Exploring Diverse Generation Paths via Inference-time Stiefel Activation Steering
Authors: Dongxuan Zhu
Ly Tran Ho Khanh
Andy Yat-Ming Cheung
Man-Chung Yue
Viet Anh Nguyen
Venue: ICLR 2026 Poster (2026)
Keywords: activation steering
generation diversity
manifold opimization
Primary area: optimization

Abstract: Language models often default to a narrow set of high-probability outputs, leaving their generation paths homogeneous and prone to mode collapse. Sampling-based strategies inject randomness but still struggle to guarantee diversity across multiple concurrent generation runs. We address this limitation by introducing STARS (**ST**iefel-based **A**ctivation Steering for Diverse **R**ea**S**oning), a training-free, inference-time intervention method that transforms activation steering into an exploration engine. At each token, STARS collects the hidden activations of concurrent generation runs and optimizes multiple additive steering directions jointly on the Stiefel manifold. STARS maximizes the geometric volume of the steered activations, while the Stiefel manifold induces orthogonality of the steering interventions. This formulation explicitly promotes divergent activation vectors of concurrent generation runs, and implicitly promotes divergent generation trajectories. This manifold optimization formulation can be solved using a Riemannian gradient descent algorithm with convergence guarantees, but this algorithm is too time-consuming for real-time inference. To guarantee low latency, we further design a lightweight one-step update with an aggressive, closed-form stepsize. For test case generation and scientific discovery benchmarks, STARS consistently outperforms standard sampling methods, achieving greater diversity without sacrificing qualitative performance.

Published as a conference paper at ICLR 2026
EXPLORING DIVERSE GENERATION PATHS VIA INFERENCE-TIME STIEFEL ACTIVATION STEERING
Dongxuan Zhu"
Ly Tran Ho Khanh"
The Chinese University of Hong Kong dxzhuêse cuhk. edu. hk
The Chinese University of Hong Kong hokhanhlytranee cuhk edu. hk
Andy Yat-Ming Cheung"
Man-Chung Yue
Hong Kong Polytechnic University andy. www.schhhuunuluuuuueee edu.
Hong Kong Polytechnic University manchung, yueêpolyu. edu ck
Viet Anh Nguyen
The Chinese University of Hong Kong nguyenn se. cuhk. edu. hk "Theee authors contributed equally.
ABSTRACT
Language models often default to narrow set of high-probability outputs, leav- ing their generation paths homogeneous and prone mode collapse. Sampling- based strategies inject randomness but still struggle to guarantee diversity across multiple concurrent generation runs. We address this limitation by introducing STARS (STiefel-based Activation Steering for Diverse ReaSoning). a training- free, inference-time intervention method that transforms activation steering into exploration engine. At each token, STARS collects the hidden activations of con- current generation runs and optimizes multiple additive steering directions jointly on the Stiefel manifold. STARS maximizes the geometric volume of the steered activations, while the Stiefel manifold induces orthogonality of the steering in- terventions. This formulation explicitly promotes divergent activation vectors of concurrent generation runs, and implicitly promotes divergent generation trajec- tories. This manifold optimization formulation can be solved using Riemannian gradient descent algorithm with convergence guarantees, but this algorithm is too cone-consuming for real-time inference. To guarantee low latency, further design lightweight one-step update with an aggressive, closed-form stepsize. For test case generation and scientific discovery benchmarks, STARS consistently outperforms standard sampling methods, achieving greater diversity without sac- rificing qualitative performance.
INTRODUCTION
Language models (LMs) have achieved impressive performance across broad spectrum of tasks (Achiam al., 2023 Nam al., 2024 Shao et al. 2024 Yang al.) 2025a powerful
emerging paradigm reframes complex problems as a search problem: an LM generates diverse pool of candidate solutions. and selector then identifies the best one. This best-of-N N strategy has driven significant gains in reasoning, coding, and planning (Lightman et al. 2023 Ni et al.) (023). However, the diversity of the candidate pool fundamentally bottlenecks its efficacy. When multiple despite stochastic sampling. converge the same high-probability line of thought, they follow nearly identical latent trajectories. The selector is left with paraphrases, unable to uncover better solution, and performance saturates regardless of how much the sampling budget is increased.
The consequences of this diversity collapse are profound and far-reaching. For reasoning tasks, maintaining distinct trajectories is crucial for exploring the solution space and avoiding premature convergence on a suboptimal line of argument. In mathematical reasoning, a diversity collapse oc- curs when model becomes trapped in single plausible but flawed line of argument, unable to

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backtrack and explore alternative proof strategies. For open-ended applications, such scientific discovery or creative writing, diverse outputs are not just a means to an end, but are intrinsically valuable, offering genuinely different perspectives rather than mere paraphrases of single idea. For safety and aligoment, a lack of diversity prevents us from discovering varied failure modes, adversarial attacks, unintended model behaviors during red-teaming. Therefore, increasing gen- eration diversity is not an incremental improvement but fundamental step toward unlocking the full reasoning and creative potential of LMs.
Existing methods for inducing diversity primarily perturb the token-level sampling process. Tech- niques such as temperature sampling (Ackley et al. 1985). nucleus sampling (Holtzman et al. 2020), beam search (Xie et al., (023), and self-speculative decoding (Naik et al. 2024) adjust token prob- abilities during decoding. However, these decoders operate locally and myopically; they are unco- ordinated across parallel runs and lack a global objective for diversity. This often leads to diversity collapse, a failure mode where superficially distinct sequences still follow nearly identical under- lying chains of thought (Yun et al.) 2025 Dang et al., 2025). Other approaches promote diversity during training by modifying the learning objective or data, often with reinforcement learning (e.g..
Slocum et al., 2025; Chen et al. 2025a; Lanchantin et al. 2025 Chung et al. (025). While promis- ing, these methods demand access to the full training pipeline, impose substantial computational costs, and their benefits can be fragile across domains (Ling et al. 2025). Consequently, enhancing generation diversity at inference time, without retraining, remains a critical and practical goal.
Recent advances in activation steering have opened new avenues for controlling LMs' behavior during inference (Turner et al. [2024] van der Weij et al. 2024). Activation vectors represent the model's working memory and thought space, in which it assembles features, subgoals, and circuits over time. If multiple runs occupy nearly the same region in this space, surface-level stochastic- ity has little impact (Wang et al. [205aa] Troshin et al. 2025). This phenomenon suggests a more structural intervention: actively diversifying the hidden trajectories themselves. Prior work on acti- vation steering has shown that carefully crafted directions can control attributes or concepts without retraining (Subramani et al. 2022. Turner et al. 2024 Wang et al. 2025b). Recent work has inves- tigated adaptive activation steering techniques. including methods that adjust the seeering-strengee parameter (Rodriguez et al.) 2025a b) and approaches that optimize the position of the steering to- ken (Hedström et al. 2025). However, existing steering techniques push a single generation run toward fixed. predefined direction, making them ill-suited for exploration.
A major technical obstacle to inference-time diversification is reconciling this principled objective with practical constraints. An effective inference-time intervention must be lightweight enough to run at every token without prohibitive latency, yet powerful enough to produce meaningful diversity. Furthermore, steering must preserve fluency and factuality, as naively aggressive diversification can degrade generation quality, producing meaningless or irrelevant text. This paper introduces a novel approach that repurposes activation steering, originally designed for targeted control, into a dynamic engine for exploration and diversity. Instead of steering a single run toward a fixed concept, we steer multiple concurrent runs away from each other. Our training- frre approach intervenes at each token to push the models' internal states into distinct regions of the activation space, thereby promoting divergent and structurally distinet reasoning paths.
Contributions. We summarize our contributions as follows:
We propose STARS (Stiefel-based Activation Steering for Diverse ReaSoning), a framework that steers LM activations at inference time diversify reasoning trajectories. At each token, STARS extracts the hidden activations from all sequences, then computes the orthogonal steering directions that maximize the total volume spanned by the intervened activations.
In Section we cast the mannian optimization problem on the Stiefel manifold. We then propose a Riemannian gradient descent with a theoretical convergence guarantee.
objective with orthogonality constraints Rie-
we develop practical and lightweight one-step update algorithm. By pairing spe- cific initialization scheme with the activation matrix as a computationally--Fi search direction, we replace the intractable exact line search with a well-posed quadratic approximation. This enables us to derive an aggressive, moseed-ormm step size, guaranteeing low latency and allowing steering at every token during inference time.
In Section 5,
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Notations. We use I for the identity matrix of appropriate size. We use Vε(V) for the Euclidean gradient of the function l while grad ((V) is for the Riemannian gradient. For matrix E RAX,, we use ||A|| - VTr(A] A] for the Frobenius norm. The spectral norm ||A||2 is defined as the largest singular value of A. We use R++ A {xr E R : 0} for the set of all strictly positive reals.
RELATED WORKS
Diversity of LM Generation. Recent studies underscore the fragility of diversity in LM- LM-generated outputs, with mode collapse increasingly recognized as a recurrent failure mode wherein responses become homogenized over time across inputs (Shumailov et al., 2024). Empirical evidence in- dicates that LMs frequently produce similar responses even when the prompts are systematically varied along demographic dimensions designed to generate distinct perspectives (Park al. 2024). In response, test-time scaling methods have been proposed to enhance output diversity, thereby im- proving downstream performance. A prevalent approach entails sampling multiple independent can- didate solutions, often via Chain-of- Thouuhh prompting and its variants, and subsequently applying selection mechanisms to identify the highest-quality response (Wei et al.] 2022 Zhang et al. 2022 Wang & Zhou 2024). These candidates are filtered or ranked using reward models or heuristic cri- teria policies such as Best-of-N sampling or beam search, to extract superior outputs from a diverse candidate set (Vijayakumar et al.] 2016 Lightman et al. 2023). Complementary prompting-based techniques explicitly incentivize variation in reasoning styles or perspectives, including persona- based prompting (Cheng et al. 2023) and step-by-step recall prompting (Hayati et al., 2023). In
parallel, Chhngg et (2025:: Ismayilzada et al. (2025): Deshpande et al. (2025) investigate post- training interventions aimed fostering output diversity.
Activation Steering directly intervenes the model's behavior by extracting steering vectors within specific layers of the LM's transformer architecture (Panickssery al.) 2023] Turner et al.) 2024
Wang et al. 2025b: Chen et al., 2025b). Typical steering methods are for target behavior or pur- pose, where steering vectors can computed directly or contrastively from positive and negative examples. For example, to align model behavior with human preferences, Panicksserr et al. (2023) averages the differential residual-stream activations between matched pairs of positively and neg- atively labeled exemplars of target behavior. [Stoloo et al. (2025), instead, compute the steering vectors as the difference activations between inputs with and without instruction to guide mod- els to follow constraints even without explicit instructions. Instead of adding the steering vector to the residual stream, Postmus Abreu (2024) projects the activations to ellipsoidal regions by a conceptor matrix, which captures the principal directions and variances of set of neural activation vectors. addition, some works employ sparse autoencoders to identify steering directions from
sparse latent representations (O' (O'Briene et al. 2024 He et al. 2025 Yang et al.) 2025b).
While a recent paper Khanh et al. (2025) also focuses on inference-time activation steering for diverse generations, this paper differs in three ways. First, this paper enforces orthogonality and formulates volume maximization on the (scaled) Stiefel manifold, whereas Khanh et al. (2025) maximizes activation volume on product of spheres. Second, this paper performs token-level steering by recomputing directions at each token position, while Khanh et al. (2025) updates only synchronization anchors. Third, make token-level steering practical, this paper develops lightweight one one-step update with a closed-form stepsize, thereby creating an algorithmically distinct pathway from the iterative Riemannian optimization routines emphasized in KKanh al. (2055)
3 STIEFEL ACTIVATION STEERING FRAMEWORK FOR INFERENCE-TIME DIVERSE GENERATION
In this section, we present STARS, our proposed framework illustrated in Figure | The key idea is to promote diverse generation paths from a language model during inference. Given a query, we generate N output sequences in parallel and extract their final hidden-state vectors from specified layer. For each path, we compute an additive steering vector and apply it to the corresponding hidden state, resulting in new of steered representations that serve input to the next layer.
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Generate
Final answers
token in parallel
Token
@n.uwer
YFS
Question
Token
EOS?
@n.werr
NO
Token
Compute steering vectors
answer y
Figure 1: Schematic overview of our STARS framework. Given input question, the model gen- erates N candidate sequences in parallel. At each decoding step, hidden states from a designated layer are extracted and used to recompute the steering vectors, which then guide the generation of the next token. This iterative process continues until the end-of-sequence (EOS) token is produced. Paths that emit EOS are dropped from the active
3.1 ATTENTION BASED STEERING INTERVENTION
Here, we rigorously introduce where the activation steering is applied in transformer-based LMs. Transformer architectures (Vaswani et al., 2017) have become the foundation of modern LMs, achieving remarkable performance across diverse natural language processing tasks. We consider an LM with M heads per layer, exhibiting following per-block information flow for layer
M
(++) = -FFN(MHA(ZO)) FFN
+
W/ (Atttn)(z=))))) 1=1
where 20)) is the input of the specific layer I, with Wi denoting the output projection matrix and Attn, denoting the single-head attention transformation for each head j I...., M. Here, FFN, MHA denote the feed-forward and multi-head attention, respectively. The direct sum operator θ concatenates the outputs from all attention heads before applying the feed-forward transformation. The multi-head attention mechanism is compelling target for activation steering because its in- dividual heads are functionally specialized. As demonstrated by Clark et al. (2019), specific heads learn model interpretable linguistic phenomena, such distinct syntactic dependencies and coref- erence relations. Following (Li et al. 2023) Chen et al. [202]] Ma et al.) 2025). we add the steering vector the output of each attention head, giving the following edited information flow:
PFN(MMHA'(> FFN A W3 (Attny(z")) + ol.)))
where ((-j) is the steering vector for head at layer L. For notational clarity, we denote the un- modified head output as (1,))) = Attny(x:!) Each (1.) E Rd lives the head dimension d,, E Rd with d = dr x M. Similarly, the steering vectors are stacked as ((() = [o.(1.) (..; E Rd.
and concatenating across all M heads yields ((()) 1,(..))...;:16666.).))))
