Taming Polysemanticity in LLMs: Theory-Grounded Feature Recovery via Sparse Autoencoders
Authors: Siyu Chen
Heejune Sheen
Xuyuan Xiong
Tianhao Wang
Zhuoran Yang
Venue: ICLR 2026 Poster (2026)
Keywords: sparse autoencoder; training dynamics; superposition; feature learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We present a theoretically grounded sparse‐autoencoder training algorithm that provably recovers underlying features while outperforming existing benchmark methods

Abstract: We study the challenge of achieving theoretically grounded feature recovery using Sparse Autoencoders (SAEs) for the interpretation of Large Language Models. 
Existing SAE training algorithms often lack rigorous mathematical guarantees and suffer from practical limitations such as hyperparameter sensitivity and instability. We rethink this problem from the perspective of neuron activation frequencies, and through controlled experiments, we identify a striking phenomenon we term neuron resonance: neurons reliably learn monosemantic features when their activation frequency matches the feature's occurrence frequency in the data.
Building on this finding, we introduce a new SAE training algorithm based on ``bias adaptation'',  a technique that adaptively adjusts neural network bias parameters to ensure appropriate activation sparsity. We theoretically prove that this algorithm correctly recovers all monosemantic features when input data is sampled from our proposed statistical model. Furthermore, we develop an improved empirical variant, Group Bias Adaptation (GBA), and demonstrate its superior performance against benchmark methods when applied to LLMs with up to 2 billion parameters. This work represents a foundational step in demystifying SAE training by providing the first SAE algorithm with theoretical recovery guarantees and practical effectiveness for LLM interpretation.

Published as a conference paper at ICLR 2026
TAMING POLYSEMANTICITY IN LLMs: THEORY- GROUNDED FEATURE RECOVERY VIA SPARSE AU- TOENCODERS
Siyu Chen Heejune Sheen"
Xuyuan Xiong
Tianhao Wang
Zhuoran Yang*
Department of Statistics and Data Science Yale University
i Antai College of Economics and Management, Shanghai Jiao Tong University
$ Toyota Technological Institute at Chicago
(siyu. (siyu.chen.c sc3226, heejune sheen, zhuoran yang Sbyale..dd xxy20210sjjuu edu.en tianhao wangettic..
ABSTRACT
We study the challenge of achieving theoretically grounded feature recovery us- ing Sparse Autoencoders (SAEs) for the interpretation of Large Language Mod- els. Existing SAE training algorithms often lack rigorous mathematical guaran- tees and suffer from practical limitations such as hyperparameter sensitivity and instability. We rethink this problem from the perspective of neuron activation fre- quencies, and through controlled experiments, we identify a striking phenomenon we term neuron resonance: neurons reliably learn monosemantic features when their activation frequency matches the feature's occurrence frequency in the data. Building on this finding, we introduce a new SAE training algorithm based on bias adaptation, a technique that adaptively adjusts neural network bias param- eters ensure appropriate activation sparsity. We theoretically prove that this algorithm correctly recovers all monosemantic features when input data is sam- pled from our proposed statistical model. Furthermore, we develop an improved empirical variant, Group Bias Adaptation (GBA), and demonstrate its superior performance against benchmark methods when applied to LLMs with up to bil- lion parameters. This work represents foundational step in demystifying SAE training by providing the first SAE algorithm with thcoretical recovery guaran- tees and practical effectiveness for LLM interpretation. Code is made available _BAA.
https://github.com
INTRODUCTION
Large Language Models (LLMs) have demonstrated remarkable capabilities across diverse tasks. is found that LLMS encode vast amounts of information by superposition (Lu et al., 2024; Xiong et al., 2024; Elhage et al., 2022; Bengio et al., 2013) (013)-accking multiple concepts into the same weight or activation directions maximize capacity. This efficiency comes at a cost: individual neurons (or activation vectors) become polysemantic (Scherlis et al., 2022), meaning they respond to several monosemantic features at once, making interpretation challenging.
Dictionary learning has recently been applied to disentangle polysemantic LLM representations, with Sparse Autoencoders (SAEs) emerging as leading approach (Cunningham et al., 2023; Bricken et al., 2023; Templeton et al., 2024; Gao et al., 2024; Rajamanoharan et al., 2024b). SAE encodes an LLM's internal activation x E IRdd into a high-dimensional, sparse code (z) E R. with M > d, then decodes f (2) x By enforcing sparsity-so only a few components of z are nonzero-each active neuron ideally reflects a single interpretable feature. Empirically, SAEs have revealed such monosemantic features in models like Pythia-70M (Cunningham et al., 2023) and Claude 3.5 Sonnet (Templeton et al., 2024).
Despite these promising empirical advances, existing studies SAEs still lack rigorous guaran- tees regarding feature recovery. Popular SAE training algorithms, which typically minimize a loss function of the form [(x, f) ||ær 1 X R(2) where R(z) is sparsity regularizer, involve

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Figure I: Illustration SAE architecture and neuron resonance (left) and a demo neuron (right) learned using GBA. Left: SAE architecture and the resonance phenomenon cecnenoonnnnuoons successfully learn features when their activation frequency p matches the feature occurrence frequency f. Right: that activates for the concept "class".
hyperparameters like A. For instance, methods employing L regularization for R(2) = || and E {0, 1}. Other strong candidates include the TopK activation method (Makhzani & Frey, 2013; Gao et al., 2024) and gated SAE (Rajamanoharan et al., 2024a). However, these methods exhibit specific drawbacks. For example, L1 regularization is sensitive in the hyperparameter À and often leads to activation shrinkage, where the magnitudes of the learned features are systematically un- derestimated (Tibshirani, 1996). TopK approaches, while enforcing hard sparsity constraint, often overlook the fact that different inputs may require varying numbers of active features, and also suffer from inconsistency across random seeds (Paulo & Belrose, 2025), which means that they yield sets of learned features that are sensitive to the random initialization (Paulo & Belrose, 2025).
This landscape motivates to address fundamental questions concerning the reliability and theo- retical underpinnings of feature recovery with SAES:
What enables neurons to successfully recover features? Can design training algorithm that provably recovers features while being practical for modern L.Ms??
Let us consider what makes SAE training successful. In ideally trained SAE, each neuron learns a distinct monosemantic feature and activates precisely when that feature appears in the input. Thus, the neuron will have an activation frequency p-the fraction of inputs for which activates that matches the occurrence frequency f of its corresponding feature in the data. This observation raises natural question: if we control neurons activate with frequency p matching a feature's 's frequuecy f. will they reliably learn that feature? Moreover, since we typically cannot know feature's ' frequency ff advance, what conditions on a neuron's ' activation frequency p enable it to learn feature with unknown frequency f?
To investigate these questions, we conducted controlled experiments on synthetic data with known feature frequencies. Our experiments reveal striking phenomenon we term neuron resonance: Neurons reliably recover features when their activation rate matches the feature's 's frequency in the data. Like radio tuning to specific frequency for clear signal, SAE neurons must "resonate" the right activation rate to capture their target features. Importantly, our theory shows that successful learning requires only that p fall within resonance band around f, not exact match. This flexibility enables practical feature discovery: even without knowing f in advance, we can recover features by ensuring neurons' activation frequencies cover a diverse range.
The resonance principle reveals fundamental yet intuitive correspondence: common features re- quire frequently active neurons, while rare features need selective, infrequently-frrnng neurons. Based on this, we develop Group Bias Adaptation (GBA), an algorithm that creates multiple groups of neurons with geometrically-spacee target activation frequencies (e.g., 10%, 5%..... Each neuron computes 2m o ((w]]( (I: bpre) + bm), where Wmm E IRd is the weight vector, bm E IR is the bias, òpre E Rd is the shared pre-bias, and 6 is the activation function (e.g., ReLU). GBA dynamically adjusts these biases match the target frequencies: decreasing bias if fires too fre- quently to increase selectivity, and increasing bias when rarely fires to encourage activation. The direct frequency control across diverse activation ranges ensures comprehensive feature recovery while cireumventing the hyperparameter sensitivity and dead neuron problems in existing methods.
We thus provide affirmative answers to both fundamental questions posed earlier through the follow- ing contributions. First, we discover and investigate the neuron resonance phenomenon, revealing the principle that governs successful feature learning in SAEs from the view neuron activation frequeney. Theoretically. we justify the resonance principle by rigorously showing that neurons with appropriate activation frequencies can provably recover all monosemantic features when data follows well-defined statistical model. To our best knowledge, this provides the first dynamical
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analysis and learning guarantee for SAE training. Empirically, scale GBA to Qwen2.5-1.5p and Gemma2-22 2B on Pile datasets and demonstrate its superiority: (i) achieving the Pareto frontier in reconstruction-spariity tradcoff comparable to TopK, (ii) significantly higher cross-seed consis- lency than TopK, (iii) competitive performance on SAEBench (Karvonen et al., 2025) interpretabil- ity metrics while maintaining 99% neuron aliveness, and (iv) remarkable consistency and robustness through ablation study, requiring only simple hyperparameter rules without dataset-specific tuning.
Related works. Our work builds on long history of sparse dictionary learning (Olshausen & Field, 1996; Spielman et al., 2012; Bruckstein et al., 2009). Recently, SAEs have been increasingly used for LLM interpretation. revealing monosemantic features and eircuit patterns (Bricken al., 2023: Dunefiky et al., 2024; Ameisen et al., 2025), cross-run feature stability (Papadimitriou et al., 2025), and behavior steering (Shu et al., 2025). A detailed discussion deferred to SA.
Notations. Let R+ denote the set of non-negative real numbers. We use standard Big-O and small- notation and use 2 to hide polylog(n) factor for sufficiently large n. We denote by [n] the set (1, .....,,,,.... .n. for positive integer
PRELIMINARIES
A model for feature recovery. As motivating example, consider how model processes "The detective found a muddy footprint near the broken window." The internal representation mixes monosemantie features:
x= == h1 + 122 +..., where "muddy footprint",
"broken window".
Here, h1, h2 2 O are nonnegative coefficients, where negative values would imply contradictory concepts. We formalize this as follows: Let V E Rx be feature matrix where each row i, is a monosemantic feature. For data points, each row Ie of data matrix X E R Nxdd is an 8-sparse mixture of features with nonnegative coefficients collected in E NRx,,
X HV E Nxd.
(2.1)
We focus on the superposition regime where n > d, meaning features are necessarily linearly de- pendent (Arora et al.. 2018; Olah et al., 2020; Elhage al., 2022). Our goal to recover V from X without knowing H-a a common challenge in model interpretation.
SAE architecture. We follow Gao et al. (2024); Cunningham et al. (2023) and use threc-layer neural network for SAE with tied encoding and decoding weights. Let M be the width of the SAE, and for input x E Rdd its output is
[x:; (x; = Σ mm1 amm Wom ( w. bpon)) + bm) + Upre-
(2.2)
where θ = [Wm, am, dm, bpre J-M denotes the trainable parameters. For each neuron m E [M]: Wm E RR is the tied encoder/decoder weight, an E IR is the output scale, bm E R is the bias, and bpre E IRd centers the input. The pre-activation is Yn = w bpre) + bm- and neuron m activated when yo. > O. When activated, neuron contributes 0. Wm -6(ym) ) to the reconstruction, where the tied weight Wma serves as both detector (encoder) and reconstructor (decoder).
Existing SAE training methods. Prior methods minimize reconstruction loss ( Q) M (x; θ) æ11. with sparsity constraints. L1 SAE adds penalty > Emm. |/wm///2 ((ym) but suffers from shrinkage bias (Tibshirani, 1996). TopK SAE (Makhzani & Frey, 2013; Gao et al., 2024) retains only K largest activations but exhibits extreme seed sensitivity (Paulo & Belrose, 2025). Both methods have significant limitations detailed in the introduction.
RETHINKING How SAES LEARN: NEURON RESONANCE
We rethink SAE training from the perspective of neuron activation frequency: how should a neuron's activation frequency p relate to feature's occurrence frequency f for reliable feature learning?
To investigate this, we conducted controlled experiments using synthetic data generated from (2.1). We construct s-sparse coefficient matrices H, which have uniform feature occurrence frequency f = nn. The selected features can be viewed as the "concepts" " in each data point, and f reflects how often each concept appears in the data. We generate the feature matrix V by randomly sampling
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vectors from the unit sphere in R mimicking independent features in nigh-dimensional space. To study the relationship between neuron activation frequencies p and feature occurrence frequencies f. we train set of SAEs while systematically controlling p through dynamic bias adaptation. More details can be found in $D.1 and the bias adaptation can be found in $4. We measure feature learning success using the Feature Recovery Rate (FRR), which quantifies the percentage of features learned by at least one FAR Heatmap (a VA:nn Thggelationhh between d. and FRR-i shown in Figure 2.
(see
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Figure 2: Feature Recovery Rate (FRR) for varying activation frequencies and dimensions d. Left: Light superposition (d > vn, 128). Right: Heavy superposition (d < vn, n 65536). In the left panel, the high-FRR region forms a wide band in p above f. In the right panel, it collapses into narrowing diagonal band, where must track tightly. The contrast between wide and narrow resonance bands provides empirical evidence for the theoretical phase transition at d z /ne showing that the same feature frequency. f yields different learning tolerances depending on superposition level. Here, μ stands for 10 bb and m stands for 10-3.
Neuron resonance phenomenon. The results reveal a striking pattern we term neuron resonance: neurons successfully learn features when their activation frequency falls within a specific band around the feature's 's occurrence frequency f. The width of this resonance band depends critically on the degree of superposition. In heavy superposition where d < /n (right panel), the band is narrow, requiring p to closely match f. light superposition where > vn (left panel), particularly when d >> n, the band widens significantly. This widening is intuitive: when d > features become nearly orthogonal and easier to separate, allowing neurons with imperfect frequency matching to still converge to individual features due to reduced interference. Since real-world data typically exhibits heavy superposition (n > d), we expect the resonance phenomenon to persist with a narrow band similar to the right panel of Figure 2. Here, we set 3, M 512 (left) and M 262k (right).
In $6, we theoretically characterize a feasible activation frequency range for faithful feature recovery. A feature with occurrence frequency f is learned when neurons' activation frequency p lies in the resonance band f S S.S min [V], d } (up to logarithmic factors). With f = s/n, a phase transition occurs d < /n: light superposition (d > /n) yields wider band p X vT, while heavy superposition (d < /n) constrains it to S df. narrowing as d' decreases. This phase transition and narrowing band in heavy superposition perfectly matches our empirical findings in Figure 2.
Motivation for frequency-awan training. Existing methods cannot directly control neuron acti- vation frequencies. They achieve this by imposing sparsity constraints: L2 SAE uses penalty terms while TopK SAE limits the number of active neurons per input. The resonance principle indicates that optimal feature learning requires aligning neuron activation frequencies with the natural fea- ture frequency distribution--ragiingg -raginng from fighfrreueecc features (e.g.. common function words) to low-frequency features (e.g., domain-specific terminology). This insight motivates our Group Bias Adaptation algorithm in the next section.
4 ALGORITHM: GROUP BIAS ADAPTATION
From resonance principle to algorithm design. The neuron resonance phenomenon ($3) reveals that successful feature learning requires matching neuron activation frequencies to feature occur- rence rates. This insight motivates our Group Bias Adaptation (GBA) algorithm, which operational- izes the resonance principle through two key design choices:
1. Direct frequency control: Instead of relying on indirect penalties (L₂) or fixed constraints (TopK), we directly control each neuron's 's activation frequency through adaptive bias adjustment. When a neuron fires too frequently, we decrease its bias to make it more selective; when it rarely fires, increase its bias make more active.
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2. Multiple frequency bands: We partition neurons into groups with geometrically-spaeed target activation frequencies (e.g.. 10%, 5%, 2.3%, ...), creating spectrum of "resonance bands" that automatically covers the diverse feature frequency mage-from common features to rare, spe- cialized ones. Then we use the previously described adaptive bias adjustment within each group maintain the desired activation frequency.
These design principles ensure: (i) sufficient sparsity for interpretability while avoiding dead neu- rons by controlling the lowest activation frequencies, and (ii) smooth training dynamics via adaptive bias adjustment while maintaining efficient control. The complete algorithm is presented below.
G
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Activation Threshold (-h)
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Data
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Neuron Groups
Activation Frequency
Figure 3: Illustration of Group Bias Adaptation (GBA). Left: Neurons are partitioned into K groups with geometrically-ssace target activation frequencies (TAFs) from 10% to 0.01 %. creating resonance bands that match the natural feature frequency distribution. Right: Bias adaptation mech- anism-if if neuron over-activates (fm p&), we decrease its bias to make more selective; if under-activates (pm- < <), we increase its bias using the group baseline Fa to make more sensitive.
Neuron grouping strategy. To cover the diverse feature spectrum, partition the M neurons into K groups (default K = 10) with peameetiicallyssaacee target activation frequencies (TAFs). Specifically, fix the decaying ratio pa/P*+1, yielding TAFs from 10% down to 0.01%. This geometric spacing naturally matches the long-tail distribution of feature frequencies in language- from common words to rare technical terms (see SC.3 for detailed justification). Each group contains M/ K neurons sharing the same TAF pk within the group.
Algorithm Group Bias Adaptation (GBA)
Input: data X, initialization ( neuron groups and desired target activation frequencies [Gr.P&]K>1 a first-order optimization algorithm Opt
2: Hyperparameters: T, L, B, 7+. 7-, and
3: For all m E [M), initialize buffer Bm.
4: For t = I....,T:
5:
Forward pass, backward with reconstruction loss, optimizer step with fixed biases.
6:
Sample bini-batch X, E L row-normalize, and compute: ((a) t (xe ()--1) Backward and optimizer step (exclude biases): (+)) + Opt(O(t-1)) (b(t-1),, VC())
7:
Append pre-activations buffers: Bm +- Bm U (Vre) 1......... LmmLL } for all m
8:
9:
> Bias adaptation: when buffers reach size B, update biases and clear huffers.
10:
If |B| 2 B then
11:
Compute per-neuron activation frequency Pm and max pre-activation I'm in buffer. Set Pm + |Bml-1 EEEB...B..... 1 > 0) and I'm +- max max.eB... y, 0} for m E [M] D Average positive max pre-activation I'm for each group as group baseline Tk-
12:
13:
14:
Set Fr + -(EmeG 1( > "0 "EmeG,
for k E [K]
Adjust biases based on target activation frequency Pk-
16:
For each group K 1,..,, and each neuron m G
17:
If Pm > Pk. set by +- maxx(bm,
7- 7-Tm, -13
18:
If Pm < e, set 6m f- min{bm 74TR, Clear buffers: set Bm +- Ø for all m
19:
20:
Return the final SAE parameters (TT)
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Tracking activation frequencies. We measure each neuron's empirical activation frequency ing buffered pre-activations. For neuron m with weight Wm and bias bm. the pre-activation is ym (2) = w (x prre) + bm. During training, we accumulate B samples in buffer and compute the empirical frequency Pm as shown in Algorithm line 12. While the algorithm shows storing full pre-activations for clarity, the implementation is memory-efficient: we only track each neuron's maximum pre-activation I'm and activation count, updating these statistics incrementally.
Adaptive bias updates. Biases adapt to maintain target activation frequencies through feedback control. For neuron m in group k, we compare its empirical frequency Pm to its target pa:
1. If Pm > Pk (over-active): decrease bias bm + 6m - 7-Tm to make the neuron more selective 2. If Pm << (under-active): increase bias by.. +- bm + 7 to make it more sensitive
Here I'm is the neuron's max pre-activation for proportional decrease of over-active neurons, while Fy is the group average for boosting under-active neurons. We clamp biases to [-1, 0] to prevent extreme values, which should not be loss of generality since inputs are normalized. To give a sense of how the bias adapts, we find the rates 7+ 7 0.01 provide smooth adjustment to the bias without oscillations in the loss when we perform one adaptation step against every 50 optimizer steps with batch size L = 512.
Summary. GBA integrates with standard SAE training through periodic bias adaptation. The train- ing alternates between: (1) gradient phase where a standard optimizer (Adam/Adam updates weights W and output scales a, and (2) an adaptation phase that adjusts biases when the buffer reaches B samples. Crucially, biases are excluded from gradient updates and controlled only through the adaptation mechanism. Algorithm 1 presents the complete procedure. This design ensures each neuron finds its resonant features through frequency matching. The groups create "resonance bands" covering the feature spectrum, while adaptive bias control maintains target frequencies despite train- ing dynamics. Features naturally migrate to neurons with matching activation rates.
EXPERIMENTAL EVALUATIONS
To demonstrate the effectiveness of our proposed method, we conduct experiments on the Qwen2.5- 1.5B base model (Yang et al., 2024) using Pile Github and Pile Wikipedia datasets (Gao et al., 2020). We train SAEs with 66k hidden neurons attached to the MLP outputs at layers 2, 13, and 26. We evaluate each method using two metrics: (1) reconstruction loss and (2) average fraction of activated neurons per input. All methods employ JumpReLU activation (Erichson et al., 2019; Rajamanoharan et al., 2024b) for optimal performance. We compare GBA against three baselines: L1, TopK, and Bias Adaptation (BA)-a single-group variant of GBA with fixed target activation frequency p. Additional details and comparisons between ReI.U and JumpRel.U are in $D.
Evaluation metrics. Beyond reconstruction loss and sparsity, we use the following metrics through- out our experiments (see $C.4 for formal definitions). Feature Recovery Rate (FRR) is the fraction of ground-truth features v; such that at least one learned neuron has cosine similarity above a threshold Talign: FRR- = + Ei 1 3mmm m E [M] Kwm, vz) / /[//////lllllll/
2 Talign]: To assess cross-seed consistency, Maximun Cosine Similarity (MCS) computes, for each neuron in one run, the highest cosine similarity with any neuron in another run. The Neuron Z-score Z = (om.max pmm/ /mm quantifies selective activation: high values indicate neurons that fire sharply for specific inputs rela- tive to their baseline. Here, @mn is the maximum activation of neuron m across inputs, and jmm and Sm are the mean and standard deviation of its activations, respectively, In GBA, neurons are partitioned into K groups whose target activation frequencies form geometric sequence between the Highest Target Frequency (HTF) P1 and Lowest Target Frequency (LTF) Pk. Finally, the top- selection rule restricts analysis to the top fraction of neurons ranked by scalar metric (e.g., maximum activation or Z-score), focusing evaluation on the most active or significant neurons.
Toy example: why frequency-aware training matters. To isolate the benefit of frequency- aware training, we construct synthetic dataset with n 128 features in d = 42 dimensions where data points are imbalanced: half contain s = 3 active features (sparse) and half contain s = 20 (dense), so all features share the same occurrence frequency f Rs 0.09. TopK SAES assume a fixed sparsity K per input. Since in real data, one never knows the true number of active features, we test TopK with K E {20. 30, 50), which are all reasonable guesses for the average sparsity level. As shown in Table 1, TopK with K 20 achieves near- wear-perfect recovery, but performance degrades sharply as K increases (c.g., FRR at MCS > 0.9 drops from 98.4% to 23.4% at K = 50). In contrast, GBA with full frequency coverage (HTF= 0.5, LTF= 0.001) achieves 100% FRR without any tuning

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needed or strong prior knowledge of f. This fact demonstrates that frequency- aware training adapts to varying sparsity levels without much manual tuning. See $K.4 for details.
Method
FRR ( (Talign 2 0.8)
FRR (Talign 2 0.9)
TopK (K = 20)
100.0%
98.4%
TopK (K = =30)
98.4%
24.2%
TopK (K =50)
94.5%
23.4%
GBA (full coverage)
100.0%
100.0% %
GBA (misspecified)
38.3%
3.9%
Table I: Feature Recovery Rate on imbalanced synthetic data (n = 128, d = 42, M 8192). TopK's FRR degrades when K does not match the data sparsity, while GBA with proper frequency coverage achieves perfect recovery.
L.2 Reconstrrctionn Loss
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Github-Layer
Github-Layer
Wiki-Layer 26
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0.02% 0.1% 0.5%
2%
0.02% 0.1% 0.0%
2%
10%
0.02% 0.1% 0.0% 2% 10%
TooK - GBA -
BA
Fraction of Activated Neuron
Figure 4: Curve for reconstruction loss and sparsity (average fraction of neurons activated per data point). All experiments are conducted using an SAE with 66k neurons. For TopK, we vary K E [50, 600). For Lı, vary the penalty coefficient À E [0.001, 0.1]. For BA (non-grouped), vary the target frequency p E [0.003, 0.1.. For GBA, we sample within the range K E 20), Pi E [0.05, 0.5], and px E [10-1,5 ,,5 10 "
Reconstruction loss and activation sparsity frontier. We first compare the normalized & re- construction loss against the average fraction of activated neurons across different methods. The results are presented in Figure 4, where each benchmark method (TopK, L1, BA) involves varying sparsity-related tuning parameters. Our method performs comparably to the best-performing bench- mark, TopK- Specifically, when these methods have the same average fraction of activated neurons, GBA's recon- struction (yellow star) is comparable to TopK's 's best curve while significantly outperforming both the L1 penalty method and the non non-groupee grogggoupd variant BA. The consistent superiority over BA across all experiments provides strong evidence that the grouping mechanism is crucial for achieving both optimal performance.
the lowest reconstruction loss among all methods for given sparsity level.
Top Neurons Max Activation
Top Neurons Z-Score
Neuron Percentage
Neuron Percentage
MM0
X0
GBAA
GBA
0.3 %
0.05 %
10
10
Max Casine Similarity Threshold
Max Cosine Similarity Threshold
Figure 5: Fraction of neurons that have max cosine similarity exceeding threshold for Github- Layer 26, where the max cosine similarity is evaluated for neurons from 6 different runs initialized with different seeds. We take Max Activation and Z-Score the selecting criteria and plot within subset of neurons that rank top-a under the criteria in all the neurons with a in (0.3%, 0.05%) (i.c.. t0p-200 and top-30 neurons out of 66k).
Consistency of recovered features. Furthermore, we assess the consistency of the learned fea- tures across independent runs with different random seeds. Since ground truth features are unavail- able, consistency serves as proxy for the reliability of the training method. For each neuron in one run, we compute its Maximum Cosine Similarity (MCS) with neurons from another run; a high MCS indicates that a feature is consistently recovered. To avoid the influence of rarely activated neurons, we restrict our analysis to the top-a neurons selected based on maximum activation or Z-score. The results presented in Figure 5 and Table 2, and the key findings shown follows:
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1. As noted in prior work. TopK is seed-sensitive (Paulo & Belrose, 2025). In experiments, GBA yields a higher percentage of neurons with high MCS. To quantify this effect, Table reports the fraction of consistent neurons (MCS > 0.9) under different selection criteria; GBA consistently exceeds TopK. Variability across seeds is small: all runs show tight fluctuations over the (o) = 15 pairwise comparisons.
2. The L1 penalty -based SAE is generally more consistent than TopK, and results confirm this trend: across most selection criteria, L1 achieves higher consistency than both TopK and GBA. However, when focusing on the most active neurons (top-0.05% by activation). GBA surpasses
L1, Suggesting stronger recovery of the most salient features.
Additional scatter-plot analyses of neuron-level metrics (Z-score maximum activation, activation fraction, and MCS) for both GBA and TopK are provided in $K.3, Key findings include: (i) neurons with higher Z-scores also exhibit higher MCS, confirming that selectively activated neurons are con- sistently recovered across seeds; (ii) GBA captures infrequent yet salient features more effectively than TopK, whose high-Z-score neurons concentrate at higher activation fractions.
Combined with the reconstruction-spariii fron- tier in Figure in our experimental setting. the proposed GBA method achieves the Pareto fron- tier in terms of reconstruction fidelity, activation sparsity, and feature consistency.
Top-a Neurons
GBA (ours)
TopK
10%
0.0366 0.0010
0.0317 0.0006
25%
0.0146 + 0.0004
0.0127 = 0.0002
Interpretability. To further evaluate the inter-
50%
0.0073 + 0,0002
0.0063 = 0.0001
pretability of the learned features, we employ suite of metrics by Karvonen et al. (2025). To ensure a fair comparison, we retrain GBA SAE with 66k neurons on Gemma2-2B (Team et al., 2024) residual stream after layer 12, and com- pare it against six baselines provided in SAE- Bench: Standard, GatedSAE (Rajamanoharan et al., 2024a), Botch Matryoshka (Bussmann al., 2025), TopK. and JumpReLU SAE (Ra- jamanoharan et al., 2024b), all with Lo between 300 400. The results are summarized in Table 3, demonstrating the competitive interpretability of GBA across all metrics. GBA achieves the best performance in 4 out of 9 metrics, particularly excelling in Explained Variance, Absorption Score, and Alive Fraction. This evidence further sup- ports our thcorectical claim to be made in $6 that GBA can reliably recover monosemantic features. More visualizations and detailed explanations the interpretability metrics are available in $K.5.
100%
0,0037 TL 0.0001
0.0032 1 0.0001
Table 2: Fraction neurons with Maximum Co- sine Similarity (MCS) > 0.9 across different se- lection percentiles based on the top selection rule in maximum activation. Results averaged over (9) = 15 pairwise comparisons from 6 ran- dom seeds with standard deviations shown. GBA achieves higher consistency than TopK for all a.
GBA (ours)
Metric
Standard
GatedSAE
BatchTopK
Matryoshka
Topk
JumpReLU
Lo
468.9
408.7
339.4
338.1
334.4
309.0
305.3
Explained Variance
0.816
0.871
0.859
0.840
0.859
0.902
0.855
Absorption Score
0.1355
0.0696
0.0347
0.01588
0.0269
0.0041
0.0424
SCR Metric t
0.228
0.294
0.268
0.331
0.332
0.296
0.309
Sparse Probing f
0.958
0.959
0.957
0,957
0.958
0.956
0,958
TPP Metric
0.026
0.086
0.267
0.312
0.213
0.184
0.099
Alive Fraction
0.743
0.918
0.770
0.828
0.872
0.970
0.711
RAVEL Disent.
0.709
0.764
0.729
0.742
0.725
0.737
0.737
RAVEL Cause f
0.670
0.749
0.697
0.710
0.700
0.694
0.731
RAVEL Isolation f
0.748
0.778
0.761
0.773
0.749
0.779
0.742
Table 3: Performance comparison of SAF models with To between 300 400 SAF.Bnnhh. Arrows indicate whether higher (f) or lower (4) values are better. Bold indicates best performance, and underline indicates second best. GBA (ours) achieves the best performance in 4 out of 9 metrics, particularly excelling in Explained Variance and Absorption Score.
Ablation study on GBA hyperparameters. To assess GBA's sensitivity to hyperparameters, we perform ablation study varying the number of groups K, Highest Target Frequency (HTF) P1, and Lowest Target Frequency (LTF) Px. as shown Figure 6. The left panel reveals a key pattern: HTF increases, performance stabilizes- scatter points converge and align with TopK's curve,
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Published as a conference paper at ICLR 2026
especially for K 2 10. Additionally, since low HTF values (e.g., 0.05) hinder recovery of frequent features, it results in higher reconstruction loss. The middle and right panels again confirm that both reconstruction loss and sparsity stabilize when K 2 10, demonstrating insensitivity to the exact number of groups. We observe a slight increase in loss when increasing the number of groups K in the middle panel. This is not detrimental but rather reflects tradeoff: with few groups (e.g., K = 3), many neurons are assigned high target frequencies, resulting in denser activations (right panel) and thus lower reconstruction loss at the expense of interpretability.
Fraction of Activated Neurans
1.2 Reconstrratioo Lass
HTF=0005
HTF=0.1
HTF=0.3
HTF=0.5
13
#Groups
3
20
Topk
10.05 0.03
15.05 00001
Reconstruction
12.05 0.0031
R.A U.11
0.15
1
10.1 0001
0.1
(HTTE) LTF))
0.05
Fraction of Activated Neurons
4Groups
4Groups
Figure 6: Ablation study for GBA terms of K, HTF, and LTF for Github-Layer 26. For each run, we partition neurons into K groups with target frequencies as a geometric sequence between HTF and LTF. IITFs: [0.05, 0.1, 0.3, 0.53: LTFs: {10- 10 10101000000 5 x 10 10. Left: Loss vs sparsity grouped by HTF. Different colors represent different K values; dots of the same color correspond to different LTFs. Middle & Right: Loss and sparsity for varying K, where each curve represents a pair of HTF
and LTF. Results show GBA stabilizes when HTF 0.5 and K 2 10.
Simple rule for hyperparameter selection. These results establish that GBA is nearly tuning-free free with simple selection rule: (1) HTF 0.5 the default upper bound, since randomly initialized neurons with zero bias fire 50% of the time; (2) set LTF= = 10-3 to 10-4 to cover rare features while preventing dead neurons; (3) use a large number of groups for better frequency coverage and stable performance. This principled setup eliminates the need for dataset-specific tuning- in stark contrast to searching TopK's 's K across 66k neurons or tuning L1 's penalty coefficient A.
NEURON RESONANCE: A THEORETICAL PERSPECTIVE
The neuron resonance phenomenon observed in $3 raises a fundamental question: How does fre- quency matching enable reliable feature recovery, and what determines the resonance band? We provide a theoretical analysis that justifies this phenomenon with precise recovery conditions.
To rigorously analyze the neuron resonance phenomenon, we study simplified variant of Algo- rithm that captures its core mechanism. We consider the Bias Adaptation (BA) algorithm, which essentially GBA with single neuron group and all neurons share fixed target activation fre- quency p. The SAE is trained via spherical gradient descent (weight updates normalized to unit sphere). This single-group setting isolates the activation frequency factor from other dynamics, helping understand how neurons with frequency p selectively learn features with similar occur- rence frequency.
For the data model 12.1), we assume V has i..d NO 1) entries and simplify H to have exactly 8-sparse rows: each row l has uniform random support Se with |.A|| = & for a constant s and entries He, = 1/v8 for i St, zero otherwise. This simplified H structure is only for presentation convenience; our analysis captures more general coefficient matrices (see $B.1). The following theorem characterizes the conditions under which BA can recover all features with high probability.
Theorem 6.1. Consider the simplified data model X HV with data size N. feature size and feature dimension d. We train an SAE with M neurons using the BA algorithm with spherical gradient descent, target frequency p, and learning rate n Z (p Under certain regularity conditions on the SAE model ($B.2.1), for any small constants S (0, such that
Network Width:
M 2 n (11-)))
(6.1)
Frequency Range: n -1 SPS min f - -1++ 1)/2 n 2---(11- 2/144) d
(6.2)
with probability at least 1 - n - every feature E [n] is recovered by at least one neuron m;
(T) within T =1 iterations in the sense that (wiT), vz) /lulll2 2 -o().

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Published as a conference paper at ICLR 2026
See $B for the full version of the theorem with detailed assumptions and the proof is in SG. To our best knowledge, Theorem 6.1 provides the first provable guarantee that an SAE training algorithm recover monosemantic features within constant number of iterations. The theorem relies V being Gaussian for technical convenience; we discuss the role of this assumption and its empirical robustness in SC.1. Our results both synthetic ($3) and real L.M data ($5) confirm that the BA and GBA algorithms work well when V is non- non-Gaussian. Moreover, although the theorem only analyzes for a single group with frequency p for clarity of presentation, we can easily extend it to GBA with multiple groups under the same regularity conditions on the SAE model. See $B for detailed discussions.
Interpreting the theorem. The theorem reveals two critical factors for successful feature recovery:
1. Network width: The required width M Σ n p =/((/)==)² shows that M scales linearly with the number of features n but exponentially with sparsity when p is fixed. The linear scaling with n is intuitive, as each neuron can learn at most one feature. This exponential dependency arises from the challenge of distinguishing features when they co-occur in the same data points. Figure 7(a) experimentally validates this exponential scaling. This result highlights the benefit of overparameterization in SAE training.
Activation frequency range: The condition on p translates to a "resonance band", where fea- tures are most effectively learned when the neuron's 's activation frequency p falls into the band. The upper bound depends on both the superposition level (controlled by d/ n) and the feature sparsity 8. Figure 7(b) visualizes these resonance bands for different sparsity levels. Notably, in our simplified data model, the feature occurrence frequency is = s/n = ((n so for a large constant 8, we can rewrite the condition as f S p S min (VT, fd), as mentioned in $3.
FRR Heatmap 100, 384)
Theoretical Learnable Region
1.0
1052
-0.6
0.0
M
-0.8
FRR
0.5
Tigh-oocerteeen (I goog))
occureeeee group)
Highenccu rejoo (4 group)
0.2
384
0.0
Lim-orrurmme groms)
0.2
0.4 flow hhggh
0.6
0.8
1.5
S
log d/logn
Figure 7: (a) Network width scaling: Heatmap of Feature Recovery Rate (FRR) with respect to (M, s) for the GBA algorithm with M axis in log scale, showing exponential dependency on (b) Resonance bands: Thcoretical learnable region (yellow) for different sparsity values, demonstrating the transition d = /n between heavy and light superposition regimes. For large s, the upper bound approaches min { [T.4f) df) with f = 6(m ). (c) Feature imbalance: FRR vs. relative occurrence from/ / Shigh. showing GBA's advantage over BA in handling imbalanced feature frequencies. All exper-
iments use (n, d) (384, 100). For (c), we use 3 and M 1024.
This theorem rigorously justifies the neuron resonance phenomenon by proving that neurons with frequency p optimally recover features within a specific frequency band. This insight motivates the GBA algorithm's 'ss multi-group design: by creating groups with geometrically decaying target frequencies, we ensure coverage of the entire feasible frequency range, enabling recovery of features with diverse occurrence patterns.
GBA handles imbalanced features. As an extension to the discussion above and to build connec- tion to the GBA algorithm, we compare the analyzed BA algorithm with GBA (with 4 groups) on data with imbalanced feature frequencies. To demonstrate the effectiveness of GBA, we construct dataset with features divided into two groups of equal size: one group with high occurrence fre- quency fligh and the other with low frequency flow. We vary the imbalance ratio flww/figh while keeping the average frequency fixed. The results in Figure 7(c) show that GBA significantly outper- forms BA as the imbalance increases, i.e., faw/ fow//high < 0.3, highlighting GBA's ability to recover features across wide frequency spectrum, and flexibility to handle real-world data with diverse feature occurrence patterns.
Reproducibility. The anonymous source code this project is available in the supplementary for both data processing and model training. The assumptions and proofs to the main theory can be found in $B and SG, respectively.
