Determinantal Beam Search

• Determinantal Beam Search is an advanced algorithm for neural sequence generation that optimizes diversity using determinantal point processes and kernel-based interactions.
• It refines traditional beam search by incorporating a log-determinant objective to model pairwise candidate repulsion, ensuring selection of varied yet high-quality outputs.
• The method is applied in translation, dialogue, and summarization tasks, although its NP-hard optimization necessitates efficient approximations.

Determinantal beam search is an algorithmic refinement of traditional beam search for neural sequence generation that explicitly optimizes the diversity of output candidates via subset selection methods grounded in determinantal point processes (DPPs). Whereas conventional beam search ranks candidates solely by their individual likelihood and is prone to selecting sequences differing by trivial edits, determinantal beam search instead models intra-set interactions, favoring sets of sequences that are both high-probability and diverse. This is achieved by maximizing a subdeterminant objective rooted in setwise kernel functions, generalizing the inference procedure to encode pairwise candidate repulsion and n-gram coverage in the selected output set.

1. Motivation and Algorithmic Distinction

Traditional beam search is used for sequence model decoding by maintaining a beam of the $k$ most likely incomplete sequences at each step. The score for each candidate in the beam is a scalar log-probability, and the algorithm proceeds by greedily considering local extensions based on this score. Empirically, this produces sets of outputs with high overlap; for example, machine translation systems often output $k$ sentences differing by one or two words.

Determinantal beam search reframes beam search as a subset selection problem where the objective function is not simply the sum or product of individual scores, but a log-determinant over a matrix encoding pairwise candidate interactions. The central idea is to select a beam $Y_t \subset B_t$, $|Y_t| = k$, at decoding step $t$ that maximizes the subdeterminant:

$$Y_t = \arg\max_{Y' \subset B_t, |Y'|=k} \log \det(D_{Y'} + w \cdot K_{Y'})$$

where $D$ is a diagonal matrix of candidate scores (e.g., $D_{ii} = p(y_{1:t}^{(i)}|x)$), $K$ is a kernel matrix quantifying similarity, and $w$ tunes the diversity penalty (Meister et al., 2021). Setting $w=0$ recovers classical beam search.

2. Theoretical Framework: Determinantal Point Processes

Determinantal point processes (DPPs) are probabilistic models favoring subsets with high quality and repulsion (diversity) between items. In determinantal beam search, the selection of beams is interpreted as a k-DPP subset maximization involving an L-ensemble matrix $L = D + wK$. The diagonal entries of $L$ favor individually strong candidates; the off-diagonals encode similarity, with high similarity reducing the likelihood of both candidates being chosen.

For text generation, kernels such as the string subsequence kernel

$$K(s, t) = \sum_{u \in V^n} \sum_{i: u=s[i]} \lambda^{l(i)} \sum_{j: u=t[j]} \lambda^{l(j)}$$

encourage n-gram coverage and penalize sequences sharing many subsequences. Normalization ensures invariance to length:

$$K_{norm}(s, t) = \frac{K(s, t)}{\sqrt{K(s, s) \cdot K(t, t)}}$$

This structure ensures that the algorithm avoids sets with redundant candidates, directly addressing the overlap prevalent in vanilla beam search outputs.

3. Implementation Details

The determinantal beam search algorithm proceeds as follows:

1. Initialization: $Y_0 = \{\text{bos}\}$.
2. Expansion: For each timestep $t$, form candidate set $B_t$ by extending all sequences in $Y_{t-1}$ with every possible token. Sequences ending with eos remain unchanged.
3. Matrix Construction: Compute $D$ and $K$ on $B_t$ (quality and similarity, respectively).
4. Subset Selection: Select $Y_t$ by maximization of $\log \det(D_{Y'} + wK_{Y'})$.
   • Exact maximization is NP-hard; efficient approximate greedy algorithms and incremental Cholesky updates are used.
5. Decoding Continuation: Repeat expansion and selection until termination.

Empirical code implementations utilize kernel libraries for string similarity and fast linear algebra operations for determinant calculations.

4. Empirical Evaluation and Performance

Experiments in neural machine translation (e.g. WMT’14 En–Fr, WMT’19 De–En) demonstrate that determinantal beam search achieves sets of outputs with higher n-gram diversity (measured by the ratio $d_n = \text{unique n-grams} / \text{total n-grams}$ across the set) compared to diverse beam search, stochastic beam search, and standard beam search (Meister et al., 2021). Median and max BLEU scores remain competitive, indicating no significant degradation in individual output quality.

Determinantal beam search differs from methods that inject stochasticity or post-hoc diversity penalties (e.g., DBS, SBS). The determinantal kernel explicitly optimizes for diversity during candidate selection. Temperature scaling in SBS, for comparison, is less principled and can produce inferior diversity gains.

5. Applications and Broader Implications

Determinantal beam search has broad utility in sequence generation tasks where diverse, representative outputs are needed:

• Neural Machine Translation: Multi-output settings benefiting from non-redundant translations.
• Dialogue Generation and Storytelling: Expanding the variety of plausible completions.
• Summarization: Multiple non-overlapping summaries for the same source.
• Structured Prediction and Recommendations: Selecting diverse panels or proposals.
• Data Summarization and Optimization: Any problem reducible to subset selection requiring quality and diversity.

The DPP-based framework is general and extensible to structured prediction domains beyond text. It provides a foundation for future research on integrating more informative similarity kernels and efficient approximate inference.

6. Limitations and Future Directions

The main computational bottleneck is the NP-hardness of exact log-determinant maximization. Approximations such as greedy submodular maximization are necessary and may not always yield the globally optimal beam. The quality-diversity trade-off depends on the kernel choice and interaction weight $w$; poor tuning may either undercut diversity or degrade output quality.

Scalability to very large beams or outputs with complex structural similarity remains a challenge. Research directions include integrating cognitive-inspired regularizers (e.g., uniform information density (Meister et al., 2020)) or dynamic nucleus pruning (Shaham et al., 2021), though current evidence suggests simple deterministic beam search remains competitive on key benchmarks.

7. Relationship to Other Diversity-Promoting Algorithms

Compared to diversity-promoting RL-based decoding using DPPs (Wang et al., 2019), determinantal beam search does not alter the training objective but instead optimizes the decoding subset directly. Connections to optimality certificates and length rewards (Huang et al., 2018) are largely orthogonal; determinantal beam search can be adapted to incorporate rigorous length normalization or bounded length rewards to prevent degenerate outputs.

A plausible implication is that hybrid approaches combining determinantal beam objectives with regularization (e.g., UID constraints) may offer further improvement, especially for tasks requiring both linguistic naturalness and high diversity in generated outputs.