting over a correlation graph of financial indicators. Given recent market history, we generate future price trajectories as graph signals, capturing the market uncertainty and rare tail events that point forecasts and statistical baselines often miss. The second is resource allocation in wireless networks under ergodic quality-of-service (QoS) requirements, where optimal policies are inherently probabilistic and are realized through time-sharing. Treating solution samples from an expert primal–dual algorithm as graph signals on channel (interference) graphs, we learn to sample from the expert allocation distribution given channel and user-state information.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A brief summary of our contributions is as follows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We develop a graph signal generative modeling approach that combines denoising diffusion models with GNNs to sample stochastic graph signals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We parametrize the reverse diffusion process with a novel U-GNN architecture, whose building blocks are GNN layers with stride and zero-padded pooling, that generalizes image-convolutional U-Nets to graph-structured signals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We apply our approach to stock price forecasting and optimal wireless resource allocation, with extensive numerical results in both domains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The rest of the paper is organized as follows. Section II presents the conditional graph signal generative diffusion framework. Section III introduces GNNs with built-in stride and pooling. Section IV assembles the U-GNN architecture. Section V validates the method on wireless resource allocation and financial forecasting. Section VI concludes the paper.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Graph Signal Generative Diffusion Models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider a weighted graph 𝒢 = ( 𝒱 , ℰ , 𝒲 ) {\\\\mathcal{G}}=({\\\\mathcal{V}},{\\\\mathcal{E}},{\\\\mathcal{W}}) on N N nodes, with node set 𝒱 = { 1 , … , N } {\\\\mathcal{V}}=\\\\{1,\\\\ldots,N\\\\} , edge set ℰ ⊆ 𝒱 × 𝒱 {\\\\mathcal{E}}\\\\subseteq{\\\\mathcal{V}}\\\\times{\\\\mathcal{V}} , and weight map 𝒲 : ℰ → ℝ {\\\\mathcal{W}}:{\\\\mathcal{E}}\\\\to{\\\\mathbb{R}} . A graph signal assigns features to the nodes and is collected in a matrix 𝐗 = [ 𝐱 1 , … , 𝐱 F ] ∈ ℝ N × F {\\\\mathbf{X}}=[{\\\\mathbf{x}}^{1},\\\\ldots,{\\\\mathbf{x}}^{F}]\\\\in{\\\\mathbb{R}}^{N\\\\times F} , whose f f th column 𝐱 f ∈ ℝ N {\\\\mathbf{x}}^{f}\\\\in{\\\\mathbb{R}}^{N} gathers the f f th feature across all nodes. We endow 𝒢 {\\\\mathcal{G}} with a graph shift operator (GSO) 𝐒 ∈ ℝ N × N {\\\\mathbf{S}}\\\\in{\\\\mathbb{R}}^{N\\\\times N} , a sparse matrix with [ 𝐒 ] m , n = 0 [{\\\\mathbf{S}}]_{m,n}=0 whenever m ≠ n m\\\\neq n and ( m , n ) ∉ ℰ (m,n)\\\\notin{\\\\mathcal{E}} . We note that this GSO definition allows the diagonal entries to be chosen freely. The GSO encodes the graph topology, and successive products (shifts) 𝐒𝐗 {\\\\mathbf{S}}{\\\\mathbf{X}} propagate signal values along the edges. Common GSO choices are the (weighted) adjacency matrix 𝐀 {\\\\mathbf{A}} and the graph Laplacian 𝐋 = diag ( 𝐀𝟏 N ) − 𝐀 {\\\\mathbf{L}}=\\\\mathrm{diag}({\\\\mathbf{A}}{\\\\mathbf{1}}_{N})-{\\\\mathbf{A}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We address the conditional generation of graph signals. The target is a graph signal 𝐱 0 ∈ ℝ N {\\\\mathbf{x}}_{0}\\\\in{\\\\mathbb{R}}^{N} on a known graph 𝒢 {\\\\mathcal{G}} . The conditioning collects the topology, through the GSO 𝐒 {\\\\mathbf{S}} , and auxiliary node states 𝐮 ∈ ℝ N × U {\\\\mathbf{u}}\\\\in{\\\\mathbb{R}}^{N\\\\times U} (side information) into the variable 𝝃 = ( 𝐒 , 𝐮 ) \\\\boldsymbol{\\\\xi}=({\\\\mathbf{S}},{\\\\mathbf{u}}) . Let q 𝝃 q_{\\\\boldsymbol{\\\\xi}} denote the distribution of the conditioning and q 𝐱 | 𝝃 ( ⋅ | 𝝃 ) q_{{\\\\mathbf{x}}|\\\\boldsymbol{\\\\xi}}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi}) the law of the target given 𝝃 \\\\boldsymbol{\\\\xi} . These define the joint data distribution q data = q 𝐱 | 𝝃 q 𝝃 q_{\\\\mathrm{data}}=q_{{\\\\mathbf{x}}|\\\\boldsymbol{\\\\xi}}\\\\,q_{\\\\boldsymbol{\\\\xi}} , from which we draw pairs ( 𝐱 0 , 𝝃 ) ({\\\\mathbf{x}}_{0},\\\\boldsymbol{\\\\xi}) . We seek a parametric generator p 𝜽 ⋆ ( ⋅ | 𝝃 ) = p ( ⋅ | 𝝃 ; 𝜽 ∗ ) p_{\\\\boldsymbol{\\\\theta}^{\\\\star}}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi})=p(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi};\\\\boldsymbol{\\\\theta}^{*}) whose conditional law matches q 𝐱 | 𝝃 ( ⋅ | 𝝃 ) q_{{\\\\mathbf{x}}|\\\\boldsymbol{\\\\xi}}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi}) for q 𝝃 q_{\\\\boldsymbol{\\\\xi}} -almost every 𝝃 \\\\boldsymbol{\\\\xi} , i.e.,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The conditional laws are unknown in closed form. We assume samples ( 𝐱 0 , 𝝃 ) ∼ q data ({\\\\mathbf{x}}_{0},\\\\boldsymbol{\\\\xi})\\\\sim q_{\\\\mathrm{data}} are available. Therefore, we realize the optimal generator p 𝜽 ⋆ p_{\\\\boldsymbol{\\\\theta}^{\\\\star}} as a conditional denoising diffusion model, trained on a dataset { ( 𝐱 0 ( i ) , 𝝃 ( i ) ) } i = 1 | 𝒟 | ∼ q data \\\\{({\\\\mathbf{x}}^{(i)}_{0},\\\\boldsymbol{\\\\xi}^{(i)})\\\\}_{i=1}^{|{\\\\mathcal{D}}|}\\\\sim q_{\\\\mathrm{data}} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Denoising Diffusion Models for Graph Signals\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A denoising diffusion model pairs a fixed forward process q ( 𝐱 1 : K | 𝐱 0 ) q({\\\\mathbf{x}}_{1:K}\\\\,|\\\\,{\\\\mathbf{x}}_{0}) , which noises clean targets toward a tractable reference prior, with a learned reverse process p 𝜽 ( 𝐱 k − 1 | 𝐱 k ) p_{\\\\boldsymbol{\\\\theta}}({\\\\mathbf{x}}_{k-1}\\\\,|\\\\,{\\\\mathbf{x}}_{k}) , which denoises prior samples back toward the data. Conditional generation requires no additional machinery. We condition the reverse process on 𝝃 \\\\boldsymbol{\\\\xi} as p 𝜽 ( 𝐱 k − 1 | 𝐱 k ; 𝝃 ) p_{\\\\boldsymbol{\\\\theta}}({\\\\mathbf{x}}_{k-1}\\\\,|\\\\,{\\\\mathbf{x}}_{k};\\\\,\\\\boldsymbol{\\\\xi}) , leaving the forward process and the training objective unchanged. Fig. 1 illustrates this construction, which we detail next.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Forward noising process. The forward process is a Markov chain that adds noise to the target 𝐱 0 {\\\\mathbf{x}}_{0} over K K steps. Given ( 𝐱 0 , 𝝃 ) ∼ q 0 = q data ({\\\\mathbf{x}}_{0},\\\\boldsymbol{\\\\xi})\\\\sim q_{0}=q_{\\\\mathrm{data}} with 𝝃 \\\\boldsymbol{\\\\xi} held fixed, we corrupt 𝐱 0 {\\\\mathbf{x}}_{0} with white noise on an increasing schedule { β k } k = 1 K \\\\{\\\\beta_{k}\\\\}_{k=1}^{K} (e.g., linear), producing iterates { 𝐱 k } k = 1 K \\\\{{\\\\mathbf{x}}_{k}\\\\}_{k=1}^{K} through the Gaussian transitions,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since the drift is linear and the noise increments are independent, the conditionals remain Gaussian at each step k k , i.e.,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with α k ≔ 1 − β k \\\\alpha_{k}\\\\coloneqq 1-\\\\beta_{k} and α ¯ k ≔ ∏ m = 1 k α m \\\\bar{\\\\alpha}_{k}\\\\coloneqq\\\\prod_{m=1}^{k}\\\\alpha_{m} for k = 1 , … , K k=1,\\\\ldots,K . A standard reparametrization of ( 3 ) expresses 𝐱 k {\\\\mathbf{x}}_{k} as a deterministic map of ( 𝐱 0 , k ) ({\\\\mathbf{x}}_{0},k) plus independent standard noise [ 13 ] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and it permits drawing any iterate 𝐱 k ∼ q k ( ⋅ | 𝐱 0 , 𝝃 ) {\\\\mathbf{x}}_{k}\\\\sim q_{k}(\\\\cdot\\\\,|\\\\,{\\\\mathbf{x}}_{0},\\\\boldsymbol{\\\\xi}) directly, without simulating the chain in ( 2 ). For a suitable schedule and a large enough K K , the process drives 𝐱 0 ∼ q 0 ( ⋅ | 𝝃 ) {\\\\mathbf{x}}_{0}\\\\sim q_{0}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi}) toward a Gaussian prior, with 𝐱 K ∼ q K ≈ 𝒩 ( 𝟎 , 𝐈 ) {\\\\mathbf{x}}_{K}\\\\sim q_{K}\\\\approx\\\\mathcal{N}({\\\\mathbf{0}},{\\\\mathbf{I}}) for every 𝝃 \\\\boldsymbol{\\\\xi} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Backward (reverse) denoising process. Denoising diffusion fixes p K ( ⋅ | 𝝃 ) = q K ≈ 𝒩 ( 𝟎 , 𝐈 ) p_{K}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi})=q_{K}\\\\approx\\\\mathcal{N}({\\\\mathbf{0}},{\\\\mathbf{I}}) and learns a reverse Markov chain whose kernels p 𝜽 ( 𝐱 k − 1 | 𝐱 k ; 𝝃 ) p_{\\\\boldsymbol{\\\\theta}}({\\\\mathbf{x}}_{k-1}\\\\,|\\\\,{\\\\mathbf{x}}_{k};\\\\boldsymbol{\\\\xi}) approximate the time reversal of the forward chain. Consistent with the continuous-time limit, we model each reverse step as Gaussian transitions,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As is standard, we fix the covariance 𝚺 𝜽 = σ k 2 𝐈 \\\\boldsymbol{\\\\Sigma}_{\\\\boldsymbol{\\\\theta}}\\\\!=\\\\!\\\\sigma_{k}^{2}{\\\\mathbf{I}} for a predefined σ k \\\\sigma_{k} tied to the schedule β k \\\\beta_{k} and learn only the mean 𝝁 𝜽 \\\\boldsymbol{\\\\mu}_{\\\\boldsymbol{\\\\theta}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training maximizes the evidence lower bound (ELBO) on the log-likelihood log p 𝜽 ( ⋅ | 𝝃 ) \\\\log p_{\\\\boldsymbol{\\\\theta}}(\\\\cdot{\\\\,\\\\big|\\\\,}\\\\boldsymbol{\\\\xi}) . Under the fixed covariance and the reparametrization ( 4 ), this is equivalent to a noise-prediction objective. Omitting its per-step ELBO weights matches, and often improves upon, the weighted objective in practice [ 13 ] . We therefore minimize the following objective,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the expectation is over ( 𝐱 0 , 𝝃 ) ∼ q data ({\\\\mathbf{x}}_{0},\\\\boldsymbol{\\\\xi})\\\\sim q_{\\\\mathrm{data}} and diffusion steps k ∼ unif { 1 , … , K } k\\\\sim\\\\mathrm{unif}\\\\{1,\\\\ldots,K\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By the reparametrization ( 4 ), the trained noise predictor ϵ 𝜽 ⋆ ( 𝐱 k , k ; 𝝃 ) \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}^{\\\\star}}({\\\\mathbf{x}}_{k},k;\\\\boldsymbol{\\\\xi}) yields both a graph signal denoiser and the mean of the reverse kernel in ( 5 ),\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus ϵ 𝜽 ⋆ \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}^{\\\\star}} and the fixed 𝚺 𝜽 = σ k 2 𝐈 \\\\boldsymbol{\\\\Sigma}_{\\\\boldsymbol{\\\\theta}}\\\\!=\\\\!\\\\sigma_{k}^{2}{\\\\mathbf{I}} specify ( 5 ) completely. For sampling, we construct a family of generally non-Markovian reverse processes [ 32 ] that preserve the forward marginals ( 3 ),\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with 𝐰 ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\\\mathbf{w}}\\\\sim\\\\mathcal{N}({\\\\mathbf{0}},{\\\\mathbf{I}}) . A single factor η ∈ [ 0 , 1 ] \\\\eta\\\\in[0,1] tunes the noise scale σ k \\\\sigma_{k} [cf. ( 29 )]. Each step draws 𝐱 k − 1 {\\\\mathbf{x}}_{k-1} exactly from the reverse kernel ( 5 ) at η = 1 \\\\eta=1 , and smaller values yield near-deterministic and usually faster sampling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At inference, we initialize 𝐱 K ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\\\mathbf{x}}_{K}\\\\sim\\\\mathcal{N}({\\\\mathbf{0}},{\\\\mathbf{I}}) and iterate ( 9 ) for k = K , … , 1 k=K,\\\\ldots,1 to draw samples 𝐱 0 ∼ p 𝜽 ⋆ ( ⋅ | 𝝃 ) {\\\\mathbf{x}}_{0}\\\\sim p_{\\\\boldsymbol{\\\\theta}^{\\\\star}}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi}) . These samples are distributed approximately according to the conditional data distribution q 𝐱 | 𝝃 ( ⋅ | 𝝃 ) q_{{\\\\mathbf{x}}|\\\\boldsymbol{\\\\xi}}(\\\\cdot\\\\,|\\\\,\\\\boldsymbol{\\\\xi}) for the given conditioning 𝝃 = ( 𝐒 , 𝐮 ) \\\\boldsymbol{\\\\xi}=({\\\\mathbf{S}},{\\\\mathbf{u}}) , thereby realizing the conditional generative model sought in ( 1 ). To reduce the number of model evaluations, we run an accelerated variant of ( 9 ) (see Appendix A ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"U-Nets have been instrumental in image generative models. Accordingly, we particularize ϵ 𝜽 \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}} to a U-Graph Neural Network (U-GNN) that tailors U-Nets for graph signal denoising.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Pooling & Stride in Graph Neural Networks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first review graph filters, convolutions, and a standard GNN layer. We then propose a GNN module with built-in pooling through selection matrices and stride, which Section IV arranges into the U-GNN encoder–decoder pipeline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Graph filters and convolutions. Graph filters are the building blocks of GNNs and process input graph signals through successive graph shifts. A single shift 𝐒𝐱 {\\\\mathbf{S}}{\\\\mathbf{x}} mixes each node’s value with those of its one-hop neighbors, and k k repeated shifts yield 𝐒 k 𝐱 {\\\\mathbf{S}}^{k}{\\\\mathbf{x}} , which aggregates at each node the signals within its k k -hop neighborhood. A linear shift-invariant graph filter of order K K is the matrix polynomial 𝐡 ( 𝐒 ) ≔ ∑ k = 0 K h k 𝐒 k {\\\\mathbf{h}}({\\\\mathbf{S}})\\\\coloneqq\\\\sum_{k=0}^{K}h_{k}{\\\\mathbf{S}}^{k} with taps (coefficients) 𝐡 = [ h 0 , … , h K ] {\\\\mathbf{h}}=[h_{0},\\\\ldots,h_{K}] . It processes an input signal 𝐱 {\\\\mathbf{x}} by graph convolution,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and produces an output graph signal 𝐲 ∈ ℝ N {\\\\mathbf{y}}\\\\in{\\\\mathbb{R}}^{N} as a weighted sum of the shifted signals 𝐒 k 𝐱 {\\\\mathbf{S}}^{k}{\\\\mathbf{x}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Graph neural networks. A GNN stacks graph convolutional layers, each composing a graph filter with a pointwise nonlinearity [ 8 ] . The ℓ \\\\ell th layer of an L L -layer GNN reads\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for ℓ = 1 , … , L \\\\ell=1,\\\\ldots,L . Here, each scalar tap of ( 10 ) becomes a MIMO filter bank 𝚯 ℓ , k \\\\boldsymbol{\\\\Theta}_{\\\\ell,k} that maps the F ℓ − 1 F_{\\\\ell-1} input features in 𝐗 ℓ − 1 {\\\\mathbf{X}}_{\\\\ell-1} to F ℓ F_{\\\\ell} output features within K K -hop neighborhoods, with learnable coefficients Θ ℓ = { 𝚯 ℓ , k } k = 0 K \\\\Theta_{\\\\ell}=\\\\{\\\\boldsymbol{\\\\Theta}_{\\\\ell,k}\\\\}_{k=0}^{K} and a pointwise nonlinearity σ ℓ \\\\sigma_{\\\\ell} , e.g., a ReLU. Stacking these layers defines the Θ \\\\Theta -parametrized GNN 𝚽 ( ⋅ , 𝐒 ; Θ ) = 𝚽 L ( ⋅ , 𝐒 ; Θ L ) ∘ ⋯ ∘ 𝚽 1 ( ⋅ , 𝐒 ; Θ 1 ) \\\\boldsymbol{\\\\Phi}(\\\\cdot,{\\\\mathbf{S}};\\\\Theta)=\\\\boldsymbol{\\\\Phi}_{L}(\\\\cdot,{\\\\mathbf{S}};\\\\Theta_{L})\\\\circ\\\\cdots\\\\circ\\\\boldsymbol{\\\\Phi}_{1}(\\\\cdot,{\\\\mathbf{S}};\\\\Theta_{1}) , with trainable parameters Θ = { Θ ℓ } ℓ = 1 L \\\\Theta=\\\\{\\\\Theta_{\\\\ell}\\\\}_{\\\\ell=1}^{L} . It maps an input 𝐗 0 ≔ 𝐗 ∈ ℝ N × F 0 {\\\\mathbf{X}}_{0}\\\\coloneqq{\\\\mathbf{X}}\\\\in{\\\\mathbb{R}}^{N\\\\times F_{0}} to an output 𝐗 L ≕ 𝐘 ∈ ℝ N × F L {\\\\mathbf{X}}_{L}\\\\eqqcolon{\\\\mathbf{Y}}\\\\in{\\\\mathbb{R}}^{N\\\\times F_{L}} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The GNN in ( 12 ) is general-purpose but not tailored to diffusion-based generation. Since graph convolutions already extract features on arbitrary graphs, adapting the U-Net backbone additionally requires only node (un)pooling, which we introduce next together with a stride on the graph shift.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pooling. Consider a signal 𝐗 ∈ ℝ N × F {\\\\mathbf{X}}\\\\in{\\\\mathbb{R}}^{N\\\\times F} on N N nodes. A sampling (selection) matrix 𝐃 ∈ { 0 , 1 } N ′ × N {\\\\mathbf{D}}\\\\in\\\\{0,1\\\\}^{N^{\\\\prime}\\\\times N} with N ′ < N N^{\\\\prime}<N retains the rows of 𝐈 N {\\\\mathbf{I}}_{N} indexed by a sampling set Ω ⊆ 𝒱 \\\\Omega\\\\subseteq{\\\\mathcal{V}} with | Ω | = N ′ |\\\\Omega|=N^{\\\\prime} . 1 1 1 We present 𝐃 {\\\\mathbf{D}} as binary for exposition. We relax it to a soft selection mask during training and binarize it to pick N ′ N^{\\\\prime} nodes at inference (Appendix C ). We learn Ω \\\\Omega , and hence 𝐃 {\\\\mathbf{D}} , end-to-end, so it is input-dependent rather than fixed (Section IV-B ). We call ρ ≥ 1 \\\\rho\\\\geq 1 with N ′ = ⌊ N / ρ ⌋ N^{\\\\prime}=\\\\lfloor N/\\\\rho\\\\rfloor the down-sampling (pooling) factor that sets the resolution reduction. By construction, 𝐃 {\\\\mathbf{D}} has orthonormal rows, 𝐃𝐃 ⊤ = 𝐈 N ′ {\\\\mathbf{D}}{\\\\mathbf{D}}^{\\\\top}={\\\\mathbf{I}}_{N^{\\\\prime}} , and 𝐃 ⊤ 𝐃 {\\\\mathbf{D}}^{\\\\top}{\\\\mathbf{D}} is a diagonal mask with ones on Ω \\\\Omega . The forward map 𝐙 = 𝐃𝐗 ∈ ℝ N ′ × F {\\\\mathbf{Z}}={\\\\mathbf{D}}{\\\\mathbf{X}}\\\\in{\\\\mathbb{R}}^{N^{\\\\prime}\\\\times F} down-samples 𝐗 {\\\\mathbf{X}} . Its adjoint 𝐃 ⊤ 𝐙 {\\\\mathbf{D}}^{\\\\top}{\\\\mathbf{Z}} up-samples back to N N nodes by zero-padding , i.e., it restores the active entries on Ω \\\\Omega and inserts zeros on the inactive nodes 𝒱 ∖ Ω {\\\\mathcal{V}}\\\\setminus\\\\Omega .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A graph convolutional layer with pooling processes a down-sampled signal 𝐙 ℓ − 1 ∈ ℝ N ′ × F ℓ − 1 {\\\\mathbf{Z}}_{\\\\ell-1}\\\\in{\\\\mathbb{R}}^{N^{\\\\prime}\\\\times F_{\\\\ell-1}} into 𝐙 ℓ ∈ ℝ N ′ × F ℓ {\\\\mathbf{Z}}_{\\\\ell}\\\\in{\\\\mathbb{R}}^{N^{\\\\prime}\\\\times F_{\\\\ell}} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We call ( 13 ) a lift–filter–reduce layer. We lift 𝐙 ℓ − 1 {\\\\mathbf{Z}}_{\\\\ell-1} to the full vertex set by zero-padding through 𝐃 ⊤ {\\\\mathbf{D}}^{\\\\top} , filter on 𝐒 {\\\\mathbf{S}} , and reduce the result back to the active set through 𝐃 {\\\\mathbf{D}} . Because the inactive nodes are zero, each active node aggregates only the active values in its neighborhood on 𝐒 {\\\\mathbf{S}} , which can be few under heavy pooling and motivates the stride introduced next.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stride. Modern CNNs, and U-Net variants in particular, often replace fixed pooling rules such as max- or sum-pooling with strided or dilated convolutions for learned down-sampling and receptive-field growth, e.g., [ 15 , 45 ] . A dilated convolution spaces its taps by a fixed step, so each tap reads from a point several positions away rather than from an immediate neighbor. On a graph, the natural spacing is the hop count. Thus, we introduce a stride (dilation) parameter γ ∈ ℤ + \\\\gamma\\\\in\\\\mathbb{Z}^{+} and replace the unit shift 𝐒 {\\\\mathbf{S}} with the γ \\\\gamma -hop shift 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} . The order- K K filter then aggregates from the dilated hops { 0 , γ , 2 γ , … , K γ } \\\\{0,\\\\gamma,2\\\\gamma,\\\\ldots,K\\\\gamma\\\\} rather than the consecutive hops { 0 , 1 , … , K } \\\\{0,1,\\\\ldots,K\\\\} . This yields the strided graph convolutional layer with pooling ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which recovers the plain layer ( 11 ) when 𝐃 = 𝐈 N {\\\\mathbf{D}}={\\\\mathbf{I}}_{N} ( ρ = 1 \\\\rho=1 ) and γ = 1 \\\\gamma=1 . The pooling factor ρ \\\\rho and stride γ \\\\gamma serve complementary roles. Through 𝐃 {\\\\mathbf{D}} , the factor ρ \\\\rho sets how many nodes survive and hence the sparsity of the zero-padded signal. Through 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} , the stride γ \\\\gamma sets how far each tap reaches [cf. Remark 1 ].\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The strided GNN layer ( 14 ) keeps the lift–filter–reduce form of ( 13 ), filtering on 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} instead of 𝐒 {\\\\mathbf{S}} . The computations remain on the original vertex set, and 𝐒 {\\\\mathbf{S}} is reused at every resolution without explicitly forming 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} in our implementation. Instead, we filter through γ K \\\\gamma K successive unit shifts and tap the signal every γ \\\\gamma hops. Thus, each layer reduces to sparse matrix–vector products, which we realize with the sparse routines of the PyG library. We refer to Appendix B for details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reduced GSOs. The strided layer ( 14 ) also admits an equivalent form that operates directly on the N ′ N^{\\\\prime} -dimensional signals. Pushing 𝐃 {\\\\mathbf{D}} and 𝐃 ⊤ {\\\\mathbf{D}}^{\\\\top} through the sum defines the k k -hop reduced graph shift operators (GSOs) ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"whose one-hop instance 𝐒 ( 1 ) {\\\\mathbf{S}}^{(1)} is the support of a coarse subgraph on Ω \\\\Omega . The family { 𝐒 ( k ) } k = 0 K \\\\{{\\\\mathbf{S}}^{(k)}\\\\}_{k=0}^{K} then processes the low-dimensional signals directly,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two forms are equal because row selection commutes with the pointwise nonlinearity, 𝐃 σ ℓ ( 𝐌 ) = σ ℓ ( 𝐃𝐌 ) {\\\\mathbf{D}}\\\\,\\\\sigma_{\\\\ell}({\\\\mathbf{M}})=\\\\sigma_{\\\\ell}({\\\\mathbf{D}}{\\\\mathbf{M}}) , and they differ only in the domain of computation. The reduced form avoids computation on the inactive nodes but is ill-suited to our setting. The reduced GSOs do not compose, i.e., 𝐒 ( m + n ) ≠ 𝐒 ( m ) 𝐒 ( n ) {\\\\mathbf{S}}^{(m+n)}\\\\neq{\\\\mathbf{S}}^{(m)}{\\\\mathbf{S}}^{(n)} , since 𝐃 ⊤ 𝐃 ≠ 𝐈 N {\\\\mathbf{D}}^{\\\\top}{\\\\mathbf{D}}\\\\neq{\\\\mathbf{I}}_{N} in general. Therefore, each 𝐒 ( k ) {\\\\mathbf{S}}^{(k)} must be formed separately rather than by powering 𝐒 ( 1 ) {\\\\mathbf{S}}^{(1)} . Moreover, 𝐃 {\\\\mathbf{D}} is input-dependent, so the family { 𝐒 ( k ) } \\\\{{\\\\mathbf{S}}^{(k)}\\\\} is rebuilt on every forward pass, negating any gains from pre-computing. We therefore adopt the lift–filter–reduce form.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GNNs with pooling and stride. Stacking L L strided graph convolutional layers with pooling [cf. ( 14 )] defines the ( Θ , 𝐃 ) (\\\\Theta,{\\\\mathbf{D}}) -parametrized GNN module 𝚽 ( ⋅ , 𝐒 ; Θ , 𝐃 ) = 𝚽 L ( ⋅ , 𝐒 γ ; Θ L , 𝐃 ) ∘ ⋯ ∘ 𝚽 1 ( ⋅ , 𝐒 γ ; Θ 1 , 𝐃 ) \\\\boldsymbol{\\\\Phi}(\\\\cdot,{\\\\mathbf{S}};\\\\Theta,{\\\\mathbf{D}})=\\\\boldsymbol{\\\\Phi}_{L}(\\\\cdot,{\\\\mathbf{S}}^{\\\\gamma};\\\\Theta_{L},{\\\\mathbf{D}})\\\\circ\\\\cdots\\\\circ\\\\boldsymbol{\\\\Phi}_{1}(\\\\cdot,{\\\\mathbf{S}}^{\\\\gamma};\\\\Theta_{1},{\\\\mathbf{D}}) . It maps an input 𝐙 0 ≔ 𝐗 ∈ ℝ N ′ × F 0 {\\\\mathbf{Z}}_{0}\\\\coloneqq{\\\\mathbf{X}}\\\\in{\\\\mathbb{R}}^{N^{\\\\prime}\\\\times F_{0}} to an output 𝐙 L ≕ 𝐘 ∈ ℝ N ′ × F L {\\\\mathbf{Z}}_{L}\\\\eqqcolon{\\\\mathbf{Y}}\\\\in{\\\\mathbb{R}}^{N^{\\\\prime}\\\\times F_{L}} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with parameters Θ = { Θ ℓ } ℓ = 1 L \\\\Theta=\\\\{\\\\Theta_{\\\\ell}\\\\}_{\\\\ell=1}^{L} and 𝐃 ∈ { 0 , 1 } N ′ × N {\\\\mathbf{D}}\\\\in\\\\{0,1\\\\}^{N^{\\\\prime}\\\\times N} . This GNN module is the main processing unit of each block in the U-GNN architecture, which is introduced next.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In both views, the stride γ \\\\gamma offsets the sparsity introduced by down-sampling with 𝐃 {\\\\mathbf{D}} . Under lift–filter–reduce, a stride such as γ = ⌊ ρ ⌋ \\\\gamma=\\\\lfloor\\\\sqrt{\\\\rho}\\\\,\\\\rfloor keeps nodes from aggregating predominantly zeros. Under the reduced GSO, projecting through 𝐃 {\\\\mathbf{D}} may isolate nodes in the induced subgraph, and 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} with γ > 1 \\\\gamma>1 restores connectivity. A suitable value depends on the average connectivity and the uniformity of 𝐃 {\\\\mathbf{D}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A variant of ( 16 ) instead learns a genuine polynomial filter in the one-hop reduced GSO 𝐒 ( 1 ) = 𝐃 𝐒 γ 𝐃 ⊤ {\\\\mathbf{S}}^{(1)}={\\\\mathbf{D}}\\\\,{\\\\mathbf{S}}^{\\\\gamma}\\\\,{\\\\mathbf{D}}^{\\\\top} , the principal submatrix of 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} on Ω \\\\Omega . This is no longer equivalent to ( 14 ), since sampling and filtering commute only at first order. Powering 𝐒 ( 1 ) {\\\\mathbf{S}}^{(1)} confines every hop to Ω \\\\Omega , whereas the family 𝐒 ( k ) {\\\\mathbf{S}}^{(k)} allows walks to traverse inactive nodes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV U-Graph Neural Networks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The GNN in ( 17 ) processes graph signals at a single node resolution fixed by the sampling matrix 𝐃 {\\\\mathbf{D}} . To build the multi-resolution encoder–decoder hierarchy of a U-Net on top of this module, we first define the resolutions through depth-dependent selection matrices { 𝐂 b } b = 1 B \\\\{{\\\\mathbf{C}}_{b}\\\\}_{b=1}^{B} , where 𝐂 b ∈ { 0 , 1 } N b × N b − 1 {\\\\mathbf{C}}_{b}\\\\in\\\\{0,1\\\\}^{N_{b}\\\\times N_{b-1}} selects N b N_{b} nodes from the N b − 1 N_{b-1} nodes active at the preceding depth, with N 0 = N N_{0}=N and N = N 1 ≥ N 2 ≥ ⋯ ≥ N B N=N_{1}\\\\geq N_{2}\\\\geq\\\\cdots\\\\geq N_{B} . We form the composite (nested) sampling matrices,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which relate the depth- b b nodes to the full N N -node domain and induce a nested hierarchy of sampling sets Ω B ⊆ ⋯ ⊆ Ω 1 ⊆ 𝒱 \\\\Omega_{B}\\\\subseteq\\\\cdots\\\\subseteq\\\\Omega_{1}\\\\subseteq{\\\\mathcal{V}} . We set 𝐃 1 = 𝐂 1 = 𝐈 N {\\\\mathbf{D}}_{1}={\\\\mathbf{C}}_{1}={\\\\mathbf{I}}_{N} by convention, so the first depth operates at full resolution. Here, 𝐂 b {\\\\mathbf{C}}_{b} and 𝐃 b {\\\\mathbf{D}}_{b} carry the per-level and cumulative down-sampling factors ρ b ≔ N b − 1 / N b \\\\rho_{b}\\\\coloneqq N_{b-1}/N_{b} ( ρ 1 = 1 \\\\rho_{1}=1 ) and ρ ¯ b ≔ ∏ i = 1 b ρ i = N / N b \\\\bar{\\\\rho}_{b}\\\\coloneqq\\\\prod_{i=1}^{b}\\\\rho_{i}=N/N_{b} , respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We call the resulting architecture a U-Graph Neural Network (U-GNN) . It spans B B resolution levels N = N 1 ≥ ⋯ ≥ N B N=N_{1}\\\\geq\\\\cdots\\\\geq N_{B} , with B − 1 B-1 encoder–decoder block pairs and a bottleneck block at the coarsest resolution N B N_{B} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present the U-GNN in two stages. We first fix a pure encoder–decoder cascade whose blocks are single encoding, decoding, and bottleneck GNN modules [cf. ( 17 )], written 𝚽 b E \\\\boldsymbol{\\\\Phi}^{{\\\\text{E}}}_{b} , 𝚽 b D \\\\boldsymbol{\\\\Phi}^{{\\\\text{D}}}_{b} , or 𝚽 B B \\\\boldsymbol{\\\\Phi}^{{\\\\text{B}}}_{B} with weights Θ b E \\\\Theta^{{\\\\text{E}}}_{b} , Θ b D \\\\Theta^{{\\\\text{D}}}_{b} , and Θ B B \\\\Theta^{{\\\\text{B}}}_{B} , respectively, along with the signal flow among them in ( 19 )–( 21 ). These equations include the skip connections, take the selection matrices 𝐂 b {\\\\mathbf{C}}_{b} as given, and omit the node-state and diffusion-step embeddings. We then complete the blocks with projection (fusion) layers and, on the encoder side, node-selection heads (Sections IV-A and IV-B ). These only enlarge the block interiors and parameter sets, leaving the composition here unchanged. The U-GNN overview in Fig. 2 and the complementary block interface details in Fig. 7 (deferred to Appendix E due to limited space) may be consulted throughout this section.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The encoder blocks form an encoding path 𝚽 E = 𝚽 B − 1 E ∘ ⋯ ∘ 𝚽 1 E \\\\boldsymbol{\\\\Phi}^{{\\\\text{E}}}=\\\\boldsymbol{\\\\Phi}^{{\\\\text{E}}}_{B-1}\\\\circ\\\\cdots\\\\circ\\\\boldsymbol{\\\\Phi}^{{\\\\text{E}}}_{1} , and the decoder blocks a decoding path 𝚽 D = 𝚽 1 D ∘ ⋯ ∘ 𝚽 B − 1 D \\\\boldsymbol{\\\\Phi}^{{\\\\text{D}}}=\\\\boldsymbol{\\\\Phi}^{{\\\\text{D}}}_{1}\\\\circ\\\\cdots\\\\circ\\\\boldsymbol{\\\\Phi}^{{\\\\text{D}}}_{B-1} . The encoding path extracts graph convolutional features at progressively coarser resolutions, while the decoding path reverses this progression by restoring resolution and reintroducing fine details through skip connections.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We describe each path in turn, starting from the input features 𝐙 0 {\\\\mathbf{Z}}_{0} . Each encoder block maps its input to an encoded signal 𝐙 b ∈ ℝ N b × F b {\\\\mathbf{Z}}_{b}\\\\in{\\\\mathbb{R}}^{N_{b}\\\\times F_{b}} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for b = 1 , … , B − 1 b=1,\\\\ldots,B-1 , where 𝐂 b {\\\\mathbf{C}}_{b} down-samples the depth- ( b − 1 ) (b{-}1) feature onto the N b N_{b} nodes processed at depth b b . At the top level, 𝐂 1 = 𝐈 N {\\\\mathbf{C}}_{1}={\\\\mathbf{I}}_{N} and 𝚽 1 E \\\\boldsymbol{\\\\Phi}^{{\\\\text{E}}}_{1} operates at full resolution N 1 = N N_{1}=N .