3.2 ORTHOGONAL STEERING FRAMEWORK
At inference, we exploit the mechanism (I) to diversify decoding trajectories by steering at each token. Concretely, generate N response trajectories and, at each token position т, collect their hidden states (1) ,..,..... 1 E Id from the layer For each trajectory, we compute a steering
(8) vector f and form modified hidden states h + + u....... 11
+ which are then propagated through subsequent layers along their respective paths. We focus on a pre-determined layer and hence omit the superscript / in what follows; also omit the token index for notational simplicity.
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The central idea of STARS is to maximize the geometrie notion of dispersion for the set {h₃ the+ug 1 in the activation space. A canonical dispersion metric is the volume the parallelepiped spanned by these vectors. Defining H [ha.....hu hNl E R and V [ uN] E R the volume of the parallelotope by (h4 + 1/ in the activation space is equal to the square root of the determinant of (H + V) (M + Pyle 1962). To encourage diversity between different generations, we require the steering vectors to be orthogonal with each other, i.e., 1. " 0 when i { j. The orthogonal constraint regularizes the steering vectors U; ensuring each run receives a distinct direction and preventing the trivial solution where all es align. We further magnitude hyperparameter a > O to control the magnitude of e; that is, ||v:l for all i. A too big a may break the meaningful information h4, leading generation collapse. The optimization problem for computing steer vectors is:
user-set
min ((V) _ log ()) VTV=
For problem to be meaningful, there must exist at least feasible V with finite objective value. Since the domain of - log det is the set of non-singular matrices, this requires the existence V) is positive definite. The following proposition confirms this, whose proof is constructive and yields practical initialization for problem 2
of V satisfying V V al and (H + +V) (H
Proposition 1 (Existence of full-rank solution). Let H E dx N be any matrix with rank S d- Then there exists V Rdx with V = al such that (H VV) TH +V) positive definite.
The assumption d 2 r r N is mild in our setting. In practice, the activation dimension d ranges from 1536 (e.g., OWEN-2.5-1.5B)
to 2 (e.g., LLAMA-3, 1.1MA--..-40BB. whereas the number of responses N usually small (e.g.. 4, 8, or 16). Consequently, substantially exceeds r + N under typical configurations since < N. Proposition also gives us initialization for V. described in Algorithm[[ The computational cost of Algorithm| iss dominated by the singular value decomposition (SVD) of the matrix H. A classical method, such as the Golub-Reinsch Reinsch algorithm, can compute SVD in O + SdN2 + 9N3) (Golub & van Loan 2013 Section 8.6.4).
Algorithm 1 Initialization for V
Require: Incumbent activation matrix Hl RdxN magnitude 1Q+
1: Compute the SVD of H as H = QEWT and r = rank(H), where Q E Rdxd E diag(O1..... Fr, .,......)) 0) E dx and W RNN
2: Set Q2 as the last d - columns of Q.
3: Randomly select N columns (f1,...,1y UN] of Q2.
4: return Vo= = Va(ui,.... uN). E, W.
RIEMANNIAN GRADIENT DESCENT OVER STIEFEL MANIFOLD
In this section, we present Riemannian optimization approach for finding steering vectors based on problem (2). The feasible region of problem 2 is the scaled Stiefel manifold in R4xN with magnitude parameter a > 0:
St N, (V E R VTV a/}-
Since St,d, N, a)) is nonconvex, the classical projected gradient descent theory for convex sets does not apply here. A naive scheme that takes step along the Fuclidean gradient of E(V) and then projects back onto the constraint set may actually increase the objective. Even local convergence can be difficult to guarantee, since the projection may distort the step direction enough to slow or stall progress unless the step size is chosen extremely small. To circumvent these issues, we adopt Riemannian optimization approach to problem (2). thereby exploiting the manifold's differential- geometric structure to design efficient algorithms.
Riemannian gradient descent resembles projected gradient descent in the Euclidean setting, and each iteration consists of two main steps: first, computing a descent direction in the tangent space and choosing suitable step size generate tangent vector; second, retracting this tangent vector back
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onto the manifold. At iterate & with incumbent solution V& St _N, a.)), descent direction Sy. is computed in the tangent space at Vg. The tangent space at point V& E St(d,N,ar is
TV. Sud, N, (x) = (U RdNN V 0}.
A tangent space is subspace of the Fuclidean space Rd consisting of tangent vectors that represent the directions of feasible curves passing through the point and remaining on the man- ifold. We equip the tangent space with the Frobenius inner product (U1, = Tr[u U2] for U1,U2 E Tv, St(d, N,a). To obtain descent direction Sk, one can first compute the Euclidean gradient V((V)) A -2(H + VWI(H + (H + Vo)]=1, and then project it onto the tan- gent space Sudd, a)) via the orthogonal projection Projrv (U) 2 U - V1 Vi U+0 for U E IddN,, which gives the Riemannian gradient grad ((V&)) = Proizv, (Ve(V))). The negative Riemannian gradient S₁: - grad ((Vi)) serves as our descent direction. Indeed, any direction Sp E TV, St(d, N, a)) with (Sk, grad ((Va)) < 0 is also descent direction at point V/2
2a
The second step is to update the iterate Vi by moving along the direction S₃ with a stepsize 1j > 0. Because a standard Euclidean update V2 + 1/S5 takes the solution off the manifold, we will use a retraction, an operation that generalizes the concept a straight-line step to a curved space. translates the tangent vector 7 into point the manifold, ensuring the new iterate Vi+1 remains feasible. We employ the polar decomposition- baased retraction
(v) & ValV + Ua + +UTU)-1/2
(3)
The full update step is V = Rv, (n))), where the stepsize > 0 is determined using back- tracking line search to ensure sufficient decrease in the objective function. Ensuring sufficient decrease also prevents H + V& from becoming rank-deficient, which could render E(V)) ill-posed. We now summarize the complete procedure as Algorithm 2
Algorithm 2 Riemannian gradient descent with line-search
Require: Hidden activation matrix H RdxN max iterations K, magnitude R4+, line search parameters E (0, 1),c E (0. 1). fj E R++
Initialize Vo via Algorithm
2: for k ==00 0 to K do
3: Compute the Euclidean gradient V((V), <- -2 + VI) (H
4: Compute the Riemannian direction S +- VV((k) + V2 V. (VAV))
2a
5: Initialize + f
6: while True do Compute Va-1 + va(Vi + (aI Sx)- 8: if (VVi.. ((V++1) 2 Sxl1 then break else 17 < pr) end
7:
9: end while
end for
11:
return Vi
Since require decrease in function value during each step by line search in Algorithm 2 and initialize Vo to make H + Vo full rank, the iteration sequence { [V/} falls into the compact sublevel set C {V E St(d, N, a) E(V) < {(V))) which is the intersection of sublevel set and the manifold. Let a define the smallest singular value of H + over Since H + V is always full- rank for all V E C and C is a compact set, we can conclude that is bounded away from 0, i.e.,
O. Furthermore, we define D A maxyec [|Ve()))l < 2 / of ([H va) oo, which an upper bound on [|V over the sublevel set C.
The following theorem shows the convergence guarantee of Algorithm 2
Theorem (Convergence Rate of Algorithm Pp. Let (Vi)a21 be the sequence generated from Al- gorithm 2 with p. c (0, 1) and 0 < f Val D. Then for all K 2 1 have
min 0<k<K
||grad ((Vi)) O

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On Riemannian manifold, V is stationary point of smooth function e if and only if its Rie- mannian gradient vanishes, that is, grad {(V) = (Proposition 4.6 Boumall 2023). Theorem therefore shows that Algorithm 2 converges to the critical point of problem 2 at rate O(1/K). Despite the favorable theoretical guarantee, Algorithm has multiple practical drawbacks. Firstly, T indicates that the initial stepsize vj depends on the quantity D, which is challenging timate. Secondly, each iteration is expensive because it requires computing matrix inverses, square roots, and inverse square roots, each with complexity ((N3). Further, if the matrices H/ Vi are ill-conditioned, the repeated inversion and square-root computations can be numerically unsta- ble. Additionally, the backtracking line search requires multiple evaluations of the log-determinant objective. When the stepsize shrinks excessively, progress becomes very slow, and poorly chosen parameters can lead to significant computational waste. We will explore a more computationally friendly procedure in the next section.
ONE-STEP UPDATE WITH GREEDY STEPSIZE
In this section, we propose an efficient method for finding a suitable set of steering vectors. Instead of an iterative algorithm, we employ a one-step update with a carefully designed search direction and greedy stepsize. For a given point V E St(d, N, a) and search-direction S E TySt(d, N, a), the exact line search determines the stepsize by minimizing the following objective function
min ((n _ - log det (RRv())) (H + Rv (nt))) 7200
(4)
where (-(()) is the polar retraction defined in (3. As solving this problem is intractable due the nonlincarities in the degermetrmnnat and the retraction, we instead work with a quadratic approxi- mation of (()).
Proposition 2 (Quadratic approximation of C(n)). For any activation H, steering vectors V and search direction S, we define A (H + v) (II + v). D2 Tr[A-' "HIS S H)] and
D2 1 (HTSTS + S SVSTT) + TH S ())) The function L(n)
has the following second-order expansion:
1 D2n2 - Diu - log det A + 0
((n)
2
Neglecting the higher-order term (TT²), we consider the quadratically approximated stepsize:
1 D2n2 -1
rf arg min 720 2
(5)
For the minimization problem (5) to be well-posed and yield finite stepsize, D2 must be positive;
otherwise, the approximated objective would be unbounded below as 7) + cc. When D2 > 0, the optimal stepsize is given by 1γ* max(0, D2) To ensure a meaningful update (0 < nn < oc), we choose specific combination of Vo and search direction as follows: (i) We choose Vo from the output of Algorithm I (ii) We choose the search direction as S H.
Next, we show that is descent direction in the Riemannian sense at the point Vo chosen above.
Proposition 3 (Descent direction). Let E dx and let Vo be given from Algorithm 7 Then H is descent direction at point Vg in the Riemannian sense, i.e., (H, grad ((Vo)) < 0.
We now proceed to demonstrate that the above choice of (V,, S) yields meaningful stepsize n)" Moreover, we can compute 17" in an analytical form of the singular values H.
Proposition 4 (Positivesness of 1f Let H E Rdx N and let Vo be given from Algorithm Let r be ...,,,,,, the rank of H and oi,..., Jr be the singular values of H. Then we have
0f (a) + a)
D = Σ
and D₂
4
0.
In this case, the optimal stepsize is 17¹
D1 D2
0.
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Now are ready to propose Algorithm heuristic one-step update procedure for solving prob- lem Given the activation matrix H, Line 1 of the algorithm first calls Algorithm [ to obtain a feasible initialization Vi along with the SVD of H. Then Line 2 computes the stepsize 17" using the singular values of H. Finally, Lines 3 implements the polar retraction 33), where the inverse matrix square root is efficiently computed via the SVD of H. Algorithm 3 is one-step update, and it is hopeless to establish any theoretical convergence guarantees for it. However, Appendix (B.7 presents extensive empirical results to show that Algorithm 3 achieves the optimality gap of approximately 2% in single update, while using only about 3% of the run time of Algorithm
Algorithm 3 One-step update with greedy stepsize
Require: Incumbent activation matrix H E R of rank r, magnitude R++ 1: Compute Vo, Σ and W via Algorithm !