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CONTENTS
1 Introduction
2 Preliminaries
3 Rethinking How SAEs Learn: Neuron Resonance
4 Algorithm: Group Bias Adaptation
5 Experimental Evaluations
Neuron Resonance: A Theoretical Perspective
Related Works
16
Formal Theory of (AA Training Dynamics
17
B.1
Data Model
17
B.2
SAE Dynamics with Bias Adaptation
18
B.2.1
Simplification for Theoretical Analysis
18
B.2.2
Main Theorem Training Dynamics
19
B.3
Details on ReLU-like Activation
B.4
Proof Overview
22
B.4.1
Good Initialization with Wide Network
22
B.4.2
Preactivations Approximately Gaussian
22
B.4.3
Weight Decomposition and Concentration under Sparsity
23
B.4.4 State Recursion and Convergence
24
C Supplementary Discussions
24
C.I Discussions for Gaussian Feature Assumption
24
C.2 Details on TopK and L1 Training Methods
25
C.3 Discussions for Geometric Spacing of Target Activation Frequencies
26
C.4 Evaluation Metrics
26
Additional Experiments Details
29
D.I Synthetic Experimental Setup
29
D.2 Additional Details for $5
D.3 Comparison between JumpRel.U and ReL.U activation
30
Good Initialization and Gaussian Conditioning
30
E.1
Initialization Properties
E.2 Rewriting the Gradient Descent Iteration
32
E.3 Gaussian Conditioning
34
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E,4
Additional Proofs
38
F
Concentrations Results for the SAE Dynamics
39
F.1
Isolation of Gaussian Component
40
F.2 Sparse Activation
41
F3 Second Order Concentration
F.4 First Order Concentration
45
F.5
Mon-Gaussian Error Propogation
47
SAE Dynamics Analysis: Proof of Theorem B.2
48
G.1 A General Version of the Theorem
48
G.2 Concentration Results Combined
50
G.3 A Two-State Alignment Recursion
53
G.4 Simplifying the Conditions of Theorem G.13
55
H Proofs for Concentration Results
57
H.1 Proofs for Non-Gaussian Components
58
H.1.1 Proof of Theorem F.1
58
H.1.2
Additional Proofs for Theorem F.1
58
H.2
Proofs for Concentration Lemmas
60
H.2.1
Concentration for Ideal Activations
60
H.2.2
Activations with Non-Gaussian Component: Proof of Theorem F.3
63
H.2.3
Concentration for | ( b1 Proof of Theorem F.4
64
H.2.4
Concentration for || (p({((( 0 1.-1; b:)
] Proof of Theorem F.6
72
H.2.5
Concentration for (=r, E ( Eyt: be)): Proof of Theorem F.7
74
H.2.6
Concentration for (z,, F Tp (F 0 wx)1; Proof of Theorem F.9
76
H.2.7
Concentration for 0 (FFut + Oe wz-1;bb). b,): Proof of Theorem F.10
78
H.3
Propogation of the Non-Gaussian Error
79
H.3.1 Error Analysis for AE; Proof of Theorem F.12
79
H.3.2 Error Analysis for AF; Proof of Theorem F.13
80
H.4
Proofs for Technical Lemmas
81
H.4.1 Proof of Theorem F.5
H.4.2 Proof of Theorem F.8
81
H.4.3 Proof of Theorem F.ll
82
I Proofs for SAE Dynamics Analysis
83
I11
Proof of Theorem G.2
83
I.2 Proofs for Concentration Results Combined
84
1.2.1
Proof of Theorem G.5
84
1.2.2
Proof of Theorem G.6
85
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I.2.3
Proof of Theorem G.7
86
1.2.4
Proof of Theorem G.8
87
L.3
Proofs for Recursion Analysis
88
I.3.1
Proof of Theorem G.10
88
I.3.2
Proof of Theorem G.12
90
1.
Proofs for Condition Simplification
93
1.4.1
Proof of Theorem G.13
93
1.4.2
Proof of Theorem G.14
94
1.4.3
Proof of Theorem G.15
95
I.5
Proofs for Technical Lemmas
96
1.5.1
Proof of Theorem 1.1
96
1,5.2
Proof of Theorem 1.2
97
1.5.3
Proof of Theorem 1.4
97
J
Auxiliary Lemmas
98
J.1
Concentration Inequalities
98
J.2
Efron-Stein Inequalities
99
K Additional Discussions and Results
102
K.I
Clarification on Neuron Resonance
102
K.2
Clarification that GBA is not a Hierarchical SAE
103
K.3
Additional results on neuron analysis
103
K.4 Why Frequency-Aware Training? A Toy Example
104
K.5 Additional SAEBench Evaluation
106
L The Use of Large Language Model
109
A RELATED WORKS
SAE Training Methods. Many methods have been proposed to train SAEs, addressing the trade- off between reconstruction fidelity and sparsity-induced interpretability from various perspectives. One canonical approach is imposing an L1 penalty on the activations Bricken et al. (2023). Al- though L2 natural surrogate for enforcing Lo sparsity, it typically suffers from activation shrink- age Tibshirani (1996). Several works have attempted to overcome this drawback through alterna- tive techniques Wright & Sharkey (2024); Taggart (2024); Rajamanoharan al. (2024a); Konda et al. (2014). In particular, Rajamanoharan et al. (2024b) proposed the JumpReL. activation, which achieves state-of-the performance despite requiring backpropagation with pseudo-derivatives be- cause of the non-smooth nature of JumpReLU and the need for tuning the kernel density estimation bandwidth. Another representative example
use of TopK activation Makhzani Frey (2013), which has proven effective when scaled to large models Gao et al. (2024). However, it has been observed that features learned via TopK activation are quite sensitive to the random seed Paulo & Belrose (2025), raising concerns about their reliability.
Sparse Dictionary Learning. Beyond SAE training methods, there is long history of research on sparse dictionary learning (SDL) dating back to Olshausen & Field (1996); Kreutz- Kreutz-Delgade et al. (2003). Numerous techniques have been developed for applications in signal processing and com-
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puter vision (Bruckstein et al., 2009; Rubinstein et al., 2010). For example, Spielman et al. (2012)
proposed a polynomial-timg algorithm that can accurately recover both the dictionary and its coeffi- cient matrix, under the assumption of sparsity in the coefficients.
Using SAEs for Model Interpretation. In recent years, SAES have gained attention for model interpretation, particularly in the context of large language models (LLMs) (Bricken et al., 2023; Paulo & Belrose, 2025). Notably, Bricken et al. (2023); Dunefsky et al. (2024); Ameisen et al. (2025) have identified several interesting features and circuít patterns learned by SAEs or their variants. Beyond detecting monosemantic features, Papadimitriou et al. (2025) found that groups of SAE- learned features remain remarkably stable across different training runs and encode cross- semantics. Additionally, the potential of SAE activations for steering model behavior has been explored (Ameisen et al., 2025; Shu et al., 2025).
B FORMAL THEORY OF SAE TRAINING DYNAMICS
In this section, we present a formal theory of SAE training dynamics, providing rigorous guarantees for feature recovery when data follows well-defined statistical model. However, before delving into the details of the theory, we first need to answer the following fundamental questions:
What is the precise statistical model for data generation should consider?
What does it mean to recover features, and under what conditions feature recovery even possible?
In this section, we will
Formalize the statistical model for data generation. State the feature recovery problem and define identifiability of features, which is pre-requisite for any recovery guarantee Present the full set of assumptions and result on SAE training dynamics.
Notations. Let R denote the set of non-negative real numbers. For two sets A and B, we denote by A B the disjoint union of A and B. We denote by 1 the all-ones vector, whose dimension will be clear from context. In the remaining of the section, we abuse the notation and use Z b to denote that a 2 b+ Oloog log n/ log n) for sufficiently large n, which differs from what we use in the main text.
B. DATA MODEL
We consider the data model = HV from (2.1), where data matrix X E Nxd a sparse, nonnegative combination of monosemantic features E Ruxd with coefficients H E NNn Our statistical framework requires the following decomposable data conditions:
Definition B.1 (Decomposable Data). We say that the data matrix X E NR is decomposable if there exists a positive integer n N, a nonnegative matrix H E RRN and a feature matrix V E Rxd such that X = HV. Moreover, each row of H has unit E2 norm and the &a norm of each row of V is @(vd). Furthermore, the coefficient matrix H E RNx satisfies the following three conditions:
(HI) Row-wise sparsity: maxte[N] [|oe, with O(1)
(H2) Non-degeneracy: For every i E [n]. | = O(1.
(H3) Low co-occurrence: P2 :2:=mm: maxifj
(H,, * 0}, ((H., # 0))/ & n1/2
In addition, we further assume that the feature matrix E R x satisfies:
(VI) Incoherence: For all i + j. (e,, ())s/([|u;ll |0y (2)) = o(1.
These conditions ensure feature recoverability: nonnegativity removes sign ambiguity since op- posite directions yield contradictory concepts; row-wise sparsity (H1) limits each data point to features, essential for sparse recovery; non con-degeyyracy (H2) ensures sufficient feature magnitude
when present; low co-occurrence (H3) and incoherence (V1) guarantee features are distinguishable
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by occurrence pattern direction- generalizing the orthogonality assumption common in sparse recovery (Marques et al., 2018; Candès & Plan, 2009). All these conditions will be used our theoretical analysis of SAE training dynamics.
Feature recovery problem. Note that the bilinear representation X = HV has two intrinsic ambiguities: (i) feature permutatoo- features leaves HV unchanged; (ii) feature scal- ing scallng features while inversely scaling coefficients preserves the product. With the data model, we can now define the feature recovery problem: given data X generated from an unknown decomposable pair ( H, V), the goal is to learn an SAE such that for each feature in V, there exists a neuron my the SAE with weight vector Wm; satisfying
(Wm, U4) / || 2 1 (11)
This means each feature is closely approximated by at least one neuron, up to a small error.
B.2 SAE DYNAMICS WITH BIAS ADAPTATION
In the following, we first introduce a Bias Adaptation (BA) algorithm, which is simplified version of the GBA algorithm with only one group of neurons and fixed target activation frequency (TAF)
p. Then, we provide theoretical results on the training dynamics of BA, which is accompanied by synthetic experiments to validate the theoretical findings.
B.2.1 SIMPLIFICATION FOR THEORETICAL ANALYSIS
We make several simplifications to the setup of SAE to facilitate theoretical analysis.
Decomposible data with Gaussian features. We assume that the data matrix X E RNvxn decomposable in the sense of Definition B.1. Moreover, we assume that the feature matrix V E Raxd has i.idd entries following (0, Such a choice of V satisfies the incoherence condition (VI).
SAE model. We consider simplified version of the SAE model [(x; () in (2.2), where the only trainable parameters the weights {wm) 1. m==1
(Small outpul scale) We assume that the output scale Om and is sufficiently small. When computing the gradient, we rescale the VZ(()) back to its original scale by multiplying -!
(Fixed pre-bias) We fix the pre-bias bpre 0, the data matrix X is centered.
(ReLU-like smooth activation) We use a smooth, ReLU-like activation function & (see Defini- tion B.3 for details). One example is the softplus activation ((x) = log(1 + exp(x)).
(Fixed bias) For each neuron m E [M]. we fix the bias bn, b < throughout training, where is negative scalar whose value will be specified later. This fixed bias will determine the target activation frequency (TAF) p of all neurons via p = ( where o(-) is the tail probability function for Gaussian distribution. We will detail the intuition behind this choice later.
These simplifications help isolate the aspects of feature recovery and make the analysis more tractable.
Bias Adaptation (BA) algorithm. Recall that Bias Adaptation (BA) algorithm is special case of GBA algorithm with only one group of neurons and a fixed TAF As our goal is to systematically understand how neurons with a specific TAF p can recover features with similar occurrence fre- quency, it reasonable to try using a version of GBA algorithm with single group and a fixed TAF p. We introduce the algorithm as follows. Here we determine the value of p implicitly by choosing fixed bias b O, and they are related by p (--)), where (+-) is the tail probability function for Gaussian distribution. That is, (t)) = P(Z 2 t) for (0,
Given the data matrix X and the SAE model fæ: θ), we compute the loss function as Liee( Avgrex fx:; θ) - x: x=|3). and its gradient with respect to the weights {w, IM m= Since only the directions of the features { 1 e matter, adopt spherical gradient descent update the weights.
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(0) That is, starting from the initial weights {wom ) m uniformly sampled from the unit sphere Sd-1,
for any & 2 1, in the (-th iteration, we update each Wmm.
(t-1)
by
(f-1)
(t)
BA: w
Wm (t-1)
+ 17 9m
where
Im
lim -a a-40
V V Mmm foee ( (O(t-))).
(B.1)
1m +1 fm ||2
(2)
Here, 9m is the rescaled negative gradient of the loss function (-) with respect to the weight Wm neuron m at iteration We will show that, under proper conditions, for any feature o,, there exists at least one neuron m, E [M] such that the alignment between wm; and v; is arbitrarily close to one when T is sufficiently large.
(t)
Before we proceed to the main theoretical results, we make several remarks on the above simplifi- cations for thcoretical analysis and their implications.
Fixed bias is without loss of generality. As consider Gaussian features and always normalize with to the unit sphere, it can be shown using the Gaussian conditioning tech- nique that the pre- antivactiss remain approximately Gaussian,
ReLU
Shifted ELU
Softplus
i.e. (xz) = (wm xze + b ~ ((b,1) 1) for a constant num- ber of iterations t. See $B.4 for details. Therefore, to achieve the desired TAF p, it is without loss of generality to fix the bias < 0 such that ((-b) = p. which means that the pre- activations of each neuron will be non-negative for approxi- mately fraction of the N data points throughout the training.
2
Figure 8: Smooth ReLU-like activa- tions
Smooth ReLU-like activation approximates ReLU. We choose smooth activation function for technical convenience. For definition, we defer to Definition B.3. These activations can be viewed a smooth approximation the ReLU function, illustrated in Figure 8. This class of activations encompasses functions like Softplus and shifted ELU, and closely resembles the standard ReL.U activation function. We believe that the standard ReL.U activation, but we leave this as future work.
refined analysis can also be applied to
Small output scale decouples neuron dynamics. Following a common paradigm in the litera- ture (see e.g. Lee al. (2024); Chen et al. (2025)), assume that the output scale of the SAE sufficiently small. The benefit of this condition that it decouples the dynamics among the neurons, making the analysis more tractable. Specifically, the rescaled negative gradient of the loss ((O) given by
Im - (())=
Σ
( b)ar v' 9) 4
( b() (B.2)
where we define -) :) and vmm( () as
c v) Φbu w) + d'(u v)
Vi (x; θ) = 0' J, b) wT (x; θ) (w + b) f(x; θ)
Here, (p 18 + is a decoupled term that depends only on each individual neuron's weight and bias, while t/m Rd - Bd is coupling term that captures the interaction between the neuron and the rest of the network. Since the scale of (a; +) proportional a, this coupling term negligible when a is small. As result, when is infinitesimally small, each neuron m evolves independently of the other neurons. Furthermore, thanks to the decoupled dynamics, the restriction to a single group with a fixed TAF p does not result in any loss of generality, as the analysis of multiple groups is a straightforward extension.
B.2.2 MAIN THEOREM ON TRAINING DYNAMICS
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Intuitively, h is critical threshold such that, among all data points containing Ui, at least polylog(n)- fraction of them have coefficients no smaller than h₁. other words, h; reflects the magnitude of coefficients associated with feature 8ú within the subset of data points where u; is Published as a conference paper at ICLR 2026
hr
Intuitively, to recover feature c, it has appear in sufficiently many data points with sufficiently large coefficients. To charac- terize this intuition, we introduce two key quantities based the coefficient matrix H. First, for each feature index i E [n). let D; (1 [M] Hi, # 0} be the set of data indices that con- lain feature ey. The occurrence of the feature vú is thus given by [D:I/N. We define the maximum feature occurrence as the largest occurrence among all features, i.e.,
4
h.
2
1/v3
0
0.2
0.4
Figure 9: Relationship between 8, h. and h; with different concentration level in HI's zero entries' empirical distribu- tion (shadow). A less concen- trated H leads to larger h, and h4.
L/N Σ 1ilii + #0) [E[M]
P1 = max = max :E(m]
To ensure each feature v appears in sufficiently many data points.
we require that the occurrence of each feature is comparable to Pr, i... (Pal/(p)N) is not too small for each E [n].
Second, to measure the magnitude of coefficients associated with each feature, we define the cut-off level for the feature
h4 max 121- I E 1(11 h3 polylog(n)-1)
(B.3)
[D;] [D.]
present. Thus, h, can effectively be viewed as a notion of "signal strength" for feature U; and we should require that h; is not too small for each E [N].
Furthermore, we additionally introduce a global quantity called the concentration coefficient h4 (H), whose definition is technical and deferred to (G.1) in the appendix. Intuitively, h, character- izes the global concentration level of nonzero entries in H. For now we can intuitively understand it the variance of the nonzero entries in H, and thus h, will increase when the nonzero entries in H are less concentrated.
With these definitions, we now ready to state the main theorem the training dynamics.
Theorem B.2. Let X HV be decomposable in the satisfying all the conditions therein. and further assume that V E R has i..d. entries following (0, For this X, train the SAF with BA algorithm given in (B.1). Let S, (0,1) be any small constants. We assume that the number of neurons M is sufficiently large:
of Definition B.1 with H Nxn
log M
82
+ 1
Network Width:
2 max rEIn
(B.4)
log n
2(1 =)2h2 log
Moreover, we assume that the learning rate n satisfies logn 2 (b2/2 log N) and that the bias < 0 set to satisfy the following condition:
2
(1 h? 2(1 + (11+=)?23
log d log
Bias Range:
max
<)
(B.5)
2 logn
(2
Furthermore, assume the coefficient matrix H satisfies the following feature balance condition:
D N
log log(n) log(n)
Feature Balance:
2 polylog(n)-1,
h
Vi E [n].
(B.6)
Then, it holds with probability least 1-n over the randomness of fartfor any feature E [n], there exists at least one unique neuron such that after at most T = 5-1 iterations, the alignment (T)
between the weights neuron m, and the feature vector v; satisfies (w0(2 v.)/://l /1/ o(1)
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See $G for a detailed proof of this theorem. Theorem B.2 shows that under appropriate conditions, BA provably recovers all monosemantic features within a constant number of iterations. These conditions include that (i) the network is sufficiently wide compared to the number of features as specified in (B,4), (ii) the bias b is chosen within a certain range as specified in (B.5), and (iii) the co- efficient matrix H satisfies the feature balance condition in (B.6), ensuring that each feature appears frequently enough with sufficiently large coefficients. To our best knowledge, this theorem is the first theoretical result that proves SAE training algorithm can provably recover all monosemantic features.
Going from one group to multiple groups. The analysis of BA with a single group can be nat- urally extended to the case of multiple groups in GBA thanks the decoupled dynamics among neurons, Specifically, as we have shown in (B.2), each neuron m evolves independently when all neurons' output scale am is sufficiently small. Therefore, if we have K groups of neurons, each with M/K neurons and bias by for group k E [K] such that the TAF pæ: = ( then the same analysis can be applied each group separately, and we will derive the same conditions as in The- orem B.2 for each group. As result, to learn features of a certain occurrence frequency f. we just need ensure that at least one group l has TAF pt: and the corresponding bias by satisfying the Bias Range condition in (B.5).
Specializing Theorem B.2 to Theorem 6.1. To relate the above theorem back to the one presented in the main text:
We first note that the decomposable data condition in Definition 1.1 is always satisfied when is designed to have exactly 8-sparse rows: each row has uniform random support S₃ with |Se] for constant and entries Hz,; = =//6 for i S₁, zero otherwise.
Moreover, the feature balance condition in (B.6) is also satisfied with high probability in this case because each feature appears in s/n fraction of data points, and |Di| = *N/n for each
E [n]] by the law of large numbers, and so is p1 s/n.
Finally, in this case, the concentration coefficient h, defined in (B.3) is equal 1/Vs, as the nonzero entries are all equal to 1/vs.
Hence, the remaining conditions to checked are the network width condition (B.4) and the bias range condition in (B.5). We now invoke Theorem G.I, which states that if H has every entries belonging to {0, 1/5).. then h. = 1/v8 as well. Therefore, by substituting h2 = h, = 1/ /s into (B.4) and (B.5), multiplying both sides of the inequalities by log n, taking exponential, and using the fact that
b²
θ (ppolylognn)
exp
by the Gaussian tail estimate for Vlog n recover the conditions in Theorem 6.1.
>> (guaranteed by the bias range condition), we
B.3 DETAILS RELU-LIKE ACTIVATION
In this section, we provide the omitted details for $6. We give a formal definition of ReLU-like activations.
Definition B.3 (ReLU-like Activation). For the activation function & : - R. we define up as <(x) = ((x; 6(x) + x
We say that é is ReLU-like if satisfies the following:
1. (Lipschitzness) The activation function & is continuously differentiable, 1-Lipschitz, and m- smooth with m Oppoyylog(n)) Furthermore, plx)) is "i-Lisschit with 72 O(polylog(n))
2. (Monotonicity) The activation function ó is non-decreasing, and moreover, &' d'(x) > Co for some constant Co and all x 2
3. (Diminishing Tail) There exists a threshold Kg = O n)) -1/2) and a sufficiently large con- stant Co > 0 such that for all x < -Ko, max (oxxl(((x)) 16' (z)), Ix (x))) S n ~o.
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Lipschitznss. Under the above assumptions, note that p(z; b) is L-Lipschitz in with L (72 + (b/m) = O(polylog(n)) > 1. The Lipschitz property the function yp is pivotal in our analysis since it enables control
propagation across iterations. However, this property depends on the smoothness of the activation function Φ condition that the standard ReL.U does not satisfy. Fortunately, many common activation functions-
as softplus, noisy ReL.U, and shifted ELU (with the limit at -00 set to d)-do satisfy this smoothness requirement. In particular, with large smoothness parameter 71 polylog(n), we can use smooth activation function well approximate the ReLU function. For instance, we can take ((x) 7i log(1 e> for some hi = (nlylog(n) as smooth approximation of the ReLU activation function.
Monotonicity. The monotonicity property ensures that neurons with large pre-activations, which indicate a good alignment with the underlying features, will also have large post-activations. This then guarantees a continuous growth of the corresponding neuron weights.
Diminishing Tail. The diminishing tail condition ensures that both the activation function ó and its derivative o' are negligibly small when the input is below the threshold -oo- This property
suppresses unwanted neuron activations, thereby promoting sparsity in the activations key factor in the successful training of the SAE.
B.4 PROOF OVERVIEW
In the following, we provide an overview of the key steps in the proof of Theorem B.2.
B.4.1 GOOD INITIALIZATION WITH WIDE NETWORK
By planting a large pool of i.i.d. random neurons initialization, we can-with overwhelming prob- ability-(1) (1) assign to each feature one neuron whose inner product with is already very large, and (2) simultaneously ensure that this same neuron has only weak correlations with all the other features. Concretely, we prove that if M grows fast enough relative to then there exists choice of distinct neurons {m,}, such that
InitCond-1: (2 wal) 2 (1 (1-E) 2
InitCond-2: max (uy, (0)0))) S v + +) V log
(B.7)
These two properties together ensure a good initialization for the neuron m. dedicated to feature 2. With M > n²", deduce that neuron Wm, aligns exclusively with feature v. In fact, M increases the separation between the two thresholds also increases, so (0) ever more strongly aligned with its own feature v, than with any other v the start. This widening margin precisely captures the benign over-parumeterization effect: having many neurons actually promotes clean, feature-specific initialization. See Theorem E.1 for more details.
B.4.2 PREACTIVATIONS ARE APPROXIMATELY GAUSSIAN
We give a brief overview of how we deal with the challenge of tracking the highly nonlinear dynam- ies in (B. With an abuse of notation, let us denote by wi and one neuron's weight and bias after iteration t. For the first step. the preactivations are Gaussian, i.e., w/ Ze + by ~ (bo. 1). For later steps, we expand the gradient descent update for the neuron weights W, iteration t. Let denote by fP: (1 Ty; b--1)) )){[()] and g: X for the gradient computed in (B.2) at iteration f. By the gradient formula in (B.2), we have
Wt
X or + Ao wo A
and Xw,
Σ Nr Xge + do Xwo T=1
(B.8)
for some coefficient Xy. Let recall the decomposition X HV. The first equality in (B.8) indicates that w only contains information of V through the (t - 1)-dimensional projection &b = span(y? H) = For the second equality, the most recent component Xg. = HVg. in the pre- activations contains information from new gradient direction-thhe direction of projecting ge
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the orthogonal space of G {wo, 91 .....-1991 which we denote g. Data X's projection onto this new direction can be decomposed as
Xg: H (Φ Vg + +V g
where T Vgg is independent of all the previous updates, the projection is orthogonal to both + and G. Therefore, V Vor is high-dimensional independent Gaussian vector plus a low-dimensional coupling term &V $Vg 9f The argument holds true for all iteration steps, and if < d n, ap- proximately have Wy ~ [bbb, 1) thanks to the normalization of the weight wt. This argument can be made rigorous by use of the Gaussian conditioning technique (Wu & Zhou, 2023; Bayati & Montanari, 2011; Montanari & Wu, 2023) in the formal proof. See $E.3 for details.
B.4.3 WEIGHT DECOMPOSITION AND CONCENTRATION UNDER SPARSITY
For one neuron dedicated to the target feature U; and satisfying the initialization conditions in (B.7), decompose the weight wi into two directions: 1) the projection of we onto the 2-dimensional subspace spanned by wp and u;; 2) the projection of to, onto the orthogonal space wr-. We define
(we, u1)
Be 1|w 1|2.
04
||0:122
Using ar and Br, one can compute the first and second moments of the post-activation (P: under the decomposition of the pre-activations (into high-dimensional Gaussian component and a low- dimensional coupling term) obtained by the Gaussian conditioning technique. The post-activation Pe then gives rise to the next-step W++1, and thus obtain induced recursion over at and Be. As a more concrete example, let us take learning rate 1j = oC, and we can express Ot as
(wi, uz)
0. X y
a-1,1
||1,|12
| Telll
Recall that X HV. By splitting of V [ in the and splitting of H [H-i, Hi) in the column, we have
11:112 H. 9t + vg V.. H.
||
H
2. TVI H_
Q-1,
||1il12 ||X selll
||X pell2
|0l112 ||
Pelle
Signal
Noise
Here, we explicitly separate the signal from the noise. Our goal is to steer the neuron toward the direction of 04 while treating gradient contributions from other features as noise.
Controlling Moment of the Activations. To proceed, we must tightly control both the signal and noise terms in the nu- merator and the denominator. Concretely, this means bounding the first moment of the activation P (which enters the numera- tor) and its second moment (which controls the denominator), all while respecting the sparsity structure of spt. A core diffi- culty stems from the pre-activation
W
W
V
a
W ||
Xw, H We.
whose entries are independent-even under Gaussian approximation- because the feature rows He,: are correlated. This correlation invalidates the assumptions of classical con- centration inequalities, such as Bernstein's, s, and the problem only worsens once we apply the nonlinear activation. More- classical concentration techniques based on the bounded- differences property, such as McDiarmid's inequality (McDi- armid et al., 1989), are not applicable here. This is because the offers a uniform bound on the impact of each individual input change on the output, and it fails to capture that the activations
Wo
Visualization of the projection of W, onto 1) the sub- space spanned by wo and v, and 2) the orthogonal space w,!'
Figure 10:
property only
wwrrss-remainnng
most of the time.
To overcome these dependencies, we invoke the Efron-Sterl inequality (Boucheron et al., 2003). inequality, which requires each individual input change
Unlike McDiarmid's
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to have uniformly small impact on f. Efron-Stein only demands bound on the total conditional variance, namely
E ((x)))
S V
where 20()) denotes the vector obtained by replacing the ith coordinate of with an independent copy. This suraance-based condition is far more flexible in the presence of both correlation and nonlinearity, allowing us to derive the sharp moment bounds we need.
B.4.4 STATE RECURSION AND CONVERGENCE
We track at iteration the alignment 1- and the orthogonal component 3r of the neuron weight W/-- By exploiting the "Gaussian-like" concentration of the pre-activation Xw, H V wt and applying the refined Efron-Stein inequality to handle both feature correlations and the nonlinearity, one obtains the coupled recurrences
(1+ (1+0((1) + Ae ((-)) a-1.1- -
E..
S Se a-
Be
De- Pe-1 a--11111 + +G)
11111
Here, A & N/ and (--b) denotes the Gaussian tail probability beyond the threshold -b, which captures the activation sparsity. For clarity, we focus on the noiseless regime (i.e.. assume &. = 0) so that noise contributions are neglected. We now elaborate on these recursions in detail:
Recall that require Be < a-1,1 since the neuron should eventually converge exclusively in the direction of the target feature. In our framework, the minimal growth rate of the ratio 31/0-1,t is intrinsically controlled by A Ö(p N) ). By the definition of pt, this ratio is inherently larger than 1. Thus, to prevent an unbounded escalation B1/0-1,1 we must restrict
A to, at most, a polylogarithnn scale, i.e., A (().
2. If we additionally set 4 (--b)/
d for some (0, then the map a a a is contractive. Hence 1.1 grows from its initialization (dd-1/2) to 1-o(1) in O(1) steps. and the growth rate much faster than that of 3e/ 3//o-1,t thanks the sparsity condition ((-b)/(p1d) < 1.
From the above discussions, we already justify the inclusion of the Individual Feature Occurrence condition
[D.] 2 polylog(n) in (B.6) and part of the Bias Range condition 2
h²2
2
N
s)
+ ()b) < d'-s /n = Ô(d-s (pid)) in (B.5). The remaining conditions can be derived based on more careful analysis, including the noise term & and the initialization conditions (B.7).
logd d log n
C SUPPLEMENTARY DISCUSSIONS
C.I DISCUSSIONS FOR GAUSSIAN FEATURE ASSUMPTION
In our current theory, we assume that the features are Gaussian distributed. This is primarily used to obtain clean, closed-form concentration bounds
1. 1$B.4.1) Inner products between different feature directions, so we can control interference between features when multiple are active, and
2. ($B.4.2) Pre-activations y w x and their sparsity under the data model X HV.
In specific, employ the Gaussian conditioning technique, which allows to decompose a high dimensional Gaussian random vector into components that are explicitly dependent on the condi- tioning event and components that are independent Gaussian noise. Theoretically extending these probabilistic techniques to non-Gaussian settings non-trivial task and out of the scope of this paper.
One generalization of the Gaussian conditioning technique involves using features uniformly sam- pled from the unit sphere in our synthetic experiment ($3). This setup better reflects real-world
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scenarios where features in LLMs often normalized through noyrrnoomaalization Despite the different feature distribution, we still observe the resonance phenomenon predicted by our theory. This is evident when inspecting the Feature Recovery Rate (FRR) in Figure 2, plotted for varying activation frequencies and dimensions d. This observation suggests that our theory robust to the specific distribution of features.
In addition, the empirical results on real LLM activations further confirm that the proposed GBA method works well when V's rows are just the features learned by the LLM, which are non-Gaussian.
C.2 DETAILS ON TOPK AND L1 TRAINING METHODS
We provide here more details the training methods used in our experiments, including the Sparse Autoencoder (SAE) with TopK activation and SAE with L1 regularization.
Sparse Autoencoder (SAE) with TopK activation. In an SAE with TopK activation, sparsity is enforced by selecting only the K neurons with the highest activation values in the hidden layer. Let y W bpre) + b be the pre-activation values of the hidden layer. Let Φ(y) be the activations after applying standard activation function. The TopK selection mechanism, denoted as Sx(-), operates on Φ(y). For a vector v E MR,, (v( produces a vector v' E IR such that:
U/ (0
if 0g is among the K largest values in v.
o/
otherwise
for j E [M]. The post-activation in a TopK SAE is:
Sx (<(W(x bpre) + b)):
which by definition is K-sparse. The reconstructed output is:
diag(a) WT + bpre.
Let θ = (W, Upre; b, a) be the parameters of the SAE. The loss function for the TopK SAE is the reconstruction loss:
Lree (2;; || 2
Sparse Autoencoder (SAE) with L1 regularization. In an SAE with L1 regularization, sparsity encouraged by adding penalty term to the reconstruction loss, proportional to the sum of the absolute values of the hidden layer activations. Let W(x prre) + b be the pre-activation values of the hidden layer. Let <(y) <(W(x bpre) + b) be the activations after applying standard activation function; these are the hidden layer representations that will be encouraged towards sparsity. The reconstructed output is:
diag(a) T + bpre-
The loss function for the LI SAE, L(x; θ), incorporates both the reconstruction error and the L1 penalty the hidden activations
C(x; θ)
|| 21 + A Σ lasl 11155122
where À is the sparsity penalty parameter that controls the strength of the regularization, m is the number of neurons in the hidden layer, and w. is the j-th row of the weight matrix W.
JumpReLU. In our real-data experiments, we also consider the JumpReLU activation, a non- smooth, non-monotonic function. Conceptually, it behaves like ReLU for positive inputs but intro- duces a sharp jump for sufficiently large inputs. In our implementation, we adopt a simplified scalar form adapted neuron pre-activation wT bm:
[0,
W. + bm < 0, if wTw + bm 2 0.
JumpReLU(ww dm) =
w TT..
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This activation acts as hard thresholded identity: it passes the neuron's 's response only when the pre- activation crosses bias-controlled threshold. Although JumpReLU does not satisfy the smoothness or Lipschitz conditions required in theory (see Definition B.3), it is empirically effective and included our experimental comparisons $5. To train SAEs with JumpRel.U activation, we follow Rajamanoharan et al. (2024b) and use straight-through estimators for the gradient of JumpRel.U with respect to the bias bm Specifically, for small constant > O, we approximate the gradients
@JumpReLU(; b) 0. if + <b<<0 <0 By
@JumpReLU(; b) [. if 1 by U
1, if fby+b20*
The approximation follows the logic: the gradient with respect to is in essence the gradient of Heaviside step function, which can be approximated by smoothed version over a small interval around the threshold. Note that for GBA method, we do not apply any gradient for the bias; instead, we update the bias through the frequency control mechanism described in $4.
Activation sparsity. For both the L1 and TopK SAE, we define the sparsity the number of non- zero entries in the latent 2, i..., ||2||o-
Minor notational discrepancy. In the main text and above definition we express the activation as σf fi + bm whereas in the definition above the JumpReLU activation is indeed bivariate function of w w and bm- This slight difference is purely notational and does not affect the un- derlying functionality or the definition of activation sparsity. For simplicity, we always stick to σ( w.,, m x + bm) in the main text, even for the JumpReLU activation.
C.3 DISCUSSIONS FOR GEOMETRIC SPACING OF TARGET ACTIVATION FREQUENCIES
In our Group Bias Adaptation (GBA) method, we assign neurons into K groups with Target Activa- tion Frequencies (TAFs) {Pi } k= that are geometrically spaced between a Highest Target Frequency (HTF) and Lowest Target Frequency (LTF). This design choice motivated by several theoretical and practical considerations:
1. Theoretic guided group allocation. The theoretical resonance condition depends on p being within at least a multiplicative band around f (can be even wider though if we have less superposition). A geometric grid guarantees that for any feature frequency f within [pK.Pıl there exists some group with TAF pı within a constant factor of f, regardless of the exact exponent of the empirical feature distribution.
2. Better coverage in log-frequency space. Geometric spacing minimizes the number of groups K needed to cover wide frequency range [px, Pi] to just logarithmic in the ratio P1/PK. Empirically, we show in Figure that having K > 10 groups is sufficient for covering much of the frequency spectrum. However, other spacing (such as Zipfian) would potentially require many more groups to achieve similar coverage, as the frequency decay is slower than geometric. This would dilute the number of neurons per group, and we might risk missing features in that frequency range.
C.4 EVALUATION METRICS
We explain here the details of the evaluation metrics used in our experiments to assess how well the GBA algorithm recovers the underlying features.
We first introduce the maximum activation and neuron Z-score, which are used to measure the quality of the learned neurons. Then, we introduce the notion of Max Cosine Similarity (MCS) and Feature Recovery Rate (FRR), which are used to measure the quality of the alignment between the learned neurons and the ground-truth features, or the consistency of the learned features across different runs. We also introduce the neuron percentage, constructed from the MCS, which is used generate Figure 5.
We introduce maximum activation and neuron Z-score of neuron as follows.
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Maximum activation. Unless specified, we define the maximum activation of neuron the maximum of its pre-activations over the validation set:
Maximum Activation(m)
max ae Validation Set Um (2), where Um (m)))) = wmn (2c bprc) + dm-
(C.1)
Note that the maximum activation is computed based on the tokens in the validation set, which is a held-out dataset separate from the training data used for evaluation purposes. It maps each neuron to scalar, characterizing the maximum pre-activation the neuron across all validation tokens.
Neuron Z-score. Let ((+) denote the neuron's activation function (e.g., ReLU, or JumpReLU). For each neuron m and a minibatch {rr we define its post-activation responses as
Φm.i of w (4; bpre) + bm))
= ,,
where Wm RR is the neuron's weight vector and bm IR is its bias. We can compute the mean and standard deviation of these activations in the minibatch as
Pm
Pm.i:
8y
(dmi Hmm)
B
The Z-score of neuron m on data point zj is defined as
Zu (4m,i pm))/m Sm IR.
We can also take the maximum of the Z-scores over the batch:
7max
(#m.max - /mm)/8m: where
Φm,max
max Om, 15i5 B
(C.2)
A large value of Z (or Zmax > 0) indicates that on some input the neuron's 's activation Φm, lies multiple standard deviations above its mean. Thus, when Z is large, neuron m is well- learned to sensitively detect certain data points within the batch. More specifically, when Zmax is large, the two following conditions hold:
There exists some z; within the batch such that Øm, >> Mms i.c., the neuron's activation 6m. "spikes" " for input I,. Low Baseline Variability: Within the whole batch, the neuron's activation Φm, is relatively stable, i.e., the standard deviation 8m is moderate.
Strong Selectivity:
As result, Zmax serves quantitative measure of the neuron's 's specificity the batch of data. When generating Figure 5, we use the maximum Z-score of each neuron across the whole validation set select a subset of neurons.
Next, we introduce the Max Cosine Similarity (MCS) and Feature Recovery Rate (FRR) metrics, which are used measure the quality of the alignment between the learned neurons and the ground- truth features, or the consistency of the learned features across different runs.