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each decoder block receives the coarser decoded signal 𝐘 b + 1 ∈ ℝ N b + 1 × F b + 1 {\\\\mathbf{Y}}_{b+1}\\\\in{\\\\mathbb{R}}^{N_{b+1}\\\\times F_{b+1}} from depth b + 1 b+1 , up-samples it onto the N b N_{b} -node support through 𝐂 b + 1 ⊤ {\\\\mathbf{C}}^{\\\\top}_{b+1} , and combines it with the matched encoder feature 𝐙 b {\\\\mathbf{Z}}_{b} through a skip connection,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for b = B − 1 , … , 1 b=B-1,\\\\ldots,1 , where [ ⋅ ; ⋅ ] [\\\\,\\\\cdot\\\\,;\\\\,\\\\cdot\\\\,] denotes the skip combination. The decoded signal 𝐘 b {\\\\mathbf{Y}}_{b} then passes to the next decoder block or to the read-out layer at b = 1 b=1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two paths connect at depth B B through the bottleneck block, which acts as a B B th encoder block,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The bottleneck output 𝐘 B {\\\\mathbf{Y}}_{B} enters the decoding path ( 20 ) at depth b = B − 1 b=B-1 , and the final decoder output 𝐘 1 ∈ ℝ N × F 0 {\\\\mathbf{Y}}_{1}\\\\in{\\\\mathbb{R}}^{N\\\\times F_{0}} is mapped by a read-out layer to the model target, i.e., the noise prediction ϵ 𝜽 \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}} here. We note that all blocks share the same 𝐒 {\\\\mathbf{S}} , and a depth-dependent stride γ b \\\\gamma_{b} is implicit in the notation [cf. ( 14 )]. Appendix B specifies how γ b \\\\gamma_{b} is set and how each block evaluates its strided convolutions on the shared 𝐒 {\\\\mathbf{S}} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Input Processing and Block-Level Conditioning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The U-GNN realizes the denoiser ϵ 𝜽 ( 𝐱 k , k ; 𝝃 ) \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}}({\\\\mathbf{x}}_{k},k;\\\\boldsymbol{\\\\xi}) with conditioning 𝝃 = ( 𝐒 , 𝐮 ) \\\\boldsymbol{\\\\xi}=({\\\\mathbf{S}},{\\\\mathbf{u}}) . It incorporates the conditioning along two routes. The graph enters as the shared shift operator 𝐒 {\\\\mathbf{S}} of every GNN module [cf. Section III ], while the node states 𝐮 {\\\\mathbf{u}} enter as side information, embedded and fused into each block together with an embedding of the diffusion step k k . Note that the step k k is a model input and not part of the conditioning 𝝃 \\\\boldsymbol{\\\\xi} . We now detail the embedding pipeline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A read-in layer 𝚷 0 \\\\boldsymbol{\\\\Pi}_{0} maps the noisy signal 𝐱 k ∈ ℝ N {\\\\mathbf{x}}_{k}\\\\in{\\\\mathbb{R}}^{N} to the initial node embeddings 𝐙 0 ∈ ℝ N × F 0 {\\\\mathbf{Z}}_{0}\\\\in{\\\\mathbb{R}}^{N\\\\times F_{0}} that enter the first encoder block. The node states 𝐮 ∈ ℝ N × U {\\\\mathbf{u}}\\\\in{\\\\mathbb{R}}^{N\\\\times U} and the step k k are embedded separately into global representations shared across all blocks. A node-wise MLP 𝚷 0 U \\\\boldsymbol{\\\\Pi}^{{\\\\text{U}}}_{0} (a shared per-node MLP with no cross-node mixing) maps 𝐮 {\\\\mathbf{u}} to per-node embeddings 𝐔 0 = 𝚷 0 U ( 𝐮 ) ∈ ℝ N × F 0 {\\\\mathbf{U}}_{0}=\\\\boldsymbol{\\\\Pi}^{{\\\\text{U}}}_{0}({\\\\mathbf{u}})\\\\in{\\\\mathbb{R}}^{N\\\\times F_{0}} . A diffusion-step encoder 𝚷 0 k \\\\boldsymbol{\\\\Pi}^{{\\\\text{k}}}_{0} , a sinusoidal positional encoding followed by a node-wise MLP, maps the scalar k k to an F 0 F_{0} -dimensional vector broadcast to all nodes as 𝐊 0 = 𝟏 N 𝚷 0 k ( k ) ⊤ ∈ ℝ N × F 0 {\\\\mathbf{K}}_{0}={\\\\mathbf{1}}_{N}\\\\,\\\\boldsymbol{\\\\Pi}^{{\\\\text{k}}}_{0}(k)^{\\\\top}\\\\in{\\\\mathbb{R}}^{N\\\\times F_{0}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At each depth b b , a fusion layer 𝚷 b \\\\boldsymbol{\\\\Pi}_{b} merges the main signal path with these global embeddings before the GNN module. The block input 𝐕 b {\\\\mathbf{V}}_{b} and the down-sampled embeddings 𝐃 b [ 𝐔 0 ; 𝐊 0 ] {\\\\mathbf{D}}_{b}\\\\,[{\\\\mathbf{U}}_{0};\\\\,{\\\\mathbf{K}}_{0}] are each projected to a common width, merged along the feature dimension, and mapped to F b F_{b} channels,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The factor 𝐃 b {\\\\mathbf{D}}_{b} restricts the global embeddings to the N b N_{b} nodes active at depth b b , matching the support of 𝐕 b {\\\\mathbf{V}}_{b} , and the per-input projection reconciles their differing feature (channel) widths. The block input depends on the path. For b b th encoder block, 𝐕 b = 𝐂 b 𝐙 b − 1 {\\\\mathbf{V}}_{b}={\\\\mathbf{C}}_{b}\\\\,{\\\\mathbf{Z}}_{b-1} down-samples the depth- ( b − 1 ) (b{-}1) feature onto the N b N_{b} active nodes. For b b th decoder block, the coarser output 𝐘 b + 1 ∈ ℝ N b + 1 × F b + 1 {\\\\mathbf{Y}}_{b+1}\\\\in{\\\\mathbb{R}}^{N_{b+1}\\\\times F_{b+1}} is up-sampled through 𝐂 b + 1 ⊤ {\\\\mathbf{C}}_{b+1}^{\\\\top} , concatenated with the encoder skip 𝐙 b {\\\\mathbf{Z}}_{b} along the feature dimension, and mapped to F b F_{b} channels by a learned skip-projection 𝚷 b skip : ℝ F b + 1 + F b → ℝ F b \\\\boldsymbol{\\\\Pi}^{\\\\mathrm{skip}}_{b}:{\\\\mathbb{R}}^{F_{b+1}+F_{b}}\\\\to{\\\\mathbb{R}}^{F_{b}} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fusion layer 𝚷 b \\\\boldsymbol{\\\\Pi}_{b} is shared between the matched encoder and decoder blocks, so its weights appear in both Θ b E \\\\Theta^{{\\\\text{E}}}_{b} and Θ b D \\\\Theta^{{\\\\text{D}}}_{b} , while the GNN module and skip-projection remain block-specific. The node-state and diffusion-step embeddings enter each block by input substitution. In ( 19 ) and ( 21 ), the module input 𝐂 b 𝐙 b − 1 {\\\\mathbf{C}}_{b}\\\\,{\\\\mathbf{Z}}_{b-1} is replaced by the fused 𝐏 b {\\\\mathbf{P}}_{b} . In ( 20 ), the skip combination [ 𝐂 b + 1 ⊤ 𝐘 b + 1 ; 𝐙 b ] [{\\\\mathbf{C}}^{\\\\top}_{b+1}{\\\\mathbf{Y}}_{b+1};\\\\,{\\\\mathbf{Z}}_{b}] is first mapped by 𝚷 b skip \\\\boldsymbol{\\\\Pi}^{\\\\mathrm{skip}}_{b} [cf. ( 23 )] and then fused into 𝐏 b {\\\\mathbf{P}}_{b} . The GNN modules and the block composition remain unchanged, and only the module inputs differ. Figure 2 shows this separation, with the fusion drawn as the purple projection 𝚷 b \\\\boldsymbol{\\\\Pi}_{b} preceding each green GNN module, and Fig. 7 (Appendix E ) details the block interface. Finally, a read-out layer 𝚷 out \\\\boldsymbol{\\\\Pi}_{\\\\mathrm{out}} , a node-wise MLP, maps the final decoder output 𝐘 1 ∈ ℝ N × F 0 {\\\\mathbf{Y}}_{1}\\\\in{\\\\mathbb{R}}^{N\\\\times F_{0}} to the prediction ϵ 𝜽 ∈ ℝ N \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}}\\\\in{\\\\mathbb{R}}^{N} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Down-sampling (Pooling) via Learned Node Selection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selection matrices 𝐂 b {\\\\mathbf{C}}_{b} and their compositions 𝐃 b {\\\\mathbf{D}}_{b} determine the active nodes at each depth. Rather than fixing them in advance, we learn them end-to-end by letting each encoder block score its own GNN output and select the active nodes propagated to the next depth. The node-selection head of encoder block b b derives 𝐂 b + 1 {\\\\mathbf{C}}_{b+1} from the encoded feature 𝐙 b {\\\\mathbf{Z}}_{b} , for b = 1 , … , B − 1 b=1,\\\\ldots,B-1 , producing 𝐂 2 , … , 𝐂 B {\\\\mathbf{C}}_{2},\\\\ldots,{\\\\mathbf{C}}_{B} . The top level performs no selection since 𝐂 1 = 𝐃 1 = 𝐈 N {\\\\mathbf{C}}_{1}={\\\\mathbf{D}}_{1}={\\\\mathbf{I}}_{N} , and the deepest selection 𝐂 B {\\\\mathbf{C}}_{B} down-samples into the bottleneck, which itself does not further down-sample.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We describe the selection operation using two index systems. Global indices label nodes in 𝒱 {\\\\mathcal{V}} , with the active set Ω b ⊆ 𝒱 \\\\Omega_{b}\\\\subseteq{\\\\mathcal{V}} , | Ω b | = N b |\\\\Omega_{b}|=N_{b} , nested as in the beginning of Section IV . Local indices [ N b ] ≔ { 1 , … , N b } [N_{b}]\\\\coloneqq\\\\{1,\\\\ldots,N_{b}\\\\} enumerate only the active nodes in a fixed order. The composite matrix 𝐃 b ∈ { 0 , 1 } N b × N {\\\\mathbf{D}}_{b}\\\\in\\\\{0,1\\\\}^{N_{b}\\\\times N} maps a global-indexed signal to its local form, and 𝐃 b ⊤ {\\\\mathbf{D}}_{b}^{\\\\top} scatters a local-indexed signal back to the N N -node domain. The selection below acts in the local space [ N b ] [N_{b}] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each encoder block is equipped with a head that scores the active nodes and keeps the highest-scoring ones. Concretely, a node-wise MLP 𝚿 b \\\\boldsymbol{\\\\Psi}_{b} , with parameters 𝚯 b sel \\\\boldsymbol{\\\\Theta}^{\\\\mathrm{sel}}_{b} , takes the block output 𝐙 b {\\\\mathbf{Z}}_{b} , along with the node and step embeddings restricted to the active nodes, and returns a per-node score vector,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The dependence on 𝐃 b [ 𝐔 0 ; 𝐊 0 ] {\\\\mathbf{D}}_{b}\\\\,[{\\\\mathbf{U}}_{0};\\\\,{\\\\mathbf{K}}_{0}] , as in ( 22 ), allows the selection to adapt to the node states and the diffusion step. Given a down-sampling factor ρ b + 1 > 1 \\\\rho_{b+1}>1 , we keep the N b + 1 = ⌊ N b / ρ b + 1 ⌋ N_{b+1}=\\\\lfloor N_{b}/\\\\rho_{b+1}\\\\rfloor highest-scoring nodes,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selection matrix 𝐂 b + 1 = [ 𝐈 N b ] 𝒮 b , : ∈ { 0 , 1 } N b + 1 × N b {\\\\mathbf{C}}_{b+1}=\\\\big[{\\\\mathbf{I}}_{N_{b}}\\\\big]_{\\\\mathcal{S}_{b},\\\\,:}\\\\in\\\\{0,1\\\\}^{N_{b+1}\\\\times N_{b}} keeps the rows of 𝐈 N b {\\\\mathbf{I}}_{N_{b}} indexed by 𝒮 b \\\\mathcal{S}_{b} , and the composite matrix updates as 𝐃 b + 1 = 𝐂 b + 1 𝐃 b {\\\\mathbf{D}}_{b+1}={\\\\mathbf{C}}_{b+1}{\\\\mathbf{D}}_{b} [cf. ( 18 )]. The retained global set Ω b + 1 ⊆ Ω b \\\\Omega_{b+1}\\\\subseteq\\\\Omega_{b} collects the nonzero columns of 𝐃 b + 1 {\\\\mathbf{D}}_{b+1} , so 𝒮 b \\\\mathcal{S}_{b} and Ω b + 1 \\\\Omega_{b+1} express the same selection in local and global coordinates, respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The decoder performs no selection. At each depth, the encoder-produced 𝐂 b + 1 {\\\\mathbf{C}}_{b+1} is reused by the matched decoder for up-sampling through 𝐂 b + 1 ⊤ {\\\\mathbf{C}}_{b+1}^{\\\\top} [cf. ( 23 )], while 𝐃 b {\\\\mathbf{D}}_{b} parametrizes the depth- b b GNN modules [cf. ( 19 )–( 21 )].\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Top-K operation in ( 25 ) is non-differentiable in 𝐯 b {\\\\mathbf{v}}_{b} . Therefore, we train the head 𝚿 b \\\\boldsymbol{\\\\Psi}_{b} with a straight-through estimator (STE) [ 1 ] that keeps the hard selection 𝒮 b \\\\mathcal{S}_{b} in the forward pass, while gradients reach the scores 𝐯 b {\\\\mathbf{v}}_{b} through a differentiable sigmoid surrogate of the binary selection mask. Appendix C details the surrogate mask, and Appendix D integrates classical fixed-pooling rules with the U-GNN.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Numerical Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the proposed U-GNN denoiser on stock price forecasting for the S&P 500 index and wireless resource allocation (WRA) in a power control setup. Both use the same depth- B = 4 B{=}4 U-GNN backbone ϵ 𝜽 \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}} , diffusion noise schedules, and optimizers. They differ only in the signal and conditioning interface that each task dictates—a multi-step return trajectory for forecasting versus a single power allocation for resource allocation. Table I summarizes the model, diffusion, and optimization hyperparameters. We detail the shared and application-specific configurations in Appendix E .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-A Stock Price Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We cast stock price forecasting as a spatio-temporal generative task. Given recent market history for a panel of stocks, the goal is to sample from the conditional distribution of their short-horizon future log-return trajectories.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. We use daily S&P 500 data obtained with the open-source yfinance library. After aligning all stocks on a common trading day axis and dropping those with insufficient coverage, we retain N = 468 N=468 stocks across 11 11 GICS sectors over 2353 2353 trading days (2016-09-06 to 2026-02-13). Let 𝐫 ( t ) ∈ ℝ N {\\\\mathbf{r}}^{(t)}\\\\in{\\\\mathbb{R}}^{N} collect the daily log returns on trading day t t , where [ 𝐫 ( t ) ] i [{\\\\mathbf{r}}^{(t)}]_{i} is the log ratio of stock i i ’s consecutive closing prices. For each stock and day, we gather U = 12 U=12 market features into 𝐮 ( t ) ∈ ℝ N × U {\\\\mathbf{u}}^{(t)}\\\\in{\\\\mathbb{R}}^{N\\\\times U} , comprising the open, high, low, and standardized closing prices; the log return 𝐫 ( t ) {\\\\mathbf{r}}^{(t)} and its trailing moving averages; the log-volume; and two technical indicators, RSI and MACD.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We build a static, undirected graph 𝒢 {\\\\mathcal{G}} from long-term company fundamentals (e.g., market capitalization and price-to-earnings ratios). Its adjacency 𝐖 {\\\\mathbf{W}} combines the rank correlation between two stocks’ fundamental profiles with a same-sector bonus. We threshold and spectrally normalize 𝐖 {\\\\mathbf{W}} to obtain the GSO 𝐒 {\\\\mathbf{S}} , which is shared across all forecast windows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We form data samples with a sliding window of length ( T h + T p ) (T_{h}+T_{p}) over the panel. In each window, the first T h T_{h} days are the conditioning history, stacked into 𝐮 = [ 𝐮 ( t − T h + 1 ) , ⋯ , 𝐮 ( t ) ] ∈ ℝ N × T h × U {\\\\mathbf{u}}=[\\\\,{\\\\mathbf{u}}^{(t-T_{h}+1)},\\\\ \\\\cdots,\\\\ {\\\\mathbf{u}}^{(t)}\\\\,]\\\\in{\\\\mathbb{R}}^{N\\\\times T_{h}\\\\times U} , and the next T p T_{p} days are the forecast target, stacked into the clean graph signal 𝐱 0 = [ 𝐫 ( t + 1 ) , ⋯ , 𝐫 ( t + T p ) ] ∈ ℝ N × T p {\\\\mathbf{x}}_{0}=[\\\\,{\\\\mathbf{r}}^{(t+1)},\\\\ \\\\cdots,\\\\ {\\\\mathbf{r}}^{(t+T_{p})}\\\\,]\\\\in{\\\\mathbb{R}}^{N\\\\times T_{p}} . We reversibly instance-normalize (RevIN) each target window [ 17 ] , using statistics from its conditioning block alone, and we invert this normalization on the DDIM-generated samples before scoring. The U-GNN treats the T p T_{p} horizon steps as a temporal axis. It applies graph convolutions on 𝐒 {\\\\mathbf{S}} at each step, couples the steps through interleaved temporal-convolution layers, and fuses a per-node summary of the T h T_{h} history steps into the conditioning context by cross-attention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We set T h = 20 T_{h}=20 and T p = 5 T_{p}=5 , yielding 2328 2328 windows in total. We partition them using an interleaved chronological scheme. We split the trading-day axis into 10 10 equal chunks. Within each chunk, we assign the earliest 80 % 80\\\\% days to training, the next 10 % 10\\\\% to validation, and the final 10 % 10\\\\% to testing, then pool the matching parts across chunks. This ordering keeps each test block later than its training block, so the model is never fitted on days that follow its test period. Interleaving also spreads anomalous regimes, such as COVID-19, across all three splits rather than concentrating them in one, which mitigates the test distribution shift that a single contiguous split would create.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results. We assess the U-GNN forecasts qualitatively through the sample trajectories of Fig. 3 and quantitatively through the accuracy and distributional fidelity metrics of Table II , both computed on the held-out test split with 100 100 sampled trajectories per window. We benchmark U-GNN against a geometric random walk (GRW), in which each stock’s future log-returns are drawn i.i.d. from a fixed per-stock Gaussian. Its drift and volatility are estimated once on the training set, with shrinkage toward the market-wide statistics, and they are held fixed at inference, independently of the conditioning window.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"U-GNN outperforms the GRW on nearly every metric, at both the return and price scales. It reduces price CRPS, RMSE, and MAE by 22 % 22\\\\% , 28 % 28\\\\% , and 21 % 21\\\\% , respectively, and recovers directional information unavailable to a random walk, raising majority-vote direction accuracy above chance ( 0.53 0.53 vs. 0.49 0.49 ). Beyond point and interval accuracy, U-GNN reproduces stylized facts of returns. It substantially shrinks the volatility-clustering gap ( 0.06 0.06 vs. 0.10 0.10 ) and reduces the momentum gap. The one exception is excess kurtosis, where it overshoots the empirical heavy tails. This overshoot is the artifact of a desirable property. The GRW confines nearly all of its mass to a band around the mean, whereas U-GNN places non-negligible probability on the rare, far-from-mean moves that real markets exhibit. At the reported checkpoint it allocates somewhat more mass to these extremes than the data warrant, and we read this gap as a fidelity–sharpness trade-off rather than a fixed limitation. Checkpoints with a near-vanishing kurtosis gap are attainable, but only by sacrificing sharpness. Sharp forecasts emerge early in training and distributional fidelity improves later, with the reported model sitting toward the sharper end (Fig. 4(d) ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These dynamics are evident in the individual forecasts of Fig. 3 . The sampled trajectories spread out from the last observed price, and the spread grows over the forecast days, while their mean stays close to the realized path. Each sample tends to track the recent behavior of its stock, picking up its trend and volatility rather than flattening into a straight line or spreading evenly in all directions. A few paths break from the bundle to trace larger excursions, which is the generative model anticipating the occasional market swings noted above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 4 assesses the distributional fidelity of U-GNN samples against those of GRW along four axes. Fig. 4(a) shows that U-GNN matches the autocorrelation signature of stock returns, which is near-zero for r r and persistent for r 2 r^{2} , with the latter implying strong volatility clustering. Fig. 4(b) pertains to the cross-sectional structure, where U-GNN tracks the leading eigenvalues of the return covariance C ^ \\\\hat{C} that capture the dominant market directions, whereas the GRW baseline flattens them. The remaining two plots take a score-based view. Each assigns every evaluation window a scalar score, then compares the resulting distribution to the real one through the 1 1 -Wasserstein ( W 1 W_{1} ) distance between their empirical CDFs. They differ only in which model supplies the score.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 4(c) uses the GRW score, the analytic NLL per (stock, step) in nats, for which an i.i.d. Gaussian is lower-bounded by its differential entropy 1 2 ln ( 2 π e ) ≈ 1.42 \\\\tfrac{1}{2}\\\\ln(2\\\\pi e)\\\\approx 1.42 (dotted line). Scored by the homoskedastic Gaussian that generated them, the GRW samples collapse onto a near-degenerate spike essentially at this floor, so every window is equally surprising, and the score carries no volatility information. The real returns, instead, spread out with a rightward skew, as calm windows fall below the floor and heavy-tailed ones, beyond the reach of the GRW, are pushed into the tail. U-GNN’s samples sit just left of the floor (mean ≈ 1.25 \\\\approx 1.25 ), which the GRW scorer even rates as more typical than an i.i.d. draw. Even under the baseline’s own likelihood, U-GNN lands closer to the real distribution ( W 1 = 0.14 W_{1}{=}0.14 ) than GRW’s own samples do ( W 1 = 0.23 W_{1}{=}0.23 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 4(d) instead scores each window by a uniformly weighted version of U-GNN’s training loss in ( 6 ), estimated by a stratified Monte Carlo average over 64 64 strata of the diffusion step (one noise draw each). This orders windows by denoising difficulty, and hence by volatility. U-GNN’s samples cluster lowest (mean ≈ 0.28 \\\\approx 0.28 ), while the real data lies just to their right (mean ≈ 0.32 \\\\approx 0.32 ). GRW paths form a distinct high-error mode (mean ≈ 0.46 \\\\approx 0.46 ), since memoryless Gaussian increments are hard to denoise for a model that expects temporal and cross-sectional structures. Here, U-GNN nearly coincides with the real data ( W 1 = 0.04 W_{1}{=}0.04 ), while GRW sits farther away ( W 1 = 0.14 W_{1}{=}0.14 ). Neither score is impartial on its own, but read together, they corroborate that U-GNN better captures the real distribution. While U-GNN’s score isolates GRW samples as unrealistic, GRW ranks U-GNN above the real data. That said, the temporal nature of stock data may favor attention-based architectures suited to long-range dependencies. We leave the integration of a graph transformer with our U-GNN backbone to future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-B Wireless Resource Allocation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We study optimal power control in multi-user interference networks. The setup closely follows our preliminary work on diffusion policies for resource allocation [ 36 , 37 ] and the related state-augmented formulations of [ 39 ] . We summarize it below and collect the simulation parameters in Table III .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. We consider wireless networks of N = 400 N=400 transmitter–receiver (tx–rx) pairs dropped uniformly at random over a square service area. The pairs share a single channel of bandwidth W W with noise PSD N 0 N_{0} and per-transmitter power budget P max P_{\\\\max} (Table III ). The channel gain h i j , t ∈ ℝ + h_{ij,t}\\\\in{\\\\mathbb{R}}_{+} from transmitter i i to receiver j j at step t t combines a static large-scale component with a unit-mean Rayleigh fast-fading component. The large-scale component follows a dual-slope path-loss law with log-normal shadowing and fixes the long-term average gain h i j h_{ij} . We collect the long-term and instantaneous gains in 𝐇 , 𝐇 t ∈ ℝ + N × N {\\\\mathbf{H}},{\\\\mathbf{H}}_{t}\\\\in{\\\\mathbb{R}}^{N\\\\times N}_{+} , with [ 𝐇 ] i j = h i j [{\\\\mathbf{H}}]_{ij}=h_{ij} and [ 𝐇 t ] i j = h i j , t [{\\\\mathbf{H}}_{t}]_{ij}=h_{ij,t} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The policy and the instantaneous gains 𝐇 t {\\\\mathbf{H}}_{t} evolve on separate timescales. The fast-fading process generates T T realizations of 𝐇 t {\\\\mathbf{H}}_{t} , while the policy emits one allocation per slot and holds it constant over a window of T 0 T_{0} realizations, giving S = T / T 0 S=T/T_{0} slots per network (Table III ). The network configuration 𝐇 {\\\\mathbf{H}} stays fixed over the operation window and is the only channel state available at allocation time. The policy is conditioned on 𝐇 {\\\\mathbf{H}} via the GSO 𝐒 {\\\\mathbf{S}} and node-state 𝐮 {\\\\mathbf{u}} (defined below), while the instantaneous gains 𝐇 t {\\\\mathbf{H}}_{t} are observed only through the rates they induce.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We model each configuration as a draw 𝐇 ∼ q 𝐇 {\\\\mathbf{H}}\\\\sim q_{{\\\\mathbf{H}}} from a family of network geometries and large-scale fading. Sweeping the area side length R R over four values yields four user densities, each contributing 32 32 networks for a dataset of 128 128 configurations in total, which we split 5 : 1 : 2 5{:}1{:}2 into training, validation, and testing (Table III ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given an allocation 𝐩 {\\\\mathbf{p}} and a channel realization 𝐇 ~ \\\\widetilde{{\\\\mathbf{H}}} , the rate of the receiver j j is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate a policy by its ergodic rate, estimated by averaging the instantaneous rates over the fading realizations,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In ( 27 ), the expectation is over the policy allocations 𝐩 ∼ 𝝅 ( ⋅ | 𝐇 ) {\\\\mathbf{p}}\\\\sim\\\\boldsymbol{\\\\pi}(\\\\cdot\\\\,|\\\\,{\\\\mathbf{H}}) and the fast-fading channel 𝐇 ~ ∼ q 𝐇 ~ | 𝐇 \\\\widetilde{{\\\\mathbf{H}}}\\\\sim q_{\\\\widetilde{{\\\\mathbf{H}}}|{\\\\mathbf{H}}} . The allocation 𝐩 s {\\\\mathbf{p}}_{s} is drawn once per slot s = ⌈ t / T 0 ⌉ s=\\\\lceil t/T_{0}\\\\rceil and held fixed across it, while 𝐇 t {\\\\mathbf{H}}_{t} is the channel realized at step t t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given 𝐇 {\\\\mathbf{H}} , an optimal policy 𝝅 ⋆ ( 𝐇 ) = 𝝅 ⋆ ( ⋅ | 𝐇 ) \\\\boldsymbol{\\\\pi}^{\\\\star}({\\\\mathbf{H}})=\\\\boldsymbol{\\\\pi}^{\\\\star}(\\\\cdot{\\\\,\\\\big|\\\\,}{\\\\mathbf{H}}) maximizes the ergodic sum-rate subject to a per-user minimum-rate (QoS) requirement f min = 0.6 f_{\\\\min}=0.6 bits/s/Hz, i.e.,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The diffusion targets are produced by an expert primal–dual (PD) algorithm that solves ( 28 ) via Lagrangian dual (sub)gradient ascent. We train a separate expert per density group, parametrized by a shallow GNN, across all networks. We then collect 200 200 allocations per training network from its converged primal iterates, giving 16000 16000 conditional samples that the diffusion model (U-GNN) learns to reproduce given ( 𝐒 , 𝐮 ) ({\\\\mathbf{S}},{\\\\mathbf{u}}) . Running the same expert across all splits, including the validation and test networks held out from U-GNN, yields the reference allocations against which we benchmark the U-GNN policy. Since the expert solves each network directly rather than generalizing, it is a near-ideal reference that U-GNN can only approach, so the comparison favors the expert by design. At inference, U-GNN thus samples directly from this distribution of near-optimal, feasible allocations rather than re-solving the optimization per network. We refer to [ 37 ] for the expert’s algorithmic details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We represent 𝐇 {\\\\mathbf{H}} as a directed, weighted channel graph on N N nodes, one per tx–rx pair. A self-loop carries the direct-link gain h i i h_{ii} , and an edge carries the interference gain h i j h_{ij} , i ≠ j i\\\\neq j , which is generally asymmetric, i.e., h i j ≠ h j i h_{ij}\\\\neq h_{ji} . We sparsify this graph into the GSO 𝐒 {\\\\mathbf{S}} by retaining, for each receiver, its self-loop and the incoming edges from the m = 10 m=10 strongest interferers, followed by a log-normalization that compresses the wide dynamic range of the channel gains. We also derive a two-feature node-state vector 𝐮 {\\\\mathbf{u}} that collects each node’s direct-link strength and its aggregate incoming interference under full-power transmission. Both 𝐒 {\\\\mathbf{S}} and 𝐮 {\\\\mathbf{u}} are deterministic functions of 𝐇 {\\\\mathbf{H}} , so they form the conditioning input 𝝃 = ( 𝐒 , 𝐮 ) \\\\boldsymbol{\\\\xi}=({\\\\mathbf{S}},{\\\\mathbf{u}}) of Section II , and q 𝐇 q_{{\\\\mathbf{H}}} induces its distribution q 𝝃 q_{\\\\boldsymbol{\\\\xi}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The diffusion variable is the rescaled allocation 𝐱 0 = 𝐩 / P max − 1 2 ∈ [ − 1 2 , 1 2 ] N {\\\\mathbf{x}}_{0}={\\\\mathbf{p}}/P_{\\\\max}-\\\\tfrac{1}{2}\\\\in[-\\\\tfrac{1}{2},\\\\tfrac{1}{2}]^{N} , viewed as a scalar graph signal on 𝐒 {\\\\mathbf{S}} . We realize the U-GNN policy, which is trained to imitate the optimal policy 𝝅 ⋆ \\\\boldsymbol{\\\\pi}^{\\\\star} , through the conditional diffusion generator p 𝜽 ⋆ ( ⋅ | 𝐒 , 𝐮 ) p_{\\\\boldsymbol{\\\\theta}^{\\\\star}}(\\\\cdot\\\\,|\\\\,{\\\\mathbf{S}},{\\\\mathbf{u}}) of Section II . That is, given 𝐇 {\\\\mathbf{H}} , we sample a batch of S = 100 S=100 rescaled allocations 𝐱 0 ∼ p 𝜽 ⋆ ( ⋅ | 𝐒 , 𝐮 ) {\\\\mathbf{x}}_{0}\\\\sim p_{\\\\boldsymbol{\\\\theta}^{\\\\star}}(\\\\cdot\\\\,|\\\\,{\\\\mathbf{S}},{\\\\mathbf{u}}) and map them to power allocations 𝐩 = clip [ 0 , P max ] ( P max ( 𝐱 0 + 1 2 ) ) {\\\\mathbf{p}}=\\\\mathrm{clip}_{[0,P_{\\\\max}]}\\\\!\\\\big(P_{\\\\max}({\\\\mathbf{x}}_{0}+\\\\tfrac{1}{2})\\\\big) . We place these allocations in random order as { 𝐩 s } s = 1 S \\\\{{\\\\mathbf{p}}_{s}\\\\}_{s=1}^{S} and execute them sequentially, with slot s s transmitting 𝐩 s {\\\\mathbf{p}}_{s} over its T 0 T_{0} -realization window.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare the U-GNN policy against two deterministic baselines that forgo time-sharing. For every slot s s and network 𝐇 {\\\\mathbf{H}} , full power (FP) sets 𝐩 s = P max 𝟏 N {\\\\mathbf{p}}_{s}=P_{\\\\max}\\\\mathbf{1}_{N} , and average power (AP) is the expert’s per-network mean allocation, i.e., 𝐩 s ( 𝐇 ) ≈ 𝔼 𝐩 ∼ 𝝅 ⋆ ( ⋅ | 𝐇 ) [ 𝐩 ] {\\\\mathbf{p}}_{s}({\\\\mathbf{H}})\\\\approx\\\\mathbb{E}_{{\\\\mathbf{p}}\\\\sim\\\\boldsymbol{\\\\pi}^{\\\\star}(\\\\cdot\\\\,|\\\\,{\\\\mathbf{H}})}\\\\,\\\\big[{\\\\mathbf{p}}\\\\big] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results. We report in Table IV the ergodic-rate performance of the U-GNN policy, expert (PD) policy, and baselines. The U-GNN diffusion policy closely matches the expert across the full distribution. Its 5th- and 10th-percentiles (p 5 5 , p 10 10 ) rates reach 0.77 0.77 and 0.92 0.92 bits/s/Hz, trailing the expert by only 5.3 % 5.3\\\\% and 0.9 % 0.9\\\\% . The p 5 5 rate already exceeds the f min = 0.6 f_{\\\\min}=0.6 bits/s/Hz floor, and only the bottom ∼ 2 % {\\\\sim}2\\\\% of users fall slightly short. U-GNN’s gap to the expert is largest at p 1 1 , at 17.8 % 17.8\\\\% ( 0.55 0.55 vs. 0.66 0.66 bits/s/Hz), whereas FP and AP baselines trail by 87.9 % 87.9\\\\% and 71.8 % 71.8\\\\% at this percentile, making U-GNN’s degradation minor by comparison. FP attains the highest mean rate at 3.14 3.14 bits/s/Hz while collapsing its p 1 1 rate to 0.08 0.08 bits/s/Hz and lowering feasibility to 88.0 % 88.0\\\\% . U-GNN instead sustains a mean rate of 2.81 2.81 bits/s/Hz, on par with the expert ( 2.84 2.84 ) and AP ( 2.85 2.85 ). AP matches this mean rate but still leaves 7.3 % 7.3\\\\% of users below f min f_{\\\\min} , so collapsing the policy to a deterministic mean allocation is not enough.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"U-GNN’s value rests jointly on its stochastic formulation and generative sampling. The stochastic formulation treats the policy as a distribution over allocations, sustaining feasibility through randomization where the deterministic AP fails. The expert shares this formulation but reaches it through thousands of primal–dual iterations per network, whereas U-GNN bypasses online optimization and attains comparable performance in a single DDIM sampling pass.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 5 illustrates why the optimal policy must be stochastic and how U-GNN realizes it through time-sharing . The optimal (expert) policy is multi-modal, activating a different subset of the mutually interfering tx–rx pairs in each slot. No single allocation serves all users, yet, alternating them across slots renders the time-averaged rate 𝐟 ( 𝝅 ⋆ , 𝐇 ) {\\\\mathbf{f}}(\\\\boldsymbol{\\\\pi}^{\\\\star},{\\\\mathbf{H}}) feasible for every receiver. Crucially, this rate averages the per-slot rates and must not be conflated with the rate of the single mean allocation 𝔼 [ 𝐩 ] \\\\mathbb{E}[{\\\\mathbf{p}}] , which activates every interferer at once and remains infeasible. This is exactly the failure mode of the AP baseline in Table IV .