2: Compute D1 + Ei= बें D2 4 Er Ei=l
d. and 17" D1/D2
3: Compute Vi va + (a + (1
4: return V
NUMERICAL EXPERIMENTS
We evaluate our framework STARS on two benchmarks that require generating diverse outputs: the test case generation benchmark TESTEVAL (Wang et al.) 2024) and the scientific discovery benchmark I,i ve Rench (Ruan al. 2025), allowing us to demonstrate the effectiveness of approach across different domains that benefit from output diversity. For TESTEVAL, we N = 20 as in the original benchmark (Wang et al. 2024); for LiveIdeaBench. we set = 4. The hyperparameter in Algorithm IS set to
C H where C > 0 is scaling constant and | ||2 is the spectral norm of matrix H. We denote configurations by STARS C to indicate the choice of C. All the used prompts are presented in Appendix B The details of steering layer selection is in Appendix B.2
6.1 TEST CASE GENERATION
Following the TESTEVAL benchmark (Wang et al. 2024). we evaluate the generation of test cases on the overall coverage task. After generation, all test cases must undergo correctness check. Syntactic correctness determines if the generated test case is free of syntax errors, while execution correctness evaluates if the test sertion correctness evaluates whether the generated test case contains correct test assertions. The overall coverage for program is computed by the proportion of lines/branches in the program that have been covered by at least one test case, while line/branch cor measures the line/branch cov- erage with a subset of the generated test cases of size k. The coverage metrics are batch-level by construction: for each program, we consider the union of coverage over the N generated tests and compute the fraction lines/branches covered by at least one test. The correctness metrics are computed per-test but averaged over the full set of generated tests.
be executed successfully without any runtime errors. As-
Table demonstrates that the STARS method consistently outperforms the temperature sampling approach (labeled as SAMPLING) across all temperature settings and evaluation metrics on both the GEMMA- 1.5-2B-IT and QWEN3-1. 7B 1. models. STARS 0.5 (STARS run with = 0.5) achieves particularly strong performance, yielding substantial improvements in overall coverage as well as line- and branch-level coverage at both A = and = 5. The trend is consistent across both model families. Notably, on QWEN3-1.7B, STARS improves execution correctness by more than sampling at all temperatures, while also delivering strong coverage gains. Overall, the results indi- cate that STARS provides substantially better balance between diversity and functional correctness than temperature sampling, with STARS_0.5 emerging as the most effective configuration.
6.2 SCIENTIFIC DISCOVERY
We assess LMs' idea- dieaa-generttin capabilities using the LiveIdeaBent dataset, which contains 1 118 keywords. Following Guilford's creativity theory (Ruan al. 2025), methodology evalu-
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Table 1: Experimental results on TESTEVAL across different models and sampling temperatures. All the numbers are in percentages. Best-performing methods are highlighted in bold.
Correctness t
Overall cov. f line
Line coc@k f
Branch coo f
T
Method
syntax
exe
assert
branch
k
A =
k =
ht =
SAMPLING
89,90
1.02
0.95
5.36
5.06
5.32
5.35
4,98
5.04
1.0
STARS.0.1
86.48
1.43
1.17
10.03
9.32
9.97
10.02
9.12
9.25
STARS 0.5
79.76
3.83
3.45
34.76
30.93
33.81
34.45
29.58
30.40
SAMPLING
91.79
1.19
0.00
3.54
3.37
3.46
3.51
3.21
3.30
GEMMA S.NMA-1.11251 1.1--1111111111
0.8
STARS 0.1
87.83
1.45
1.12
10.77
9.88
10.64
10.72
9.65
9.78
STARS 0.5
79.43
3.79
3,45
32.96
30.02
31.70
32.59
28.23
29.43
SAMPLING
93.45
1.29
0.00
3.01
2.84
2.94
2.99
2.70
2.77
0.6
STARS.0.1
89.19
1.71
1.43
12.10
11.22
12.03
12.10
10.99
11.15
STARS.0 00.
79.24
3.86
3.24
34.07
30.38
32.04
33.22
28.00
29.46
SAMPLING
95,40
1.36
1.26
2.33
2.31
2.29
2.32
2.20
2.27
0.4
STARS 0.1
90.43
1.48
1.43
8.44
7.69
8.41
8.44
7.58
7.63
STARS_0.5
80.31
4.38
4.00
40.03
35.96
38.39
39.36
33.73
35.07
SAMPLING
95.64
1.36
1.36
1.44
1.41
1.37
1.42
1.27
1.35
0.2
STARS.0.1
91.52
1.45
1.36
7.28
6.50
7.23
7.25
6.36
6.40
STARS.0.5
78.93
4.38
4.02
39.03
35.05
37.38
38.67
32.94
34.39
SAMPLING
18.07
11.88
4.19
59.95
55.05
57.18
58.66
51.31
53.39
1.0
STARS
61.10
39.90
19.17
86.77
81.10
80.54
83.62
71.71
76.36
STARS 0.5
72.21
41.86
30.79
91.05
86.81
82.18
87.31
73.59
81.02
SAMPLING
13.86
8.17
2.83
37.00
33.73
36.15
36.47
32.43
33.04
QWEN3-1 1,7B
0.8
STARS 61.10
39.90
19.17
86.77
81.10
80.54
83.62
71.71
76.36
STARS.0.2 .05
71.55
41.31
29.07
92.22
87.86
83.00
88.42
74.39
82.14
SAMPLING
11.69
6.19
2.26
23.92
21.79
22.76
23.24
20.35
20.89
0.6
STARS 1011 61.57
41.10
19.43
86.
80.95
81,08
83,9
72.24
76.53
STARS 505
72.79
43.02
31.05
90.73
86.35
82.39
87.25
73.77
80.88
SAMPLING
9,00
3.83
1.52
13.56
12.26
13.29
13.43
11.86
12,08
0.4
STARS.0.1
62.69
41.60
19.74
87.14
81.17
81.82
84.46
73.15
77.13
STARS_0.5
73.88
42.79
30.74
91.30
87.01
82.95
87.37
74.40
81.00
SAMPLING
8.17
2.69
1.17
4.71
4.16
4.70
4.71
4.13
4.14
0.2
STARS 0.1
62.93
41.11
18.81
86.25
80.22
80.67
83.60
71.72
75.92
STARS 0.5
73.40
41.81
29.95
91.35
87.13
82.95
87.61
74.28
81.32
ates these five core dimensions, with each metric scored on a 0-10 scale. Flexibility is computed as the 30th percentile of the average scores across the other four metrics. Fluency measures both the quantity and diversity of ideas, capturing the ability to produce diverse responses. All metrics are judged by GPT-4.1-MIN with the specific prompts in Appendix B.T
Table presents experimental results evaluating the performance of the QWEN2.5-3B. INSTRUCT and D.AAMA--.2-3B-INSTTEE
models the L.veTdeaBench dataset. The ex- perimental setup extracts hidden states at layer 20, evaluates performance across temperature values ranging from 0.2 1.0, and C E {0.1, performance across most temperature settings. In terms of fluency (a metric reflecting diversity), our method achieves 5.09, nearly twice the temperature sampling (2.68) T 0.2. The results pattern: while standard sampling shows declining performance as
STARS consistently achieves the highest average
reveal a rumperature-dependeer temperature decreases (from 5.71 T 1.0 to 5.01 ble performance across the temperature range. This suggests that STARS provides more robust text generation capabilities, particularly in low-temperature regimes where standard sampling typically suffers from reduced diversity. We provide case study in Appendix B.9 that shows that STARS generates more creative ideas than standard temperature sampling. Table 14 in Appendix B.8 re- ports the average per-question runtime across models and tasks for Algorithm and temperature sampling. The results indicate that Algorithm incurs only a small absolute overhead, averaging no more than 2 seconds per question.
0.2), STARS methods exhibit more sta-

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Published as a conference paper at ICLR 2026
Table Experimental results on Li LivedeeBerch across different models and sampling temper- atures. All values are reported on a scale with maximum score of 10. Best-performing methods highlighted in bold.
Method
Originality
Feasibility
Clarity
Fluency
Flexibility
Avg.
SAMPLING
6.22
5.50
5.78
5.42
5.61
5.71
1.0
STARS 0.1
6.26
5.64
5.79
5.32
5.71
5.74
STARS 0.5
6.30
5.45
5.75
5.81
5.70
5.80
QWEN2.5 33 -B--NSTRUCT
SAMPLING
6.20
5.55
5.76
5.02
5.50
5.60
STARS STARS .05
6.14
5.57
5,80
5.45
5.67
5.73
6.13
5.62
5.70
5.96
5.74
5,83
SAMPLING
6.07
5.63
5.68
4.51
5.38
5.45
0.6
STARS 0.1
6.17
5.59
5.73
5.19
5,55
5.64
STARS
6.01
5.68
5.72
5.17
5,48
5.61
SAMPLING
6.03
5.56
5.75
3.57
5.14
5,21
0.4
STARS
6.09
5.70
5.76
4.61
5,47
5.53
STARS_0.5
5.99
5.70
5.73
5.02
5,50
5,59
SAMPLING
6.04
5.69
5.74
2.68
4.92
5.01
0.2
STARS 0.1
6.06
5.61
5.79
4.20
5.30
5.39
STARS 0.5
6.04
5.62
5.68
5.09
5.52
5.59
SAMPLING
6.51
5.34
5.89
5.41
5,66
5.76
1.0
STARS.0.1
6.57
5.31
5.89
5.45
5.68
5.78
LLAMA-3 3.2-33--BBBBBB INSTRUCT
STARS.0.3
6.60
5.32
5.85
5.60
5.71
5.82
SAMPLING
6.47
5.39
5.92
5,07
5,58
5.69
STARS.0.1
6.49
5.39
5.11
5.59
STARS 0.5
6.56
5.36
5.91
5.22
5.62
5.73
SAMPLING
6.43
5.42
5.99
4.59
5.46
5.58
0.6
STARS 0.1
6.45
5.46
6.01
4.78
5.57
5.65
STARS 0.5
6.53
5.42
5.97
4.81
5,58
5.66
SAMPLING
6.41
5.42
6.00
4.17
5.35
5.47
0.4
STARS.0.
6.44
5.46
6.02
4.23
5,43
5.52
STARS.0.
6.50
5.44
5.99
4.41
5,48
5.56
SAMPLING
6.42
5.45
6.03
3.27
5.19
5.27
0.2
STARS 0.1
6.46
5.48
6.03
3.60
5.27
5.37
STARS .0.5
6.51
5.46
6.02
4.04
5.39
5.48
7 CONCLUSION
We introduced STARS, a training-free, inference-time method that steers activations every token across multiple concurrent generation paths. By jointly optimizing orthogonal steering directions on the Stiefel manifold, STARS maximizes the geometric volume of the hidden activations from these runs, explicitly promoting divergent internal states and implicitly inducing diverse trajectories. While a full Riemannian gradient descent offers convergence guarantees, it too slow for real-time use; our lightweight one-step update with closed-form aggressive stepsize enables low-latency, per-token intervention. Across test case generation and scientific discovery benchmarks, STARS substantially increases diversity without degrading quality, outperforming baselines.
ACKNOWLEDGEMENT
Man-Chumg Yue is supported in part by the Hong Kong Research Grants Council under the GRF project 17309423. Viet Anh Nguyen gratefully acknowledges the support from the CUHK's Improvement on Competitiveness in Hiring New Faculties Funding Scheme, UGC ECS Grant 24210924, and UGC GRF Grant 14208625.
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Published as a conference paper at ICLR 2026
ETHICS STATEMENT
This work focuses on methods for improving the diversity of outputs from large language mod- els through inference-time activation steering. Our study does not involve collecting notating human data, and all benchmarks employed are publicly available (e.g., TESTEVAL, LiveIdeaBench). The primary contribution algorithmic, and no new risks of data misuse or privacy violations are introduced.
10 REPRODUCIBILITY STATEMENT
Our code implementation is publicly available at htttttt:/ggtthiiie All experiments use random seed of 42 and conducted on NVIDIA RTX A5000 24GB GPUs. We specify experimental details and hyperparameters in Section 6 and Appendix B Our theoretical contributions are accompanied by complete proof' Appendix
STTARS
11 LLM USAGE STATEMENT
We used large language models exclusively for polishing the writing of this manuscript, specifically to improve clarity and readability. No parts of the technical contributions, mathematical derivations, experimental design, technical results were generated by language models.
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A PROOFS OF MAIN RESULTS
This section contains the proofs of all technical results presented in the main paper.
Proof of Proposition L Let Q1 E Raxr be an orthonormal basis of the column space of H. then we can extend Q1 an orthogonal matrix Q [Q1, Q2] E Rdxd where the columns of Q2 E Rdx(d-r) span the orthogonal complement the column space of И. Since r+ N, we can take V any N columns of the matrix vaQ2- Then one can verify that V V = aa and V H 0. Also, we
have
(H V) + V) H + V V ( +V H H +a
which means (H + T (H + V) is positive definite, hence full rank.
The following lemma generalizes the second-order boundness of polar retraction from the standard Stiefel manifold the scaled Stiefel manifold.
Lemma (Second-order boundness of polar retraction). For any V E St(d, N, a) and U StSt N. a) with | I Vo, it holds that
(U) - +V+U) = Va LU?..
We also have [U)(U) Vl & IV +U V F for all V St(d,
Proof of Lemmal
7 By the definition of the polar retraction, we have
[Rv(U)) - (V + U lI +U)(aI+
(V U)Ilr
$ ||V + U + -1 1l|
Let UTU SEST be spectral decomposition of U with diag()), AN). For the second term in the product, we have
ViValal + + 1/2 -1 = [I(1 + - - U)-12 111
-I([ + SEST - ) 1|
=
N
Σ
(1 + 11)-1//2-11 112
i=1
Σ X
4
UU.
4a
Hence when | S Vo. it holds that
[U) (V + U)lle s IIV + UII+ + -U -I|e
< ||V + U|
|
2a
< Va
where use the fact that IIV + Ulr || ||]|| S 22a..
Now we prove the second statement. The convex hull of St(d, N. a) is given by co St(d, N. =
(V BaxN | < a), where ||V|lop denotes the spectral norm of V. We claim that Rv
Projsudd.N.o) (V + U) Projesstid.N.a U). Since U E TvSt(d, N, N,o), have (V +
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(VT(++U) aI which implies that all singular values of V U at least vā. Hence, claim follows. For V, V E St(d, N, a),, and U TySt(d, N, a)), we can write
[Ry(U) F
St(d,N,)) (V + U)- Projec St(d,N.a) (V)le
I IV +U - -
since projection onto closed convex set is non-expansive.