Max Cosine Similarity (MCS) for synthetic data. Rd, we define
For each neuron m with weight vector um
(Wm, vt) |mm 12 ||1/||2
MCS(m) max xE(n
E [-1,1].
By definition, (C) if and only if w. coincides with one of the true features Vç.
Max Cosine Similarity (MCS) for real data. For real data, we do not have access to the ground-truth features, we define the MCS as the maximum cosine similarity between neurons across different runs. This definition is used in Figure 5. Specifically, consider the trained neurons weights W E R Mx for j = 1, ..... J where J is the number of runs with different random seeds. We fix the first run as the host run and compute the MCS for the m-th neuron in the host run with respect to the j-th run with j > 2 follows:
MCS(m..) maxfcco(W<() (())))
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Here, the term inside the max is the cosine similarity between the m-th neuron in the host run and all neurons in the j-th run, which is an M -idiinnniinnal vector. The maximum taken outside can be interpreted as finding the best match for the m-th neuron in the host run. Now, given threshold r for the MCS value, i.e., the x-axis in Figure 5, we define neuron m to have an MCS above the threshold if MCS(m, j) 2 for all j 2 2. We require this condition hold for all runs j 2 because if the algorithm learns a consistent feature, it should be present no matter which random seed is used. When this is the case, neuron in the host run can find a corresponding neuron in each of the other runs that has cosine similarity above the threshold 7. Thus, by computing MCS for all the neurons in the host run, we evaluate the consistency of the learned features across different runs.
Neuron percentage in Figure 5. Recall that we call the first run of the algorithm the host run. Under the definition of MCS, Figure we plot the neuron percentage as a function of the MCS threshold т. In particular, for any threshold (x-axis in Figure 5), we compute the fraction of neurons in the host run that have an MCS above the threshold across all runs. That is, we define
M
Neuron Percentage(r) = M
Σ m=1
(MCS(m, 2 T, Vj 2).
(C.3)
By definition, this quantity computes the fraction of neurons in the host run that have an MCS above the threshold across all runs j 2 2. If this quantity is large, the algorithm is able to produce consistent results across different runs with different random seeds. Moreover, because a considerable portion of the neurons of SAE are rarely activated, instead of enumerating over all neurons as in (C.3), we can also consider the neuron percentage over a subset of neurons, denoted by M [M]. Then, focusing on M, we define the neuron percentage as
Neuron Percentage M)
Σ ((CCS(m..) 2T, ( 2). ME.
(C.4)
M
In particular, in Figure we choose M to be the top-a subset of neurons in terms of the maximum activations or neuron Z-score in the host run, which are defined in (C.1) and (C.2), respectively. Note that these two metrics are computed based on the validation dataset. The y-axis in Figure is computed as in (C.4) with these two versions of M.
The notion of Feature Recovery Rate (FRR) is only used for synthetic data, where we have access to the ground-truth features.
Feature Recovery Rate (FRR). For one monosemantic feature U4 we say it is recovered if there exists neuron m E [M] such that the cosine similarity between the neuron and the feature is above a certain threshold Talign:
0
0/3
1, = (
1
if m E [M] such that (00m wt)l/1bt//2 2
Falign
otherwise.
Then the Feature Recovery Rate is
FRR = n2 Σ1.
1 1,
E [0, 1].
In words, FRR is the fraction of ground-truth features v/ that have been recovered, i.e., aligned to at least one learned neuron. Here, we find the following way to define the threshold Talign useful:
Figure 11:
An illustration of the learnable region surrounding the feature. Any neuron weight within the cone has cosine sim- ilarity ahove the threshold with the feature.
Talign cos G arccos max i#$
(Ut, vz) ||ei|| |
(C.5)
Intuitively, the angle given by arccos in (C.5) is the smallest angle among all pairs of features v, and vy in V. which is denoted by 0 in Figure 11. Then, if neuron exhibits a cosine similarity above the threshold Falign with a feature v. then it lies within the cone centered at U; with angle 0/3. See Figure 11 for illustration. By choice of Taliga. these cones associated to all monosemantic features lie in the d --diennnnoo sphere without overlapping, ensuring that each neuron exceeding the threshold is uniquely aligned with single feature.
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D ADDITIONAL EXPERIMENTS DETAILS
We provide additional experimental results and implementation details that complement the main findings presented in the paper.
D.I SYNTHETIC EXPERIMENTAL SETUP
We generate synthetic data X HV satisfying decomposable conditions outlined in Definition B.1. In the default setting, each row of H contains exactly nonzero entries, each with value 1/ 1/v8, and the support of each row is chosen independently at random. We implement the BA algorithm with fixed TAF p. where the SAE adopts the ReLU activation. We fix the output scale am 1 for all m E [M] and the pre-bias 6prc- = 0, and initialize the weights wh uniformly on the unit sphere gd-1 with bias (0) O.
In synthetic experiments, we use Spherical Gaussian features. For each sample I, (j E [N], we randomly sample indices (with replacement) from [n] to form multi-set Sy. The corresponding features are then combined with a weight 1/v/ to construct the reconstruction target:
Σ w//33.
I
EE
To evaluate feature learning of neuron m, we use the Max Cosine Similarity (MCS) metric. For any neuron m, MCS is defined max]g[m] [wmm///mmlll U// [| Thus, MCS measures how well a neuron aligns with the most aligned feature in V. We say a neuron is aligned with some feature if the MCS for that neuron exceeds a certain threshold. To evaluate overall feature recovery, use the Feature Recovery Rate (FRR) metric, defined as the proportion of features that are aligned with least one neuron, See $C.4 for more details on these metrics and the choice of thresholds.
D.2 ADDITIONAL DETAILS FOR
Data and model details. We choose the subsets of Github and Rikipedia_en of P iile (Gao al., 2020) without copyright as our datasets. The Github dataset is a collection of 1.2 billion tokens from public Gi Hit repositories, while the Wikipedi a en subset contains 1.5 billion tokens from English Wikipedia a articles. We use the first 99.8k rows from each dataset for train- ing and the next 0.2k rows for validation. Each row in the dataset is truncated to the first 1024 tokens after tokenization. Therefore, the total number of tokens roughly N 100m. We use the Swen2.5-22 5B base model (Yang al., 2024) our LLM, which has 1.5 billion parameters and MLP output dimension 1536. We attach an SAE to the output of the LLM's MLP output at layer 2, 13, and 26 with M 66k neurons, resulting in three different SAEs for each dataset. The dimen- sion d of the input data points is equal to = 1536. We use the JumpReLU activation (Erichson al., 2019; Rajamanoharan al., 2024b) for all training methods.
Training details. We train the SAEs using methods such as GBA, TopK, LI, and BA, where BA simply GBA with one group. For all these methods, we use the AdamW optimizer with learning rate of 10-4 and a weight decay of 10 Since the sentences are truncated to 1024 with padding token removed, we set the batch size to 8192 tokens and buffer size of B 40k tokens. Each run can be completed using a single NVIDIA A100 GPU with 80GB memory, and we train 8 epochs for each method. The hyperparameters of each method are as follows:
For GBA, we set K E (3, 10, 20). P1 E {0.05, 0.1, 0.3,0.5). and pk E {10-1, 10-3,5 x 10-3), where K is the number of groups, Pı is the target frequency of the first group, and pk is the target frequency of the last group. In addition, we have [pи]eeкk] form geometric sequence.
2,3 10-3x000 10-33 and vary For the BA method, we set the HTF to be from (10-1,3 10-2 20-2,00-2,33
the choice. The other parameters are the same GBA.
For TopK method, we implement two versions
the pre-activation TopK and the post-activation TopK. See SC.2 for details. We vary the value of K in {50, 100, 200, 300, 400, 500, 600)-
For L1 method, vary the penalty parameter An
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D.3 COMPARISON BETWEEN JUMPRELU AND RELU ACTIVATION
For the SAE trained on the Github dataset at layer 26, we compare the performance between J and the standard ReLU activations across all methods considered in this paper. As shown in Figure 12, the sparsity-loss frontiers for TopK and LI methods are nearly identical un- der both activations. However, the GBA method demonstrates marked improvement when using Jump activation. With ReLU, decreasing the neuron bias also reduces the output magnitude. Thus more neurons are needed to compensate for the loss of output magnitude, which leads to a less sparse model, which degrades the sparsity-loss frontier. In contrast, JumpRel.U decouples the neu- ron output magnitude from its bias bils-only the activation frequency is influenced seluenced-yyeldding a more robust sparsity- performance.
0.4
pupt-jumpteeujggeaatt
Relliree act
-G+ Tupt-)).mpPe (ppotwrrt
0.4
-Gh Tipt-Re Uiposbart
0.4
Topk tump
L1 1J-mmpp8LL)
L1 Rel
0.3
ale GB4-JunoReWU
0.3
ReLL
0.3
+ T humphctutttt
L1+RelU
0.2
0.2
0.2
GBA+JumpReLJ
0.1
0.1
0.1
lower is better
lower better
lower is better oe
0.2
5
20
0.2
1
20
0.2
1
5
20
Figure 12: Comparison of sparsity-loss frontier between JumpReLU and ReLU activations. The left and middle plots show the sparsity-loss frontier with JumpReLU and ReLU activations, respectively. The right plot combination of the two, where the faded plots represent the sparsity-loss frontier of the ReL.U activation.
Bias clamping to prevent over-pparsification During the bias scheduling subroutine of the GBA algorithm (Algorithm I), we enforce clamp on the bias values, restricting them to the range [-1, 0]. This constraint serves two primary purposes. The upper bound of 0 ensures that a neuron only activated when the input data exhibits a sufficient alignment with the neuron's 's weight vector. Con- sequently, allowing negative bias values (b,,. < 0) effectively prevents excessive or premature acti- vation of neurons.
The lower bound of implemented to avoid ova--deactivation and the emergence of a reinforcing loop. We have observed experimentally that when the pre-bias (épre) significantly deviates from zero, certain neurons may develop weights that are in opposition to the pre-bias to compensate for this drift. As these compensatory neurons are more likely to be activated by the initial pre-bias, the GBA algorithm might inadvertently continue to deactivate them by further reducing their bias (bm)- This deactivation would then necessitate increase in the neuron's 's weight to maintain its influence, leading to a counterproductive cycle of deactivation and weight growth.
By limiting the bias to be less than 1, effectively interrupt this reinforcing loop and promote training stability. The rationale behind choosing -1 as the lower bound stems from the fact that our input data is normalized. This normalization typically results in pre-activation values that are significantly smaller than 1, with values approaching 1 only when the data strongly activates specific neurons. Therefore, lower bias bound of -1 provides sufficient range for deactivation without causing the problematic feedback loop. This clamping strategy has been shown to significantly enhance the stability of the training process.
E GOOD INITIALIZATION AND GAUSSIAN CONDITIONING
In this section, we provide proofs for two important lemmas: Theorem E. on the initialization properties and Theorem E.2 on the Gaussian conditioning. These lemmas provide the necessary foundation for analyzing the SAE training dynamics, enabling us to isolate and control the relevant sources of randomness throughout the analysis.
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E. INITIALIZATION PROPERTIES
If we initialize the network with a sufficiently large number of neurons M, then for each neuron. there must exist feature that aligns well with it. However, the question is how many neurons we need to achieve sufficiently large alignment and with all features of interest simultaneously. Theorem E.1 provides an answer this question. particular, we prove that when M is sufficiently large, for each feature ti, we can find a neuron m, that aligns well with it (InitCond-I) while maintaining small alignment with all other features (InitCond-2).
Lemma E.1 (Good initialization). Given n i.i.d features {ui)i=1 with is ~ NN Ia) and weights M {wm m== independently initialized from the uniform distribution on the unit sphere, then for any constants e E (0, 1) and с > O such that n upper bound exp(-n with probability at least (0) 1 n== over the randomness of both {u:]==1 and {wm Im=1. one can select a sequence of neurons [m,)== satisfying the following properties:
1. For any E [n], we have
InitCond-1: (Ue, (2)) (1- 2og(M))))))
2. For any i [n]. when conditioned the selection of neuron my which aligns well with feature 0, in the sense of InitCond-l, the distribution of the remaining features {uylizi remains unchanged, iee, they independently drawn from (0, Ia).
3. For any i [n], when conditioned on selecting neuron my with probability at least n 1- over the randomness of {1g hi4s. we have
InitCond-2 (uz. (0!) I v2(1+e) 22 n, Vj #i
Proof of Theorem E.I. We present the proof by constructing such m1, my,..... my explicitly. Suppose we provided with n features v1, U2,..., Un. and M neurons with initial weights (0)
w W2 ,..., W We first put all the pair-wise alignments (v,, (m) > into a matrix R
(0)
where Aim (vi, w/)) for i [n] and m [M]. The algorithm execute follows for going from to n:
Randomly divide the M neurons into n disjoint groups MI, M2,...... such that each group M; contains M/n neurons.
2. For each M,, find the neuron m, as the that maximizes the alignment with feature e. i.e.,
m. argmaa Aim = argmax((;, (0)) mEM,
mEM:
By construction, we know that the selection of m; is independent of the selection of my for i + j. It is not hard see that the distribution of (e,, wil) is the same (up to scaling) as the distribution of the first coordinate of random vector uniformly distributed on the unit sphere. Therefore, for each i [n]. each group (Asm)) J,, iid sampled from the following distribution:
1 l2
Arm MEM.
where Z% ~ ~10, 1), Vk E [d].
7
By the concentration for Chi-square distribution, Theorem J.1, we know that the denominator and also the norm of ||ol|e satisfies
P
z d 2 2lgg 5-1 2log 5-1) s 88
A=1
P
[1: d > VVllog 5-1 + 2log 58.
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To proceed, label each d-dimensional random vector as Z(..) -2 (Z(
Zz Z where
the superscript (i, m) corresponds to feature i and neuron m. Applying a union bound over all M/n pairs of (i,m) m) and choosing
n -/MM for some universal constant c, we deduce that with probability at least - n= the following holds for all i E [n] and m E [M]]
2 m) (d CVdlgg(MM) -0 C'log(nM)) 1/2
Aim
++CCVloognMM + C' C'log(nM))) 1/2
where C' is universal constant. Moreover, by property of the maximum of Gaussian random vari- ables in Theorem J.4, it holds that
(M/n) (M/n)a-6/4
_ (8.m) 2 (12
2 exp(
(E.1)
MEM,
3 TT 3/Tlog(M/n)
Here, we divide by 2 because
dog((M)) C' loog( (o) 1/2
A zith ((im).
> 1 - ((i,m)
C+CCVlog(nbll C' Colo(nn 1/2 1 8/2 2
for small constant e. Consequently, by multiplying both sides of the inequality inside P(.) in (E.1) by 1-c/2 we can recast the probability statement so that the maximum of A.mm over all M: exceeds (1 - =) /
By taking a union bound for E [n]. the probability of successfully finding sequence of neurons mı, m.....; my satisfying A,m > (1 e) log (M(())) for all i [n] and m [M] is least
P Vi [n] max A,m ()- 2log(M/))) 21 21-n exp (M/n)s-52/ mEM;
n
3 log (M/g))
where we can safely take to some constant as the failure probability is exponentially small in given that M 2 n To this end, we conclude that with probability least n we can find sequence non-overlapping neurons m1, m2,...; m. such that Ai,m > (1 - e)--v)22l0g (og((n))) for all i E [n].
Observe that the selection of each neuron my is done independently for each feature. Consequently, when we condition on the selection of m,, the distribution for the remaining features (uy (Uj)izi re- mains unchanged. This proves the second statement.
remains analyze the probability that Aj.m < < V2 for all j [n] and A j. By the second statement, we know that when conditioned on neuron m, the collection {Aj.m bis (for any fixed i) consists of (n 1) independent and identically distributed random variables with distribution (0, 1). Thus, we can apply the tail probability for the maximum of Gaussian random variables in Theorem J.2 to obtain
P max Aim, > V2 +s) . llg S 122(t() jElljus
< -1-4e
Thus, prove the last argument for Theorem E.
A direct corollary of Theorem E.1 is that InitCond-1 and InitCond-2 hold simultaneously for all i [n] and j A with probability at least
n n 4= < 1 n -F after taking union bound over the success of InitCond-2 for all i E [n]. These two conditions together imply that the neuron m; exclusively focuses on feature v; at initialization, which is crucial for developing a o(1) alignment with feature during training.
E.2 REWRITING THE GRADIENT DESCENT ITERATION
Single neuron analysis. the previous Theorem E.I, we have shown a correspondence between each feature and a neuron m, such that the initial weight of neuron m, aligns well with feature 2 while maintaining small alignments with all other features. In other words, m, is the neuron that most likely to leam feature t; during training. As the neuron dynamics are decoupled under the small output scale assumption, we only need to analyze the dynamics of neuron m; to understand how feature en is learned.
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Notation. In the following, denote by the feature of interest and by w, the weight of the corresponding neuron at iteration t. Let the maximum number of steps considered and the time step t ranges from 0 to T. For the sake of notational convenience, of interest by w--1 v and the normalization u-1 = e/|le|22- Meanwhile, Wh = W0 is the initialization that is already normalized to unit length. Here, the bar notation indicates that the vector is normalized to unit length throughout the whole proof.
also denote the feature
Reformulating the iteration. In this section, reformulate the gradient descent update (B.1) isolate the contribution of specific feature from the remaining features. Recall that the data matrix is given by X = HV. where I E g is the weight matrix and V E _gxd is the feature matrix. The gradient descent update (B.1) with gradient explicit (B.2) is
Modified BA: W2 W/-1 79h
where
St Σ
| + 17
which can be written in terms of and as:
y/ = V WW,-1,
by = A.H)))) 4 = H ( (Hyl::
(E.2)
Wi = V + 17 11 We = w/ ////lll2.
Here, the meaning of these quantities are given as follows:
Ut Rd is the projection of the normalized weight vector onto all the features, which refer to as the feature pre-activation.
b IR is the bias term updated by bias adaptation algorithm A.(-) that depends the feature preactivation and time
E I" is the feature post-activation that aggregates the post-activation information from all the data points back the feature space.
Wi E Rdd is the unnormalized weight vector after one step of gradient descent update, and Id is the normalized weight vector.
In our analysis, as the bias is fixed, (-) always returns the same bias value. However, keep this general form which can be useful for adapting the current proof framework to handle more complex bias adaptation algorithms. Note that yo( by) E R obtained from the gradient calculation in (B.2) is not exactly the post-activation (recall definition sp(;; b) = o(x + 6) + o' + b)æ, where ó the actual activation function. However, in the following proof, we will abuse the notation and refer to yp( Hy; by) as the post-activation for brevity.
H2
H2 ... H
H
Hg # 0
H1 Hip ... Hig-n
Hn H
H3 Hm ". Hg
Hzn
Hgi Hzg ". Hag-n
Hagn
Hy Hig Hy
Hyn
: :
%. H
V
HyI
Hya
Hrin
Hyy
Hn
". Hx
Hyn
"
H
Hyi
Hyz
H
Hmmen
[8g
Hy:
E
The weight matrix H is splitted into matrices F. and F by row according whether the corresponding entries in the i-th column nonzero entries i-th column H are collected as vector
(a)
(b) Isolating the i-th fea- from feature matrix
not. The
Figure 13: Illustration of the split of matrices H and V.
Without loss of generality, suppose is the i-th feature. To isolate the contribution from feature of interest v from the remaining features, we decompose the weight matrix into three parts: (i) 6: the non-zero entries of the i-th column, (ii) F: the rows with non-zero entries in the i-th column,
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and (iii) E: the remaining rows with zero entries in the i-th column. Formally, suppose is the i-th feature, then we decompose H as follows:
0 (Ha: Hai # 0) REIN F = (Hxg HA + 0) xE(T) E (Hæj H = 0) (E(N)
(E.3)
Notably, the rows of E and F do not include the i-th column of H, it is already isolated vector See Figure 13a for an illustration this decomposition.
Using the above decomposition, we can rewrite the actual projection of the weights w each data point as
W Interleave (IF; ([F E] V- + [0: 0] v (T w(-1)
= Interleave (F;; FI] Ut.-i + [0; 0) u w.-1).
where [E; F is the vertical concatenation of E and V- the feature matrix with the i-th row removed, and Vt.-; = V_, is the vector yr with the i-th entry removed. The interleave operation simply restores the original order of the rows in H. Therefore, we can rewrite the original Me in (F.2) as
(E.4) In order to avoid overcomplicated subscripts, we let V denote the feature matrix V-, with the i-th row removed, and let refer the original i-th row of V. See Figure 13b for an illustration of this decomposition. We also rewrite Vt.- as ye. and following the above notation, we still have V = Vi--1. Now with (E.4), we can explicitly separate the contribution feature v from the remaining features in the gradient descent iteration (E.2) and obtain the following equivalent iteration:
dt == 4 (HH = FT ((Ey,,(()b)) + F (FF(
Gradient Descent Iteration
feature pre-activation:
V/1 = Vwi-1, Wz-1 = Wt 11/11//11111 122
bias scheduling:
b. = Ar(bi-1, Eyc. Fy: + 0 (2-1),
feature post-activation:
ET be) + F ((Fy + +0. w; be),
weight update: w. V "u v0 (Fy : 0 v wi-1:bb + W 1
(E.5)
Note that the notation in (E.5) self-consistent with E, F, defined in (E.3) and V, v defined below (E.4). We will keep using this notation throughout the rest of the proof.
E.3 GAUSSIAN CONDITIONING
Since both the feature of interest v and each row of the feature matrix follow Gaussian distribu- tions, leverage the properties of Gaussian distributions to simplify the dynamics. However, the coupling between different iterations prohibits a direct application of Gaussian properties. This challenge motivates us explicitly split the intermediate variables in (E.5) into two components: (i) a coupling component that lies in the subspace spanned by the previous intermediate variables, and (ii) independent component that is orthogonal to this subspace. We can then apply Gaussian concentration arguments to the orthogonal component to simplify the dynamics.
Additional notation. To achieve this, we introduce some additional notations. Let us define P Pw-r.-,- I as the projection of z onto the subspace spanned by {w-1.... W2-1}, and P! - x- P.
w 1t-1 I. as the orthogonal projection. In the following, we use the notations wi- P_ w-1-t-1 We
to denote the new direction induced by w/, and we define ui- = P ur in a similar manner (note that dn starts from 1). Note that when < U1 is empty and PL P becomes the identity mapping. Also, we enforce w-1 -h u.
In the following, use the trick of Gaussian conditioning (Wu & Zhou, 2023; Bayati & Montanari, 2011; Montanari & Wu, 2023) to simplify the dynamics in (E.5). Specifically, we will define an al- ternative dynamics that is distributionally equivalent to the original one, where for each iteration, two
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new independent Gaussian vectors are introduced to replace the original Gaussian components com- ing from the V matrix. To make the presentation clearer, we will denote the variables in the original dynamics in (E.5) by (Vt, We, Ue, by) and the variables in the alternative dynamics by (Dt. We, Ue be) in the following proofs.
Lemma E.2 (Alternative dynamics), For any E N, let 2-1, 20. ..... 2, and 21...., 2, be sequences of ii.dd random vectors from (0,,n-1) and (00, la-1), respectively, with mutual independence. In addition i:t and a are also independent of the initialization wi and the feature of interest v. Consider the following alternative iteration for (Wt. W.):
P.l. Dt 7
2r +
dr,
(E.6)
Wr
Σ
(Pat..
Rr, Ut)
( Ut)
|00 12
w
01.
- 2
10+ 12
|
|| 1 1|2
E 1i 1/2 +
co Fij + 0 0.00 w; be) +
where we define the alignment
w)
W
OlT,
with We
[ ll2
In addition, (b., we) in the alternative dynamics are updated by the same formula as in (E.5):
b, = (b, Fūr + u.v TT @. U = (pEūib) + F + 0 v Wr bz).
(E.7)
Then, conditioned on w- = v (the same as our previous definition of and wi = wy being the initialization of the neuron weight, the alternative dynamics (Tr, Wr ū, b. from (E.6) and (E.7) and the original dynamics (yr, Wr, Ur by) from follow the same distribution.
Proof of Theorem E.2. To show that the trajectory from (E.6) and (E.7) follow the same distribution as the trajectory from (E.5), we first decompose the iteration in (E.5) in the following lemma.
Lemma E.3 (Decomposition). For the iteration in (E.5), define the alignment between the weight vector We and the weight direction wi as OrTY (w,, w w/) //1l l|2, Then, we have the following decomposition for the preactivation vector Ye 1m-1;
[-1
l 1l2
u
Σ
TL V uir
Σ 0,
Vt
ar.-1
2
|u ll2
12"
and the following decomposition for the unnormalized weight vector Wi E IRd;
1-1
w
(u us)
|/w
w
Wt
Mier
V
|| |
112
ut)
| l|2
| lle
|| ll2
| ll2
P.l
VT
ll2 + vo
(Fye + 0 v (x:1:bb) + + 11
|
Proof.
See $E.4 for the proof of Theorem E.3.
With the above decomposition, if we do the following substitution for y and wP in the above lemma:
P P Zt P.L.
V
PLl
P. 11-11
VT
ut | 112
I 21
the assertion in Theorem E.2 follows immediately. The following proof is devoted to showing that the substitution does not change the joint distribution of the whole dynamics. To show that, we just need verify that for each iteration when conditioned on all the history up to ieeraionn & - 1, the
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two newly introduced vectors Pul Vw, /|[ 2/2 and Pu VT us /1/a ||2 still follow a standard Gaussian distribution and are independent of all the history.
To proceed, we denote the original iteration in (E.5) by (yt, wt, ue by) and the alternative iteration in (E.6) and (E.7) by (Dt, Wr, Me Tt). Following explicitly from the decomposition in Theorem E.3 and the construction in (E.6), in both iterations.
further derive the following dependency between the variables
Lemma E.4. For each iteration (we, we) in (E.5), it holds for any 2 1 that
t-1
U E a
w w-1:0
P.l
Wi
P.l
V
u
u1
112
1
|| [e lle
7=-1
r=1
t-1
W2 a w-1:0
pl
W
pl w-1ir-1
uT
uirr
1
7=-1
where (X) denotes the o-algebra generated by the random variable X. For the Gaussian condi- tioning iteration (üt, Wx) in (E.6) and (E.7), it holds for any t 2 that
ūe E w 1:00 {Pa a - D (=+)==)),
We E f PPa 2 3/)/=1 (=+)=
Proof. See $E.4 for proof of Theorem E.4.
The message of the above lemma is intuitive: each iteration only inserts new randomness coming from
PL ame-t [w 1l2
and
Pa ------
V
11
for the original iteration, and from
PLl - 41- and P.L. P. P Zt
for the alternative iteration. Using the dependency results, we next prove the equivalence between the trajectory (w_1. Wo. (Tr, Wr, ū, Te, /= from the Gaussian conditioning and the trajectory {w_,, Wo, (Ur, W/; Mr, b-)+ from the original iteration by considering the conditional distribu- tion of the newly introduced randomness at each iteration. Let us define A, as a realization of the random variables (w-1:0. 1:0, 1:1-1 Z1:e)
W 1111
P1 V
P. 1or
V
112
|
ll2
r=1
By property of the Gaussian ensembles, it holds that
pil V
w/
{
(
(w-1.o0
pal.
w 2-1
pl --i-r-
V
u
A
usst |ww
11
u1.r
w 12 1=-1
l lla ==1
d p.l V. [w 112
wi
(
w-1:00 { ussr
p.l V w
t-1
pl W-1---1
V
u
A
|
[|2
| |ut
r=-1
d p. 21 ((w- Uis 2-1.1-1, Z11) Ar).
(E.8)
where V, _ Viis is independent copy of V and is independent of all the histories. Here, the first equality holds because Ma:t Vw /|| ||2 is orthogonal to any of the previous row/column space that we have conditioned on. In particular,
pll V
is orthogonal to {P Pair V wi l
in the column space of V since wi is )i-L
Mase orthogonal to w for any 7 < t.
I
112
P...
V
w || 12
is orthogonal to {P.... VT
m.. 1 in the row space of V since P.L. is
Mist projecting to the space orthogonal uf for any t.
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Moreover, V is also independent of w--1 and the initialization Wp. See Figure 14 for more intuitive explanation. Therefore, the conditional distribution of P. Vw, / 2l2 is the same as that of an (n - ( -idmeenioon Gaussian vectors. Hence, we able replace V by independent copy V. For the second equality, we can set 2n V/wz /||w> 22 which is again a Gaussian vector independent of all the histories. Similarly, let Be be realization of (# 1:0; 2- Z1
w
t-1
w- w-11::
{ P
PL W-1--1
VT
I l2 -=-1
T=1
we similarly have for PaL
w-iit-1 VTu /l///luu M that
P w-1a-1
VT
ut 11 ll2
w-1:0-
Pul V
Wi
t-1
p
V
t-1
B
| 1l2
-1
2
r=-1
r=l
1--a-12 1:0, 2-1:1-1, 11 Bi..
(E.9)
To this end, it can be concluded that
The initializations (w-1, wo) and (w-1, Wa) are the same.
2. By (E.8) and (E.9), we have the same conditional distributions for the updates of
(PuL. Mide Vwz // 12. P.l. -- Vu,f//lee (12) and those of (P 2t, P.l. 1 =c), which
means the conditional distributions of (ye, w/) and (Wt. ih) given the past are the same.
3. The updates of (b,, uz) and those of (b,, ū.) are also the same.
We hence conclude that the joint distribution for the two iterations are the same for any time Consequently, we obtain that
[ Wo, (Tr, Wy, Cr br) d {w_,. wo, (Br, Wr Uy by b-1
This completes the proof.
Since the alternative dynamics Theorem E.2 are distributionally equivalent to the original dynam- ics, work exclusively with the alternative formulation below. We emphasize the following key point when running the alternative dynamics for steps:
The randomness in the alternative dynamics comes from the initialization Wo the feature of interest and the random vectors 2-1.T and št..
Since the system is rotation-invarian without loss generality, we fix the direction of the initial- ization Wp in the following analysis, and only consider the randomness over v, 1:1,7, and TT.T.
Remark. In fact, the iteration in (E.6) is reformulation of (E.5) obtained by decomposing the random matrix V into its projections along the row spaces ut us-,.. and column spaces wt, wa ..... and then replacing the corresponding components by the following rules:
w
wh
W
2g
Pul. 21 f- P...
|
P W.
Z2
11
7
P.L W-
PL
11
1
W-10
For detailed explanation, we refer interested readers to The- orem E.3 and its following discussions. In essence, the terms on the right-hand side combine to reconstruct the matrix V, as illustrated in Figure 14. A crucial property is that these terms are orthogonal in direction; within Gaussian ensemble, such orthogonality implies their mutual independence. This decou- pling randomness across iterations considerably simplifies the subsequent analysis.
Illustration sian conditioning. After removing the feature of interest the remaining part of V are sliced into P and P 1-1-1-1- E. that are orthogonal to each other.
Figure 14:
Gaus-
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Rewriting the initial conditions under the alternative dy-
namics. Let us now specify the randomness in equation (E.5) by describing the distributions of the vector and the matrix V. In the absence of any conditioning the initialization, v and V have i..dd standard normal entries. However, the neuron selected for analysis is not arbitrary: it must satisfy the initialization conditions detailed in Theorem E. We first restate these conditions in the following more concise form:
(v, wp) 2 (1 2lgg
= I +=) 2 logn 1
where b indicates that every element of a is no greater than the corresponding element of b. In fact, these two conditions induce correlation among V, and the initialization Woo Under the alternative dynamics in (E.6) and (E.7), we can reformulate these conditions without involving V as follows:
InitCond-1: a- ||u/|2 So.
InitCond-2: y1 = --1102-1 00,020 51 1, (E.10)
where
(n= - 2)V/2logM//)
:== V log
(E.11)
Here, we recall that a-1.0 = (v, wo) ||l|| and 00,0 (wo wo) 11 112.
Decoupling the randomness. In the following analysis, we can safely decouple the randomness in and Wo from the randomness in 2-1:7 and T by definition of the alternative dynamics. No- tably, the second initial condition in (E. only couples 2-1 and 2g we treat a-1.0 and 00.0 as deterministic quantities when conditioning e and wy- In fact, if condition on and Wh the second condition can be satisfied with probability at least n - by Theorem E.I.
Rewriting the alignment recurrence under the alternative dynamics. Under the reformulation (E.6), the alignment we are interested in is 1- = (v, / position of wt, only the terms the direction w-1 = w-1 = v contribute to the inner product (u, wz). Therefore, the alignment can be expressed
Note that in the decom-
(2-1, uz) ||0|2 0
TT(T((( (p(FFr we 0. v wi b;)) + 1
(E.12)
||w/||2
This formula will be useful the later proof.
E.4 ADDITIONAL PROOFS
Proof of Theorem E.3. The proof follows from a direct decomposition of the preactivation vector yo and the unnormalized weight vector wt. By a direct decomposition of V /wz, we have
(ut
Vwl
V = P.l. usi
Vwe
+ ue
1(1 2 1) + Pan. Vwy
(V
uT 7
P.l. V ut
1( 2 1)
||u|
11
i P.l. V Vwe
+
ul
|/
112
|/ l|e 1(t 2 1).
1| 11
|1
12
Here, (i) follows from the fact that for any 1,....
VJu Ur = wi - w0 40( Fyy 0.0 v 0r br)
T span(w-
which is orthogonal to w, In (ii), we use the fact that
up - w. = V Wi - V P Pomt-144
=-v0l = (F
0 T w.11;b) -n
1 -
V P Punce de E span(w-1a-1)
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Therefore, (VT w (we, )
(mg) w.!)
| ||l
-|aa, 212.
Using the above result, we derive for the preactivation vector yr that
W2-1)
8/ = V
Voy
22 11
ar,1-1
Purer
Vw=
+
u
Iw
|| ll2 1(T 11)
| 2
| 112
| ll2
IM
|/0
1/2
u
0,
P.
V
0r
oir
11
l/2
And also for the unnormalized weight vector wt, we have
wi u (PFy h + + w)-1ibb) W1-1
P 1a V ue
{V (V us)
V u PL la |
W
( ue)
|| ll2
||
+
(PL
ud)
+
Mir
|0
10
ll2
u la
| lle
|| 20 112
Therefore, we complete the proof of Theorem E.3.
Proof of Theorem E.4. Recall that
u E (EEyi;b))) F
(p 0. w,-1ib.).
This implies that us can be expressed as a function of only. This also holds for ūt. For each iteration (Me, wx) in (E.5), holds by the explicit decomposition in Theorem E.3 that
E
w- 1---h 11s-1
p uI
V
W
Wt E a w-1:11, U1:4s
p uur
|!w
P
Pulna
V
11
(E.13)
7=-1
where oX) denotes the r-algebra generated by the random variable X. For the Gaussian condi- tioning iteration (ūe, w)) in (E.6) and (E.7), it also holds that
Th E a - Un {P.t. uy:- 2 >-1--1))
We E a (w-11-111 Tna, (P 2 21--111 P.t.
1 2t).
Notably, for un (only depending on y1) we have
W-1 (w-1, Wo)
W-1 E w P.t.
91 =
| |
||w-1||
by the definition that Pe the identity mapping and w w-1. Similarly, wi is also measur- able by
w1 E 1:0;
P.l. 11-1
P.
VT
u |
|w
11
This verifies the base case for Now recursively apply the dependency results in (E.13) for = 2,3.... and obtain the desired conclusion. This completes the proof of Theorem E.4. [
F CONCENTRATIONS RESULTS FOR THE SAE DYNAMICS
Notation. In the following proofs, we use the blue color box to highlight the definitions that are used in the proofs for readers' convenience, and use the olive color box to highlight different versions
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of the conditions in (B.5) and (B.6) to inform the readers how the conditions evolve throughout the proof. We use N1 to denote the number rows in matrix E and N₂ to denote the number of rows in matrix F. In the statement of lemma, use 4, C 0 to denote some universal constants that may change from line to line. We redefine
HHsll N
El=1 1(H,,) + (H,, - 0)
P1 max max :E/m]
Em=1 1(H,, = 0)
(F.1)
N [HH, o OHH, #0)
21=1
P2 = max (F
Σ 1HH,, +0)
Compared to the original definition in the main text, add an additional term in the definition of We remark that this is not an issue as
Σ 1H,, 0) 1 (14, -0) E HH 0)
max ||H tf N i S ma H Mo/N
max
maxie|m| H Hloo/N
Since we assume the main theorem that max [()) H..loo/ < 1, we have
Σ (HH, * 0) HH,, (1++ max iE[n]
|H
max 2O1J
1 1H,, =
N
The two terms in the definition #1 are only different up to factor of + o(1), and hence we can safely stick to the new definition of #1 in the proof. Consequently, P1 2 max;e([-1]] E ilo.oo/// / In addition, N1 2 (1 -P1)N. By assuming P1 I 1/2, we have N1 N/2. We use notation = = x + to indicate = E [x y, y).
P2 (ax,g(n--]] F., F.,.lo////
Initialization conditions. the following analysis, we focus on a single neuron whose initializa- tion satisfies the conditions in (E.10) for given feature of interest, For clarity, we restate the initialization conditions:
InitCond-1:
a-10 /ell2 2 Co.
InitCond-2: y1 = --108-11 + 00,0 1 S1
where
:= - 8) e) 2ogg(M/)) :== 2lggn
Once InitCond-1 satisfied fixed wi and v, it remains to ensure that the Gaussian vectors =-1 and 2g satisfy InitCond-2. In the subsequent analysis,
sometimes relax InitCond-2 leverage the standard Gaussian properties of 2_1 and 2g. In fact, if an event & holds with probabiity 1 p without enforcing InitCond-2, then the joint event that both InitCond-2 and & hold occurs with probability at least p n by a union bound. For this reason, unless otherwise specified, we
to
Roadmap. In SF11, we decompose the pre-activation y, into two parts: the Gaussian component ve. which aggregates independent Gaussian contributions and captures the nominal dynamics, and the non-Gaussian component Ay,, which accounts for deviations induced by cross-iteration cou- pling that is typically non -on-Gausiinn. Using this decomposition, in $F.2 we demonstrate that only small fraction of the training examples activate the neuron phenomenon refer as sparse activation.