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 6 examines the size-transferability of the U-GNN policy. The model is trained only on networks of the native size | 𝒱 | = 400 |{\\\\mathcal{V}}|=400 tx–rx pairs (diamonds) with mixed densities. It is then evaluated without retraining on networks ranging from 200 200 to 1600 1600 pairs at similar densities. The ergodic 5 5 th-percentile rate stays essentially flat and above the f min f_{\\\\min} floor across this range for every density. Denser networks ( 11.9 11.9 vs. 6.6 6.6 pairs/km 2 ) attain lower tail rates due to stronger interference, but their ordering and stability are preserved across all sizes. This indicates that the permutation-equivariant GNN backbone is also scalable. Similar observations hold for the mean and additional tail rates, which are not shown here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"VI Conclusion & Future Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We proposed a denoising diffusion framework for generating graph signals. We cast this as a denoising process and parametrized it with a U-GNN, a graph-domain adaptation of the U-Net. At its core, a pooling mechanism formalizes down/up-sampling as learned node selection via nested selection matrices and zero-padding. We demonstrated the framework on wireless resource allocation and stock-price forecasting. Future directions include designing graph-aware and latent diffusion processes, and integrating graph transformers into the U-GNN backbone for spatio-temporal tasks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base DDIM sampler ( 9 ) deferred the specification of its noise scale σ k \\\\sigma_{k} . We provide it here, where it also governs the accelerated sampler used in our experiments. A factor η ∈ [ 0 , 1 ] \\\\eta\\\\in[0,1] controls the per-step stochasticity through\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which interpolates between the ancestral DDPM update at η = 1 \\\\eta=1 and the deterministic DDIM map at η = 0 \\\\eta=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because the DDIM updates in ( 9 ) preserve the marginals of the forward process, we can subsample the denoising steps to accelerate inference without retraining. Concretely, we train ϵ 𝜽 ⋆ \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}^{\\\\star}} on the full grid of K K steps but sample along only an increasing subset { τ 1 , … , τ K ′ } ⊂ { 1 , … , K } \\\\{\\\\tau_{1},\\\\ldots,\\\\tau_{K^{\\\\prime}}\\\\}\\\\subset\\\\{1,\\\\ldots,K\\\\} with τ K ′ = K \\\\tau_{K^{\\\\prime}}=K and K ′ ≪ K K^{\\\\prime}\\\\ll K , and we set τ 0 ≔ 0 \\\\tau_{0}\\\\coloneqq 0 . We draw 𝐱 τ K ′ = 𝐱 K ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\\\mathbf{x}}_{\\\\tau_{K^{\\\\prime}}}={\\\\mathbf{x}}_{K}\\\\sim\\\\mathcal{N}({\\\\mathbf{0}},{\\\\mathbf{I}}) and iterate the update ( 9 ) over this sub-grid,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for k = K ′ , … , 1 k=K^{\\\\prime},\\\\ldots,1 , and 𝐰 ∼ 𝒩 ( 𝟎 , 𝐈 ) {\\\\mathbf{w}}\\\\sim\\\\mathcal{N}({\\\\mathbf{0}},{\\\\mathbf{I}}) . We note that both 𝐱 ^ 0 = 𝐱 ^ 0 ( 𝐱 τ k , τ k ; 𝝃 ) \\\\widehat{{\\\\mathbf{x}}}_{0}=\\\\widehat{{\\\\mathbf{x}}}_{0}({\\\\mathbf{x}}_{\\\\tau_{k}},\\\\tau_{k};\\\\boldsymbol{\\\\xi}) [cf. ( 7 )] and ϵ 𝜽 ⋆ = ϵ 𝜽 ⋆ ( 𝐱 τ k , τ k ; 𝝃 ) \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}^{\\\\star}}=\\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}^{\\\\star}}({\\\\mathbf{x}}_{\\\\tau_{k}},\\\\tau_{k};\\\\boldsymbol{\\\\xi}) are evaluated at the current step τ k \\\\tau_{k} . The noise scale σ τ k \\\\sigma_{\\\\tau_{k}} follows from ( 29 ) under the substitution ( k , k − 1 ) ↦ ( τ k , τ k − 1 ) (k,k{-}1)\\\\mapsto(\\\\tau_{k},\\\\tau_{k-1}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A literal evaluation of the strided graph convolution in ( 14 ) would form the operator 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} and accumulate its powers ( 𝐒 γ ) k ({\\\\mathbf{S}}^{\\\\gamma})^{k} , k = 0 , … , K k=0,\\\\ldots,K . For γ > 1 \\\\gamma>1 , this densifies 𝐒 {\\\\mathbf{S}} and forfeits its sparsity, much like the reduced GSOs 𝐒 ( k ) {\\\\mathbf{S}}^{(k)} of ( 15 ) do. We instead exploit ( 𝐒 γ ) k = 𝐒 γ k ({\\\\mathbf{S}}^{\\\\gamma})^{k}={\\\\mathbf{S}}^{\\\\gamma k} and evaluate the layer through γ K \\\\gamma K successive sparse products with the unit shift 𝐒 {\\\\mathbf{S}} , tapping the signal every γ \\\\gamma shifts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Starting from the lifted signal 𝐙 ~ 0 = 𝐃 ⊤ 𝐙 ℓ − 1 \\\\widetilde{{\\\\mathbf{Z}}}_{0}={\\\\mathbf{D}}^{\\\\top}{\\\\mathbf{Z}}_{\\\\ell-1} , we form 𝐙 ~ j = 𝐒 𝐙 ~ j − 1 \\\\widetilde{{\\\\mathbf{Z}}}_{j}={\\\\mathbf{S}}\\\\,\\\\widetilde{{\\\\mathbf{Z}}}_{j-1} for j = 1 , … , γ K j=1,\\\\ldots,\\\\gamma K and accumulate the tapped terms 𝐙 ~ γ k 𝚯 ℓ , k \\\\widetilde{{\\\\mathbf{Z}}}_{\\\\gamma k}\\\\,\\\\boldsymbol{\\\\Theta}_{\\\\ell,k} , k = 0 , … , K k=0,\\\\ldots,K . We then apply the pointwise nonlinearity σ ℓ \\\\sigma_{\\\\ell} and the pooling 𝐃 {\\\\mathbf{D}} . The untapped shifts, j ∉ { 0 , γ , … , γ K } j\\\\notin\\\\{0,\\\\gamma,\\\\ldots,\\\\gamma K\\\\} , only advance the walk to the next tapped hop. Algorithm 1 summarizes the procedure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, we never materialize the reduced GSOs. The shift 𝐒 {\\\\mathbf{S}} is a sparse weighted aggregation, while the unpooling 𝐃 ⊤ {\\\\mathbf{D}}^{\\\\top} and pooling 𝐃 {\\\\mathbf{D}} are index scatter and gather operations. Only the running buffer 𝐙 ~ j \\\\widetilde{{\\\\mathbf{Z}}}_{j} and one accumulator are kept, each of size N × F N\\\\times F , so each layer runs in roughly 𝒪 ( γ K | ℰ | F ) \\\\mathcal{O}(\\\\gamma K\\\\,|{\\\\mathcal{E}}|\\\\,F) aggregation time and 𝒪 ( N F ) \\\\mathcal{O}(NF) memory, where | ℰ | |{\\\\mathcal{E}}| is the number of edges of 𝒢 {\\\\mathcal{G}} . Forming the dense 𝐒 γ {\\\\mathbf{S}}^{\\\\gamma} explicitly would instead cost 𝒪 ( N 2 ) \\\\mathcal{O}(N^{2}) memory. The filter keeps K + 1 K+1 taps for any γ \\\\gamma , so the stride adds shifts but no parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The stride enters only as a multiplier on the shift count and is tied to the pooling ratio. When the GNN module sits in a depth- b b block of the U-GNN, it instantiates ( 14 ) with 𝐃 = 𝐃 b {\\\\mathbf{D}}={\\\\mathbf{D}}_{b} , and all of its layers share the stride γ b = min { ⌊ ρ ¯ b ⌋ , γ max } \\\\gamma_{b}=\\\\min\\\\{\\\\lfloor\\\\sqrt{\\\\bar{\\\\rho}_{b}}\\\\,\\\\rfloor,\\\\,\\\\gamma_{\\\\max}\\\\} , where ρ ¯ b = ∏ i = 1 b ρ i = N / N b \\\\bar{\\\\rho}_{b}=\\\\prod_{i=1}^{b}\\\\rho_{i}=N/N_{b} is the cumulative down-sampling factor [cf. ( 18 )]. This is the rule γ = ⌊ ρ ⌋ \\\\gamma=\\\\lfloor\\\\sqrt{\\\\rho}\\\\,\\\\rfloor of Remark 1 applied at depth b b with ρ = ρ ¯ b \\\\rho=\\\\bar{\\\\rho}_{b} and capped at γ max \\\\gamma_{\\\\max} , so the total reach γ b K \\\\gamma_{b}K does not span the whole graph and we avoid oversmoothing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To train the score head 𝚿 b \\\\boldsymbol{\\\\Psi}_{b} by backpropagation, we apply a straight-through estimator (STE) [ 1 ] to the binary selection mask. That is, the forward pass selects a hard mask, while gradients reach 𝐯 b {\\\\mathbf{v}}_{b} through a differentiable sigmoid surrogate. Moreover, we employ a Gumbel-perturbed Top-K rule to encourage exploration during training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stochastic selection. Let 𝐡 b ∈ { 0 , 1 } N b {\\\\mathbf{h}}_{b}\\\\in\\\\{0,1\\\\}^{N_{b}} be the Top-K indicator, with [ 𝐡 b ] n = 𝟙 { n ∈ 𝒮 b } [{\\\\mathbf{h}}_{b}]_{n}=\\\\mathbbm{1}\\\\{n\\\\in\\\\mathcal{S}_{b}\\\\} . At inference, 𝒮 b \\\\mathcal{S}_{b} is the deterministic Top-K of ( 25 ). During training, we draw 𝒮 b \\\\mathcal{S}_{b} stochastically to encourage exploration of the discrete selection. Given i.i.d. standard Gumbel variates [ 𝐠 b ] n = − log ( − log u n ) [{\\\\mathbf{g}}_{b}]_{n}=-\\\\log(-\\\\log u_{n}) , u n ∼ Uniform ( 0 , 1 ) u_{n}\\\\sim\\\\mathrm{Uniform}(0,1) , and an exploration scale ε ∈ ℝ + \\\\varepsilon\\\\in{\\\\mathbb{R}}_{+} , we perturb the scores before ranking,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that Top-K is invariant to positive scaling, i.e., TopK ( 𝐯 b + ε 𝐠 b , N b + 1 ) = TopK ( 𝐯 b / ε + 𝐠 b , N b + 1 ) \\\\mathrm{TopK}({\\\\mathbf{v}}_{b}+\\\\varepsilon{\\\\mathbf{g}}_{b},\\\\,N_{b+1})=\\\\mathrm{TopK}({\\\\mathbf{v}}_{b}/\\\\varepsilon+{\\\\mathbf{g}}_{b},\\\\,N_{b+1}) . By the Gumbel-Top-K trick [ 19 ] , ( 31 ) therefore draws 𝒮 b \\\\mathcal{S}_{b} without replacement with probabilities proportional to e [ 𝐯 b ] n / ε e^{[{\\\\mathbf{v}}_{b}]_{n}/\\\\varepsilon} . The scale ε \\\\varepsilon acts as a sampling temperature, and as ε → 0 \\\\varepsilon\\\\to 0 , the sample concentrates on the deterministic Top-K, which we recover at inference when the perturbation vanishes at ε = 0 \\\\varepsilon=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Straight-through gradient. For the backward pass, we form a surrogate from the unperturbed scores. Given a temperature τ ∈ ℝ + \\\\tau\\\\in{\\\\mathbb{R}}_{+} and the logistic sigmoid ς ( ⋅ ) \\\\varsigma(\\\\cdot) , we set 𝐡 ~ b = ς ( 𝐯 b / τ ) ∈ ( 0 , 1 ) N b \\\\widetilde{{\\\\mathbf{h}}}_{b}=\\\\varsigma({\\\\mathbf{v}}_{b}/\\\\tau)\\\\in(0,1)^{N_{b}} . The training mask is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where sg ( ⋅ ) \\\\mathrm{sg}(\\\\cdot) is the stop-gradient operator. The soft terms cancel in value, thus 𝐦 b = 𝐡 b {\\\\mathbf{m}}_{b}={\\\\mathbf{h}}_{b} in the forward pass. In the backward pass, sg ( 𝐡 ~ b ) \\\\mathrm{sg}(\\\\tilde{{\\\\mathbf{h}}}_{b}) is constant, so the gradient reaches 𝐯 b {\\\\mathbf{v}}_{b} only through 𝐡 ~ b \\\\tilde{{\\\\mathbf{h}}}_{b} . We note that because the surrogate uses the unperturbed scores, the Gumbel noise affects only which nodes enter 𝒮 b \\\\mathcal{S}_{b} , not the surrogate gradient at any node.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Gated down-sampling. We insert this mask into the encoder down-sampling. During training, the depth- ( b + 1 ) (b{+}1) input 𝐕 b + 1 = 𝐂 b + 1 𝐙 b {\\\\mathbf{V}}_{b+1}={\\\\mathbf{C}}_{b+1}\\\\,{\\\\mathbf{Z}}_{b} of ( 22 ) becomes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ⊙ \\\\odot scales each row of 𝐙 b {\\\\mathbf{Z}}_{b} by the corresponding entry of 𝐦 b {\\\\mathbf{m}}_{b} . Since 𝐦 b = 𝐡 b {\\\\mathbf{m}}_{b}={\\\\mathbf{h}}_{b} in the forward pass, the selected rows pass through unchanged, and 𝐂 b + 1 {\\\\mathbf{C}}_{b+1} discards the rest. At inference, the STE branch is dropped ( 𝐦 b = 𝐡 b {\\\\mathbf{m}}_{b}={\\\\mathbf{h}}_{b} ), and ( 33 ) reverts to the plain down-sampling via 𝐂 b + 1 𝐙 b {\\\\mathbf{C}}_{b+1}\\\\,{\\\\mathbf{Z}}_{b} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Schedules. We anneal both τ \\\\tau and ε \\\\varepsilon linearly over training. Let t t and T T denote the current and total number of training epochs, with T = 5000 T=5000 as reported in Table I . The schedule uses a warm-up of T w T_{\\\\mathrm{w}} epochs followed by an anneal of T a T_{\\\\mathrm{a}} epochs,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where r ( t ) = min { max { ( t − T w ) / T a , 0 } , 1 } r(t)=\\\\min\\\\{\\\\max\\\\{(t-T_{\\\\mathrm{w}})/T_{\\\\mathrm{a}},\\\\,0\\\\},\\\\,1\\\\} . We set T w = 0.02 T T_{\\\\mathrm{w}}=0.02\\\\,T , T a = 0.73 T T_{\\\\mathrm{a}}=0.73\\\\,T , ( τ 0 , τ min ) = ( 1 , 0.5 ) (\\\\tau_{0},\\\\tau_{\\\\min})=(1,0.5) , and ( ε 0 , ε min ) = ( 1 , 0 ) (\\\\varepsilon_{0},\\\\varepsilon_{\\\\min})=(1,0) . This way, both ε \\\\varepsilon and τ \\\\tau reach their floors at epoch T w + T a = 0.75 T T_{\\\\mathrm{w}}+T_{\\\\mathrm{a}}=0.75\\\\,T , after which selection is (approximately) deterministic, and the surrogate sharpness is fixed. These values worked well in our experiments, but we found the results to be robust, as any choice that explores broadly early and becomes deterministic well before training ends gave comparable performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We remark that the STE in ( 32 ) supplies a surrogate gradient for the binary mask, and not a differentiable relaxation of the hard Top-K rule. In the forward pass, 𝐂 b + 1 {\\\\mathbf{C}}_{b+1} removes the unselected rows, so each training iteration passes diffusion-loss gradients only to the score entries of the nodes in 𝒮 b \\\\mathcal{S}_{b} . The head nonetheless learns a global ranking for two reasons. First, while ε > 0 \\\\varepsilon>0 , the stochastic selection varies 𝒮 b \\\\mathcal{S}_{b} across iterations, so the gradient reaches a broader set of nodes over training. Second, the head shares its parameters across all candidate nodes, so each update reshapes the scoring function at every node in subsequent passes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CNNs with pooling couple neighborhood aggregation and spatial down-sampling into a single operation that summarizes each pixel window, e.g., by its average or maximum, at a reduced resolution. This coupling exploits the intrinsic geometry of the regular grid. For a general graph, the GSO 𝐒 {\\\\mathbf{S}} supplies the analogous structural information, which we use to define pooling neighborhoods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our design decouples these two operations. The strided graph convolution in ( 14 ) already returns a nonlinear summary of each node’s neighborhood. In reduced indices, this is the K K -hop reach of { 𝐒 ( k ) } k = 0 K \\\\{{\\\\mathbf{S}}^{(k)}\\\\}_{k=0}^{K} , and in the lifted domain, it is the γ K \\\\gamma K -hop reach of 𝐒 {\\\\mathbf{S}} after 𝐃 b ⊤ {\\\\mathbf{D}}_{b}^{\\\\top} lifts the signal and before 𝐃 b {\\\\mathbf{D}}_{b} reduces it back [cf. ( 15 )]. The learned selectors 𝐂 b {\\\\mathbf{C}}_{b} in turn perform the node-space down-sampling [cf. ( 25 )]. An explicit sum- or max-pooling over graph neighborhoods is therefore optional rather than essential. We describe it next for completeness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pooling over the full graph. For a given 𝐒 {\\\\mathbf{S}} and pooling range κ ∈ ℤ + \\\\kappa\\\\in\\\\mathbb{Z}^{+} , we define the binary reachability matrix\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that each node reaches itself. From 𝐑 = 𝐑 ( 𝐒 ) {\\\\mathbf{R}}={\\\\mathbf{R}}({\\\\mathbf{S}}) we read off, for each node n ∈ 𝒱 n\\\\in{\\\\mathcal{V}} , the index set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"of nodes aggregated at n n . Let φ : ℝ ⋅ × F → ℝ F \\\\varphi\\\\colon{\\\\mathbb{R}}^{\\\\cdot\\\\times F}\\\\to{\\\\mathbb{R}}^{F} be a pooling map that is invariant to permutations of its rows (the pooled nodes) and acts per feature, e.g., [ φ sum ( 𝐗 ) ] f = ∑ i [ 𝐗 ] i , f [\\\\varphi_{\\\\mathrm{sum}}({\\\\mathbf{X}})]_{f}=\\\\sum_{i}[{\\\\mathbf{X}}]_{i,f} for sum-pooling and [ φ max ( 𝐗 ) ] f = max i [ 𝐗 ] i , f [\\\\varphi_{\\\\max}({\\\\mathbf{X}})]_{f}=\\\\max_{i}[{\\\\mathbf{X}}]_{i,f} for max-pooling. In analogy with the graph-filter notation 𝐇 ( 𝐒 ) 𝐗 {\\\\mathbf{H}}({\\\\mathbf{S}}){\\\\mathbf{X}} , we write 𝐑 ( 𝐒 ) 𝐗 {\\\\mathbf{R}}({\\\\mathbf{S}}){\\\\mathbf{X}} for the pooled signal,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that for sum-pooling, this operator form coincides with the matrix product 𝐑 ( 𝐒 ) 𝐗 {\\\\mathbf{R}}({\\\\mathbf{S}}){\\\\mathbf{X}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pooling at reduced resolutions. Pooling at the resolutions of the U-GNN follows the reduced-GSO construction. We define the node-reduced counterpart of 𝐑 {\\\\mathbf{R}} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which intersects the κ b \\\\kappa_{b} -reach of 𝐒 {\\\\mathbf{S}} with the active set Ω b \\\\Omega_{b} . Like the reduced GSOs in ( 15 ), 𝐑 b {\\\\mathbf{R}}_{b} is indexed by the local labels [ N b ] [N_{b}] . Also, the range κ b \\\\kappa_{b} is left implicit at depth b b , just as the stride γ b \\\\gamma_{b} is in ( 14 ). We define the set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which collects the active nodes within κ b \\\\kappa_{b} hops of n n on 𝐒 {\\\\mathbf{S}} , including n n itself. We then pool a down-sampled signal 𝐙 ∈ ℝ N b × F {\\\\mathbf{Z}}\\\\in{\\\\mathbb{R}}^{N_{b}\\\\times F} as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In ( 40 ), 𝐑 b ( 𝐒 ) 𝐙 {\\\\mathbf{R}}_{b}({\\\\mathbf{S}}){\\\\mathbf{Z}} admits a lift–pool–reduce form mirroring that of the strided convolutions. One lifts 𝐙 {\\\\mathbf{Z}} to the full vertex set, pools over the κ b \\\\kappa_{b} -hop neighborhoods of 𝐒 {\\\\mathbf{S}} , and reduces back to Ω b \\\\Omega_{b} through 𝐃 b {\\\\mathbf{D}}_{b} . The lift assigns each inactive node the identity element of φ \\\\varphi , so that it leaves the pooled output unchanged. This element is 0 for sum-pooling, for which the plain 𝐃 b ⊤ {\\\\mathbf{D}}_{b}^{\\\\top} suffices, and − ∞ -\\\\infty (a large negative constant in practice) for max-pooling, supplied by a modified lift 𝐃 ~ b ⊤ \\\\widetilde{{\\\\mathbf{D}}}_{b}^{\\\\top} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optional placement in the U-GNN. At each encoder depth b = 1 , … , B − 1 b=1,\\\\ldots,B-1 , the encoder output 𝐙 b {\\\\mathbf{Z}}_{b} is first pooled as 𝐑 b 𝐙 b {\\\\mathbf{R}}_{b}{\\\\mathbf{Z}}_{b} and then down-sampled by 𝐂 b + 1 𝐑 b 𝐙 b {\\\\mathbf{C}}_{b+1}{\\\\mathbf{R}}_{b}{\\\\mathbf{Z}}_{b} , which feeds the block at depth b + 1 b+1 [cf. ( 19 )]. The node selectors use the pooled features as well, so we replace 𝐙 b {\\\\mathbf{Z}}_{b} with 𝐑 b 𝐙 b {\\\\mathbf{R}}_{b}{\\\\mathbf{Z}}_{b} in ( 24 ). Both stages fold into the definition of the encoding GNN module. Note that the bottleneck at depth B B performs no node-space down-sampling or pooling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The case for explicit pooling is weak in our design since the graph convolutional layers in ( 14 ) and selector heads in ( 24 ) already provide nonlinear neighborhood aggregation and learned down-sampling, respectively. In our experiments, max-pooling added computational overhead without a noticeable gain. Hence, we adopt ( 14 ) as the default GNN operation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Max-pooling admits a compositional shortcut. By associativity and the telescoping of iterated neighborhoods, the κ \\\\kappa -reach maximum equals κ \\\\kappa applications of the unit-reach maximum. This avoids forming 𝐑 ( 𝐒 ) {\\\\mathbf{R}}({\\\\mathbf{S}}) explicitly and reduces the cost to κ \\\\kappa sparse scatter_max calls. The same property holds in the reduced domain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a regular graph with uniform down-sampling, a natural CNN analog sets the pooling range κ b = κ ⌈ N / N b ⌉ \\\\kappa_{b}=\\\\kappa\\\\lceil N/N_{b}\\\\rceil , proportional to the effective down-sampling at depth b b , so that pooling regions cover comparable numbers of active nodes across resolutions. For an arbitrary graph, the tension that governs the stride γ \\\\gamma reappears [cf. Remark 1 ]. Namely, a too small κ b \\\\kappa_{b} collapses 𝐧 b ( n ) {\\\\mathbf{n}}_{b}(n) to { n } \\\\{n\\\\} and reduces pooling to the identity, whereas a too large κ b \\\\kappa_{b} pools broadly over Ω b \\\\Omega_{b} and may oversmooth the representation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This appendix expands Table I . In particular, Appendix E.1 details the denoiser, diffusion process, and training shared across the two tasks, and Appendix E.2 specifies the signal and conditioning interface particular to each.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.1 Shared Denoiser, Diffusion, and Training\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Backbone. The denoiser ϵ 𝜽 \\\\boldsymbol{\\\\epsilon}_{\\\\boldsymbol{\\\\theta}} is a U-GNN spanning B = 4 B{=}4 node resolutions. Each of its 3 3 down-sampling levels reduces the active node set by a factor 2 2 , for an 2 3 = 8 × 2^{3}{=}8\\\\times reduction overall and four resolutions of N N , N / 2 N/2 , N / 4 N/4 , and N / 8 N/8 nodes. Each reduction is realized by a learned node-selection pooling operator, made differentiable with a straight-through estimator (Appendix C ). To counteract the increasing sparsity of the coarsened graphs, the graph convolutions are strided, capped at γ max = 2 \\\\gamma_{\\\\max}{=}2 (Appendix B ). The feature (channel) width is uniform at 64 64 channels across all levels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each encoding and decoding level applies 2 2 GNN layers of order K = 2 K{=}2 over powers of the augmented adjacency 𝐈 + 𝐀 ^ {\\\\mathbf{I}}+\\\\hat{{\\\\mathbf{A}}} , with channel-wise layer normalization (LayerNorm), a rectified linear unit (ReLU) nonlinearity, and dropout 0.1 0.1 ; these layers are strided variants of the topology-adaptive graph convolution (TAGConv), following its PyG implementation . A 2 2 -layer bottleneck processes the coarsest summary, and the decoder path restores full resolution while concatenating the encoder skip features at each level. The output head applies LayerNorm, a sigmoid linear unit (SiLU) nonlinearity, and a linear map. Fig. 7 complements Fig. 2 and shows a detailed schematic of one encoder–decoder block pair at depth b b .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conditioning and fusion. The diffusion step k k is mapped to a 128 128 -dimensional sinusoidal embedding followed by a multilayer perceptron (MLP), and the node conditioning to a 128 128 -dimensional representation. Both are projected to the level width 64 64 at every level, where the signal path and the projected step embedding are concatenated and linearly mapped back to 64 64 channels ( 128 → 64 128{\\\\to}64 ). The conditioning is then merged in a task-specific way: by concatenation into the same projection for WRA, so that it maps 192 192 channels down to 64 64 over the signal, step, and conditioning; and by two-head cross-attention for S&P 500—queries from the signal path, keys and values from the conditioning context—which leaves the projection from 128 128 channels down to 64 64 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Diffusion. The forward process applies a linear β \\\\beta schedule from 10 − 4 10^{-4} to 2 × 10 − 2 2\\\\times 10^{-2} over 500 500 steps. At inference, we sample using 5 × 5\\\\times -accelerated denoising diffusion implicit models (DDIM) with 100 100 steps and stochasticity parameter η = 0.2 \\\\eta{=}0.2 (Appendix A ). We draw 100 100 samples per conditioning input to form the predictive ensemble.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimization. We train for 5000 5000 epochs with AdamW (learning rate 10 − 4 10^{-4} , ( β 1 , β 2 ) = ( 0.9 , 0.98 ) (\\\\beta_{1},\\\\beta_{2}){=}(0.9,0.98) , weight decay 10 − 4 10^{-4} ), gradient-norm clipping at 1.0 1.0 , and automatic mixed precision (AMP). The learning rate follows a per-step cosine schedule with a linear warm-up over the first 1 % 1\\\\% of training, then a cosine decay to a floor of 5 % 5\\\\% of the peak over 0.8 0.8 of training, after which it is held constant. The node-selector temperature and exploration noise are annealed over roughly the first 75 % 75\\\\% of training, moving the selector from soft selection toward nearly hard selection (Appendix C ). We retain the best checkpoint under a task-specific composite validation criterion. Both models were trained on a single NVIDIA RTX 3090. The S&P 500 model completed training in ≈ 58.5 {\\\\approx}58.5 GPU hours, and the WRA model did so in ≈ 33 {\\\\approx}33 GPU hours.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.2 Application-Specific Knobs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stock-price forecasting (S&P 500). The diffused signal is the future log-return trajectory 𝐱 0 ∈ ℝ N × T p {\\\\mathbf{x}}_{0}\\\\in{\\\\mathbb{R}}^{N\\\\times T_{p}} over the N = 468 N{=}468 stocks. The pooling cascade coarsens the node set as 468 → 234 → 117 → 58 468{\\\\to}234{\\\\to}117{\\\\to}58 across the four levels. Because returns are unbounded, the predicted clean signal is left unclipped.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The conditioning comprises 12 12 market features over the past T h T_{h} -day window. A per-node temporal encoder summarizes this window and projects the T h T_{h} history steps onto the T p T_{p} horizon steps, forming the per-level context that the cross-attention fusion above consumes. This encoder interleaves dilated 1-D convolutions with two-head temporal self-attention. A lightweight in-block temporal mixer further couples the T p T_{p} horizon steps. The target channel is additionally normalized by reversible instance normalization (RevIN) [ 17 ] , using statistics from the conditioning window. RevIN uses a blend weight of 0.7 0.7 and a scale correction of 1.107 1.107 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model has 1.44 M 1.44\\\\mathrm{M} trainable parameters and is trained on batches of 64 64 sliding windows. We retain the checkpoint that minimizes a composite of the validation noise loss, its train–validation gap, and continuous ranked probability score (CRPS). This checkpoint is reached near epoch 4500 4500 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Wireless resource allocation (WRA). The diffused signal is the centered allocation 𝐱 0 = 𝐩 / P max − 1 2 ∈ [ − 1 2 , 1 2 ] N {\\\\mathbf{x}}_{0}={\\\\mathbf{p}}/P_{\\\\max}-\\\\tfrac{1}{2}\\\\in[-\\\\tfrac{1}{2},\\\\tfrac{1}{2}]^{N} over the N = 400 N{=}400 transmitter–receiver (tx–rx) pairs. The pooling cascade coarsens the node set as 400 → 200 → 100 → 50 400{\\\\to}200{\\\\to}100{\\\\to}50 . The powers are recovered as 𝐩 = clip [ 0 , P max ] ( P max ( 𝐱 0 + 1 2 ) ) {\\\\mathbf{p}}=\\\\mathrm{clip}_{[0,P_{\\\\max}]}\\\\!\\\\big(P_{\\\\max}({\\\\mathbf{x}}_{0}+\\\\tfrac{1}{2})\\\\big) . In contrast to the stock model, the allocation is a single snapshot with no time axis, so T = 1 T{=}1 . Thus, the temporal encoder and in-block mixer are therefore disabled, and RevIN is not used.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 2 2 per-node conditioning features are concatenated into the level fusion described above. These features are the direct-link gain and the aggregate cross-link interference. Because the support is bounded, the predicted clean signal is clipped to the valid power range at each reverse step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Target allocations are supplied by a primal–dual expert. The expert is a shallow GNN with 3 3 layers, hidden width 64 64 , and K = 2 K{=}2 hops, trained per density group with dual-ascent step 0.2 0.2 . As supervision for the U-GNN, we collect 200 200 allocations per training network from its converged primal iterates. This yields 16000 16000 samples in total. Running the same expert on the held-out networks yields the reference allocations used for evaluation. Theoretical and algorithmic details appear in [ 37 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model has 0.63 M 0.63\\\\mathrm{M} trainable parameters and is trained on batches of 800 800 network graph and resource allocation signal pairs. We retain the checkpoint that minimizes a composite of the mean rate-feasibility gap and the 1 % 1\\\\% and 5 % 5\\\\% rate gaps, averaged over the four density groups. This checkpoint is reached at epoch 1600 1600 .\"\n }\n]","isConversational":false}}},{"rowIdx":498,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06818"},"doi":{"kind":"string","value":"10.18653/v1/2021.naacl-industry.33"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Ad Headline Generation using Self-Critical Masked Language Model"},"authors":{"kind":"list like","value":["Yashal Shakti Kanungo","Sumit Negi","Aruna Rajan"],"string":"[\n \"Yashal Shakti Kanungo\",\n \"Sumit Negi\",\n \"Aruna Rajan\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06818v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"There are a various types of ads. A set of example ads that showcase products selected by sellers along with headlines that advertise them are shown in Figure 1 . Sellers create multiple ad campaigns for multiple products, bid in an auction to advertise and pay for clicks on the ad."},{"type":"para","text":"An E-Commerce product catalog may have millions of products which can be advertised. To ease the ad headline writing process, humans resort to programmatically padding keywords, or repasting the retail catalog content in the advertisement."},{"type":"para","text":"Templated creatives such as “Save Now on …\" or “Buy more (product) of (brand)\" save the creative effort but fail to create any excitement or brand identity in the minds of shoppers. High quality headlines are more attractive to shoppers and offer better value proposition. In this paper, we describe how we built a Natural Language Generation (NLG) system to generate instantaneous, attractive and brand identity building headlines for advertisements that intend to promote a wide range of products offered by a brand."},{"type":"para","text":"The content associated with a retail product has challenging characteristics. Some product titles have poor structure, grammatical issues, or partial phrases. The product titles also include varying number of product features such as “Hyper Tough 18V Cordless Drill, 3/8 inch Chuck, Variable Speed, with 1.2Ah Nickel Cadmium Battery, Charger, Bit Holder LED Light\" along with titles such as “ZIPIT Grillz Backpack, Camo Grey\" ."},{"type":"para","text":"The generated headlines need to capture the information present in the retail attributes and at the same time be different and uniquely attractive. Advertisers select multiple related products that are advertised as part of a single ad campaign. The ad campaign headline is then shared across all of these related products. Thus, the headline also needs to generalize the shared characteristics of the products and cannot be specific to a single product within the campaign."},{"type":"para","text":"The key contributions of our work are:"},{"type":"para","text":"We use Masked Language Model (MLM) for the generation of advertisement headlines using multiple products at the same time. Extensive test-set metrics, quality and grammar audits show that the proposed model outperforms all the baselines and the human-submitted headlines in terms of quality and grammar."},{"type":"para","text":"The novel usage of RL for the training of MLM allows us to directly optimize the MLM for improved headline quality metrics without changing inference setup or latency. Our method can also be applied to any other NLG task such as summarization, translation etc."},{"type":"para","text":"Our model reduces the extensive effort and time that is required to manually create headlines and has low latency."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"Natural Language Understanding (NLU) using Language Models (LM) has observed great leaps in recent years. LMs have evolved from using word level models Joulin et al. ( 2016 ) to to a variety of extensions to the Transformer Vaswani et al. ( 2017 ) . The BERT Devlin et al. ( 2019 ) employs Transformer in a pre-training setting and introduced the MLM training objective."},{"type":"para","text":"Ramachandran et al. ( 2016 ) first demonstrated textual generation by using auto-regressive prediction in a seq2seq architecture. Transformer based auto-regressive methods such as GPT2 Radford et al. ( 2019 ) and BART Lewis et al. ( 2019 ) which predict one word at a time have also shown good results. Zhu et al. ( 2020 ) concatenated BERT representations with the Encoder and Decoder layers of another LM to incorporate pre-trained LM. Another model Dong et al. ( 2019 ) combines BERT-based Transformer Encoder with attention masking from the Transformer decoder. Rothe et al. ( 2019 ) combined pre-trained BERT Encoder with GPT decoder for NLG."},{"type":"para","text":"Ranzato et al. ( 2016 ) framed NLG as an RL problem and the generation quality as a reward. The Self-Critical Sequence Training (SCST) approach Rennie et al. ( 2017 ) replaces the learned baseline from other approaches Bahdanau et al. ( 2017 ) with the model’s own inference time algorithm to normalize the rewards."},{"type":"para","text":"For advertising, recent works Xu et al. ( 2019 ); Hughes et al. ( 2019 ) have combined LSTM based pointer network See et al. ( 2017 ) with RL methods to generate advertisement headlines. While these methods improve the results, they fail to utilize extensive pre-training of Transformer based models and their various well-demonstrated advantages."},{"type":"para","text":"Our method extends BERT based generation Dong et al. ( 2019 ) by using Self-Critical policy gradient method Rennie et al. ( 2017 ) and jointly conditioning the generated sentence on multiple products at the same time. This allows us to use pre-trained BERT based LMs that can be trained to optimize various inference time metrics that are typically non-differentiable such as BLEU, Rouge, Readability etc."},{"type":"heading","level":1,"text":"3 Self-Critical Masked Language Model"},{"type":"heading","level":2,"text":"3.1 Masked Language Model"},{"type":"para","text":"The BERT model takes an unlabeled input sequence x = ( x 1 , x 2 , … , x | x | ) x=(x_{1},x_{2},...,x_{|x|}) and randomly masks some positions M x M_{x} by replacing them with a special mask token [MASK] , to produce a sequence like ( x 1 , [MASK] , … , x | x | ) (x_{1},\\textnormal{[MASK]},...,x_{|x|}) . All the tokens are embedded and added to special positional embeddings. It then uses N N identical Transformer layers to generate contextualized representation, with each layer employing self-attention by taking in the output of the previous layer. To compute self-attention, the output of the previous layer is projected into triplets of vectors named Query, Key and Value ( Q , K , V Q,K,V ) of dimensions d d . The attention A A is then given as:"},{"type":"para","text":"After the final Transformer layer the model uses a feed forward layer followed by a softmax over the vocabulary to predict the masked tokens. The MLM loss for the sequence x x is then calculated as:"},{"type":"para","text":"where ( x m ′ ∈ x ∖ M x ) (x_{m^{\\prime}}\\in x\\setminus M_{x}) represents all the tokens in x x that are not masked and m ∈ M x m\\in M_{x} are all the masked positions."},{"type":"heading","level":2,"text":"3.2 Encoding multiple products and common headline for Proposed MLM"},{"type":"para","text":"During training, for a given advertising campaign, our model takes as input it’s headline x h = ( x 1 h , … , x | x h | h ) x^{h}=(x^{h}_{1},...,x^{h}_{|x^{h}|}) and a set P P of one or more products. Each product p p is represented by its title x p = ( x 1 p , … , x | x p | p ) x^{p}=(x^{p}_{1},...,x^{p}_{|x^{p}|}) . The titles and the headline are tokenized to sub-word tokens."