The following proposition shows that with proper stepsize, one can expect a decrease in the objec- tive value with an Riemannian gradient step.
Proposition 5 (Sufficient descent). Let V E C. If0 < 1 < vajD. D, then it holds that
8(Rv(- grad ((V))) - {(V) 2 -C(nllrrad e(V) where C(n) 1 pa nn D
maxved [[([[)] < C0. 1 1
supvec,|ull==1 / < 00 and L2 = L + D.
Proof of Proposition For simplicity, we define P - H + V. Let be the smallest singular value of M over the set C. Since M is always full-rank for all V E and C is compact set, is bounded away from 0, that is, σ > 0. Noting that
VE(V) = -2P(PTPPP
must have
1 o² VEe
D é max VEC
[V(V))
|||||||(PTTT P) 1)-111
|P|| <
VEC
Since grad ((V) = Projr, (V((V)) it also holds that maxvec l|grad ((V)lr S D. By basic computation, the Euclidean Hessian of E evaluated in the direction U is
PTPPPUUUY U)(P P)-1 2U p)-1 One can also show that there exits I4 > 0 such that for all V E e,
v²((V)[U] = 2P P) (U
T P
pT
(6)
[|V s LT.
sup
||0']| x=
Thus, ((V) is -moothh on C. For any S TvSt(d,N, a), Chen al. 2021 that with L2 A Ln + D. one has
Lemma 2.4) shows
EiR (S)) < (VV) + (grad ((V), Rv - V2 + L2 2
(Rv(S) - -
= [(V + (grad (V)),) + (grad Rv), - (V + S)) + T.2
|| (S) - -11
2
S {VV + (grad (V), S) + Iggrad ((V)) (V + 5)]r + L
( VI
(7)
Since 1) < va/D and maxvec llgrad [(V)ll= & D, we must have Il grad e(V)lle < Va. Lemma then asserts that
1/² J Igaa (V)), (n grad ((V)) (V + 1 grad ((V))l
and
[Ry grad ((V)) - V S nllgrad ((V)lP..
Then, take S = =- grad ((V) in 7 and we have
g-n ((V)) < ((V) (grad S)) + Iggaad ((V)l Ry (S) - (V S)Ile
+ L2 [R
-VI
2
S {(V) nllrad (V) + D 1
T221²
Igrad ((V) + I|rrad ((V)
Va
2
{(V) C(n)llgrad (V)
where C(n) n|| ( Pa + f/3 )n] The proof is completed.
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Published as a conference paper at ICLR 2026
Now are ready to prove Theorem
Proof of Theorem 7 The backtracking line-search procedure in Algorithm 2 gives that for each k there exits constant C such that
(8) We first prove that Ck is bounded away from 0. Since grad {(V) Projry (V((V)) it also holds that maxvee llgrad [(V)l I D and thus, lggrad ((V&)llr I D for all k. Hence, va/ /lgrd ((V&)]lF must has a lower bound va/D. Note that S n S Va/D, we always have Ingrad ((Vi)lle S Va. Proposition guarantees
((Va) - 2(VR+1) 2 CRlISAlie.
((Vi.) ((V++1) C(r) ((V)))
On the other hand, if the backtracking procedure does not terminate for a certain value 1,, then {(V+) ((V+aa) cyllgrad ((V+))
If both are true simultaneously, then we must have en > C(n). that is,
2(1 c)
?> 2D + L2va
Thus, the backtracking procedure terminates when 17 is below this value. Hence, we can conclude that the returned 17 satisfies
2 min 2p - c)va
2D L2va }.
which implies that C₁ 2 Cmin A c (7,
2p(1-c)vā for all k.
2D+Lava
Based on a standard telescoping sum argument, we get
((V)) e" 2 ((V)) - {(vx)
K-1
(((vx) {(V++1))
C Illaaad ((2)
Cmin
Igrad ((Vx)ll
Cmin
K
min k=
||rra E(V2)
which completes the proof.
Proof of Proposition
2
From the Taylor expansion
(1+ =1 -
we have
"
"? ST
(aI + 5)-
aa-1/111
2a
which gives the following expansion of (v(S)):
(n(n)) = va(v + ()) + +) =V + 20
V
Thus, have
(II + (v T (H + Ry (nS))
n12 V S + (H+ + +
1
=(H + V +
VS 5 + S
2n
20
=(1 V + +V)
S H)
(H VS S S SV H) + 0
2a
17

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Published as a conference paper at ICLR 2026
Since log det (I+ Tr Tr +(1 for [(7) it holds that [(1) =
log dee(((H (enddl((+++r(// ( (M+ RR(1))))
Tf² (H
log de (H V) + +H S
H)
VS S +0
2a
,²
log det A log det / + - + S H)
(S H) + 0
2a
log det A 17 Tr|A (H S+5
=
11²
+
Tr(A VS S ST H)]
20
n
Tri(((A (H S S H)) + 0(1))
2
-log del A - Di7 + 2 2291 + +
where D1 = Tr(A-I(H + STH)}, D2 = : THAA- + ST STSTT) H)) + Tr(((A- + S H)) 3I. The proof completed.
Proof of Proposition
With Vo from Algorithm 1 have H T Vo 0 and Vo Vo aT. Then,
(H Vo) T Vo + Vo (H + Vo) H,
Projtv, (H + Vo) = H + Vo Vo
2a
which means that H is valid tangent vector at Vo. Then have
( H. grad ((V)) ) = (HI ve(V)) = - 11 (H Vo) + + +V)) 11) 2( H, H ((H + Vo) +V)) <0,
since H Vy is of full rank.
Proof of Proposition Let H QER and Han be the SVD of H, where Q E Rdxd
E RdxN and R E NxN Then we have
A [( Vo) TO)THH + Vol-
(H H +
+V
H V.
H Vo + Vo)
TV,
=HTT H +al)
=RRERR +al)
= = RTT
which gives that
= RE R(E -(E* - RT,
and
With S = H we have
Dj 2
-TH[R(E2 +al)
=2TH((E² -al)
of
= 2
> 0.
o +a
18

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Published as a conference paper at ICLR 2026
Noting that S= Vol H 0, for D2 have
D2 = s + (I" S ST H))
4TT:((A H)
-4Tr[(>² - +al) -a
of
c 0)
The positiveness of 17' comes from D1, D2 > 0. The proof is completed.
B EXPERIMENTAL DETAILS
B.1 EXPERIMENT SETTINGS
B.1.1 TEST CASE GENERATION
This is the prompt for generating the test cases.
Test Case Generation Prompt
Please write test method for the function func_name given the following program under test and function description. Your answer should only contain one test input. Program under test: program Function description for func name: description
Your test method should begin with:
def rest__. func sann__nme] () solution = Solution ()
B.1.2 SCIENTIFIC DISCOVERY
We use the following prompts to generate ideas and evaluate the quality in the scientific discovery task.
Idea Generation Prompt Example
I'll be submitting your next responses to "Good Scientific Idea" expert review panel. If they consider your idea to good one, you 'll receive reward. Your assigned keyword is: keywords, You may provide background information. The idea MUST be concisely expressed within 100 words total (including any background information).
Note: good scientific ideas should be original (novel contribution), feasible (technically implementable), clearly articulated, and address meaningful problems in the field.
19

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Published as a conference paper at ICLR 2026
Judge Prompt Example
You are an extremely demanding scientific reviewer with the highest critical standards, like those at Nature or Science. When evaluating scientific ideas, assess them on three key dimensions:
Originality: Novel contribution unexplored existing problems.
innovative approaches
Feasibility: Technical implementation and practicality.
Clarity: How well-articulated and easy to understand the idea is.
Your response should consist of two parts: a brief text analysis (under 100 words), followed by JSON score block:
"originality" <acore_l_to_10x "feasibility" <score_1_to_le "clarity": <score 1 to_
(role: user, content: prompt)
Fluency Judge Prompt Example
Here are four ideas submitted to the "Good Scientific Ideas" Competition, all related to
keyword:
Idea 1:
{{A}]
Idea 2:
{BB)
Idea 3:
[[C))
Idea 4:
{(D}
Question:
Evaluate the similarity between the four ideas that both relate to keyword. Please choose
the best answer:
A. Completely different ideas addressing different problems, despite relating to the same keyword.
B. Different ideas but addressing similar problems.
C. Similar ideas addressing similar or identical problems.
D. Academically identical ideas with the same core approach and problem statement.
ONLY ANSWER A/B/C/D, DO NOT EXPLAIN.
B.2 EVALUATION ACROSS STEERING LAYERS
We the random seed to 42 for all experiments. We conduct experiments investigate the impact of modifying the steering layers the performance of the proposed algorithm. For simplicity, we sample 10% of each dataset select the optimal layer for steering.
To identify optimal steering layers, we systematically evaluated layers 0-17 for Gemma-1 1. 1--b--it and layers 0-27 for Qwen3-1.7b. For each layer, we calculated the mean values across five evalu- ation metrics: Syntax Correctness, Executable Correctness, Assertion Correctness, Overall Cover- age Line, and Overall Coverage Branch. Subsequently, we ranked the layers by their performance on each metric and computed the average rank across all metrics. The layer achieving the low- est average rank was designated as exhibiting superior overall performance and selected for further experiments. See Table 3 and Table 4 for the ranking results.
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Published as a conference paper at ICLR 2026
Table 3: Layer ranking for model Gemma-1.1-2be based multiple evaluation metrics. The Average represents the average ranking across these metrics, with lower values indicating better overall performance.
Correctness
Overall Coverage Branch
Layer
Average
Syntax
Executable
Assertion
Line
3
2
1
2
2
1.6
7
5
5
3.8
15
6
6
6
4
4
5.2
14
2
2
6
11
3
8
7
10
7
2
8
4
4
9
10
7
17
5
14
14
7.8
8
4
3
3
16
16
8,4
9
1
8
7
14
14
8.8
1
8
12
12
7
6
9
5
12
13
5
9.6
0
13
7
9
9
12
10
15
11
10.4
14
11
10
10
15
15
12.2
10
15
15
12
12.2
6
18
16
16
9
6
13
10
17
17
17
18
16
17
12
16
18
18
16
18
17.2
Table 5 shows the results when extracting the hidden states from layers 10 and 30 with C' E (0.1,055)
21

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Published as a conference paper at ICLR 2026
Table 4: Layer ranking for model Qwen3-1.7h based multiple evaluation metries. The Average represents the average ranking across these metrics, with lower values indicating better overall per- formance.
Correctness Executable
Overall Coverage Branch
Layer
Average
Syntax
Assertion
Line
6
4
5
1
2.6
2
3.4
2
5
3
4
3
2
3.4
3
6
2
3
10
5.8
9
9
11
8
5
4
7.4
24
12
4
7
10
8.2
15
7
9
14
6
8.6
11
6
5
12
13
9.4
12
3
14
12
9.4
20
14
12
19
9.8
21
16
8
9
14
11
11.6
0
10
7
6
22
21
13.2
11
2
13
17
13.4
7
19
10
11
15
12
13.4
14
9
11
16
13.4
17
13
13
15
16
15
14.4
23
21
20
14.6
26
18
15
18
13
14
15.6
8
19
20
16.4
5
22
16
17
18
18
18.2
23
24
20
22
20
19
21
16
17
23
21
24
24
21.8
19
25
22
23
23
23
23.2
25
27
24
24
22
23.6
22
21
26
26
26
26
25
18
26
25
25
25
25
25.2
13
20
28
28
28
28
26.4
27
28
27
27
27.2
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Published as a conference paper at ICLR 2026
Table 5: Experimental results on 10% Live Ideeedeeeccc dataset across different sampling tem- peratures Qwen2.5-3B 3B-IIntruct All values reported scale with maximum score of 10. Best-performing methods are highlighted in bold.
Method
Originality
Feasibility
Clarity
Fluency
Flexibility
Sampling
6.22
5.42
STARS 00.1
10
6.34
5.58
5,80
5.17
5.60
5.70
STARS_0.1
5.27
STARS.0.5
10
6.26
5.46
5,81
5,70
5.68
5.78
STARS U.S
6.30
5.54
3,80
5.27
5.71
Sampling
6.20
5.55
5,76
5.02
5.50
5.60
0.8
STARS.0.1 30
6.11
5.62
5.82
4,97
5.55
5.61
STARS.0.5
6.21
5.51
5,80
5.17
5.66
$0.
6.21
5.58
5.87
4.97
5.55
5.63
Sampling
6.07
5.63
5.68
4.51
STARS 0.1 10
6.13
5.56
5.85
4.58
5.40
5.51
STARS,0.
STARS.0..