F.1 ISOLATION OF GAUSSIAN COMPONENT
As is discussed $E.3, the key step in our analysis is to isolate the Gaussian component from the non-Gaussian component. In the following, decompose yt, which is the alignments between the weight and all features, into the Gaussian component that contains weighted sum i.idd Gaussian vectors, and non- Gnn-Gassiaan part whose f--nrrm be bounded by tracking the evolution of the
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dynamics. Recall the definition of y/ in (E.6), use the fact that Put.. 2x decompose Y as
P Pui 2T to
|10 l|2
u
artt- Put 2x +
Σ
yt
ar,111
| ll2
1 l|2
||:0 l|2
u
Σ
Pai 27).
ar.-- +
ar,11
| ll2
11
We can thus define the Gaussian component y and the non-Gaussian component Ayt as
t-1
[ 1l2
1-1 Σ
y
ar,-1
Zr
Ayr
Σ
a,,t1
1 l
-ant-1 Pu Zr. (F:2)
In the above, the Gaussian component vt Σーz 07 is obtained by summing independent Gaussian vectors 2-1; -..-.. ... 7t-1 with weights (t,,--1- Conditional these coefficients, vi is simply a standard Gaussian vector independent of the learned directions wi:-11 and U14-1- In contrast, the non-Gaussian component Ay, quantifies the deviation of the true feature pre-activation from y due to tross-iteration coupling.
In the sequel, let us recall the form of a,,1-1 in (E.12) and define 3
1-1 u.) |10122 T0 (p(Fu 0- (rr-1;b)) + 11 a 1
1|w/
(F33
h- Σ 22,1-1 -IIPPl 7.11
3--1:= ==
Here, Q-11, is the alignment between Wi and the feature of interest w-1, and Bg is the norm of the projection of w, onto the subspace orthogonal to both w- and Wp- Tracking 0-1,1 quanti- fies how far the neuron has progressed from its initialization wo toward the feature direction w- Ideally, we want O--1, + 1, indicating strong alignment with the feature while remaining con- fined to the plane spanned by w-1 and tig In contrast, 3r measures the extent to which the neuron drifts away from that plane due to the influence of irrelevant features. We can build an interesting connection between the non- Gaussian component Ay, and 9 1 as stated in the following lemma.
Lemma F. (Upper bound the non-Gaussian component Ay:). Suppose T S Vd and d E (nt/es, nt ) for some universal constant ei > 1. For all = ,...... T, holds with probability
at least n - for some universal constants c, C > 0 that
| s Cd 13_
Proof. See $H.1.1 for detailed proof.
F.2 SPARSE ACTIVATION
Before we move on to studying the evolution of a-1,1 and 3r defined in (F.3), we first present con- centration results for the neuron's activation frequency. To leverage the benefits of sparse activation, we analyze how the scheduled bias b, induces sparsity in the neuron.
will first study the ideal case where Ay.. 0, and then move on to the real case in Theorem F2 where replace yt with y/ in Theorem F.3. For more generality, we present a full version in Theorem H.4 and derive Theorem F.2 as a direct corollary. In the following, recall that e, is the 1-th row of matrix E, which is submatrix of H defined in (E.3). We study the activation frequency of the neuron on the set of data that does not contain the feature (i.e., the contained in F.).
Concentration for ideal activation.
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Corollary F.2 (Concentration for ideal activation). Let et be the I-th row of matrix E. For No as the threshold defined in Definition B.3, denote by b, b, + No- Let ye
= 1- = - Q
with 2, being the i..dd standard Gaussian vectors. It holds for all t S T S n. 04-1 = (0-1,t-1,.... - E S by E R and any 5 E (exp(-n/4), that with probability at least 1 - over the randomness of 2-1, the following holds:
N (ee uE + > 0) s C. ( + P sn + pis pislog(o-1))
Σ
(F.4)
N1
Proof. This a direct corollary of Theorem H.4.
Here, neuron is considered active when its ideal pre-activation e, b, exceeds the threshold -Ko- In the idealized setting (i.e., as N1 + oc, and yi ~ (N n-1)), the expected activation frequency is exactly b(-b,), making the ((-b;) term tight. The additional terms in the bound capture the empirical fluctuations in the activation frequency due to data coupling. In particular, the parameter quantifies the maximum fraction of data coupled through a single feature, thereby governing the fluctuation term. A key point note that a E S also depends on the randomness of 2-1:7, hence how to approximate yt with random Gaussian vector is not straightforward. In the proof, we decouple the dependence of y on at-1 by proving concentration result for all ay-1 that form covering net of St, and then take union bound over the covering net of size n o(1), This gives rise to the t llog n. factor in the bound when taking the logarithm of the covering number.
Efron-Stein inequality for handling data correlation. In proving the lemma, we use a refined version of the Efron-Stein inequality (Boucheron et al., 2003) to overcome challenges caused by data correlation. In our setting, two data points may be correlated they share the same feature, which violates the independence assumption required by classical concentration results such as Bernstein's inequality.
Traditional techniques based the -iffeedddiffee property for example, McDiarmid's in- equality (McDiarmid et al., 1989) would treat the left-hand side (LHS) of (F.4) as a function
(1) yt (n 1))
of (n 1) variables, where yi (i) is the i-th coordinate of yt. Since altering single coordinate of yi has the same effect as modifying each feature influences at most pr Ni fraction of the terms in the sum on the LHS, we obtain the houfeed-differences property
projection W. onto a single feature, and because
If (1)...., 2 (i)...., Consequently, McDiarmid's inequality would yield a fluctuation bound of order
... 37 (n 1))
vē (n 1))1 S Pr.
which is clearly suboptimal. Unlike McDiarmid's bounded-differrnces inequality, which requires each individual input change have uniformly small impact f. Effon-Stein only demands weaker bound on the variance incurred by altering one coordinate. We defer interested readers to 1 for detailed proof.
Concentration for original activation. To fully characterize the behavior of the activation, we also need to take into account the Gau-asssi component Ay. This gives rise the following lemma.
Lemma F.3 (Activation with non- Gaussian component). Following the setup of Theorem F22, sup- pose b, < -2. Then for all t < T S n", a 1 E St and b, R. it holds with probability at least 1 n over the randomness of z- 1-1 that
N₁ Σ
[(e 91 +b > 0) < C. ( + ptst log(n) + +p p1 Ayel
N
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Proof. See $H.2.2 for detailed proof.
The fluctuation term in the upper bound now depends on both Pi and the lg norm of the non- Gaussian Aye. This is because larger || can shift the pre-activations further away from the ideal Gaussian case, thereby in the worst case, increasing the activation frequency.
Concentration for 1-1,1 and Bt.- We next aim to characterize the evolution of the parameters a-1,t and Br defined in (F.3). Note that in the formula of a-11c
(2 1, wx) + ||ll2 H (p y + +..0" 1.-1;bb -
a-1,111
||wel12
we can decompose the first term in the numerator as follows:
ET (x-1,u)) E (Evi be)) + (2-1,
+0. (;. be))
according to the defintion of in (E.5). Here, E and F are the submatrices of H defined in (E.3), where E corresponds the rows containing the feature of interest v, and F corresponds the rows containing v. To this end, we just need to control
(Zr, E b.))), (2+, F (F 0..00 02 w;:1;
(F.5)
for general r E [-1 T] and then specialize to = -1 Note that the above two terms for general т will also be used in computing the norm of ||w:|12 later. Let just consider simplfied case where 2, is independent of Ye (which does not hold in general). To control the fluctuation of the above terms, it is important to compute the second-order moments with respect the randomness of 2,- As concrete example, for the first term, we have the second-order moment computed as
E ()) E Eyy b,)) = ||ET <pEyr b
The second-order moment of the second term can be computed similarly. Therefore, as a first step, we will focus on the follwoing two terms:
|E (EE b.)1. ( +0 w b,)
(F.6
In $F.3, we will first present concentration results for the second-order terms in (F.6) and then use them to derive the concentration results for the two first-order terms in (F.5). In addition, will also derive the concentration result for the term 0 (T((FFr + 0 T. w)b1;bb) as in the numerator of Q-1.111.
F.3 SECOND ORDER CONCENTRATION
In this subsection, we present concentration results for the second-order terms with respect to the Gaussian component of defined in (F.2):
| ((yyts
and |FFTT(Fu) + + w;
(F.7)
We will bridge the gap between these two terms and the original terms in (F.5) by using the analysis of the oonGGaussian component Аÿ, in $E... For now, let us focus on the two terms in (F.7). We now present our concentration result formally in the following lemma.
Lemma F.4 (Second-order concentration for E-related term). Under Definition B.3, let b, + Ko < O, and assume further that -b, = O (VVlog n) and -b; < S1, with Çi defined in (E.11) as required by InitCond-2. Suppose P1 < 1 1-11/Ci for some universal constant C1 > 0. Then with probability least -n-- - over the randomness of standard Gaussian vectors z 2-1:T, it holds for all L < T with T < n that
1
NZ || |T TTTTTp(F.) (FFat:b.)) ((E)) < CT. (logn) .,²
+ CL2 2. .(((..)) By
+ (Tool))
1 (he.br)
(he, hv
l 1 (he,
(F.8
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where Ey denotes the empirical average over 1,l' [N), h denotes the l-th of H. L 722 + 1bch1, and En is the event such that 2-1 and 2g satisfy InitCond-2. Here we define K₂ as
1/4
1/4
bc
be
K+:=
n |br
+
p2sn by b
N2, +
2 // + 3
(F.9)
+ ①b by + Ms,eSe
+ (p28) 1/44) ( (Ilog(n)) 1/4 + n 1/4 p2 8 log(n),
1
In the above definition, we let ag and a for any positive q and time t 2 be the smallest real values in [0, 1] such that the following inequalities hold:
1
b
b
max JEIn]
S
$
(F.10)
D
IED;
9-1 H
+
112
+
9
9
9
9
b, + H1
b +
max JE(n
Dj
V1 -
n22
(F.11)
LED
Here = ( [N] the j-th column, and &. Ci + 1(t 2) C(3--1 10-1,-11 (0.11.0)) (o-.ool)/llog(n)) Hogg(nt)
huj # 0) is the of row indices in matrix H that has non-zero entries in
with the value
S1 in InitCond-2 and B = V E -
Proof. See $H.2.3 for detailed proof.
Validity of the definition of h and fi... The definitions of hi. and fg.1 are valid as the right- hand sides (RHSs) the above two inequalities are strictly increasing in terms of fig and fig1. respectively, under the condition -b, < G1.
is also strietly decreasing in terms of Hig. Therefore, the composition of the two functions is strietly increasing in terms of hy...
To see this for hy we note that (-) is strictly decreasing function, while HP
To see this for fg observe that &e 2 <1 > -c = -b, since the bias is fixed at 6, = in the current algorithm. Moreover, the derivative of the right-hand side of the inequality in (F.11) with respect to
fggs is
dr + (
by + (--1111-
by + 2CI 9-1
Ty x (
Si (-bea
(D
p
0. (F.12)
x
(1 - x2)3/2
Therefore, the definitions of hn and hage the smallest real values satisfying the inequalities in (F.10) and (F.1I) are valid.
Heuristic derivation for (Evt b,) The first term involves the submatrix E. Before present the concentration result, let us derive heuristically what the concentration result should look like. Let us denote by e, the 1-th row of matrix F. We can compute the expectation of the squared norm as
N
N? E (( bo)l
Σ
[[ yf: ylee yt: bo)] (et, ev).
E
NO
1F=1
we assume 0: are fixed, then of is just a standard Gaussian vector, and
(eT vi (7 vi) N [
(et,er)
(e))er)
This fact enables direct upper bound on the expectation, detailed in Theorem F.5.
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Lemma F.5. Let b b + Ro O. Suppose [o(xb)]] I (nVd) L(x + b) 1(x > -b) for
some L > and Co > 0 under Defimition B.3. For two independent z, ~ N0, 1) and E (0, 1),
it holds that
Rlo(x; + V1- 12 2;; 2; CL(V + C(I.+ +((l) 015 (
1
Proof. See $H.4.1 for a detailed proof.
By relaxing the e,, ey of E the corresponding rows hu, hy of H, derive the second term in the concentration result (F.8). The first fluctuation term is obtained again via the Efron-Stein inequality, which needs careful analysis up the 4-th moment. In particular, uniform bound over the sphere S' for 0t-1, which gives rise to the dependency on i in the definition of Kg in (F9).
also apply
We now turn to the second term in (F.7,, which is || (F +0 "0. (; b,)
Lemma F.6 (Second-order concentration for F-related term). Under Definition B.3. suppose by S Ki and I. "/2 + [b.lm. For all t S S n², holds with probability least n over the randomness of standard Gaussian vectors 2-1:7 that
F (f + 10.0 T w.-1:bb)l s CL2 p2 (01101Ba21, 11-11 P2n + p2t logn, N2
where 0 |101//22.
Proof. See $H.2.4 for a detailed proof.
F,4 FIRST ORDER CONCENTRATION
In this subsection, continue present the concentration results on the first order terms specified in (F.5). Let's s first consider the concentration for (2r, E" (EEy;;
Heuristic derivation for (z., E ((TT((E b.)). Let us recall that of -t=l1 ar and we can rewrite the term as
N₂ Σ aJ 2, p(e. ve:
(2 E ((Eyt;
1=1
for being the I-th row of matrix E. -t, St and by the fact that ||ez||a = for E [Ni], we have
Moreover, we have for any fixed a,1
(e) ãr, vT) N ((0)
ar+-1)
(F.13)
where j E In 1] is the entry index of the vectors. Hence, the term we are interested in should be close to
Ni Σ
E
s.e L [ + v a 5) (((:bb)] Nı -..1- (1bb)),
where we define
(b) Ew-
Building this intuition, the following lemma provides the concentration result in detail.
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Lemma F.7 (First-order concentration for E-related term). Under the condition of Theorem F.4, let = + byl For all & T S n", it holds with probability at least 1 - over the randomness of standard Gaussian vectors 2-1:7 that
N1 (Zr, E ((Eyt: 0r,1-1 (1(b))
S CCar,--ilog(r) (V spi (pii((1) ((bb,))t sprtlog(n))
+ C ,-11-11 VIIET ( (Fyt; b log(n).
N1
Proof. See $H.2.5 for detailed proof.
In the above lemma, we bound the deviation of the first-order term (z,, E (p by)) from its expectation 0r,--1 (b+) by some Pi and (((8:))-epenndet fluctuation terms. The dependence on ((bb,) is consistent with the intuition that sparser activation which avoids unnecessary activations on other features except the one of interest, often leads to less fluctuation. The following lemma provides upper and lower bound for G (b,).
Lemma F,8 (Upper and lower bounds for @1 (b:)). Suppose Definition B.3 holds and let b, b + Kg < 0, = 122 + ( If [el = w(1,, and No lb, = 0(1), then
Co (bbe)) S (b.) s 2 (l.))
4
Proof.
See $H.4.2 for detailed proof.
The message from Theorem F.8 is quite straightforward: the expectation term & (bx) is the same order the activation sparsity level ()b,
Heuristic derivation for (-r, FT (F + u.0 1; Similar the previous case, still
use the approximation in (F.13) except that this time each row fi of F has norm / 87, and have
S 2r, vi) N (1 08) (1.
1
ar.e-1)
1 1)
ar.e-1
This leads the following approximation:
(z,, F o(Fyt + v w.)1ib))
a 1 0f EE (xpp( 0f + 010 ( wi-1;bb)
We now present the formal concentration result for (Zr, FT (p +0v w.-lib;)) in the following lemma.
Lemma F. (First-order concentration for F-reeated term). Under Definition B.3, suppose b,
by + Ko I O and let L + |& lblm. For all v < t < T with T S n", it holds with probability at least n - over the randomness of standard Gaussian vectors 2-1:7 that
No Σ 1=1
N₂ (2r, T p(Fvt + +0. Tww
a
0f EENNCOD (aply -1.02 0,0 w)-1;bb)
< CLara-1 - (log(n) + -1001200111-11)) /p28 (l))(1)))8/2
C - VILTT p(Fy p(FF + 0 wr-1; be)l1 llg(n)).
N₂
Proof. See HH.2.6 for detailed proof.
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Heuristic derivation for TT (Pyyt + 8 v" "i The last term need to control is p(Fy, + 9. + i;-1:b;). b.). Using the Gaussian approximation fi yi ~ N 1 - 8P) as in the
previous case, we have
N₂
0 ('Ft + V - (.b1;
Σ
01 En- w...n/oo//////// of + 010 bi)
For our convenience, let define
N₂ E E 00: po.l-ll111 - 0g x x+ Os ((-1;bb))
(F.14)
and it follows that 0 (F + 0..0 To wi-1;b6 N v./Vdd. Lastly, present the concentration for 0* 'pyy + u wt-1;be).
Lemma F.10 (First-order concentration for signal term). Under Definition B.3, suppose b, b, + Ko S and let ?/2 + |b.l For all I S T S n., it holds with probability at least n over the randomness of standard Gaussian vectors 2-1:T that
1 A (( + b;) N2
N
& CL (L(ool)g()) + || [1012a-11511) (log())
VdN2
Proof. See $H.2.7 for detailed proof.
Lastly, we provide useful bound for the term th, defined in (F.14) in the following lemma, which related to the strength of the weight vector for the feature of interest. To quantify the strength, make the following definition
N₂ 2₂
b
101
Q
T
(F.15)
Vda
N₂
Lemma E.11 (Bounds for the signal term). Under Definition B.3, holds for di defined Theo- rem F.10 that
c-10-Q. N2do S Noh, CLP N2da-1111
Proof. See $H.4.3 for a detailed proof.
F.5 NON-GAUSSIAN ERROR PROPOGATION
In the following, let define the following error terms
A Fr = F. cp( (Fy, b.)) - E. 4 EFyt be),
AF: = F (F + +0 0.v T m. - F FT (( - 0 w;
Aipit sp( Fy: 0 (w sp Fut 0.00 w. b.).
The last piece of the puzzle is control the enror propagation in the dynamics due to the non- Gaussian component Ay, in the pre-activation. Let recall the error terms
A Fr = F o Fay: b.) F (p (Fat: by) AF, = FT (F + -.
;.-1;b)) - F (F +0. To. b.)1;
We interested in how the error Aye propagates through the nonlinear function y in the update.
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Lemma F.12 (Error propogation for AE Under Definition B.3 the activation function, let
by = by + Kg, L = )/2 + beln and suppose by < -2. For all t S ' S n², it holds with probability at least - n " over the randomness of standard Gaussian vectors 2-1:T that
| S CLNN ((V ((vpp1((-b)) + spi tl |1Ayelll + / [bel -11
+CNIVS(2 + (b.l) (0 d)=
and the l2 norm of A are bounded as || I (72 + (6ch)) P1 A1.lllAuull.
Proof. See $H.3.1 for detailed proof.
In the above lemma, incorporate the sparsity in the activation obtain more refined bound for | l12. Next, we also present the error bound for A
Lemma F.13 (Error propogation for FF) Define Aipr. = sp( Fy.. + 0 U Wh 1 be) - s + v
w be). The following bounds hold:
1. |A S V |
2. |A S P2N22 |Aye 12-
3. || S N2L ||Ayel12.
Proof. See $H.3.2 for detailed proof.
G SAE DYNAMICS ANALYSIS: PROOF OF THEOREM B.2
In the sequel, we will first state more general version of Theorem B.2, accompanied by the full details on the related definitions and assumptions that are mentioned in the main text. Then we will present the proof of the theorem.
G.1 A GENERAL VERSION OF THE THEOREM
In the follwoing, we first state the definition of the concentration coefficient h. and general version of the main theorem. Then, we present the rigorous definition of the ReLU-like activation function.
Details concentration parameters ht. To measure the magnitude of coefficients associated with each feature, we recall in the definition of the cut-off level for feature in (B.3) as
1 h: = max h < 1: [Di EIED. 11 2 h3 2 polylog(n)-13
To measure the concentration level of the global coefficients across all features, we define the con- centration coefficient f,, follows. We first recall the definitions of h and fly from Theorem F.4 (with = 1 for any q > 1). In particular, Ngg. and 1g.11 are defined as the smallest numbers satisfying the following inequalities:
b.
b
max JE[n]
S
D
IED
9-1 H?
+
12 Fig
+
9
9
9
1
max JEIn]
1
Σ IED,
+
by H1,551
b,
[D,]
11-1/
112
Here, D, f1 Hi # 0} is the set of row indices in matrix H that has non-zer entries in the j-th column, and 4 = 2(1 + E) /log that is formally defined in (E.11), Here, o(-) is the tail probability function of the standard Gaussian distribution, i.e., ((x) f /222 /2/ du. The definitions of h and f1 are valid as the hight-hand sides (RHSs) of the above two inequalities are strictly increasing in terms of hig and hg, respectively. We defer readers to the discussion under
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Theorem F.4. We define the concentration coefficient for the weight matrix H, denoted by h., as the smallest number such that
Hi, Hi,5 Hv 1 +
1 h
max max 12 12 S he:
+
S n
18
DjP
1 + h
LUED,
(G.1)
In fact, the RHS of the last inequality in (G.1) is also strictly increasing in terms of h,, and hence the definition is valid. In the extreme case where H does not have any diversity its nonzero entries, have the following simple relationship between 8,, 8, and s:
Proposition G.1 (Concentrated coefficient H). If Hij E {0, 1/v for all I [N] and j [m], then h, = h4 = 1/
In this extreme case, every row of has exactly non-zero entries, and the non-zero entries are all equal to 1/v8. In the following, let us define 0 = ||0 1 and (x = 1D,1-'EEa (H,, 2 a) for æ E [0, 1]. The following proposition relates and 8. to the sparsity s through inequalities that must be satisfied.
Proposition G.2 (General coefficient). Recall the definitions of in (B.3) and h. in (G.1). Suppose the bias b < -V3, then for any feature E [n]] sattisfyng the conditions in (B.5) and (B.6) and that 03 Q (ht), we have the following inequalities:
h4 2 V0 -uhh))
h4 2 1/V8,
Proof. See $1.1 for detailed proof.
General version Theorem B.2. the following, we will let s, = 1/h2. To ensure consistency in the notation, will also define 1/h₂ for h, defined in (B.3). We give version of Theorem B.2 in the following theorem, which will be formally proved in the remaining part of this section.
general
Theorem G.3. For feature E [n], let us take some small constant e E (0, 1) and define Q( as
b/ b/Vlgg n
Q = E
(1 - E) / M - 1)
Suppose
log 7 log n
62/2 - log
log n
For any feature i E (6), consider the following joint conditions for p2, d, Q and bias b < 0 with respect to constant parameter E (0, 1):
|| ll P 2 polylog(n)-1
Individual Feature Occurrence:
log, (pzz 1 Z max -logg, Q()) 1
log, ((:) ]
Limited Feature Co-ocurrence:
b²
1
h
h-
Bias Range: 1 1 2 logn max
+
(1 + h2) log, ((1)
(3h² + 1) log, Q
2
2
(1+h,(1+s) + -(1 h?)
log,, 21) 1 1-111 - log,, d log, Q
Here x Z, means x 2 y O (log log(n)/ hoo///lggnn)......./m Then with probability at least - n 4e over the randonness of the features V, for any feature such that there exists some constant &; satisfying the above conditions, there exists at least one unique neuron m, and after at most T; = max((2<;) steps of training, have (wT,, v2) /
o(1).
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Relationship between Theorem B.2 and Theorem G.3. The main difference between Theo- rem B.2 and Theorem G.3 is that the latter allows Q to have a larger range of values, while the former requires Q(+) = (1,)))) to be strictly larger than polylog(n) -1 A direct consequence of this restriction in Theorem B.2 is that the range of M is smaller compared to that Theorem G.3, However, the conditions in Theorem G.3 have ((+) and p2, coupled together, which makes diffi- cult to gain a clear understanding, while in Theorem B.2, we decouple the conditions by enforcing the range of Q(1), Specifically,
1. The condition Q(" polylog(n) is equivalent to
S h4
v (1og, M 1)
by recalling the definition of h, This gives the range of M as in (B.4) if we require all the features be learned simultaneously. In fact, if the condition satisfied for only a subset of features, our theorem still holds on that subset of features.
2. The individual feature occurrence condition is the same in both theorems, and the limited feature co-occurrence condition in Theorem G.3 will reduce to P2 < 112--o((1) which is already implied by the data condition in Definition B.1.
3. The bias range condition in Theorem G.3 will reduce the version in Theorem B.2 by removing the terms that involve Q(+) as log (og(n)/ log(n) gap is already enforced by the 2 notation.
Moreover, we assume that 2 3 as mandated in Theorem B.2. Since s, S by Theorem G.2, if 8, the following incquality
1 12 2logn
62
2 s.
1 (V2(1+e)+
[) V 1) log, Q(i)
cannot hold, because the right-hand side would exceed 1.
Roadmap for the proof of Theorem G.3. The remaining part of this section is organized as follows:
Concentration simplification: In $G.2, we will combine the concentration results derived in $F derive explicitly the simplified concentration results for the atomic terms in (F.3) for the evolution of a-1,t and 3r.
Conditions for strong alignment: In $G.3, we formulate set of conditions Cond. to Coid. Cond.(I) and CIId.(II) that will yield simple two-state recursion. Building upon these conditions, we further identify (ovd.(iv) to Cond.(vi) that will guarantee strong alignment Q-117 1- o(1) with only T O(1) steps of training.
Conditions simplification: In $G.4, we further simplified the series of conditions into a more concise form in (G.12), which yields the full list of conditions in Theorem G.3.
Notation. Following the convention in SF, we let by = by + Ko where Ko O((lo n) -1/2) is
defined in Definition B.3. Recall the definition C1 = 2(1 + =) =lolo and Ço = (1 E) 2 log n in (E.11) for some small constant (0, 1). We let C' be universal constant that may vary from line to line.
G.2 CONCENTRATION RESULTS COMBINED
We now combine the concentration results for the second-order terms in Theorem F.4 and Theo- rem F.6 under the assumption that t logn < n. In particular, by taking the square root of the upper bounds in these lemmas and noting that || = O(d) holds with probability at least 1 n (see Theorem J.1), we can express the combined square-root upper bound
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& VstlognK, pi (((bel) Bis Φ (1bil)
(hr,hrr (tes, hv)
1+ (hi, (hi
+V 1 + p2vn..
We formally state the combination of the above two lemmas in the following corollary.
Corollary G.4 (Second-orde concentration combined). Then under the conditions t tlog < n, -b, @(Vlog < S1. P1 < 1, it holds for all t T S n' with probability at least 1 n over the randomness of standard Gaussian vectors 2 2-1:7 and v that
+ b,) + || (F3g 0 11-1; S CLNPIE.
Here, the constant C' hides some factors from using the inequality va vb S 2( We refrain from detailed proof here. With the second order concentration results in Theorem G.4, we can now derive the first-order concentration results for the terms (Zr, ue) based on Theorem F.7 and Theorem F.8. To further simplify the concentration bound, we impose the additional condition ((bba)) > Ls (t log(n))3 which in particular holds if ((bb.) > n polylog(n). This require- ment is reasonable because ensures that the neuron is not activated too rarely compared to the average occurrence frequency (s/1)) of the features.
Lemma G.5 (First-order concentration combined). If ((bbe) > Lspı (tog()))()))), -b, O(oog < <1, Kn by|= 0(1), for all I. < T S n., it holds with probability at least 1 - n over the randomness of standard Gaussian vectors 2-1:T that
(Zr, Ue) = 11( (1 + o(1) + CN / P2s(t log log da
1CNp:LLtlogn & + CLN /log n- (V sp + Vspi 3--1) 91-1,
where & is defined in Theorem G.4.
Proof. See 11.2.1 for a detailed proof.
In order to derive the recursion for 1,r r in (F.3), need control the numerator
a-1,///we|2 = (v, wt) (=- uz) + ||ul|2 0J (FFye + ";-1; b.) + n a-11111
||ll|2
Using Theorem G.5 and the concentration for the second term in Theorem F.10, we derive the following lemma for (v, wrr///uil..
Lemma G.6 (Concentration for numerator in a-recursion). Suppose pid(stlogn) > ((bb)) >
Lsp (o)) -be = O(vogn < Ç1. Kig byl = 0(1). Ts log n
B1-1 < 1, and
Vda- > 1. Furthermore, assume that
Co02, >> max Lpi pos((tog Ld ((b.). L s/ttlgnpp npi da da N₂
E.
Be-
Lpi
N
a
1 & N2dCoT2Q, Then it holds with probability at least - n over the randomness of standard Gaussian vectors 2-1:7 and that
(v, wr)
(1 + (1))
||l||
Proof. See $1.2.2 for a detailed proof.
Now that we have characterized the "numerator" for o-recursion. It remains to control the "denu- merator" ||w/|2- In what follows, we will decompose the norm ||we]|2 into two parts: the projection onto the subspace spanned by w- 1:0 and the projection onto the orthogonal compliment of this sub- space. For Pa P we being the projection onto the orthogonal complement of the subspace spanned by w-1:00 have the following bound.
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n- >> .((b,)) > Lsp((tlo((n)), -b, ((Vlog n) < S1. No bel O(1), vts log n by Bt- < 1, Vda 1, (p2s(t log n/)3/2 1, and n << ((b().).. Then, for all - T S Vd, it holds with probability at least - n over the randommess of standard Gaussian vectors 1.r and that
Lemma G.7. Suppose a log
P 1 welle S CNLLpiVd(e++ vdB--1).
Proof. See $1.2.3 for a detailed proof.
For Pw- we being the projection onto the subspace spanned by w--100 we have the following bound.
Lemma G.8. Suppose p1 (st logn)- p((b,)) (tlggt)g)))) -b, @(Vlogn) Si,
kol = ((), vls log b.l..1111111111 < 1, and Vdax-1,-11 > 1. Furthermore, assume for some
constant Co > 0 that
Nº N
& 1.--1.11
C >> max Lp (2g Ldd (d LV llggnn np1
Lp
a a-1,11
!f < NydCyfQQANN (byl), then it holds with probability at least 1-n over the randommess of standard Gaussian vectors 1:7 and that
Pw will2 = (1+)(1)) (Y
(Now.t-1(((bo))
Proof. See $1.2.4 for a detailed proof.
Combining the results from Theorems G.7 and G.8, obtain the upper bound for ||wella
Lemma G.9. Suppose pid((t log n) >> ((brl) >> Lspı (t log(n)) -b, O(Vlogn), kolbel = O(1), Vls log n Br-1 < 1, Vdc > 1 and /p25(1 log n) 3/2 & 1. Furthermore, assume for some constant Co > 0 that
N2 CoFRQ >> max Lpi (pss(tog Ld Ld' LV totlognp np1 N
& Lpi
da da-1
a-1-1
Ifr1-1 < NodCoO=Q,AN&( bel). then it holds for all t S T S dd with probability at least over the randomness of standard Gaussian vectors 1.T and v that
12
||wll2 S (1+ V (No)) + (NN) + CN VIE..
Proof of Theorem G.9. By the triangle inequality, it holds that
||wel22 $ P will2 || ||
wellz
$1 {1+ (N (Noo,t- CNLLIVV + Vdßt-1).
By condition N2 Co022 > Lp1 a- and the lower bound No,; C shown in Theorem F.11. we have Noà, > CNLp1d3,-3 satisfied and can be absorbed into the upper bound of |u wi 1|2. Hence, conclude the proof.
8
When derive the above lemmas step by step, collect all the conditions used in the final Theorem G.9. In the following proof, we will be focusing on the conditions listed in the statement of this lemma.
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G.3 A STO-STAAT ALIGNMENT RECURSION
From now on, we adhere to the fact that the bias remains fixed throughout the dynamics. Thus, we drop the time index in b, (writing it simply as b) and define b + ko- To further simplify the conditions in the previous section, we have the following lemma.
Lemma G.10. Consider fixing the bias to be b < 0 and b + Ko < O. Suppose InitCond-I and InitCond-2 hold. With the following conditions at initialization:
(i) - O(vlogn) < C1, kolo| O(1), logg)))) 17 & N2dC,02Q1
(ii) pid(sT 1o ((bb) > Lsp1 (T log(n))
(ii) No C00211 > max Lpı p2s (T 3/2 Ld ((66), LVT lognpt np1
E
-a-1.
If for some time step t - T vd we have
(I) 0-1 1.00 Tollon 3
(II)
No N C, > Lp1 a-1,1-1
P-1
Let us define
CL. Coo2. N2/N'
Ag
E. = (
S CVTo 1lt 2 2)
Ag
d
Q
Va a-110
for some sufficiently large constant C > 0. Under the above conditions, we have the following
conclusions:
(1). All the conditions in Theorem G.9 hold t ST;
(2). Then with probability at least - over the randomness of standard Gaussian vectors 2- 2-1:T and v, we have the following two-state alignment recursion:
Two-State Alignment Recursion
Be S d (Er a-1,1
32-1 3--l)):
1 & (1+ o(1) (1+
+ +
(l(b)
pid a-11-1 + E.)
a-1,t-1
a-14
Proof. See $1.3.1 1 for a detailed proof.
From the above lemma,
obtain the following observations:
The ratio x, //pp)pppdd pid controls the growth of the alignment 1. order for the align- ment to grow faster, we need a smaller activation frequency ((b)), i.e., a larger bias [b] in the absolute value.
The term A controls the growth of the ratio 31/a-1,1- By definition, we know that A 2 1.
The maximum alignment achievable is o(1).
Therefore, the best we can do is set A as close to possible while exploiting small ratio ((b))//id to ensure that the alignment a-1,1 goes to 1 before the ratio Br B blows up. Since 3g O, we have 31/a
X E.. This means we also need a small initial value E to avoid large ratio $1/0-1,1 at the beginning. In the sequel, we quantitatively analyze the evolution of the =b
above recursions. Before proceed, by definition Q. N Ei ((; ) we that Q; is nondecreasing in a-1,-11 Therefore, we have the following fact:
Fact G.11. If a-1-11 0-111 then Q: 2 Q2 and d & X2-
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Expanding the recursions. Let define To + 1 as the minimum of t such that either of the following conditions fails:
(1) Cond.(I) Cond.(II);
(2)Con..(2). 1-1, 2 a-111
(3) t < log(n).
In other word, To is the stopping time up which all the conditions above hold. We have d I A2 by Theorem G.11 and the definition X >a/Qu. To obtain a simple recursion for a-1, we take
G G= = (1 + (1)) + A + CdavETy (log m)3/2
(G.2)
vda-1.0
vd
1 (21(/b/)
Here, we take the (11) term above to be the maximum of all the o(1) terms in the recursion for 1-1,1 for any t & To. For 2 S t S To. we have from substracting C1 from both sides of the recursion for 1-1 that
C S (1 d(1) +
((bb)
1
1
(13)) a-10
& + (VB((logg)
n//ggn))///../// 1(1 2))
0.-11
Pid
0-1,1-1 Vd
(lag5T (1220((0))
(1+ + V,a-1,0
+
Vd
Pid
-G1
(11((b))
-c). V2 v2stsTo
(G.3)
Mid
a-11t-1
Using the fact that
1
(1
11
50111+011 +
C1
1-1,1
Pid
vd
X1 (16))
A1E1
(G.4)
Pid
Vd
Q-1,0
we obtain that
1
Ao (1b((
6-1
\ ((61)
A1ε1
1 +
AS To.
(G.5)
1,t
pid
pid
vd
a-1.0
In the above formula, we can extend to allow as
(1D(((b) XIEI
1
X (1b)
A
x1+0(1)+
+
< C1 +
Pid +
0-1,1
Pid
Vd
a-1.0
Vd
a-
For the ratio 81/a-1,1, we use the fact that de S A2 for 2 S S To and also that
E
&
+ (ogg)()
2 2 To
Vd
a-100
to expand the recursion for 81/a-1. as follows:
Be x
1( &
1
(lggs
Σn
2/ xg-r+1 + + 31
a-14
a-10
a-11
7=2
s
To
(
&
+CVsT log()33/))
-1- x-11.
A1Ei
Vd a-100
Vda-1.0
1
((o+
& + CVe18 log(n)3/2).
vi To.
(G.6)
Vd
a-1.0
where in the second inequality, we use the fact that A2 2 1 and the recursion for the ratio that
3
AιEi
$ X1
(G.7)
a-11
Vda-1.0
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Also in the last equality of (G.6), for = clearly upper bounds the right-hand side of (G.7). Using the results derived in (G.5) and (G.6), we have the following statement. Now, building upon the results derived in (G.5) and (G.6), we have the following lemma, which summarizes the additional conditions needed to ensure that the alignment 1-11, be driven to (1)-
relax the condition allow the right-hand side
Lemma G.12. Let E (0,1) be constans. Take < C' log log n (s log d) for some sufficiently large constant C' > Suppose InitCond-1 and InitCond-2, Cond.(i) to (oid. hold. Under the following conditions
(iv) Ao @(polylog(n))
(v) 2o - =0--d===<(lblp/p
(vi) S/Q1 <
llg..).
there exists time 1 < ((25)-1 1)) A To such that a-11, = 1- -o(1), where To the stopping time before and at which To To (o-Cool.())) () to (o-Cond.(3) hold.
Proof. See $1.3.2 for a detailed proof of the lemma.
Since t* S To. Cond.()) and CIId. hold for all < automatically. In summary, in Theo- rem G.12, we have shown that under Cond.(i) to Cond.(vi), the alignment a-1,4 can be driven to (1) in constant time steps.
G.4 SIMPLIPYING THE CONDITIONS THEOREM G.13
To finish the proof of Theorem B.2, it remains to simplify the conditions in Theorem G.12. As first step, we have the following lemma.
Lemma G.13. Under Ini(Cond-1, InitCond-2, and Definition B.3, Cond.(i) Cond. (vi) hold upon the following conditions for some constant E (0,1) and e = C" log log n/(ss log d) for some sufficiently large constant C' > 0:
Q1 d ((b1) Ao
Pid > maxfd V f1,
Ls log(n)3
(G.8)
A O(polylog(n)), n-' < N4(()).
Proof.
See $ for a detailed proof.
Next, will plug in the definition of ç into the above condition obtain the statement in Theo- rem B.2. In what follows, let us define h4 as the smallest number such that
Hi Hi, Hv
1 h2
mhx R2 1 $ 13
Σ
b
S n
[B|
1 + h?