},{"type":"para","text":"To encode using the model that only accepts a single product, we simply append ‘ [EOS] ’ ∈ 𝕍 \\in\\mathbb{V} to both the title and the headline and concatenate their tokens. The entire concatenated sequence is prepended with ‘ [SOS] ’ ∈ 𝕍 \\in\\mathbb{V} ."},{"type":"para","text":"We encode multiple products by concatenating the tokens from different products using a special token ‘ [P_SEP] ’ ∈ 𝕍 \\in\\mathbb{V} . We replace a token ‘ [UNUSED_0] ’ ∈ 𝕍 \\in\\mathbb{V} that remains unused during pre-training, with this special token during multi-product fine-tuning. This makes a distinction between different titles as well as the source and target sub-sequences. It also yields individual embeddings for each product for other tasks."},{"type":"para","text":"Only the tokens from the headline x h x^{h} are randomly masked with token ‘ [MASK] ’ ∈ 𝕍 \\in\\mathbb{V} . We discuss results for the model that additionally also masks the source tokens in section 5.1 ."},{"type":"para","text":"The complete process for an example such that all products in the ad have two tokens and the headline has 4 tokens is illustrated in Figure 2 ."},{"type":"para","text":"We also experimented with adding of category based embeddings. The category labels for each product such as “Cell Phones and Accessories\" are tokenized to subword units, encoded using the same embedding matrix as that of the title tokens, averaged and added to the title token embeddings."},{"type":"heading","level":2,"text":"3.3 Generation using Self-Critical Masked Language Model"},{"type":"para","text":"The BERT MLM framework with multi-directional attention discussed in Section 3.1 cannot be used for auto-regressive generation directly. This is because, during training, the masked headline words may condition on the future words which are not available during auto-regressive inference. For MLM auto-regressive generation, we employ masked attention Dong et al. ( 2019 ) that modifies the attention from equation 1 as below:"},{"type":"para","text":"where Φ i j \\Phi_{ij} represents the attention mask between the positions i i and j j . The elements are set to 0 if attention is allowed and − ∞ -\\infty if it is not allowed. Figure 3 illustrates the attention mask for headline generation using multiple input products."},{"type":"para","text":"The BERT MLM uses log-likelihood (Equation 2 ) of masked words during training to optimize the model parameters. The likelihood is predicted using other ground-truth words during training and other predicted words during inference. This causes exposure bias Ranzato et al. ( 2016 ); Rennie et al. ( 2017 ) and accumulates error during inference. Moreover, the training is optimized for log-likelihood, while we actually care about other more evolved measures of headline quality such as overlap metrics BLEU Papineni et al. ( 2002 ) and ROUGE Lin ( 2004 ) ."},{"type":"para","text":"To overcome these issues and improve the quality of the generated headlines, we frame the MLM as an RL problem. The model is an ‘agent’ that takes the ‘action’ of predicting masked words and updates the ‘state’ such as the self-attention weights. The MLM follows a policy π θ \\pi_{\\theta} defined by the parameters θ \\theta of the model. It receives a reward that is proportional to the quality of the generated headline. This quality may either be the overlap with ground truth headlines that have been approved by internal subject-matter-experts or be predicted by another model. Our goal is to maximize the reward corresponding to a generated headline x ^ h \\hat{x}^{h} during training, with the tokens at some masked positions M x h M_{x^{h}} sampled from the model."},{"type":"para","text":"We thus minimize the negative expected reward defined by any reward function r ( ⋅ ) r(\\cdot) for headline quality r ( x ^ h ) r(\\hat{x}^{h}) as:"},{"type":"para","text":"We can compute the gradient ∇ θ ℒ R L \\nabla_{\\theta}\\mathcal{L}_{RL} using the REINFORCE algorithm Williams ( 1992 ) . It is defined as:"},{"type":"para","text":"where,"},{"type":"para","text":"such that M x ^ h M_{\\hat{x}^{h}} are the masked positions and x ^ h ∖ M x ^ h \\hat{x}^{h}\\setminus M_{\\hat{x}^{h}} are all the unmasked tokens."},{"type":"para","text":"To reduce the variance without changing the expected gradient, the algorithm proposes to use a baseline b b that does not depend on the generated headline x ^ h \\hat{x}^{h} . b b is used to normalize the reward along with P P from equation 6 as:"},{"type":"para","text":"A single Monte-Carlo sample for each set of products and headline can be used to approximate the gradient. Using the definition of P P from equation 6 , we have the approximate gradient:"},{"type":"para","text":"Instead of using other models to estimate the expected baseline reward Ranzato et al. ( 2016 ); Bahdanau et al. ( 2017 ) , we employ Self-Critical training Rennie et al. ( 2017 ) that involves generating two headlines using the same underlying MLM. The first headline x ^ h \\hat{x}^{h} is generated by sampling from the vocabulary distributions generated by the model for the masked tokens. The second headline z ^ h \\hat{z}^{h} is generated using the inference time strategy, which uses the token with the maximum probability at each step rather than sampling. The difference in the reward achieved by these two headlines is used to compute the gradient:"},{"type":"para","text":"where P P is defined by equation 6 ."},{"type":"para","text":"Thus, this method maximizes both the reward of the headlines generated by MLM and the likelihood of correct words by incorporating both the likelihood and the reward in the loss function."},{"type":"heading","level":2,"text":"3.4 Inference"},{"type":"para","text":"During inference, we generate the headline auto-regressively using beam search until we reach the predetermined max length or each beam generates the end token. We have employed a modified version of Length Normalization Wu et al. ( 2016 ) to better adapt to our headline lengths and training setup. This is necessary as the default beam search setup uses the log probability of each word to select the best headline. However, this biases the results as longer headlines would have lower probability of generation. We thus use the following normalized scores for each word to select the best headline:"},{"type":"para","text":"where α \\alpha is the length normalization coefficient and x ^ i h \\hat{x}^{h}_{i} is the i t h i^{th} word of the generated headline in each beam. We also include additional Regular Expression based post-processing to remove extra spaces around various symbols such as ‘-,+()’ etc."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":2,"text":"4.1 Training and Inference"},{"type":"para","text":"We used over 500,000 ad campaigns that were created on Amazon by sellers who have signed-up for advertising. Each campaign contains a set of related products along with an ad headline. We only selected the campaigns that contained English headlines and products with English titles. They were also de-duplicated to only have unique products-headline pairs. The mean product title length is 19.6 words and the mean headline length is 6.16 words. The entire dataset was divided into train (85%), validation (5%) and test (10%) sets. For training, we only selected the campaigns that comply with ad policies as verified by internal experts."},{"type":"para","text":"We use HuggingFace Wolf et al. ( 2020 ) implementation of Transformer BERT ‘Large’ models as the base for our experiments. The models are pre-trained on WikiPedia and BookCorpus Devlin et al. ( 2019 ); Dong et al. ( 2019 ) . We first fine-tune the pre-trained model for up-to 15 epochs with early stopping using ℒ M L M \\mathcal{L}_{MLM} and Adam Kingma and Ba ( 2014 ) . We then further fine-tune the model for another 15 epochs with early stopping using Adam with ∇ ℒ S C _ M L M \\nabla\\mathcal{L}_{SC\\_MLM} (Equation 9 ). We use the Rouge L F1 Lin ( 2004 ) overlap with the approved headlines as the headline quality reward. For a fair comparison, the MLM-only model is fine-tuned for upto 30 epochs."},{"type":"para","text":"The model training is very time expensive with a single fine-tuning sub-experiment of 30 epochs taking over 20 days on an Nvidia v100. We thus only performed the essential experiments that help to determine the contribution of different sub-experiments and proposals. We estimated post-experiment that a single fine-tuning sub-experiment of 30 epochs would consume approximately 150 kWh of energy based on the GPU’s power draw."},{"type":"heading","level":2,"text":"4.2 Baseline"},{"type":"para","text":"We used a Pointer Network See et al. ( 2017 ) based bi-LSTM with intra-decoder and temporal attention. We also used Self-Critical training with the bi-LSTM, similar to other ad headline generation methods Xu et al. ( 2019 ); Hughes et al. ( 2019 ) methods for a fair comparison to Self-Critical MLM."},{"type":"heading","level":2,"text":"4.3 Ablations"},{"type":"para","text":"We trained a model with the same architecture, number of parameters and input as the proposed models but without MLM pre-training and separately without Self-Critical loss to study the impact of the proposals."},{"type":"para","text":"We also trained a model with MLM pre-training but fine-tuning only using the primary first product from each campaign instead of using all the products. This is interesting since some of the campaigns are cohesive to a degree with similar products and using only one product improves training time and inference latency."},{"type":"para","text":"We also report overlap metrics for model that does not use length normalization and post-processing discussed in equation 10 . We also include results for model that uses BERT Base as the base model instead of BERT Large."},{"type":"heading","level":1,"text":"5 Results"},{"type":"heading","level":2,"text":"5.1 Overlap with Approved Headlines"},{"type":"para","text":"The first evaluation criterion we adopt is overlap Sharma et al. ( 2017 ) of model headlines with subject-matter-experts approved human-submitted headlines from the test set (Table 1 )."},{"type":"para","text":"Masking the source product title words reduces the performance as the titles and headlines do not follow the same sentence structure and distribution. Adding product category embedding reduces performance and our hypothesis is that this is because the base model cannot be pre-trained with these embeddings. Only using one title achieves lesser but respectable performance, highlighting the efficacy of multi-product conditioning."},{"type":"para","text":"“No pre-training of MLM\" highlights the advantage of using non-pretrained Transformer based architecture over bi-LSTM. ‘Proposed MLM’ shows the advantage of using pre-training, BERT Large and only masking the headline. ‘Proposed Self-Critical MLM’ achieves the best scores across all the metrics and highlights the applicability of our proposed approach."},{"type":"heading","level":2,"text":"5.2 Quality and Grammar Audits"},{"type":"para","text":"We also conducted large scale crowd-sourced evaluation studies of the headlines with over 150,000 judgments. All headlines are shuffled and each headline is rated by 3 random and double-blind crowd-sourced auditors. The quality is judged on a 3-point scale of [1. Incorrect or Irrelevant, 2. Correct, 3. Correct and Attractive] and we use the mode of the 3 judgments."},{"type":"para","text":"In this double-blind audit, the auditors were not aware of the source of the headlines and we were not aware of the identity or demographics of any auditor. More details about the workforce may be found in the platform documentation (Ground Truth, 2021 ) . In order to determine the compensation for the crowd-sourced workers, we used the guideline provided by the crowd-sourcing platform to “choose a price consistent with the approximate time it takes to complete a task\" (Visible in the Console while creating the Labeling ( 2021 ) job). We thus first conducted an internal audit by volunteers across our organization to determine the time required to complete the task (average 21.59s) and then used the remuneration recommended for the corresponding time range ($0.12 for 20s - 22s)."},{"type":"para","text":"Table 2 summarizes the quality audits. The SC-biLSTM model performed worse compared to human-submitted headlines. The proposed SC-MLM model achieves the highest average rating and the most number of perfectly rated headlines. Using just a single product does produce correct headlines with 8% faster inference latency but fails to produce attractive headlines due to lack of input from multiple products."},{"type":"para","text":"We also conducted Grammar specific audits (N ≈ \\approx 10000) in which the grammar of the headlines is judged independently. 98.13% of SC-MLM and 98.12% of MLM generated headlines were judged to have correct grammar against 93.14% of human submitted headlines."},{"type":"para","text":"Table 3 shows a sample of headlines for campaigns in the blind test-set. Excessive keyword stuffing in source product titles does hamper headline quality at times and post-filtering using beam search score helps to filter them out. We do observe cases where both the models generate the same headline. This is an artifact of the fact that both the models share the first 15 epochs. The SC-MLM model generates more descriptive headlines and both models are able to abstract the product qualities."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"Ad headline generation is a difficult problem owing to the varying nature of retail product attributes. A lot of historical methods focus on template based creation of ad headlines that are not very expressive."},{"type":"para","text":"We demonstrated a new NLG based method to generate headlines for multiple products. Our method achieves highest score in overlap metrics, quality audits and grammar audits compared to the baselines and human-submitted headlines."},{"type":"para","text":"Masked Language Models were relatively unexplored for ad headline generation and we were able to demonstrate their utility. We further extended the performance of the model by using Reinforcement Learning. The method only changes the training procedure without impacting inference latency. Thus, our work contributes to both SOTA and practical business applications."},{"type":"para","text":"The approach can also be used for any other NLG task."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There are a various types of ads. A set of example ads that showcase products selected by sellers along with headlines that advertise them are shown in Figure 1 . Sellers create multiple ad campaigns for multiple products, bid in an auction to advertise and pay for clicks on the ad.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An E-Commerce product catalog may have millions of products which can be advertised. To ease the ad headline writing process, humans resort to programmatically padding keywords, or repasting the retail catalog content in the advertisement.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Templated creatives such as “Save Now on …\\\" or “Buy more (product) of (brand)\\\" save the creative effort but fail to create any excitement or brand identity in the minds of shoppers. High quality headlines are more attractive to shoppers and offer better value proposition. In this paper, we describe how we built a Natural Language Generation (NLG) system to generate instantaneous, attractive and brand identity building headlines for advertisements that intend to promote a wide range of products offered by a brand.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The content associated with a retail product has challenging characteristics. Some product titles have poor structure, grammatical issues, or partial phrases. The product titles also include varying number of product features such as “Hyper Tough 18V Cordless Drill, 3/8 inch Chuck, Variable Speed, with 1.2Ah Nickel Cadmium Battery, Charger, Bit Holder LED Light\\\" along with titles such as “ZIPIT Grillz Backpack, Camo Grey\\\" .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The generated headlines need to capture the information present in the retail attributes and at the same time be different and uniquely attractive. Advertisers select multiple related products that are advertised as part of a single ad campaign. The ad campaign headline is then shared across all of these related products. Thus, the headline also needs to generalize the shared characteristics of the products and cannot be specific to a single product within the campaign.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The key contributions of our work are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use Masked Language Model (MLM) for the generation of advertisement headlines using multiple products at the same time. Extensive test-set metrics, quality and grammar audits show that the proposed model outperforms all the baselines and the human-submitted headlines in terms of quality and grammar.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The novel usage of RL for the training of MLM allows us to directly optimize the MLM for improved headline quality metrics without changing inference setup or latency. Our method can also be applied to any other NLG task such as summarization, translation etc.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our model reduces the extensive effort and time that is required to manually create headlines and has low latency.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Natural Language Understanding (NLU) using Language Models (LM) has observed great leaps in recent years. LMs have evolved from using word level models Joulin et al. ( 2016 ) to to a variety of extensions to the Transformer Vaswani et al. ( 2017 ) . The BERT Devlin et al. ( 2019 ) employs Transformer in a pre-training setting and introduced the MLM training objective.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ramachandran et al. ( 2016 ) first demonstrated textual generation by using auto-regressive prediction in a seq2seq architecture. Transformer based auto-regressive methods such as GPT2 Radford et al. ( 2019 ) and BART Lewis et al. ( 2019 ) which predict one word at a time have also shown good results. Zhu et al. ( 2020 ) concatenated BERT representations with the Encoder and Decoder layers of another LM to incorporate pre-trained LM. Another model Dong et al. ( 2019 ) combines BERT-based Transformer Encoder with attention masking from the Transformer decoder. Rothe et al. ( 2019 ) combined pre-trained BERT Encoder with GPT decoder for NLG.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ranzato et al. ( 2016 ) framed NLG as an RL problem and the generation quality as a reward. The Self-Critical Sequence Training (SCST) approach Rennie et al. ( 2017 ) replaces the learned baseline from other approaches Bahdanau et al. ( 2017 ) with the model’s own inference time algorithm to normalize the rewards.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For advertising, recent works Xu et al. ( 2019 ); Hughes et al. ( 2019 ) have combined LSTM based pointer network See et al. ( 2017 ) with RL methods to generate advertisement headlines. While these methods improve the results, they fail to utilize extensive pre-training of Transformer based models and their various well-demonstrated advantages.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our method extends BERT based generation Dong et al. ( 2019 ) by using Self-Critical policy gradient method Rennie et al. ( 2017 ) and jointly conditioning the generated sentence on multiple products at the same time. This allows us to use pre-trained BERT based LMs that can be trained to optimize various inference time metrics that are typically non-differentiable such as BLEU, Rouge, Readability etc.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Self-Critical Masked Language Model\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Masked Language Model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The BERT model takes an unlabeled input sequence x = ( x 1 , x 2 , … , x | x | ) x=(x_{1},x_{2},...,x_{|x|}) and randomly masks some positions M x M_{x} by replacing them with a special mask token [MASK] , to produce a sequence like ( x 1 , [MASK] , … , x | x | ) (x_{1},\\\\textnormal{[MASK]},...,x_{|x|}) . All the tokens are embedded and added to special positional embeddings. It then uses N N identical Transformer layers to generate contextualized representation, with each layer employing self-attention by taking in the output of the previous layer. To compute self-attention, the output of the previous layer is projected into triplets of vectors named Query, Key and Value ( Q , K , V Q,K,V ) of dimensions d d . The attention A A is then given as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After the final Transformer layer the model uses a feed forward layer followed by a softmax over the vocabulary to predict the masked tokens. The MLM loss for the sequence x x is then calculated as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ( x m ′ ∈ x ∖ M x ) (x_{m^{\\\\prime}}\\\\in x\\\\setminus M_{x}) represents all the tokens in x x that are not masked and m ∈ M x m\\\\in M_{x} are all the masked positions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Encoding multiple products and common headline for Proposed MLM\"\n },\n {\n \"type\": \"para\",\n \"text\": \"During training, for a given advertising campaign, our model takes as input it’s headline x h = ( x 1 h , … , x | x h | h ) x^{h}=(x^{h}_{1},...,x^{h}_{|x^{h}|}) and a set P P of one or more products. Each product p p is represented by its title x p = ( x 1 p , … , x | x p | p ) x^{p}=(x^{p}_{1},...,x^{p}_{|x^{p}|}) . The titles and the headline are tokenized to sub-word tokens.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To encode using the model that only accepts a single product, we simply append ‘ [EOS] ’ ∈ 𝕍 \\\\in\\\\mathbb{V} to both the title and the headline and concatenate their tokens. The entire concatenated sequence is prepended with ‘ [SOS] ’ ∈ 𝕍 \\\\in\\\\mathbb{V} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We encode multiple products by concatenating the tokens from different products using a special token ‘ [P_SEP] ’ ∈ 𝕍 \\\\in\\\\mathbb{V} . We replace a token ‘ [UNUSED_0] ’ ∈ 𝕍 \\\\in\\\\mathbb{V} that remains unused during pre-training, with this special token during multi-product fine-tuning. This makes a distinction between different titles as well as the source and target sub-sequences. It also yields individual embeddings for each product for other tasks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Only the tokens from the headline x h x^{h} are randomly masked with token ‘ [MASK] ’ ∈ 𝕍 \\\\in\\\\mathbb{V} . We discuss results for the model that additionally also masks the source tokens in section 5.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The complete process for an example such that all products in the ad have two tokens and the headline has 4 tokens is illustrated in Figure 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also experimented with adding of category based embeddings. The category labels for each product such as “Cell Phones and Accessories\\\" are tokenized to subword units, encoded using the same embedding matrix as that of the title tokens, averaged and added to the title token embeddings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Generation using Self-Critical Masked Language Model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The BERT MLM framework with multi-directional attention discussed in Section 3.1 cannot be used for auto-regressive generation directly. This is because, during training, the masked headline words may condition on the future words which are not available during auto-regressive inference. For MLM auto-regressive generation, we employ masked attention Dong et al. ( 2019 ) that modifies the attention from equation 1 as below:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where Φ i j \\\\Phi_{ij} represents the attention mask between the positions i i and j j . The elements are set to 0 if attention is allowed and − ∞ -\\\\infty if it is not allowed. Figure 3 illustrates the attention mask for headline generation using multiple input products.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The BERT MLM uses log-likelihood (Equation 2 ) of masked words during training to optimize the model parameters. The likelihood is predicted using other ground-truth words during training and other predicted words during inference. This causes exposure bias Ranzato et al. ( 2016 ); Rennie et al. ( 2017 ) and accumulates error during inference. Moreover, the training is optimized for log-likelihood, while we actually care about other more evolved measures of headline quality such as overlap metrics BLEU Papineni et al. ( 2002 ) and ROUGE Lin ( 2004 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To overcome these issues and improve the quality of the generated headlines, we frame the MLM as an RL problem. The model is an ‘agent’ that takes the ‘action’ of predicting masked words and updates the ‘state’ such as the self-attention weights. The MLM follows a policy π θ \\\\pi_{\\\\theta} defined by the parameters θ \\\\theta of the model. It receives a reward that is proportional to the quality of the generated headline. This quality may either be the overlap with ground truth headlines that have been approved by internal subject-matter-experts or be predicted by another model. Our goal is to maximize the reward corresponding to a generated headline x ^ h \\\\hat{x}^{h} during training, with the tokens at some masked positions M x h M_{x^{h}} sampled from the model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We thus minimize the negative expected reward defined by any reward function r ( ⋅ ) r(\\\\cdot) for headline quality r ( x ^ h ) r(\\\\hat{x}^{h}) as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can compute the gradient ∇ θ ℒ R L \\\\nabla_{\\\\theta}\\\\mathcal{L}_{RL} using the REINFORCE algorithm Williams ( 1992 ) . It is defined as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"such that M x ^ h M_{\\\\hat{x}^{h}} are the masked positions and x ^ h ∖ M x ^ h \\\\hat{x}^{h}\\\\setminus M_{\\\\hat{x}^{h}} are all the unmasked tokens.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To reduce the variance without changing the expected gradient, the algorithm proposes to use a baseline b b that does not depend on the generated headline x ^ h \\\\hat{x}^{h} . b b is used to normalize the reward along with P P from equation 6 as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A single Monte-Carlo sample for each set of products and headline can be used to approximate the gradient. Using the definition of P P from equation 6 , we have the approximate gradient:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instead of using other models to estimate the expected baseline reward Ranzato et al. ( 2016 ); Bahdanau et al. ( 2017 ) , we employ Self-Critical training Rennie et al. ( 2017 ) that involves generating two headlines using the same underlying MLM. The first headline x ^ h \\\\hat{x}^{h} is generated by sampling from the vocabulary distributions generated by the model for the masked tokens. The second headline z ^ h \\\\hat{z}^{h} is generated using the inference time strategy, which uses the token with the maximum probability at each step rather than sampling. The difference in the reward achieved by these two headlines is used to compute the gradient:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where P P is defined by equation 6 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, this method maximizes both the reward of the headlines generated by MLM and the likelihood of correct words by incorporating both the likelihood and the reward in the loss function.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Inference\"\n },\n {\n \"type\": \"para\",\n \"text\": \"During inference, we generate the headline auto-regressively using beam search until we reach the predetermined max length or each beam generates the end token. We have employed a modified version of Length Normalization Wu et al. ( 2016 ) to better adapt to our headline lengths and training setup. This is necessary as the default beam search setup uses the log probability of each word to select the best headline. However, this biases the results as longer headlines would have lower probability of generation. We thus use the following normalized scores for each word to select the best headline:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α \\\\alpha is the length normalization coefficient and x ^ i h \\\\hat{x}^{h}_{i} is the i t h i^{th} word of the generated headline in each beam. We also include additional Regular Expression based post-processing to remove extra spaces around various symbols such as ‘-,+()’ etc.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Training and Inference\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We used over 500,000 ad campaigns that were created on Amazon by sellers who have signed-up for advertising. Each campaign contains a set of related products along with an ad headline. We only selected the campaigns that contained English headlines and products with English titles. They were also de-duplicated to only have unique products-headline pairs. The mean product title length is 19.6 words and the mean headline length is 6.16 words. The entire dataset was divided into train (85%), validation (5%) and test (10%) sets. For training, we only selected the campaigns that comply with ad policies as verified by internal experts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use HuggingFace Wolf et al. ( 2020 ) implementation of Transformer BERT ‘Large’ models as the base for our experiments. The models are pre-trained on WikiPedia and BookCorpus Devlin et al. ( 2019 ); Dong et al. ( 2019 ) . We first fine-tune the pre-trained model for up-to 15 epochs with early stopping using ℒ M L M \\\\mathcal{L}_{MLM} and Adam Kingma and Ba ( 2014 ) . We then further fine-tune the model for another 15 epochs with early stopping using Adam with ∇ ℒ S C _ M L M \\\\nabla\\\\mathcal{L}_{SC\\\\_MLM} (Equation 9 ). We use the Rouge L F1 Lin ( 2004 ) overlap with the approved headlines as the headline quality reward. For a fair comparison, the MLM-only model is fine-tuned for upto 30 epochs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model training is very time expensive with a single fine-tuning sub-experiment of 30 epochs taking over 20 days on an Nvidia v100. We thus only performed the essential experiments that help to determine the contribution of different sub-experiments and proposals. We estimated post-experiment that a single fine-tuning sub-experiment of 30 epochs would consume approximately 150 kWh of energy based on the GPU’s power draw.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Baseline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We used a Pointer Network See et al. ( 2017 ) based bi-LSTM with intra-decoder and temporal attention. We also used Self-Critical training with the bi-LSTM, similar to other ad headline generation methods Xu et al. ( 2019 ); Hughes et al. ( 2019 ) methods for a fair comparison to Self-Critical MLM.