10
6.21
5.52
5,75
4.81
5.45
5.55
STARS.C 0.5 30
6.14
5.55
5.71
4.86
5.47
5.55
Sampling
6.03
5.56
5.75
3.57
5.14
5.21
5.54
0.4
STARS 0 30
6.06
5.71
5.82
4.03
5.26
5.37
STARS.0.5
6.15
5.46
5,76
4.76
5.51
STARS 0.5
6.01
5.61
5.82
4.79
5.51
5.59
Sampling
6.04
5.69
5.74
2.68
5.01
STARS 0.1 10
6.07
5.58
5.78
3,06
5.01
5.10
0.2
STARS.0.1
6.14
5.64
5.77
3,26
STARS.0.5
6.05
5.54
5,78
4,00
5.25
5.32
S0.RS.0.5
6.17
5.56
5.79
4.41
5.41
5.47
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Published as a conference paper at ICLR 2026
B.3 COMPARE INTERVENTION POSITIONS
A critical design choice in activation steering is determining where in the model architecture to apply the intervention. We investigate two natural intervention points within each transformer layer: (1) before the output projection of the attention mechanism (o_proj), which captures the attended representations before they are projected back to the residual dimension, and (2) the output of the MLP block, which represents the processed information after the feed -orward transformation.
Table O shows the experimental results on TESTEVAL with Gemma-1.1-22 across different tem- peratures and two intervene positions (attention and residual stream).
Table 6: Experimental results on TESTEVAL across different sampling temperatures. All the num- bers are percentages. Best-performing methods are highlighted in bold. STARS RES means applying STARS to the residual stream.
Correctness f
Overall cov. f
Line corak f
Branch cov 1
Method
syntax
exe
assert
line
branch
k = 1
k
k: ==
h: = 5
SAMPLING
89.90
0.00
0.00
0.00
0.00
0.00
0.00
STARS RES.OE RES.0.1
87.71
6.02
5,93
33.54
30.10
31.54
32.55
27.64
28.94
1.0
STARS RES .05
26.40
0.02
0.02
0.05
0.00
0.05
0.05
0.00
0.00
STARS 1
86.48
1.43
1.17
10.03
9.32
9.97
10.02
9.12
9.25
STARS_0..
79.76
3.83
3.45
34.76
30.93
33.81
34.45
29.58
30.40
SAMPLING
91.79
1.19
0.00
1.44
1.41
1.37
1.42
1.27
1.35
STARS RES 0.1
87.71
6.02
5.93
33.54
30.10
31.54
32.55
27.64
28.94
STARS RES 0.5
28.81
0.00
0.00
0.00
0.00
0.00
GEMMA- 1.1 1.1111111111 2B-1IT
STARS 0.1
87.83
1.45
1.12
10.77
9.88
10.64
10.72
9.65
9.78
STARS.0.5
79.43
3.79
3.45
32.96
30.02
31.70
32.59
28.23
29.42
SAMPLING
93.45
1.29
0.00
3.01
2.84
2.94
2.77
2.70
2.77
STARS STARS_RES.0 RES_O. 10.1
90.74
6.52
6.36
33.23
29.73
31.09
32.46
26.94
28.61
STARS STARS_RES_O.. .05.
24.83
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.6
ST.RS_0.1
89.19
1.71
1.43
12.10
11.22
12.03
12.10
10.99
11.15
STARS
79.24
3.86
3.24
34.07
30.38
32.04
33.22
28.00
29.46
SAMPLING
95.40
1.36
1.26
2.33
2.31
2.29
2.32
2.20
2.27
STARS RES_O. 10.
90.83
6.38
32.45
29.32
30.90
31.77
27.35
0.4
STARS RES_REE 00.5
27.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
STARS 0.1
90.43
1.48
1.43
8.44
7.69
8.41
8.44
7.58
7.63
STARS 0.5
80.31
4.38
4.00
40.03
35.96
38.39
39.35
33.73
35.07
SAMPLING
95.64
1.36
1.36
1.44
1.41
1.37
1.42
1.27
1.35
STARS_RES.O. R
90.67
6.33
6.21
31.09
27.62
30.21
30.60
26.33
26.92
STARS RE5
28.14
0.00
0.03
0.00
0.03
0.00
STARS STARS
91.52
1.45
1.26
7.28
6.50
7.23
7.25
6.36
6.40
78.93
4.38
4.02
39.03
35.05
37.38
38.67
32.94
34.39
These findings indicate that while MLP-based interventions can be effective at moderate steering strengths for improving execution correctness, attention- baatid interventions (before o-proj) offer better robustness and scalability for maximizing test coverage. The attention mechanism appears to provide a more stable intervention point that tolerates stronger perturbations without catastrophic degradation output quality. This may be because intervening before the output projection allows the model's subsequent layers to better adapt and compensate for the perturbation. Based on these results, we adopt attention-base intervention (before o o.proj) as our default approach throughout the paper.
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Published as a conference paper at ICLR 2026
B.4 ADDITIONAL BASELINE METHODS
To further validate the effectiveness of our approach, we compare against additional baseline meth- ods that provide alternative strategies for controlling model behavior during generation. Specifically, we evaluate two natural baselines. The first one is RAND, which applies randomly sampled steer- ing vectors explore the impact arbitrary directional interventions. The second one HMEAN, which uses steering vectors computed as the mean offset from mean point to test whether spreading activations uniformly from a central location improves diversity. Specifically, with a of activa-
tions {he 1.W we first compute the mean for the i-th path v; [[A- h--h
N Em h4. With a scalar a > 0, the steering vector
Table 7 shows the experimental results on TESTEVAL with GEMMA-1.1-24 1-1-T across five tempera- ture settings T {0.2, 0.4, 0.6, 0.8, 1.0]. We compare vanilla temperature sampling against RAND and MEAN baselines at two steering strengths (0.1 and 0.5), as well as our proposed STARS method.
Table 7: Experimental results on TESTEVAL across different sampling temperatures. All the num- bers in percentages. Best-performing methods are highlighted in bold.
Correctness
Overall cov.
Line cor
Branch
cor
T
Method
syntax
assert
line
branch
k
k
k ==]
k: 5
SAMPLING
89.90
1.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
RAND 00.1
89.81
1.14
1.10
4.89
4.54
4.83
4.88
4,40
4,45
RAND 0.5
89.98
1.07
0.98
6.48
6.14
6.44
6.46
6.05
6.10
1.0
MEAN 00.1
90.64
1.10
1.02
4.96
4.56
4.92
4.95
4.44
4.51
MEAN 0.5
90.76
1.12
4.51
4.20
STARS 0.1
86.48
1.43
1.17
10.03
9.32
9.97
10.02
9.12
9,25
STARS
79.76
3,83
3,45
34.76
30.93
33.81
34.45
29.58
30.40
SAMPLING
91.79
1.19
0.00
1.44
1.41
1.37
1.42
1.27
1.35
RAND 92.02
1.26
1.05
5.19
4.72
5.13
5.17
4.59
4.68
RAND_0.2
91.43
1.26
1.14
4.73
4.38
4.71
4.72
4.31
4.34
0.8
MEAN 0.1
92.10
1.26
1.24
3.94
3.54
3.89
3.92
3.41
3.47
MEAN 0.5
92.43
1.14
1.07
5.34
5.07
5.29
5.33
4.94
5.02
STARS 0.1
87.83
1.45
1.12
10.77
9.88
10.64
10.72
9.65
9.78
STARS
79.43
3.79
3.45
32.96
30.02
31.70
32.59
28.23
29.42
GEMMA-I 1.14-11111 21--BB-I
SAMPLING
93.45
1.29
0.00
3.01
2.84
2.94
2.77
2.70
2.77
RAND 0.1
93.17
1.10
1.07
2.78
2.70
2.72
2.77
2.58
2.66
RAND 0.5
93.00
0.90
0.86
2.78
2.64
2.74
2.75
2.52
2.60
0.6
MEAN.0.1
94.67
2.77
2.77
2.55
MEAN
94.33
1.12
1.10
3.21
2.96
3.18
3.19
2.90
2,92
STARS_0.
89.19
1.71
1.43
12.10
11.22
12.03
12.10
10.99
11.15
ST.
79.24
3.86
3.24
34.07
30.38
32.04
33.22
28.00
29.46
SAMPLING
95.40
1.36
1.26
2.33
2.31
2.29
2.32
2.20
2.27
RAND 0.1
95.14
1.19
1.14
1.92
1.88
1.88
1.91
1.77
1.83
RAND 0.5
94.67
1.24
1.19
2.23
2.08
2.20
2.23
1.98
2.04
MEAN_0.1
95.90
2.33
2.24
MEAN 00.5
95.93
1.07
1.07
1.41
1.36
1.40
1.41
1.29
1.32
STARS.0. 00.1
90.43
1.48
1.43
8.44
7.69
8.41
8.44
7.58
7.63
STARS.0.5
80.31
4.38
4.00
40.03
35.96
38.39
39.35
33.73
35.07
SAMPLING
95.64
1.36
1.36
1.44
1.41
1.37
1.42
1.27
1.35
RAND_0.1
96.24
1.33
1.31
1.89
1.88
1.84
1.87
1.77
1.81
RAND_0.5
95.86
1.21
1.14
1.89
1.88
1.83
1.88
1.77
1.85
0.2
MEAN_0.1
96.57
1.45
1.40
1.89
1.88
1.87
1.77
MEAN 0.5
96.74
1.29
1.29
1.41
1.36
1.41
1.41
1.31
1.33
STARS 0.1
91.52
1.45
1.26
7.28
6.50
7.23
7.25
6.36
6.40
STARS 0.5
78.93
4.38
4.02
39.03
35.05
37.38
38.67
32.94
34.39
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Published as a conference paper at ICLR 2026
Table 8 presents the experimental results TESTEVAL with QWEN3-1.78 1.7B across five temperature settings T E {0.2, 0.4, 0.6, 0.8, 1.0). We compare vanilla temperature sampling against RAND and MEAN baselines, well proposed STARS method.
Table 8: Experimental results TESTEVAL across different sampling temperatures. All the num- bers are percentages. Best-performing methods are highlighted in bold.
Correctness exe
Overall cov. f line
Line cor
Branch cov f
T
Method
syntax
assert
branch
k:
5
k:
k:
SAMPLING
18.07
11.88
4.19
59.95
55.05
57.18
58.66
51.31
53.39
RAND 0.1
17.93
12.12
4.57
56.87
52.23
54.15
55.40
48.45
50.25
RAND 0.5
22.19
15.71
6.19
71.67
66.07
68.32
70.04
61.45
63.81
1.0
MEAN_0.1
35.05
24.55
9.02
68.48
63.31
64.90
66.38
58.44
60.69
MEAN_0.5
83.05
58.14
24.21
74.30
69.24
69.53
71.62
62.10
65.23
STARS.0.1
61.10
39.90
19.17
86.77
81.10
80.54
83.62
71.71
76.36
STARS 5055
72.21
41.86
30.79
91.05
86.81
82.18
87.31
73.59
81.02
SAMPLING
13.86
8.17
2.83
37.06
33.73
36.14
36.47
32.43
33.04
RAND .011
15.52
10.05
3.69
.9.98
43.74
46.11
47.17
41.03
42.58
RAND 00.
20.21
14.14
5.50
67.79
62.06
64.93
66.93
58.08
60.83
0.8
MEAN.CP 0.1
33.86
23.60
8.33
63.22
58.59
60.08
61.72
54.19
56.49
MEAN.0.5
85,95
59.98
25,43
71.53
66.64
67.29
69.14
60.24
63.14
STARS 0.1
61.10
39.90
19.17
86.77
81.10
80.54
83.62
71.71
76.36
STARS 0.5
71.55
41.31
29.07
92.22
87.86
83.00
88.42
74.39
82.14
QWEN3-1 1.7B
SAMPLING
11.69
6.19
2.26
23.90
21.79
22.76
23.24
20.35
20.89
RAND.0.1
13.43
7.93
2.88
42.20
38.38
41.11
41.70
36.80
37.73
RAND_0.5
20.19
13.33
4.90
63.51
57.62
61.58
62.90
54.89
56.65
0.6
MEAN_0.1
30.62
20.10
7.14
51.70
47.43
49.95
50.77
44.79
46.06
MEAN 0.5
88.10
61.00
25.43
69.92
64.79
66.17
68.08
59.13
61.84
STARS 0.1
61.57
41.10
19.43
86.84
80.95
81.08
83.99
72.24
76.53
STARS 0.5
72.79
43.02
31.05
90.73
86.35
82.39
87.25
73.77
80.88
SAMPLING
9.00
3.83
1.52
13.56
12.26
13.29
13.43
11.86
12.08
RAND 001
11.74
6.36
2.57
30.53
27.76
29.53
29.81
26.40
26.81
RAND 0.5
19.07
12.83
4.88
59.58
53.91
58.25
58.89
51.94
52.97
0.4
MEAN 0.1
28.10
18.86
6.48
39.73
36.11
38.71
39.12
34.62
35.24
MEAN.0.5
89.36
62.40
27.05
66.51
61.23
63.89
65.28
57.18
59.34
STARS_0
62.69
41.60
19.74
87.14
81.17
81.82
84.46
73.15
77.13
STARS_0. 73.88
42.79
30.74
91.30
87.01
82.95
87.37
74.40
81.00
SAMPLING
8.17
2.69
1.17
4.41
4.16
4.70
4.71
4.12
4.14
RAND 0.1
10.74
5.17
2.05
24.70
22.24
24.06
24.42
21.39
21.89
RAND 0.5
18.21
12.29
4.57
58.35
52.85
57.31
58.03
51.22
52.36
0.2
MEAN 0.1
27.74
17.90
6.26
29.13
26.36
28.82
28.91
25.89
26.01
MEAN.0.5
91.10
63.00
27.60
62.30
$7.07
60.88
61.54
54.87
55.85
STARS 1.1
62.93
41.11
18.81
86.25
80.22
80.67
83.60
71.72
75.92
STARS.0. 0.5
73.40
41.81
29.95
91.35
87.13
82.95
87.61
74.28
81.32
Table shows the experimental results on the full LiveIdeaBench dataset with QWEN2.5- 3B-INS INSTRUCT across different sampling temperatures T {0.2, 0.4, 0.6, 0.8, 1.0). We compare vanilla temperature sampling against RAND and MEAN baselines, as well as our proposed method, STARS. The activation is extracted layer 20.