LP'ED
(G.9)
where lig,, and 1 are defined in Theorem F.4. The definition is valid as the right-hand sides of both inequalities are increasing in h.. In addition, notice that h, < 1 always holds, as h, gives the trival upper bounds for all the incqualities in (G.9). In fact, the quantity h, characterize
the concentration level for the empirical distribution of (H1 hED,
Lemma G.14. (1 (1 + h2) = (1) for h defined (G.9), it holds that
1-12 (hi, bu (1u, (mrs < Cnpp +(( + 21p28. + (hr,
Bi Φ (T
(G.10)
Proof. See $1.4.2 for detailed proof.
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Next, we also upper bound K1 in terms of h..
Lemma G.15. Under the conditions that /ih./1bl < 1 for some small constant (0,1) and ((b) 2 Pr, it holds for some sufficiently large constant C > 0 that
c='K1 - (12 [6]) 1/4 ((b) मा + (p2sn(B)) ((bb) का
(1
( b))
(42 (logn) 1/4 n I-
+n//1p22 p28 p2slogn.
Proof. See $1.4.3 for a detailed proof.
In the following, let us take s = O(polylog(n)), L O(polylog(n)) and
d n
P1 "1
P2 22
((b))
1423
(G.11)
Using the above configurations, we have by the Mill's ratio that
2(1 r3) log + O(log log(n).
In the following, we use the notation x 2 y to denote that x S y Olog (ng(n)/ log n), and to denote that x S y and y S x. Consequently, we have [b]/ Vlooolgn- = /2(1 - xz). and
C1 bb|
2(1 (1++))llggn (1+
2
2( x3) log
1 233
With Theorems G.14 and G.15, we can now upper bound č as
log, K1 S max 1 F3 1
1 22 3(xs 1)
(V1 23 V2h, (1 + =))²
T2
x2
36? +
8h + 4
h2
4
In addition, using Theorem G.14, the second term the definition of &ı is upper bounded as
log, Pi ((bb
E1,1
[6]]
(h1, hur)
(Ne, mers) S max
33 + h
1
82
21
1 + (hr, huy)
1 h?
2
2
Therefore, &ι is upper bounded
log, E S max {
33
1
1-
1
-22
3(03 - 1) (VI - 33 - -2h,, (1 + +E))²
3h
+
1'
4
8h2 +
1 h2
32 1
33
(1 h?? ) 1)
33 -22 +1 -
1
22
22
2(1 + h2?)
2
2
Plugging this bound into the first inequality in (G.8), we have the following reformulation:
log, Q ~ 23 - (1 - s) Z log, ET,
x3 2
where we note that e Oloo log(n)/ log and can be ignored in the context of ~ notation. Therefore, we just need to solve the following inequality system:
Z3
1
1
32
3(x3 1)
(VI- 23 (1 + +))²
log,, Q1 2 max
+
4
3h3
+
8h3 +
1-h2
J2
1
53
- 1:33 1)
+ -
32
22
32
1 4
2
2(1 h?)
2
0 0 2タ 2 (1
5)=o + log, Q1
0 $
2,
Solving this inequality system, we arrive at the following conditions that ensures (G.8):
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1 - log, Q1). logn
N2
12 12 2 max
-4
log Q1,
2 0, <'<NNI
2
Pi N
0 0. r3 x min
1
h12
+ (1
logn Q1,
3
h2
+ (3h + 1) logn Q1
(G.12)
2
2
4
4
1 1 (v22h + + - (1 - 1:?)) logn Q1) (1 s)00 + logn Q1
Now, the first condition involving z2 log, (p5 can be transformed into the Limited Fea- ture Co-occurrence condition in Theorem G.3. The second condition log, p1 Z 0 can be transformed into the Individual Feature Occurrence condition in Theorem G.3 by noting that N₂ H llo for feature of interest. The third condition 17 < ( can be transformed into
log 2 - log N - log ((bl) + Olog log n),
where the second term on the right-hand side can be further upper bounded as
B log ((b)) & 2 + Olog log n) 1 2
122 + Ollg log n + [b)ro rip + A = + (log log n),
1²2
2
where we use the Mill's ration in the first inequality and the fact that Ko = O 1)1122 and b < 2og in the second inequality. Therefore, sufficient condition will be
log 7 log n log n
6² log N
The last condition involving
33 16/2 + Olog log(n))
1612 + (log log(n) + 16/Ko + kg)
6²
log
log n
log
can be transformed into the Limited Feature Co-oceurrence condition in Theorem G.3. Lastly, we remind the readers that Q1 is also lower bounded as a function of X3, which is shown in the following proposition.
Proposition G.16. Under InitCond-1 and the reparameterizatiion in (G.11), we have
b/ 1///
N₂
Q 2 i (1 (1- log, M - where Q(x):=
:- N₂
Σ
1 ((t 2 x)
1=1
the tail function for the empirical distribution of f:.
Proof of Theorem G.16. By InitCond-1, we have Vda 2 (1 (1-=) (1og)/2log(M// Recall the definition of Qt in (F.15), we have by the non-increasing property of Q(-) that
Q -
(11-))/20g(M (1
b
ddo-10
(1-e)V2log(M//))
This completes the proof of Theorem 1.
Note that using the lower bound Q1 only strengthens the conditions in (G.12). Hence, directly plug in the lower bound of Q1 into all the conditions in (G.12), and this gives us the final statement of Theorem G.3.
H PROOFS FOR CONCENTRATION RESULTS
In this section, we provide proof for the concentration results presented in the previous section. We first provide proofs for Theorem F.I that controls the norm the non-Gaussian component Αyr- Then give the proof for the concentrations of the second-order and first-order terms in the decomposition the alignment recursion. Finally, we provide the proof for the error propagation in the dynamics due to the non-Gaussian component Ay,-
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H.1 PROOFS FOR NON-GAUSSIAN COMPONENTS
In this subsection, we provide the proofs that are related to the Gaussian & non-Gaussian compo- nents. In particular, we provide the proof for Theorem F.1 that controls the norm of the non-Gaussian component Ayh-
H.1.1 PROOF THEOREM F.1
By definition Ay in (F.2,, we further define
[-1 0r,1-1
Avi (1)
Ayl (2)
Σ
Pa Zr:
|
||
1
(2) The proof Theorem F.l then based on the bounding the & norm of Ay:") (1) and Ayt (2) respectively. To proceed with controlling the norm | (1) via the following lemma.
and thus Ay, Avi" (1) + Avi
12. first
control the ratio ||w=
Lemma H.1 (Ratio I ///1////l // 112). Take some total step T S vd and suppose d E (n n for some universal constant e₁ (0, 1). For all = 1, ,,...,I T, holds with probubility at least 1 n for some universal constant C >> 0 that
| Juu ll
vdl < C(logn)
W |
d S CVdlogg
Proof. See 1.21.2 for a detailed proof.
With Theorem H.I, we can now control l norm of Aut (1) and Ay respectively with following two lemmas.
Lemma H.2 (l2 norm of Ay; ). Under the conditions in Theorem H.I, for all = 1,...,? T, holds with probability at least n= for some universal constant c, C 0 that
(d C /Vlogn)
|| w- Th 1 S Al (d CVdlogn) P| w
W1 w1-1
Proof. See $H.1.2 for a detailed proof.
Lemma H.3 (le norm of ).) Under the conditions in Theorem H.1. for all I....
holds with probability at least ne for some universal constants c, C >> 0 that
A 2) s C(I + -lgg))
| w- 10 Tt
Proof. See HH.1.2 for detailed proof.
Combining Theorem H.2 and Theorem H.3, we complete the proof of Theorem F.I by additionally noting that
A 112 < 2 13 + A| 11 2(d + CVVllog C(t log n)) || w;-1 l12
As the first term d|| P.L 1_. 1 11 d3L_1 is the leading term, we conclude the proof of Theo- rem F.I.
H.1.2 ADDITIONAL. PROOFS FOR THEOREM F.
Proof of Theorem H.l. Recall from (E.6) that
1-1
t-1
||
w
w. Σ
(PL uii
žz, u.)
wi
Σ
(u up)
+ pL
- 1|utt112
10
|
llz
|w ll
w-
==1
v0 up( (Fy: 0. 8.0 TT
Wi be) w
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Applying projection P P to both sides, we have
Wi Pat 1: We Pal 2 || 12,
which implies that ||,, |l2/ /// 1|2 follows standard Gaussian distribution. Therefore, ||u; /12/ 1B/lluz 1l ~ x - 1) and we have by the concentration in Theorem J.1 that with probability at least n for all E [T].
22112- Note that 2, is independent of a (w-1x-1) and
log((Tn)) S C' CV dlog(n), $ T + 2/ 2 +
where the last inequality holds by conditions T S v and d E (n1/os, n Therefore, we conclude that with probability at least 1 n for all 1 [T],
| 1
Vdl s C(lgg))
|u- 12
This completes the proof of Theorem H.1.
Proof of Theorem H.2. By definition of Ay; we have
11A - P w-10 11
1 = Σ a 7
|
d)
t-1 d
2
|
sup
|u=
7
r=1
==
S CCVllogn P W 12:
where the first equality holds by 1 - = according to the definition of dr and the second inequality holds by Theorem H.1 with probability at least 1 n
Proof of Theorem H.3. By rewriting the definition of Ay! we have
f-1 ΣΣ T=ljer
U
Ay
ai
|
We note that when conditioned on fa-,r-1) T-1 and u1-7-1, the random variables ||= 23bs.
of
for any S S s t - are i.i.d. standard Gaussian. Let denote the filtration F -alaa,--1 - 11) Therefore, we have
Av (
Σ
- |
2
1=T
where (=/)==] are i.i.d. standard Gaussian independent of the filtration F. Hence,
A 2
12
- (=,)²
T=1j=r
Using the concentration of x² distribution in Theorem J.1 gives
t-1t-1 af
| 2 -
2 C
Σ
E
v log(n) + C Σ a log(n(t) [F |
ne
j=1j=r
j=r
Each term inside the probability can be upper bounded by
0z ) 5 Vi. (1
2211-11 - og.-1)= = Vl. PpL
w-10 11
(-11)
2 T
2 11-1
- = p
W W--1//
ΣΣ
2
j S St|lpd T1
w-110 11-1
10110
r=1j=r
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Therefore, conclude that when conditioning on F, it holds with probability least 1 n= and for 1...., that
|| 2 - C(++ Vtlogn + n) || w- w S C(t log n) I| w-10 -
where C' is universal constant that changes from line to line. Here, also use the condition that T S n. Now, since for any event in the filtration F, the failure probability is at most n n we can safely remove the conditioning and conclude the proof of Theorem H.3.
H.2 PROOFS FOR CONCENTRATION LEMMAS
In this subsection, we first provide formal lemma that characterizes the sparsity of the activations when tuning the bias by be some negative value. Building upon this result, we then provide the proofs for the concentration results concerning the recursion of the alignment.
H.2.1 CONCENTRATION FOR IDEAL ACTIVATIONS
The statement of the following lemma slightly generalize beyond the settings in (F:2) for technical convenience. Specifically, want understand how the neuron's activation frequency concen- trates around ( As we have the coefficient matrix H decomposed into E' and F' we want have general result that be applied all of them. Therefore, we consider general sparse weight matrix G in the following lemma.
Lemma H.4 (Concentration for Activations). Let G E IR be a nonnegative weight matrix whose rows (gi (e(r)) satisfy [[ ll2 1, and assume that G is sparse in both rows and columns:
For every coordinate E [n], the ith column satisfies [|Gr,llo S for some p E [n (n_1, ,1]
For every row E [L], we have l1a/lo S s.
For any integer S ne (with some fixed constant 0), define
y1
0 27:
where the vectors E Rd -for==1, vectors, and the coefficients Ot-1 = (ar,t-1) E St belong to the unit sphere in R++1, Next, let
1) are independent standard Gaussian random
by E TR be an arbitrary bias and let 02 E RR++1 and 5 = (si)sE(L) E TR be fixed vectors. For each neuron E [L]. define its shifted bias by
bi by SJ 1T--00
Then, for any failure probability 8 E (exp(-n/4), 1), there exists universal constant C. > 0 such that with probability least 1- 8 (over the randomness of the Gaussian vectors {2+ }+=1 the following holds simultaneously for all choices of 0:-1 St and by E IR:
1gT b SC L 1=1
Σ
((b..) + pst log(n(1 + (10:1e01@:: ps log(o-1)
L
1=1
where (..) denotes the standard Gaussian tail probability. In particular, if ai-1 and by are also fixed, then with probability at least 1 - 5 it holds that
L Σ ((b,)) + ps log(ō-))
I L v1 5o
Reduction to Theorem F.2 We remark that when take G to be the weight matrix F,, I. to be N1, n be n p to be Pl, b, be b, and letting 0, = O, we directly obtain Theorem F.2 as a special case. In the remaining of this subsection, we will present the proof of this lemma.
Proof of Theorem H.4. In the following proof, we will use C to denote universal constants that change from line line.
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Step I: Concentration for fixed ag-1, by and d1. When fixing at-1, by and 0, note that
bet b, - 1
also fixed and the only randomness comes from the Gaussian vectors 2-1, 20: ................................ 2t-1- In partic- ular, B/+ (0, in) since ||ar-1ll2 1 by assumption. In the sequel, the discussion will be focused on one time step and we omit the subscript for simplicity. The following is a table of the notations will use in the proof:
5/ = D(11 0r,t-12r b = b: -00 10
y x- yt
1i +- bi,
t 04-1
0t--1 (a,,1-1) S
y(
y() is the vector with the i-th coordinate
replaced by an independent copy v/ (0,
Z = レ= Σ 1 (g y > br.1)
Z
Z = L 1 (g (d) > by,1)
Table 4: Summary of notations used in the proof of Theorem H.4.
b1). To study the concentration of Z, need to analyze the fluctuations when we change one coordinate y. This leads us to the definition of ( Table with the corresponding Z(9) L Eí 1 (gt y()) dr.)) Let us also define the Exceedance-
Define Z - (9
Perturbed Variance (EPV) as follows:
V = Σ -Z)
z 200)
In the definition of EPV, we only count the contribution from the i-th coordinate of when Z exceeds its perturbed counterpart Z Next, we show that V4 actually controlled by Z itself up to small factor. In particular, for the term inside the expectation in the definition of V4, we have
Σ 22 1(2 200) = Σ ( 2)² 1 (wv 1/1
199 #0) 1( y > bui))
(gr, + #0) ( y > b.t1) = ps7.
where
in the first identity, we use the fact that Z is monotone in the i-th coordinate y: due to the nonnegativity of the weight matrix G.
In the first inequality, we use the fact that 0 < 1 y > b..)) - 1 > b,.)) < 1l (ggTT T y > br.z) thanks to the condition > 1/1 which is guaranteed by the condition 7 > z
the Cauchy-Sehwarz inequality with the fact that Σl_ 11 (gr, * 0) 1 y > b..) < EL 1( * 0) S pL. Then, by also noting that each is also s-sparse, obtain the last equality.
In the last line,
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Meanwhile, the mean of Z is simply
= ((b.,),, where use the fact that l|
1 by assumption and g y (O 1). Invoking Theorem J.5, conclude that for fixed and by, we have with probability at least - 6,
Z $ E|Z| + CV PCCCppssEE22 ps logó Cpslogo SC
(1
Σ + (b..) + ps pogg(())))
(H.1)
for some universal constant C > Here, we directly apply the inequality vab I a a b for a, b in the last inequality. In the following, will apply union bound 04-1, by to extend the above bound to arbitrary choices of a and by.
Step II: Union bound over at-1 and by the following argument, we will also drop the sub- script t. Since Z is a function of a and b, we use the following notation:
Z(a,b)
1 Σ Σ
Σ
a-gl b- saa saaT))
sufficient to construct covering net for the pair (a,, Since the Gaussian vectors 2, are unbounded, we first introduce truncation step in our covering argument. By applying the Chernoff bound for Gaussian tails and then taking a union bound over all indices T = -1,0, ..,t.. - 1, we deduce that with probability at least
1 (t + 1)n exp(-n/2) 2 - -cxp(_nn/4)//
we have
max
|2 S Vn.
In what follows we condition on this high-irohheee eigh-probability event.
For St, we take uniform covering net on the sphere, denoted by Na. such that for any a, there exists a' E Na satisfying ||a - a'|loo I c. The covering number is upper bounded by [W S c-
See for example Example 5.8 Wainwright (2019). To proceed, let us define
= (t 1)- (v stn + ||
The intuition for this definition is that μ represents the Lipschitz constant of Σ a+g 24 + sta T with respect to any perturbation on a the -orm. For b, leveraging the Gaussian tail property, define the following covering net with size at most 4ue 4:
No = (k: μ c k Z, k [1 12c- []} U uf-o0}
There are three special points in No: -oo, the minimal finite point bmin - p e, and the maximal point bmax [2e-
e. For any a E St and b E R, we pick à = argmin ||Q
a'Ilso and 6- argmax (V/ E Nb D' a c). Therefore, we have by the monotonicity of the indicator function that
1 Σ
Σ ar gi 2t > GāT 1pp00pp- ...)) D()
Z(a b) S
(H.2)
L
1=1
On the other hand, for b = = - á T using the definition of â and b, it holds that
((b)) < ④bb 3 E) 11- Amin S by bmax) + 1 -0))
+ - To) 0)- (be = bmax).
The above inequality holds by considering three cases:
When by d-- bmax). we have by close to by up to an approximation error of 3p c, where one μ e comes from the approximation between a and & and the other 2p- comes from the approximation between by and by
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When bj -0,, simply upper bound the tail probability by 1.
When by mmax. we have ((;)) @bmmax qa T v) 22 qa T 0).
Next, characterize in each case the approximation error between {(bi) and the bound given above, which are (b; - 3p c), 1, and $(2μu Q.a 0) respectively. In particular,
For the first case be E [ max) we have the approximation error (bb 3) (b,)) directly bounded by 3μe by Lipschitz continuity of the Gaussian tail function.
For the second case b, -oo, it must hold that b, < buin + ju, and the approximation error is thus upper bounded by - ((b,,)) = 1 ((b, - qa 0) I exp(- (Ibiill (1+
<)p))/2) I
For the third case by = bmax. it must hold that b, > bmax > 2p.. Hence, the approximation error is upper bounded by ((2u sia ToaT ((b,,)) o22 qo 0) z (2p n)² /2) < exp(-1²/4),
Combining these three cases, conclude that
+ (x) S (b:) exp(-1²/4) + 3..
(H.3)
If we choose the covering net parameter = pp then the upper bound can be simplified as & (bb) < (bb)) + exp(-n/4) + 3p. Since is least 1/n, we further conclude that ((b)) I
((by) + 4p given that exp(-n/4) < 1/n S p. Lastly, note that the log cardinality of the joint
covering net is upper bounded by
log(IW.)) + log((N6)) S z Alog(= + log(4 1) S CL log(n11 + 1|111==11011)c))
(H.4)
given that PP 1 > (nqu)-1 Here, for the last inequality,
the fact that log(a)
log((t + 1)(v + ||000=01000000)) [o0) S C Clog(n(1 + ||||| (())))))) since n2 for some constant > 0 and S n. We can also apply a similar argument for every t < n.. This only increases the
size of the covering net by a factor n², Combining (H.1), (H.2) and (H.3) with the log cardinality
(H.4), we conclude that with probability at least 1 - 8 for all a, b, and > exp(-n/ that
I.
Z(a--1, bb) S 2(0r. C.
I
(bb..) + pst log(n(1 + || + ps log(6
SC. (7, ((bi,)) + pst log(n(1 ||||==0|01| ps log(6 log((-1)
where the second inequality, we apply a union bound on the joint covering net for a and b and also for all & < nf. In the last inequality, we just need a change in the constant factor C' to absorb the approximation error 4p for the approximation error ((b,,)) ((b..). Here, the lower bound 8 > cxp(-n/4) is to ensure that the good event 1,00. concludes the proof of Theorem H.4.
S Vtni holds true. This
H.2.2 ACTIVATIONS WITH NON-GAUSSI COMPONENT: PROOF OF THEOREM F.3
In the following proof, we will use C to denote universal constants that change from line to line. Let us denote by b, = b, + Ko as the shifted bias. Let us pick @: > be specified later. For any I [N1], the neuron is activated only if either of the following two conditions hold:
1. e, vt + b, Pc:
2. el yi + by a: and eJ Ayj. > Dt-
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For the first case, by Theorem F.2, we have with probability at least 1 5 that
1(e Vi + bb 21) C. (Ø(-b: 21) p1stlog(n) + pislog(o-
(H.5)
For the second case, we only need to control N1 E I(e Au > 2t). We have the following upper bound
N;
n-1
1
Ay: > e1) S
(1ell
Σ
Ayi, (FI, #0>>
N
N1
1=1
N
Avt. 1(I * 00) > o.)
N1
1=1
M
Avi 1BBi, #0) P1 e
|
(H.6)
NoO
where the first inequality holds by the by the fact that ||er||2 = The second inequality follows from fact that 1(r > a) S x/a for any >> 0 and 0. The last inequality holds by noting that Elloo P1. Combining (H.5) and (H.6), we conclude that with probability at least 1 n
inequality and the following equality holds
N1 I(e y > b Σ
s C. (00(-,, - @i) + pistlog(n))
+ 11
Pi ei
(H.7)
Let us pick e 1b.1-1 Note that by assumption by < 2, we have b 0: 3/2 and by
the Mills ratio inequality (x 33) < ((z)/p(z) x-- - - -5 for > 0, where
p(x) = exp(-x2/1 /2) 2n is the density standard Gaussian distribution, we have
3( [b]] et)
(17y1 - e:)²
b(-b. - et) s
exp(
(H.8)
2m (0rl @c)
2
1 exp
[6,13 (1 + 3(6b) 13 1111111110
2
[6g]
< (b)),
[bl )1 1b2 exp
2 [ae
(1bgl
where in the last inequality, note that the highlighted ratios bounded by universal constant. Combining (H.7) and (H.8), we conclude the proof of Theorem F.3.
H.2.3 CONCENTRATION FOR | ; PROOF THEOREM F.4
When treating {ar,e- | and b, to be deterministic, it follows that yテ NN 1). When condi- tioned on the good event E, we always have ||vv" I (1 + c) e) 1 log(nf). In the following, use y replace yテ for notation simplicity. We use yj denote the j-th coordinate of y. Let
be b, Kig
Good event on bounded Gaussian vectors. Let Eg denote the event that InitCond-2 is satisfied by the vectors 2:0 Throughout the proof, C' will denote universal constant whose value may change from line to line. Fix a time step t 2 (we omit the subscript for notational simplicity). Define the "good event"
| S (1+ c) log(nt))
max 1,0....-1
Then, by Theorem J.2 (applied to the i.d. standard Gaussian vectors 2-1:11), we have
P(&)) 2
(n > n
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Good event the activation sparsity. Let define S3 {1 [Ni] Eg, # 0}. It holds that |Sy] I N1p1. In addition, we define event &g as
Vj [n m--1 Va--1 E vhe ER
by + E(,)U1) ) |S, lo2st log(n) Σ Φ IES;
&
[ LES,
V V1-E% E7,
To show that Eg holds with high probability, let us define E as the submatrix E by keeping the rows indexed by Sj while removing the j-th column. We also normalize each row of to have f--norm equal to one. We then have
1. | ll S and |lo S E (HHs 1 # 0) |S, (s2 where the last inequality holds by definition of p2.
2. It holds that
b, Eng 00. V1 Ei,
|S Σ JES,
1 y > = |S,
Σ 1 (e + leS,
En
where či is the 1-th row of E and y-j is the vector y with the j-th coordinate removed.
In the following, we use žr.j to denote the j-th coordinate of 2r, and Vj = E.= 0r,1-127-5- We denote by žr.-j the vector 2, with the j-th coordinate removed. Therefore, we invoke Theorem H.4 with the configurations
G + p p2, P2.
20.1 (2-1111000.....--111))
a Et 1- V Et,.b by f- bb/ E and 2, t 2x.-5
obtain that with probability at least 8/n over the randomness of standard Gaussian vectors Z- 1-1t-11-- and for fixed I, 0z-1, b, and 8 = (z- 1.j: 20.jo...3 ; Rt)1,
Dy + Ei,iVs)
Σ [eTT by > 0) sC. ES
Σ +
+ +-Sylp2s log(nd
v E2,
YES,
b, + H1,jUl) V1 -H?,
SC.
+ |Sylps log(no-1))
(H.9)
LED,
where C' is a universal constant independent of f, 04-1, by and 1. Here, in the last inequality, we define D, = {I E [N] Hi, * 0} the set of rows matrix H that have nonzero j-th coordinate. Since E just submatrix of H, adding more rows to the summation does not decrease the target value in the second inequality. Note that 2-1=-11,-- are independent of 2-1:t-1.j- We thus conclude that the above bound holds with probability at least 1 1-6/n over the randomness of 2-1:1-1- Further applying the union bound for all j e in 1]. we conclude that (H.9) holds with probability at least 1 - 5 for all [n- 1].
Note that the randomness discussed above is only over 2-1:4-1- We invoke a covering argument over ag-1 E S and be R similar to the proof of Theorem H.4. Since the argument is largely the same, will not repeat it here. The size of the covering net is n and can pick d c-O(t+1) in (H.9), which gives us the upper bound in the definition of & with probability at least - n n
Refined upper bound on y. We work with fixed time step & and aim to bound every coordinate yj for j E [n - 1). Here, we recall definitions
Vj 0 žnjo 8-1
Σ
where BI-1 represents the form of the component of W in the subspace orthogonal to w-0--- (Recall that the coefficients {a,,1-
[ arise when projecting w;-1 onto the orthonormal basis
w
w
W
|
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To leverage InitCond-2, we make a change of basis for the first two directions, namely, replace
w-1: w
| }
with
{ (wo w]. where t = Co,0 w-1 - 1.0
Note that is orthogonal to Wig The projection of w onto the direction satisfies
[ T |o0, a-1,t-1 - 2-1.0 -0, S |a-1,t-1| |0-1,//-
Since Wo, w. and {wo w [|==] form orthonormal basis, the component of wt-1 orthogonal to wo is bounded by 3 + a-1,t-1 + |a-1,ol. Moreover, we can also decompose y/ into the new basis as follows:
y1 (Wo, w(-1) (a-1,0 2-1 00,0 + (20, (1-1)(a0,03 --110---------3
Σ
a,,t- 2
[Wo, wt-1) 91 (0o, wz-1> ( 2-1 a-1,0 z0)
ar,t- 2T
Under InitCond-2 the first term, (wo, W.-1) 1) M1, is bounded by <i. Moreover, since both
40.0 2-1 a-110 20 and {
have their entries bounded by 2(1 + e) llg((n)) on the good event & the contribution from the subspace orthogonal to Wi bounded by
C (B:-1 10-1,--1| |a= Volo(nnt).
Thus, by the triangle inequality, for every coordinate we have under event En and E that
yj S C + C (B--1 + 1 Vtlog(nt) = C.
(H.10)
Good event the Bernstein concentration. In the following. we will another good event control the upper bound in the definition of Eg. Consider the function (- sry)// /1 x²)9 for 2 1. We demonstrate that this function is Lipschitz continuous and monotonically increasing on the interval
[0. 1] if yj -b, by taking the derivative with respect b
d 20 dæ
cl by + zy
VI x )² q
b. + ryy 9-1
p(
b + zy
y/ (-
(1 (1-x x1-32/3/2 > 0.
VI-
VI
Using the upper bound for y specified in (H.10), we can define the critical value by as the smallest real number such that the following inequality holds:
b
H1,33) H!?
11
max jE
Σ LED,
be + Hi
I T
by + Nig
[D
[Dy
HP
LED,
4.
(H.11)
As will only be using q f3, 4) in the following proof, we define the event Eg as the event such that for all E (3.4) 4), a-11 E ES be E IR and j E [n - 1].
by H1,58y (E 1(8))
Es:
H1,
#=1
LED
b + HI.yij
by +hg llogn)) 9 )] + 2 (0--))
C
E
H1?
12?
where C' is universal constant independent of t, a;-1, b, and G. To show the event Eg holds with high probability, we can apply the Bernstein concentration inequality in Theorem J.3 for the bounded random variables
b H1
D
H2
LED,
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That is, for fixed ag-1, by and with probability at least 1 over the randomness of 2-1:8-1, have
by + H1,
1(8)) ((6)
/ HPp
LED;
2 2log 8-1,
ΣE E
1
by + Hi.
() 1 ((E)nE))
[D5]
V 1 11?,3
LED;
b.
H1,j3s
ht + Nigeee
E
+
+ ②b
log(0
==1
Dy
IED;
V Hi,
3
ng
Moreover, we have for the second moment term that
by + Hi ))
2
E
1( ni)
D
JED
by + HesWis
((E)
Σ x6 MED,
by + Hr
R
Dy1
V1 -1P
V 33
LED
b.
(1)
by
E
H
IED
where in the first inequality, we invoke the upper bound in (H.11). Using the fact that v b S a for a, b 2 0, we derive that
T
Hi 9 (E6) 1(8)
+
Dj
Hi
LED,
n-1
b. + H1.j
1
(e(--)
by + Myge ) 1og(6-1)
SC.
+
2
j=1
D2
IED;
H2
3
n2.00
(H.
Now, we apply the covering argument a4-1 St and b, R similar to the proof of Theo- rem H.4. The size of the covering net is 0(++1) and we can pick 8 = n re-0((+1) in (H. which gives us the upper bound the definition of Eg with P(Ex) 2 1 1-n- n
The Perturbed Variance. Given the good events Eo, E1, Eg, and Es, we define
N;
Zur,
where Zr plee yi be) p(eTu(b)) (er, er) 16 n &
NZ
11'=1
(H.13)
For concentration of Z, we consider the following Perturbed Variance (PV) defined as
Σ Z z(0) | w)
V:=E
where the perturbed term Z is defined follows:
Ni
z = =(eT (0; (0 (er, ev) 1())) c(0).
where
2
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and (6) is given by replacing the i-th coordinate of 2, by an independent (0 1) random variable. In addition, the good events 12 defined similarly
Here, v0 Σイ= 07, (1)
I-=o
(2) to e,, but using 2= - instead of 1-1-111- We begin by noting the elementary inequality (a (a-b)
b)² -
2a² + 2b Thus, we obtain
1f E1, {BEss #0% V Ev,, $ )0))²
V
E
Z
y
N
(I)
N1
2 [ Fas + 20 Bv, + o0)
+
N1
E
( 1.11
y/
1=1
1,1=1
where the upper bound is obtained by the following reasoning. For each perturbed quantity z(0,, we have
Z - 2
ZI
-(-)) 1(E,, + #0V Ev, 00..
N? 111=1
Note that the difference Z,,- (1) is nonzero only when at least one of the vectors et or ey has nonzero ith coordinate. The two terms (I) and (II) correspond to the contributions from the original and the perturbed parts, respectively. what follows we focus on an upper bound for the term (I); the term (II) can be estimated by a completely analogous argument.
'.1'
Controlling Term (I). Due to the I-Lippshitz continuity of 4p with L = =2 + bb /1,, on the good event E the absolute value of yeec y; bt) bounded by lsp(e, y: b.)) I kp(0;:b)) L le vl. which can be further bounded as
Is(eT v.bb)l S (d co + LVs lullo S CLvts ts log(n) == B
where we used that 5 nf. |exll S Vs. and that (d v <15 LVLs s log(n). Note that the same bound holds for spec] (00;b by) on the corresponding good event E(¹) For Zr defined in (3.13), we first upper bound ple, bi) (pee y: b.) by
sfee y; bi) b.))) S 1eT + b, > 0) 1() + b, > 0) + +B + (dV 2c0
where recall that if e, y + b, O, the neuron is deemed activated and its output is bounded above by B. Otherwise, by Definition B.3, the activation is bounded by (d V n)- Note that the term (d V n) B- can be made arbitrarily small as Oo is some large constant no less than 4. Therefore, we just keep the first term above. Secondly, the inner product (e), ev) is upper bounded by
# 0)- 1 (Err, # 0) as |l < 1. Lastly, the indicator 1 E,,, * V Er,, # 0) can be upper bounded by 1 (E,; + 0) - 1(Er, + 0). For (I), we then have
CB N
1 <
E
(eT v by b0>0 0) 0) 1 1(evv + tb 0)
====
((1 #0) + Ff + 03) FFF # #0} F # #0)) (n 11
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Due to symmetry in the indices I and I', we can multiply the constant factor C' by 2 and obtain
CB
(1)
E
Σ
[ef y +b, > 0) 1(E,, 0) 1(E,,s # 0)
N
N1
1 [Brs * 0) 1(e b > 0))² 2 =(n==0E)) V/
CB
- ø-In-1
N1
[le] y + b > 0) 1(E,, # (E,, # #0) )
2
E
Σ
N1
N
+ by >,0b>>0)) 01) 2 I( #0) yc
[0]
Σ
where the last inequality holds by the Cauchy-Schwarz inequallity Note that for + j:
(E,, +0) 1 +0) S
HHi, + 0) (Hi, 7 0) P1P2
1=1
Using P1 to substitue one Σm 1 (e y + by 0) [(a,, # 0) 1 (Fts * 0) for i + j. we obtain
N p1
(1) 5
E
[eTyy+ - > 0) 1BE,, + 0) [(Ba) + 0)
No
N
1 # 0) 1eNv + by > 01) 2
1(((= 3
B Ni
1 [eT y + b, > 0) 1Bis 01)
+
E
N1 (Brs #0) 1 (cT + by > 01)² 1( m0
Rearranging the order of summation and using the fact that E 1(E,,s # 0) S for any fixed j. we can further simplify the terms as
2BB (I) <
P188
[eTu+bd >) 1(Ess # =))) - y)
E
N7
2B
I y +b, > 0) I(Ei, + 0))² - b)
E
(H.14)
NO
1=1
Observe that the above two terms share common structure. We define the common structure as
N₂
[e] y + > 0) 1(E,, + 0))" 1((2) m)
E
N
where g E {3, 4). Recall the definition S = (1 E [Ni] Ei + 0}. It holds that |S S Nip1. We aim to control
Σ (eTuu by 0) [(Fa, # 0) = |S Σ
[(e y + 0)
IES,
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in the following. By the definition of the good event Eg, we have
m
Σ
be + Hi.Bi
+ |S, lp2st (02)
1(
N
j=l
LED
V
His
LC-LC
n-1
by + HinWih ( + (p25t
[Dy1%
Σ
NY
(Dyl
His
j=l
IED;
n-1 Σ
1 5
b, HI.jiy ) (((=ue.) + nn m(p2stlog(nn))
ccp
(H.15)
D
VI-IIi, Hi.
IED;
where we use the Hölder's 's inequality for the second line, and in the last line, we absorb the constant factor 29-1 into the universal constant C' and use the fact that [Sy] < (Dyl S Np1 I N1P11(11 pı) NNNip p1 for all j E n - 1], where we also absorb the factor C7 into the universal constant C. By the definition of the good event Eg, it holds that
b + H1 a 1(
D
HP
LED;
H1,j b. +
9
be + My. ' (log(n)
cC.
E
+ 10
(H.16)
/
HP
IED,
To evaluate the expectation term, the Mills ratio ((x) Cp(z)) for some universal constant
C > 0, and p(x) exp(-x2/2) 2t to obtain
b. + H1.31
qbc + HLSWs) 1 (b, + HigWs S 01) + p(r + Hi, > 0)
S C
E
exp
V1 1--1
2(1 - -H(,)
9(b, + H1,,,/b
}]
by
SC. E
exp
+ +b
2(1 Hz₂)
Hi.j
1 1 1-H
b. H1.5
V
eepp
1 (g - H)
2 H3
1)
(H.17)
where the third equality holds by direct algebraic calculation for Gaussian integral. By the Mills ratio (=)/((() x -3 Cr 1 for >> and also the fact that Hi.5 E [0, 1], we conclude that the right-hand side of (H.17) is bounded by
Φ be + HLSW3
)
T
E
S C
(H.18)
HP
9-11 H
as a non-decreasing function of for z E [0. 1] by checking the derivative. We define ,y.. as the smallest real number such that the following inequality holds:
Similar to the previous argument, we also have 1
b
be
Σ
S
5n.w
(H.19)
D
1 Hi
8-1 12
LED,
Plugging (H.18) and (H. into (H. have that
b, + Hig
1(33
Dj
1 HP
iED;
b
by + Ny.
(eo))((oogg tlogg()))
SC.
n2/b// Φ
+ +
(H.20)
12
n2
9
-.
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Combining (H.15) and (H.20), obtain
be
(0 be + Rgg 'llo((n) n na log(n)))
(III) S Cpi n n1/ell s
2 nig
V1- 12g
Note that we always have Ryg S
and Fgge S for 2 As both +
b
and
Φ (- -I-
(when &e -b:) are non-decreasing functions with respect æ. for the first term in the right-hand side of (H.14), we take 3 and hgge. 1 to have the following upper bound:
2B P1P28
E
Σ
Ni
(e y + by > 0) (Bi, # +0)) 1( y
N
b.
S CBBpppps
(n|b db
+log(n) + n(p2st
(H.21)
V f +
For the second term the right-hand side of (H.14), we take q 4 and obtain
2B
N
(ee y + by > 0) [(Es, # 0)) 4
N
(n (Te T
-bx
by + Nia
< CBBM
+ +
(log(n) + m(p2 st log(n))
V2 12, +
V1 11.
(H.22)
We conclude by combining (H.21) and (H.22) that
(I) SCB:M (m [n|Bal t
b.
Dt
+ p2sh by
nE
b. + hG
+
②
+ p28
(log(n) n n(p2 st llg(n))) Vo.
Similarly, (II) can be bounded by Vo- We are now ready to invoke Theorem J.9. Since V S 2Vo with probability 1, the final bound for Z 1 is then given by
7 E 2|
C log(5-1),
where the inequality holds with probability at least over the randomness of standard Gaussian vectors 2-1. Plugging in the formula for Vo. we obtain the following upper bound
IZ E < CB.P
Ielel +b
+ p2sn|bell
V
No.
+
3
+
1/2
b,
fig. ) p28 (lg(()) + n(p2 llg(n))))
+
llog 8
122
with probability 1-8. For notational convenience, we define Kr, the 1/4 power of each term inside the bracket in the above equation (see (F.9) for the definition). The fluctuation of Z is controlled by
1
E[2]
CL2 pf logn KC? log 5-
where we plug in the definition B, = L ts log n and L = 72 + 16// the Lipschitz constant for the activation function yo.
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Expectation E[Z].