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Ablations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We trained a model with the same architecture, number of parameters and input as the proposed models but without MLM pre-training and separately without Self-Critical loss to study the impact of the proposals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also trained a model with MLM pre-training but fine-tuning only using the primary first product from each campaign instead of using all the products. This is interesting since some of the campaigns are cohesive to a degree with similar products and using only one product improves training time and inference latency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also report overlap metrics for model that does not use length normalization and post-processing discussed in equation 10 . We also include results for model that uses BERT Base as the base model instead of BERT Large.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Overlap with Approved Headlines\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first evaluation criterion we adopt is overlap Sharma et al. ( 2017 ) of model headlines with subject-matter-experts approved human-submitted headlines from the test set (Table 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Masking the source product title words reduces the performance as the titles and headlines do not follow the same sentence structure and distribution. Adding product category embedding reduces performance and our hypothesis is that this is because the base model cannot be pre-trained with these embeddings. Only using one title achieves lesser but respectable performance, highlighting the efficacy of multi-product conditioning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"“No pre-training of MLM\\\" highlights the advantage of using non-pretrained Transformer based architecture over bi-LSTM. ‘Proposed MLM’ shows the advantage of using pre-training, BERT Large and only masking the headline. ‘Proposed Self-Critical MLM’ achieves the best scores across all the metrics and highlights the applicability of our proposed approach.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Quality and Grammar Audits\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also conducted large scale crowd-sourced evaluation studies of the headlines with over 150,000 judgments. All headlines are shuffled and each headline is rated by 3 random and double-blind crowd-sourced auditors. The quality is judged on a 3-point scale of [1. Incorrect or Irrelevant, 2. Correct, 3. Correct and Attractive] and we use the mode of the 3 judgments.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this double-blind audit, the auditors were not aware of the source of the headlines and we were not aware of the identity or demographics of any auditor. More details about the workforce may be found in the platform documentation (Ground Truth, 2021 ) . In order to determine the compensation for the crowd-sourced workers, we used the guideline provided by the crowd-sourcing platform to “choose a price consistent with the approximate time it takes to complete a task\\\" (Visible in the Console while creating the Labeling ( 2021 ) job). We thus first conducted an internal audit by volunteers across our organization to determine the time required to complete the task (average 21.59s) and then used the remuneration recommended for the corresponding time range ($0.12 for 20s - 22s).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 summarizes the quality audits. The SC-biLSTM model performed worse compared to human-submitted headlines. The proposed SC-MLM model achieves the highest average rating and the most number of perfectly rated headlines. Using just a single product does produce correct headlines with 8% faster inference latency but fails to produce attractive headlines due to lack of input from multiple products.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also conducted Grammar specific audits (N ≈ \\\\approx 10000) in which the grammar of the headlines is judged independently. 98.13% of SC-MLM and 98.12% of MLM generated headlines were judged to have correct grammar against 93.14% of human submitted headlines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 shows a sample of headlines for campaigns in the blind test-set. Excessive keyword stuffing in source product titles does hamper headline quality at times and post-filtering using beam search score helps to filter them out. We do observe cases where both the models generate the same headline. This is an artifact of the fact that both the models share the first 15 epochs. The SC-MLM model generates more descriptive headlines and both models are able to abstract the product qualities.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ad headline generation is a difficult problem owing to the varying nature of retail product attributes. A lot of historical methods focus on template based creation of ad headlines that are not very expressive.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrated a new NLG based method to generate headlines for multiple products. Our method achieves highest score in overlap metrics, quality audits and grammar audits compared to the baselines and human-submitted headlines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Masked Language Models were relatively unexplored for ad headline generation and we were able to demonstrate their utility. We further extended the performance of the model by using Reinforcement Learning. The method only changes the training procedure without impacting inference latency. Thus, our work contributes to both SOTA and practical business applications.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The approach can also be used for any other NLG task.\"\n }\n]","isConversational":false}}},{"rowIdx":499,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.06799"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"What Predicts Correctness in Text-to-SQL? A Selective-Prediction Study"},"authors":{"kind":"list like","value":["Robert Richardson"],"string":"[\n \"Robert Richardson\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.06799v1"},"published":{"kind":"timestamp","value":"2026-07-07T00:00:00","string":"2026-07-07T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"A text-to-SQL system turns a natural-language question into a SQL query that runs against a database. These systems are increasingly used where a wrong query is costly, so a system needs to know not just how to answer but when it should decline to. Knowing when to decline requires, for each query the system produces, an estimate of how likely that query is to be correct. We use correct in its standard benchmark sense: the generated query, when executed against the database, returns the same result as a human-written reference query for the same question. Predicting correctness so a system can choose to answer or to abstain is the problem of selective prediction, and it is the focus of this paper."},{"type":"para","text":"Most uncertainty-quantification (UQ) work for text-to-SQL reads it off statistics of the model’s own outputs, without any extra model or labels. The clearest example is self-consistency: draw several queries for the same question and measure how much they agree, on the premise that a confident model repeats itself (Wang et al., 2023 ) . Related signals include the entropy of the samples’ meanings (Kuhn et al., 2023 ; Farquhar et al., 2024 ) and how often a particular query structure recurs. We call these black-box statistical signals because they look only at agreement among samples, not at whether a query’s logic actually answers the question. The alternative is verification: show a separate model the question, the schema, and the candidate query, and ask it to judge whether the query is correct (also called LLM-as-judge). We ask a direct question: on hard text-to-SQL, which of these signals actually predicts correctness, and by how much?"},{"type":"para","text":"To answer it we score each signal by its ability to rank correct queries above incorrect ones, measured as the area under the ROC curve ( auroc ), which is the probability that a randomly chosen correct query is scored above a randomly chosen incorrect one, where 0.5 0.5 is chance and 1.0 1.0 is a perfect ranking. We attach paired bootstrap confidence intervals to every comparison, so a difference between two signals can be read as significant or not. Our contribution is this execution-grounded comparison of the signals available to a practitioner, together with what it implies for deployment."},{"type":"para","text":"A correctness ceiling for black-box statistical UQ (roughly 0.61 0.61 – 0.68 0.68 auroc ), shared by string, structural, and execution self-consistency, by a schema-relevance score, and, once compared against the strongest of these, by white-box log-probability (the model’s own probability for the query it generated)."},{"type":"para","text":"Verification moves past the ceiling, with bootstrapped intervals that exclude zero, across two benchmarks, two generators, and two judge providers. Judges from different providers make different errors, and an ensemble of them is the best and best-calibrated correctness signal we obtain ( 0.82 0.82 auroc , 0.03 0.03 calibration error)."},{"type":"para","text":"A selective-prediction frontier that the self-consistency baseline cannot match."},{"type":"para","text":"A study of trained verifiers, fine-tuned on labeled examples. They do well on databases seen in training but fail to transfer (keep working on databases never seen during training), which separates an in-domain verifier from a universal one."},{"type":"para","text":"Our scope is selective prediction rather than generation. We quantify and calibrate correctness to decide whether to answer or abstain; we do not iterate to improve the query, in contrast with the self-correcting agents discussed next."},{"type":"heading","level":1,"text":"2 Related work"},{"type":"para","text":"Relevant lines of work that have a similar goal to what we are attempting to do here include self-consistency and sampling-based UQ (Wang et al., 2023 ) , semantic entropy via meaning clustering (Kuhn et al., 2023 ; Farquhar et al., 2024 ) , verbalized confidence, conformal and selective generation (Geifman & El-Yaniv, 2017 ; Angelopoulos et al., 2021 ; 2023 ) , sub-clause Platt scaling (Ramachandran & Sarawagi, 2024 ) , and conformal abstention over schema-linking branch points (Somov et al., 2025 ) . Schema linking has also been studied with encoder-side graph models (Wang et al., 2020 ; Cao et al., 2021 ) . LLM-as-judge is widely used for open-ended evaluation. Here we evaluate it specifically as a calibrated correctness predictor for SQL, against black-box baselines, and we test whether judging can be trained and whether it transfers. We draw data from the Spider (Yu et al., 2018 ) and BIRD (Li et al., 2023 ) benchmarks, and reference Spider 2.0 (Lei et al., 2025 ) for context."},{"type":"para","text":"A closely related line builds self-correcting or agentic text-to-SQL systems, in which the model generates a query and then critiques or executes it and regenerates, often over several rounds (Shinn et al., 2023 ; Chen et al., 2024 ; Pourreza & Rafiei, 2023 ; Wang et al., 2024 ) . That work shares a component with ours, since the critic in such a loop is an LLM verifier much like the judge we study, but the objective differs. A self-correcting agent uses the critic to drive iteration toward a more accurate query and reports end-task accuracy. We do not regenerate, but rather ask how well a signal predicts correctness so a system can answer or abstain, and we report calibration and risk-coverage rather than accuracy gains. The two are complementary, because even after the system corrects itself, it still reports a final query and that query still needs a calibrated estimate of whether to trust it. We check this directly (Section 4.3 ) and find that one round of self-correction barely changes accuracy and produces badly miscalibrated confidence, so it does not substitute for a calibrated verifier. Typical self-correction also relies on execution feedback, whereas our verifier reasons over the question, schema, and SQL without executing the candidate, and we find executability itself uninformative for correctness."},{"type":"heading","level":1,"text":"3 Setup"},{"type":"para","text":"Benchmark and generation. We use BIRD (Li et al., 2023 ) , which has databases with data dictionaries and external-knowledge “evidence.” We generate SQL with GPT-4o-mini ( K = 8 K=8 samples per question, temperature 0.7 0.7 , schema and evidence in the prompt) for a fixed slice of 800 questions spanning 8 databases. The slice was fixed before any analysis and does not depend on model output. It takes the questions in benchmark order from each of the 8 databases whose files we held locally, capped per database and at 800 in total. We execute every generated sample against the real database and label a query “correct” when its result set matches the reference query’s, which is the standard execution-correctness criterion for these benchmarks. Throughout, the query we score is the modal query, the one a question’s samples most often produce. Modal-query execution accuracy is 0.451 0.451 , a hard regime with ample headroom for UQ. For breadth we also use a second generator, GPT-4.1-mini (accuracy 0.522 0.522 ), and a second benchmark, Spider’s (Yu et al., 2018 ) multi-table dev queries ( 490 490 questions over 20 20 databases), generated and executed the same way."},{"type":"para","text":"Signals. The black-box statistical signals are string self-consistency (the size of the largest cluster of identical queries among the samples), structural self-consistency (clustering by canonical query structure), execution self-consistency (clustering by result set), a schema-relevance score (the average embedding similarity between the question and the tables and columns the query uses), and query executability (whether the query runs and returns rows). The logic-aware signals are the mean sequence log-probability of the modal query (white-box) and an LLM verifier shown the question, evidence, schema, and candidate SQL. The verifiers are GPT-4o-mini and GPT-4o, which report P ( correct ) P(\\text{correct}) from the YES/NO first-token logits, and Claude-Sonnet-4.6, an independent-provider judge that reports a verbalized 0 – 100 100 probability (Anthropic does not expose token logprobs). All judges see the same input in the same order; the exact prompts are in Appendix B ."},{"type":"para","text":"Metrics. We report tie-robust auroc for correctness, with 2 , 000 2{,}000 -sample paired bootstrap confidence intervals on auroc differences. We also combine signals with a cross-fit logistic regression, which means the combiner is fit on one split and scored on a disjoint split, so it never sees its own test data. For calibration we report expected calibration error (ECE), the average gap between a score read as a probability and the empirical fraction correct at that score (Guo et al., 2017 ) . For abstention we report a risk–coverage frontier, which is the selective risk, or error rate among answered questions, as a function of coverage, the fraction answered, using both an empirical frontier and a distribution-free (Bonferroni-over-grid, δ = 0.1 \\delta=0.1 ) certificate (Angelopoulos et al., 2021 ) . All model outputs are cached, and the API experiments are inexpensive."},{"type":"heading","level":1,"text":"4 Results"},{"type":"heading","level":2,"text":"4.1 The black-box correctness ceiling"},{"type":"para","text":"Table 1 shows the results for the black box signals that do not require additional models. It shows that they are not enough to tell correct SQL from incorrect SQL at high rates. The black-box statistical signals occupy a narrow band, roughly 0.61 0.61 – 0.68 0.68 auroc , with string self-consistency the strongest at 0.675 0.675 . Structural self-consistency and execution self-consistency are weaker, and executability is essentially chance: wrong queries often run successfully and return rows."},{"type":"para","text":"White-box log-probability is also not enough in this harder multi-table setting. Its auroc is 0.669 0.669 , slightly below string self-consistency, and the paired difference is small and not significant: − 0.006 -0.006 with 95% CI [ − 0.028 , + 0.017 ] [-0.028,+0.017] . This is a useful negative result. In earlier single-table experiments, log-probability helped separate confidently wrong unanimous samples. On BIRD, where the errors involve more complex reasoning, it provides no better correctness signal than sampling agreement. The result suggests a ceiling for signals that do not directly reason about whether the SQL answers the question."},{"type":"heading","level":2,"text":"4.2 Verification moves past the ceiling"},{"type":"para","text":"We next test signals that can inspect the candidate query itself. The verifier is shown the question, evidence, schema, and generated SQL, and asked whether the SQL correctly answers the question. Unlike self-consistency or log-probability, this signal can in principle evaluate the query’s logic, deciding whether the aggregation is appropriate, whether the filters match the wording of the question, whether the grouping is correct, and whether the query computes the requested quantity."},{"type":"para","text":"Table 2 shows that verification is the first signal to move clearly past the black-box ceiling. All three LLM verifiers exceed string self-consistency, and the paired bootstrap intervals exclude zero. GPT-4o-mini gives a moderate gain, while the stronger GPT-4o and Claude judges give substantially larger gains."},{"type":"para","text":"The gain comes from the verifier itself, not from a generic combination with self-consistency. A cross-fit logistic model using GPT-4o plus string self-consistency reaches 0.754 0.754 auroc , which is below GPT-4o alone at 0.770 0.770 . In other words, once a strong verifier is available, the sampling agreement signal contributes little."},{"type":"heading","level":2,"text":"4.3 Robustness: cross-provider judges, a second generator, and self-correction"},{"type":"para","text":"A natural concern is that a verifier from the same model family as the generator might simply share the generator’s biases. To test this, we add an independent-provider judge. Claude-Sonnet-4.6, judging GPT-4o-mini’s SQL, is the strongest single verifier in this experiment, with auroc 0.776 0.776 compared with GPT-4o’s 0.770 0.770 . More importantly, the GPT-4o and Claude scores are not redundant as their correlation is only r = 0.43 r=0.43 , indicating that they make different errors."},{"type":"para","text":"The pattern is not specific to GPT-4o-mini as the generator. We regenerate the same BIRD slice with GPT-4.1-mini, which is stronger on this task, and judge the resulting modal queries with GPT-4o-mini. The verifier still improves over self-consistency, although the margin is smaller."},{"type":"para","text":"For both generators, self-consistency remains in the same performance band and the verifier moves beyond it. The smaller gain for GPT-4.1-mini is consistent with the judge being weaker than the generator: subtler errors made by the stronger generator are harder for GPT-4o-mini to detect. This suggests that the judge should be at least as strong as the generator when the goal is universal correctness prediction."},{"type":"heading","level":4,"text":"A second benchmark."},{"type":"para","text":"The same qualitative pattern appears on Spider. On Spider’s multi-table dev queries ( 490 490 questions over 20 20 databases; modal accuracy 0.620 0.620 ), string self-consistency again sits at the ceiling ( 0.618 0.618 auroc ), while a strong verifier clears it: GPT-4o reaches 0.712 0.712 auroc , a paired improvement of + 0.094 +0.094 over self-consistency with 95% CI [ + 0.029 , + 0.155 ] [+0.029,+0.155] . The weaker GPT-4o-mini judge is above the ceiling but not significantly so ( 0.669 0.669 , Δ = + 0.051 \\Delta=+0.051 , CI [ − 0.013 , + 0.114 ] [-0.013,+0.114] ). Across two benchmarks, two generators, and two judge providers, the same lesson holds: self-consistency plateaus, and a sufficiently strong verifier moves past it."},{"type":"heading","level":4,"text":"A self-correction baseline."},{"type":"para","text":"Self-correcting agents are another way to use a critic, so we also test a simple reflection baseline. Each modal query is fed back to GPT-4o-mini, which reviews it, revises it if needed, and reports a confidence. This changes the query, so it addresses a different problem from selective prediction; nevertheless, it is an important comparison because it asks whether a self-correction loop makes a separate trust signal unnecessary."},{"type":"para","text":"It does not. One reflection round improves execution accuracy only marginally, from 0.451 0.451 to 0.468 0.468 . More importantly for our purposes, the model’s self-reported confidence is badly miscalibrated. It assigns confidence at least 0.9 0.9 to 99 % 99\\% of revised queries, with mean confidence 0.97 0.97 , but its correctness signal has only 0.637 0.637 auroc and ECE 0.501 0.501 . Thus, a single round of self-correction neither closes the accuracy gap nor produces a calibrated abstention score. Regeneration and selective prediction are complementary: even after a model revises a query, the system still needs to decide whether the final query should be trusted."},{"type":"heading","level":2,"text":"4.4 Can the verifier be trained? In-distribution yes, transfer no"},{"type":"para","text":"The frozen LLM verifier is effective, but it requires an additional model call at inference time. A natural alternative is to train a cheaper verifier. We fine-tune models on examples of the form"},{"type":"para","text":"using 6 , 400 6{,}400 execution-labeled pairs. We then evaluate whether the trained verifier transfers to unseen databases. The transfer evaluation uses leave-one-database-out (LODO): one database is held out, the verifier is trained on the remaining databases, and the held-out database is scored. We repeat this for each database and report macro-averaged transfer auroc ."},{"type":"para","text":"We compare a feature classifier, a fine-tuned encoder classifier, and LoRA-tuned generative judges at two sizes. Table 4 summarizes the result."},{"type":"para","text":"There are four takeaways. First, fine-tuning works in-distribution. The trained models reach approximately 0.77 0.77 – 0.80 0.80 auroc , and the 7B generative judge gives the best in-domain result at 0.798 0.798 . For a fixed deployment that owns its schemas and can label examples from those schemas, this is a useful operating point."},{"type":"para","text":"Second, the same models transfer poorly. Across architectures and scales, the trained verifiers land near 0.66 0.66 – 0.67 0.67 on unseen databases. Scaling the generative judge from 1.5B to 7B improves the in-domain number but does not improve LODO transfer. The frozen GPT-4o judge is the exception: when macro-averaged in the same way, it scores 0.710 0.710 , and when pooled over all held-out questions it scores 0.770 0.770 . Thus the trained models are useful in-domain, but they do not yet behave like universal correctness verifiers."},{"type":"para","text":"Third, the transfer gap is not closed by the additional levers we tried. Adding Spider’s schemas to increase training diversity, distilling rationales from a strong judge, scaling the generative verifier, and cross-benchmark training all leave transfer in roughly the same band (Appendix A ). The evidence is consistent with the fine-tuned models relying on schema-specific surface regularities rather than learning a robust cross-schema reasoning procedure."},{"type":"para","text":"An input ablation of the frozen judge supports this interpretation. When the judge sees only the question and SQL, it reaches 0.692 0.692 auroc . Adding the schema changes almost nothing ( 0.688 0.688 ), while adding the evidence helps ( 0.724 0.724 ). This suggests that the frozen judge is not mainly looking up schema content; it is reasoning about whether the query answers the question. Per database, the frozen judge leads on every held-out schema (Figure 1 )."},{"type":"para","text":"Fourth, the in-domain verifier is useful for abstention. A feature classifier with no LLM call at inference gives a risk–coverage frontier competitive with the frozen GPT-4o judge and far above self-consistency. At a 20 % 20\\% risk target it answers 21 % 21\\% of questions, compared with 9 % 9\\% for the frozen judge and 1 % 1\\% for self-consistency; the corresponding areas under the risk–coverage curve are 0.340 0.340 , 0.355 0.355 , and 0.413 0.413 (lower is better). Thus, for a fixed deployment, a trained verifier is not only accurate but also the cheaper route to a useful abstention policy. For open, cross-schema use, the frozen reasoning judge remains stronger."},{"type":"heading","level":2,"text":"4.5 Calibration and selective prediction"},{"type":"para","text":"For selective prediction, a score must do more than rank correct queries above incorrect ones; it must also support a usable threshold. We therefore examine calibration and risk–coverage. Raw self-consistency and raw verifier probabilities are over-confident, but the cross-fit two-provider ensemble is well-calibrated and can be used directly as an estimate of P ( correct ) P(\\text{correct}) ."},{"type":"para","text":"As stated in Section 4.3 , the GPT-4o and Claude scores are not redundant as their correlation is only r = 0.43 r=0.43 , indicating that they make different errors. This difference is useful. A two-provider ensemble of GPT-4o and Claude reaches 0.822 0.822 auroc , improving over GPT-4o alone by + 0.052 +0.052 with 95% CI [ + 0.032 , + 0.073 ] [+0.032,+0.073] . It is the strongest correctness signal we obtain, and it is also well-calibrated after cross-fit logistic calibration (ECE 0.031 0.031 ; Section 4.5 ). The result argues against a self-agreement artifact. The best correctness signal is not merely a larger judge from the same source, but a combination of independent reasoning judgments. The OpenAI judges use YES/NO first-token logits, while Claude reports a verbalized probability; to check that the difference is not just an elicitation artifact, we also evaluate a method-matched OpenAI verbal judge, which still clears the black-box ceiling at 0.709 0.709 auroc (Appendix B )."},{"type":"para","text":"We next use the scores for abstention. For each target selective risk, we split the questions in half: the threshold is chosen on the calibration half, and coverage and risk are reported on the disjoint test half. Thus, no threshold is evaluated on the same data that selected it. Table 6 reports the resulting empirical frontier."},{"type":"para","text":"Self-consistency cannot form a low-risk subset at any threshold in this split: its most confident region is still too error-prone. The logic-aware ensemble, by contrast, yields useful operating points. At a target risk of 0.30 0.30 , for example, it answers 27 % 27\\% of questions with held-out selective risk 0.24 0.24 ."},{"type":"para","text":"We also compute a distribution-free Bonferroni-over-grid certificate. This certificate is conservative in the present regime because the base generator accuracy is only 0.45 0.45 . It fires only at the loosest target, certifying the all-signals score at α = 0.40 \\alpha=0.40 with 16 % 16\\% coverage and 22 % 22\\% risk. We therefore treat the empirical frontier as the practical result and the certificate as a conservative lower bound. A tighter certificate would likely require a stronger base generator or more calibration data, rather than a different confidence score."},{"type":"heading","level":2,"text":"4.6 Where the errors are: computation, not schema linking"},{"type":"para","text":"The preceding results show that verification works and schema relevance does not. To understand why, we parse each gold query into structural features and examine where the generator fails. The pattern is clear: the difficult cases are not mainly the ones with more joins. They are the ones that require computation and composition."},{"type":"para","text":"Join count barely changes accuracy: queries with 0, 1, 2, and 3 or more joins have accuracies 0.48 0.48 , 0.45 0.45 , 0.41 0.41 , and 0.45 0.45 , respectively. A logistic model over the same features ranks arithmetic, ORDER BY, and DISTINCT as the strongest independent error predictors, while join and aggregate indicators have negative coefficients. The benchmark’s own difficulty labels agree with this reading: accuracy is 0.544 0.544 on simple questions, 0.332 0.332 on moderate questions, and 0.287 0.287 on challenging questions."},{"type":"para","text":"This explains why schema relevance is a weak correctness signal. The generator often identifies the relevant tables and columns, but fails on ratios, conditional logic, grouping, ranking, or nested structure. A schema-relevance score cannot see those failures. A verifier can, because it evaluates whether the computation expressed by the SQL matches the question."},{"type":"para","text":"To see whether the verifier earns its advantage where the errors actually are, columns 5–7 of Table 7 compare string self-consistency and the GPT-4o verifier as correctness predictors within each feature subset. The verifier improves on self-consistency for almost every feature, and its strongest discrimination is on arithmetic questions ( auroc 0.813 0.813 ), the largest error category. The clearest exception is GROUP BY, the one subset where the verifier ( 0.609 0.609 ) does not beat self-consistency ( 0.651 0.651 ). Subqueries are a near-tie ( + 0.016 +0.016 ), but only because self-consistency is already strong there ( 0.804 0.804 ): the model’s samples disagree visibly when a query needs nesting, so sampling agreement already flags those failures."},{"type":"para","text":"The verifier’s advantage is not confined to the hardest queries. Splitting the slice into logic-heavy queries (arithmetic, subquery, CASE, or GROUP BY; n = 225 n=225 , accuracy 0.293 0.293 ) and the rest (simple; n = 575 n=575 , accuracy 0.513 0.513 ), the verifier’s edge over self-consistency is actually larger on the simple queries ( + 0.117 +0.117 , from 0.638 0.638 to 0.755 0.755 ) than on the logic-heavy ones ( + 0.073 +0.073 , from 0.690 0.690 to 0.763 0.763 ). The reason is that self-consistency is weakest precisely on the simple queries, where the model is often confidently and unanimously wrong: its samples agree, so sampling agreement reports high confidence on a query that is in fact incorrect. The verifier catches these cases because it reasons about the computation rather than counting agreement."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"para","text":"The central lesson is that correctness uncertainty for text-to-SQL is real, but it does not come from the signals that are cheapest to compute. Sampling agreement, structural agreement, execution-result agreement, schema relevance, executability, and even sequence log-probability all occupy the same broad performance band on hard multi-table data. These signals are useful diagnostics, but they mostly measure whether the generator is internally consistent, not whether the generated SQL actually answers the question. In this setting, a model can be highly self-consistent and still be consistently wrong."