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Table 9: Experimental results full Live across different sampling temperatures on 1e-.55--B-Instrru All values are reported on scale with maximum score of 10. Best- performing methods are highlighted in bold.
T
Method
Originality 1
Feasibility t
Clarity
Fluency
Flexibility t
Avg.
SAMPLING
6.13
5.55
5.81
4.83
5.46
5.56
RAND 0.1
5.82
5,97
6.18
4.79
5.57
5.67
MEAN 0.1
5.90
5.92
6.20
4.93
5.67
5.73
STARS 10.1
5.88
5.91
6.19
5.12
5.65
5.75
STARS .0.5
L0.5
5.65
5.92
6.05
5.43
5.67
5.74
SAMPLING
6.09
5.56
5.80
4.55
5.39
5.48
RAND 001
5.76
6.01
6.20
4.65
5.54
5.63
0.8
MEAN 00.1
5,86
5.94
6.22
4.75
5.56
5.67
STARS .0.1
5.83
5.96
6.24
4.84
5,59
5.69
STARS _0.5
5.56
5.94
6.06
5.27
5,57
5.68
SAMPLING
6.06
5.57
5.80
4.19
5.30
5.38
RAND 101
5.72
5,99
6.21
4.83
5,58
5.67
0.6
MEAN_O.
5.81
5,93
6.22
4.30
5,45
5.54
STARS_0.
5.79
5.96
6.23
4.57
5.52
5.61
STARS_0.2
5.51
5.97
6.08
5.01
5.50
5.62
SAMPLING
5.76
5.97
6.20
3.59
5.27
5.36
RAND_O. 001
5.71
6.03
6.23
3.75
5.31
5,41
0.4
MEAN 0.1
5,83
5.92
6.24
3.68
5,30
5,39
STARS 00.1
5.75
6,01
6.22
4,30
5,45
5,55
STARS_O. .0.5
5.48
6.01
6.10
4.96
5,55
5.62
SAMPLING
5.74
6.02
6.23
2.61
5,03
5.13
RAND.0.1
5.72
6.04
6.24
3.27
5.19
5.29
0.2
MEAN 001
5,83
5.94
6.25
2.71
5.05
5.16
STARS 0.1
5.75
5.97
6.22
3.89
5.35
5.44
STARS 0.5
5.44
5.99
6.08
4.77
5.44
5.54
Moreover, we compare our STARS method against nucleus sampling (Holtzman et al. 2020).
widely-used technique for controlling output diversity by dynamically truncating the sampling dis- tribution. Table results on TESTEVAL using Gemma-1.112ba
temperature 0.8 with two
nucleus sampling configurations (top_p = 0.9 and 0.95). While nucleus sampling achieves 92-93% syntax correctness, its line coverage reaches only 4-5%. In contrast, STARS achieves 35% line coverage and 30-31% % branch coverage, though syntax correctness drops to 78-79%. These results demonstrate that traditional sampling techniques, such as nucleus sampling, which operate solely at the token-level distribution, verse program behaviors. Our activation-basel steering approach enables fundamentally different exploration capabilities by directly manipulating the model's internal representations.
insufficient for tasks that require systematic exploration of di-
Table Experimental results on TESTEVAL to compare with nucleus sampling. All numbers are in percentages. Best-performing methods are highlighted in bold.
Top-p
Method
Correctness
cov. 1 brauch
Line comtik 1 =
Branch couft
syntax
assert
line
k = 5
k
à
NUCLEUS SAMPLING STARS_0.5
93.45
1.19
1.12
4.15
3,70
4.10
4.14
3.59
3.66
78.81
30.47
NUCLEUS SAMPLING
92.57
1.21
1.19
5.09
4.88
5.07
5.09
4.73
4.84
0.95
STARS 0.5
79.36
4.40
3.74
35.69
31.65
34.13
34.99
29.47
30.65
B.5 IMPACT OF TEMPERATURE ON STARS PERFORMANCE
We evaluate STARS's robustness temperature variations by testing across range from 0.2 to 2.0 on TESTEVAL using Gemma-l 1.1-2a- (Table ITI. Across all temperatures, temperature sam- pling achieves high syntax correctness but minimal coverage. While STARS achieves 32-40% line
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coverage and 29-36% branch coverage, trading modest syntax correctness (78-80%) for substantial diversity.
Table 11: Experimental results on TESTEVAL across different sampling temperatures. All the numbers are in percentages. Best-performing methods are highlighted in bold.
Correctness
Overall cov. t
Line @o t k
Branch
cov@k &
T
Method
syntax
assert
line
branch
k ==
k= 1
SAMPLING
68.86
1.05
0.69
11.74
10.34
11.41
11.57
9.89
10.10
2.0
STARS.0.L
68.71
1.19
1.07
13.23
11.91
13.00
13.18
11.53
11.82
STARS_0.5
58.26
1.64
1.36
16.51
14.44
16.04
16.15
13.82
13.98
SAMPLING
75.76
0.95
0.83
12.45
11.49
12.38
12.43
11.31
11.45
1.8
STARS.0. 78.00
0.93
0.81
11.86
10.71
11.81
11.86
10.54
10.68
STARS
66.76
2.29
1.90
24.51
21.48
23.11
23.97
19.79
20.76
SAMPLING
82.21
1.10
0.95
9.30
8.68
9.21
9.29
8.51
8.63
1.6
STARS_0.1
80.90
1.26
1.07
11.84
10.85
11.72
11.79
10.54
10.71
STARS 0.5
71.26
2.48
2.17
27.23
24.11
25.26
26.57
21.91
23.34
SAMPLING
85.50
1.26
1.14
8.76
8.00
8.69
8.76
7.85
7.94
GEMMA- 11..-1111B--- 11-2B--17
1.4
STARS 0.1
83.64
1.31
13.91
13.31
13.82
13.90
13.04
13.25
STARS 0.5
74.69
2.88
2.52
28.94
25.91
27.02
28.13
23.47
24.87
SAMPLING
88.48
1.36
8.70
8.63
8.66
7.69
7.78
1.2
STARS.0.J 85.29
1.43
1.26
11.32
10.43
10.99
11.28
9.89
10.33
STARS.0.2
76.95
3.93
3.60
37.04
33.78
34.72
36.16
30.91
32.66
SAMPLING
89.90
1,02
0.95
5.36
5.06
5.32
5.35
4.98
5.04
1.0
STARS.0.
86.48
1,43
1.17
10.03
9.32
9.97
10,02
9.12
9.25
STARS-0.5
79.76
3.83
3.45
34.76
30.93
33.81
34.45
29.58
30.40
SAMPLING
91.79
1.19
0.00
3.54
3.37
3,46
3.51
3.21
3.30
0.8
STARS_0.1
87.83
1.45
1.12
10.77
9.88
10.64
10.72
9.65
9.78
STARS_0.5
79.43
3.79
3.45
32.96
30.02
31.70
32.59
28.23
29.43
SAMPLING
93.45
1.29
0.00
3.01
2.84
2.94
2.99
2.70
2.77
0.6
STARS.0.1
89.19
1.71
1.43
12.10
11.22
12.03
12.10
10.99
11.15
STARS 0.5
79.24
3,86
3.24
34.07
30,38
32.04
33.22
28.00
29.46
SAMPLING
95.40
1.36
1.26
2.33
2.31
2.29
2.32
2.20
2.27
0.4
STARS 001
90.43
1.43
8.44
STARS_0.5
80.31
4.38
4.00
40.03
35.96
38.39
39.36
33.73
35.07
SAMPLING
95.64
1.36
1.36
1.41
1.37
1.42
0.2
STARS 0.1
91.52
1.45
1.36
7.28
6.50
7.23
7.25
6.36
6.40
STARS 0.5
78.93
4.38
4.02
39.03
35.05
37.38
38.67
32.94
34.39
B.6 STATISTICAL ANALYSIS ACROSS DIFFERENT RANDOM SEEDS
Table 12 shows the results on the full LiveIdeaBence dataset across different sampling tempera- tures on Qwen2.5-3B. Random seeds are 1, 2, and 42. The results demonstrate that STARS exhibits strong stability across different random seeds, as evidenced by small standard deviations (mostly 0.1) l) for most metrics, and consistently outperforms baseline sampling across all temperature set- tings.
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Table 12: Experimental results on LiveIdeaBench across sampling temperatures. Values repre- sent mean + std over three random seeds (1, 2, 42) for Qwen2.5-3B. All values are reported on scale with a maximum score of 10. Best-performing methods are highlighted in bold.
T
Method
Ori. 1
Feas.
Clar. f
Flu.
Flex.
Avg. f
SAMPLING
6.01 + 0.03
5.90 + 0.06
6.15 + 0.06
5.27 + 0.12
5.73 + 0.06
5.81 + 0.05
1.0
STARS 0.1
6.01 + 0.05
5,85 + 0.04
6.17 0.02
5.47 1 0.18
5.79 + 0.07
5,86 1 0.04
STARS 0.5
5.75 i
5.88 1 0.04
6.05 0.03
5.75
5.73 1
5.83 = 0.04
SAMPLING
5.96 + 0.01
5,95 0.05
6.14 0.01
5.00
5.67 0.04
5.74 + 0.03
0.8
STARS 0.1
5.98 1:
5.91 1 0.07
6.21 0.03
5.27 0.07
5.73 1
5,82 T 0.02
STARS.0.5
5.70 1 0.07
5.86 1 0.08
6.06 1.06:1:000
5.69 0.31
5.69 1 0.08
5.80 1 0.08
SAMPLING
5,88 - 0.04
5.90 0.04
4.49 0.05
5.52
5,59 + 0.06
0.6
STARS_0.
5.93 + 0.05
5.89 + 0.03
6.21 0.01
5.59 + 0.07
5.69 = 0.07
STARS 0.5
5.60 + 0.07
5.88 + 0.03
6.03 + 0.03
5.30 +: 0.12
5.57
5.67 + 0.04
SAMPLING
5,84 + 0.04
5,98 + 0.08
6.21 + 0.02
+.50
+
5.36 + 0.00
0.4
STARS_0.1
5.86 + 0.03
5.97 + 0.11
6.21 + 0.09
4.67 + 0.15
5.58 0.00
5.66 + 0.01
STARS 0.5
5.62 + 0.04
5.94 + 0.04
6.08 0.00
0.28 5.18
5.57 0.06
5.68 + 0.07
SAMPLING
5.83 + 0.03
6.05 + 0.04
6.26 0.06
2.60 + 0.07
5.05 0.06
5.16 + 0.03
0.2
STARS_0.1
5.83 + 0.08
5.98 + 0.06
6.25 0.04
3.99 + 0.26
5.41 0.06
5.49 + 0.06
STARS 0.5
5.56 - 0.05
5.91 + 0.04
6.07 - 0.01
5.08 + 0.15
5.54 + 0.00
+ 0.02
B.7
NUMERICAL COMPARISON OF ALGORITHM 2 AND ALGORITHME 3
This section presents a numerical comparison between the Riemannian gradient descent with line- search (Algorithm 2 and the one-step update algorithm (Algorithm 3) for solving the steering- vector computation problem 2 across different dimensional settings. We run Algorithm 2 for 100 iterations with hyperparameters 50 times. For each run, we use the final loss of Algorithm 2 as a lower bound and compute the relative gap for both methods at every iteration. Figure [ reports reports the average results across different dimensional settings. We also provide the computation time in Table [3] Notably, Algorithm 3 reduces the relative optimality gap from 7% about 2% in a single update while using only about 3% of the runtime of Algorithm
0.2, 10 and 100. We repeat all experiments for
Table 13: Computational time of Algorithm and Algorithm 3 (in seconds), average over 50 runs.
(d, N)
Algorithm
Algorithm
(1024, 8)
1.206
0.015
(1024, 20)
3.299
0.024
(2048, 8)
1.575
0.058
(2048, 20)
4.322
0.103
B.8 RUN-TIME COMPARISON
Table 14 reports the average per-question runtime across different models and tasks. The results show that Algorithm 3 introduces only minor absolute overhead (no more than 2 seconds per ques- tion on average) and modest relative overhead most cases. Thus, the additional computational burden associated with Algorithm aapears negligible in practice.
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Table 14: Runtime per question of baseline and Algorithm (in seconds).