For E|| ((Ey : be)
have
N
E| N? E
: v N?
Σ
E (1 y: be) (hh y; be) (hr. he)
1.1
c E.r
| yi bi). (h ys be)l] (hi, he)
where the first inequality, we obtain the upper bound by also adding the rows of that are not contained in the submatrix E' to the sum. Here, we use the notation
hy (Hi,I...., -1 Hi Hin--))
to denote the /-th row of И with the i-th entry removed. This structure comes from the definition (E.3) where we decompose the matrix H into submatrices E,, F and the column vector 0 as the non-zero entries H., if the feature of interest is the i-th feature. In the second inequality, we use the fact that N/N1 s C1. and define Eir the empirical expectation over L, [N]?. Invoking Theorem F.5 with L 72 + lm. be be No. conclude that
Ei [ y: b.) (h) yr bb)l] Tin,
Cha Bur' (ree Perr)
(n..(VVd + ((bel) Ber
+ ( V
+ (Ta, Bur'
(hre, hur)), 1 (hi, (hi,
CCL CL ((bee) Er,
② (bbl
V
(tes hx)
where in the first inequality, we directly apply Theorem F.5 to the expectation term, and in the second inequality, we use the fact that (hs, hyy S (/t, hy) for 1, 1 [[v1] and the fact that the term inside the expectation is non non-decreg when increasing the value of (hi, h.). Just as before, since co >> large enough, the first term is negligible, and we can absorb into the constant and focus on the second term:
/
(hu, (hr: (he, ni))
N? E[IEE (Eyi; b.) < CI² +((be)) Er Φ (buly
1 (h4, hy)
Since T :y: 1 is non-negative, the same upper bound applics to E[Z], where includes the indicator condition (.)
Finally, we plug in 8 = n - to conclude that with probability at least n holds that
Ny | (pEy b.)1 ((6)) 1() s CLP
(logn) K2?
(hr, hy
+ CL2 20((b))) Eir
(hes hw)
1 (hr,
Note that the joint event 1n E,) holds with probability at least - n- discussed earlier. Therefore, we can safely drop the indicator 1 17))=,,) in the above inequality. This completes the proof of Theorem F.4.
H.2.4 CONCENTRATION FOR ||FT ((F +0 w,-1;b,)11
b:)1 PROOF THEOREM F.6
In the following proof, we will use C' to denote universal constants that change from line to line. Let fix (0,,t-))-=1 and b. Then yf (0, Tn-1). For simplicity, will denote y by y in the following. Let us define the good event
|| S (1 vC) 22ggnn)))
max 1......
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It then follows from Theorem J.2 that P(E) vol 2t log(nt) & In particular,
(nt) S n, and also lleloo $ (1 +
(e( y + 0,e b.)1; (()) S (72 + [b) + v
log(()) (alpt/))+llllllaaaaa1_)) 0111o]22a111-1)) + (n V co
-= Br,
where the inequality holds by the Lipschitz continuity of in Definition B.3 and also the fact that + Ko I for the bias. Deñin
2
Σ
(U, (li y 010 - m.-1;b,) (f y + Ovv w.-1;b.) 1(E).
N 1,1=1
Using the Cauchy-Schwandd inequality, we have
N₂ (0(ST y 010 Th- we-1;bb)² + plJe y + Orr T Twe-1:b)))) I((i,Jr) #0) 1(8)
2
1,1=1
N₂ p(s y - 0br1)))21(( (()),
2p2 N₂
Σ
(H.23)
where the first inequality follows from S a² + b2, and the second inequality follows from the fact that (fi, fv is nonzero for most N2p2 terms when going over I' by definition (F.1). Next, we concentrate the right-hand side of (H.23). Note that by the Lipschitz continuity of i, we have
1x(fi y 010 w;-1;bb) b.)l S (72 (6 yl + 01leel2aa-1--1)) (n "
By the Cauchy-Schward inequality, we further obtain
rlfi y + 14 w.-lib,)² < C(12 + (b.l)) v) + (011u1)2a-14-1))}} C(n d) 200 -200
(H.24)
To this end, we apply the Cauchy-Schwarz inequality again to obtain that
N₂ 2º
N₂ N₂
R-1
ST
+ 11 s /2 11v112-
Under the good event E, have llvlla (1 22 In fact, llv1 ~ x2 1s and we apply the concentration inequality for the chi-squared distribution in Theorem J.1 to obtain that with probability least 1 ō, it holds over the randomness of y that
No
1)
Σ 1=1
u)
s Cp2 (n ( log 6-g
No
Nº
Applying a union bound over fa,,--1 11=11 and b, similar to Theorem H.4, and since Z is uniformly bounded, we conclude that with probability least 1 n= it holds for all & S n² that
EUT
U (()) s Cp2 (n log(n))-
(H.25)
N₂
Combining (H.23), (H.24), and (H.25), we conclude that with probability at least 1 - n- it holds for all S n" that
Z 3 C(12 + (b |101 1l + P2n + p2t logn
As the good event & holds with sufficiently high probability if we choose large enough in the definition of ε, similar bound holds for the original quantity |[FTT (p(Fy, + 0 w. b, 112 This completes the proof of Theorem F.6.
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H.2.5 CONCENTRATION FOR (2r, E (((ut: :(Eyt:b))) PROOF OF THEOREM F.7
In the following proof, we will use C' denote universal constants that change from line line. When treating {0r,5-1)==1_ and b, to be deterministic, we have 2y d ar,--14 + / 11-1a a-- 2, where 2 ~ (O0, 1) and is independent of y/- In the following, we use replace y/- and a to replace a 1 for notational simplicity. Therefore, the concentration we consider can be reduced to the concentration of
N₁ W E ((yr + VI- a² N, (2, ET ((y be)),
Firstly, note that when conditioned on y. (z, E (( is a gaussian random variable with mean zero and variance | "Eyy; be)1 it holds with probability at least 1 - over the randomness of y that
M (2, [e, E "'a,E"s(EEri ((E,,b)) Ni V22IET (eEy b1 log N₁
where the second order term has already been handled Theorem F.4. Similar to the proof of Theo- rem H.4, and S n to obtain that with probability at least n n it holds for all (r, t) that
covering argument over {ar,t-1 (0r,t-1)/=-p g1+1, by R, -1, O, -........
1 N1
C VIET (Ey; b,) flog(n).
(2, E p(Ey; N₁
Now it remains to control the first term. Define good event
E (1I1|>> S (1+ v log(m())
In fact, the above good event can be directly implied by the following good event:
m E=
max 1.0...... ||=||0> ||--ll
(1 + vcl 2 log(nt))
For notational simplicity, we will just focus on the latter definition of the good event. It follows from Theorem J.2 that P(E) 2 (tn)-o 2 - - Let us define
2 = Ni (v, E (EEr 1(8), and V 12- 2 [
where z(0 (1))), E p b.))) 1()6) and y is given by replacing the i-th coordinate y with an independent copy 5/ ~ NO, 1). Note that this is equivalent to replacing the i-th coordinate
of each 2, with an independent copy 2 Thus, the good event & can be also changed to g() accordingly. Next, we show how to control the variance V. Let us define
Z eT y (ee y: b.) ((E) and ( = ej ())) pleT ())) (((())
for any / E [N.]. On the joint event E g()) U u 1)), we have by the Lipschitzness of y in Definition B.3 that
Z C(12 + (ele))tttgggt =: Br, Vi E [N1].
(H.26)
This bounds also holds for all 2 for E [n - 1]. By reformulation, we obtain for the joint event EUε, U. U that
1
Σ E(a- -10) 2
Zi 2((0) 1(E,, + f0) 2
(Z - 20)
NY
N7
N,
Nr
<
P1
Σ
(Z - 2009) 1(E,s v
2pr
(zt + (z())2) 1(E1, #0)
N1
N₂
1=1
1=1
2p1
Bi
Σ
[1(e] y +b > 0) + 1 (0)) +b, > 0) +2, (n vd) 1(E,, # #0),
N
1=1
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where the first inequality holds by the Cauchy-Schwarr inequality, the second one holds by (a 6)2 & 2(a + b²), and the last line holds by Definition B.3 and the upper bound in (H.26). Since co some sufficiently large constant,
safely ignore the term involving (n V d) -o in the sequel (when invoking a constant factor C). Taking a summation over 1, , n taking the conditional expectation, obtain that
1 on both sides and
Cpi N1
n-1 Ni (1(e) y > +) y + > 1h>0)1a 1)]) ( #0.
V S
BB
ΣΣ
Let us define
sly) 2p1 B² n-1 N₁
Σ Σ
1 y + b, > 0) 1(E,, + 0).
N1
Therefore, the moment generating function of is controlled by
[[ex)(AV)) < (exp(Ag(v)) exp(AE(o(v))) wi)] S E (exp(Ao(V) exp()a(1))))
for À > 0. Here, the last inequality follows from the Jensen's 's inequality. To this end, we notice that y is non-decreasing functions of y. Then by Theorem J.10, we have that Eleep(>ggy)) (xp(Ag((())
S [leex((2g(())]. Therefore, we just need to focus on the moment generating function of (v). Note that since et is s-sparse, with probability at least 1 - å over the randomness of y, we have
2sp1 B N1
N
g(v) $
Σ
+ be > 0) s C, (o((b.)) ) + P1 log 1e
where in the last inequality, we invoke Theorem H.4. This can be transformed into the following tail bound
E [AV)(AV)] < E (exp(2)((w)) where ((((v) > Cspi ((.)) + +o) S <exp C
v
and any > 0. In particular, for V+ and V- defined in (J.2), we always have 0 < V+ < V and O V_= < V. With the sub-exponential tail bound, we now invoke Condition of Theorem J.8 to conclude that with probability at least 1 - 8 over the randomness of y,
Z E| I CB.(V spi 2 (b:)) log 1+ + pls loglog
(H.27)
Since Z is Lipschitz over {- 11-1, ) and { we follow a similar covering argument over ] with T < n.. Note that the failure probability of the joint event & U g()) U U g((-1) is at most ni-c. In addition, we set 8 "(n -cg"") in (H.27), where is the approximation error in the covering argument in the infinity norm. By a union bound of the covering net of size ===
the balls {8--1)
will obtain failure probability at most n- well. By decreasing the constant e slightly (up to 2), we can combine the two failure probabilities to obtain that for all t < n", it holds with probability at least - - that
Z - E E & CB, (V spi ((b.)) t log(n) sp1 +loo(n)).
Next, let us evaluate the expectation E[Z]. By definition,
B[Z]
E N
[vy
ET
((Ey bi)l = N E[ E ((Eg) 1())
<
E (v, ET Tp((ppE(pn b, b,)) P(E).
N₂
Since (EE) < Thile (y. E "EEy b is most C(b) + (m (2))) for some universal constant C by the Lipschitzness of &p given by Definition B.3. We can pick c in the definition of to be sufficiently large, Thereby, the approximation error in the expectation is negligible. We thus just need to evaluate
N1
1 E El,,, E (EE b.)) Ni
1
Σ
BleT v plel us be)l (... (x)) (b.).
No
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Hence, we conclude that for all t and t n², it holds with probability least 1 n= that
1 N1 (2r, E. ((Er;bb)))) ar,1-1 (b((b) ar,e-1 CB. sp1 ((be))) =log(n) sp1 llg(n))
C
V21LET ((E; b.)1 (l)g()).
1 ,,1-1 Ni
Plugging in the definition of B, C(12 +
we complete the proof of Theorem E.7.
H.2.6 CONCENTRATION FOR (Zr, F op + 0- w;-1; PROOF OF THEOREM F.9
In this proof, we will show the concentration for the term N₂ (2+ F 8 (F + 0 v' w(-1;b()). Similar to the proof of Theorem F.7, when fixing {ar,t-1)r=-- 1- and {br,)l=1, we have Vi ~
(0, n-1)- For simplicity, we will denote y/ by y in the following. Note that 2, & ar,--111 + a - where (O, 4n-1) is independent of y. In the sequel, also simplify a,,11 to a. Therefore, the concentration we consider can be reduced to
1 No
a. 2 Fu, ((FF - 0 w(-)))))))) + VI a²
((Fy+ 0- wt wi-1;b))).
The concentration for the second part follows directly from the Gaussian tail bound. That said, with probability least 1 - 8, it holds that
1 N₂ (2, F (FyF + 0-00 wi-1: S
N2 221 (Fyy + 0 uT m 1og8-1,
where the right-hann hand side be controlled by Theorem F6. Then by covering argument {a-,--1)==-1 and b, similar to Theorem H.4 (with proper truncation of the random variables that yields a sufficiently small error probability), we conclude that with probability at least n holds for all = 1...., T and -1,
1 that
N₂ Kz, F p(Fy 0 ot w.)-ibb))
C N₂
V T wt 1i llg(()).
To control the first term, define good event
E= { max
|2 -
(1 vo) v log(nt))
On this good event, |1 < (1 + 2t
22 log(nt) and this good event holds with probability
least 1 - (In)- 2 - -c We define
w. b.);;b,) and = Σ zZ z u).
Z
N₂ (y. FT (Fyy + a. T
where ( = (y(() FT ((F((( + oT.,1;; b.)) (8))). Here, we define y
Σ 03 (K) with (0 given by replacing the i-th coordinate zj with independent copy,
and {() is the event defined with respect to (1) Let us define
Z1 = fi y. p(T y + 0: (
z = fl ( ppff (0 + 0,0 w.-1;bb) (()()), ((E
where fi the [-th row of F. On the joint event (()) U g(n-1), we have by the Lipschitz continuity of yo in Definition B.3 that
1Zil ((122 + (blm)) (Vtlog(n) [|01|2-111-1) (n /tlog(n) Be,
where we also use the fact that b, + Ko S 0 for the bias. Note that the (n V d) - term is negligible when Co is sufficiently large. For notation simplicity, define by by Ko
Oev W This
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bound also holds for z
On the joint event & U g(2) U... ..m.11) we have
N2
(Z - =0)
1
Z - 2/)
1
Z1
)/ 1(Fi, * 01
N
N2
1=1
N₂
2₂
P2
(Z- - 2, 2/ 1 #0) v
2p2
Σ
( + () #0)
2 1=1
2₂
|=1
N2
2p2
b. > 0) + 1fi y + b,g > 0) + (n (FFi, + 0), (1
N₂
1=1
where the first inequality holds by the Cauchy-Schwarz inequality, the second one holds by (a b)² I 2(a² + b)), and the last line holds by Definition B.3 and the upper bound for ZI and z
We can also ignore the 2B, (n V d)= term by multiplying some universal constant. Taking summation over = I,..., n2 1 on both sides with the conditional expectation, we obtain
Bp VS
No.
(1 y + b. > + B[ICTT + bu > 01] (Fi, # 0).
N2
1=1
Let us take
Bp n
No
g(v)
1 y + Tu.1 > 0) (Fi, #
N2
Cp2sBB B²
N₂
Σ
1(f + bis 0) - Cp
N2
Then we have by the monotonicity of g and Theorem J.10 that E(exp(AV) S E(cxp(2Xg(v)) for all À > O. Invoking Theorem J.9 for this bounded variance, we obtain that with probability at least 1 - 8 over the randomness of y, holds that
Z E[Z]| S CBiyP25 P28 log 1.
By a covering argument over {ar,,-)) ||ا and b, similar to Theorem H.4, we conclude that IZ E|Z] < /22s flog(n) with probability at least - n for all = I,..., T and T =
- -1, 0, t - 1. In addition, the approximation error
1 Eky, F ((Fy+ m.-1ib)))l x /PEE)
B[Z]
N2
by the Cauchy-Schward inequality and the fact that fiT yP y+ Orv T "w,-i 1; be) has bounded second moment. Therefore, by taking sufficiently large in the definition of the good event E, we can make this approximation error negligible. Morcover, we also have
Blft uP(fi y + fre b.)l:bb]] / 11-0f 0g E.. [ 88. + 0,0" m.-1;bb)]
Combining everything, we conclude that with probability at least 1 - n it holds for all t 1,..., T and T = -1,
1 that
No Σ 1=1
1
(2r, FT "(Fvi + +0. T.. w..1;bb))
a / - of EEWN(O (2p(V1 0f T wi-1;bb)
N₂
N₂ V 1 - a v ||F p(Fy + 0. T w. b1) (l)g(n)
C
+ -Cara-1 (72 + lb.h)) (Vtlog(n) + [101 /p2s (eas--ttoog (tlog(n))3/2
This completes the proof of Theorem F.9.
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H.2.7 CONCENTRATION FOR 0 (( fo wc-1;bb): PROOF OF THEOREM F.10
In the following, we will use C to denote universal constants that change from line line. Let fi denote the (-th row of F. Let us first fix (er.e-1)<=- and br. Then vi ~ (N(0, (n-)). In the sequel, we will simplify y +- Vr- Let define the good event
3
max 1.0,.., 1--00,,,,,111, 11211l00 (1+ v log(nt).
It then follows from Theorem J.2 that P(E) I (nt) I n and also 1 S (1 + vc) / lot(nt) &. In particular,
(ff y 0,0 0 wx-1;bb) (E) s (12 + (b)))((( vOlv 2t log(nt) + (|llla--1001) (n d)-co co == By.
where the last inequality holds by noting that b() is 72 + 1by -Lipscchitzz by Definition B.3, and also the fact that b, b, no I O. The target function study is
N2 0i ys b. (()), where b,,1= = bit + 0111111120011
Σ Z N2 1=1
Let ( be the vector obtained by replacing the i-th element of ye with an independent standard Gaussian random variable (i). The good event g(, is defined similarly. Define 7 as the correspondence of Z with y and g(t) Let us define variance V B - and V- Notice that this V upper bounds both V+ EE (Z 2 EI ) 1(Z < }. Note that when changing one coordinate in y. the total number of terms affected in Z is most N2p2 by definition (F.1). It then holds by the Cauchy- -chharz inequality that
Z y
Cp2 n-1 N2
of [[-[[(([]] v: v( p(fi ;b,b, 1(8)) 1n
V
N2
CB3 P2
No ΣΣ 0f 1 (fi(1) #),
n-1
N₂ ==1
where in the second inequality, the indicator is included since the term will be zero if fi(i) 0. Additionally, we invoke the bound B, to upper bound the +p(.) term. Let us define
( = CB n-1 No
CB.
N2 ΣΣ o8 ((f((t) #0) 20 <
11012.
N₂
By Theorem J.10, know that the MGF of V be upper bounded by E(exp(AV)) < E ((x)(())()))) Thanks to the bounded variance, invoking Theorem J.9, we conclude that with probability at least 1 over the randomness of y. it holds that
Z - E|Z] < CBB1000
P25 N₂
log(8-)).
Next, we invoke a union covering argument over the ball St-1 for 07,1-1 and also for by Since Z Lipschitz and bounded, the approximation error can be made sufficiently small. Therefore, conclude that with probability at least n= it holds for all t S n" that
7
E
CB.
1|
P28
llg(n))
N₂
Similar to previous proof, the error in E[Z] and N 'B 'B sufficiently small we choose a large e in the definition of the good event E. Consequently we just need plug in the expectatin
(Fy + 0 v b;)1 can be made
N₂
1
T p(Fyt + +..v w.)l;bb)) N₂
E,
V.- (0.p))00ppp111111111111 07 + 0 11 bbb
N₂
1=1
This completes the proof of Theorem F.10O
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H.3 PROPOGATION OF THE CON-GAUSSIAN ERROR
In this subsection, we analyze how to Non-Gaussian error AУЛ propagates through the nonlinear activation.
H.3.1 ERROR ANALYSIS FOR AE; PROOF OF THEOREM F.12
In the following proof, we will use C' denote universal constants that change from line to line.
Bounding E By definition of A Er, we have
A = |ET ((((tt + Ay)): E (( bi)li S vs (lo(E(y + by)):b)) - ;(
N₂ > 1=1
S V5((2 + (6el))
lep Ayrl 1(e ye + > 0 v e vi + + > 0)
N1
2(2 (611 (n vd 1le y + b. 0 el vt v/ + b. 0). V8. Σ
where b, be + Ni the shifted bias. The first inequality follows from the fact that |eell1 I /s as each row et is s-sparse. The second inequality holds by splitting the summation into two parts. For the first part (1: e/ y/ + b > 0 V ej Vi + b > 0} where the neuron is activated, we have the term bounded by the Lipschitz continuity of &P times the pre-activation difference le, Ayel- Here, recall from Definition B.3 that & is (72 + J-Lisshizz continuous. For the second part {I: ej Yt b; < 0 el yr + by < 0} where the neuron is inactive, we simply apply the upper bound on o Definition B.3 as (2 + [6,1) (n V d) no, Note that co can be chosen to be sufficiently large constant. Thus, we just need to focus on the first part. Using the Cauchy-Schwar inequality twice, we have
N
N;
le Ayd (e Y + b > 0) -
elllz 1|4 o 1( + 0)]l2 (e, y - b, > 0)
1 [e Ve by Σ Ay 1 (e # 0)122
(H.28)
where x: o is the Hadamard product between vectors and y- Note that the second term on the right hand side can be further bounded by
N I o [e: + 0)13 = ΣΣ Avi. 1(E,, + S P1N1 || Σ 1=1
N1 n-1
(H.29)
==
Plugging (H.29) back into (H.28), and invoking Theorem F.3, conclude that with probability at least - n for all /. < nf,
Ayal 1(e De +b, > 0) SNN (0((-b,) p pist log(n) + pi (2 Ay, 11) P1 Al
SCN ((V + P logn) ||Aurl12 + P |bel 1l
Note that the ideal activation Σn, Ile yt + b > 0) has an upper bound Theorem F.2 even tighter than the one we use above. Therefore, we just need to double the above error term. Thereby, we conclude that
|| CNi(12 + (6cl)) ((Vsp) (p11)((b))) spı Vllog) ||Ayella VSpilbel. |l411
+CNVV [bel) (n V aw
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Bounding | vector x E R Ni
The proof is similar to bounding | 11- Again, notice that for any test
N1
2
n1
N1 BB + 0))
N1 Et S P1N111
|
Σ
Here, the first inequality holds by the Cauchy-Schwand inequality while the second inequality holds by the sparsity assumption on the columns of E and also the fact that Σ Ei = ||e: = 1. Thereby, it holds for | 12 that
N1 ej
A NiNloll(EE(t Ay));be) ((Eut; PINN +bmm)
Ay
N1
S N + m Σ
aall
11 o 1e # 0)113 z (22 + 1bll))
where the second inequality holds by the Lipschitz continuity of yp and the third incquality follows from the Cauchy- Schwar- inequality. The last inequality holds by invoking (H.29). Hence, complete the proof of Theorem F.12.
H.3.2 ERROR ANALYSIS FOR AF; PROOF OF THEOREM F.13
In the following proof, we will use to denote universal constants that change from line to line. Let fi be the I-th row of matrix F. Note that
|| S v (F + Aur) + (i =F(Fut + 0 + we-1;
be)ll
N2
Vo((2 + (b.lm) Σ 1sT Ayal S
A
> lfflll2
S V + 1
where the first inequality follows from the fact that ||fi| S / by the Hölder's inequality for s- sparse fi with [|filla 1, the second inequality follows from the Lipschitzness of yp and the third inequality. In the last inequality, we use the fact that
inequality follows from the [ S 1. Next, we turn to the bound for ||AFel2. For any test vector RRV we have
n
N
2
n-1
| x
S
| = P2N211=112.
(H.30)
where recall that P2 maxxe(n-1]] Fl lo/ Since AF, F ApF., have || ||3 < 22N22l Next, we use the same Lipschitzness of yp to upper bound App; as
N₂
N
R
||ppR (12 + 12l1))
|fT Ay. 2 s (72 16
Σ
Avi. 1(F,, # 0)
< (12 + )
A (Fr,s # 0) S N202(12 + 12 || (H.31)
1=1
where the Cauchy-Scheee -chhwaz inequality in the second inequality, the fact that [| < 1 in the third inequality, and the definition of p2 in the last inequality. Combining (H.30) and (H.31), we conclude that A Frll2 S P2N2(12 + (bll)) ||Ayll2- This completes the proof of Theorem F.13.
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H.4 PROOFS FOR TECHNICAL LEMMAS
H.4.1 PROOF OF THEOREM F.5
We invoke the upper bound (p(x;)) S Vn +)x 1(x > -b) to obtain that
(1p(2:b)(((s v 22 b; (n d) 200 + 2(n d) 00 L- Ex + b) 1(x > -b]
+ L² B + 1(x b) (LX + V1 1--22+ b) 1(xx + V1 122 > b)]
Note that Bfx 1( > -b]] = -((b)) for any b by explicit calculation, where p(z) =
exp(-x" --² /2)/22//2/ 2 is the standard Gaussian density function. Therefore, we have
B[(2 b) 1(x > -b] = 1 > D)] 6 P(r > -b) = =p( - ( = F(b)),
We note that the function F(x) monotonically decreasing for all x E TR. To see this, we take the derivative of F(x) and using the fact that p((xr) (x)(x)
where we define F(x) =
and E (x) = -p(x), which gives us
F'(x) -(x) x (z) xp(x) = - (xx) + xp(xx) xp(x) = -b()) < 0. (H.32)
In particular, function F F(x) is always positive for any I E IR as lim F(x) = by the Mills
ratio ,im (<f())/((r)) = Therefore, F((D)) S F()) 112 and the first two terms involving (n V do are negligible. For the last term, by marginalizing we have
Bl(x + b) 1(x b) (1x + V1 - 22 + b)- 1(xx +
- 11-222 > -b]]
(x ++b b) 1(x > -b) v1 - 12. 1x + b
b+ 1.5 vi 13
+ 1x
= E
+ +p(
1
v1
E (z b) 1(x -) V1 - 12 F(
b + ie
Since F(x) is monotonically decreasing, we can upper bound the expectation by just plugging in
x = - to obtain that
(I) < E (1r + b) 1(x> > b)j v1 V r( (-DV 1 = 2 F(T) F (( 1
2.
Next, we prove that F(x:) < (x)))) for all x > 0. For any x > 0, we have F'(x) (())) by (H.32), and (xxx) = -p(x). Therefore,
F'(x)
(x)
(0 p(0)
π = 2
22
('(x)
p(x)
where we use the fact that 4 (x))/(()) is monotonically decreasing. Noting that lim F(x)
and ,om ((x) O. we thus conclude that F(x) < (2(()) for all æ > O. Consequently,
1 < <4VI- ((8)) + (TlV 1-
2 1 2. F(b) F
Therefore, we conclude the proof of this proposition.
H.4.2 PROOF OF THEOREM F.8
Proof of Theorem F.8. Note that
(b.) = Ex be)c]]
< L. E[1 + by > () + bx)azl + B||a|
1(x + b, S 50] (d nm
[b
b,
L,
exp(- ((bb.) /2T
exp( exp(-b//2) + C(dv
/2T
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Here, the last inequality holds by the following integral calculation:
l zp(z)dr= =p(z)dr==(0))
JL =²p(z)dr= = bp(b) - (b(b)
= p(b).
for the standard normal distribution p(=) exp(-x" 22) For b. < O, the first and the last term cancel in the bracket, and conclude that & (b.) S ([b,)( as (d v n))o can be sufficienlty small. On the other hand, using the condition (0 bn) 2 2Φl (o + b) 2 Carlx + by) for x 2 -b, by Definition B.3, have
P (6t) 2 Ca + b, > 0) - by)ar] + Blpp(;; by jæ 1( b-b S b.)] (n
Here, we recall definition ş(æ;;) b) = 6(x + b) + x o'(x b). Therefore, (0xx; b) 2 ol + b) for
x > 0. By Definition B.3, we know that ''(x + b) ž 0 for all x. Since -be > 0, we have for
a E bz b,] that
p( b.) 2 +b) 2 (n vd) "
where the last inequality holds by the monotonicity of Φ. Therefore, conclude that
Co th (((b)))
(br) 2 Co (ar + by > (x + b,)aa C - (n V d) c 2
2
Since make Kig lbal |b.| log-polynomially small, e.g., Ko (log(n V vd)) for [bg]
(log(n v d) we have 24 (b.():)) 2 ((bb,)) 2 *(b..)
2 This completes the proof.
H.4.3 PROOF OF THEOREM F.11
Proof of Theorem F.I.. Lower bounding the signal term. Let us lower bound the signal term. Note that by the monotonicity assumption in Definition B.3,
p(x;b()) 1a>- 6(x + b.) + (x + bb) bb) b. 2 Co..
For (-be, be), have bl(;bb)) 2 pp( b(-b;;b)= 2 - (dV Together, conclude that
N2
O1 pl(11[1 0fx + 01 VVda Vda-it--ibb))
E B
1=1
No
O1 11 1 x V1 0) Co (6(11-08 0 @idda
E N(0.1)
1=1
V2(dVn)-
N2
b. Vo 11-11111))
+
Coff Vol N2(dV
V1-0p 0f
1=1
1
((1)
N₂
bb
(A) >
Cobf Vda-1.1111
2
Vda-1.-11
1=1
where in the second inequality, it follows from the direct calculation of the integral of the Gaussian that (o (x > a)æ] p(a) 0 with p(a) being the density of AO 1) at a. The -(d V n on the right-hand side negligible. Note that the indicator is selecting the larger half of 0r. and we can thereby obtain the following lower bound
C- CoVda-1,t1 TQ.., where Q
10,
b
0
0 1
N₂ 1=1
vda
N₂
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Upper bounding the signal term. To arrive at upper bound, the fact that yp(x; bt) 1 (d (dV (x <
-b:) + Lx (x b)) to obtain that
N2
B N(0.) 0: v0(1 fxx O1 V da dodad-.i 1.:0,)))1::bb)
1=1
B N(0.1) 0.1 x+ 011(x+
liv 1,1-1 +
> >0)
(VI- 0fx 0 0iVda
V1- 0f
+ N2(dVn)
2₂
S CL
Σ 1=1
0 1
02 + + Vda- S CCNNDRV da
where the last second inequality holds by noting that Ei1(x > a)æ] = p(la)) S 1, and the last one holds by noting that Vdo--11-11 > 1.
I PROOFS FOR SAE DYNAMICS ANALYSIS
In this section, we provide supplementary proofs for the results used in the proof of the main theorem in SG.
1.1 PROOF OF THEOREM G.2
Let us first prove that there must exists some E [n] such that 03 2 1/s. Since the total sum
Eje(n LIED Hi, Σn || N, and there most Ns non-zero entries in the weight
matrix H, we have the average
Hi H = Σ G 1Hz
N 1 Ns
> 0)
5
On the other hand, we also have
VL プ DDj H
max 07. _j(mn )
Σn (Dyl
thus follows that there exists some i [n]] such that 0z 2 1/8.
Proof of the first inequality. By definition of h.., have l13 2 12 for = 4. To prove the
upper bound on h,, we just need to show that n2 2 03 for any E [n]. Let us consider the kernel function in the definition hy.::
f(x) +
9
In particular, we aim to show that f(.) is convex for [0, 1]. given by
The second derivative of f(x) is
6(
(x = r(
3 1-11
2) x + 9
82
9-1
1 + 2)
7/2
Using the property that < <-v3 33 we conclude that f (x) > 0 for x E [0, 1), and f is convex. Now, by definition of fig... we have
f f(h
max JE(n]
1
Σ
f(H?)) 2 max f
Σ Hi.)
f(09). jE[n)
max
(I.I)
DDj
JE(n]
Dr
IED;
LED;
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where the second inequality follows from the convexity of f(z) and Jensen's inequality. Moreover, the first derivative of f(x)) is given by
D -=
f( -(x)==pp -pl
2 7 + 3/2 > 0.
1
Therefore, we have by (I.1) that f 2 03 for any j [n] and q 4. Consequently, h 2 A2. Z max]e[[n] 03 1/s. This proves the first incquality.
Proof of the second inequality.
Since we have by definition of 8 that
||: 12
0 = D2
S (1 (ha)) 12 + @(h)) 1 @.(h.) + h,,
follows from the condition 0: > (hs) that
h 2 V 08 1/2).
This completes the proof of the third inequality. Hence, we have completed the proof of Theo- rem G.2.
1.2 PROOFS FOR CONCENTRATION RESULTS COMBINED
In the following, we present the proof's of the lemmas and propositions used in $G.2,
1.2.1 PROOF OF THEOREM G.5
From ((bb,) >> Lspi log n)), we deduce that t tlgg < n. since Lspin >> > ( (recalling that P1 2 n )) Hence, we can directly apply Theorem G.4 in what follows. Using the bound Theorem F.7 together with Theorem F.8, we further assume + ((bel) >> Tssp1 (t log n) then the desired concentration result is obtained as follows:
(Z,, ET ((Eti; = (1 o(1) Na..-- + - oz 1
VIEE <( / b01l3 log(n))
(I.2)
Here, we use the fact that |N - < PI < 1, where < can also be deduced from the condition ((b+e) > Lspı(2 (t log(n)) For the concentration result for (zr, FT (F + I w.-1; be)) in Theorem F.9, the Stein's lemma to derive that
N₂
N
Σ
aa. aa.--1l11-1 08 E.-N/O.1) (op(v 1-- + 0,v u. w.-1;bb)]
N
No N2
No
Σ
(1 (1-07) 0f)
BE (0'( -0x + o,0 T w.-1;bb)
N
-=1
< pp =b (()))))))))
(I.3)
where in the second inequality
the Lipschitzness of &p and in the last inequality we
Lsp (tlogn) 3 < (bb.)) Moreover, we have
N₂ N (Vtlog(m) + || pp25 (t (tlog(n))32
L La
< L|a (togps( + p pisloo(tlon -la- < <(((( (0.,-11) + P (2 3
do-1,t-10,.-1
(I.4)
where in the first inequality, we use N2/N < P1 by definition and in the second inequality, we use the fact Pr (t m)² < pr (1llogn)) n & ((b..) under the condition Lsp1 (1 logn)² < (((b.))-
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Moreover, by Theorem F.8, we know that Φ1 (b)b) 2((((b)))) Consequently, by combining (1.3)
and (1.4) with the upper bound in Theorem F.9, we have
(2r, FT ((Fye + 1.00.. wr-1ibb))
+ CC of V ||FT ((Fy+OF+++ + -- log(n)
+ CNLP1 p1 (1 log n) 3/2
dlar,-10-1,--
(1.5)
Let us consider the good event with respect to some universal constant C >> 0:
E: (11211100 S CVlog(tn), VT ST}.
As we increase the constant C, the failure probability of the event & be made polynomially small, e.g., 1 - n for some other constant > (See Theorem J.2). Conditioned on the success of this event, we have for the non- Ga-Gaussiin components that
(2+, AE,) + (2r, AF) S C (lo((tn) (1 + |[
COLNVLog(()) (Vspi ((b,,)) /spptlogri vd3,-1+ /sptlb,ld33
CCL.Voog((A P1 sd
CCLN (v sp (aoaa(((b)))) 6e1)) /spa [ b,lld3
(I.6)
where in the second inequality, we invoke Theorem F.12 12 and Theorem F.13 to bound the é of the еπrо terms, and also the fact that is at most polynomial in n. In the last inequality, we use the fact that A & vd3,-1 by Theorem F.I. Now, we combine the derived concentration results in (I.2), (I.5) and (I.6) with 1 a S and the upper bound for |E (Fy be) + (FFy 0 0.0 we-1; 1 in Theorem G.4 to obtain that
(Zr, un) (Zr, E (e be) + (Er, F ((Pyr + 0 v 11 + (zr, AE + (Zr, AF))
-Na o1)) + CNLP1 /
d00,,0-0011--11
+ LNN & + CLN v Vlo n (v (111)4((6))))) + Vopi (b.1d
Hence, we complete the proof of the Theorem G.5.
1.2.2 PROOF OF THEOREM G.6
Recall by definition of Wi, (v, w/) /|/ll/ can be decomposed into
(v, wr) | (=-1, u.) + |ella 0 ((Fy: + 0- v. w)-1;b)) a--1--
(I.7)
Taking -1 in Theorem G.5, we have
(z-1, NN-1,--1i( (1 + o(1)) + CN CN logg) d/ 110-111-11
- CC tlog n & + CLN Vlogn (V (1b + /sptlbelds?_))
(I.8)
Moreover, by a direct decomposition of the second term, we have
||l||200 ((Fy + +0 (;)1;bb)) = || (Fyt + 0 v ()-1;bb) + ||l|100 APF.
= 1| p(Fy; + 0 (x-1;bb) = ||||||00|| A.
Notice that ||o||2 Vi. (1 C1 with probability at least 1 - - by concentration of x² random variables (see Theorem J.1). By Theorem F.13, we have |A < Vo2N2L 1|Agel22 S V Therefore,
|| SCV. / N₂00 L P2N221 CLN
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Now, combining the concentration results for H PF Vi + 0 u. bi) in Theorem F.10, obtain that
|lll2 0 up Fyy 0. 0-v0 +0.0" (t-1ibe)
= =1 + CNLLLLVP (ogg))32 11-1,-1 CC pr
(I.9
Furthermore, we have by Theorem R.11 that Nita 2 CoPQQ N2da-1,111 Under the conditions
N2 CoRQs >> max Lei (ppyppsstlognn) ((be)) LV f tlo da
&
N
we conclude by also noting that Vdo-1,1- > that
3/2 11-1,111, Na-1,--1((0b), CN Ex Nuhe >> max CNL n
Now plug (1.9) and (1.8) into (I.7) obtain
(v, we)
= (1 o + t a-11-1
|
+ V ((be))) Vpidp2s-1 + v /pdd(b/BB--)) - 9--1- (I.10)
Finally, under the conditions Vlslog b st logn F( Pid, we have
Vdpis logislogm n. (v 4 ((bl)) + Viddps- + / 3 S Cpıd.
Here, use the fact that P2 log < 1, which can be deduced from the following inequality under the condition N Co02Qu Qu > Lpı pptt/psstoog
N2 CoT > Lpv Lpi//2s(tlogg) 2 P V p2 logn.
N
Moreover, under the condition
B N₂ Co > CLp1
N
a-1,t1
conclude that the second line of (1.10) be upper bounded by o(Ne))). Hence, the proof of Theorem G.6 completed.
1.2.3 PROOF THEOREM G.7
Recall from the definition of we that
(nt,us)
Il ||2
| (we 1 1--011
1)11
Σ (z,, Uz) -(Pu 2r, un)
| ll
||
l|z
+ || 1 2e112 || 113.