},{"type":"para","text":"Verification changes the picture because it evaluates the candidate query at the level where many errors occur. The difficult cases in our study are not primarily failures to mention the right table or to produce executable SQL. They are failures of computation and composition: ratios, percentages, conditional logic, grouping, ranking, nesting, and distinctness. A confidence signal that only sees agreement among samples has little access to those errors. A verifier, by contrast, can compare the question to the computation expressed by the SQL. This explains why verifier scores move past the black-box ceiling while schema-relevance and self-consistency do not."},{"type":"para","text":"The results suggest two different deployment regimes. In a fixed deployment, where the system owner has historical questions and execution labels from the same schemas, a trained verifier is attractive. It is cheap at inference time and performs well in-domain, giving useful abstention frontiers without requiring an additional large-model call. In an open or cross-schema setting, however, fine-tuned verifiers are much less reliable. Across the transfer experiments we tried, small and medium trained models improved in-domain performance but did not close the gap to a large frozen judge on unseen schemas. For that setting, the evidence points toward using a strong reasoning model as the verifier, and toward combining independent-provider judges when the cost is justified."},{"type":"para","text":"The multi-provider result is practically important. The strongest single verifier is already well above the black-box baselines, but the best score comes from combining GPT-4o and Claude. Their errors are only moderately correlated, and the calibrated ensemble improves both discrimination and reliability. This suggests that verification uncertainty is not merely a property of one model family. Independent judges appear to notice different failures, which makes the ensemble a better trust signal than either judge alone."},{"type":"para","text":"The abstention results should be read as empirical operating points rather than as a claim that the problem is solved. The calibrated ensemble gives useful risk–coverage frontiers, while self-consistency cannot form a low-risk subset on this split. The distribution-free certificate is more conservative, firing only at the loosest target. This is not surprising in a regime where the base generator is correct less than half the time. A stronger generator, more calibration data, or both should make certified coverage more useful. The present result is that a logic-aware score creates a usable empirical frontier and a conservative certified lower bound; sampling agreement does not."},{"type":"para","text":"Finally, the paper separates two questions that are often conflated. One question is whether a critic can improve a query by driving a self-correction loop. Another is whether a final query, after whatever generation or correction process produced it, should be trusted. Our self-correction baseline shows why these are different. One reflection round barely improves accuracy and produces severely overconfident self-reported confidence. Even systems that use self-correction still need calibrated selective prediction at the end of the pipeline."},{"type":"heading","level":1,"text":"6 Limitations"},{"type":"para","text":"The main experiments use two OpenAI generators and two judge providers. This is enough to test whether the pattern survives a stronger generator and an independent verifier, but it is not a complete survey of current text-to-SQL systems. An open-weight generator would be a useful additional check, especially because generation errors may differ across model families."},{"type":"para","text":"The verifier experiments also leave open the question of scale. We find that fine-tuned small and medium verifiers perform well in-domain but do not transfer reliably to unseen schemas. Scaling a generative verifier to 7B improves the in-domain result but does not close the transfer gap. This does not rule out the possibility that a larger fine-tuned verifier, or a verifier trained on a much broader collection of schemas and tasks, could generalize better."},{"type":"para","text":"Our distribution-free certificates are conservative. This is partly a sample-size issue and partly a base-accuracy issue: when the generator is correct only about 45% of the time, a very strong confidence score is needed to certify high coverage at low risk. The empirical risk–coverage curves are therefore the more informative deployment summary in the present study, while the certificates should be viewed as conservative lower bounds."},{"type":"para","text":"Finally, the study focuses on deciding whether to trust a generated query, not on improving the query. Verifier-guided reranking, verifier-guided regeneration, and interactive clarification are natural next steps. The present results suggest that such systems should not rely on sampling confidence alone; they should include a component that explicitly evaluates the logic of the candidate SQL."},{"type":"heading","level":1,"text":"7 Conclusion"},{"type":"para","text":"On hard text-to-SQL, the signals that most directly predict correctness are not sampling agreement, structural agreement, executability, schema relevance, or log-probability. Those signals plateau in a relatively narrow band. Verification moves past that ceiling because it can reason about whether the SQL expresses the computation requested by the question. Independent-provider verifiers make different errors and combine into the strongest and best-calibrated correctness signal we observe, enabling empirical abstention where self-consistency cannot. Trained verifiers are useful in-domain, but cross-schema transfer remains difficult; for open settings, a large frozen reasoning judge is currently the more reliable option. The practical message is simple: if a system needs to decide whether to trust generated SQL, it should pay for a logic-aware verifier, and it should calibrate that verifier for selective prediction."},{"type":"para","text":"Each row is an attempt to make a fine-tuned verifier transfer to unseen schemas; none closes the gap to the frozen reasoning judge ( 0.710 0.710 macro, 0.770 0.770 pooled). Schema diversity adds Spider’s 20 schemas (28 in all); reasoning distillation trains a 1.5B model on a strong judge’s rationales plus the verdict, versus verdict-only; cross-benchmark trains on one benchmark and tests on the other."},{"type":"para","text":"Every verifier sees the same content, presented in this order: the schema, the question, the evidence string (when available), and the candidate SQL, followed by the elicitation request. Two elicitation formats are used. The logit format (the OpenAI judges) asks for a one-word verdict and reads P ( correct ) P(\\text{correct}) from the first-token YES/NO probabilities; the verbalized format (Claude, which does not expose token logprobs, and the method-matched OpenAI verbal judge) asks for a numeric probability and divides it by 100 100 ."},{"type":"para","text":"Logit (YES/NO) judge. System: “You are a strict SQL reviewer. Decide whether the candidate SQL CORRECTLY answers the question (right tables, columns, conditions, aggregation, and result). Answer with exactly one word: YES or NO.” The candidate block ends with “Is it correct? Answer YES or NO.”"},{"type":"para","text":"Verbalized (0–100) judge. System: “You are a strict SQL reviewer. Decide whether the candidate SQL correctly answers the question. Respond with ONLY an integer from 0 to 100: the probability (percent) that it is correct. Output only the number.” The candidate block ends with “Probability (0-100) that the SQL is correct:” The returned integer divided by 100 100 is the correctness probability."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A text-to-SQL system turns a natural-language question into a SQL query that runs against a database. These systems are increasingly used where a wrong query is costly, so a system needs to know not just how to answer but when it should decline to. Knowing when to decline requires, for each query the system produces, an estimate of how likely that query is to be correct. We use correct in its standard benchmark sense: the generated query, when executed against the database, returns the same result as a human-written reference query for the same question. Predicting correctness so a system can choose to answer or to abstain is the problem of selective prediction, and it is the focus of this paper.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Most uncertainty-quantification (UQ) work for text-to-SQL reads it off statistics of the model’s own outputs, without any extra model or labels. The clearest example is self-consistency: draw several queries for the same question and measure how much they agree, on the premise that a confident model repeats itself (Wang et al., 2023 ) . Related signals include the entropy of the samples’ meanings (Kuhn et al., 2023 ; Farquhar et al., 2024 ) and how often a particular query structure recurs. We call these black-box statistical signals because they look only at agreement among samples, not at whether a query’s logic actually answers the question. The alternative is verification: show a separate model the question, the schema, and the candidate query, and ask it to judge whether the query is correct (also called LLM-as-judge). We ask a direct question: on hard text-to-SQL, which of these signals actually predicts correctness, and by how much?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To answer it we score each signal by its ability to rank correct queries above incorrect ones, measured as the area under the ROC curve ( auroc ), which is the probability that a randomly chosen correct query is scored above a randomly chosen incorrect one, where 0.5 0.5 is chance and 1.0 1.0 is a perfect ranking. We attach paired bootstrap confidence intervals to every comparison, so a difference between two signals can be read as significant or not. Our contribution is this execution-grounded comparison of the signals available to a practitioner, together with what it implies for deployment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A correctness ceiling for black-box statistical UQ (roughly 0.61 0.61 – 0.68 0.68 auroc ), shared by string, structural, and execution self-consistency, by a schema-relevance score, and, once compared against the strongest of these, by white-box log-probability (the model’s own probability for the query it generated).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Verification moves past the ceiling, with bootstrapped intervals that exclude zero, across two benchmarks, two generators, and two judge providers. Judges from different providers make different errors, and an ensemble of them is the best and best-calibrated correctness signal we obtain ( 0.82 0.82 auroc , 0.03 0.03 calibration error).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A selective-prediction frontier that the self-consistency baseline cannot match.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A study of trained verifiers, fine-tuned on labeled examples. They do well on databases seen in training but fail to transfer (keep working on databases never seen during training), which separates an in-domain verifier from a universal one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our scope is selective prediction rather than generation. We quantify and calibrate correctness to decide whether to answer or abstain; we do not iterate to improve the query, in contrast with the self-correcting agents discussed next.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Relevant lines of work that have a similar goal to what we are attempting to do here include self-consistency and sampling-based UQ (Wang et al., 2023 ) , semantic entropy via meaning clustering (Kuhn et al., 2023 ; Farquhar et al., 2024 ) , verbalized confidence, conformal and selective generation (Geifman & El-Yaniv, 2017 ; Angelopoulos et al., 2021 ; 2023 ) , sub-clause Platt scaling (Ramachandran & Sarawagi, 2024 ) , and conformal abstention over schema-linking branch points (Somov et al., 2025 ) . Schema linking has also been studied with encoder-side graph models (Wang et al., 2020 ; Cao et al., 2021 ) . LLM-as-judge is widely used for open-ended evaluation. Here we evaluate it specifically as a calibrated correctness predictor for SQL, against black-box baselines, and we test whether judging can be trained and whether it transfers. We draw data from the Spider (Yu et al., 2018 ) and BIRD (Li et al., 2023 ) benchmarks, and reference Spider 2.0 (Lei et al., 2025 ) for context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A closely related line builds self-correcting or agentic text-to-SQL systems, in which the model generates a query and then critiques or executes it and regenerates, often over several rounds (Shinn et al., 2023 ; Chen et al., 2024 ; Pourreza & Rafiei, 2023 ; Wang et al., 2024 ) . That work shares a component with ours, since the critic in such a loop is an LLM verifier much like the judge we study, but the objective differs. A self-correcting agent uses the critic to drive iteration toward a more accurate query and reports end-task accuracy. We do not regenerate, but rather ask how well a signal predicts correctness so a system can answer or abstain, and we report calibration and risk-coverage rather than accuracy gains. The two are complementary, because even after the system corrects itself, it still reports a final query and that query still needs a calibrated estimate of whether to trust it. We check this directly (Section 4.3 ) and find that one round of self-correction barely changes accuracy and produces badly miscalibrated confidence, so it does not substitute for a calibrated verifier. Typical self-correction also relies on execution feedback, whereas our verifier reasons over the question, schema, and SQL without executing the candidate, and we find executability itself uninformative for correctness.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Benchmark and generation. We use BIRD (Li et al., 2023 ) , which has databases with data dictionaries and external-knowledge “evidence.” We generate SQL with GPT-4o-mini ( K = 8 K=8 samples per question, temperature 0.7 0.7 , schema and evidence in the prompt) for a fixed slice of 800 questions spanning 8 databases. The slice was fixed before any analysis and does not depend on model output. It takes the questions in benchmark order from each of the 8 databases whose files we held locally, capped per database and at 800 in total. We execute every generated sample against the real database and label a query “correct” when its result set matches the reference query’s, which is the standard execution-correctness criterion for these benchmarks. Throughout, the query we score is the modal query, the one a question’s samples most often produce. Modal-query execution accuracy is 0.451 0.451 , a hard regime with ample headroom for UQ. For breadth we also use a second generator, GPT-4.1-mini (accuracy 0.522 0.522 ), and a second benchmark, Spider’s (Yu et al., 2018 ) multi-table dev queries ( 490 490 questions over 20 20 databases), generated and executed the same way.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Signals. The black-box statistical signals are string self-consistency (the size of the largest cluster of identical queries among the samples), structural self-consistency (clustering by canonical query structure), execution self-consistency (clustering by result set), a schema-relevance score (the average embedding similarity between the question and the tables and columns the query uses), and query executability (whether the query runs and returns rows). The logic-aware signals are the mean sequence log-probability of the modal query (white-box) and an LLM verifier shown the question, evidence, schema, and candidate SQL. The verifiers are GPT-4o-mini and GPT-4o, which report P ( correct ) P(\\\\text{correct}) from the YES/NO first-token logits, and Claude-Sonnet-4.6, an independent-provider judge that reports a verbalized 0 – 100 100 probability (Anthropic does not expose token logprobs). All judges see the same input in the same order; the exact prompts are in Appendix B .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metrics. We report tie-robust auroc for correctness, with 2 , 000 2{,}000 -sample paired bootstrap confidence intervals on auroc differences. We also combine signals with a cross-fit logistic regression, which means the combiner is fit on one split and scored on a disjoint split, so it never sees its own test data. For calibration we report expected calibration error (ECE), the average gap between a score read as a probability and the empirical fraction correct at that score (Guo et al., 2017 ) . For abstention we report a risk–coverage frontier, which is the selective risk, or error rate among answered questions, as a function of coverage, the fraction answered, using both an empirical frontier and a distribution-free (Bonferroni-over-grid, δ = 0.1 \\\\delta=0.1 ) certificate (Angelopoulos et al., 2021 ) . All model outputs are cached, and the API experiments are inexpensive.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 The black-box correctness ceiling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 shows the results for the black box signals that do not require additional models. It shows that they are not enough to tell correct SQL from incorrect SQL at high rates. The black-box statistical signals occupy a narrow band, roughly 0.61 0.61 – 0.68 0.68 auroc , with string self-consistency the strongest at 0.675 0.675 . Structural self-consistency and execution self-consistency are weaker, and executability is essentially chance: wrong queries often run successfully and return rows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"White-box log-probability is also not enough in this harder multi-table setting. Its auroc is 0.669 0.669 , slightly below string self-consistency, and the paired difference is small and not significant: − 0.006 -0.006 with 95% CI [ − 0.028 , + 0.017 ] [-0.028,+0.017] . This is a useful negative result. In earlier single-table experiments, log-probability helped separate confidently wrong unanimous samples. On BIRD, where the errors involve more complex reasoning, it provides no better correctness signal than sampling agreement. The result suggests a ceiling for signals that do not directly reason about whether the SQL answers the question.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Verification moves past the ceiling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next test signals that can inspect the candidate query itself. The verifier is shown the question, evidence, schema, and generated SQL, and asked whether the SQL correctly answers the question. Unlike self-consistency or log-probability, this signal can in principle evaluate the query’s logic, deciding whether the aggregation is appropriate, whether the filters match the wording of the question, whether the grouping is correct, and whether the query computes the requested quantity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 shows that verification is the first signal to move clearly past the black-box ceiling. All three LLM verifiers exceed string self-consistency, and the paired bootstrap intervals exclude zero. GPT-4o-mini gives a moderate gain, while the stronger GPT-4o and Claude judges give substantially larger gains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The gain comes from the verifier itself, not from a generic combination with self-consistency. A cross-fit logistic model using GPT-4o plus string self-consistency reaches 0.754 0.754 auroc , which is below GPT-4o alone at 0.770 0.770 . In other words, once a strong verifier is available, the sampling agreement signal contributes little.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Robustness: cross-provider judges, a second generator, and self-correction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A natural concern is that a verifier from the same model family as the generator might simply share the generator’s biases. To test this, we add an independent-provider judge. Claude-Sonnet-4.6, judging GPT-4o-mini’s SQL, is the strongest single verifier in this experiment, with auroc 0.776 0.776 compared with GPT-4o’s 0.770 0.770 . More importantly, the GPT-4o and Claude scores are not redundant as their correlation is only r = 0.43 r=0.43 , indicating that they make different errors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pattern is not specific to GPT-4o-mini as the generator. We regenerate the same BIRD slice with GPT-4.1-mini, which is stronger on this task, and judge the resulting modal queries with GPT-4o-mini. The verifier still improves over self-consistency, although the margin is smaller.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For both generators, self-consistency remains in the same performance band and the verifier moves beyond it. The smaller gain for GPT-4.1-mini is consistent with the judge being weaker than the generator: subtler errors made by the stronger generator are harder for GPT-4o-mini to detect. This suggests that the judge should be at least as strong as the generator when the goal is universal correctness prediction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A second benchmark.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The same qualitative pattern appears on Spider. On Spider’s multi-table dev queries ( 490 490 questions over 20 20 databases; modal accuracy 0.620 0.620 ), string self-consistency again sits at the ceiling ( 0.618 0.618 auroc ), while a strong verifier clears it: GPT-4o reaches 0.712 0.712 auroc , a paired improvement of + 0.094 +0.094 over self-consistency with 95% CI [ + 0.029 , + 0.155 ] [+0.029,+0.155] . The weaker GPT-4o-mini judge is above the ceiling but not significantly so ( 0.669 0.669 , Δ = + 0.051 \\\\Delta=+0.051 , CI [ − 0.013 , + 0.114 ] [-0.013,+0.114] ). Across two benchmarks, two generators, and two judge providers, the same lesson holds: self-consistency plateaus, and a sufficiently strong verifier moves past it.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A self-correction baseline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Self-correcting agents are another way to use a critic, so we also test a simple reflection baseline. Each modal query is fed back to GPT-4o-mini, which reviews it, revises it if needed, and reports a confidence. This changes the query, so it addresses a different problem from selective prediction; nevertheless, it is an important comparison because it asks whether a self-correction loop makes a separate trust signal unnecessary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It does not. One reflection round improves execution accuracy only marginally, from 0.451 0.451 to 0.468 0.468 . More importantly for our purposes, the model’s self-reported confidence is badly miscalibrated. It assigns confidence at least 0.9 0.9 to 99 % 99\\\\% of revised queries, with mean confidence 0.97 0.97 , but its correctness signal has only 0.637 0.637 auroc and ECE 0.501 0.501 . Thus, a single round of self-correction neither closes the accuracy gap nor produces a calibrated abstention score. Regeneration and selective prediction are complementary: even after a model revises a query, the system still needs to decide whether the final query should be trusted.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Can the verifier be trained? In-distribution yes, transfer no\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The frozen LLM verifier is effective, but it requires an additional model call at inference time. A natural alternative is to train a cheaper verifier. We fine-tune models on examples of the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"using 6 , 400 6{,}400 execution-labeled pairs. We then evaluate whether the trained verifier transfers to unseen databases. The transfer evaluation uses leave-one-database-out (LODO): one database is held out, the verifier is trained on the remaining databases, and the held-out database is scored. We repeat this for each database and report macro-averaged transfer auroc .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare a feature classifier, a fine-tuned encoder classifier, and LoRA-tuned generative judges at two sizes. Table 4 summarizes the result.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There are four takeaways. First, fine-tuning works in-distribution. The trained models reach approximately 0.77 0.77 – 0.80 0.80 auroc , and the 7B generative judge gives the best in-domain result at 0.798 0.798 . For a fixed deployment that owns its schemas and can label examples from those schemas, this is a useful operating point.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, the same models transfer poorly. Across architectures and scales, the trained verifiers land near 0.66 0.66 – 0.67 0.67 on unseen databases. Scaling the generative judge from 1.5B to 7B improves the in-domain number but does not improve LODO transfer. The frozen GPT-4o judge is the exception: when macro-averaged in the same way, it scores 0.710 0.710 , and when pooled over all held-out questions it scores 0.770 0.770 . Thus the trained models are useful in-domain, but they do not yet behave like universal correctness verifiers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Third, the transfer gap is not closed by the additional levers we tried. Adding Spider’s schemas to increase training diversity, distilling rationales from a strong judge, scaling the generative verifier, and cross-benchmark training all leave transfer in roughly the same band (Appendix A ). The evidence is consistent with the fine-tuned models relying on schema-specific surface regularities rather than learning a robust cross-schema reasoning procedure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An input ablation of the frozen judge supports this interpretation. When the judge sees only the question and SQL, it reaches 0.692 0.692 auroc . Adding the schema changes almost nothing ( 0.688 0.688 ), while adding the evidence helps ( 0.724 0.724 ). This suggests that the frozen judge is not mainly looking up schema content; it is reasoning about whether the query answers the question. Per database, the frozen judge leads on every held-out schema (Figure 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fourth, the in-domain verifier is useful for abstention. A feature classifier with no LLM call at inference gives a risk–coverage frontier competitive with the frozen GPT-4o judge and far above self-consistency. At a 20 % 20\\\\% risk target it answers 21 % 21\\\\% of questions, compared with 9 % 9\\\\% for the frozen judge and 1 % 1\\\\% for self-consistency; the corresponding areas under the risk–coverage curve are 0.340 0.340 , 0.355 0.355 , and 0.413 0.413 (lower is better). Thus, for a fixed deployment, a trained verifier is not only accurate but also the cheaper route to a useful abstention policy. For open, cross-schema use, the frozen reasoning judge remains stronger.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Calibration and selective prediction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For selective prediction, a score must do more than rank correct queries above incorrect ones; it must also support a usable threshold. We therefore examine calibration and risk–coverage. Raw self-consistency and raw verifier probabilities are over-confident, but the cross-fit two-provider ensemble is well-calibrated and can be used directly as an estimate of P ( correct ) P(\\\\text{correct}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As stated in Section 4.3 , the GPT-4o and Claude scores are not redundant as their correlation is only r = 0.43 r=0.43 , indicating that they make different errors. This difference is useful. A two-provider ensemble of GPT-4o and Claude reaches 0.822 0.822 auroc , improving over GPT-4o alone by + 0.052 +0.052 with 95% CI [ + 0.032 , + 0.073 ] [+0.032,+0.073] . It is the strongest correctness signal we obtain, and it is also well-calibrated after cross-fit logistic calibration (ECE 0.031 0.031 ; Section 4.5 ). The result argues against a self-agreement artifact. The best correctness signal is not merely a larger judge from the same source, but a combination of independent reasoning judgments. The OpenAI judges use YES/NO first-token logits, while Claude reports a verbalized probability; to check that the difference is not just an elicitation artifact, we also evaluate a method-matched OpenAI verbal judge, which still clears the black-box ceiling at 0.709 0.709 auroc (Appendix B ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next use the scores for abstention. For each target selective risk, we split the questions in half: the threshold is chosen on the calibration half, and coverage and risk are reported on the disjoint test half. Thus, no threshold is evaluated on the same data that selected it. Table 6 reports the resulting empirical frontier.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Self-consistency cannot form a low-risk subset at any threshold in this split: its most confident region is still too error-prone. The logic-aware ensemble, by contrast, yields useful operating points. At a target risk of 0.30 0.30 , for example, it answers 27 % 27\\\\% of questions with held-out selective risk 0.24 0.24 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also compute a distribution-free Bonferroni-over-grid certificate. This certificate is conservative in the present regime because the base generator accuracy is only 0.45 0.45 . It fires only at the loosest target, certifying the all-signals score at α = 0.40 \\\\alpha=0.40 with 16 % 16\\\\% coverage and 22 % 22\\\\% risk. We therefore treat the empirical frontier as the practical result and the certificate as a conservative lower bound. A tighter certificate would likely require a stronger base generator or more calibration data, rather than a different confidence score.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 Where the errors are: computation, not schema linking\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The preceding results show that verification works and schema relevance does not. To understand why, we parse each gold query into structural features and examine where the generator fails. The pattern is clear: the difficult cases are not mainly the ones with more joins. They are the ones that require computation and composition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Join count barely changes accuracy: queries with 0, 1, 2, and 3 or more joins have accuracies 0.48 0.48 , 0.45 0.45 , 0.41 0.41 , and 0.45 0.45 , respectively. A logistic model over the same features ranks arithmetic, ORDER BY, and DISTINCT as the strongest independent error predictors, while join and aggregate indicators have negative coefficients. The benchmark’s own difficulty labels agree with this reading: accuracy is 0.544 0.544 on simple questions, 0.332 0.332 on moderate questions, and 0.287 0.287 on challenging questions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This explains why schema relevance is a weak correctness signal. The generator often identifies the relevant tables and columns, but fails on ratios, conditional logic, grouping, ranking, or nested structure. A schema-relevance score cannot see those failures. A verifier can, because it evaluates whether the computation expressed by the SQL matches the question.