Task
Model
Temperature Sampling
Algorithm 3
GEMMA-1.1-2B-IT QWEN3-1.7B
4.53
4.63
Test Case Generation
9.01
9.97
QWEN2.5-3B-INSTER 1.-.MMA-..-3--ISTTTT-
3.02
5.01
Scientific Discovery
4.21
4.33
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Relative Objective Gap
Algorithm Algorithm
19 Iteration (log scale)
99
(a) d 1024, N 8
Relative Objective Gap
8
Algorithm 2
Algorithm 3
6-
2
0
9
19
49
99
Iteration (log scale)
(b) d 1024, 20
Relative Objective Gap
Algorithm 2
Algorithm 3
2
19 Iteration (log scale)
(c) d = 2048, N 8
Relative Objective Gap
Algorithm 2 Algorithm
2
9 19
49
99
Iteration (log scale)
(d) d 2048, 20
Figure 2: Optimality gap comparison of Algorithm and Algorithm 3 The y-axis is the average relative gap (in percentage) across runs, and the x-axis denotes the iteration number on a logarithmic scale. The standard deviation is very small relative to the average relative gap and is therefore not visually apparent.
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B.9 CASE STUDY
Here we provide generation example from IdiveddeeBench dataset with the keyword periodic table. The four paths of SAMPLING are
Develop interactive periodic table app that machine learning predict chemical reactions based on user input of elements. This tool would enhance chemistry education by providing real-time predictions and explanations, making complex concepts more accessi- ble and engaging for students.
Develop interactive periodic table app that allows users to visualize and manipulate chemical elements based on their electronic configurations. This tool would help educators and students better understand atomic structure and bonding principles through dynamic visualizations, potentially enhancing learning outcomes and engagement.
Develop an interactive periodic table app that uses Al to predict chemical reactions based user input of elements. This tool would enhance chemistry education by providing real- time reaction predictions, aiding students in understanding complex chemical interactions.
Develop an interactive periodic table app that uses augmented reality to visualize chemical elements in real-world environments. This tool would enhance learning by allowing users to see how elements interact in their surroundings, potentially sparking new insights into chemistry and materials science.
The four generations from STARS
Designing periodic table with quantum states as elements could revolutionize our under- standing of chemical reactions and materials science. By mapping quantum states along- side traditional elements, can predict new compounds and optimize energy storage solutions. This innovative approach would bridge the gap between quantum mechanics and chemistry, offering unprecedented insights into material properties and potential ap- plications in renewable energy technologies. This concept aims to explore the relationship between quantum states and chemical elements, potentially leading to breakthroughs in materials science and energy technology.
Develop an Al system that predicts chemical reactions using the periodic table as foun- dational dataset. This system would analyze patterns and relationships between elements based on their positions in the periodic table, aiming to improve reaction prediction accu- racy and potentially discover new chemical compounds.
A periodic table app that allows users to visualize and interact with elements in real-time, enhancing understanding of chemical properties and periodic trends. This idea involves developing an educational tool for chemistry enthusiasts and students. By creating an in- teractive periodic table app, users can explore the relationships between elements, their atomic structures, and how they relate to each other based on their positions in the table. This could include features like real-time element animations, comparisons of physical properties, and visual representations of chemical reactions involving different elements. The app would also allow for customization, enabling users to choose specific elements or groups to study in depth.
Utilizing the periodic table for personalized medicine by mapping genetic variations to cor- responding medical treatments. This idea aims to revolutionize personalized medicine by creating comprehensive map that links specific genetic variations with effective medi- cal treatments, based on the periodic table's 's structure of elements and their properties. This could significantly improve treatment efficacy and reduce side effects by tailoring therapies to individual genetic profiles.
To summarize, temperature sampling collapses to near-duplicates of an "interactive periodic-table app with AI reaction prediction/AR" theme. whereas STARS yields four conceptually distinct but still topical and grammatical ideas: (i) a quantum-state -uggtettegeeteedd periodic table for materials dis- covery; (ii) an AI reaction-prede prediction system grounded in periodic structure; (ii) an educational interactive table; and (iv) a (cross-domain) "periodic-table of personalized medicine" mapping geno- types to treatments.
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B.9.1 QUALITATIVE ANALYSIS OF SIMPLE PROMPT ' HOUSE'
Here we provide a simple example with the prompt 'A house'. We test with QWEN2.5-3B and temperature 0.6, and max_tokens 48.
With N 4, the temperature sampling gives
Sure! What would you like to know about house? Here are some common questions and information related to houses: 1. **Types of * There are many types of houses including detached houses (single-family homes), semi-detached houses
Sure! A house is a structure designed to provide shelter and living space for people or animals. Houses can vary greatly in size, style, and features depending their intended use, location, and the preferences of the homeowner. Here are some
Sure! I'd be happy to discuss various aspects of house. What specifically would you like to know about it? Here are a few topics we could explore: **Types of h Different styles such as modern, traditional,
Sure! A house is structure designed for human habitation that typically includes rooms for living, sleeping, and sometimes eating or entertaining. Houses can vary greatly in size, style, and features depending on their location, the culture of the region
With N = 8, the temperature sampling gives
A house is structure designed to provide shelter and living space for humans animals. It can vary greatly size, style, and location, from simple cottages in rural areas to luxurious mansions in urban centers. Houses typically consist of
A house a structure designed to provide shelter for people or animals. It can vary greatly in size, style, and location, but generally includes rooms for living, sleeping, cating, and sometimes storage. Houses can be made of various materials
A house is a structure designed for human habitation, typically consisting of one or more rooms and roof to protect from weather conditions. Houses can vary greatly in size, style, and location, depending on the region, culture, and personal
Sure, I can describe typical house for you! A house generally consists of several key com- ponents: **Structure**: This includes the foundation, walls, and roof. The foundation is what supports the entire structure and ensures it stays stable
Sure, I can provide information about house! Do you have specific question or detail in mind? Here are some common topics related to houses: 1. Types of houses (e.g., single- family homes, apartments, townhouses)
A house is a structure designed for human habitation. It typically includes rooms for living, sleeping, eating, and other activities. Houses can vary greatly in size, style, and location depending on the region and culture. They often include features
Sure, can provide information about house! Do you have any specific questions or details in mind? For example: 1. What type of house are you referring (e.g., single- family home, apartment, townhouse)? 2
A house is structure designed to provide shelter for people and their belongings. It can vary greatly in size, style, and location, but typically includes rooms for living, sleeping, eating, and other activities. Houses can be made from various
When N = 4, the STARS gives
A house is a structure designed for human habitation. It can vary greatly in size, style, and location depending on the region and culture. Houses typically include rooms for living, sleeping, eating, and working, as well as facilities like
Sure! I'd be happy to help describe house. Could you please provide more details? For example, what type of house (modern, traditional, etc.), where it located, or any other specific features you'd ' like
33

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Published as a conference paper at ICLR 2026
A house is structure building that serves dwelling for people other living beings. It can be made of various materials such as wood, brick, stone, and more recently, with the advancement of technology, also includes eco-friendly
Sure! Could you please provide more details about the house? What specific information would you like to know discuss about it? For example, do you want to know about its size, location, features, or history?
When N = 8, the STARS gives
How can I assist you about house? Do you want information houses, house prices, or perhaps you need help with something related to building or decorating a house? Feel free ask any specific questions you might have, and
Hello!
Sure! can describe house for you. Would you like details about a specific type of house general information? If you have particular question in mind, feel free to ask!
Sure! house is a structure where people live and can be made of various materials such wood, brick, even bamboo. It can also come in different sizes and styles. If you's d like more details about houses or have any
A house is object, structure designed provide shelter and comfort for individuals and families. Describing it more comprehensively, can share some information: A house typically consists of walls, floors, roof, and windows. It
A house is more than just a structure; it's often a home where memories and stories are made. can certainly help you generate ideas facts about houses if you're 'e interested in that topic. However, to be more specific, could
Let me describe house: A house is structure where people live. could be small house or a large mansion. The materials used to build a house can vary widely, from wood, brick, stone, and other natural materials
Sure, I can provide information about houses if you give me more details such as the loca- tion or type of house you'le interested in. For example, are you looking for specific type of house, or do you have any other questions?
Sure. A house is very good start for conversation, but I need more context to provide useful response. Could you clarify if this what you wanted to ask about? you have a specific question need assistance with something
We now increase C from 1.0 (as in the previous samples) to 3.0. The following are the 4-generation results of STARS with C 3.0. With stronger steering, STARS yields more diverse responses, including different starting modes.
Sure, I can describe a house for you. You want me to create a description or do something with the house? Or would you like provide more details on what kind of house you have in mind? If you give me more context
Here is some information about a house: Location: A house can be found in many places around the world, and you can choose to describe specific location if needed. Size: A house can vary in size. I can give you more
A house is an object which can refer to many things depending on the context. If you're thinking about house in the context of "house and buildings," then there could be many houses around the world. If you're
For example, let me describe house: A house has four walls made of brick and mortar, each wall facing different directions. The roof over the house is made of tiles or metal sheets. Houses can vary in size, shape, and
B.9.2 UNIFORM DISTRIBUTION RECOVERY IN DICE ROLLING
We the dice experiment with the following prompt:
34

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Published as a conference paper at ICLR 2026
Roll A Die Prompt
Roll die.
Output ONLY one integer from to 6. Do not add any explanation text.
Output format: < number
For each generation, the output logits are passed through softmax to obtain probabilities, and the probability P(token = d) for each digit is recorded by indexing the corresponding token ID. The ex- periment ran 10 iterations, each with 20 rolls (200 total samples). Table presentts the probability distributions obtained from temperature sampling and STARS. The KI. column provides the KI. divergence between the sampled distribution and the uniform distribution. Under standard tempera- ture sampling, the model concentrates most of its probability mass on just two outcomes: and 3. In contrast, increase the STARS strength from 7, the probability mass becomes much more spread out across all six faces, although it does not become fully uniform. We believe this behavior warrants further investigation.
Table 15: Probability distributions across dice outcomes (1-6) under temperature sampling and STARS with varying C. KI. divergence measures the distance from the uniform distribution (lower better).