(I.11)
Lemma L.1 Assume that T S Vd and d (n= ni) for some universal constant e₁ E (0, 1).
Then there exist universal constants c, C. > O such that with probability at least n over the randomness of i.i.d. standard Gaussian vectors 2-1:T, for all E [T']
(ut ui)
w/w 1/
Σ (Pn.. Zr, uz) 2
| 14- 11 ||4 11 s Cd ||u 12
|
|
112
Proof. See 1 for a detailed proof.
Lemma 1.2 (Upper Bound for |lu 112). If t logn & n. -b, = O(Vlogn) P1 & 1, it holds with probability at least n- for all t < T < Vd that
||u||2 - CNT.pp vd.3-1)-
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Proof. See $1.5.2 for a detailed proof.
Combining Theorems I.1 and 1.2, it holds with probability at least n= for all Vd.
(ut, we)
||00
Σ (Pan 2r, ux)
+
|| 1.P. Eel1 ||u, l12
| ll2
|| ll2
S Cvd. |uel22 S CNN.pp Valee vdBi-1).
(I.12)
It remains to upper bound Σ (Zr, u,)2.
Recall that B V a² 1 -0
Σ r= af 7- Using Theorem G.5, we conclude that
(2r, 2 (NN))-)(()))- CNNC:)))/toNo h dla-1,e-118-11
+ CONpiLtylogr + CLN Vtoggn (V (pspid0((b.))) + /spilbeld (c-1)))))1)))) B2-1
S LN logn & CLN
(L.13)
where in the first inequality, the 3r terms in the first line is obtained by the Pythagorean sum with respect 0,,1-1 for 1, 1. In the second line, an additional vt factor is added the
upper bound for (2,, uz) since V - x? S Vt max,= [ry In the last incquality, we use the conditions (t log n) < 1, ((bbl) < pid((st and /st log bel 3 < upper bound all the terms containing B by CLNp1d3i-1 Plugging (I.12) and (I.13) into (1.11), we obtain
|| well2 S CVd- ||uell2
C C
(Zr, uz)² + +1 S CNLpiValEs VdB.-1) 17 BB-11
Here, we use the fact that lo I Vd, which is implied by the condition pid(stlogn) ((bb,)) >> Lsp1( (t log(n))3. Lastly, by condition 1 < (N((),)) and the fact that Lp d > + ((,1)) by assumption, we can absorb the 7) - 3,-1 term into the CN Lp term. Hence, we complete the proof of Theorem G.7.
1.2.4 PROOF OF THEOREM G.8
Recall by definition of wo that
(v. wx)2
Pw n.0 Welle wolle V
+ ((20, un) + n - a0,0-1)
By Theorem G.6, we already have (v, wx)/ (1 N1l) It remains to characterize (zo, u.). We have by Theorem G.5 that
(zo, u.) ( o(1) + LNN
100.--0-1,--1
LCNN:LV/tlggn & (v dd + (1 B
-Non;-17((b)) o + + +CNp.L & CLNp,
3/2 100.-10-11--1
Here, in the last term we use the condition lo.. < 1 to upper bound bel 1, and p d(t log > ((bbe)) upper bound v (o <
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Cp1d. d. Note that the fluctuation terms similar to the for (2-1, in the proof of Theo- rem G.6. Specifically, under the same conditions
N2 Co > max (LLV/22stlog)) L Lytlognps da-1.1-1 a-1,111
&
Lp Be-1
N
we have
Nú. > CNL max{ PI (I)pps((logn)) -00 Aiy prd3e-1j.
Thus, we conclude that (zo, ue) = 100 (b:) + (1)) + (N)) Thus,
/ (1,,12) wx)2 ((20, un) + 77 a0,t-1)
|P wellz = V ||u1|2
(Ni (1 o(1))) + (Nao,,- (
+ o(1)) = o(Nv)) + 7
= (1+) V (N +
Here, the last inequality holds by also noting that 17- << (+((b.)) s (N)) This completes the proof.
1.3 PROOPS FOR RECURSION ANALYSIS
I.3.1 PROOF OF THEOREM G.10
What need to prove here is that all the conditions in Theorem G.9 hold for the current time step t if the conditions in Theorem G.10 hold. This is because the conditions in Theorem G.9 are the union of the conditions in Theorems G.4 to G.8. In the following, check all the listed conditions one by one.
Step Checking all conditions in Theorem G.9. For the first step, we divide the conditions in Theorem G.9 into three groups.
Group 1: Implication of Cond.(i) and Cond.(I) We first notice that since S conditions
b = O( O(Vlogn) < G1, kolbel = 0(1), vp2s(t log n)3/2 1 "f < No(((be)) A N2dC
guaranteed by Co)d.(i). Here, need be careful about condition 1/ 17 < NzdCn
as Q. is function of i, and what we directly have in Cind.(i) is for Q1 only. By definition Qt =
E 1(0) vda-1,e-1)> we note that Qi is nondecreasing in 1-1,1-1- Therefore, we have the
-b
following fact:
Fact L.3, Ifa-11111 2 a-1,0 then Qi 2 Q1.
In fact, the condition a-11-1 2 a-10 is automatically guaranteed by Cond.(). Therefore, the condition 17 & N2dCyfPQ, will hold for all successive & as long as holds for and a-14-1 2 a-1.0 Meanwhile, we also have by the same reasoning that
Vda 2 vda-1.0 >>
where the last inequality is guaranteed by InitCond-1. The condition vts log 1 & 1 is guaranteed by Cond.(I) well.
Group 2: Implication of Cond. to (ii)..
The direct implication of Cond. that logn) > ((bl) > (spr(t
Similarly, the direct implication CIId.(II) and Cond. that
N₂ Co02Q, > max Lpi pos//ossll Ld ((be))),.)). Lpi
B--1
N
a-11-1
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Here, the fact that T and the monotonicity of Qe in Theorem 1.3. It remains to check whether Ny Co > Lyttog np1 da 1,t-1 holds.
&
Group 3: Implication of (in, Cond.(I) and Cond.(II). To verify this inequality N2 CyCfQ > L./ log RP11 -., we just need to show that f1/0-1,t-1 S CC1/a-111 for some universal constant C > 0, the corresponding inequality for the latter is already guaranteed by Coid. Recall the definition of & in Theorem G.4, the ratio fe/0-1,1-1 is given by
Et
Vstlog(n) KG pi ((b()) Err (boly
(1. (hi, (hi,hr)
+ P2vn
& 0-
Itehe,
Vp2d.
0-1,1-1
(I.14)
We obtain the above formula by the nonnegativity of a-1,4-1 guaranteed by Cond.(I). Proposition L.4. If. S V2log ni for some universal constant 6. >> 0, then for 2,
K2 1. (Kg + Cylogn (Bt-1 + 10-1,1-11 + la-1,,l))-
Proof.
See $1.5.3 for a detailed proof.
Combining (1.14), Theorem 1.4 and the fact that a a- 2 12-1100 2 1.0 by Cond.(I), we have
V5l² log(n)(( + CVoll Cogl))o 3/2 Be-1 11-1+2) +2)
Q 1-1-11
a-1,11
11-1,1-1
D((be)) Eer ( (1oely
1- (-(bk
(he, + P2v7
P1
+
V22d
01-1,
E1 + Cvst log(n)3/22
a-1,t11 +2). -1-1
31-1
(L.15)
0.110
where in the second inequality, we directly plug in the definition of &1 with in (I.14) and use the fact that 1-1,--1 2 P to upper bound the first term in the right-hand side. Furthermore, for each term Cond.(II, we have the following relationship:
N₂
N P1, -
L=1(1),
where the first inequality holds by direct definition of p1 in (F.1), the second equality holds by noting that 0 S 1, Q S and Co is universal constant, and the last inequality holds by ??. Together, have the following implication:
N2 CoPQQ Lp1 a-1-1
8-1
B <
N
a-12-1
Therefore, we can further simplify the upper bound in (1.15) to
CV5t2
(og()) 1(1 2 2).
(I.16)
0-1111
a-10
Using (I.16), in order for condition ensure
N₂ N
C > I.ttlognp do & 1.-1 to hold, we just need to
N₂ CoO22 > L L np:
E1
N₂ Co0RQ > t stogm))))
da_10'' The first one clearly given by Coid, and the second one is satisfied because we have by using (iid.(ii) and Coid. that
N
N
N2
CoP > Ld Ld (((6)) > L'd Iid PTs( log(n))* 2 CLd vtt5
N
Here, the first inequality holds by the second condition in Cond. the second inequality holds by (iid and the last inequality holds by noting that we are considering any S T. The last inequality shows that the last condition N Co02Q, > L./Flog ni1 do 1.1 also holds automati- cally under the conditions in Theorem 10 To this end, we have shown that all the conditions in Theorem G.9 hold for / if the conditions in Theorem 10 are satisfied.
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Step II: Deriving the recursion. As have shown in the previous step, all the conditions in Theorem G.9 hold for if the conditions in Theorem G.10 hold. Therefore, we can safely apply all the concentration results derived in $G.2. We next show how to use the previous derived concen- tration result on (v, wa) [|/l22 || Pa 1. We ll, and || w-110 wx ll2 to control the recursion of 3 and 1/a-1,t- 1,t- Since 3r is the projection of wi onto the Pw-ro direction, and a- 1,e is the projection of wy onto the direction, we have
3x
||u112 || Well2
- CLPIVa(E: vdBI-1)
a-1,t
(w,wz)
C.PQQ NN/NN do 11--1111
CLp
{ ( + CVol* log(n))/2 1 1(12 2)
)
B
CoBQu N2/N
Vd a-110
a a-1,111
where in the first inequality, we use the upper bound for || P walla in Theorem G.7 and the lower bound for (v. w ||ll| 2 in Theorem G.6 as (u, wa)/ /t/l////l22 2 (1 - o(1))N ž &NCgBQQ N2/N. 1 by the lower bound of ths in Theorem F.11. The second inequality holds by plugging in the upper bound for G./a-1r--1 in (1.16). Similarly, we have by definition of a. that
1
|101|2 /|w/l22
(1 + o(1)) Vv2 + (i + CLp vdg. -1-60 o(1) the
a-1,t
(v, wi)
(1 o())
VICOBRQ, N2/N da da-1,-1)2 + (CLL(1D1)) + CLLi Vag. CofQi: N2/NN da-1,1-1
5 CLpr
(((D) Pid a-1411
f + CVLPlog(n))) 1(1 2 2
+ Va a0-10
CoPQu NN/N (1 ((1).
where in the second inequality, we plug in the lower bound for th and the upper bound for Pi (bx) in Theorem F.8. The last inequality holds by the triangle inequality and the upper bound for & 1,1--1
in (1. This completes the proof of Theorem G.10.
1.3.2 PROOF THEOREM G.
In the following proof, let us take Tj max max{(2s)-1,1) As our goal is to establish that (G.5) and (G.6) holds for all < Ti, we just need to show that Cond.(I) and CIId hold for all < Tú as they are the only conditions that might be violated over time, and the other conditions only depend on the initial conditions.
Initial step. For 1. 1, we have a-1,1-1 1-1,0 and 3.-1 foo = 0. Hence, Cond.(I) and (II hold trivially. Before start the proof, we first derive some useful inequalities.
Useful inequalities. For Å1 we have by Cvn and Cond. that
\ CLpi
pid'-s
λoEi
dc Vslogn
<
<
(L.17)
CaT2, N2/N
((b6))
Q1
Using the above two inequalities, we have by (G.4) that
a-1,1 $ +(+0(1) + X1 ((T)
S =1+0(1) + (1112-++1)++--
+1) S 3
1/2-5
Pid
Vd
a-1.0
In fact, we have the ratio 1--10//-11 as
a-10
< (3+ + 1/2-5 a-0 S (3d 1/2 +d Colog M << 1.
(L.18)
Q-1,1
Here, we use the fact that a-1.0 = O( M) with sufficiently high probability 1 n= and M = (nly(n). The above inequality demonstrates that 1-1 is guaranteed to grow in the first step. Thus, by definition of Q₂ in (F.15), we conclude that
1 N₂
1
-b
1 (bd 1/2 +d-9) 1/2 =: Q2.
+d-9)=:0%
Q2
Vdaa111
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where we take [3d 1/2 + d d) Oly d d and denote the right-hand side of the above inequality as Q2- Since 8: E [0. 1], we have
1 N2 N2
02 v² 0² 12 + (1 Q2) v² = Q2(1 2²) v² 7 Q2 2 1 v2 Σ 2 Σ o Q3
8²
where the last inequality holds from ?? that 6² = ((polylog(n)-1) >> v². the sequel, we will
use Q2 2 /2/220 as the lower bound for Q2. By definition of Ti, we have T₁ = (2x) V = (1). In addition, for X2. we have
do
2do O(polylog(n)),
d2
(I.19)
Q2
02
where in the inequality we use the lower bound for Q2 and in the last equality we 0² ((polylog(n) in ?? and λo = Oppoyylog(n)) in Civd.
We now have for the coefficient A₂ ((5))/l)/( (pıd) that
x2 [b) Ao ((0) Qid=s
Q2 sd=
Pid
Q2pid
where the second identity holds from Cond.(v) and the last inequality holds by noting that Q-1.11 2 a-1.0 by the first step's s calculation in (1.18) and using the monotonicity of Q₁ in Theorem L.3. Next, we upper bound the quantity C1 in (G.2):
C1 (1 o(1)+ + d
C (log 3/2
1
+
vda-1,0
Vd
1 Ao ( (bl)//d
$ (1+0(1)) + A161
CV8polylog(n) vd
1
+
Vda-1,0
1- d
< {(1+0(1)+ + Vslogn/ (1+ -1))
d=
where in the first inequality, we use the fact that x2 S a1 by the fact Q2 > Q2, and we invoke the upper bound To S log and A2 = =(polylog(n)) in (I.19). In the last inequality, we use the previous bound for λ in (1. together with the fact that v V,a-1.0 2 1 by InitCond-1.
Induction step. Suppose the induction hypothesis holds for 1.2....tt t. We will show that Cond. and CIId. hold for + S Ti as well. To this end, evident that a is always growing before reaching C₁ which evident from (G.3) by noting that 02 (0))(())// pid & d=s <.
We first look at the recursion of a-1. By (G.5), the ratio --1.0/ a- 1, is bounded by
a--10
(100(11))
1-1
An ((D)(
A1ε1
<
+
C2a-1,0
O--1,
Pid
Mid
Vd
I <d-s(1--),
(d-s+
Vodlog
d=
(1 o(1)) Vd
Colog M Cd <dd-s(t- + 1/2+.
The first term on the right-hand side is decaying exponentially fast with respect to t. The second term is much smaller than 1 1/T3 given that To < log by definition. Therefore, both terms much smaller than 1/T3- This implies the first condition in Cond.(I) holds for + 1.
Next, look at the conditions involving S.. By previous analysis on Tj and the upper bound in (I.19), we obtain
Ti-1 < (polylog(n)) (2) - -ppolylog(n)).
By recursion of 3./a_11 1.t in (G.6), we have
31
(CD A1)
__ CVSI (og])og log(n)3/2)
&
0 0-
Vd
a-1.0
polylog(n)
+ CCv (V&pooyoog(n)
Vd
a--
( losgnn n
d=
Cv8 polyyoogg())
d polylog(n)
polyyog(n))
Vd
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Here, the second inequality holds by the upper bound for X/1-1 and also the fact that To + X S 2 log n since X 2 and To Σ log n. In the second inequality, we use the upper bound for 21ς1 in (I.17) and the fact that 11. 2 1 by InitCond-1. The last inequality holds because < 1/2 by definition. Using the above inequality with the fact that a-1,t t I 1, we obtain
Bg d= polylog(n)
Tos log n/6
where the last inequality holds by noting that both To and [ö] are at most O(polylog(n)). This implies that the second condition in CInd. holds for t + 1.
Eventually, for (II, we have
CaPQ. N CLpı
Qt Ao
Q2 Ay
8²
Z
((polylog(n)-1)
2Ao
Therefore, the left-hand side of the above inequality is also much larger than Be/0-1,1- To this end, we have finished the induction step and proved that Cond.(I) and CIId.(II) hold for all t S Ti.
Final step. According the recursion in (G.5), let consider the real value t" that satisfies
)24((11)) Pid
A1 ((b))
AIS1
log(d) ~
(I.20)
Pid
Vd
a-1,0
for some small constant > ( to be determined later. We first note that we can obtain the SA 1/2 factor by the inequality for A in (1.17) that
(2))(16) Pid
(18((
Pid S (1
A Vd
(12(16)
1/2/2-c
+
and
d +
Vs log (I.21)
Pid
Pid
Vd
Using the above inequality (1.21), and taking logarithm of both sides with base d for (G.5), we have for t* that
d-1/2-e) + logad 1--100
co log log d
1r loga +
2
Vølogni
log
which implics that
1 d-s +
1 ) a-1.0
Co log log d log d
1/2
t" < logg
loga
<
= (2x) V1 Tí.
d-1/2-- Vōlogn
S 1/2
(I.22)
In the second inequality, we use the fact that by InitCond-1,
1 log log (M/n)
log, a-1.0
log, (1|=112) log.((1 - =) 2log(M/n)) <
2 log d
Therefore, we can take op to be small enough but still on constant level such that
log (ooggo((Mmm)))
co log
2
2 2log d
2
log d
This justifies the second inequality in (I.22). Thus, there must exists some time t S Tj such that (I.20) holds. For this time we already have
1
(2
Ao >2
t-1
\1 ((b))
A1E1 Vd
1 +0 +0 +11+000))
1-1,1
Pid
Pid
a-10
This implies that 1-1,1 =1 1- o(1).
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a-1,1- An additional step is needed to show that a-1,t-1 a-1,1 for all t 2 2 and before t* is reached. This is required because we want to ensure that before time t², we always have a-1,t-1 2 a-1,1, and the stopping time To will not prohibit us from reaching t². In fact, we have by (G.3) that
Checking 0-1,6-1
(200(10))
(21 - Cr.
a-11
Pid
a-
Therefore, the ratio a 0-1.1-1/-111 is bounded by
a-1,1
pid (Aoe((1))
t-2
((--Ci--,))) C1a-1,1-
0-1,1-1
We consider two cases. If Cia-1,11 2 1, can just stop the gradient at 1 and obtain Q-1,1 1 1-o(1) since C1 [+)( In this case, we reach strong alignment in just one step. another case where C1a-1,1 < 1, since (2)((())
d=s, we have the above ratio strictly upper bounded by 1. Hence, the condition a-1,6-1 2 a-1,1 holds for all 2 2 and before t is reached.
In both cases, we have shown that a-1,t-1 2 a-1,1 hold for 2 S t S t*. As we have shown that CInd.(I) and (II hold for all f S Ti from the induction step, t* < Ti from the final step, and Ti S log(n) by definition, we conclude that (i-CCond.()) to Ti-CCond.()) in the definition the stopping time To hold for all t² In other words, have shown that To 2 t"
Thus, we complete the proof of Theorem G.12.
1.4 PROOFS FOR CONDITION SIMPLIFICATION
1.4.1 PROOF THEOREM G.13
Let us take t* as the maximum number of iterations considered. the following, we first provide sufficient condition for (iid, Cond.(v) and Cvid.(vi) to hold. Then, give reformulation of Cond.(i), Cond. and Cond.(iv).
A sufficient condition for (ii) hold is given by
Q
((bb)) Pid
max (2g((og)))
Vlogn E1
(I.23)
Ao
under the condition 1, 2 1. On the other hand, we note that Cond.(v) and Cond.(vi) can be reformulated as
Q1 d s Ao
((b) Pid
> d' VVllooo n E
(1.24)
Since d' V log n E > d= < we can safely delete the second recall as follows:
Vlogn- E1, we can safely delete the last term in (I.23.. Also by noting that
(I.23). Furthermore, by definition of {1. which
(he, (hr,hr)) hrr (hi, hv) + v p2d aa-1ol + Payn,
a = V K + PTT
2((b)) E 4 (BB))
+ (he, hv)
we conclude that & 2 Vp2da-1.5 1.0 2 Pp2. Therefore,
d' V'sloon E 2 d' /p25 log n 2 (22g(logm)317
where in the last inequality, we use the definition = C' log log n/ log d) 2 log log n/ log d. Therefore, the first term in (1.23) also be deleted. In summary, Cond (iii) is automatically implied by (I.24).
A reformulation Cond. (ii) gives
((b)) Pid
L.s log(n)3
>
(I.25)
log n
d
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summarize Ci), Ciid, Cond.(v) and Cond.(vi) into condition as Published as a conference paper at ICLR 2026
In the following, will simplify the above condition. Note that
((bb))(ppd) = Q1/Aod d < (slogn)
holds by using X = (polylog(n)) according to Cond.(iv) and X dd ((b))/ accord- ing to Cvnd.(v). Therefore, we can safely remove the first inequality in (1.25).
In the following, we aim to remove the condition vp25(t" log 1g
<< in Cond.(i. As Q > d' log n Ei by Cond. we conclude that Ç < Qı/Ao < 1. By definition of Ei, this condition directly implies that P2 < n 1-12. Therefore, we can safely delete the condition /P25(t" logn)3/ < 1 in (i)d
To this end, follows:
Q1 A Pid
((b)
max V Ei, Ls log(n)3
d=
and (i) and Covd.(iv) can be summarized into
A = O(polylog(n)), Ki = O((logn)113) 17 < N (fdd (())).
Note that in the last condition, we have pıd/Ao >> Hence, we only need to keep 77 < (b)). This completes the proof of Theorem G.13.
((b) according to the first equality in (I.4.I).
1.4.2 PROOF THEEREM G.14
To prove this lemma, need upper bound the expectation term the left-hand side of (G. Recall that E,r is given by uniformly samples (, l' from [NV] and that (h,, hy) S 1 always holds. We can upper bound the expectation term as follows:
Byy ② [Taly
(hr,
hw) hv) (Nu, (ur
(he, bv) + (he, he
1 (he, hvy
N2
L.EDD,
Σ
@ (1öl
(he,
(hr ho)
1 (1 llo =)
N2
+ Ybe hpy)
j=1 L,'ED)
$ (T0
hw),hw) + (hr, hv)
11(l/ loo 2 2)
j=1 L.IED
H4 My 1+ Hig Hy.
n
(T
Σ
Σ j=1 LLED,
1(1la helloo 2 2),
N2
where the identity, we split the summation according to whether how many non-zero entries are shared between two rows hy and by in the H matrix. In the last inequality, drop the indicator for the case || hyrlas = and use the fact that (-)) < 1 for the case || hy llso 2 2. For the first term, we use the fact that (DyI/N < p for all j E [n] obtain
Σ Σ (1O
a
1 - H1g
$ (T)
H1 + HI, He
No J=LLLED,
1 H1, Hu
D21
LIIED,
S npp $ ((bbl)
1 112
1 n2
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where the last inequality holds by the definition of h. in (G.9). In addition, the second term is upper bounded by
N
N 1 (Hly #0) 1(1rr + 0)
Σ
Σ jEILLED)
1(11
by loo 22)
N
I=1
3c|m]:
P=1
Hi. su
Hi,
< max le [ N
Σ
Σ
1HHr + 0) [HHv #0)
L.E mlitas HL. 40
P=1
max N N
Σ
N. P2 S p1P2.
LEE miatt "1,300 1,550
In the first inequality, we notice that if | o lo 2 2, then there must exist two different feature indices i * such that both hy by are non-zero at these two indices. This is indeed reflected in the constraints Hi, # O, H1s * 0 and the two indicators 1 (Hy * 0) (Hr * 0). Therefore, summing over all posible (i.j) pairs gives an upper bound for the second term. In the second inequality, we change the average over to be the maximum over I, and in the third inequality, we use the definition of P2 and P1 (F.1) to upper bound sum of the double indicator term. The last inequality holds by noting that each row by is s-sparse. Combining the above two bounds, we obtain that
Err ((Bi)))
(uu, Jur) I+ I+(h, tu) (be, hwr)
S <npt $ ((bil)]
1
h
V
P11228
1
h
n.
S Cnpf 2(bb,))
i
+ P1P28
where in the last inequality, we use the Mills ratio
@
(vvlly 1 12
22 < [brlg]
h2
1
1 - h2 by
exp
1 h2
2m
1 + 12
2
C|b. -1 1
12
i+a?
1-A2
( 1
exp
2T
2
and the above inequalities hold as long as (1 n12)(11 + h2) is on a constant level. Therefore, we have proved Theorem G.14.
1.4.3 PROOF OF THEOREM G.15
Recall the definition KG, in (F.9) as
1/4
1/4
b
b
K1 == n |b|
+
|2sm|bba
3 Rig
+ 4
2 R2 +
+ o
b + 1i
+ (At
(log(n)) 1/4 + 1/4
p2 s log(n).
12
To upper bound the above terms, let us consider the following inequality for any E (0, 1) and
[b] > 1:
1
2
1
1
7313133
b
(T(bl)
S
-exp
exp
2T
2
2
b
2T
2
+1
1
)
72
2
<
exp
2T -1
<
((6)))
(1.26)
(v2T)rr²
2
16/2
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where in the first and the last inequalities, the Mills' ratio bound that x/(2² 1) S (<))p(()) $ x for all > 0. Now, we can apply (I.26) to upper bound the first term in Kr
1/4
|b| 4
S C((nbb)) /4 2(bb) + < C(n ((n|b)) 1/ (b]))
(I.27)
3 14,
+
where the last inequality holds because h, S Fis,- by definition. Similarly, we can upper bound the second term as
1/4
p2sn|bl
S
C(p2sn
((bb)) t4 I C(p2sn/(b)
(bb an?+1
(I.28)
Here, the third term also follows from the above inequality as
b + 11
b+ h-
S CC 4((b)
1-M?
(I.29)
12
h
where in the first inequality, we use the derivative in (F.12) and the fact that /// 0(1) 1, which is given by the definition G (1 + e)2ylog n in (E.11), to conclude that increasing h4,1 to he will only increase the value of the whole term. the second inequality, we apply (I.26) with
1 - h. b V1 h
E (0.
Here, we claim T E (0, 1) because by condition hih, < 1 - v for some constant v > we have 1 / > and by noting that ///lb] 1, we have
1 h VI Tr
v V1- h1 S
In addition, since [6] < 22llg n by condition ((b)) 2 PI 2 n Consequently, obtain that
we also have G//6 > 1.
h - 1//0 1ラ
h 1-1-1
I
1
v
as 1. Therefore, we can apply (I.26) with 7 (h-1 (i G//bo//V22-11 (0, in the last inequality in (I.29). Now, we can combine (I.27), (I.28) and (1.29) to obtain the desired result in Theorem G.15.
1.5 PROOFS FOR TECHNICAL LEMMAS
I.5.1 PROOF OF THEOREM I.1
By Cauchy-Schwartz,
Schwartz, it holds that
1-1 (Pu 2r, u2 < Σ Pul. 2z 112 ||u,
T=1
One thing to be noted is that is independent of the filtration o(u1-)). Consiquently, when con- ditioned on U1:7, [PP z, 11 ~ x². By the concentration of x² distribution in Theorem J.1 with union bound over all [T'. we obtain that with probability at least 1 n ne for some universal constant c, C that
||Pu,, 0thn.=11: =11 < T log(nT) + Clo((TT) < C(t + +lggn, V [t 1]. t [T].
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Therefore, have that with probability at least 1 7:
Σ (Pa 2r, uz)² - C(t² tlogn) 1luell2.
Vt [T].
(I.30)
For the second term, follows from Theorem H.1 that with probability at least 1
Σ (u7 ue)
| 1|2
1-1
(u ue) | ll2
Cd.
Cd 1 1I 4el12. vt E (T).
(I.31)
| l
For the last term
| 1---,-,,2113
21113 || 113, we also note that & is independent of the filtration
(w-1:11)). Therefore, I|P Eel12 ~ Xd-1+1 We have by Theorem J.1 with union bound over all E [7]. and with probability at least - that
| -----1 2112 < d - 1 Cv 1) log(nl)) + C'log(nT) S Cd, e [T]. (I.32)
Combining (1.31) and (1.32), we have with probability at least n and for all f [
[u u4)
||= ||2
+ || |
2012 || 12 S Cd 2el1 || 12) = Cd ||
| la
| 112
(1.33)
Now we combine (I.30) and (I.33) to obtain the desired result Theorem I.1.
I.5.2
PROOF OF THEOREM 1.2
Recall that
U = E (pyt FT ((Fut FT
(.-1;b,) + AE, + AF, +0.0 +
By the triangular inequality, we have
| 12 S 22/11ET
(FFTt(Fut + +. -. i. 1 & by) b.)
By Theorem G.4, we have
VI x (Fut b,) + ||F ((Fy) + 0 T.. S CIN
By Theorems F.12 and F.13, and the fact that N₁ N and N₂ - Np1, derive that
1 || < CLD,NVAB..11 CLP1p2 d33-1 S CLNp1 Vd3 This completes the proof of Theorem I.2.
I.5.3 PROOF THEOREM 1.4
We recall from the definition of KG, that
1/4
1/4
b
K,,
n |b| Φ
p2sn|blab
1 12 +
/3 V2 1i
b+ 1ig
+ (p22))/4
(tlog(n)) 1/4
+ 1/4
s st log(n).
The terms that implicitly change with / is Ge and li,,.- Recall that
c + 1 (t 2) C(B-1 1 1a-1.-11 la-1
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with <1 -2((1 + 2lgg Moreover, by definition of ha,t, rewrite the term as
1/4
b
b
Φb
max Jelnl
(I.34)
[D]
11-11
LED;
To understand how the change in Çe affects the term, take the derivatives for positive power q:
d
V1 (-(-v1=a)>
b+ b+ 25 )'
b b+ 9-1 v1
r(
b b+ xς
ÇI O.
Φ dç
v1 v1-12/
x²
v1 -1-1" x2
c=<
Here, have the second derivative larger than 0 since ( v2(1 vllggn by assumption. Now, our goal is to upper bound the derivative with respect to Ç. We discuss in two cases:
For x E
[[1 + (1/4//////211 1]. we have b x E [[6 + +0/2,,22 C). 2 (1+ 1/V2)/2 and
thus
d
V1 x² )
b+ x5
b+14 x6
b + xς
② dç
5 -
V1 x²
V1 x²
b+ fr
sup
p(2)
q supp(z) .2 0(1).
221111+1//22//1
b b+ )
=20
For (1, + M/C)/2], have v1 x2a >
V1 (1 1b//)2/44
VI 1- (1+ 1/14111222244 2//44= -()).. Thus
d $ dç
(- (
b b+ xς
59
S
0 (VVlog) n).
x2
VI- V1
Therefore, we conclude that for = 1, it holds for any Hi, E [0, that
(-V- V1-Hi. -1R, H7,
b+ H1.5Se
b+ H1.51 < CV/logn (Bt-1 + 10-1,-11 + la-
(L.35)
V1 H
Since ||~||| - llvlla s 1|l-~vla S 1/ yoloo for any x, E R. by the triangle incquality. we conclude that the same upper bound in (I.35) holds for each j (I.34) as well. Therefore, the same upper bound also holds after taking the maximum over j E [n] in (L334. Therefore, we obtain that
K, (K₃ + CVllgn (Bi-1 + |a 10-111-1 la-1,o)))-
This completes the proof of Theorem I.4.
J AUXILIARY LEMMAS
J.1 CONCENTRATION INEQUALITIES
Lemma J.1 (Chi-square concentration, Lemma 1 in Laurent & Massart (2000)). Let XI,..., X,, be independent random variables such that X; ~ N 1) for all i. Let a E R be a vector with nonnegative entries. Then the following holds for any d (0,1):
P [
axx?
|alla 1oll + 2||a|lac log 8-1 S..
Lemma J.2 (Tail probability for the maximum Gaussian random variables). Let XI...., X, be o² -ssbgauussann random variables with mean Then for any 1.
t
max X logn +t S exp EI.....
202
In particular, if X1,.. X,, are independent standard normal random variables, then for any e >> 1,
max 11.....
X, 2222oln
S n 1-2
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Lemma J.3 (Bernstein's 's inequality). Let Xi...., .......,,,, Xn be independent random variables with [Xx B[X,]| S C for all i [n]. Then for any ā E (0, 1),
n-
Var[X,] log
C Colo 1
P X; EX;
1=
2 8.
3n
Lemma J.4. Let ,I,...,X, Xn be independent standard normal random variables and define M,
maxisisn X,. Then for any fixed (0, 1) and all sufficiently large with 2(1 -)) e)² log 2 1 that
n
PP MM S (1 (1-e) (1-g))2logn) S exp
3 log n
Proof. Since X XI...., Xn L.d. ((, 1), it holds for any
IPMM, S x) (<)))"
where (xx) is the standard normal tail distribution function. In order to upper bound (1 - (x)))" when x 22 log n, we use well-known lower bound for the Gaussian tail. Specifically, for all x > 0 (see, e.g., Ledoux & Talagrand (2013) or Boucheron et al. (2013)),
(xx) 2 (x) + 2
Hence, further applying the fact that A(x) S exp( (x)),, we get
P(M, I x) 1 (1 - A (x)) & exp( (-n(()))))
Now, for (1 - c) / log n, we have
(1-c)V2log 1 2 1- 1-2(1- - <)² log n
x
V2
3(1 - t) Vlonn'
where the inequality holds for sufficiently large n such that 2(1 - <)² log v 2 1. Thus,
(xx)
(1-e)²
3(1
V
n logn
Substituting this lower bound into our earlier inequality gives
P(M, s (1 - c)) logn
3(1 c) n log n (1-))
I exp
n 2r-
exp
3/ log
This completes the proof.
J.2 EFRON-STEIN INEQUALITIES
Let Z be function of independent random variables XI,..., X, with domain X::
7 f Xn),
(J.1)
where f n + IR is measurable function. Let X . X be independent copies of X1,.... Define the modified versions of Z where one coordinate replaced by its independent copy:
X
( = f(X Xi-,, X
Xx).
Define the deviation terms:
V. E
Σ -2- 200 200
X X
i=1
V_ = E
Σ - 200 112 < 200
X
(J.2)
The following lemma is borrowed from Theorem in Boucheron et al. (2003) for the case where V, is dominated by some linear transformation of Z.
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Lemma J.5 (Efron-Stec Stein for dominated variance). For Z and V+ defined in (J.1) and (J.2), respec- rively. suppose that there exist positive constants and such that V+ Σ aZ + b. Then there is universal constant C 0 such that for any E (0, 1). with probability least 1 8.
S R. 7 C V(a E [ + log((1/)) + log( (og(1/6))
The following lemma is borrowed from Theorem 2 in Boucheron et al. (2003).
Lemma J.6 (Efron-Stein for the moment generating function), For all 0 > 0 and X E (0,)
A0
AV+
log E exp (A(Z E[i]))] - 1- 00 On the other hand, for all 0 > 0 and À E 10,1/0).
log E
exp
X0
AV_
log E exp (-X(Z - B[2]))] - 1 X
log E
exp
0
The following lemma is borrowed from Lemma 11 Boucheron et (2003) for transforming the upper bound on moment generating function (MGF) bound into an exponential tail bound.
Lemma J.7. Suppose for any X (0,1/a), there exists constant V such that:
X²V
log Ixgg B(Z)))
T- Xa Xa
Then there exists some universal constant C. such that for any ō E (0, with probability at least
1 - 6;
Z - EE S C. VV log(1/0) + C. log(1/6).
With the above lemmas, we can derive the following Efron-Stein inequality for sub-exponential variance.
Lemma J.8 (Efron-Stein inequality for sub-exponential variance). Suppose either of the following two conditions is satisfied:
1. The variance V+ for Z satisfies that E[exp(AV4]] < E(exp(AVV4) any X >> 0, where
V4 is o-subexponential with a E (0, 1):
Q(v) = (V V + v) exp(-v/a). when V_ exceeds some threshold V > 0.
2. The moment generating function of V+ satisfies
Aa
log Elcxp( ([(cp((VV))) S = + 1 aA for some V 0,0 < < and any 0 < X <a 11/2
Then, with probability at least
holds that
E| S C /Voo((--1) + C Volog(8-1).
Similarly, V_ satisfies either of the two conditions, then with probability at least 1 8, it holds that
E|Z] Z C VVlog(5-1)) + C valog(ō
Proof. We just prove the first condition and the second condition can be implied by the proof. We explicit calculate the MGF for V+. The case for V- can be handled similarly. Take parameter 1123, we have for the moment generating function of V+ that
/ (xp(X-v)) Q(w)dd))
Elexp(AV+)] exp(AV).
lim Q(v)+ 1-40-
0+
1+> 1 cxp(-(a" (aa -a- 2) u)dd
<xxp(AV)
J0_
x.a exx((AV) 1+ -->= exp(AV) (1+ i
A
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where use 04 and 0_ to denote the limit from the right and left side of 0, respectively. Here, in the first line, we use integration by parts to obtain an integration term with respect to the tail probability (v)). In the final line, we have the denominator E (0, 1). Taking the logarithm both side, we obtain that
as since a- 1/2 a-1 for
Aa
log Dogepee((AVV) s AV log(1 Aa/(1 aA)) S XV
1 ax
Now, apply Theorem J.6 with À replaced by A/0 for some & E (al, ! to obtain that
log E(exp(x(AA -E|Z -
log
exp
AV
)
S 1- 00 (V+ - ax/0
X
X0
(V + +a)
<
(1 (0)(1 a\/0)
Note that such a exists since À < a 11/2 1/2, In particular, we have by the constraint on that al < v A-1 Let just pick vā and further restrict ourselves a a-122/2 to obtain that
\V + X² +a) log Elxp((AA - E(ZM - (1- - AVa22 S
(1 - 2\va)
Now, we invoke Theorem J.7 and conclude that there exists universal constant C > 0 such that
Z ET s C V(V + a) log(0-1) + C va log(6 2C (VV log( va log(6-
1).
similar bound holds for the lower tail with the condition on V_. Hence, we complete the proof.