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To see whether the verifier earns its advantage where the errors actually are, columns 5–7 of Table 7 compare string self-consistency and the GPT-4o verifier as correctness predictors within each feature subset. The verifier improves on self-consistency for almost every feature, and its strongest discrimination is on arithmetic questions ( auroc 0.813 0.813 ), the largest error category. The clearest exception is GROUP BY, the one subset where the verifier ( 0.609 0.609 ) does not beat self-consistency ( 0.651 0.651 ). Subqueries are a near-tie ( + 0.016 +0.016 ), but only because self-consistency is already strong there ( 0.804 0.804 ): the model’s samples disagree visibly when a query needs nesting, so sampling agreement already flags those failures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The verifier’s advantage is not confined to the hardest queries. Splitting the slice into logic-heavy queries (arithmetic, subquery, CASE, or GROUP BY; n = 225 n=225 , accuracy 0.293 0.293 ) and the rest (simple; n = 575 n=575 , accuracy 0.513 0.513 ), the verifier’s edge over self-consistency is actually larger on the simple queries ( + 0.117 +0.117 , from 0.638 0.638 to 0.755 0.755 ) than on the logic-heavy ones ( + 0.073 +0.073 , from 0.690 0.690 to 0.763 0.763 ). The reason is that self-consistency is weakest precisely on the simple queries, where the model is often confidently and unanimously wrong: its samples agree, so sampling agreement reports high confidence on a query that is in fact incorrect. The verifier catches these cases because it reasons about the computation rather than counting agreement.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The central lesson is that correctness uncertainty for text-to-SQL is real, but it does not come from the signals that are cheapest to compute. Sampling agreement, structural agreement, execution-result agreement, schema relevance, executability, and even sequence log-probability all occupy the same broad performance band on hard multi-table data. These signals are useful diagnostics, but they mostly measure whether the generator is internally consistent, not whether the generated SQL actually answers the question. In this setting, a model can be highly self-consistent and still be consistently wrong.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Verification changes the picture because it evaluates the candidate query at the level where many errors occur. The difficult cases in our study are not primarily failures to mention the right table or to produce executable SQL. They are failures of computation and composition: ratios, percentages, conditional logic, grouping, ranking, nesting, and distinctness. A confidence signal that only sees agreement among samples has little access to those errors. A verifier, by contrast, can compare the question to the computation expressed by the SQL. This explains why verifier scores move past the black-box ceiling while schema-relevance and self-consistency do not.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results suggest two different deployment regimes. In a fixed deployment, where the system owner has historical questions and execution labels from the same schemas, a trained verifier is attractive. It is cheap at inference time and performs well in-domain, giving useful abstention frontiers without requiring an additional large-model call. In an open or cross-schema setting, however, fine-tuned verifiers are much less reliable. Across the transfer experiments we tried, small and medium trained models improved in-domain performance but did not close the gap to a large frozen judge on unseen schemas. For that setting, the evidence points toward using a strong reasoning model as the verifier, and toward combining independent-provider judges when the cost is justified.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The multi-provider result is practically important. The strongest single verifier is already well above the black-box baselines, but the best score comes from combining GPT-4o and Claude. Their errors are only moderately correlated, and the calibrated ensemble improves both discrimination and reliability. This suggests that verification uncertainty is not merely a property of one model family. Independent judges appear to notice different failures, which makes the ensemble a better trust signal than either judge alone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The abstention results should be read as empirical operating points rather than as a claim that the problem is solved. The calibrated ensemble gives useful risk–coverage frontiers, while self-consistency cannot form a low-risk subset on this split. The distribution-free certificate is more conservative, firing only at the loosest target. This is not surprising in a regime where the base generator is correct less than half the time. A stronger generator, more calibration data, or both should make certified coverage more useful. The present result is that a logic-aware score creates a usable empirical frontier and a conservative certified lower bound; sampling agreement does not.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the paper separates two questions that are often conflated. One question is whether a critic can improve a query by driving a self-correction loop. Another is whether a final query, after whatever generation or correction process produced it, should be trusted. Our self-correction baseline shows why these are different. One reflection round barely improves accuracy and produces severely overconfident self-reported confidence. Even systems that use self-correction still need calibrated selective prediction at the end of the pipeline.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The main experiments use two OpenAI generators and two judge providers. This is enough to test whether the pattern survives a stronger generator and an independent verifier, but it is not a complete survey of current text-to-SQL systems. An open-weight generator would be a useful additional check, especially because generation errors may differ across model families.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The verifier experiments also leave open the question of scale. We find that fine-tuned small and medium verifiers perform well in-domain but do not transfer reliably to unseen schemas. Scaling a generative verifier to 7B improves the in-domain result but does not close the transfer gap. This does not rule out the possibility that a larger fine-tuned verifier, or a verifier trained on a much broader collection of schemas and tasks, could generalize better.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our distribution-free certificates are conservative. This is partly a sample-size issue and partly a base-accuracy issue: when the generator is correct only about 45% of the time, a very strong confidence score is needed to certify high coverage at low risk. The empirical risk–coverage curves are therefore the more informative deployment summary in the present study, while the certificates should be viewed as conservative lower bounds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the study focuses on deciding whether to trust a generated query, not on improving the query. Verifier-guided reranking, verifier-guided regeneration, and interactive clarification are natural next steps. The present results suggest that such systems should not rely on sampling confidence alone; they should include a component that explicitly evaluates the logic of the candidate SQL.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On hard text-to-SQL, the signals that most directly predict correctness are not sampling agreement, structural agreement, executability, schema relevance, or log-probability. Those signals plateau in a relatively narrow band. Verification moves past that ceiling because it can reason about whether the SQL expresses the computation requested by the question. Independent-provider verifiers make different errors and combine into the strongest and best-calibrated correctness signal we observe, enabling empirical abstention where self-consistency cannot. Trained verifiers are useful in-domain, but cross-schema transfer remains difficult; for open settings, a large frozen reasoning judge is currently the more reliable option. The practical message is simple: if a system needs to decide whether to trust generated SQL, it should pay for a logic-aware verifier, and it should calibrate that verifier for selective prediction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each row is an attempt to make a fine-tuned verifier transfer to unseen schemas; none closes the gap to the frozen reasoning judge ( 0.710 0.710 macro, 0.770 0.770 pooled). Schema diversity adds Spider’s 20 schemas (28 in all); reasoning distillation trains a 1.5B model on a strong judge’s rationales plus the verdict, versus verdict-only; cross-benchmark trains on one benchmark and tests on the other.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every verifier sees the same content, presented in this order: the schema, the question, the evidence string (when available), and the candidate SQL, followed by the elicitation request. Two elicitation formats are used. The logit format (the OpenAI judges) asks for a one-word verdict and reads P ( correct ) P(\\\\text{correct}) from the first-token YES/NO probabilities; the verbalized format (Claude, which does not expose token logprobs, and the method-matched OpenAI verbal judge) asks for a numeric probability and divides it by 100 100 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Logit (YES/NO) judge. System: “You are a strict SQL reviewer. Decide whether the candidate SQL CORRECTLY answers the question (right tables, columns, conditions, aggregation, and result). Answer with exactly one word: YES or NO.” The candidate block ends with “Is it correct? Answer YES or NO.”\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Verbalized (0–100) judge. System: “You are a strict SQL reviewer. Decide whether the candidate SQL correctly answers the question. Respond with ONLY an integer from 0 to 100: the probability (percent) that it is correct. Output only the number.” The candidate block ends with “Probability (0-100) that the SQL is correct:” The returned integer divided by 100 100 is the correctness probability.\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":4,"numItemsPerPage":100,"numTotalItems":1000,"offset":400,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NTkxODM3MiwianRpIjoiYTU5NDE3MzMtMjc0OS00ZjhjLWFjNWUtYjhmOTZjNDJjYjA4Iiwic3ViIjoiL2RhdGFzZXRzL1ByaW1lSW50ZWxsZWN0L2Rlc2h1ZmZsZS1wYXBlcnMtdjEtY29ycHVzIiwiZXhwIjoxNzg1OTIxOTcyLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.efn9MCaMFusDZRB97xVWC0yx0vMWSAgfJAkNWEsrTKWJoLdbPd24-KRx-bHlyGpH5mKZCP1af0VZko52m1tcAw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1000,"numBytesParquet":49695546}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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arxiv | 2607.10430 | null | 1 | Emergent Generalization by Representation Learning in Artificial Neural Networks | [
"Hardik Rajpal",
"Dan Goodman"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10430v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "With the advent of large-scale neural recordings, vast amounts of high-dimensional neural data recorded at high temporal resolution have become available from a variety of different brain regions and specie... |
arxiv | 2607.10420 | null | 1 | Is Model Instability just Noise to be Tolerated or a Property that can be Managed? | [
"Amirali Rayegan",
"Lunxiao Li",
"Tim Menzies"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10420v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Software analytics is an influential research area with much practical value [ 1 , 7 , 10 , 30 , 63 , 21 ] . By mining historical project data, teams allocate testing effort, prioritize high-risk modules,... |
arxiv | 2607.10413 | null | 1 | SPORT: Structure-Aware Prototype Disentanglement for Incomplete Multi-View Clustering | [
"Yaoyuan Guo",
"Zhibin Gu",
"Songhe Feng",
"Yuhui Zheng",
"Bing Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10413v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "With the rapid development of knowledge discovery and data mining techniques, data are increasingly collected and stored in multiple modalities. Such multi-source data are referred to as multi-view data. ... |
arxiv | 2607.10391 | null | 1 | Vertical Fusion: Condensing Internal Representations for Robust ViT Classification | [
"Francesco Di Salvo",
"Shyam Nandan Rai",
"Hamed Damirchi",
"Ignacio Meza De la Jara",
"Sebastian Doerrich",
"Marco Lents",
"Christian Ledig"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10391v1 | 2026-07-11T00:00:00 | [
{
"type": "para",
"text": "[1] organization=University of Bamberg, city=Bamberg, state=Bavaria, country=Germany"
},
{
"type": "para",
"text": "[2] organization=AIML, The University of Adelaide, city=Adelaide, state=South Australia, country=Australia"
},
{
"type": "heading",
"level": ... |
arxiv | 2607.10365 | null | 1 | Gradient-Skipping Relevance Propagation for Efficient Explainability of Vision Transformers | [
"Christopher Buratti",
"Michele Marchetti",
"Federica Parlapiano",
"Davide Traini",
"Domenico Ursino",
"Luca Virgili"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10365v1 | 2026-07-11T00:00:00 | [
{
"type": "para",
"text": "Keywords : Vision Transformers; Explainable AI; Gradient based Explanation; Relevance Propagation Method; Computer Vision"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Vision Transformers (ViTs) have emerged... |
arxiv | 2607.11948 | null | 1 | Ontology-Amplified Distillation and Contextuality Auditing for Sovereign Enterprise Language Models: A Combined Proof-of-Mechanism and Negative-Results Method Study | [
"Thanh Luong Tuan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11948v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Financial institutions in Vietnam operate under data-residency and sector-specific rules that restrict where customer data may be processed [National Assembly of Vietnam, 2025 , Government of Vietnam, 202... |
arxiv | 2607.10285 | null | 1 | Interpreting learning dynamics of autoencoders: Transient scaling and emerging concepts of the Ising model | [
"Max Weinmann",
"Miriam Klopotek"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10285v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Machines automate many physically demanding tasks, such as farm or factory work; formal tasks, such as calculations and thought experiments using computer simulations; and, more recently, increasingly abs... |
arxiv | 2607.11944 | null | 1 | MAGE: Understanding Stability-Performance Trade-offs in Multi-component Prompt Optimization | [
"Prateek Singh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11944v1 | 2026-07-11T00:00:00 | [
{
"type": "para",
"text": "Mage : Understanding Stability–Performance Trade-offs in Multi-component Prompt Optimization"
},
{
"type": "para",
"text": "Prateek Singh prateek29singh@gmail.com"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",... |
arxiv | 2607.11942 | null | 1 | How Query Visibility Changes KV-Cache Compression Rankings: A Matched-Budget Audit | [
"Daming Luo",
"Christy Liang",
"Junyu Xuan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11942v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Long-context inference is memory-bound: the KV cache of an 8B model at 128k tokens exceeds the weights themselves. A large literature therefore prunes the cache—score every cached token, evict the bottom ... |
arxiv | 2607.10202 | null | 1 | Two Confounds in Cross-Model Value Comparison: Response Determinism and the Access Harness | [
"Hong-In Won",
"Jinseok Jang",
"Hyoseop Kim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10202v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Language models are now routinely compared across providers for their “value dispositions”—the side they take when a prompt poses a value trade-off with no rule-governed answer. A growing practice treats ... |
arxiv | 2607.10188 | null | 2 | BiLoG-Net: A Bi-Context Location-Guided Network for Breast Mass Segmentation and Malignancy Classification in Mammography | [
"Abu Fatema Mohammad Abdun Noor",
"Md Imam Ahasan",
"Md Samiul Ahasan",
"Kah Ong Michael Goh",
"S M Hasan Mahmud",
"Raihana Zannat"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10188v2 | 2026-07-11T00:00:00 | [
{
"type": "para",
"text": "Keywords: Mammography; breast cancer detection; U-Net; multi-task learning; computer-aided diagnosis."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The latest official data shows that breast cancer is now mo... |
arxiv | 2607.10187 | null | 1 | Knowledge-Conditioned, Single-Pass LLM Synthesis of Executable Unity Game Scenes: A Compiler Error Census across 26 Goal Playable Concepts | [
"Hugh Xuechen Liu",
"Kıvanç Tatar"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10187v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) have become a practical tool for game content generation. Demos and tutorials routinely show LLMs writing Unity C# scripts, placing game objects, and implementing mechanics on... |
arxiv | 2607.10169 | null | 1 | Beyond Euclidean Clipping: Overcoming Exploration Collapse in LLM RL via Riemannian Isometric Policy Optimization | [
"Zhicheng Cai",
"Xinyuan Guo",
"Hanlin Wu",
"Mingxuan Wang",
"Wei-Ying Ma",
"Ya-Qin Zhang",
"Hao Zhou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10169v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning (RL) has emerged as a central paradigm for enhancing the reasoning capabilities of large language models (LLMs) (Guo et al. , 2025 ; Comanici et al. , 2025 ; OpenAI, 2024 ) . It has... |
arxiv | 2607.11937 | null | 1 | Mirror Horizon: Viable Path Entropy as a Measure of Bounded Reflection | [
"Tiantian Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11937v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models are usually evaluated by loss, accuracy, pass@k, benchmark score, or reward. These quantities are important, but they collapse a system’s internal capacity into a single outcome stat... |
arxiv | 2607.10139 | null | 1 | LLMs as a Jury: Cross-Model Consensus Can Outperform Process Reward Models for LLM Reasoning | [
"Ning Liu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10139v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern reasoning systems spend compute at inference time by sampling many candidate solutions and selecting among them (Wei et al. , 2022 ; Brown et al. , 2024 ; Snell et al. , 2025 ; Muennighoff et al. ,... |
arxiv | 2607.10137 | null | 1 | RDQ: Residual Distribution Quantization for Large Language Models | [
"Prateek Singh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10137v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Deploying large language models (LLMs) on edge hardware demands 3–4-bit weight quantization Lin et al. ( 2024 ); Frantar et al. ( 2023 ) . While 4-bit methods approach FP16 quality, performance degrades s... |
arxiv | 2607.10134 | null | 1 | LeRoPE: Learnable RoPE Frequencies Improve Language Modeling | [
"Petros Karypis",
"Sean O'Brien",
"Shreyas Kadekodi",
"Rui Zhu",
"Julian McAuley"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10134v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Positional encodings are typically included in Transformer-based models (Vaswani et al. , 2017 ) because the self-attention operation on its own does not directly carry positional signal. Rotary Positiona... |
arxiv | 2607.10128 | null | 1 | Energy-guided Recursive Model | [
"Yifei Zhao",
"Ying Tang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10128v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Test-time computation is now a central route to stronger reasoning, but its success depends on how extra computation is converted into a final answer. For autoregressive language models, this conversion i... |
arxiv | 2607.10127 | null | 1 | GAE: Graph-Augmented Evolution for Scientific Discovery via Reinforcement Optimization | [
"Xuanzhou Chen",
"Taoli Cheng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10127v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The discovery of new algorithms and programs is a central frontier in AI-assisted science. Recent work follows a simple but powerful recipe: combine a large language model with programmatic evaluation and... |
arxiv | 2607.10116 | null | 1 | When Data Imbalance Helps: Robust Generalization Through Shortcut Saturation | [
"Cheng-Ting Chou",
"Duc Binh Hoang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10116v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Spurious correlations are one of the central failure modes of learned models (Geirhos et al. , 2020 ; Shah et al. , 2020 ) . When a feature is highly predictive in training but unreliable at test time, mo... |
arxiv | 2607.10068 | null | 1 | Error Aware Distribution Prediction for Lightweight Implicit Neural Representations | [
"Zhimin Li",
"Jake D. Balla",
"Joshua A. Levine"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10068v1 | 2026-07-11T00:00:00 | [
{
"type": "para",
"text": "1275 \\vgtccategory Research"
},
{
"type": "para",
"text": "Error and uncertainty fields for classification- and regression-based methods on the aneurism dataset. From left to right:cross-entropy(CE), cross-entropy+ mean-squared-error (CE+MSE), mean-squared-error (MSE)... |
arxiv | 2607.10044 | null | 1 | FlashTrie: A GPU-Accelerated Constrained Beam Search for Generative Retrieval | [
"Dakshitha Anandakumar",
"Anurag Mukkara",
"Wenxiang Hu",
"Jiusheng Chen",
"M Akash Kumar",
"Ting Ye",
"Qiang Lou",
"Jian Jiao"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10044v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "FlashTrie: A GPU-Accelerated Constrained Beam Search for Generative Retrieval"
},
{
"type": "para",
"text": "Dakshitha Anandakumar 1 Anurag Mukkara 2 Wenxiang Hu 1 Jiusheng Chen 1 M Akash Kumar 1 Ting Ye 1 Qiang Lou 1 Jian Jiao 1 1 Microsoft, Redmond, WA, USA 2 Nvidia,... |
arxiv | 2607.10039 | null | 1 | Are We Ready for AI-Driven Discovery? AI Verification Before the Next Fundamental Physics Breakthrough | [
"Gaia Grosso",
"Vinicius Mikuni",
"Lukas Heinrich"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10039v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "Are We Ready for AI-Driven Discovery? AI Verification Before the Next Fundamental Physics Breakthrough"
},
{
"type": "para",
"text": "Gaia Grosso ⋆ , † \\star,\\dagger ,1,2,3 , Vinicius Mikuni ⋆ , ‡ \\star,\\ddagger , 4 , Lukas Heinrich ∘ \\circ ,5,6"
},
{
"typ... |
arxiv | 2607.10023 | null | 1 | Local Multimodal Music Alignment from Global Supervision | [
"Irmak Bukey",
"Zachary Novack",
"Jongmin Jung",
"Dasaem Jeong",
"Chris Donahue"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10023v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Many multimodal music understanding tasks involve localized correspondences between data modalities [ 3 , 6 , 32 ] . For example, we may want to understand how performance audio aligns to an image of the ... |
arxiv | 2607.10021 | null | 1 | A Symbolic Neural CPU for Quantization-Simulated Writeback and Interpretable Program Execution | [
"Jose Luis Lima de Jesus Silva"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10021v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern computers are not judged only by their final outputs. They are auditable transition systems where instructions select operations, operations act on addressable state, and the resulting trajectory c... |
arxiv | 2607.09993 | null | 1 | Beyond Bayesian Nash: Learning Minimax-Regret Equilibria for Adversarial Team Games under Asymmetric Information | [
"Naman Aggarwal",
"Jonathan P. How"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09993v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Computation of equilibria in large imperfect-information games has been a central driver of recent advances in artificial intelligence. A key challenge in such settings is determining how strategic agents... |
arxiv | 2607.09967 | null | 2 | Learning in Curved Weight Space:Exponential-Linear Weight Reparameterization for Improved Optimization | [
"Ethan Smith"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09967v2 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Standard neural network training optimizes parameters in an additive fashion with linear update steps. Yet many computations within a network are naturally relative. Doubling or halving a normalization ga... |
arxiv | 2607.09936 | null | 1 | SMETA-ZSL:Semantic Meta-Alignment for Zero-Shot Threat Classification | [
"Ivan Alejandro Montoya Sanchez",
"Anantaa Kotal",
"Aritran Piplai"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09936v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Cyber defense systems rely on large amounts of raw data, such as malware samples or logs, to train and update models, yet this data is often difficult to obtain, slow to analyze, and not always available ... |
arxiv | 2607.09906 | null | 1 | Depth-Efficient Quantum Topological Data Analysis for Regime-Specific Detection of Financial Stress | [
"Arul Rhik Mazumder",
"Shreyan Ronit Mazumder"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09906v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Financial markets exhibit complex nonlinear dynamics that defy classical statistical characterization. The 2008 financial crisis—which erased over $10 trillion in US household wealth—emerged from correlat... |
arxiv | 2607.09889 | null | 1 | Remembering Distinct Items, Not Tokens: A Learnable Dirichlet-Process Cache Between State-Space Models and Attention | [
"Siddharth Pal",
"Viktoria Rojkova"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09889v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 The problem, and the gap"
},
{
"type": "para",
"text": "A sequence model summarizes a growing history in a state of fixed size. A structured state-space model (S4 and its diagonal successors, and the selective recurrence of Mamba) does this with a ... |
arxiv | 2607.09662 | null | 1 | PHINN-EEG: Topological Time-Series Analysis of Dream-State EEG -- Dynamic Betti Curves for Dream Content Classification and Topology-Conditioned Neural Signal Synthesis | [
"Ren Takahashi",
"Emre Yusuf",
"Jayabrata Bhaduri"
] | CC0-1.0 | https://arxiv.org/html/2607.09662v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Dreams are rare structural events in the brain’s electrical record. The DREAM database — a landmark multi-laboratory aggregation spanning 3,191 awakenings from 263 participants across 20 published studies... |
arxiv | 2607.09616 | null | 1 | LLM for EDA in Front-End Design: Challenges and Opportunities | [
"Kangwei Xu",
"Bing Li",
"Ulf Schlichtmann"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09616v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "by"
},
{
"type": "para",
"text": "ACM Reference Format: Kangwei Xu, Bing Li, and Ulf Schlichtmann. 2026. LLM for EDA in Front-End Design: Challenges and Opportunities. In 63rd ACM/IEEE Design Automation Conference (DAC ’26), July 26 – July 29, 2026, Long Beach, CA, USA... |
arxiv | 2607.09528 | null | 1 | TSAI-MetaFraud: A Benchmark Dataset for Financial Fraud Transaction and Behavioral Risk Detection in Metaverse Ecosystems | [
"Refat Ishrak Hemel",
"Ehsan Hallaji",
"Roozbeh Razavi-Far"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09528v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The rapid development of metaverse technologies is transforming virtual environments into persistent digital ecosystems where users socialize, collaborate, trade virtual assets, and participate in increas... |
arxiv | 2607.09502 | null | 1 | All Explanations are Wrong, But Many Are Useful: Exploring the Rashomon Explanation Set with Large Language Models | [
"Pan Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09502v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "Abstract"
},
{
"type": "para",
"text": "Explaining machine-learning models is increasingly important for decision-making and consumer trust, yet it is widely believed to come at a cost: existing Explainable AI (XAI) methods suffer from a persistent accuracy–explainabil... |
arxiv | 2607.09833 | null | 1 | Generative Testing of Automated Speech Recognition Systems | [
"Yanis Xabier Wilbrand Peña",
"Oliver Weißl",
"Andrea Stocco"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09833v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Automatic speech recognition (ASR) has become a central interface between humans and machines, enabling applications such as voice assistants, transcription services, and accessibility tools (Ahlawat et ... |
arxiv | 2607.09832 | null | 1 | A Strong Balanced-Softmax Classifier-Retraining Baseline for Long-Tailed Recognition | [
"Juan Terven",
"Diana Margarita Córdova Esparza",
"Julio Alejandro Romero Gonzalez",
"Edgar Arturo Chávez Urbiola",
"Francisco Javier Willars Rodriguez",
"Juan Bautista Hurtado Ramos",
"Alfonso Ramirez Pedraza"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09832v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "Keywords Long-tailed recognition ⋅ \\cdot Imbalanced classification ⋅ \\cdot Classifier retraining ⋅ \\cdot Balanced Softmax ⋅ \\cdot Few-shot recognition ⋅ \\cdot Decision boundaries ⋅ \\cdot Classifier bias"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduc... |
arxiv | 2607.09424 | null | 2 | A Sovereign, Open-Source Foundation Model for German and English | [
"The Soofi-Team",
" :",
"Benedikt Droste",
"David Fitzek",
"Ruben Härle",
"Lukas Helff",
"Maximilian Idahl",
"Alex Jude",
"Abbas Goher Khan",
"Maurice Kraus",
"Timm Ruland",
"Richard Rutmann",
"Sebastian Sztwiertnia",
"Markus Frey",
"Daniil Gurgurov",
"Jan Pfister",
"Tom Röhr",
"Se... | CC-BY-4.0 | https://arxiv.org/html/2607.09424v2 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "A Sovereign, Open-Source Foundation Model for German and English"
},
{
"type": "para",
"text": "Soofi S Pretraining Report v1.0"
},
{
"type": "para",
"text": "The Soofi-Team *"
},
{
"type": "para",
"text": "Core Team: Benedikt Droste 10 , David ... |
arxiv | 2607.09402 | null | 1 | Data-Efficient Deep Learning: Empirical Guidelines for Training Set Size Estimation in Inertial Sensor Classification | [
"Ofir Kruzel",
"Itzik Klien"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09402v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "[orcid=0009-0000-4015-0853] \\cormark [1]"
},
{
"type": "para",
"text": "[orcid=0000-0001-7846-0654]"
},
{
"type": "para",
"text": "1]organization=The Autonomous Navigation and Sensor Fusion Lab, the Hatter Department of Marine Technologies, University of H... |
arxiv | 2607.09336 | null | 1 | Shortcut Trajectory Planning for Efficient Offline Reinforcement Learning | [
"Guanquan Wang",
"Yoshimasa Tsuruoka"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09336v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Diffusion-based trajectory planners have shown strong performance in offline reinforcement learning by modeling behavior trajectories as generative distributions and using planning-time sampling for decis... |