1
2
3
4
5
KL
Sampling
0,000056
0.405526
0.450347
0,032215
0.111854
0.000001
0.71
STARS 3.0
0.103587
0.452858
0.366496
0.023672
0.046169
0.007218
0.56
STARS 5.0
0.380046
0.245687
00108884
0.111879
0.103058
0.050445
0.20
STARS_7A 7.0
0.355143
0.130725
0.192331
0.184511
0.103763
0.033526
0.18
35


Reviews:
[{"review_id": "52VqBHwork", "reviewer": "Reviewer_D4n2", "summary": "The paper proposes an inference-time method (STAR) to diversify LLM generation by steering hidden activations from multiple concurrent decoding runs such that they maximise the geometric volume spanned by them. \n\nThe authors present a Riemannian gradient-descent solution with convergence guarantees to find the optimal steering vectors according to this objective. Based on the insight that it is too computationally heavy for acceptable latency, they propose a lightweight one-step update as approximation for real-time decoding. \n\nThe empirical evaluation is on TESTEVAL (test-case generation) using Gemma-1.1-2b-it and QWEN3-1.7B, as well as on LiveIdeaBench (scientific idea generation) using QWEN2.5-3B-INSTRUCT. STAR is compared to temperature sampling as a baseline for temperatures varying between 0.2 and 1.0. STAR significantly outperforms this baseline in these evaluations.", "questions": "- Line 61: “[…] and their benefits can be fragile across domains” -> Source?\n- Line 66: “If multiple runs occupy nearly the same region in this space, surface-level stochasticity has little impact” -> Source? \n- Figure 1: How do the particularities of ending the generations work? Does one of the N candidate sequences have to output a EOS token or all of them simultaneously or is a sequence outputting the EOS token simply removed from the algorithm until every sequence eventually outputs an EOS token?\n- Section 3.1: Have the authors tried to other locations for their intervention, e.g, the residual stream that Panickssery et al. (2023) intervene on?\n- Section 3.2: How is the pre-determined layer $l$ chosen?\n- Algorithm 1, requirements: what is $\\mathbb{R}_{++}$? \n- Section 4: why do the authors resort to riemannian gradient descent if the retraction step is still necessary to ensure $V_{k+1}$ is feasible? Could you not do some sort of projected gradient descent using the Euclidean gradient directly, i.e. moving a step along the Euclidean gradient and then projecting back to the manifold of feasible solutions?\n- Theorem 1: is a Riemannian gradient of 0 a necessary condition for a minimum of the objective function? Maybe a small comment on this for readers unfamiliar with Riemannian gradient descent could be helpful", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "- The paper is very well and pedagogically written\n- The method is an inference-time method and as such training free, saving compute overhead and making it accessible \n- The theoretical part of the paper is strong, from guaranteeing existence of a solution to providing an algorithm for finding the optimal solution and deriving convergence guarantees\n- The paper proposes a practical algorithm to approximate the algorithm guaranteed to find the optimal solution with a more lightweight approach for real-time low latency decoding\n- Compared to the baseline that is included in the paper, the proposed method performs strongly\n\nMinor:\n- I appreciate the comment in lines 214 to 216 on the realism of the assumption necessary in Proposition 1.", "weaknesses": "- One key element of the method, namely the constraint on $V$ to be an orthogonal matrix, is not motivated too well. Only in line 201 it briefly says “To encourage diversity between different generations, we require the steering vectors to be orthogonal with each other, i.e. […]”.  But why does this constraint ensure diversity? Is it not rather the orthogonality of the columns of the resulting $H + V$ that would maximise diversity and the objective? \n- In the experiments, the comparison to baselines is weak. \n    - The only baseline the proposed method is compared against in the experiments is temperature sampling. At the bare minimum, the 2 baselines/ablations that I would need to see to lean towards accept would be 1) adding random vectors of same magnitude $\\alpha$ (how much better than random additions are the directions you find?) 2) $v_j=(h_j - M_j * mean(H, dim=1))*s$, where $mean$ across $dim=1$ corresponds to mean across the $N$ vectors and $M_j$ is $-1$ when $H - mean$ is negative in said dimension and $+1$ when $H-mean$ is positive, while $s$ is scaling $v_j$ to be of magnitude $\\alpha$ (how much better than just pushing away from the mean is the proposed steering direction).   Ideally, it would also be good to see comparisons against other methods mentioned in lines 52-62, in particular also to a training based method among those mentioned in lines 59 - 60.  Another good baseline would be to compare to the objective of maximising variance between vectors $h_j$ rather than the volume. \n    - The only baseline that the model is compared against (temperature sampling) is only exploring temperatures between 0.2 and 1.0, even though in Table 1 for QWEN3-1.7B (all metrics) and in Table 2 for QWEN2.5-3B-INSTRUCT (all metrics except feasibility) have a positive trend for increasing temperature (in absolute terms and in relative terms to the proposed method). Higher temperatures should thus be tried to see if similar performance can be achieved through this. \n- Also, in the experiments only one model is evaluated for Section 6.2. As someone having worked on activation steering myself I have found that the effectiveness of steering methods can vary a lot by model family so would always advise to evaluate on at least 2 different models from different model families. Furthermore, weirdly enough the one model evaluated in Section 6.2 is different from the 2 models evaluated in Section 6.1, raising doubts about what the results would have looked like for the models evaluated on the other task. \n- Given that the authors put an emphasis on the need for the one-step update for smaller latency, a comparison of added latency in Section 6 would be good to characterise the trade-off between improved performance and (latency) cost of the proposed method over simple baselines such as temperature sampling\n\nMinor:\n- I would suggest using a different acronym than STAR, since there is a fairly well-known paper aiming to improve LLM generations with an almost identical acronym (*STaR: Self-Taught Reasoner - Bootstrapping Reasoning With Reasoning* by Zelikman et al. (2022)) \n- Typo: Line 217 is basically a repetition of lines 215/216 \n- Line 266/267: a small explanation why sufficient decrease prevents rank-deficiency would be helpful\n- More depth could be added to the discussion in Section 6.1, e.g. by commenting on the performance of the baseline (temperature sampling), e.g. why it catastrophically fails at the task for Gemma-1.1-2b-it (incl. why higher temperature does not increase coverage but rather decreases it after T=0.6) or by explaining the u-shaped performance of the proposed method wrt temperature for most metrics for Gemma-1.1-2b-it", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GSFp51WrF6", "reviewer": "Reviewer_BKyG", "summary": "The paper explores methods for steering large language model activations at inference time, with the goal of diversifying reasoning trajectories. \nThe work is situated within the area of activation and steering vectors for LLMs. \nThe authors propose adding a learned steering vector to the output of an attention head, \nwith the vector being initialized and subsequently updated to minimize the objective described in Equation 2.\n\nIn Section 4, the paper introduces an initialization approach based on singular value decomposition (SVD), \nfollowed by refinement using Riemannian gradient descent. \nThis combination allows the model to identify meaningful steering directions while maintaining stability in optimization. \nIn Section 5, the authors further propose a simplified, single-step variant of the method, aimed at improving computational efficiency.\n\nThe experimental results compare the proposed approach against a single diffusion-based baseline, \nalthough the specific diffusion method used is not clearly specified. \nOverall, the paper presents an interesting and technically sound contribution to the growing field of activation steering, \nthough the evaluation setup could benefit from additional baselines and clarity regarding the comparison method.", "questions": "- Could the authors include comparisons with other existing steering methods to better contextualize their approach? \n- Additionally, have the authors considered testing their method against alternative diffusion-based approaches to provide a more complete evaluation?", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper addresses an important and timely topic—enhancing diversity in large language model reasoning. \n- The idea of using training-free steering vectors is particularly interesting, as it offers a lightweight and potentially generalizable approach to influencing model behavior without fine-tuning.", "weaknesses": "- The evaluation is somewhat limited, relying primarily on a single “sampling” method as the baseline. Comparing only against one approach makes it difficult to assess the broader effectiveness of the proposed method. \n- Additionally, the reported results, while suggestive, are not particularly strong or conclusive. A more comprehensive experimental section would strengthen the paper’s empirical claims.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5SIPClFOkF", "reviewer": "Reviewer_ZBFD", "summary": "This work proposes a method to increase diversity in LLM generations. The proposed STAR approach optimizes a steering vector at inference time, trying to maximize the volume spanned by $N$ tokens at the same timestep. By repeating this procedure at each decoding step, STAR obtains diverse sequences which still preserve good quality. The authors propose to obtain the steering vectors by means of Riemannian optimization, also deriving a fast 1-step approach alternative. The experiments show that STAR is more effective than standard temperature-based sampling.", "questions": "**Q1:** One aspect that I would like to discuss with the authors is the fact that STAR tries to maximize the volume spanned by the activation vectors of the $\\tau$-th token of $N$ sequences. Isn't this implying somehow that there is a temporal correspondence across sentences? Otherwise, always pushing for orthogonality and max volume at each timestep might lead to sub-optimal quality sequences, right? Do the authors have ideas on how to modify STAR so it takes the whole _trajectory_ (tokens for $t\\leq \\tau$) into account. Could this lead to more meaningful, and at the same time diverse, sequences?\n\n**Q2:** L201: _To encourage diversity between different generations, we require the steering vectors to be orthogonal with each other_\n\nIs this truly required? Do we need orthogonal tokens at each timestep to ensure diversity? For a bias related example, one could argue that diverse sentences are those that convey the same message, but using all possible choices of gender. This does not imply that all tokens must be orthogonal. Also, why would gender options be orthogonal? In my opinion, this assertion is very strong. Very interested in knowing the authors' arguments.\n\n**Q3:** L203: _A too big $\\alpha$ may break the meaningful information in $h_i$, leading to generation collapse._ \n\nUsing an unbounded scaling parameter for vector-based addition has been shown to not respect the activation distributions. Recent work by  [Rodriguez et al. ICLR 2025](https://arxiv.org/abs/2410.23054v1) showed that respecting distributions is key, allowing interpolation between original and intervened (linearly mapped) activations, thus having a bounded scaling between 0 and 1. In further work,  [Rodriguez et al. NeurIPS 2025](https://arxiv.org/abs/2503.10679) show how to do that with gradient descent. I suggest considering a steering approach similar to the provided works, effectively avoiding the scaling problem. I completely understand that this is a fundamental change in your work, so I am not directly asking for this, but feel the authors should at least discuss on how these approaches could improve their work.\n\n**Q4:** L206: In Eq.2 $\\alpha$ is a parameter set by the user? It might be read as it is optimized jointly with $v_i$, which is not after reading the rest of the paper. Maybe I would clarify how $\\alpha$ is chosen upfront. \nUpdate: after reading the full manuscript, I see $\\alpha$ is implicitly set by the user through $C$. I still believe that this aspect should be stated earlier on in the manuscript. \n\n**Q5:** L224: In algorithm 1, $\\alpha$ is required but never used. Consider removing it, or fixing the algorithm if $\\alpha$ was not included by mistake.\n\n**Q6:** The results in Table 2 show very close numbers across methods. Could the authors provide statistical significance for this experiment? Probably the standard deviation across 3-4 runs would be enough. This would help the reader understand the real benefit of each method.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "**S1:** I believe that studying diversity is somehow lacking in the current research efforts. The applications of \"diversity increase\" are underexplored, and could lead to improvements of generative models both at inference (reduce bias, increase creativity, etc.) and at training time. This work tackles the topic at its core, which is refreshing.\n\n**S2:** This work proposes to use steering to induce diversity. To the best of my knowledge, this is an unexplored application of steering, which I deeply appreciated. The proposed approach trying to maximize the volume, although having practical limitations, is of interest to the community and can spark new research on this topic. \n\n**S3:** The approach using Riemannian optimization to maximize volume while preserving the manifold of intervention vectors is sensible and well explained. I have suggested some clarifications, but overall I am confident about the proposed approach. \n\n**S4:** The text is well written and easy to follow. The mathematical notation is clean.", "weaknesses": "**W1:** STAR applies to a single layer of the model by construction. This is a fundamental drawback in my opinion. First, the best layer must be found in advance, as the authors have done in Tables 3,4. Second, previous work has shown that intervening carefully on all layers is more effective ([Rodriguez et al. NeurIPS 2025](https://arxiv.org/abs/2503.10679)). Additionally, while being a common choice, intervening at the output of attention layers is less effective than intervening on the residual path. The latter is typically the choice when the intervention is only applied to 1 layer. I believe more discussion about the layer choice (beyond the experiment in Tables 3,4) is required, pointing out the pros/cons of the choices made. \n\nNB. Tables 3,4 are not referenced in the text.\n\n**W2:** STAR uses the same scaling for all tokens. I understand the underlying reason for that, scaling being hard to tune or adapt per token. It is true that the scaling is \"normalized\" by $||H||_2^2$ but still _fixed_ and not adaptive to what the generation requires. Indeed, it might happen that tokens don't need to deviate from each other at a specific time step. Have the authors considered some adaptive steering such as MERA ([Hedström et al., ICML 2025](https://arxiv.org/abs/2510.13290))? \n\nAdditionally, could $\\alpha$ be optimized together with $v_i$ in Eq. 2?\n\n**W3:** No samples are provided. As a reader, I was expecting a subjective analysis of the sentences obtained using sampling temperature, and using STAR. Also how do these sentences differ when $N$ increases. How different, and at the same time correct, they are from a subjective point of view? I encourage the authors to share generation samples, and include them in the manuscript. For example, I am really curious to see what are the generations for simple prompts like _\"A house\"_ while increasing $N$.\n\n**W4:** I encourage the authors to include generation timings of STAR compared to $N$ generations using standard temperature-based sampling. I believe this is a critical aspect that has been overlooked in the manuscript.\n\n**W5:** The experimental section is limited. I understand that it is hard to find suitable experiments for diversity (given how underexplored this area is). I may suggest leveraging the comment on L48: _For safety and alignment, a lack of diversity prevents us from discovering varied failure modes_. I believe this is an important aspect to tackle with methods like STAR. I think this work would benefit from having an experiment showing how increasing diversity reduces bias, or shows better performance for minority groups (eg. talks about different genders with more parity).\n\n---\n\n### **Recommendations:**\n\n> Please take the following as just recommendations, feel free to comment on them or pushback if you feel they are not justified.\n\n**R1:** I suggest the authors to emphasize why Riemannian optimization is useful in this setting, and which is the manifold we are trying to _preserve_ while optimizing. The latter is defined by the the constraing $V^{\\top}V=\\alpha I$, which might not be evident as the text is now. For the former, I defer to the authors' to complete, but probably commenting on Riemannian opt. being faster and better behaved than using Euclidean gradients + projections in this setting.\n\n**R2:** I also suggest the authors to run at least some test applying STAR at the residual path of some layer of choice, and compare to applying STAR at the attention output. With residual path I mean right after the sum with the residual connection. This is usually easier to capture as the input to the next Transformer block.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "t7KUSY1fz1", "reviewer": "Reviewer_a1Ka", "summary": "This paper studies batch inference with the goal of maximizing diversity amongst the generated responses. The author's propose a training-free inference-time intervation method, cointed STAR, which modifies intermediate activations. Namely, for each of the N generations, STAR collects the hidden activations across each of the N generations and solves optimization problem to learn a set of perturbation vectors that when added to the activations, maximizes the volume of the corresponding set of N hidden states. The authors show that this optimization problem can be solved using a Riemannian gradient descent algorithm and provide convergence guarantees, along with a more loghtweight one-step update algorithm. Experiments using the one-step optimization algorithm across test-case generation and scientific discovery benchmarks show that this method outperforms standard sampling methods.", "questions": "See Weaknesses.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper is well-written and organized. The authors do a good job of motivating the problem at hand, and I believe the problem studied is of interest to the ML community\n- The authors complement their experimental results with theory, proving the feasibility of their proposed optimization procedure.", "weaknesses": "My biggest issue with this paper is its lack of clarity. I summarize my concerns below. \n\n- **Some of the algorithmic details were unclear to me**. For a particular layer, are the hidden states for all the tokens modified or just one token? Is steering done only once on a pre-determined layer (with the modified hidden states propagated forward)? Or, do you do steering at several different layers? How do you pick the layers for steering? \n\n- **The evaluation metrics are unclear to me**. In particular, the method proposed by the authors produces a collection of N responses. However, it seems to me that the metrics in Tables 1 and 2 are for specific responses? If so, how are these metrics aggregated across the N responses and shouldn't you be using a metric that evaluates the collection of N responses as a whole? Overall, the authors should make very clear whether the metrics in Table 1 and 2 are batch- or  individual-level. In my opinion, this is the biggest weakness, because the authors should be using a batch-level diversity metric (since that's what their method optimizes for) along with an individual-level quality metric. \n\nMy other concern is with the lack of comparisons to the related methods the authors summarize in Section 2. It would be great if the authors can explain why they didn't compare their method with these ones (and include this explanation in the final version).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)