L
Lemma J.9 (Efron-Stein inequality for bounded variance). Suppose that max V-J S Vo with probability at least 1 cxp(-a) for some > n and Vo > 02 for some universal constant C1 > 0. Also assume that max(V+, V-} } is uniformly bounded by Vi with V1 I n for some universal constant cg > 0. Then, with probability at least 1- holds that
Z E C log(ō 1)+ + Va Vi log(6
Proof. By Theorem J.6, we have for the moment generating function (MGF) of V+ that
X0
AV+
log Ejexp(A(Z -[())) <
log E
exp
X0
0
X0
AVo
AV1
<
log exp
+ eexp
P(V+ vo))
X0
X
(AVo + 0xxp
X(V1 Vo)
<
X0
In the following, take 0 = 2X(V1 Vo) /a, and the above upper bound can be simplified
2²
Vo exp(-a/2)
Elex(exp\\(Z - E[(]])] <
1 - 2X - Vol/a
2X3(11 Vo) /
X²
(ap(- (1o 212(11 Vo)/a
A/2(V Vo)/a
Similarly for V_, we also have
2²
exp(-a
log E(exp(( X(Z E <
Vi
1- A 2
vo)/a
2X Vo)/a
Therefore, in the following, we only need to consider the upper tail and the lower tail can be directly implied. As long 1/(2>2(V, Vo)/a) is polynomially in n, will have (alllaa///22(((()
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Vo)/a) I Vo. Take be the deviation of Z from its mean, i.e., t 1z E[Z]| have By
Lemma Boucheron et al. (2003), we conclude by using the Chernoff bound that
x 2V
log P(Z B|Z] 1)
<
inf
1- -V-VV((VV Vo)/a 1
Ae 0/2(V Vo))
t
2(4Vo 1v20
1( Vo)/3)
where the last inequality holds as long as l satisfies
(1+ Lv 2a - (Vi Vo)
-1/2
2
(I.3)
2%
Vo
The lower bound holds similarly. A sufficient condition for (J.3) to hold is
mexp(-a/44
t
22a (Vi Vo)
This condition will be automatically satisfied if we pick C (VVo log(6-1) + Va (V1 va-1(Vi Vo) log(8 ()) Therefore, we conclude that with probability least 1 8, holds that
1z - E|2] S C- log(O v V log(d-1))
This completes the proof.
Lemma J.10. Let w = (W1, .2...., Wa) be a random vector, and let (() denote the vector where the i-th coordinate unchanged. Suppose that f Rd R and RR + R are both nundecreasinghoonncreeae functions with respect to the coordinate ws. Then, we have the inequality:
replaced by independent copy w/, while all other coordinates remain
E [[[(w)g(() z ((((())(
Proof of Theorem J.10. By the monotonicity of and g. we have:
(f(w) fww())) (g(w) g(w(()) 2
Expanding the product and taking expectations, obtain:
[[[(w](())] - [[f(w)g(((()) - ([f(ww))) [9(w)] + ([f((())))( 0.
By the symmetry of expectations, the first and last terms are equal, and the second and third terms also equal, so we obtain the desired inequality:
E [[f(w)g(w) 2 [[[(w)g(w(1))}
This completes the proof.
K ADDITIONAL DISCUSSIONS AND RESULTS
K.1 CLARIFICATION ON NEURON RESONANCE
Intuitively, once a neuron has already learned a single feature, then its activation frequency will match the feature's 's occurrence frequency. Our work, however, is about the reverse direction under concrete training procedure:
a group of neurons are trained activate at frequency p. under suitable conditions these neurons can provably recover features with frequency lying in corresponding "resonance band".
This is a non-trivial statement because at random initialization, the neurons do not align with any features, and the training dynamics must guide them to do so if they are tuned at the right frequency. This result justifies that a more active way training Sparse Autoencoders (SAEs) is plausible, and provides theoretical foundation for GBA algorithm.
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K.2 CLARIFICATION THAT GBA IS NOT HIERARCHICAL SAE
One might wonder whether GBA with multiple frequency groups is essentially the same as Ma- tryoshka SAEs (Bussmann al., 2025) or Hierarchical TopK SAEs (Balagansky et al., 2025) that are designed to recover hierarchical features. However, we would like to clarify that GBA is fun- damentally different from these models. esentially because GBA uses single loss for all groups, while hierarchical SAEs or Matryoshka SAEs use multiple losses, each applied to a subset of the neurons.
As consequence, one limitation of GBA is that it is not guaranteed to separate hierarchical features. This does not violate our theoretical analysis, since we assume that the feature coefficient matrix H has uniformly random supports, which rules out hierarchical feature structures.
To make this distinction more concrete, consider the following minimal example. Let a, b E Rd be unit vectors with b * ta, and suppose the dataset consists only of the two inputs
{ + b, bb
i... the high-level feature a always co-occurs with either b or -6. Under a single global reconstruc- tion loss (the structure used by GBA), there is no penalty for representing the data by the two basis vectors a + b and aligned with a + b and b, without ever isolating a or b individually. By contrast, hierarchical or multi-loss SAES (e.g., Matryoshka) impose intermediate reconstruction objectives or capacity con- straints at different levels. A high-level group that must explain the input with very few neurons (or with its own reconstruction loss) is incentivized to capture the common component a rather than the signed combinations = 6. Thus the multi-loss design can force decomposition that separates high- and low-level factors.
themselves. In other words, the model can minimize loss by learning neurons
Furthermore, this distinction explains why GBA is primarily compared against other single-loss SAES (TopK, JumpReLU, L1) in our experiments, rather than Matryoshka. Notably, despite not being designed for hierarchical feature recovery, GBA still demonstrates competitive performance on SAEBench compared to more advanced models like Matryoshka, as shown in Table 3. We believe that extending GBA handle hierarchical feature decomposition is an interesting direction for future work, and it remains an insteresting research question on how to apply frequency- aware training that context.
K.3 ADDITIONAL RESULTS ON NEURON ANALYSIS
We further provide additional studies on the neurons learned by the GBA and TopK methods in terms of the three metrics used in the main experiment: maximum activation, Z-score, and maximum cosine similarity across different runs with different random seeds. All the other setup is the same as in Figure 4. These metries are computed based on the validation part of Github dataset, with the hook position at the MLP output of layer 26. For the Z-score, we compute the largest value among the tokens in the validation set, and for the maximum cosine similarity, we compute the smaller value among the two additional runs. See $C.4 for rigorous definitions of these metrics. In addition, for each neuron m, we also compute the activation fraction (or activation rate), which is defined as the fraction of tokens where pre-activations of neuron m are non-negative.
Thus, for each neuron m, we have four metrics: maximum activation, Z-score, maximum cosine similarity, and activation fraction. We generate scatter plots by plotting the Z-score against the other three metrics. The results for GBA and TopK are presented in Figure 15 and Figure 16, respectively.
Z-score v.s. maximum activation. In Figure 15 (left), we present the scatter plot of the Z-score versus the maximum activation of neurons, which is shown in the logarithmic scale with base 10. We observe an almost linear relationship between the two metrics, indicating that neurons with higher Z-scores also exhibit higher maximum activations. Notably, at the upper end of the distribution, subset of neurons attains even higher Z-scores. This behavior suggests that these neurons capture "cleaner" feature and fire exclusively when the feature is present. By the definition of the Z- score, for neurons with the same maximum activation, a higher Z-score implies a lower variance. In other words, these neurons' activations tend to be bimodal predominantly near baseline when the feature is absent and significantly higher when the feature is present. This is consistent with the dashboard results for individual neurons shown in Figure 17.
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2-Senne Mae Activation
2-Gcore us Activation Traction
2-Scor ve Maxi Codre Siilartty
Activatoo Reapuenic Moggan
No Contre Samia
250m Bogast
2.5crre
2-sarr
Figure 15: Scatter plots for all SAE neurons illustrating neuron properties for the GBA method: Z-score versus Maximum Activation, Fraction of Non-negative Pre-Activations (i.., activation fre- quency), and Maximum Cosine Similarity across different with different random seed. The 66k- neuron SAE trained on Hub dataset with hook at the MLP output layer 26.
Z-score v.s. activation fraction. In Figure 15 (middle), we present a scatter plot of the Z-score versus the activation fraction, which is shown in the logarithmic scale with base 10. Neurons with higher Z-scores generally exhibit an activation fraction near 10-3 to 10-1. This suggests that they predominantly capture infrequent salient features. Comparing the TopK method, observe that the GBA method more effective in capturing infrequent features, which is primarily due to the fact that purposefully assign groups with both high and low target activation frequency. Additionally, the neuron grouping mechanism effectively adapts to diverse feature occurrence fre- quencies, underscoring the adaptivity of approach. Additionally, with activation frequencies exceeding the HTF of 0.1 (by our default configurations). This behavior facilitated by the bias-clamping mechanism, which prevents biases from becoming excessively negative, as discussed in $4.
observe several neurons
On the contrary, we observe in Figure 16 (middle) that TopK is good at capturing frequent features, but for infrequent features, it has very low Z-scores compared to the GBA method. This fact again underscores the effectiveness of our approach in capturing infrequent features.
Z-score V.s. MCS. Figure 15 (right), we present scatter plot of the Z-score versus maximum cosine similarity across different runs with distinct random seeds. Recall that higher maximum cosine similarity indicates more consistent feature recovery, and we observe that neurons with higher Z-scores tend to exhibit higher levels of consistency. This result supports the effectiveness of GBA in reliably extracting salient features.
2.Score Max Actrustion
2-Sccre Activation fraction
2:Scrre Siilarrty
Ald wodonn Pooower 0ag000
Mix Cosine Sartayy
255eem 01g
7.5c00e
Figure 16: Scatter plots SAE neurons illustrating neuron properties trained using TopK with 300. The other configurations are the same as Figure 15
K.4 WHY TREQUENCY-AWAN TRAINING? A TOY EXAMPLE
To illustrate the benefits of frequency-aware training, we present a simple toy example, where we will show that standard TopK/L1 SAEs can fail to recover the ground-truth features.
Data Generation. We randomly sample n 128 features (, g with 42 dimen- sions. We consider the dataset to be heavily imbalanced in the sense that different data could contain dramatically different number of active features. Speicifically, consider types of data points:
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NEURON ALIONMENT
ACTIVATIONS
5df
50.871%
MAX 80
3Th
+6.02
12h
5thin- scorttstaattt Iistassce Tann, kdbt)/nnnn
CORRELATEO NEURONS
Cluss Enterr wilaarllsss cless Contorthäteer Dotun
Ermor (A pass Claa EnpafStranntchaacte
10345
ut-
clas-"saut-cell
1222
10:011
041
LE1
314
</divele
wes---lcui-cell
CORRELATED FEATURES
68
-30
-045
Itcl
101
111
113
PONISE TVAnEtv-nsstntttre Instantiate Jen Class MeAtttsthlle
Elass
-6.8
0:41
161
Zlap
113
12 ias Hartor flags Static
6333
1. class EeErrcyp::
several
Undividat
TAccsss nt altew Class CAttent
Class R innicles BEtesl extest Tesiiisee
jaanausel
Jeni K Wa greecriptter
254
class fart class Anttititar
-23
-25
Figure 17: Feature dashboard for neuron 4688 in the GBA-SAE model trained on Pile Github at layer 26's s MLP output position. This neuron exhibits clear bimodal activation pattemn, and is activated before outputting the "class" token.
NCT
OENSITY 28.270%
TOP ACTTVTTTTONNS
SECT
FON VINIIT MEsS volloriiooooo der
NACK DUSSEr PHPCIC SANrviSO?
Resta EsSian TOCES
POSTVELOG11S
4090
+3500
-U00
0/5
152
439
015
SARdEEEiI Receciary NORE platfore txectom char
435
topar
0/3
ran text! not Mindee teozers
Servi 899011
Figure 18: Feature dashboard for neuron 10984 GBA-SAE model trained on Pile Github at layer 26's MLP output position. This neuron activates uniquely when the model is going to generate names of operating system such as "Ubuntu", "Windows", and "macOS". This is good example of neuron that captures concept rather than specific token.
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1. Type A: with probability 0.5, we sample = 3 features uniformly at random from {us) take their sum and fg-normalize as the data point 2.
2. Type B: with probability 0.5, we sample 8 = 20 features uniformly at random from {or 1i=1- take their sum and Eg nz-normllize as the data point z.
Thus, Type A data points are sparse combinations of a few features, while Type B data points are denser combinations of many features. This specific data generation resemble the real-world scenario where different data points could contain different number of active features. Note that the only challenge here the imbalance in the number of active features, while the feature frequencies are all uniform
(3 20) /2 x 0.09.
Why it could be a challenge for standard SAFs. Standard SAEs like TopK regularized SAEs typically assume fixed sparsity level across all data points. For example, Topk SAE with K 10 assumes that each data point can be represented by activating only 10 neurons. While this K value could be suitable for learing Type data points (which only need active features), too small for Type B data points (which need 20 active features). Hence, the model could struggle to learn real features from Type data points.
Training Setup. We consider training TopK SAEs our baseline method and make comparison with our proposed GBA method. For all SAEs, we set the number of neurons to be M = 8192. We consider the following configurations:
1. TopK SAE:
We train TopK SAEs with M 8192 neurons and sweep K {20,30) 50)- These values correspond to different sparsity assumptions: K = 20 matches exactly the sparsity of the dense (Type B) samples, while K 30 and K = 50 represent over- estimates. This setup evaluates the model's robustness to fixed-sparsity constraints when the true data sparsity varies significantly.
2. GBA SAE: We train two varients of GBA SAEs: (i) Full coverage: We set HTF=0.5, LTF=0.001 just like in our main experiments with 10 groups This setup ensures that the target frequencies perfectly cover the feature frequency f 0.09, allowing GBA's effectiveness when the frequency range is well-specified. (ii) Misspecified cov- erage: We set HTF=0.01, LTF=0.001 with 10 groups, which does not cover the feature frequency f = 0.09. This setup tests GBA's robustness to imperfect frequency range spec- ifications.
assess
Results. We present the results in Table (in the main text). We observe that:
TopK SAE's performance is highly sensitive to K: When K = 20, the model perfectly matches the sparsity of Type B samples, achieving a perfect FRR of 100% at MCS Z 0.8 and 98.4% at MCS > 0.9. However, as K increases to 30 and 50, the FRR drops significantly, especially at the higher MCS threshold of 0.9. This indicates that to recover features accurately, the sparsity level must be carefully tuned to the data distribution, which may not be feasible in practice.
GBA SAE with full coverage excels: The GBA model with full frequency coverage achieves perfect FRR of 100% at both MCS thresholds, demonstrating its ability to adap- tively learn features across varying sparsity levels in the data. The 100% FRR not sur- prising here predicted by theoretical analysis. On the other hand, if misspecify the frequency coverage, the FRR drops significantly. These results again validate that is indeed the frequency-aware training that enables effective feature recovery.
K.5 ADDITIONAL SAEBENCH EVALUATION
We provide additional comparison of GBA with other SAE variants SAEBench (Karvonen al., 2025) in Table 3.
For GBA, always fix HTF 0.5 and number of groups K 20. To create GBA run with different Ln sparsity, we vary the LTF from 10 to 10 and also also vary the number of neurons assigned each group and 7+, the coefficient that controls how much we increase the bias for dead
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neurons in each group (See line 18 of Algorithm 1). Specifically, we allocate more neurons to groups with lower target frequencies while also decrease y if we want to achieve a higher overall sparsity level Lg- For this experiment, we train groups of GBA models with final Lo values ranging from 132.9 to 694.9.
Why we do not target the very high sparsity (extremely low Lo) regime? In our sparsest run (Lo = 132.9), we set LTF = 10-4, 7+ = 10-4 and allocated a linearly increasing number of neurons groups with lower target frequencies. Empirically, we find that
1. Further decreasing the LTF, 7+ or assigning more neurons to low-frequency groups dramati- cally exacerbates the percentage of dead neurons, with the fraction of dead neurons exceeding 50%. This behavior is expected, as extremely low target frequencies (e.g.. < 10 (0-4) are difficult to maintain stable given limited batch sizes, making neurons prone to becoming permanently inactive. This constraint is fundamentally different from TopK methods, where sparsity can be globally hardcoded by the parameter K.
2. Further tuning these hyperparameters does not effectively reduce the Lo value. One evidence is from our ablation studies in Figure 6, where the lowest achievable sparsity is about 0.2%, which corresponds to Lo = 132 for our setup with 66k neurons.
3. The 100 ~ 700 region of Ln already covers a wide range of sparsity levels that are of practical interest, whereas similar range of Tap has also been used in prior SAE works (Gao al., 2024).
Consequently, to evaluate the performance of GBA in reasonable configuration setting, do not make further attempt to push GBA into the extremely low Lo regime in this experiment.
We provide the comparison results of GBA with other SAE varients in Figure 19, Here, we only train the GBA models with different Lo values, and the results of the other SAE models are taken from Karvonen et al. (2025). where the SAE baselines are all trained with the same data and LLM architecture as ours.
GBA method demonstrates distinct performance profile compared to standard Sparse Autoencoder (SAE) baselines. Key observations include:
1. Explained Variance: GBA consistently achieves the highest explained variance across all scru- tinized Lo levels (ranging from 100 to 400), significantly outperforming Standard and PAnneal baselines while maintaining slight edge over BatchTopK and Matryoshka models.
2. Absorption Score:
GBA achieves the lowest Absorption Score (approaching 10-2 on the log scale), which is significantly better than Standard, TopK, and GatedSAE (hovering around 10-2). This indicates that GBA features are consistently activated when their corresponding input features are present, demonstrating effective feature capture. We add more discussions on this point later.
3. SCR Score: GBA significantly underperforms the Spurious Correlation Removal (SCR) metric when we push the Lo value to 132.9. One possible reason for such a sharp drop is that many neurons in the model are allocated to extremely low-frequency groups (with target frequencies below 10 10-3), and the model can fail to capture spurious features that are used for the SCR task.
4. Sparse Probing:
In the sparse probing classification task, GBA remains competitive with an accuracy of approximately 0.953 0.957. However, it is marginally outperformed by other varients, which achieve slightly higher linear separability at similar sparsity levels.
5. TPP Score: GBA shows competitive performance on the TPP metric compared to other SAE varients, and is on par with BatchTopK and outperform JumpReLU and GatedSAE, demon- strating its ability capture features with high causal disentanglement quality.
6. Alive Fraction:
The method exhibits exceptional stability with a near-perfect alive fraction (Nz 1.0) for Lo values above 300. However, for lower Lo values (e.g., 132.9), the alive fraction decreases to around 0.75. The reason for this drop lies in how we configure GBA for achieving low Lo values, which involves setting very low target frequencies for some groups, assigning more neurons to low- www-reuuency groups, and using a small 7+ value. All these factors contribute to increased likelihood of neurons becoming permanently inactive.
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Explained Variance
Absorption Score
SCR Score
0.4
Explained Variance
0.90
Absorption Scure.
10-1
SCR Scure
0.3
0.85
0.80
10-2
0.2
0.75
0.1
L0
L0
Sparse Probing
TPP Score
Alive Fraction
0.960
Sparse Probing
Alive Fraction
0.958
TPP Scure
10-1
0.8
0.956
0.6
0.954
200
400
600
200
400
600
200
400
600
L0
L.
1.0
RAVEL Disentanglement
RAVEL Cause
RAVEL Isolation
RAVEL Disentanglemeen aanngeeeennn
0.750
RAVEL Cause
0.5
RAVEL Isolation
0.78
0.76
0.725
0,70
0.74
0.700
0.65
0.72
0.675
0.60
0.70
200
400
600
200
400
600
200
400
600
L.
1.0
1.0
Standard
BatchTopK
GatedSAE
JumpRelu
GBA
TopK
MatrycshhkaBatchIopk
SAEBench Evaluation Results. We compare GBA with other SAE variants across mul- tiple metries. Here, we constrain the comparison to SAE models with Lo values between 100 and 500 to ensure fair evaluation. For SCR (Marks al., 2024) and TPP metrics, take the aver- age of the scores over Top-20 and Top-50 neurons scores evaluated for these numbers tend be more stable (Karvonen al., 2025) while avoiding biases from too limited neuron counts. All metrics except Absorption Score are better when higher, while Absorption Score is better when lower. GBA demonstrates competitive performance across Explained Variance, Absorption Score, TPP Score, Alive Fraction (for high Lo), and RAVEL Isolation metrics.
Figure 19:
7. RAVEL Metrics: GBA performs competitively on the RAVEL suite, particularly RAVEL Isolation and RAVEL Disentanglement. The performance trend improves as Lu increases, sug- gesting the features recovered are causally distinct and well-separated.
The tables for similar Lo SAE comparison are also available in Table and Table for Lo around 100 and 600, respectively.
Further Discussion on Absorption Score. GBA demonstrates particularly strong performance on the Absorption Score, achieving values significantly lower than other methods. To ensure this low score reflects genuine feature disentanglement rather than trivial artifacts, we additionally evaluate the Mean F1 score for the first-letter prediction task, following Chanin al. (2024). Formally, the F1 score is defined as the harmonic mean of precision and recall:
Precision Recall
F Precision + Recall
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where Precision TP+FP and Recall TP+PN- with TP, FP, and FN denoting true positives, false positives, and false negatives, respectively. This metric evaluates the utility of the top-k neu- rons-sclected via cosine similarity with a task-specific probe-in in predicting the first letter token. high Mean Fl score confirms that the unabsorbed features are semantically meaningful.
IP
As shown Figure 20, GBA consistently achieves Mean Fl scores in the range of 0.6-0.7 across different sparsity levels. Such Mean F1 Score matches and is even slightly better than those reported (Chanin et al., 2024). This suggests that GBA successfully recovers accurate, task- relevant features, thereby validating that its low Absorption Score reflects genuine disentanglement rather than artifacts. Furthermore, the Mean Fl score increases only mildly with k, implying that the top neuron alone captures most of the predictive power for this task a strong indicator of minimal feature absorption.
F1 Score vs k (k-Sparse Probing)
0.70
Mean Fl Score
0.68
0.66
L0=100
10=200
0.64
L0-300
L0=400
1
2
10
(Number of SAE Features)
Figure 20: Mean F1 Score Comparison. We compare the Mean Fl score of the top-k neurons for the first letter prediction task in our trained GBA SAEs with different Lo values. Notably, across different sparsity levels, find that GBA consistently achieves Mcan Fl scores around 0.6 0.7, while standard SAEs usually have Mean Fl scores below for these Lo levels as reported (Chanin et 2024). This indicates that GBA indeed learns SAE latents that are accurate in capturing meaningful features and the low Absorption Score is not due to trivial artifacts.
GBA
Metric
GatedSAE
TopK
BatchTopK
Matryoshka
JumpReT.U (ours)
Standard
Lo
175.2
173.4
168.6
166.7
162.5
132.9
125.3
Explained Variance
0.812
0.816
0.816
0.793
0.812
0.859
0.730
Absorption Score
0,2657
0.0838
0.0424
0.0083
0.0821
0.0022
0,3529
SCR Metric t
0.323
0.321
0.349
0.391
0.303
0.107
0.209
Sparse Probing
0.957
0.956
0.956
0.957
0.958
0.954
0.954
TPP Metric
0.072
0.100
0.094
0.209
0.105
0.144
0.018
Alive Fraction 1
0.890
0.910
0.856
0.897
0.719
0.723
0.719
RAVEL Disent,
0.730
0.709
0.714
0.739
0.728
0.710
0.665
RAVEL Cause t
0.710
0.677
0.676
0.735
0.710
0.658
0.603
RAVEL Isolation
0.751
0.742
0.752
0.743
0.747
0.762
0.727
Table 5: Performance comparison of SAE models with Lo between 100 ~ 200 on SAEBench. Arrows indicate whether higher (f) lower (1) values are better. Bold indicates best performance, and underline indicates second best. GBA achieves the best performance in 3 out of metrics, and for TPP metric GBA also second best.
L THE USE OF LARGE LANGUAGE MODEL
We acknowledge the of large language model (LLM) primarily improve the grammar and clarity of this manuscript. The LLM was also used to assist with debugging and generating boiler- plate code snippets, which reviewed and validated by the authors.
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GBA
Metric
Standard
BatchTopK
Matryoshka
GatedSAE
TopK
JumpReLU
(ours)
Lo
835.1
695.6
694.9
675.7
662.3
655.7
605.2
Explained Variance
0.852
0.902
0.926
0.891
0.898
0.906
0.906
Absorption Score +
0.0873
0.0724
0.0044
0.0157
0.0351
0.0274
0.0052
SCR Metric t
0.239
0.242
0.235
0,306
0.254
0.230
0.329
Sparse Probing
0.958
0.956
0.960
0.957
0.958
0,959
0,959
TPP Metric
0.025
0.340
0,209
0.365
0.086
0.328
0.159
Alive Fraction T
0.750
0.601
0.997
0.531
0.913
0.718
0.584
RAVEL Discnt.
0.731
0.751
0.748
0.746
0.767
0.749
0.755
RAVEL Cause f
0.680
0.725
0.724
0.706
0.768
0.723
0.734
RAVEL Isolation f
0.781
0.777
0.772
0.785
0.766
0.775
0.775
Table 6: Performance comparison of SAE models with Tag between 600 ~ 850 on SAF.Bench. Arrows indicate whether higher (1) or lower (L) values are better. Bold indicates best performance, and underline indicates second best.
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Reviews:
[{"review_id": "LGjc9qSJ7n", "reviewer": "Reviewer_fF2q", "summary": "The paper aims to provide theoretical grounding for understanding and improving monosemantic feature recovery in Sparse AutoEncoders (SAEs). First, authors show how the phenomenon of \"neuron resonance\" -- i.e., where SAEs best recover features when their feature (neuron) activation frequency falls within a certain \"resonance band\" around the ground-truth monosemantic feature occurrence frequency -- manifests in a toy setting, then theoretically characterize this phenomenon more generally. The primary contribution is a novel training algorithm based on neuron resonance, group bias adaptation (GBA), which provably (under a given statistical model) recovers monosemantic features. Empirically, GBA achieves performance that is competitive with prior SAE baselines across a variety of common evaluations, while substantially improving the consistency of learned features across seeds and hyperparameter settings and reducing the proportion of dead features.", "questions": "**Essential missing information required for full assessment** (as discussed above, I cannot find any definitions of the following in either the main paper or appendix, making it impossible to interpret key results):\n- Missing from section 3 (and appendices):\n    - What are m and $\\mu$ appearing on the y-axis of fig 2?\n- Missing from sec 5 (and appendices):\n    - What are $\\alpha$ and the corresponding \"top $\\alpha$ selection rule\"?\n    - What are Highest Target Frequency (HTF) and Lowest Target Frequency (LTF)? \n\nGeneral questions:\n- In sec 3:\n    - What/where is the supposed \"phase transition\" at $d = \\sqrt{n}$? Wouldn't this correspond to the rightmost column of the right heatmap (where $d \\approx \\sqrt{n} = 256$), which doesn't seem to exhibit a \"phase transition\" with respect to the preceding columns? I would appreciate it if authors could provide: \n        - (a) an explanation of what, precisely, \"phase transition\" is intended to mean here; \n        - (b) a clear criterion for categorizing a given result as showing a phase transition, rather than simply a gradual change in feature recovery around good hyperparameter settings; \n        - (c) an explanation of how neuron resonance is expected to correspond to either a phase transition or a more gradual change (such as mentioned in (b)), and how this relates to the feasible frequency range in sec 6; and \n        - (d) a description of what, specifically, in figure 2 is taken to be evidence of a phase transition. (My understanding is that the \"narrow band\" in the right $d < \\sqrt{n}$ plot is interpreted as the phase transition, where the larger high-FRR region in the left plot is *not* interpreted as such, but rather a more gradual change -- is this correct?)\n    - Sec 3 experiments are based on the setting with uniform feature occurrences, unlike the \"feature spectrum\" (multi-group) settings discussed in later sections. What would the experiments in sec 3 look like given such a feature spectrum (either discretized into groups or varying continuously over a geometric distribution)?\n- In sec 4:\n    - Is there any particular reason for partitioning neuron TAFs based on a geometric distribution rather than, e.g., a [Zipfian distribution](https://en.wikipedia.org/wiki/Zipf%27s_law) (as is often observed in the context of linguistic features)?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "- The paper provides the first (to my knowledge) theoretical treatment of feature recovery in SAEs. Given the very limited current theoretical understanding of SAE training, this is an important and timely contribution to the field, which has so far relied overwhelmingly on empirical analysis alone.\n- Experiments across a variety of settings show clear empirical benefits of GBA relative to leading baselines, in addition to their improved theoretical grounding with respect to feature recovery.", "weaknesses": "The paper's greatest weakness is poor organization and clarity.\n- **Organization:** The main paper should be relatively self-contained, including all the necessary high-level information to understand key contributions. A huge amount of critical content -- including comparison with related work, proposed evaluation metrics, and key information about experiments that are strictly necessary for understanding empirical results -- is provided only in the appendix. For instance:\n    - Introductions and basic definitions of novel evaluation metrics in appendix C.2 should be moved into corresponding sections where these terms are used (even if some lower-level details remain in appendices).\n    - The related work section (appendix A) *must* appear in the main paper. (This case in particular feels like an instance where authors might have simply moved the entirety of a necessary main-paper section into the appendix in order to save space; but this could easily have been done much more appropriately by compressing the introduction or moving some experiments/ablations to the appendix instead.)\n- **Clarity:** Even after looking through the appendix, I am still unable to find definitions of several important experimental details, new terms, etc. necessary to provide a complete assessment (see the Questions section, below).\n\nPlease note: **I am completely open to raising my rating (and confidence score) if authors are able to provide this missing information** -- given my (currently incomplete) understanding of the contributions, I feel that the paper would very likely merit a higher score once I have the necessary information to make a full assessment.", "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": "oFvsUBPQdE", "reviewer": "Reviewer_nKpu", "summary": "This paper proposes an alternative SAE training approach designed around adapting the bias of subset of neurons / SAE Latents to create latents which track features of varying frequencies. This approach is motivated by an observed phenomena called \"neuron resonance\" - neurons learn monosemantic features when their frequencies are similar. They benchmark their method against other SAEs using SAE bench.", "questions": "- SAE quality is often measured by inspection of SAE latent dashboard - max activating examples, activation frequencies etc. The absence of feature dashboards is conspicuous -> can you please share feature dashboards? Ideally, comparing features with those found by other architectures and looking for signs that wSAEs are providing a more useful lens on the analysis of model activations. It seems plausible that they might.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Clarity: The paper is mostly well written and has useful explanatory diagrams. The key points / insights and core questions are made clear to the reader. \n- Originality: Directly attempting to mediate SAE latent activation frequency is an interesting proposal. \n- Significance: Training GBA SAEs on language models and not just synthetic examples makes the results much more interesting / potentially relevant to use in the real world.", "weaknesses": "- It is trivially true that if a latent is going to track a feature, it needs to fire at a similar frequency. Other SAE approaches hope that sparsity inducing loss will match features to their underlying frequencies indirectly which has the benefit of not requiring that you set target activation frequencies. How should we decide what these target activation frequencies should be? Ideally, SAEs are as unsupervised as possible and this approach risks creating challenging hyper-parameter tuning.\n- The paper doesn't compare there SAEs to Matrioshka SAEs (\"Learning Multi-Level Features with Matryoshka Sparse Autoencoders\") which are indirectly related by virtue of learning higher and lower level features which may relate to more or less frequent features. More generally, the L0 of SAEs in Table 2 is fairly high. Ideally, use of SAE bench should show results for a range of sparsity levels. If the point of these SAEs is to avoid using multiple SAEs with different sparsity levels (and often capture features of different frequencies), this should be addressed more directly.", "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": "Ls2Q7kxqUb", "reviewer": "Reviewer_73R2", "summary": "This paper studies when SAEs can recover interpretable monosemantic features from LLM activations and introduces a frequency-aware training scheme. The authors model activations as sparse nonnegative mixtures of latent features X = H V, then analyze when features are identifiable and recoverable under this statistical framework. They observe a “neuron resonance” effect: neurons reliably learn a feature when their activation frequency p lies in a band around that feature’s occurrence frequency f. Building on this, they propose Group Bias Adaptation (GBA), which partitions SAE neurons into groups with target activation frequencies and adapts biases to match those targets. Theoretically, they prove recovery guarantees for a simplified single-group Bias Adaptation variant under decomposable data and Gaussian feature assumptions, and argue the conditions extend group-wise to GBA. Empirically, on Qwen2.5-1.5B and Gemma2-2B layers, GBA matches TopK on the reconstruction–sparsity frontier and improves cross-seed consistency over TopK, while keeping a high alive-neuron fraction and competitive SAE-Bench interpretability metrics.", "questions": "1. How sensitive are your guarantees and empirical results to violations of the Gaussian and incoherence assumptions on V? \n2. Can you provide an empirical estimate or proxy for the distribution of feature occurrence frequencies in the target models, and explain how this supports your HTF/LTF choices?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper gives a formal recovery guarantee for a Bias Adaptation variant and motivates Group Bias Adaptation from the “neuron resonance” effect, then connects the two cleanly with Algorithm 1 and the resonance-to-algorithm narrative.\n- Consistency is reported over six seeds with explicit MCS definitions and tables. GBA exceeds TopK across selection schemes, and the results seem robust.\n- SAE-Bench metrics on Gemma2-2B show GBA competitive or best on multiple axes while keeping a very high alive-neuron fraction.", "weaknesses": "- The formal guarantees assume a Gaussian feature dictionary V; the paper notes this is for convenience and claims methods work when V is non-Gaussian, but it does not provide a systematic analysis of how violations affect results. This remains under-discussed.\n- Hyperparameter choices for HTF/LTF are supported by theory and ablations, but there is no measurement or estimation of the actual feature-frequency distribution in a target LLM to motivate ranges.", "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": "emXZIvCPUG", "reviewer": "Reviewer_VpeD", "summary": "This paper introduces a novel way of selecting SAE encoder bias (or, I think sometimes JumpReLU biases too) by adjusting the bias during training to force the SAE latent to fire within a pre-determined frequency range. The paper introduces a theorem stating that in order for an SAE latent to recover an underlying feature, it needs to fire with roughly the frequency of that feature. The paper turns this into an SAE architecture, called \"Bias adaptation\" (BA) or \"Group Bias Adaptation\" (BA), and evaluates the SAE on LLMs.", "questions": "### Larger questions\n- Why do SAEs not already naturally find the correct feature firing frequencies for features? In toy models SAEs seem to naturally learn the correct features and fire them the correct times, especially for toy models with independent features like you use. It sounds like your paper is claiming this only happens if you manually set the encoder bias? This seems hard to believe.\n### Section 2\n- Tied SAEs do not usually have an encoder bias, as setting a non-zero encoder bias breaks the tied SAE's ability to accurately recover features. Why was this added? If this is being used, it seems like it should harm the SAE. encoder bias only makes sense in untied SAEs.\n### Section 3\n- L150: \"We construct s-sparse coefficient matrices H, which have uniform feature occurrence frequency\" - does this mean that every input activation has the same number of active features, or just that each feature is sampled independently with a set frequency? Does every feature have the same frequency?\n- Does the SAE have the same width as the number of true features?\n- In section 2 it says that $b_m$ is a trainable parameter, but it seems you're not training it and setting it via a different method?\n- Figure 2, right: The text says optimal recovery occurs when p = f, but this plot seems to show that optimal recovery occurs when p > f. Recovery is very low at f in this plot. \n\n### Section 4\n- Algorithm 1: Line 17 and 18, I think you've mixed up \"min\" and \"max\"\n\n### Section 5\n- L286: I don't understand this line. Up to now your SAEs have all been L1 SAEs with no mention of JumpReLU. What does it mean that \"All methods employ JumpReLU activation\"? How are you training the JumpReLU SAE? There are at least 2 different ways of training these (Deepmind's version and Anthropic's version). Is this a tied SAE? Why are you not comparing to a JumpReLU SAE if you're using JumpReLU SAEs for your method?\n- In the appendix, you say: \"JumpReLU decouples the neuron output magnitude from its bias\". Are you controlling the *JumpReLU bias* with your method or the *encoder bias*? These are different things.\n- Figure 4: Are all the SAEs you're comparing with tied SAEs? I'm confused if you're using tied SAEs at all, despite section 2 claiming you are using tied SAEs. \n- Figure 4: You using JumpReLU, a state-of-the-art method for your own technique, but comparing against non-SOTA architectures (ReLU, TopK). This is not a valid comparison.\n- Table 2: The L0 for these SAEs is unreasonably high. I have never seen anyone go above L0 of 400 for an SAE on Gemma-2-2b, yet these SAEs are all L0 600+, with your method having the highest L0. These experiments should be conducted at a reasonable L0, ideally multiple L0s.\n- Table 2: SAEBench can be very noisy, so you should run multiple seeds and report mean and stdev. The results as-is do not mean much.\n\n### Minor / formatting\n- The use of the word \"neuron\" for the hidden activation of the SAE is confusing. I initially thought this was referring to model neurons. SAE hidden neurons are typically referred to as \"features\" or \"latents\" in the literature.\n- L92: It looks like this is missing subtracting the decoder bias, based on your definition of an SAE in L132", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "- The algorithm for tuning the bias is novel and simple\n- The paper provides a theorem around when features can be recovered by an SAE", "weaknesses": "- This paper seems confused about how SAEs work currently. There is an implicit assumption that current SAEs do not naturally find the correct firing frequencies for latents, but this is easily disprovable. Toy model experiments with standard SAE architectures will show that current SAEs already naturally find the correct firing frequencies for latents.\n- It is hard to tell what the authors are concretely doing in experiments. The background only talks about tied SAEs, but adds an encoder bias which does not make sense for tied SAEs. I suspect that the paper is not using tied SAEs at all, but it's very hard to tell. It is also confusing what specific bias is being used where, for instance, the authors talk about JumpReLU GBA SAEs, where it would make sense to control the JumpReLU bias, not the encoder bias.\n- The experiments in this paper are not rigorous and confusingly do not compare to SOTA architectures. The authors train a JumpReLU version of their SAEs (JumpReLU is considered SOTA), but do not compare this to any SOTA architectures (BatchTopK or JumpReLU).\n- The SAEs where SAEBench metrics are run are very strange, with L0 dramatically higher than I have ever seen before. SAEBench also tends to be noisy, and requires multiple seeds to be run to get a good signal, but this is not done.\n- This paper is built on the idea that SAE latents need to be fire at certain frequencies to learn the underlying features, but then hardcodes arbitrary feature firing bands for the latents to fire at.\n\nOverall, I believe the claim that SAEs latents should match the underlying firing frequencies of the features they track, but do not believe that the methods described in this paper are a valuable or valid way to adjust SAE bias thresholds. I also do not believe that the experiments are done in a valid and rigorous way.", "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)