arxiv | 2607.11928 | null | 1 | Repairing Shape-Prior Shortcuts in Long-Range Single-Shot Fringe Projection Profilometry | [
"Adam Haroon",
"Cody Fleming",
"Beiwen Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11928v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualization, Project Administration"
},
{
"type": "para",
"text": "Supervision, Resources, Writing – Review & Ed... |
arxiv | 2607.09306 | null | 1 | Creativity, honesty and designed forgetting emerge in small hyperbolic language models | [
"Kwan Soo Shin",
"In Seok Kang",
"Yunkyung Min"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09306v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Main Text"
},
{
"type": "heading",
"level": 2,
"text": "1 ⋅ \\cdot Introduction"
},
{
"type": "para",
"text": "Language models are moving into the most personal role software has ever occupied: systems that persist with one user, accu... |
arxiv | 2607.09290 | null | 1 | Leveraging Interpretable Tsetlin Machine for PDF Malware Detection | [
"Rahul Jaiswal"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09290v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "In today’s digital world, the Portable Document Format (PDF) has become one of the most widely used document formats for sharing and exchanging information due to its portability, platform independence, a... |
arxiv | 2607.09287 | null | 1 | Super-Tuning: From Activation-Aware Pruning to Sparse Fine-Tuning | [
"Ivan Ilin",
"Philip Zmushko",
"Peter Richtárik"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09287v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large Language Models (LLMs) have become central to modern natural language processing, achieving strong results in a wide array of tasks including question answering, summarization, code generation, and ... |
arxiv | 2607.09277 | null | 1 | Autoregressive latent diffusion for 3D molecule generation | [
"Federico Ottomano",
"Gaopeng Ren",
"Yingzhen Li",
"Kim E. Jelfs",
"Alex M. Ganose"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09277v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Generative modeling of three-dimensional (3D) molecules is a central problem in machine learning for drug discovery and molecular design. Recent progress has largely been driven by diffusion (Hoogeboom et... |
arxiv | 2607.09266 | null | 1 | LionVote: Per-Layer Learning Rate Adaptation for Lion | [
"Kris Atallah"
] | CC0-1.0 | https://arxiv.org/html/2607.09266v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Zhao et al. (Zhao et al., 2025 ) show that applying per-layer adaptive preconditioning only to the last layer and LayerNorm parameters recovers most of Adam’s advantage over SGD on autoregressive language... |
arxiv | 2607.09236 | null | 1 | Forget Narrowly, Retain Broadly: Unlearning as an Asymmetric Generalization Problem | [
"Amit Peleg",
"Naman Deep Singh",
"Naama Pearl",
"Bibhabasu Mohapatra",
"Matthias Hein"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09236v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Although several benchmarks evaluate unlearning in LLMs [ 19 , 42 , 7 , 14 , 30 , 21 , 36 ] , one fundamental question remains open: what does it mean to forget a fact while retaining knowledge of everyth... |
arxiv | 2607.09232 | null | 1 | Temporal Knowledge Graph Forecasting under Distribution Shifts: A Synthetic Evaluation | [
"Konrad Özdemir",
"Julia Gastinger",
"Lukas Kirchdorfer",
"Heiner Stuckenschmidt"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09232v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Temporal knowledge graphs (TKGs) extend static knowledge graphs (KGs) with temporal information [ 13 ] . As such, they constitute an effective, systematic mechanism for storing event-based facts across ti... |
arxiv | 2607.09202 | null | 1 | Interference and Retention in Continual Learning | [
"Julius Störk"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09202v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "Keywords: continual learning; catastrophic forgetting; model merging; neural tangent kernel; gradient projection; rate–distortion; representation geometry"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "A mode... |
arxiv | 2607.09196 | null | 1 | Application of machine learning to monster level prediction in tabletop RPG game design | [
"Jolanta Śliwa",
"Jakub Adamczyk"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09196v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "[agh]organization=Faculty of Computer Science, AGH University of Krakow, city=Cracow, country=Poland"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Artificial intelligence (AI) and machine learning (ML) metho... |
arxiv | 2607.09167 | null | 1 | Understanding Schedule-Free Methods in Nonconvex Optimization: Rate Guarantees and Escaping Saddles | [
"Jiseok Chae",
"Donghwan Kim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09167v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "There are many first-order methods that can be used to solve a minimization problem"
},
{
"type": "para",
"text": "ranging from the established gradient descent (GD) and stochastic gradient de... |
arxiv | 2607.09817 | null | 1 | Estimation, Prediction, and Assortment Optimization for Markov Chain Choice Models with Panel Data | [
"Yalcin Akcay",
"Gerardo Berbeglia",
"Young-San Lin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09817v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Incorporating customer choice behavior into assortment planning is a central problem in revenue management. A standard roadmap for assortment planning consists of two key steps: choice model estimation an... |
arxiv | 2607.09816 | null | 1 | RUBRIC: Realism--Utility Balanced Ranking for Imbalanced Classification | [
"Yanxuan Yu",
"Dong liu",
"Renata Borovica-Gajic",
"Ying Nian Wu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09816v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "RUBRIC: Realism–Utility Balanced Ranking for Imbalanced Classification"
},
{
"type": "para",
"text": "Yanxuan Yu 1,∗ Dong Liu 2,∗ Renata Borovica-Gajic 3 Ying Nian Wu 2"
},
{
"type": "para",
"text": "1 Columbia University 2 University of California, Los Ang... |
arxiv | 2607.09061 | null | 1 | On Locality and Length Generalization in Visual Reasoning | [
"Pulkit Madan",
"Sanjay Haresh",
"Reza Ebrahimi",
"Sunny Panchal",
"Apratim Bhattacharyya",
"Roland Memisevic"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09061v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Current state-of-the-art vision models have shown human level performance on tasks such as image captioning and visual question answering (Bai et al. , 2025 ; OpenAI, 2026 ; 2026 ) . This success is built... |
arxiv | 2607.09052 | null | 1 | COBS: Cumulant Order Block Sparse Attention | [
"Alexander Tian",
"Aditya Ghai",
"Sanjit Neelam",
"Zaal Vasania",
"Akshay Mishra"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09052v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Transformer inference at long context is bottlenecked by reading the key–value (KV) cache: at each decode step, attention reads the keys and values of every past token, so decoding is limited by memory ba... |
arxiv | 2607.09042 | null | 1 | Learning More from Less: Reinforcement Learning from Hindsight | [
"Iris Xu",
"Sunshine Jiang",
"John Marangola",
"Nitish Dashora",
"Richard Li",
"Thomas Liu",
"Zexue He",
"Yuheng Zhi",
"Alex Pentland",
"Pulkit Agrawal",
"Zhang-Wei Hong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09042v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning (RL) [ 10 ] has become a standard tool for post-training large language models [ 19 , 1 ] , and is increasingly being used to fine-tune vision-language-action (VLA) models for robot... |
arxiv | 2607.09039 | null | 1 | Variable-Length Generative Protein Design via Generalized Poisson Flow | [
"Chaoran Cheng",
"Zhanghan Ni",
"Yanru Qu",
"Yuxin Chen",
"Ruihan Guo",
"Jiajun Fan",
"Ge Liu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09039v1 | 2026-07-10T00:00:00 | [
{
"type": "para",
"text": "\\ul \\newkeytheorem theorem[name=Theorem, numberwithin=section] \\newkeytheorem proposition[name=Proposition, numberlike=theorem]"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Diffusion models [ 22 , 42 ] a... |
arxiv | 2607.09020 | null | 1 | Phone Segmentation and Recognition through Phonological Activation Mapping | [
"Shikhar Bharadwaj",
"Kwanghee Choi",
"Stephen McIntosh",
"Chin-Jou Li",
"Eunjung Yeo",
"Daisuke Saito",
"Nobuaki Minematsu",
"Shinji Watanabe",
"Jian Zhu",
"David Harwath",
"David R. Mortensen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09020v1 | 2026-07-10T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Phones are the smallest independent units of speech sound, conventionally transcribed across languages with the International Phonetic Alphabet (IPA). Locating and recognizing them, i.e. , phone segmentat... |
arxiv | 2607.08996 | null | 1 | Model Agnostic Graph Prompt Learning for Crystal Property Prediction | [
"Shrimon Mukherjee",
"Kishalay Das",
"Partha Basuchowdhuri",
"Pawan Goyal",
"Niloy Ganguly"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08996v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Predicting crystal properties remains a significant challenge in materials science. Unlike molecules, which are often represented as regular graphs, crystal materials are modeled as a 3D point cloud of at... |
arxiv | 2607.08961 | null | 1 | NL-PAC: Specification Ambiguity and Certified Minimax Risk Floors in LLM-Mediated Supervision | [
"Berkay Anahtarci"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08961v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern learning pipelines increasingly use large-language-model (LLM) judgments in place of human evaluation or labeling, including in LLM-as-a-judge systems (Liu et al., 2023b ; Kim et al., 2024 ; Gu et ... |
arxiv | 2607.08960 | null | 1 | Eluna: An Agentic LLM System for Automating Warehouse Operations with Reasoning and Task Execution | [
"Ning Liu",
"Kalle Kujanpää",
"Zhaoxuan Zhu",
"P Aditya Sreekar",
"Kaiwen Liu",
"Chuanneng Sun",
"Jorge Marchena Menendez",
"Matthew Bales",
"Tianyu Yang",
"Shahnawaz Alam",
"Rose Yu",
"Baoyuan Liu",
"Kristina Klinkner",
"Shervin Malmasi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08960v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "Eluna: An Agentic LLM System for Automating Warehouse Operations with Reasoning and Task Execution"
},
{
"type": "para",
"text": "Ning Liu † † thanks: Equal contribution. Kalle Kujanpää 1 1 footnotemark: 1 Zhaoxuan Zhu 1 1 footnotemark: 1 P Aditya Sreekar 1 1 footnotem... |
arxiv | 2607.08946 | null | 1 | Training, Reading, and Editing Legible Transformers | [
"Mark Oskin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08946v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Most interpretability reconstructs meaning after training: a dense activation is probed, decomposed with a sparse autoencoder, or decoded through the vocabulary, and a human names what was found. An alter... |
arxiv | 2607.08925 | null | 1 | SafeExplorer: An Unbiased Policy Gradient for Reinforcement Learning with Recovery Interventions | [
"Elham Daneshmand",
"Majid Khadiv",
"Glen Berseth",
"Hsiu-Chin Lin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08925v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "RL policies often perform best when trained directly on the task they will be deployed on, but moving that training onto a physical robot is hard for a reason unrelated to asymptotic performance: learning... |
arxiv | 2607.08863 | null | 1 | Clean2FX: Label-conditioned modeling for clean-to-effect guitar audio transformations | [
"Oliverio Bombicci Pontelli",
"Iran R. Roman"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08863v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Effects are central to the sound of the electric guitar: they reshape timbre, dynamics, sustain, and apparent space, and have become part of both the instrument’s musical vocabulary and the broader histor... |
arxiv | 2607.08839 | null | 1 | Mixture of Probes: Learning from Privileged Modalities in Multimodal LLMs Through Probing | [
"Dominick Reilly",
"Qiyu Wu",
"Hiromi Wakaki",
"Srijan Das",
"Yuki Mistufuji"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08839v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Stemming from the success of large language models (LLMs) Touvron et al. ( 2023 ); Brown et al. ( 2020 ); Team et al. ( 2024 ) , recent multimodal large language models (MLLMs) have enabled language-based... |
arxiv | 2607.08837 | null | 1 | Prompt-Driven Exploration | [
"Sunshine Jiang",
"John Marangola",
"David Zhang",
"Raghuram Kowdeed",
"Ruiyang Luo",
"Nitish Dashora",
"Richard Li",
"Pulkit Agrawal",
"Zhang-Wei Hong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08837v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning (RL) ( kaelbling1996reinforcement, ) has become a dominant post-training paradigm for foundation models because it enables scalable self-improvement beyond supervised learning. RL f... |
arxiv | 2607.08803 | null | 2 | TheBioCollection: Unified Pre-Training Scale LLM Corpus for Biology | [
"Hyunjin Seo",
"Hyeon Hwang",
"Gyubok Lee",
"Jay Shin",
"Jimin Park",
"Taesoo Kim",
"Sanghoon Lee",
"Hongjoon Ahn",
"Sungjun Han",
"Sangwon Jung"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08803v2 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "001"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) for biology (BioLM) [ naturelm , scireasoner , txgemma , jang2026towards , logos ] aim to extend LLMs beyond general text unders... |
arxiv | 2607.09787 | null | 1 | Adversarially Guided Diffusion for LiDAR Range Image Synthesis | [
"Stavros Bouras",
"Antonios Makris",
"Alexandros Gkillas",
"Aris S. Lalos",
"Konstantinos Tserpes"
] | CC-BY-4.0 | https://arxiv.org/html/2607.09787v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Deep Learning (DL) models are increasingly deployed in real-world, safety-critical perception tasks such as semantic segmentation [ 14 ] , where reliable scene understanding is required under strict opera... |
arxiv | 2607.07425 | null | 1 | Reliable mechanistic operator recovery with biologically-informed neural networks: principles for architecture and optimisation design | [
"Rebecca M. Crossley",
"Yuan Yin",
"Sarah L. Waters",
"Ruth E. Baker"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07425v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "Reliable mechanistic operator recovery with biologically-informed neural networks: principles for architecture and optimisation design"
},
{
"type": "para",
"text": "Rebecca M. Crossley *1‡ , Yuan Yin 1‡ , Sarah L. Waters 1 and Ruth E. Baker 1"
},
{
"type": "pa... |
arxiv | 2607.07754 | null | 1 | Image classification via a quantum-inspired strategy involving a mixture of experts | [
"Kumari Jyoti",
"Rohith Babu",
"Apoorva D. Patel"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07754v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Rapid technological developments have made various types of sensors and detectors affordable and convenient. They collect huge amounts of data which need to be analysed to make decisions. Often there is n... |
arxiv | 2607.07401 | null | 1 | Heterogeneity-Adaptive Diffusion Schrodinger Bridge for PET-Guided Whole-Body MRI Translation | [
"Chengbo Wang",
"Jiacheng Yu",
"Linjie Bian",
"Ming Qi",
"Xiaosheng Liu",
"Tongtong Che",
"Jichang Zhang",
"Shuyu Li",
"Shaoli Song",
"Xiuying Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07401v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Hybrid positron emission tomography combined with magnetic resonance imaging (PET/MR) is an emerging imaging modality that enables simultaneous acquisition of ... |
arxiv | 2607.07395 | null | 1 | When Prompts Ignore Structure: Graph-Based Attribute Reasoning for Calibrated VLMs | [
"Tanay Sodha",
"Aditya Sharma",
"Ramya Hebbalaguppe",
"Vinti Agarwal",
"Pranav Murthy Yeluripaty"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07395v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "When Prompts Ignore Structure: Graph-Based Attribute Reasoning for Calibrated VLMs"
},
{
"type": "para",
"text": "Tanay Sodha † † thanks: Equal contribution. 1 and Aditya Sharma 1 1 footnotemark: 1 1 and Ramya Hebbalaguppe 2 and Vinti Agarwal 1 and Pranav Murthy Yeluri... |
arxiv | 2607.07388 | null | 1 | TF-Engram: A Train-Free Engram with SSD-Backed Memory for Large Language Models | [
"Yutang Ma",
"Kecheng Huang",
"Xikun Jiang",
"Zili Shao"
] | CC0-1.0 | https://arxiv.org/html/2607.07388v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Large Language Models (LLMs) have become the dominant architecture for natural language understanding, reasoning, code generation, and knowledge-intensive question answering (Vaswani et al. , 2017 ; Devl... |
arxiv | 2607.07386 | null | 1 | Sparse Delta Memory: Scaling the State of Linear RNNs through Sparsity | [
"Loïc Cabannes",
"Pierre-Emmanuel Mazaré",
"Gergely Szilvasy",
"Matthijs Douze",
"Maria Lomeli",
"Ilze Amanda Auzina",
"Justin Carpentier",
"Gabriel Synnaeve",
"Hervé Jégou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07386v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "[Code] https://github.com/facebookresearch/sparse-delta-memory"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "As frontier models continue to progress, they are leveraged in increasingly more complex tasks. In... |
arxiv | 2607.07382 | null | 1 | Generalist Vision-Language Models for Fast Radio Burst detection: a zero-shot benchmark against a specialized detector | [
"Raiff H. Santos",
"Amilcar R. Queiroz",
"Tharcisyo S. S. Duarte",
"K. E. L. de Farias",
"Rafael A. Batista"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07382v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Fast Radio Bursts (FRBs) are intense radio transients with durations of only a few milliseconds and predominantly of extragalactic origin. Since the discovery of the first FRB by [ 1 ] , these events have... |
arxiv | 2607.07375 | null | 1 | On Adversarial Vulnerability of Vision-Language Models through the Lens of Intermediate Spectral Subspaces | [
"Chethan Krishnamurthy Ramanaik",
"Tobias Callies",
"Michael Hecht",
"Eirini Ntoutsi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07375v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Deep neural networks, including modern vision–language models (VLMs), are known to be vulnerable to adversarial perturbations that substantially alter model predictions while remaining visually impercepti... |
arxiv | 2607.07368 | null | 1 | Multi-Agent AI Control: Distributed Attacks Hamper Per-Instance Monitors | [
"Oliver Makins",
"Orazio Angelini",
"Zohreh Shams",
"Mary Phuong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07368v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "heading",
"level": 2,
"text": "1.1 The Multi-Agent Control Problem"
},
{
"type": "para",
"text": "AI control studies how to safely deploy potentially misaligned AI agents inside an organisation, with a fo... |
arxiv | 2607.07343 | null | 1 | Latency-Aware Bid Acceptance under Operational Feasibility: A Public Benchmark with Hindsight Ceilings | [
"Aswin Chandrasekaran"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07343v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Truckload carriers and brokers face an online accept/reject decision each time a load tender arrives. The decision is constrained by latency: tenders in a typical broker workflow must be priced and accept... |
arxiv | 2607.07753 | null | 1 | A Transdiagnostic Space of Disorder Like Phenotypes in Reinforcement Learning Agents | [
"Hari Prasad"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07753v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Reinforcement-learning (RL) agents increasingly act in healthcare, transport, and human-facing assistants, where their affective stability is a safety property rather than a curiosity: value estimation that... |
arxiv | 2607.07316 | null | 1 | Mechanistic Interpretability for Neural Networks: Circuits, Sparse Features and Symbolic Reasoning | [
"Pranav Sawant",
"Jakub Krejčí"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07316v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Transformer models and large language models (LLMs) have achieved unprecedented success across a wide range of domains; however, their internal decision-making processes remain largely opaque. These syste... |
arxiv | 2607.07289 | null | 1 | Bayesian Optimization of Genetic Algorithm Hyperparameters in a Multi-Fidelity Framework for Efficient Lattice Material Design | [
"Sergei Zorkaltsev",
"Maciej Haranczyk",
"Christina Schenk"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07289v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "[1] \\fnm Christina \\sur Schenk"
},
{
"type": "para",
"text": "1] \\orgname IMDEA Materials Institute, \\orgaddress \\street Eric Kandel 2, \\city Getafe, \\postcode 28906, \\state Madrid, \\country Spain"
},
{
"type": "para",
"text": "2] \\orgdiv Departme... |
arxiv | 2607.07232 | null | 1 | DiPhon: Diffusion on Graphons for Scalable Graph Generation | [
"Sergio Rozada",
"Yiming Qin",
"Manuel Madeira",
"Pascal Frossard",
"Alejandro Ribeiro"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07232v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Graph generative models based on diffusion have become a leading paradigm for sampling unknown graph distributions given access to graph samples [ 23 , 49 ] . This has proven relevant in applications such... |
arxiv | 2607.07209 | null | 1 | Continual Learning With Participation Privacy: An Auditable Buffering-Aggregation Recipe | [
"T-H. Hubert Chan",
"Elaine Shi",
"Mengshi Zhao",
"Mingxun Zhou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07209v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern federated and streaming learning systems release intermediate model snapshots throughout training, so an adversary may observe the entire trajectory ( w ( t ) ) t ≥ 0 (w^{(t)})_{t\\geq 0} rather th... |
arxiv | 2607.07748 | null | 1 | Selective Left-Shift: Turning Test-Time Compute and Difficulty-based Curation into Training Data for Low-Resource Code Generation | [
"Didula Samaraweera",
"Anjana Supun",
"Srinath Perera"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07748v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Large Language Models (LLMs) have achieved remarkable proficiency in code generation for high-resource programming languages(HRPLs) like Python and Java (Chen et al. , 2021 ) . However, their performance... |
arxiv | 2607.07146 | null | 1 | Prior-matched evaluation of operational Earth-observation classifiers: a three-number reporting method demonstrated on Sentinel-1 internal-wave detection | [
"Joao Pinelo",
"Joao Goncalves",
"Arun Shukla",
"Adriana Santos-Ferreira"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07146v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 2,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Internal solitary waves are among the most energetic features of the ocean, and their surface signature — alternating bands of increased and reduced roughness as the wave-induced currents strain the shor... |
arxiv | 2607.07144 | 10.5281/zenodo.21237993 | 1 | Fractal KV-Cache Archives: Lossless Symbolic Storage with In-Place Retrieval for Long-Context LLM Inference | [
"Vladimir Gusev"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07144v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Serving a transformer language model over a long context is increasingly a memory problem rather than a compute problem. To generate each new token the model attends to cached key and value vectors for ev... |
arxiv | 2607.07743 | null | 1 | Architecture Generalization with MetaNCA | [
"Meet Barot",
"Daniel Berenberg",
"Sina Khajehabdollahi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07743v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "Data/Code available at: https://github.com/Mythos-Scientific/meta-nca † † ©2026 Barot, Berenberg & Khajehabdollahi. Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license."
},
{
"type": "heading",
"level": 1,
"text": "Introduction"
}... |
arxiv | 2607.07128 | null | 1 | Distributed Sparse Interventions in Language Models | [
"Maximilian S. Ernst",
"Lorenz Linhardt",
"Aaron Peikert",
"Oliver Eberle"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07128v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern language models demonstrate flexible task-solving behaviours, including the adaptive combination and execution of multiple tasks through in-context learning (ICL) [ 10 , 26 , 53 ] . However, the me... |
arxiv | 2607.07127 | null | 1 | Weight-Space Physics: Interpretable Hypernetworks for Lattice Quantum Field Theories | [
"Tobias Göbel",
"Julian R. Ebelt",
"Zier Mensch",
"Mathis Gerdes",
"Miranda C. N. Cheng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07127v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Lattice field theory (LFT) discretizes a quantum field onto a finite lattice, providing a first-principles approach to non-perturbative phenomena across high-energy and condensed matter physics. The actio... |
arxiv | 2607.07085 | null | 1 | Is Randomness Necessary for Adaptive Data Analysis? | [
"Edith Cohen",
"Haim Kaplan",
"Yishay Mansour",
"Shay Sapir",
"Uri Stemmer"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07085v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Classical statistical theory for asserting the validity of a hypothesis tells us that the description of the hypothesis should be independent from the data on which it is evaluated. In practice, however, ... |
arxiv | 2607.07083 | null | 1 | Prior-aware and Context-guided Group Sampling for Active Probabilistic Subsampling | [
"Beomgu Kang",
"Hyunseok Seo"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07083v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern technologies generate massive datasets that can require lengthy acquisition time and hinder real-time onboard processing. Many real-world applications, including Magnetic Resonance Imaging (MRI) (Y... |
arxiv | 2607.07740 | null | 2 | Jet-Long: Efficient Long-Context Extension with Dynamic Bifocal RoPE | [
"Haozhan Tang",
"Zerui Wang",
"Yuxian Gu",
"Song Han",
"Han Cai"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07740v2 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "Han Cai ( hcai@nvidia.com )."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) are now deployed in long-context applications including long-document QA, repository-level code underst... |
arxiv | 2607.06979 | null | 1 | Robust Federated Learning Under Real-World Client Churn | [
"Dhruv Garg",
"Neha Lakhani",
"Debopam Sanyal",
"Myungjin Lee",
"Alexey Tumanov",
"Ada Gavrilovska"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06979v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Federated Learning (FL) enables training shared models across large populations of user devices without collecting the raw data centrally. This paradigm has unlocked machine learning on private, on-devic... |
arxiv | 2607.06973 | null | 1 | Rethinking Multimodal Time-Series Forecasting Evaluation | [
"Haoxin Liu",
"Yichen Zhou",
"Rajat Sen",
"B. Aditya Prakash",
"Abhimanyu Das"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06973v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "redacted \\correspondingauthor Haoxin Liu <hliu763@gatech.edu> . This work was done while Haoxin Liu was a Student Researcher at Google Research. This work is accepted by The International Conference on Machine Learning (ICML) 2026."
},
{
"type": "heading",
"level": 1,... |
arxiv | 2607.06856 | null | 1 | Gen4U: Unifying Video Generation and Understanding via Diffusion | [
"Michael King",
"Aravindh Mahendran",
"Matthew Koichi Grimes",
"Fedor Kitashov",
"Adham Elarabawy",
"Pedro Velez",
"Maks Ovsjanikov",
"Viorica Pătrăucean"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06856v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Current paradigms in visual representation learning struggle to reconcile geometry with semantics. Contrastive methods and large Vision-Language Models (VLMs) (Radford et al. , 2021 ; Beyer et al. , 2024 ... |
arxiv | 2607.06855 | null | 1 | Geometric Self-Distillation for Reasoning Generalization | [
"Josip Jukić",
"Ivan Titov"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06855v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "On-policy distillation (OPD) has emerged as a practical post-training recipe for large language models (LLMs), providing dense teacher supervision on trajectories generated by the student itself (Agarwal ... |
arxiv | 2607.06841 | null | 1 | Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling | [
"Robert Gruhlke",
"Julius Berner",
"David Sommer",
"Lorenz Richter"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06841v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Sampling from a complex, high-dimensional probability density"
},
{
"type": "para",
"text": "where the unnormalized density ρ target \\rho_{\\mathrm{target}} can be evaluated pointwise but the... |
arxiv | 2607.06839 | null | 1 | LEMUR 2: Unlocking Neural Network Diversity for AI | [
"Tolgay Atinc Uzun",
"Waleed Khalid",
"Saif U Din",
"Sai Revanth Mulukuledu",
"Akashdeep Singh",
"Chandini Vysyaraju",
"Raghuvir Duvvuri",
"Avi Goyal",
"Yashkumar Rajeshbhai Lukhi",
"Muhammad A. Hussain",
"Krunal Jesani",
"Usha Shrestha",
"Yash Mittal",
"Roman Kochnev",
"Pritam Kadam",
... | CC-BY-4.0 | https://arxiv.org/html/2607.06839v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Neural networks underpin numerous breakthroughs in artificial intelligence, delivering state-of-the-art results in fields such as computer vision and natural language processing. Their growing complexity,... |
arxiv | 2607.06833 | null | 1 | Generative Diffusion Models of Stochastic Graph Signals | [
"Yiğit Berkay Uslu",
"Samar Hadou",
"Sergio Rozada",
"Shirin Saeedi Bidokhti",
"Alejandro Ribeiro"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06833v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Signals defined over irregular, graph-structured domains are pervasive, spanning application areas such as recommender systems [ 41 , 24 , 23 ] , wireless communications [ 7 , 31 , 46 , 42 , 4 , 39 ] , an... |
arxiv | 2607.06818 | 10.18653/v1/2021.naacl-industry.33 | 1 | Ad Headline Generation using Self-Critical Masked Language Model | [
"Yashal Shakti Kanungo",
"Sumit Negi",
"Aruna Rajan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06818v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "There are a various types of ads. A set of example ads that showcase products selected by sellers along with headlines that advertise them are shown in Figure 1 . Sellers create multiple ad campaigns for ... |
arxiv | 2607.06799 | null | 1 | What Predicts Correctness in Text-to-SQL? A Selective-Prediction Study | [
"Robert Richardson"
] | CC-BY-4.0 | https://arxiv.org/html/2607.06799v1 | 2026-07-07T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "A text-to-SQL system turns a natural-language question into a SQL query that runs against a database. These systems are increasingly used where a wrong query is costly, so a system needs to know not just ... |
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